Letter to the Editors
The Persian Tarof in Mathematics A. AZARANG
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.
was prevailed upon by my colleague’s recent brief proof of the AM-GM inequality in [1], to give some comments on this proof. In my opinion this proof should not just be considered as another simple proof of this inequality, for it has some conspicuous advantages over the other proofs of this inequality in the literature. First, it recasts one of the best understood statements regarding the positive real numbers (i.e., a B x B b), which is written by everyone who is dealing with mathematics in any way more than thousands of times in each one’s life, for different purposes, into a useful unprecedented statement (i.e., a + b C x + ab/x). In fact, the latter trivial inequality is the whole proof of the AM-GM inequality in [1]. Second, this trivial inequality leads us to infinitely many nontrivial inequalities in two variables, in a trivial way. For example, by taking x ¼ pffiffiffiffiffiffiffiffiffiffiffi nþm ; where m, n are positive integers, we get a þ bnffiffiffiffiffiffiffiffiffiffiffi amp pffiffiffiffiffiffiffiffiffiffiffi b nþm am bn þ nþm an bm ; or if in general we take x = f(a, b), where f is a suitable function, then we get nontrivial inequalities in terms of a, b. Also, if we assume that 0 \ a B x \ y B b, then we can easily show that
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1 lnð yÞ lnðxÞ ðx þ yÞ þ ab a þ b: 2 yx ab a tR y
To see this, recall that t þ þ b; for a t b: This ab immediately implies that x ðt þ t Þdt ðy xÞða þ bÞ; hence we get 12 ðy2 x 2 Þ þ abðlnð yÞ lnðxÞÞ ðy xÞ ða þ bÞ; and if we divide both sides of the latter inequality by y - x, we get the above inequality (Note, it seems that one cannot easily obtain this inequality by using the Mean Value Theorem). Third, as mentioned in the remark in [1], it
settles the equality case in the AM-GM inequality automatically. Finally, as for the title of my letter, when the other day I was discussing these comments with Professor Karamzadeh, although he quite approved the comments, when I said that I had better send these comments to The Mathematical Intelligencer as a letter to the Editors, he suddenly tried to dissuade me from doing this. He later told me, ‘‘Don’t you ever care what the other people think when they find out a colleague of mine who also happens to be an ex-student of mine, is somehow praising my work? Or doesn’t this reflect an all-too-high opinion of ourselves?’’ And he went on, saying that, ‘‘In contrast to the expression ‘a pompous politician’ we should not let the corresponding expression ‘a pompous mathematician’ find any place in the literature.’’ And at the end, while he was walking away, he reminded me of a quoted saying of Albert Einstein (‘‘Most people say that it is the intellect which makes a great scientist. They are wrong: it is character’’), but I replied without delay, ‘‘Don’t you think that Einestein is, in a way, doing a kind of Persian tarof in this statement?’’ But he just laughed it off. What he did to dissuade me is a Persian custom, which is called ‘‘tarof’’ in Farsi (i.e., when offered something in one’s favor, you must refuse, even when you would like to accept it), but he was doing this in an excessive way (note, a person with this habit is called ‘‘tarofi’’ in Persian). But, I as an Iranian must admit, I did not know, as yet, that the Persian tarof applies even to mathematical results, too, and at the same time I will be glad, if my letter is published, that with the help of mathematics the meaning of the Persian tarof would become clear to many people, even to non-Persians who know a little mathematics. Therefore, with all due respect to Karamzadeh, I decided to try publishing this letter in The Mathematical Intelligencer, in order to, at least, let others know more about our Persian tarof and that the Persians are not the only people who do tarof. (Note, Einstein is not an Iranian, although I wish he were.) Its seems, I am doing a kind of Persian tarof inside those parentheses, aren’t I?
REFERENCE
[1] O. A. S. Karamzadeh, One-line proof of the AM-GM inequality, Mathematical Intelligencer 33 (2) (2011), 3. Department of Mathematics Chamran University Ahvaz Iran e-mail:
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DOI 10.1007/s00283-011-9252-1
Letter to the Editors
Are Financial Jobs Morally Unjustifiable? JANUSZ KONIECZNY
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.
n the Summer 2011 Viewpoint, Jonathan Korman passes a moral judgment on the mathematicians (and others) who work for banks (Korman, J., ‘‘Finance and Mathematics: A Lack of Debate’’, The Mathematical Intelligencer 33 (2011) no. 2, 4–6). ‘‘These jobs are morally unjustifiable,’’ he declares. The reason is that in J. Korman’s opinion (which he apparently considers to be a self-evident truth) the banks constitute ‘‘a destructive financial industry,’’ and they are ‘‘known for looting the public.’’ Many of the mathematicians working for banks ‘‘feel no moral issue,’’ J. Korman divines, just like Adolf Eichmann, who, while sending millions of Jews to gas chambers, ‘‘was incapable of thinking about the moral consequences of his actions.’’ There is a mitigating factor, though. Mathematics departments are ‘‘complicit in setting the trap’’ and they are ‘‘handing over many students to the banks.’’ So, J. Korman adds, ‘‘it is not fair to put the burden only on individuals.’’ After all, he clarifies, we hold Eichmann ‘‘responsible for his crimes, but the system in which he operated had its role.’’ Controversy is welcome in Viewpoint, but the editors may exercise their right not to print some views. The comparison of a large group of mathematicians (and others) to Nazi murderers may well be one of many things that is not worth printing.
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Department of Mathematics University of Mary Washington Fredericksburg, VA 22401 USA e-mail:
[email protected] Editors’ Note: As we read Dr. Korman’s Viewpoint article (vol. 33, no. 2, 4–6), he compares work in finance to Nazi atrocities not in the nature of the deeds, which are plainly dissimilar, but in the way the moral issues are left unexamined. Thus his subtitle: ‘‘A Lack of Debate’’.
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DOI 10.1007/s00283-011-9254-z
Divertimentum Ornithologicum Pedro Poitevin After Jorge Luis Borges’s Argumentum Ornithologicum.
A synchrony of wings across the sky is quavering its feathered beats of flight. Their number is too high to count—I try to estimate it but I can’t: the night is dark, the birds are black, my eyes are weak. Certainly less than N but more than k, I tell myself, but then, in an oblique arrow of thought, I argue with dismay that if k is too small, then k + 1 can’t be enough, and so, inductively, all of God’s natural numbers fail—there’s none determining how many birds I see. I entertain that He might not exist, but N being hyperfinite I resist.
Department of Mathematics Salem State University Salem, MA 01945 USA e-mail:
[email protected] Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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DOI 10.1007/s00283-011-9249-9
Elements: A Love Song Manya Raman Sundstro¨m To H.W. Lenstra Not even this life will perish entirely, laying down my pen at the end of the day, famished, I reach for a slice of dark bread and a cup of tea, still simmering. Death stops at nothing, but I have this, these words from Euclid, derived from Eudoxus, commensurate, as his units were called, deliberately, not measured. In these lines he speaks to me, across ages burned to the ground by greed, gunpowder, gusts of wind fed by the malicious storms of misunderstanding. Alexandria burning. In these lines he sings to me, past the hum of my mother sleeping, the din of students demanding merit, the timbre of requirement rendered mute against a dark, dank silence. I stand in the wake of this, the tea finished, and the bread now turned to crumbs, a few of which are caught in my freshly ironed cuff, steamed with dignity, sturdy and unruffled. And I think, whatever forces pull at me, at least I have this—these truths: searing and tender, rugged and binding, love-like and whole, as I was once, in her arms, now dying. Mathematics is not life, but it fulfills a life, just as my mother will pass and leave me wanting. I turn to Euclid again, as a friend, there, listening.
Department of Science and Mathematics Education Umea˚ University 90187 Umea˚ Sweden e-mail:
[email protected] 4
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DOI 10.1007/s00283-011-9246-z
Minesweeper May Not Be NP-Complete but Is Hard Nonetheless ALLAN SCOTT, ULRIKE STEGE,
n volume 22 of The Mathematical Intelligencer, Richard Kaye published an article entitled ‘‘Minesweeper is NPComplete.’’ We point out an oversight in Kaye’s analysis of this well-known game. As a consequence, his NPcompleteness proof does not prove the game to be hard. We present here an improved model of the game, which we use to show that the game is indeed a hard problem; in fact, we show that it is co-NP-complete. We explain why our result does prove hardness of Minesweeper. We also take the opportunity to discuss the open ‘‘NP = co-NP?’’ question, and explain why NP-completeness of the Minesweeper game under our formulation may not hold, and indeed would be surprising. Minesweeper1 is a computer game played on an (n 9 m)board of squares. These squares start covered, and some of them contain mines. The goal of the game is to locate all the hidden mines without stepping on any. The number of hidden mines, k, is given to the player. Figure 1 presents an example for n = m = 4 and k = 5. During the game, the player can make moves of two sorts: either flag a square as containing a mine, or open a square. If the square that is opened contains a mine, the player loses the game. If the player opens a square that does not contain a mine, a digit between 0 and 8 is revealed. This digit is the number of adjacent squares that contain mines. Whenever the player flags a square, the number k is reduced by 1. After
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ROOIJ
all the mines are flagged and all the squares without mines are opened, the player wins. (Minesweeper, as offered by Microsoft, also displays a counter for the time played. Because time does not affect the outcome (win or loss) of the game, we ignore this parameter in our analysis.) How do people play Minesweeper? Different people may have different strategies, but the game has to go roughly as follows: At any stage of the game, the player starts by systematically trying to infer the content of some covered square(s) from the available information. If the player determines that some square must be empty he or she opens it, and if the player determines that some square must contain a mine the player flags it. (The flags serve solely as a memory support for the player. Throughout our analysis we take for granted that the flagging is done, and done correctly.) If at some point the player is unable to infer the content of any square, he or she is forced to guess and to open a covered square thought likely to be safe. If the player is lucky, the square is safe (empty) and the game continues. This process is repeated until either all mines are located (i.e., all squares not containing a mine are opened and the player wins), or the player opens a square containing a mine, in which case the player loses. In this article, we investigate the complexity of the deterministic part of playing Minesweeper. In our analysis, we assume that the player is given a consistent Minesweeper
1
Minesweeper is a trademark of Microsoft. The game comes as a standard feature of Microsoft Windows. The authors have no relationship with Microsoft, and nothing in this article should be seen as a comment on any of Microsoft’s products.
2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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DOI 10.1007/s00283-011-9256-x
Figure 1. Example of a Minesweeper game. The squares of the (4 9 4)-board are labeled for easy referencing (top left). The locations of the hidden mines are shown (top right). At the beginning of the game, the player only knows that k = 5 mines are hidden and every square of the board is covered. A possible game sequence (bottom). At each stage, the player solves a version of Minesweeper Inference. If an inference can be made, the appropriate action is taken: if for a covered square the content ‘‘no mine’’ can be inferred, the square is uncovered, and the number of mines adjacent to it is revealed; if, on the other hand, it can be inferred that a covered square contains a mine, the square is flagged, and the mine counter k is decremented. Otherwise, the player makes a guess. Note that here the player is lucky in guessing.
board configuration. That is, when completely revealed, the board presented has in each square either a mine or a number corresponding to the correct number of neighboring mines. Further, we assume an ideal player who makes no mistakes in reasoning. Note that these assumptions imply that the player can only lose (try to open a square containing a mine) when forced to guess. Therefore, our optimal player who follows the strategy described previously will avoid guessing whenever possible. However, to do so she must be able to decide 6
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whether it is possible to make progress on the board without guessing. This problem is what we call the Minesweeper Inference problem.
Minesweeper Inference The Minesweeper Inference problem comes in two versions, a search version and a decision version. We consider first the search version. Its input is the information that is available to
the player at a particular time during the Minesweeper game. Its output is a covered square s whose content (i.e., mine or not) can be inferred from the available information if such a covered square exists; otherwise, the problem outputs the statement ‘‘The player is forced to guess.’’ Minesweeper Inference (search version) Input: A consistent board configuration with some revealed digits and some correctly flagged mine locations derivable from the revealed numbers. The number of hidden mines, k C 0. Output: A covered square s and its content (either ‘‘mine’’ or ‘‘no mine’’), if there exists a covered and non-flagged square whose content can be inferred from the available information. Otherwise output ‘‘The player is forced to guess.’’ Note that Kaye’s article established that it is NP-hard (a notion treated below) to verify that a board is consistent. This means that Minesweeper Inference is what is called a promise problem. Readers curious about this technical point are referred to Oded Goldreich’s survey ‘‘On Promise Problems (a survey in memory of Shimon Even [1935–2004]).’’ We next introduce the decision version of Minesweeper Inference, as this is the version that we analyze. The decision version has the same input as the search version, but simply asks whether any safe covered square exists. Minesweeper Inference (decision version) Input: A consistent board configuration, with some revealed digits and some correctly flagged mine locations derivable from the revealed numbers. The number of hidden mines, k C 0. Question: Does there exist at least one covered and nonflagged square whose content (either ‘‘mine’’ or ‘‘no mine’’) can be inferred from the available information? Figure 1 illustrates how playing Minesweeper involves solving successive instances of the Minesweeper Inference problem. For each of the shown configurations, the player first solves the decision version, returning either a Yes or No answer. If the answer is Yes, the player subsequently solves the search problem (for one or more squares). If the answer is No, the player makes a guess. Note that the player naturally assumes that the board she is playing is consistent. Therefore, the assumption of a
consistent board configuration in the input of Minesweeper Inference is realistic. The creation of a consistent Minesweeper board configuration is not a hard problem: just select some squares to contain mines, and then for each cell not containing a mine compute the number of neighboring mines. However, given only a board configuration, telling whether it is consistent is indeed hard, as we will soon see.
Minesweeper Consistency In his article, Kaye presented a different analysis of the Minesweeper game. He considered another decision problem, called Minesweeper Consistency. Minesweeper Consistency Input: A board configuration, with some digits and flagged mine locations. Question: Does there exist a placement of mines that is consistent with the visible digits and flagged locations? Kaye argued that the ability to solve Minesweeper Consistency yields a way of solving what we call the Minesweeper Inference problem. Let us describe his argument. (We do not keep track of k, the number of hidden mines, just because Kaye’s article did not.) Assume we have a method M for solving the Minesweeper Consistency problem. Then we can use method M to also solve Minesweeper Inference as follows: 1. We pick an arbitrary covered square, call it s. 2. We use M to solve Minesweeper Consistency for the configuration Cm, where Cm represents the present configuration with the change that s is defined to contain a mine. (a) If the answer is No for Minesweeper Consistency, then we know there is no mine in s, and we output for Minesweeper Inference square s with inferred content ‘‘no mine.’’ (b) If the answer is Yes for Minesweeper Consistency, then we use M to solve Minesweeper Consistency for each of the nine configurations C0 ; C1 ; C2 ; . . .; C8 , where Ci represents the present configuration with the change that s has as content the number ‘‘i,’’ representing the number of mines adjacent to s. • If the answer is No for Minesweeper Consistency for each such Ci, then we know there is a mine in s,
AUTHORS
......................................................................................................................................................... ALLAN SCOTT received his Ph.D. at the University of Victoria, and continues as a post-doctoral fellow at the same institution. His research focuses on computational complexity, especially fixed-parameter tractabi-lity and games.
Department of Computer Science University of Victoria Victoria, BC, V8W 3P6 Canada e-mail:
[email protected] ULRIKE STEGE, after obtaining her doctor-
ate in computer science at the ETH in Zurich, came to the University of Victoria as a postdoctoral fellow. She joined the faculty there in 2001 and is now an associate professor. Department of Computer Science University of Victoria Victoria, BC, V8W 3P6 Canada e-mail:
[email protected] 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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Figure 2. No simple (polynomial time) procedure for solving TSP is known.
and we output square s with inferred content ‘‘mine’’ for Minesweeper Inference. • If the answer is Yes for Minesweeper Consistency for at least one of the Ci, then we cannot conclude anything for square s, and we repeat Steps 1-2 for a different covered square. 3. If we run out of squares to consider, then we output that nothing can be inferred from the available information for Minesweeper Inference (i.e., ‘‘The player is forced to guess.’’). We note that, although this strategy proposed by Kaye provides one possible way of solving Minesweeper Inference, it is not necessarily the only strategy for doing so.
Is Minesweeper NP-Complete? In his article, Kaye proved that the Minesweeper Consistency problem is NP-complete. Deferring for a bit the definition and discussion of NP-completeness, we note just that it means that Minesweeper Consistency is among the hardest problems in the class NP. From this, Kaye inferred— or so the title of his article suggests—that playing the Minesweeper game is hard, too. This is the oversight we mentioned in our introduction. Minesweeper Consistency being hard does not imply that Minesweeper Inference is hard. Although it is possible to use the Minesweeper
......................................................................... after studying cognitive psychology at Radboud University Nijmegen, came to the University of Victoria for her doctorate. After three years as a post-doctorate at Eindhoven, she returned to Nijmegen, where she is an assistant professor of Artificial Intelligence as well as working at the Donders Institute.
IRIS VAN ROOIJ,
Donders Institute for Brain, Cognition, and Behavior Radboud University Nijmegen 6500 HB Nijmegen The Netherlands e-mail:
[email protected] 8
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Consistency problem to solve Minesweeper Inference, some other strategies for solving it might be more efficient. We conclude that the complexity of Minesweeper Inference, and thus also the complexity of the game, is still an open question! In this article we present a new result that puts this open question in an altogether different light. In particular, we prove that the Minesweeper Inference problem is co-NPcomplete. In the next section we will explain in detail what this means, but anticipating a little, we mention some consequences. First, our result proves what Kaye’s proof did not, viz., that the Minesweeper Inference problem (and arguably, the game) is as hard as the hardest problems in NP. But second, it also proves that the Minesweeper Inference problem cannot even be a member of the class NP—and a fortiori cannot be NP-complete—unless a famous, widely believed conjecture is false.
P, NP, co-NP, and Completeness To understand our result, one needs to know about the open question ‘‘NP = co-NP?’’ that is related to the famous million-dollar ‘‘P = NP?’’ question, for which see http://www. claymath.org/prizeproblems/pvsnp.htm for more details, and Devlin, The Millennium Problems, Basic Books, New York, 2002. The starting point is the class co-NP and its relation to P and NP. In this section we walk you through the basics using the Traveling Salesperson problem as a running example. We will return to the Minesweeper game in the following section. Consider a salesperson who wishes to know if there exists an itinerary visiting every one of n cities such that the total cost of travel is within budget. More precisely, the salesperson wants to solve the following problem: Traveling Salesperson problem (TSP) Input: A set of cities. For each pair of cities, a and b, there is a cost associated with travel from a to b. Further, there is a budget constraint B. Question: Does there exist an itinerary visiting all cities such that the total incurred cost does not exceed B? Although this may be very hard to figure out (Figure 2), it is easier to double check that a suggested itinerary is indeed within budget (Figure 3). The TSP problem is a favorite example of such discrepancy.
Figure 3. Proofs of Yes-answers to TSP can be verified in polynomial time.
In complexity theory, one often asks whether a problem can always be solved in polynomial time, that is, whether the number of steps for solution can be bounded by a polynomial in the size of the problem. If finding a solution to a problem can be done in polynomial time, the problem is said to belong to the class P. If verifying a solution (a Yes-answer) can be done in polynomial time, the problem is said to belong to the class NP. In line with intuition, the TSP is easily seen to belong to NP, but, for reasons we will recall in the following text, it seems unlikely that it is in P.
NP and co-NP Although not as famous as the ‘‘P = NP?’’ question, the ‘‘NP = co-NP?’’ question is of comparable importance to mathematics. To explain what it is about, we again consider TSP. A travel agent can either return a Yes or a No answer to the salesperson. If the answer is Yes, she can give the salesperson a possible itinerary as proof, which the salesperson can verify in polynomial time (Figure 3). But what if the answer is No? What if there does not exist any itinerary that satisfies the salesperson’s budget constraint? To prove correctness of this answer to the salesperson, the travel agent could present the cost associated with each possible itinerary, and the salesperson could then verify that none of them is within budget (Figure 4). However, this verification procedure would take a number of steps that is factorial in the number of cities, and thus not polynomial time. Problems for which No answers can be verified in polynomial time are said to belong to class co-NP. The question of whether co-NP is the same as NP is open and important. Many a problem, including TSP, may appear to require much more work for verifying No answers than Yes answers, yet the possibility of an elusive polynomial-time
verification method for No answers may not be excluded with certainty. Given that we already know TSP is in NP, it is not difficult to come up with a problem in co-NP. We simply change the question as follows: Does there not exist an itinerary that is within budget? Or, equivalently and perhaps more intuitive: Do all possible itineraries exceed the budget constraint? Let us call this problem the co-Traveling Salesperson problem (co-TSP, here the prefix ‘‘co-’’ stands for complement). If the answer is Yes for TSP then it is No for co-TSP, and if it is No for TSP then it is Yes for co-TSP. But Yes answers to TSP can be verified in polynomial time, so the same holds for No answers to co-TSP. Hence, co-TSP is a member of the class co-NP. In general, for any decision problem Q in NP there exists a complement problem co-Q in co-NP, which has the answers Yes and No reversed. It is unknown whether NP = co-NP. Mathematicians mostly believe that this is not the case; see, for example, Jon Kleinberg & Eva Tardos, Algorithm Design, Addison-Wesley, 2005. Among the problems in NP, some are known to be key, in the sense that they are the ‘‘hardest’’ problems in NP in the sense of computational complexity; they are called NPcomplete. Similarly for co-NP-complete. We will be going through the definitions in the next subsection, but let us first orient you by displaying the conclusions schematically. Figure 5 presents an overview of the possible relationships among the classes P, NP, and co-NP. There are only three possible scenarios. The first possible scenario is the one conjectured by most living mathematicians, viz., that P = NP and that NP = co-NP. The second possible scenario is that P = NP but that NP = co-NP. The last possible scenario is that P = NP, in which case NP = co-NP and thus all three classes collapse.
Figure 4. No answers to the traveling salesperson may not have simple (i.e., polynomial-time verifiable) proofs. 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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NP
co-NP
NP-complete
co-NP-complete
P
NP = co-NP NP-complete = co-NP-complete
P
these polynomial-time reductions, that every other NP problem had a polynomial-time solution too—hence that P = NP. In 1972, R. M. Karp showed that the Hamiltonian cycle problem is NP-complete. This in turn implied NPcompleteness of TSP. Analogously to NP-completeness we define co-NP-completeness: A problem Q is said to be co-NP-complete if (1) Q is a member of co-NP and (2) Q is co-NP-hard, that is, if every problem in co-NP can be polynomial-time reduced to it. Thus if any one co-NP-complete problem could be solved in polynomial time then all other problems in co-NP would be solvable in polynomial time also (this too would imply P = NP). We further know that, if we can construct a polynomial-time algorithm M that reduces some ‘‘new’’ decision problem R to a known co-NP-complete problem Q, we then can conclude that R is co-NP-hard. Namely, then we can reduce an arbitrary S in co-NP to R in polynomial time as follows: We just reduce S to Q (known possible because Q is co-NP-complete), and then reduce Q to R using method M. Now suppose that one day we could prove an NP-complete problem, say TSP, to be in co-NP. Then in particular, coTSP would have a polynomial-time reduction to TSP. Then we could start with any co-NP problem Q, get a polynomialtime reduction of co-Q to TSP by NP-completeness, and this would also be a polynomial-time reduction of Q to co-TSP, yielding in turn such a reduction to TSP: TSP would be proved co-NP-complete, and we would have to be in one of the two bottom cases of Figure 5. This is why we began the paper by expressing doubt that the problem we will prove co-NPcomplete is also in NP.
P = NP = co-NP
Minesweeper Inference Is co-NP-Complete
Figure 5. The three possible relationships among P, NP, and co-NP.
Completeness and Reductions We have used the terms NP-complete and co-NP-complete and mentioned that problems in NP or co-NP that have this property are among the hardest of their class; in particular, they are crucial in trying to prove that P = NP. What exactly does it mean for a decision problem to be NP-complete? A decision problem Q is said to be NP-complete if it is (1) in NP and (2) NP-hard. To understand what an NPhard problem is we first discuss polynomial-time reduction, a concept on which our main proof below rests. A polynomial-time reduction from a decision problem Q to a decision problem R is a polynomial-time algorithm M for transforming any given input I for Q into an input I 0 for R in such a way that the answer to I 0 is a Yes for R if and only if the answer to I is a Yes for Q. A decision problem Q is said to be NP-hard if every problem in NP can be polynomial-time reduced to Q. Thus any NP-complete problem, being NP-hard, must be as hard (in this sense) as anything in NP. Finding a polynomial-time solution for any one NP-complete problem would imply, via 10
THE MATHEMATICAL INTELLIGENCER
In this section we prove that the decision version of Minesweeper Inference is co-NP-complete. We first show that Minesweeper Inference is in co-NP. We then show that it is co-NP-hard by giving a polynomial-time reduction from a problem known to be co-NP-complete. Namely, we rely on the famous Satisfiability problem, shown to be NP-complete by Stephen A. Cook, ‘‘The Complexity of Theorem-Proving Procedures,’’ 1971. Then our task will be to reduce its complement, called Unsatisfiability, to Minesweeper Inference in polynomial time: instances of the Unsatisfiability problem must be transformed into Minesweeper boards. Minesweeper Inference Is in co-NP We now show that the Minesweeper Inference problem is a member of the class co-NP. We distinguish between two types of instances of the Minesweeper Inference problem: those for which the answer is Yes (called Yes-instances) and those for which the answer is No (called No-instances). We invoke two fictional characters, an argument-maker, let’s call her Anna, and an argument-verifier, let’s call him Vince. To prove that Minesweeper Inference is a member of co-NP, we need to show that for every No-instance there exists an argument A that Anna can make such that Vince can verify A in polynomial time; that is, Anna always has an argument A that is easy for Vince to verify. Our proof is as follows. Imagine Anna is presented with a No-instance of Minesweeper Inference. After searching for some (possibly long!)
time, she comes up with an argument A for why the given instance is indeed a No-instance. The argument has the following form: For each covered square s on the board, Anna presents two different consistent board configurations. One of these configurations assigns a mine to s, and the other one assigns to s a value that is not a mine. This effectively shows that the content of no covered square s can be inferred from the available information. Now we show how Vince can verify the correctness of Anna’s argument A in polynomial time. For every covered square s, Vince considers the two possible assignments provided by Anna and makes sure that one of them puts a mine in s and the other one does not. For each of the two possible assignments, Vince considers each square t on the board one by one and checks if its (assigned) content is consistent with the (assigned) content of its 8 neighboring squares (if it is for all t then he knows that the entire board is consistent). Why is this a polynomial-time check? There are at most nm squares on an n 9 m board, so this procedure takes on the order of 2 8 nm ¼ 16nm steps. Vince repeats the checking procedure nm times, once for each covered square s of the board. Thus, Vince requires at most 16n2m2 steps in total to verify argument A. A Polynomial-Time Reduction from Unsatisfiability To show next that Minesweeper Inference is co-NP-hard, we will give a polynomial-time reduction from the known coNP-complete problem Unsatisfiability, the complement problem of Satisfiability. Both Satisfiability and Unsatisfiability take as input a Boolean formula in conjunctive normal form. A Boolean formula is written using only ANDs, ORs, NOTs, variables, and parentheses. A Boolean formula is in conjunctive normal form if it is a conjunction of disjunctions; for example, the formula (u OR v OR w) AND ( v OR w) AND (u OR v OR x OR y) is a Boolean formula in conjunctive normal form, where z indicates the negation of z for any variable z. Each disjunction in the formula is also called a clause (e.g., (u OR v OR w) is a clause), and the variables or their negations appearing in the clauses are also called literals (e.g., v and w are literals). An assignment where each variable is given the value either TRUE or FALSE is called a truth assignment. A formula can be satisfied by assigning the values TRUE and FALSE to the variables in such a way that every clause is TRUE. Considering our example shown previously, assigning TRUE to u, w, and y, and FALSE to v and x satisfies the formula. We call a formula that can be satisfied satisfiable; otherwise, the formula is said to be unsatisfiable. An example of an unsatisfiable formula is (u OR v) AND (u OR v) AND (u OR v) AND (u OR v). No matter how we assign the values TRUE and FALSE to the variables u and v this formula is never satisfied. For a given Boolean formula F in conjunctive normal form, Satisfiability asks, ‘‘Is F satisfiable?’’ Unsatisfiability asks, ‘‘Is F unsatisfiable?’’ Unsatisfiability Input: A Boolean formula F in conjunctive normal form. Question: Is F unsatisfiable?
We now present a polynomial-time algorithm that reduces Unsatisfiability to Minesweeper Inference. This reduction takes place in two steps. First, we transform the input formula F into a formula F 0 that is more suited to our purposes while remaining unsatisfiable if and only if F is unsatisfiable. Then we show how to implement formula F 0 as a Minesweeper board BF 0 for which we can infer the content of a covered square if and only if F 0 is unsatisfiable. After the description of the reduction we prove its correctness, showing that a formula F is a Yes-instance for Unsatisfiability if and only if BF 0 is a Yes-instance for Minesweeper Inference. Transforming Formula F to Formula F 0 This transformation consists of three steps. First, we remove all clauses that are tautologies, that is, clauses that are always TRUE no matter the truth assignment. Since a clause is a disjunction of literals, it can only be a tautology if it contains both the affirmation of a variable v and its negation v. Such clauses are satisfied in all possible truth assignments and can safely be assumed absent; indeed, a formula F for which some clauses are tautologies is unsatisfiable if and only if formula F *, that is F with the tautologies removed, is unsatisfiable. Second, we add one extra clause to the tautology-free formula F *, consisting of a single new variable that does not occur in F. The purpose of this extra clause will become apparent only later, but note that it does not change the unsatisfiability of the formula: If F * is unsatisfiable, then F ** = (v) AND F * is also unsatisfiable. Conversely, any truth assignment that satisfies F and therefore F * also satisfies F ** when we additionally set v to TRUE. Third, we build F 0 from F ** in replacing every AND in F ** by its logical equivalent of ORs and NOTs according to DeMorgan’s law: for Boolean formulas F1 and F2, formula F1 AND F2 is satisfiable if and only if formula F1 OR F2 is satisfiable. We remark that all steps of the transformation from F to F 0 can be done in polynomial time. Further, F is satisfiable if and only if F 0 is satisfiable. Constructing Board BF 0 from Formula F 0 The manner in which we construct Minesweeper board BF 0 from F 0 utilizes the fact that every Boolean formula can be encoded by a Boolean circuit, resulting in circuit CF 0 (Figure 6) that is satisfiable if and only if formula F 0 is satisfiable. Specifically, CF 0 can be constructed from F 0 as follows: We first feed the extra clause and the first clause of F into an or-gate. Then we take the output of that gate and feed it into another or-gate, together with the second clause. Then that result is fed into another or-gate with the third clause, and so on, until we have included all the clauses. We must now give the prescription for building a Minesweeper board BF 0 corresponding to a CF 0 . We partition CF 0 into units (e.g., splitters, crossovers, units of wires, joints, orgates, and not-gates), and then construct for each unit a board tile, that is a partial Minesweeper board. The tiles are put together such that the resulting Minesweeper board BF 0 exactly encodes CF 0 that in turn encodes formula F 0 (Figure 7). Our construction then will ensure that CF 0 is unsatisfiable if 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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v
p
q
r
NOT OR
splitter
joint
crossover
Figure 6. A Boolean circuit representation of formula F 0 . Here, F 0 = (v) AND F with F ¼ ðp OR qÞ AND ðr OR pÞ. Equivalently, without the use of ANDs, F0 ¼ ðvÞ OR ðp OR qÞ OR ðr OR pÞ:
and only if for the constructed board a mine can be inferred. Proof of correctness of our construction is left for last. Our proof construction is similar to Kaye’s reduction from Satisfiability to Minesweeper Consistency: Like him, we construct a Minesweeper board that models a circuit representing Boolean formula. There are, however, important differences. First, our reduction is from Unsatisfiability instead of Satisfiability, which makes our proof a co-NPhardness reduction instead of an NP-hardness reduction.
Second, we reduce to Minesweeper Inference instead of Minesweeper Consistency, which makes our co-NP-hardness result relevant to the hardness of the Minesweeper game. Third, in line with the game Minesweeper, the number k of hidden mines is given along with the other information on the constructed Minesweeper board. In Kaye’s model of the game this parameter was ignored. The general setup of the Minesweeper board construction is shown in Figure 8. Analogous to the setup of the Boolean circuit, one can think of the information for each variable being sent via wires (composed of wire-tiles) and other board tiles from the input terminals on the left border of the Minesweeper board to the output terminal at the right-bottom end of the Minesweeper board. Each input variable of a circuit is transformed into a (input) terminal tile. For each such variable, a wire is transmitting its value and a wire transmitting the negation of this value is attached underneath (cf. Figure 9). Subsequently, each clause is constructed using tiles for orgates, not-gates, splitters, crossovers, wire-units, and joints (cf. Figure 10). The clauses are then ‘‘conjuncted’’ using, again, the tiles for or-gates, not-gates, crossovers, splitters, and joints. When the complete board BF 0 encoding a formula F 0 has been constructed, the number k of hidden mines is computed by adding up the number of hidden mines per gadget used in our construction. The final step in constructing our board is to take the conjunction of all the clauses according to the layout of CF 0 .
Representation of the Board Tiles In the Minesweeper board whose general layout has been described, a piece of the board will correspond to each of the many tiles. How do we construct the different tiles that go into the board? We first list general properties of the tiles, each being a partial Minesweeper board consisting of covered and uncovered squares, and then we describe the construction of tiles of the various types. 1. For each tile at least one covered square is defined as an input square, and at least one covered square is defined as
Figure 7. A sketch of the Minesweeper board for formula F 0 from Figure 6. 12
THE MATHEMATICAL INTELLIGENCER
Terminal
Terminal
Splitter
Terminal
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Not
Term nali
Terminal
Splitter
Terminal
Joint
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...
...
Term nali
Terminal
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Terminal
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... Terminal
Figure 8. General Setup.
Terminal
...
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...
Figure 9. Wires for a variable and its negation.
2.
3.
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5.
6.
an output square (with the exception of the tiles representing inputs or outputs, called ‘‘terminals,’’ which have either only exactly one output square or only exactly one input square). Each tile is constructed such that its input square(s) and output square(s) align with the output square(s) of the directly preceding and the input square(s) of the directly following tiles. Each tile is constructed such that its input square contains a mine if and only if the corresponding output square of the preceding tile does not contain a mine. Each tile is constructed such that it is impossible to derive the content of any of its covered squares without using knowledge from other tiles. Each tile is constructed such that, if all its numbers are revealed and the input/output values of its neighboring tiles are unknown, then exactly the mines that are indicated by solid squares in the neighboring tiles are derivable. Every tile is constructed such that every possible assignment of mines that is consistent with the tile’s information uses the same number of mines (this provision is needed
... ... ... ... ... ...
... ...
Splitter
Crossover
Crossover
Splitter
Crossover
Crossover
Crossover
Crossover
Crossover
Crossover
Or Joint
... ... ... ...
Figure 10. Connecting two variables using an OR-gate; the information is sent vertically down, using one splitter and crossovers.
to ensure a well-defined number of hidden mines, k, for the constructed Minesweeper board). 7. Each tile is constructed such that its width and length is divisible by three (to make it easier to plan the board layout). We distinguish between three classes of tiles: terminals, connectors, and logical operators. Figures 11–16, are 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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1 2 2 1 2 1_ 2 u 3 u 3 2 1 2 1 2 2 1
UpShifter
1 v 1 1 _1 1 2 v _ 2
uvu
2 2 _ 2 1 v 1
DownShifter
VW Figure 11. An input terminal. Output terminals are constructed symmetrically. Grey squares denote covered squares, white squares denote squares with content ‘‘0.’’ For clarity, flagged squares are represented by black squares instead of flags.
1 1
u
1 1 1
1
1 _
u
1
122222 1 11 112 3 2 1111 u 1u u2 u5 5u 2u u 1u 1 1 2 3 3 2u u 2 3 3 2 1 1 2 2 44u44 3 u 3 u 3 4u4 2 2 12322u 223 2 1 111 1u 1
1 1 1 _ 1 u 1
u 1
Figure 14. A crossover transmits information from left to right and from top to bottom.
HW HW HW
Figure 12. Top left: a horizontal wire-tile (HW). Top right: a vertical wire-tile (VW). Bottom left: A horizontal wire built out of three horizontal wire-tiles.
NOT 1 1 1 2 u 2 1 2 1
2 3 2 1 2 _ 5 4 u u _ 2 u 3 1 2 2 1 1
1 2
u
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u 2
illustrations of tiles in these classes. In these figures, covered tile squares are labeled to make it easier for the reader to assess correctness of the constructed tiles. Two squares s1 and s2 are labeled u when: s1 contains a mine if and only if s2 contains a mine. Further, two squares s1 and respectively, when: s1 does not s2 are labeled u and u, contain a mine if and only if s2 contains a mine. Below we discuss the properties of each different type of tile. THE MATHEMATICAL INTELLIGENCER
Figure 15. Realization of a splitter with one input and two outputs. The input of the splitter-kernel (SK, top) is inverted.
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Figure 13. Left: An Up-shifter moves horizontal wire tiles upwards by one row. Down-shifters, left-shifters, and rightshifters are constructed accordingly. Right: A joint connects a vertical and a horizontal wire-tile.
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SK
1. Terminals Input terminals: An input terminal is a tile of size 6 9 9 squares containing exactly 6 mines. Five of them are derivable solely by the digits (derivable mines are depicted by black squares), one needs as additional information the input value of the neighboring tile. The is the output right covered square (labeled with u) square. For each variable u in formula F 0 , we place on the left-end side of the board an input terminal. If there is a mine assignment for the Minesweeper board where the output square of an input terminal does not contain a mine, we interpret this as corresponding to a truth assignment of formula F 0 where variable u is set to TRUE, otherwise as u is set to FALSE.
1 1 1 1 _1 1 2 _ u 2 uu 3 u 1 3 _ 3 1 3 u 4u 2 3 1 2 2 2 1
2 3 3 2 3 3 2
1 2 3 u 3 1_ 2 u3 3 1 2
1 1 1 _ 2 1 _ u u 3 u 1 3 _ u4u 3 1 2 2 2
transmitted to the next tile. If, on the other hand, the input square does not contain a mine then this can be interpreted as u = FALSE being transmitted. Two or more wire tiles that are placed in sequence are also called a wire.
1 1 2 u 3 1 3 2 1
Shifters: We sometimes need shifters to ensure that input or output squares are properly aligned with neighboring tiles. Shifters come in two kinds, a horizontal shifter is a tile of size 6 9 9 and a vertical shifter is a tile of size 9 9 6. A shifter contains exactly 9 mines, 7 of which are derivable solely by the tile’s digits. Horizontal shifters can be either up-shifters or down-shifters, and vertical shifters can be either left-shifters or right-shifters. Joints: We use joints to simulate ‘‘bent’’ wires in the circuit. A joint is a tile of size 6 9 6 containing exactly 3 mines, two of which are derivable solely by the tile’s digits.
LeftShifter
UpShifter
1 1 2 v 3 12 2 2 2 1
1 1 u 1 3 2 1 1 2 _ _ u 2_ 2 v 6aa 1 b 3 2 3 4 2 2 b33 4
2 3 2 1 1 _ 3 2 1 3 a _ a 2_ a 1 a 2 a 2 2 1 3 3 1 a 3 1 2 xyz 3 3 1 1 2 3 3 3 2 1
The output square of a joint (either its top, right, or bottom covered square) contains a mine if and only if the input square (left, bottom, or top covered square) does not contain a mine.
DownShifter
Figure 16. Logical Operators: A NOT-tile (top) and an OR-tile (bottom).
Output terminal: Exactly one output terminal is placed on the right-bottom end of the constructed Minesweeper board. The output terminal is the mirror image of an input terminal (reflected across the vertical axis), with the left covered square being its input square. If there is a mine assignment where the input square of the output terminal contains a mine, then there is a truth assignment that satisfies formula F 0 . 2. Connectors Wire tiles: We simulate the wires of our Boolean circuit with wire tiles. A wire tile is of size 3 9 3 containing exactly one mine. Its location depends on its neighboring tiles. Wire tiles come in two kinds: those that transmit information from left to right and from top to bottom. The left/top covered square in a wire tile is its input square (labeled u), the right/bottom covered square is its output square (labeled u). By construction, the output square of a wire tile contains a mine if and only if the input square does not contain a mine; thus, the labeling is consistent with the truth values. In other words, if the input square of a wire tile contains a mine then this can be interpreted as u = TRUE being
Crossovers: To allow wires to cross each other we use crossovers. A crossover is a tile of size 15 9 9 containing exactly 21 mines, 14 of which are derivable solely by the tile’s digits. A crossover has two input squares—the topmost and left-most covered squares—and two output squares—the bottom-most and right-most. Output square u of a crossover tile contains a mine if and only if input square u does not contain a mine, and the other output square v contains a mine if and only if its corresponding input square v does not contain a mine. Splitters: A splitter takes one input and produces two output lines with the same value. A splitter is formed from a splitter kernel and a NOT-tile (detailed below) because, as we will see, the kernel’s outputs are actually the negation of its input. Thus, we use the NOT-tile to negate the input to the splitter kernel. A splitter kernel is a tile of size 15 9 12 containing 30 mines, 22 of which are derivable solely by the digits of the splitter kernel. It has one input square (the left covered square) and two output squares (the right-most and bottom-most covered squares). The splitter kernel is constructed such that its output squares contain mines if and only if its input square also contains a mine. 3. Logical Operators NOT-tile: We use NOT-tiles to simulate not-gates. This tile is of size 15 9 12 and contains exactly 23 mines, 16 of which are derivable solely from the tile’s digits. The leftmost covered square is the input square and the rightmost covered square is the output square. By construction, the output square of a NOT-tile contains a mine if and only if the input square also contains a mine. 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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This means that, compared to wire tiles, a NOT-tile flips the truth value fed into it. OR-tiles: To simulate or-gates, we use OR-tiles. An OR-tile is a tile of size 24 9 18 containing exactly 58 mines, 43 of which are derivable by the tile’s digits. An OR-tile has two inputs: (1) the top-most covered square (labeled by u) and (2) the left-most covered square (labeled v). Its output square is the right-most covered square (labeled a). By construction, the output square of an OR-tile does not contain a mine (corresponding to the truth value a = (u OR v) = TRUE being transmitted) if and only if neither of its input squares contains a mine (corresponding to u or v being TRUE). A sketch of the Minesweeper board constructed from the described tiles to encode a Boolean formula is given in Figure 7. Correctness of the tiles for terminals, connectors, and NOT-tile and the OR-tile can be verified by checking that the labeling of the tiles is consistent with the information in the uncovered squares (i.e., whether squares with label u indeed all either contain a mine or all do not contain a mine, and that all squares labeled u contain a mine in the latter case, but not in the former case) and checking that these tiles posses the general tile properties (listed in the previous subsection ‘‘Representation of Board Tiles’’), when properly aligned. Of these properties, the hard one to verify is Property 4, namely that it is impossible to derive the content of any of a tile’s covered squares without using knowledge from other tiles. For this, recall our previous example with Anna and Vince. If we treat the gadget as a self-contained board, we can show that the covered squares cannot be derived as Anna did—by showing that for every covered square there is at least one possible assignment (of mines to the covered squares) that puts a mine in that square, and another that leaves the square empty. To do this, consider all the possible truth assignments for the wires running through the given gadget. In all the gadgets involving just one variable—which we call u—all the covered squares correspond to u or u. The squares corresponding to u contain a mine when u is true, and are empty when u is false (vice-versa for u). Since every covered square corresponds to either u or u, considering both possible assignments (u is either true or false) immediately shows that for every covered square there is a possible mine assignment that puts a mine in that square (u true puts mines in the squares corresponding to u, u false puts mines in the squares corresponding to u), and one that leaves that square empty. This gives us Property 4 for single-variable gadgets. The property can be shown for the remaining twovariable gadgets using a similar—though more involved— technique over all four possible input combinations. Correctness of the Reduction To prove correctness of our reduction we need to show that formula F is unsatisfiable if and only if board BF 0 is a Yesinstance for Minesweeper Inference. We know from our previous discussion that formula F is unsatisfiable if and only if formula F 0 is unsatisfiable if and only if Boolean circuit CF 0 is unsatisfiable. It remains to show that CF 0 is unsatisfiable if and 16
THE MATHEMATICAL INTELLIGENCER
only if we can infer the content of at least one covered square of the minesweeper board BF 0 (viz., for a covered square in the output terminal). (It is not required that that inference be made in polynomial time.) We prove this in two steps: we show (1) CF 0 is unsatisfiable ) BF 0 is a Yes-instance for Minesweeper Inference, and (2) CF 0 is satisfiable ) BF 0 is a No-instance for Minesweeper Inference. 1. If CF 0 is unsatisfiable, then for each possible truthassignment to the variables inputted into CF 0 —simulated by all possible mine placements in the input terminals of BF 0 —CF 0 outputs FALSE. This corresponds to a mine placement in the right covered square of BF 0 ’s output terminal. But this means that the right covered square in the output terminal of BF 0 can be inferred to contain a mine. 2. If CF 0 is satisfiable, then there exists a truth-assignment to the variables—simulated by corresponding mine placements in the input terminal—such that CF 0 outputs TRUE. This corresponds to a mine placement in the left covered square in the output terminal of BF 0 . We know on the other hand that for every formula F 0 there exists a truthassignment as input to CF 0 that results in output FALSE. (This fact comes as the payoff for the sly dodge of giving F 0 above an extra variable not needed in F.) This corresponds to a mine placement for BF 0 that has a mine in the right square of the output terminal. This, however, implies that no mine in the output terminal tile can be inferred. Given Property 4 discussed previously and the manner in which the circuit and therefore the board simulates the conjunction of clauses, for no other covered square in the Minesweeper board can the content be inferred. With this proof we complete the reduction from the Unsatisfiability problem to the Minesweeper Inference problem. To verify that the reduction is a polynomial-time reduction, observe that the size of the board is proportional to the number of literals in formula F. Because Unsatisfiability is known to be a co-NP-complete problem, the presented polynomial-time reduction proves that Minesweeper Inference is co-NP-hard. Taken together with the co-NP-membership proof in the previous section, this proves that Minesweeper Inference is co-NP-complete. This means that solving the Minesweeper Inference problem—which is necessary for playing the Minesweeper game optimally—is as hard as solving any co-NP-complete problem. It also means that Minesweeper Inference itself is not NP-complete unless NP = co-NP.
Conclusion As noted by Richard Kaye, the Minesweeper game can be used to illustrate fundamental questions in the theory of computational complexity. In this article, we extended Kaye’s work and showed how Minesweeper illustrates not only the famous ‘‘P = NP?’’ question, but also the important ‘‘NP = co-NP?’’ question. In the process, we revealed several interesting properties of Minesweeper. To explain these properties, we consider in the following text several scenarios that may arise in playing the game of Minesweeper.
Say you are playing Minesweeper and, at some point, you find yourself stuck. You are unsure if you can proceed in deterministic mode or if you are forced to guess. You could, of course, consider every possible placement of mines consistent with the visible information and verify whether there exists a covered square whose content is fixed across all possible placements. In that case you could infer its content. But such an exhaustive method looks daunting, as there are already billions of possible ways of placing k = 8 mines on a 10 9 10 board. You wonder if you could perhaps devise a faster method to determine if you are forced to guess, say, one that takes at most polynomial time. Our co-NP-completeness result shows that, no matter how hard you try, you will not be able to come up with an efficient (polynomial-time) method for solving your inference problem—or at least that if you do, you will have proved P = NP. Now, suppose that a friend is looking over your shoulder at the configuration on the screen and claims to know the answer to your inference problem. There are two possibilities. Either he claims you can infer the content of another covered square, or he claims that you cannot. Of course, in both cases you would want to be convinced. What do you think will be more difficult for your friend, to convince you that you can infer something or to convince you that you cannot? Again our co-NP-completeness is relevant, this time with (what you might find to be) a counterintuitive twist. It shows that if you cannot infer anything, then your friend should easily be able to convince you. That is, he can give you a simple argument that you can verify in polynomial time. However, if you can infer something, then your friend will generally not have a simple argument that you can verify in polynomial time—at least, not unless he proves NP = co-NP. Assume that your friend tells you that you can infer something, but at the same time claims that he cannot give an argument for this fact that you can verify in polynomial time.
2
You are willing to concede he is correct, but would at least like to hear an argument for why he cannot give you an easily verifiable argument. Again you will be disappointed, because your friend cannot give you such an argument either—at least, not without proving that NP = co-NP and therefore also P = NP. Finally, our result is relevant to another paper by Kaye on Minesweeper2, where he considers playing Minesweeper on boards of infinite size. In this more recent paper, Kaye shows that playing infinite Minesweeper is co-RE complete and finds this curious, as NP-complete finite problems usually correspond to RE-complete problems in the infinite setting. In light of our result, we can now explain this discrepancy: In the finite setting Kaye’s Minesweeper Consistency problem looks for a consistent placement of mines for an arbitrary board configuration, whereas in the infinite setting his generalized consistency problem asks for a consistent mine placement that also extends the input configuration. This change means that Kaye’s co-RE complete infinite version of Minesweeper is not a generalization of his original NP-complete Minesweeper Consistency problem but of the co-NPcomplete Minesweeper Inference problem that we have presented in this article. ACKNOWLEDGMENTS
We thank Hausi, Curtis, and Brian Muller, and Denise Broekman for fruitful discussions on the Minesweeper game. Allan Scott and Ulrike Stege are supported by an NSERC discovery grant awarded to Ulrike. Allan Scott furthermore acknowledges the financial support of the University of Victoria in the form of a student fellowship. Parts of this article were written while Iris van Rooij was supported by the Department of Human-Technology Interaction and the J. F. Schouten school at TU Eindhoven. We further are grateful to our anonymous referees for helpful suggestions.
Richard Kaye, Infinite versions of Minesweeper are Turing complete. http://www.mat.bham.ac.uk/R.W.Kaye
2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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Mathematically Bent
Colin Adams, Editor
Leonhard Euler and the Seven Bridges of Ko¨nigsberg The proof is in the pudding.
COLIN ADAMS
O Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, ‘‘What is this anyway—a mathematical journal, or what?’’ Or you may ask, ‘‘Where am I?’’ Or even ‘‘Who am I?’’ This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.
â
Column editor’s address: Colin Adams,
Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, USA e-mail:
[email protected] 18
THE MATHEMATICAL INTELLIGENCER Ó 2011 Springer Science+Business Media, LLC
DOI 10.1007/s00283-011-9206-7
nce upon a time, a small boy was born in the town of Basel, Switzerland. His parents, impressed by the intelligent look in his eyes, named him Leonhard Euler, after his great uncle Leonhard, who had been smart enough to marry a countess, and now lived in a castle, giving him the right to treat his relatives like dirt. Because of some confusion about the ownership of a loaf of rye bread, Euler’s family relocated to the secluded town of Riehen, down the river from Basel. Compared to Basel, Riehen was a backwater, with fewer brawls, demonic possessions, or other family entertainments. So, to occupy themselves, couples would stroll about town. Although not generally known for their competitive nature, the townspeople soon began to consider ways to outdo one another in their strolls. The Mandelbaums skipped as they walked. The Feidelhofers took their stroll walking backward, nodding as they passed their acquaintances. Not too be outdone, the Lalleputers hopped on one foot for their entire promenade. Finally, the Blandasmoths suggested the ultimate challenge, to take a stroll that crossed each bridge exactly once. At the time, no one knew whether or not it could be done. Young Euler, just 2 at the time, came to know of this problem through his babysitter. She would often deposit him in his pram with his rattle, ostensibly to take a stroll, but actually to meet her boyfriend, who looked quite fetching in his lederhosen. As they stood on a bridge overlooking the river Wiese, little Euler, who had little interest in rattles or lederhosen, overheard the conversations of the passing strollers and quickly grasped the problem. Doodling with his saliva on the inside of the pram, he painted each bridge as the edge of a graph connecting vertices that represented the land masses. Luckily, he was a substantial drooler. He quickly realized the solution. ‘‘Gertrude,’’ he called to his babysitter, who was canoodling with her boyfriend, ‘‘you must run at once to the mayor and inform him of my solution.’’ He pointed to the saliva dripping down the inside of his pram. ‘‘For, you see, there is only one bridge in Riehen, and therefore no one can ever succeed. The only way home is to cross the same bridge twice.’’ That very day saw the birth of topology. Not in that town. Actually seven towns over, a boy playing with a worm
invented knot theory. Unfortunately, the goat he was riding tripped into a moat and topology was stillborn. It would be some time before it was reincarnated. But in the meantime, Gertrude, astonished at the acumen of her tiny charge, raced off to the mayor’s office to explain the solution. The mayor and all of the town officials were so thrilled to know the answer, they rewarded her with a huge wheel of Appenzellar cheese. She used it for her dowry and disappeared to Basel with her boyfriend, never to be seen again. It wasn’t until a day later that Euler’s parents took an evening stroll and discovered a disgusted little Leonhard still in his pram on the bridge. Word of the young prodigy’s solution quickly spread throughout the land. People whispered to one another, ‘‘Little Euler has determined that one cannot take a stroll through Riehen and return home, crossing each bridge once.’’ As the message was passed along, it became garbled, and transformed into, ‘‘Turnips do not borrow Lucifer’s nightcap in Uppland.’’ This caused quite a bit of confusion, but eventually, it all got sorted out, and residents from towns as varied as Untholm, Masterdol, and Winterflagen soon streamed to Riehen to implore Euler to determine for them whether or not one could traverse all the bridges in their towns once in an evening stroll. After a few disastrous attempts, petitioners came to realize it was too difficult to bring the bridges to Euler, so instead they would draw a picture of their town and the bridges connecting the various islands and banks. Upon seeing these rudimentary maps, Euler would laugh his high-pitched childish laugh. ‘‘Silly townspeople,’’ he would say. ‘‘You need not draw these elaborate depictions of your towns for me. I need only a graph with edges that represent the bridges and vertices that represent the land masses. Just use your saliva to draw these pictures on the inside of a pram and bring it to me.’’ And so was born the mathematical field of graph theory, or at least a rudimentary saliva-based version of it. Upon the arrival of a drool-covered pram, Euler would spend several hours studying it, and consider various paths that might traverse each edge exactly once. Then he might do a few calculations. Finally, he would make a pronouncement about his results. Sometimes his news would be good, as for the town of Masterdol, which was built on the two sides of a river connected by two bridges. Once having been made aware of Euler’s solution, the townspeople could leisurely saunter first across the one bridge to the far side of the river and then back again across the other bridge, arriving at home just in time for a brandy nightcap. But sometimes the news was bad, as happened to the town of Winterflagen, where Euler explained that yes, in fact, you could take a stroll and cross every bridge exactly once. However, to do so meant that you would finish on the opposite side of the river from whence you had begun. So you should be sure to bring a sleeping roll and plenty of victuals as there would be no going home that night. Many a distraught townsperson was compelled to move from a home in Winterflagen to take up residency downriver in Masterdol, where the stroll was possible. And so the young bridge-stroller-problem-solver spent his early years solving bridge-stroller problems with ever greater
numbers of bridges. He handled three-bridge problems and four-bridge problems. And when he solved his first fivebridge problem, his fame grew to substantial proportions. Not satisfied with the level of difficulty of the problems that were brought to him, Euler began to travel to municipalities a greater distance away. Towns up and down the Wiese River both hoped for and feared a visit from Euler. For if he determined that the stroll was possible, the town experienced tremendous economic benefit from the hordes of tourists that descended on the town, eager to try the new walk. But if a town was unfortunate enough to have no such stroll, it meant economic strangulation, as no one desired to live there any longer. Then one day, a delegation arrived from the city of Ko¨nigsberg. ‘‘Great Euler,’’ said the mayor. We have heard of your amazing abilities. We need your help immediately. For we have numerous townsfolk who have already crossed bridges once, and are now trapped in strange parts of the city, unable to return home without crossing a bridge a second time. And many are the couples that began a stroll attempting to cross each bridge exactly once and ended up in France. ‘‘Other townspeople, fearful of a similar fate, are foregoing an evening stroll altogether. This is tearing asunder the very social fabric of the town, as the public discourse that occurred amongst the various strollers has vanished. And married couples are remaining indoors, thereby increasing the population at an unsustainable rate. Please come to help us solve our bridge-stroller problem.’’ ‘‘And how many bridges have you in Ko¨nigsberg?’’ asked Euler. ‘‘Not so many,’’ replied the mayor evasively. ‘‘How many is not so many?’’ demanded Euler. ‘‘There are seven bridges in Ko¨nigsberg,’’ admitted the mayor with downcast eyes. A gasp went up from the crowd. For seven bridges was a greater number than Euler had ever tackled before. People whispered that no one could possibly solve a seven-bridge problem. Euler waved his hand to silence the crowd. ‘‘All right,’’ he said. ‘‘I will come to Ko¨nigsberg to solve your problem.’’ A cheer went up from the crowd. And so, Euler set out for Ko¨nigsberg, in what was then Prussia. This flourishing commercial town was located right at the elbow of the Pregel River. The situation was complicated by the fact the river split at Ko¨nigsberg, and the city spanned both sides of the river as well as an island and the region between the branches of the river. Here was a challenge worthy of the great Euler. Upon arriving in town, he saw the pockets of people unable to return home because they had already crossed bridges once. They cried out to him, ‘‘Please, great Euler, solve our conundrum that we might return home to feed our starving children.’’ Euler called back to them, ‘‘I just got here. Give me a break.’’ As he often did when solving a new town’s bridge-stroller problem, Euler obtained a small wooden stool and placed it on the main bridge overlooking the river. He sat and sat and sat, his chin cupped in his hand, watching the water flow under the bridge. Presently, a beautiful young maiden approached him. Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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‘‘Pardon me sir,’’ she said. ‘‘But I am lost. I know not the city of Ko¨nigsberg. I have been out strolling for six hours, and I have not crossed the same bridge twice, but I cannot seem to get home.’’ Euler looked up at her with surprise. For it were as if he had been hit between the eyes with a large stump. He sprang up and grabbed her by the shoulders. ‘‘That’s it,’’ he shouted. ‘‘Of course. You cannot get home because the vertices are all of odd degree.’’ ‘‘Degree? Vertices?’’ she said nervously. ‘‘Look,’’ said Euler, slopping some of his saliva on the bridge wall. ‘‘Here is the graph that represents the town.’’ He pointed at one of the vertices. ‘‘You see, with an even number of edges, whenever you enter a vertex, you can always leave. But with an odd number of edges at a vertex, you can enter and leave that vertex until you are down to one edge. And then you can enter, but you can never leave. Or you can leave but never re-enter. So if the goal is to return to your home, you can never succeed if there is any vertex of odd degree. And if your goal is to take a stroll
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THE MATHEMATICAL INTELLIGENCER
crossing each bridge once, and you do not care if you return home, then you must have two vertices of odd degree. But for Ko¨nigsberg, all four vertices are of odd degree. So there is no such stroll in either case.’’ ‘‘I see,’’ she said, a disconsolate look clouding her pretty brow. ‘‘So I am trapped here.’’ ‘‘So it would seem,’’ replied Euler. ‘‘But perhaps we can make the best of it.’’ And so the couple was married, and they built a house right there on the main bridge over the Pregel River, and they lived happily ever after. Well, at least on and off. And when Euler’s negative solution of the Ko¨nigsberg bridge-stroller problem was announced, it actually was the birth of topology, or at least a rudimentary saliva-based version of it. And Euler was celebrated by many, and they called him the master of us all. But in their house over the river Pregel, his wife was in charge.
Editor’s Warning: The ratio of fact to fiction in this story is precariously low.
Mathematical Communities
‘‘The Word for World Is Forest’’: A LongRange Funding Source for Women in Math in Developing Countries NEAL KOBLITZ
he Word for World Is Forest is the title of a classic work of science fiction by Ursula K. Le Guin. Written in 1972 at a time of growing awareness of both the ecological importance of forests and the rights of indigenous peoples—the two themes of the novella—it was republished by Tor Books in 2010 in the wake of the hugely successful movie Avatar, which was partly based on Le Guin’s story. What do forests have to do with mathematics? Not much.1 However, mathematicians are often lovers of the outdoors. To cite just one example, a long tradition at the Mathematisches Forschungsinstitut Oberwolfach has been that every conference make time for a Wednesday afternoon hike in the Black Forest. In addition, forests are playing a central part in the plans of the Kovalevskaia Fund to provide a source of support for women in the mathematical sciences in developing countries through the 21st century and beyond.
T
Kovalevskaia in Mexico This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of ‘‘mathematical community’’ is the broadest. We include ‘‘schools’’ of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
â Please send all submissions to Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063 USA e-mail:
[email protected] The mathematician Sofia Kovalevskaia (1850-1891), whose name my wife Ann and I gave to the small foundation we started in 1985 (see http://kovfund.org), never traveled to Mexico or anywhere else in the western hemisphere. However, she received fan mail from as far away as El Salvador, and she would have been pleased that a Latin American country, namely Mexico, has the strongest tradition of honoring her memory through activities to promote mathematical research by women. Indeed, in 1991, as far as I’m aware, Mexico was the only country to organize a major conference in commemoration of the centenary of her death. In 2005 the Mexican Mathematical Society (MMS), in collaboration with the Kovalevskaia Fund, started giving annual grants to women doctoral students and junior researchers. The organization, publicity, and selection are handled by a special MMS committee, currently consisting of Drs. Marı´a Jose´ Arroyo, Carlos Bosch, Begon˜a Ferna´ndez, and Patricia Saavedra. The grants are of varying amounts and serve different purposes. For graduate students they typically provide ‘‘bridge money’’ in the latter stages of their Ph.D. work so that they don’t have to drop out for financial reasons. Postdocs often use the awards to finance travel to conferences or to work with a collaborator at a university or research center in Europe or North America. The grants are awarded every October during the opening ceremony at the MMS National Congress. In 2009, at the Congress in Zacatecas, the Kovalevskaia Grant Committee organized a Special Session on women in mathematics in Mexico and other countries of the region. Patricia Saavedra
1
This is not really true. I am currently collaborating with a mathematician in Jamaica and some researchers in my university’s School of Forest Resources on a project to use mathematical modeling to study carbon sequestration, habitat preservation, water resources, and other ecological variables in Jamaica’s forest preserves.
Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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DOI 10.1007/s00283-011-9237-0
reported on the statistics she had compiled on Mexican women’s participation in mathematics at all levels. Among the highlights: women math students have lower attrition than their male peers, so although in the early 2000s women were only 24% of entering Master’s students, they were almost 50% of those who graduated. In 2004, 14 of the 36 math doctorates (about 40%) went to women; this percentage is somewhat higher than in the U.S. and much higher than in Great Britain, Germany, and Scandinavia. There is, however, a ‘‘glass ceiling’’; and the percentage of women among recipients of Mexican government research grants in mathematics is extremely low, even lower than the percentage in the exact sciences overall. Another speaker at the seminar was Marı´a Luisa Sandoval. She was a Kovalevskaia grant recipient in 2005, the first year it was given. She had had to take time off from graduate school to care for her aged parents, and as a result had passed the age limit for both Mexican and Spanish graduate stipends. In her remarks she emphasized that without the Kovalevskaia grant she would never have been able to complete her doctorate.
AUTHOR
......................................................................... NEAL KOBLITZ received his Ph.D. from Princeton in 1974, and since 1979 he has been at the University of Washington. He works in number theory and cryptography, and is the coinventor of elliptic curve cryptography. He has written six books, of which the last one, Random Curves: Journeys of a Mathematician (Springer 2007), is autobiographical. In addition to his mathematical works, Neal has stirred up controversy with his writings about such topics as the misuse of mathematics in cryptography (see http://ano therlook.ca) and the negative American influences on higher education in Vietnam (see http://www.math.washington.edu/*koblitz/ vn.html). He also assists his wife Ann Hibner Koblitz in directing the Kovalevskaia Fund for women in science in developing countries (see http://kovfund.org). Ann is Professor of Women and Gender Studies at Arizona State University, where she teaches such courses as Gender and Science, Women as Healers, and Feminist Theory. She has written three books: A Convergence of Lives: Sofia Kovalevskaia – Scientist, Writer, Revolutionary; Science, Women, and Revolution in Russia; and (forthcoming) Sex and Herbs and Birth Control: Fertility Regulation Across Cultures and Through the Ages. Ann is director of the Kovalevskaia Fund.
University of Washington Seattle, WA 98195 USA e-mail:
[email protected] 22
THE MATHEMATICAL INTELLIGENCER
Ann Hibner Koblitz
Mexico is not the only country where the Kovalevskaia Fund supports awards for women science students and researchers; we also have successful projects in Vietnam, Peru, and Cuba. However, Mexico is the only country where the awards are administered by the math society and are exclusively for the mathematical sciences. And the Mexican grants are also the prototype for the sort of awards we envision the Kovalevskaia Fund making far into the future in many developing countries, particularly if, as we hope, the responsibility for the Fund’s projects is eventually transferred to the American Math Society.
Fundraising by Powers of 2 The impetus to start the Mexican Kovalevskaia awards came from a donation of $2500 per year from a former student of mine and his wife, who occupies a high position at Microsoft (and wishes to remain anonymous). According to company policy concerning charitable gifts by employees, Microsoft matches this donation every year. Soon after the program started, the MMS decided to match the $5000 it was receiving from the Fund, thereby showing that Mexican mathematicians, male as well as female, are truly committed to the project. Finally, in 2010 some mathematician/cryptographer friends of mine decided to match the $5000 per year that the Fund was sending to the MMS, and the MMS matches that as well. Thus, at present a total of $20,000 per year—an amount that goes a long way in Mexico—is being awarded by the MMS Kovalevskaia Grant Committee.
And Out to Infinity The first long-range goal that Ann and I have for the Kovalevskaia Fund is that all projects be funded at their current levels in perpetuity. We believe that after we die our house, life insurance, etc., will provide a large enough endowment to do this, whether or not other donors make provisions to continue their contributions forever. However, that endowment will not be enough for any major expansion of Kovalevskaia prizes and grants. We also have a more ambitious plan. Our hope is that investments we are making now will bear fruit later in the
century, providing a major boost in the Kovalevskaia endowment. As more money becomes available, the AMS (assuming that that is the organization administering the projects) will be able to collaborate with sister societies in a number of countries in Asia, Africa, and Latin America, adding new grants and increasing the funding in places where the impact has been the greatest. Our investment is in land in yet-to-be-discovered places in the American West. We own about 450 acres (180 hectares); except for a 20-acre (8-hectare) parcel of high desert on a mountain in Arizona, the rest of the land is in the forests of western Washington: (1) A 240-acre (96-hectare) mountainside just south of the border with Canada. It includes a 2-kilometer hiking trail and 0.5 km of logging roads from which views open up of the border towns and the snow-covered Canadian Coastal Range. (2) A 43-acre (17-hectare) parcel with frontage on the scenic Mount Baker Highway. It has extensive hiking trails with breathtaking views of the Nooksack River, the South Fork valley, and the Twin Sisters mountains.2 (3) A 50-acre (20-hectare) property on a steep flank of Lookout Mountain at the south end of Lake Whatcom. We also have four lots adjacent to the forest that are zoned for residential construction. (4) A 98-acre (39-hectare) property on a narrow peninsula that juts out into the Hood Canal near Dabob Bay. The views are dramatic, and there are several areas nearby that have protected status due to their importance as a watershed and as animal habitat. As in the case of our other properties, we chose this one in large part because of the biodiversity, vitality, and overall health of the forest. During our lifetime none of these properties will be put to any economic use. There is an English proverb He who plants pears [i.e., pear trees] Plants for his heirs. More generally, a recently-logged and replanted forest (this includes most of the acreage we purchased; only about 20% consists of older trees) will not be ready for another harvest for roughly a half-century. Our best guess, however, is that the vast majority of this land will never again be logged. We think that eventually over 80% will be put in a perpetual conservation easement, and the remaining 10%-20% will be cleared for low-density housing. We are wagering that, one way or another, this land will become quite valuable later in the century. In our opinion the Pacific Northwest is the most beautiful and livable region of the continental United States. This is likely to become more so as climate change causes other
places, especially in the Southwest, to become less desirable. Specialists expect the effects of global warming to be less severe in our region than elsewhere, and some are predicting an influx of ‘‘climate refugees.’’ There will be intense pressure to allow residential development on some of the attractive land near Seattle. At the same time, people in western Washington have a tradition of environmental activism, and the state and county governments have strong anti-deforestation policies. So a compromise will have to be worked out. The logical resolution would be to allow residential rezoning of a small proportion of a landowner’s forest property, provided that the remaining part is made into a preserve. Already some of the upscale, low-density residential developments include extensive forest acreage.3 Homeowners take pride in their stewardship of that land, and they can boast to their friends of being ‘‘carbon-negative.’’ This trend is likely to grow along with increased public appreciation of the importance of forests. Current land-use policy sees little difference between keeping land in commercial forestry (to be logged at halfcentury intervals) and putting it in a conservation easement (never to be logged). Both are regarded as preserving the forest. However, this will have to change, because in the long run there is a world of difference, for two reasons. In the first place, the ecological importance of a tree, which is not the same as its value as timber, depends on its biomass, its number of leaves, and the size of its canopy. These grow roughly as a cubic or quadratic (the latter after the tree reaches full height) function of time. In other words, from an ecological standpoint trees reach peak efficiency well after the age at which they would be logged.4 In the second place, the timber industry’s model of sustainability—based on the replanting of seedlings after clearcutting—will start to fail as the climate changes. Established healthy forests are resilient. They can usually withstand temperature increase and reduced rainfall, increased windstorms and wildfires, and new insects and pathogens. A baby forest made up of newly planted seedlings is much more sensitive, and its odds of survival diminish as the climate becomes less favorable for forest growth. For both of these reasons, in the latter decades of the century mature forests will be far more viable and valuable than young ones.
In the Meantime If our calculation is correct, the future directors of the Kovalevskaia Fund will hold onto each property until it can be sold for vastly more than what we paid for it. If we’re wrong, then these properties will bring in only modest increases in the endowment and in the Kovalevskaia awards. In the meantime, the forests serve a different purpose— recreation. Maintaining and improving the forest lands—
2
This property can be viewed on http://www.youtube.com; search for ‘‘Elliptic Curve Cryptography: The Serpentine Course of a Paradigm Shift.’’ There have also been projects that combine land conservation and affordable housing; see Kendra Briechle’s study of such initiatives, available at http://www.conservationfund.org/publications/improving_nature_of_affordable_housing 4 This is a not-completely-accurate and in any case greatly oversimplified explanation of the importance of old forests. A comprehensive treatment of this and other issues can be found in D. A. Perry, R. Oren, and S. C. Hart, Forest Ecosystems, 2nd ed., The Johns Hopkins University Press, 2008. 3
Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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clearing trails, cutting back invasive species, clearing blocked culverts, hiking all over, monitoring the forest as it matures— can sometimes be hard work, but it’s also thoroughly enjoyable. Any readers of The Mathematical Intelligencer who are planning to be in the Seattle area are welcome to contact me
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THE MATHEMATICAL INTELLIGENCER
and arrange to come with us to spend an afternoon visiting one or two of our forests. We like to show them to our friends and colleagues, and we think we can offer you an experience that’s at least as stimulating as a Wednesday hike in the woods near Oberwolfach.
Mathematical Entertainments
Michael Kleber and Ravi Vakil, Editors
magic square is a square array of numbers so arranged that their sum taken in any row, any column, or along either diagonal, is the same. Figure 1a shows a famous example, the ‘‘Lo shu’’, a 3 9 3 specimen of Chinese origin dating from the 4th century BC.
Geometric Magic Squares
A
LEE SALLOWS
(a) This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
â
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-mail:
[email protected] (b)
Figure 1. The Lo shu magic square in numerical form (left) and in geometrical form (right).
The topic enjoys an extensive literature: books, articles and websites abound. By now we might expect that virtually every aspect of these curiosities had been exhaustively explored. Nevertheless, in 2001 I hit upon an innovation that has cast magic squares in an entirely new light. Viewed anew, numerical magic squares are better understood as a special instance of a wider class of geometrical magic squares. Traditional magic squares featuring numbers are then revealed as that particular case of such a ‘‘geomagic’’ square in which the elements are all one-dimensional (1-D), which is to say, they are straight-line segments of a given length. Consider, for example, a spatial equivalent of the Lo shu seen in Figure 1b, in which line segments of length 1,2,3,.. replace like-valued numbers in each cell. The three lines occupying each row, column, and diagonal can be joined head to tail so as to form or ‘‘pave’’ the same straight line segment of length 15. But just as line segments can pave longer line segments, so areas can pave larger areas, volumes can pack roomier volumes, and so on up through higher dimensions. In traditional magic squares we add numbers to form a constant sum, which is to say, we ‘‘pave’’ a one-dimensional space with onedimensional ‘‘tiles’’. What happens beyond the one-dimensional case? Figure 2 shows a 3 9 3 two-dimensional (geo)magic square, its cells occupied by nine distinct planar or 2-D shapes or ‘‘pieces’’. Any three entries in a straight line can be assembled to pave an identically shaped region known as the target, in this case a 6 9 6 square, as shown to right and below in the figure. Note how some pieces appear one way in one target, while flipped and/or rotated in another. Thin grid lines on pieces within the square help identify their precise shape and relative size.
Ó 2011 The Author(s). This article is published with open access at Springerlink.com, Volume 33, Number 4, 2011
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DOI 10.1007/s00283-011-9229-0
Figure 3. A 3-D geomagic square of order 3. The 3 pieces in any row, column or diagonal pack the same 3 9 3 9 3 cube. Figure 2. A 2-D geomagic square of order 3. The 3 pieces in any row, column or diagonal tile the same 6 9 6 square target.
Analogously, 3-D magic squares in which solid pieces combine to form a constant 3-D target can also be found. I have one before me as I write, the target of which is a cube; see Figure 3. Likewise geomagic squares using higher dimensional entries also exist, if less easy to visualize. By the dimension of a geomagic square we refer to the dimension of its entries. For a formal definition of geomagic squares, see http://www.GeomagicSquares.com/, which also includes a large gallery of geomagic squares exhibiting a rich variety of special properties. Staying with 2-D types, an array of N 9 N planar pieces is called ‘‘magic’’ when the N entries occurring in each row, column, and both main diagonals, can be fitted together jigsaw-wise to tile an identical region without gaps or overlaps. In tessellating this target, pieces may be rotated or reflected. Below we shall see that pieces may also be disjoint or disconnected. As with numerical magic squares, geomagics showing repeated entries are deemed trivial. Rotated or
AUTHOR
......................................................................... LEE SALLOWS was born in 1944 and was
raised in post-war London. He has lived in Nijmegen, The Netherlands, for the past 40 years. Until recently he worked as an electronics engineer for the Radboud University. A handful of published articles on computational wordplay and recreational mathematics are the only fruits of an idle, if occasionally inventive, life. Johannaweg 12 Nijmegen 6523 MA The Netherlands e-mail:
[email protected] 26
THE MATHEMATICAL INTELLIGENCER
reflected versions of the same specimen are counted identical. A square of size N 9 N is said to be of order N. As we have seen, every numerical magic square corresponds to a 1-D geometrical magic square written in shorthand notation. But this is not to say that numerical squares account for all possible 1-D geomagic squares. In fact they account only for that subset of 1-D squares using connected line segments. Figure 4 shows a 1-D geomagic square of order 3 that includes disjoint pieces, or pieces composed of two or more separated islands bearing a fixed spatial relation to each other. The overall shape of the compound piece is thus preserved even if moved. Here the 1-D lines have been broadened and coloured to enhance clarity, a trick that could obviously be extended so as to yield a true 2-D geomagic square sporting rectangular targets. However, the point to be made here is that Figure 3 is a 1-D geomagic square for which there exists no corresponding numerical magic square. Magic squares using numbers thus account for no more than a small fraction of all 1-D geomagic squares. There is a second way to create a geometrical analog of any numerical magic square, which is to use circular arcs or sectors of appropriate angle, rather than straight line segments. Figure 5 shows an example based on the Lo shu. Since the constant sum is 15, the smallest sector subtends an angle of 360 7 15 = 248. Clearly the target could be replaced by a
Figure 4. A one-dimensional geomagic square of order 3 using (thickened) disjoint line segments. The target is of length 12 units.
c+a
c −a −b
c+b
c −a +b
c
c + a −b
c −b
c+a +b
c −a
Figure 7. An algebraic generalization of numerical magic squares of order 3, due to the renowned French mathematician E´douard Lucas [1842–91].
Figure 5. A geometrical version of the Lo shu using circular, rather than linear segments. Here the target is a complete circle but could have been any desired fraction thereof.
regular 15-gon, the sectors then changing to 15-gon segments of corresponding size. Further possible targets may occur to the reader. As before, circular arc pieces do not have to be connected. Figure 6 shows a 3 9 3 square using disjoint arcs, their unit segments here simplified into single coloured dots. Once again, such disconnected pieces cannot be represented by
Figure 6. A semi-panmagic square of order 3 using disjoint circular arcs. The latter are represented by unit colored dots of size 360/15 = 24 degrees.
single numbers. But Figure 6 is of greater interest in demonstrating an important, if unsurprising, fact, namely that 2-D geomagic squares listen to laws different from those holding for 1-D types. As shown by the two targets at top, Figure 6 is a ‘‘semipanmagic’’ square. That is, in addition to rows, columns, and main diagonals, the target is tiled by the ‘‘broken’’ diagonals AFH and CDH. However, the impossibility of a 3 9 3 numerical semi-panmagic square is shown by Figure 7, which is Lucas’ general formula that describes the structure of every 3 9 3 numerical magic square. As with AFH in Figure 6, suppose now that a,b,c in Figure 7 are assigned values such that (c + a) + (c + a - b) + (c + a + b) = 3c = the magic constant. But then a equates to zero, which entails c + a = c - a = c, meaning repeated entries. A non-trivial solution therefore does not exist. It was in fact Lucas’s formula that first led to the idea of geomagic squares. Such algebraic formulas had long held for me a peculiar fascination. As I put it in an unpublished essay (Magic Formulae, 1980) on the topic, ‘‘Every algebraic square is like an x-ray photograph exposing a skeletal structure underlying the numbers.’’ A vague notion of finding some kind of graphical representation that would make that skeleton visible haunted me for years. Twenty years later, thinking once again about Lucas’s formula, I hit on a new approach. Suppose the three variables in the formula are each represented by a distinct planar shape. Then the entry c + a could be shown as shape c appended to shape a, whereas the entry c - a would become shape c from which shape a has been excised. And so on for
Figure 8. This pictorial representation of Figure 7 first prompted the idea of a geomagic square. Ó 2011 The Author(s), Volume 33, Number 4, 2011
27
Figure 10. A geomagic square using polyominoes of sizes 1–9. The very existence of such a square had once seemed a daring idea. In reality this is one of 1411 similar solutions, all with a same 3 9 5 target.
Figure 9. A true geomagic square derived by interpreting Lucas’s formula as a template.
the remaining entries. A back-of-the-envelope trial then led to Figure 8, in which a is a rectangle, b a semi-circle, and c a relatively larger square, three essentially arbitrary choices. This result was more effective than anticipated, the match between keys and keyholes making it easy to imagine the pieces interlocking, and thus visually obvious that the total area of any three in a straight line is the same as a rectangle of size 1 9 3, or three times the area of the central piece, in agreement with the formula. However, the fact that the three central row and three central column pieces do not fit together to complete a rectangle, as the pieces in all other cases will, now seemed a glaring flaw. The desire to find a similar square lacking this defect was then inevitable, and the idea of a geometric magic square was born. Figure 9 shows a second attempt that made good the shortcomings of the first. Note that, like Figure 9, 5 is itself a geometrical analog of Lucas’s formula, the variables a, b, and c then corresponding to circular segments of 728, 248, and 1208, respectively. And the same will go for variants of Figure 5 using alternative targets. However, although it is natural to regard all such trivial variants as essentially the same geomagic square, we should hardly describe Figures 5 and 9 as equivalent, even though they share a common algebraic ancestor. In fact a clear definition of equivalence has thus far proved an elusive quarry, a shortcoming that can sometimes reveal itself in a degree of ambiguity. The problem of how to go about producing new geometric magic squares now took centre stage. Following much deliberation on this question, two approaches gradually emerged: (1) pencil and paper methods based on algebraic 28
THE MATHEMATICAL INTELLIGENCER
templates, along the lines just mentioned, (2) in the case of squares restricted to polyforms or shapes built up from repeated atoms, brute force searches by computer. Foremost among the polyforms are polyominos (built up from unit squares), polyiamonds (equilateral triangles) and polyhexes
Figure 11. A further example of a square using pieces of consecutive size, in this case polyhexes. Here the latter have been reduced to dots and lines to produce a diagram reminiscent of a well known Chinese rendering of the Lo shu.
Figure 12. A near miss at a geomagic square of order 2. The question of whether or not there exists a fully magic solution occupied me for years.
(regular hexagons). On the aforementioned website I present a selection of some of the more interesting squares brought to light by these two methods. In most cases, the examples shown have been discovered in response to some pertinent question, such as: Does there exist a 3 9 3 square composed of nine polyominos with sizes in consecutive order? As a trophy-hunter, I found the prospect of getting such an exotic gem enticing. Often such questions entailed weeks of work before arriving at an answer. In this case, the outcome shattered every expectation. Figure 10 presents one of the 1411 different solutions, target in each case being a 3 9 5 rectangle. And if this prolixity was surprising, what to make of Figure 11, which is among 169,344 alternatives, the all using 9 polyhexes of the same size [here reduced to nodes linked by lines] and the same target? With a single exception, the 2-D squares to follow are all of size 3 9 3 or 4 9 4, larger squares being to my mind of scant interest. It is a common fallacy that the bigger the square the greater the achievement, because of the supposed difficulty of getting so many numbers to comply with the magic conditions. On the contrary, the constraints implied diminish rapidly with increasing size, as is shown by the algebraic generalization of the N 9 N numerical magic square, which can be written so that it contains N 2 – 2N cells each containing a single free variable. Turning to the other end of the scale, clearly a magic square of size 2 9 2 cannot be realized using four distinct numbers. The smallest numerical magic squares are thus of order 3, and the same is true of ‘‘semi-magic’’ squares, which are those that are magic on rows and columns only. However, Figure 12 shows a non-trivial 2 9 2 semi-magic square using 2-D pieces. It is based on a finding due to Michael Reid. Note that besides rows and columns, one diagonal is magic.
Figure 13. The first ever 2 9 2 geomagic square due to Frank Tinkelenberg of the The Netherlands. The square uses disconnected pieces and a disconnected target. Does there exist a solution using connected pieces? The question remains unanswered.
Figure 14. A deceptively simple-seeming geomagic triangle. The discovery of such specimens is a lot harder than first sight suggests.
Ó 2011 The Author(s), Volume 33, Number 4, 2011
29
Figure 15. Order 3 geomagic squares using pieces of the same area are far rarer (and thus more difficult to find) than those using unequal pieces. This one uses nine hexominoes. In searching for such specialities different target shapes must be tried. The result in this case was felicitous.
Figure 17. Here the title, Dudeney Type X, is a reference to H. E. Dudeney, the famous British counterpart to America’s Sam Loyd, and author of many wonderful puzzle books during the late 19th and early 20th centuries. Dudeney’s original work on numerical magic squares included a classification of the 880 normal squares into 12 types, depending upon how their complementary pairs, 1 and 16, 2 and 15, etc., were distributed. In his system, the above square is of Type X, or type ten.
Figure 16. This magic jigsaw puzzle is an example of what I call a ‘‘self-interlocking’’ geomagic square. The 16 pieces are no longer separated from each other within their cells, but interlock so as to pave a single square area. I had never imagined that such a structure was possible until an examination of the geometrical analogues of certain algebraic magic squares forced their existence upon me. The visual harmony of the square is a reflection of the symmetries to be found in the algebraic magic square on which it is based.
Figure 18. A 3 9 3 panmagic or nasik square, which is one in which every diagonal, broken or otherwise, is magic. In this case, the target can also be formed by any three of the four corner pieces. This square was of particular interest to me because, in the realm of numerical magic squares, panmagics of 3 9 3 are impossible. The possibility of finding 2-D panmagics of 3 9 3 was thus exciting and their initial discovery an event to celebrate. The resort to disjoint pieces is an indication of the difficulty encountered in finding it. Such nuggets are thin on the ground. In an as yet unpublished paper I prove that the nine entries in any 2-D panmagic 3 9 3 square can always be rearranged to yield 54 distinct panmagic squares, rotations and reflections not counted.
30
THE MATHEMATICAL INTELLIGENCER
Figure 20. A computer-discovered 4 9 4 geomagic square. Every magic line contains three hexominoes and one heptomino. 3 9 6 + 7 = 5 9 5, the area of the square target. Figure 19. The target here is a variation on the first ever impossible figure that depicted a similar ‘‘object’’, but using 9 rather than 15 cubes. This was invented by Oscar Reutersvard in 1934. Later, in 1958, Penrose and Penrose, unaware of Reutersvard’s work, published an equivalent figure composed of three solid beams, nowadays known as the Penrose tribar. The above is one of two solutions using the same target and similar ‘‘pieces’’ of size 1,2,..9. The idea of such a target had occurred to me long ago. Following countless failed attempts, I finally found a way to do it, eight years later.
Until very recently, every attempt to discover a fully magic 2-D square of order 2 had failed. However, following an airing of the problem with the recent launch of my website GeomagicSquares.com, Frank Tinkelenberg, a Dutch software developer, finally cracked the problem with a square using disconnected pieces and a disconnected target; see Figure 13. In any case, the extreme difficuly met with in tracking down this solution is merely further confirmation of the point just made, that the smaller the square, the greater the constraints, or the fewer the degrees of freedom. The question now remaining is whether or not there exists a 2 9 2 square using connected pieces? Meanwhile, a slightly related device can be seen in the magic triangle of Figure 14. Figure 3 illustrated the 3-D square with cubic target referred to earlier. With a little patience the precise shapes of
the pieces can be inferred, although the deficiencies of trying to present these and higher-dimensional specimens via the page will be apparent. Two dimensional squares, on the other hand, present no such difficulty, being not only almost self-explanatory, but both elegant and ornamental besides. Hence my focus on 2-D squares in the present article. Although I am no artist, in creating pictures I have taken pains to present each square to the best effect. But appearances should not distract. Fundamentally, every square is a timeless Platonic form, a constellation in the firmament of logical space consisting in a nexus of geometrical relations. The latter are of no particular significance perhaps. But for all that, they remain among the immutable and eternal patterns woven into the magic carpet of mathematics. The geomagic squares below are taken from the Gallery of Lee Sallows’s website http://www.GeomagicSquares.com/, which includes a wealth of further examples, among them Figures 15– 20, here reproduced. OPEN ACCESS
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Ó 2011 The Author(s), Volume 33, Number 4, 2011
31
Removing Magic from the Normal Distribution and the Stirling and Wallis Formulas MIKHAIL KOVALYOV
T
he Wallis formula is often written as
described in McCartin (2006) as ‘‘providing an intriguing connection between p and e.’’ To obtain (1.2) we rewrite (1.1b) as
2 2 4 4 6 6 8 8 10 10 12 12 14 14 p ¼ 1 3 3 5 5 7 7 9 9 11 11 13 13 15 2 ð1:1aÞ with the meaning that the sequence of partial products p1 ¼ 21 ; p2 ¼ 21 23 ; p3 ¼ 21 23 43 ; p4 ¼ 21 23 43 45 ; p4 ¼ 21 23 43 4 6 p 5 5 ; of the left-hand side of (1.1a) converges to 2 : Since h 2n 2 i2 2 ðn!Þ pffiffiffiffiffiffiffiffiffi ; the partial product of the first 2n terms is p2n ¼ ð2nÞ! 2nþ1 we may rewrite (1.1a) as
ðfm Þ2 pffiffiffiffiffiffi ¼ 2p; m!þ1 f2m lim
Since fm [ 0;
ð1:1bÞ
which is often used to obtain the ubiquitous Stirling formula pffiffiffiffiffiffi e m m! pffiffiffiffiffi ¼ 2p; lim m m!þ1 m m
32
ð1:2Þ
THE MATHEMATICAL INTELLIGENCER Ó 2011 Springer Science+Business Media, LLC
DOI 10.1007/s00283-011-9259-7
¼
e
mþ0:5
ð1þm1 Þ fm is positive and monotonically decreasing and as such must have a non-negative limit. To show that lim fm 6¼ 0; m!þ1
m consider the sequence gm ¼ ðm1Þf ; which has the same m mþ0:5 2 gmþ1 m [ 1; limit as the sequence fm. Since gm ¼ me 2m1 mþ1
then 22n ðn!Þ2 pffiffiffi pffiffiffi ¼ p; lim n!þ1 ð2nÞ! n
fmþ1 fm
e m m! pffiffiffiffiffi: ð1:3Þ mm m mþ0:5 m ¼ e mþ1 \1; sequence fm ¼
lim gm [ g2 [ 0 and thus
m!þ1
lim fm [ 0: Formula
m!þ1
(1.2) is obtained by simple application of the laws of limits 2 lim fm pffiffiffiffiffiffi ðfm Þ2 m!þ1 to (1.3) as follows: 2p ¼ lim f2m ¼ lim f2m ¼ m!þ1
m!þ1
lim fm : Thus (1.2) and (1.1) are equivalent in the sense
m!þ1
that each one of them implies the other. Of many applications of the Stirling formula, one is the derivation of the normal distribution as the limiting case of
the binomial distribution. In a typical derivation found in many textbooks, one considers an infinite row of cells numbered by integers and a one-dimensional random walk of a point P that starts at the cell K = 0 and at each step jumps from the point it occupies to either the right or the left adjacent cell with probability 12 :
For N sufficiently large and jK j N ; formula (1.2) allows pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi us to approximate N ! 2pN N N e N ; N K pðN K Þ 2 ! N K N K N K 2 e 2 ; which upon substitution into 2N N þKN! ! N K ! 2 ð 2 Þð 2 Þ qffiffiffiffiffi 2 2 K2N leads to the approximation 2N N þKN! ! N K ! pN e of the ð 2 Þð 2 Þ binomial distribution by the normal distribution.
The probability of finding point P at a cell K after N steps is given by the following table:
The nonzero entries are 21N multiples of the binomial coefficients N þK !N !N K ! ; jK j 6 N ; written as even functions ð 2 Þð 2 Þ of K. The probability P(K, N) of finding point P at a cell labeled K is given by ( PðK ; N Þ ¼
N! if jK j 6 N ; N K is even; N K ; ðN þK 2 !Þð 2 !Þ 0; otherwise: 2N
ð1:4Þ
AUTHOR
......................................................................... MIKHAIL KOVALYOV was born in Minsk,
Belarus, and began his education there. After emigrating in 1980, he obtained his doctorate at the Courant Institute in 1985. He was on the faculty of the University of Alberta, Canada, from 1987 to 2011, but has now moved to South Korea. Much of his research is in PDEs. Sungkyunkwan University 300 Cheoncheon-dong, Jangan-gu Suwon City South Korea e-mail:
[email protected] The given derivation of the normal distribution is based on formula (1.2) with all difficulties buried in (1.2), or equivalently in (1.1). One would assume that formulas as fundamental as (1.1) and (1.2) had an intuitive proof, yet as pointed out in Gowers (2008), all proofs of (1.1) seem to contain a non-intuitive step with an identity or an estimate magically pulled out of a hat. Attempts to find a simple intuitive proof have led to a rather large number of publications, some of which are listed in Bibliography, yet none seems to be fully intuitive. Most assume that formulas (1.1) are known and try to construct an appropriate proof. But is it possible to arrive at formulas (1.1) in a completely natural way without any magical steps? The author of this paper thinks it is. The proof to be presented was outlined in Kovalyov (2009). The intuitions appealed to are largely from probability; this seems wholly appropriate for this area. For simplicity’s sake, let us take N ¼ 2n is even :
ð2:1Þ
Then
8 k Q > ð2nÞ! ðn!Þ2 nðkjÞ > > ¼ 22nð2nÞ! > nþj 22n ðn!Þ2 ðnþkÞ!ðnkÞ! ðn!Þ2 > > j¼1 < k Pð2k; 2nÞ ¼ Pð2k; 2nÞ ¼ Q 1kj n > ¼ 22nð2nÞ! if 0 6 k 6 n; j ; > 2 > ð n! Þ 1þn > j¼1 > > : 0; if k [ n:
ð2:2Þ
Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
33
The coefficients P(2k, 2n) also satisfy n X
60:5
n0:5þe \k6n j¼1
n X
ð2nÞ! Pð2k; 2nÞ ¼ 2n ðn þ kÞ!ðn kÞ! 2 k¼n k¼n 1 1 2n þ ¼ 1: ¼ 2 2
ð2:3Þ
jk
which lead to
j
j n n 1 kj n e ;1 þ n e ; k kj Q 1 n j¼1
1þnj
2n k2 ðn!Þ2 e n ; 2 ð2nÞ! Pð2k; 2nÞ
n 22n ðn!Þ2 X pffiffiffi Pð2k; 2nÞ ð2nÞ! n k¼n |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} ¼1 due to ð2:3Þ
k 1 nþk
"
k #n k n 1þ ¼ 0:5 n n0:5þe \k6n |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} X
replacing k with
The main idea of the derivation of (1.1b) is to estimate P(2k, 2n) in (2.2) by using the rather intuitive approximations
k Y
X
n X 1 k2 pffiffiffi e n n k¼n |fflfflfflfflfflfflfflffl fflp {zfflfflfflfflfflfflfflffl ffl} ffiffi
kj
1 n 1þnj
e
60:5
" 60:5
k
2
1
1
n
0:5þe
makes it
larger
n0:5e
n0:5e #n2e
1 þ 0:5e n |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
X
16
n0:5þe \k6n
0:5n : 2n2e
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} \n
Inside the core P(2k,2n) satisfies k2
approaches p as n!þ1
1þ
n0:5e #n2e
1e \12 for large n
; and
: These in
X n0:5þe \k6n
e n ;
kn
"
e n e n
0:5þ3e
22n ðn!Þ2 k2 0:5þ3e Pð2k; 2nÞ6e n e n 6 ; if jkj 6 neþ0:5 : ð2nÞ! ð2:5Þ
turn imply (1.1). The problem is that the approximations are valid only for jjj 6 jkj n; jk jj 6 jkj n, and as k gets close to ± n the approximations lose their validity for values of j and k - j close to n. To turn these ideas into a rigorous proof we break up the set -n O k O n into the core jkj 6 neþ0:5 and two tails later, so neþ0:5 \jkj 6 n; with 0\e\0:5 to be determined jk kj j n ;1 þ e that inside the core approximations 1 n n j e n are valid, whereas the total probability of P being outside the core approaches zero as n ! þ1:
For large n the total probability of P being in one of the two tails is X X P ð2k; 2nÞ ¼ P ð2k; 2nÞ 1 jkj\n0:5þe
jkj>n0:5þe
¼2
X
2e
P ð2k; 2nÞ 6 2n n; ð2:4Þ
k>n0:5þe
and hence goes to 0 as n ! þ1: Indeed, X
P ð2k; 2nÞ ¼
k>n0:5þe
ð2nÞ! 2n 2 ðn!Þ2 |fflfflfflffl{zfflfflfflffl}
X
k Y n ðk jÞ
n0:5þe \k6n j¼1
Due to P(-2k, 2n) = P(2k, 2n) it suffices to prove (2.5) for k > 0: To do so we employ the inequality 2
e xx 6 1 þ x 6 e x ;
To prove (2.6a) notice that functions w1(x) = e - 1 - x, 2 w2(x) = 1 + x - ex-x are analytic and satisfy w1 ð0Þ ¼ w2 ð0Þ ¼ w10 ð0Þ ¼ w20 ð0Þ ¼ 0; w100 ð0Þ ¼ w200 ð0Þ ¼ 1: Thus each of them must be of the form 0.5x2 + o(x2) [ 0 for x sufficiently small. j Inequality (2.6a) applied with x ¼ kj n and x ¼ n gives us e
ð Þ 6 1 k j 6 e kj n ; n
kj kj 2 n n
j
X
k Y
k Y
X nkþj \0:5 1 ¼ 0:5 n þ j n0:5þe \k6n j¼1 n0:5þe \k6n j¼1
k nþj |ffl{zffl}
THE MATHEMATICAL INTELLIGENCER
j j 6 en : n
ð2:6bÞ ð2:6cÞ
Dividing (2.6b) by (2.6c) we obtain k
34
j 2
e n ðnÞ 6 1 þ
replacing j with k makes it larger
ð2:6aÞ x
nþj
this term is less than 0:5
for jxj 1:
e n
ðkjÞ2 n2
6
1 kj n 1 þ nj
k
j2
6 e n þ n2 ;
ð2:6dÞ
which yields 2
e
kn
k P ðkjÞ2
n2
j¼1
¼e
k
P k
nþ
j¼1
ðkjÞ2 n2
k Y 1 kj n
6
1 þ nj
j¼1
¼e
2 kn
þ
k
P
6 e j¼1
2
nk þ nj 2
1
k P j2 j¼1
n2
ð2:6eÞ
:
Using
2
Zþ1
¼
k3 ðn0:5þe Þ3 6 6 n0:5þ3e ; 6n2 n2 ¼
6 4n0:5
3 e
ex
Zþ1
2 y 2
dxdy ¼
2
e r rdr
Z2p
5e n
e
Since the first and third terms of (2.9) approach 1 as n ! þ1; we conclude that pffiffiffiffiffiffi k2 lim Pð2k; 2nÞ pne n ¼ 1; ð2:10aÞ
ð2:7aÞ
X
2 kn
# e n
6
The author would like to thank the reviewer for suggestions that improved the manuscript.
22n ðn!Þ2 pffiffiffi ð2nÞ! n
jkj6n0:5þe
" n
P
0:5
e
2 kn
ð2:7bÞ
# e
n
P
6
logarithmic constant e, Mathematical Intelligencer 20, no. 4, 25-29.
:
Pð2k; 2nÞ
Brun, V., 1951. Walliss og Brounckers formler for p (in Norwegian),
jkj6n0:5þe
Norsk matematisk tidskrift 33, 73-81. Coleman, A. J., 1951. A Simple Proof of Stirling’s Formula, American
If we take
Mathematical Monthly 58, 334-336.
1 0\e\ 6 then lim
n!þ1
X
BIBLIOGRAPHY
Brothers, H., Knox, J., 1998. New closed-form approximation to the
0:5þ3e
jkj6n0:5þe
ð2:10bÞ
ACKNOWLEDGMENTS
0:5þ3e
Pð2k; 2nÞ
1 k2 Pð2k; 2nÞ pffiffiffiffiffiffi e n ; pn
providing us with the simplest case of the Central Limit Theorem.
Pð2k; 2nÞ becomes
jkj6n0:5þe
P
d/ ¼ p:
We may rewrite (2.5) as pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi k2 np np 0:5þ3e ð2nÞ! 0:5þ3e ð2nÞ! n 6 en e n 6 Pð2k; 2nÞ np e : 22n ðn!Þ2 22n ðn!Þ2 ð2:9Þ
and consequently
;
jkj6n0:5þe
n0:5
0
n!þ1
0:5þ3e
which upon division by P
2
e r rdrd/
¼2p
jkj6n0:5þe
"
2
ey dy
1
Zþ1 Z2p 0
¼12
valid for 0 \ k 6 n0:5þe ; we may further simplify (2.6e) to (2.5). Multiplying (2.5) by p1ffiffinffi and summing over k, we obtain 2 3 X 22n ðn!Þ2 X k2 0:5þ3e 4n0:5 pffiffiffi e n 5e n 6 Pð2k; 2nÞ ð2nÞ! njkj6n0:5þe jkj6n0:5þe 2 kn
1
ZZ
Zþ1
2
ex dx
0 0 |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflffl{zfflffl}
k 3 ðn Þ 6 6 n0:5þ3e 2 2 n n
X
Zþ1
et dt5 ¼
R2
0:5þe 3
2
32
2
1
k 1 X kðk þ 1Þð2k þ 1Þ 2k 3 þ 3k 2 þ k j2 ¼ ¼ 2 n j¼1 6n2 6n2
6
from multivariable calculus; it is proved by the string of identities 4
k 1 X kðk 1Þð2k 1Þ 2k3 3k2 þ k 2 ðk jÞ ¼ ¼ n2 j¼1 6n2 3n2
6
pffiffiffi must also exist and be equal to p; thus Zþ1 proving (1.1b). pffiffiffi 2 e t dt ¼ p was imported Notice that the identity 2 22n ðn! pÞffiffiffi ð2nÞ! n
term
lim e n
ð2:8Þ
Formula, American Mathematical Monthly 93, 123-125. Feller, W., 1968. ‘‘Stirling’s Formula’’, § 2.9 in An Introduction to
0:5þ3e
n!þ1
Pð2k; 2nÞ ¼ 1
due
¼ lim e n
0:5þ3e
n!þ1
to
(2.3)
and
¼ 1;
(2.4),
jkj6n0:5þe
Zþ1 X 2 pffiffiffi 1 2 kn lim pffiffiffi e ¼ e t dt ¼ p;and as n ! þ1 n!þ1 n jkj 6 n0:5þe ; k is even
Diaconis, P., Freedman, D., 1986. An Elementary Proof of Stirling’s
1
the limits of the first and third terms in (2.7b) exist and are pffiffiffi equal to p: But this implies that the limit of the middle
Probability Theory and Its Applications, Vol. 1, 3rd ed., New York: Wiley, pp. 50-53. Feller, W., 1967. A Direct Proof of Stirling’s Formula, American Mathematical Monthly 74, 1223-1225. Gowers, T., 2008. http://gowers.wordpress.com/2008/02/01/remov ing-the-magic-from-stirlings-formula/ Kovalyov, M., 2009. Elementary Combinatorial-Probabilistic Proof of the Wallis and Stirling Formulas, Journal of Mathematics and Statistics, 5, no. 4, 408-410.
Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
35
Mavromatis, H., 1992. The Wallis Formula for p, Including Two New
Romik, D., 2000. ‘‘Stirling’s Approximation for n!: The Ultimate Short
Expressions Involving Irrationals’’, International Journal of Computer Mathematics 43, nos. 3, 4, 197-203.
Proof?’’, American Mathematical Monthly 107, 556-557. Shozo Niizeki; Makoto Araki, 2010. ‘‘Simple and Clear Proofs of
McCartin, B., 2006. e: The Master of All, Mathematical Intelligencer 28,
Stirling’s Formula’’, International Journal of Mathematical Educa-
no. 2, 10-21. Miller, S., 2008. ‘‘A Probabilistic Proof of Wallis’s Formula for p’’, American Mathematical Monthly 115, 740-745. Moritz, R. E., 1928. ‘‘An Elementary Proof of Stirling’s Formula’’, Journal of the American Statistical Association 23, no. 161, 55-57.
tion in Science and Technology 41, no. 4, 555-558. Sondow, J., 2005. ‘‘A Faster Product for p and a New Integral for ln p2’’, American Mathematical Monthly 112, 729-734. Wastlund, J., 2005. ‘‘An Elementary Proof of Wallis Product Formula for p’’, Linkoping studies in mathematics, no. 2.
Mortici, 2010. ‘‘New Improvements of the Stirling Formula’’, Applied Mathematics and Computation 217, nos. 2, 15, 699-704.
Yaglom, A. M. and Yaglom, I. M., 1953. An Elementary Derivation of the Formulas of Wallis, Leibnitz, and Euler for the Number p (in
Robbins, H., 1955. ‘‘A Remark on Stirling’s Formula’’, American
Russian), Uspekhi Matematicheskikh Nauk 8, nos. 5, 57,
Mathematical Monthly 62, 26-29.
36
THE MATHEMATICAL INTELLIGENCER
181-187.
Years Ago
David E. Rowe, Editor
An American Goes to Europe: Three Letters from Oswald Veblen to George Birkhoff in 1913/ 1914 JUNE BARROW-GREEN
Years Ago features essays by historians and mathematicians that take us back in time. Whether addressing special topics or general trends, individual mathematicians or ‘‘schools’’ (as in schools of fish), the idea is always the same: to shed new light on the mathematics of the past. Submissions are welcome.
â
Send submissions to David E. Rowe, Fachbereich 08, Institut fu¨r Mathematik, Johannes Gutenberg University, D-55099 Mainz, Germany. e-mail:
[email protected] he American mathematician Oswald Veblen spent the academic year 1913–1914 in Europe. He was 33 years old, had been a Professor at Princeton for three years, had been married for five, and had several publications to his credit. Travelling with his wife, Elizabeth, his itinerary, which began in Scandinavia, centred on the mathematical centres of Go¨ttingen, Berlin, and Paris, and included Italy and England. While he was overseas, Veblen, an inveterate and accomplished correspondent, wrote to his fellow mathematician and friend George Birkhoff, describing the people he had met and the places he had visited.1 Veblen had known Birkhoff since the early 1900s when they had both been students in Chicago. They had each written a thesis under the direction of E. H. Moore2—Veblen on the foundations of geometry (1903) and Birkhoff on asymptotic problems of ordinary differential equations (1907)—and their academic paths crossed again in 1909 when Birkhoff took a position at Princeton. Birkhoff moved to Harvard in 1912, by which time the two had been close colleagues for three years and their families had become good friends.3 Birkhoff, who was 29, had never been to Europe, but he did have a visit planned for 1916, presumably to attend the International Congress of Mathematicians (ICM) scheduled to take place in Stockholm that year. But the First World War intervened and neither the ICM nor Birkhoff’s visit materialized, and it was not until 1926 that he finally made the journey.4 Although one can only speculate about why he did not go to Europe until so late in his career, it seems likely that it was due to a combination of family circumstances and lack of funds. Birkhoff had married in 1908 and had two children by 1911, and it is known that Harvard did not pay its academic staff well at that time. Indeed, in March 1913, Birkhoff wrote to Veblen: All of the men here [i.e., Harvard], except Byerly, Jackson and myself, have undertaken a ‘‘Sabbatical’’, and their comments about the expense have made us relinquish hope of going abroad in the near future. I notice however that all have taken the opportunity when it presented itself. Professor Boˆcher regards the ‘‘half-year at full pay’’ and the ‘‘full year at half pay’’ as arrangements on an entirely different footing. The University loses by the first arrangement and the man by the
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1 Harvard University Archives, Papers of George David Birkhoff, HUG 4213.2, Box 7. Three letters from O. Veblen to G. D. Birkhoff, 7 September 1913, 25 December 1913, 17 May 1914. 2
For a discussion of E. H. Moore and his students, see Parshall and Rowe (1994, 372–393).
3
The closeness of the two families is evident from the correspondence. See, for example, the letter Birkhoff wrote to Veblen on 16 March 1913, expressing disappointment that the Veblens would not after all be coming to Cambridge. ‘‘Both of us,’’ he wrote, ‘‘have been counting on a good visit.’’ (Veblen papers, Library of Congress.) 4
See the first page of Birkhoff’s Memorandum for the International Education Board of 8 September 1926 in which he reports on his recent European trip and which is reproduced in Siegmund-Schultze (2001, 265–271).
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DOI 10.1007/s00283-011-9248-x
George Birkhoff, 1913. Courtesy of the American Mathematical Society.
Oswald Veblen, c. 1923. Courtesy of the American Mathematical Society.
second. Unfortunately, we only have the second kind! Professor Boˆcher tells me that in some of the lean years here the Professors were urged to take ‘‘Sabbaticals’’ for the sake of the University finances—in fact for a time it was almost a duty to do it. Another disadvantage of our situation is that you either take your time when you can get it, or exchange time with someone else, but cannot go when you please.5
But despite remaining on American soil, Birkhoff was not unknown to his European colleagues. European mathematicians visiting the United States—such as Borel and Volterra in 19126—often visited Harvard, and, although American journals were not always widely read in Europe, several of Birkhoff’s publications, both those published in the USA and those published outside, had attracted international attention. An article on linear difference equations (Birkhoff 1911a) attracted a two-page review in the Jahrbuch u¨ber die Fortschritte der Mathematik, ‘‘a rare honour for a young and unknown author’’ (Whittaker 1945, 122), and an article on dynamical systems had, with the help of Hadamard, been published in French in the Bulletin de la Socie´te´ Mathe´matique de France (Birkhoff 1912a). However, far more significant than either of the previously published papers is Birkhoff’s proof of Poincare´’s last geometric theorem, published a few months before Veblen’s departure for Europe (Birkhoff 1913a). The theorem, which concerns solutions to the restricted three-body problem, had been stated without proof by Poincare´ shortly before his death. Although Birkhoff’s paper would be only one of many important ones in which he took up an idea of Poincare´’s in celestial mechanics, it would be the one that mathematicians remembered as creating a sensation. Richard Courant recalled that for the first time in his memory, ‘‘the Go¨ttingers looked with admiration across the ocean’’ (Reid 1986, 270), whereas Veblen himself declared that it ‘‘brought immediate and worldwide fame to its author’’ (Veblen 1946, 282). Even Norbert Wiener, who was not known for singing Birkhoff’s praises, said that the proof had ‘‘astonished the mathematical world.’’ Wiener went
AUTHOR
......................................................................... is a senior lecturer in the history of mathematics at the Open University (UK) and she has recently had a very enjoyable period as a visiting research professor at the Poincare´ Archives, Nancy-Universite´. Following her earlier research on Poincare´ and the three-body problem, she is currently working on George Birkhoff and the development of dynamical systems theory. When she is not buried in an archive, she is either running marathons or playing tennis.
JUNE BARROW-GREEN
The Open University Milton Keynes UK e-mail:
[email protected] 5
Letter from Birkhoff to Veblen, 16 March 1913. Veblen papers, Library of Congress. It is evident from later correspondence that Birkhoff’s financial situation continued to cause him concern. For example, on 24 May 1916, E. H. Moore wrote to Veblen concerning the latter’s visit to Chicago for the dedication of Ida Noyes Hall: ‘‘Birkhoff can’t come, having had unusual and heavy expenses to meet recently. We are very sorry.’’ Veblen Papers, Library of Congress. 6 Letter from Birkhoff to Veblen, 28 November 1912. Veblen papers, Library of Congress.
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on to say that ‘‘what was even more remarkable was that Birkhoff had done his work in the United States without the benefit of any foreign training whatever.’’ As he explained, ‘‘Before 1912 it had been considered indispensable for any young American mathematician of promise to complete his training abroad. Birkhoff marks the beginning of the autonomous maturity of American mathematics.’’ (Wiener 1953, 230–231). Wiener’s last point was well made, and it was one that applied equally well to Veblen also. Veblen could not claim a publication as newsworthy as Birkhoff’s paper on Poincare´’s last theorem, but his work— much of it on another Poincare´an topic: analysis situs, or topology as it is now called—was known in Europe too, his papers being regularly abstracted in the Jahrbuch u¨ber die Fortschritte der Mathematik. Furthermore, his Projective Geometry, written jointly with J. W. Young (Veblen and Young 1910) had been enthusiastically reviewed in Germany. The reviewer, Hans Beck from the Technical University at Berlin-Charlottenburg, called it not only ‘‘an excellent book’’ but also noted that it was ‘‘unfortunately unrivalled by anything in Germany.’’ He expressed the hope that it would be ‘‘made accessible to a broader readership through a German edition’’ even though he personally found the English to be quite understandable.7 Visiting Europe in 1913, Veblen was a year behind many of his compatriots who had made the journey in 1912 to attend the ICM in Cambridge. The Congress had attracted a 60-strong delegation from the USA—only the host nation could boast a greater number of participants—led by E. H. Moore, who was elected as a Vice-President. Maxime Boˆcher from Harvard and Ernest Brown from Yale presented plenary talks, while other speakers from the USA included Harry Bateman from Bryn Mawr who, along with Brown, was an e´migre´ from Cambridge, F. R. Moulton from Chicago, and D. E. Smith from Columbia who had been heavily involved in the International Commission on the Teaching of Mathematics. For the first time at an ICM, American mathematicians had held their own with their European counterparts. Thus it is no wonder that in his letters to Birkhoff, Veblen appears rankled by the lack of interest in, or awareness of, important American mathematical work shown by some of the mathematicians he encountered in Europe. And the Germans—Hilbert was an exception—come across as more insular than most. Reading the letters, it is not hard to detect signs of the natural solidarity that Veblen felt with Birkhoff, each of them a representative of the first home-grown generation of American mathematicians.
Kristiania In the first of his letters to Birkhoff, Veblen describes his serendipitous attendance at the Third Scandinavian Congress of
Mathematicians, held in Kristiania (Oslo) from 3 to 7 September 1913. The Congress was an almost exclusively Scandinavian affair. Of the 72 official participants listed in the Proceedings (Størmer 1915, 6–8), 41 were described as coming from Norway, 14 from Sweden, 10 from Denmark, 4 from Finland, and 3 from Germany (Leipzig). This last group consisted of the Norwegian meteorologist Vilhelm Bjerknes and two of his students, one of whom was Scandinavian and the other German (to judge by the names). Although Veblen found the lectures, which were delivered in Scandinavian languages, almost impossible to follow, he had no difficulty conversing with the participants. The Swedish mathematician Go¨sta Mittag-Leffler made a particular impression on him. By then aged 67 and retired from the chair of mathematics in Stockholm, Mittag-Leffler was the doyen of the Scandinavian mathematical community. The Scandinavian (later Nordic) Congresses, which at that time took place every two years, were the product of his initiative,8 and it was at his instigation that the next ICM was being planned for Stockholm in 1916. As founder and editor-in-chief of the journal Acta Mathematica, Mittag-Leffler was always looking to extend his network of international contacts, and congresses provided him with the ideal environment in which to do so. He had never been to the United States—although at one time he had gone so far as to consider taking a position there9—and he clearly relished the unexpected chance to meet one of the rising stars of American mathematics. Veblen was evidently pleased, and was probably flattered, that MittagLeffler was prepared to invite him to Stockholm at first meeting, but for Mittag-Leffler, ever the consummate diplomat, issuing such an invitation was only natural. Among other members of the Stockholm contingent at the Congress were Helge von Koch, (of the eponymous snowflake curve), and the Hungarian-born analyst Marcel Riesz, both of whom had joined the faculty at Stockholm University in 1911 (Horva´th 2008, 744). Von Koch had succeeded to Mittag-Leffler’s chair, whereas Riesz had been invited to Stockholm by Mittag-Leffler to edit a prosopographical index, complete with portraits, of the first 35 volumes of Acta Mathematica. Although Norway as the host country provided by far the largest delegation to the Congress, Veblen mentioned only two Norwegians: the President of the Congress, Carl Størmer, renowned for his mathematical modelling of the aurora borealis, and the (by then elderly) group-theorist Ludwig Sylow. Sylow, who had been primarily responsible for the 1881 edition of Abel’s works (completed in cooperation with Sophus Lie), spoke at the opening session about the Abel manuscripts acquired by Mittag-Leffler in 1898.10
7 Ein ganz vorzu¨gliches Buch! In der Tat eine Neuerscheinung, der in Deutschland unseres Wissens leider nichts an die Seite gestellt werden kann. Wir geben daher der Hoffnung Ausdruck, daß trotz des recht lesbaren Englisch das Werk durch eine deutsche Ausgabe dem gro¨ßeren Leserpublikum zuga¨nglich gemacht werde. (Beck 1910, 84). 8 The first Scandinavian Mathematical Congress was held in 1909 Stockholm under Mittag-Leffler’s guiding hand; a list of the papers is given in the Bulletin of the American Mathematical Society 16 (1909) 155–156. The Proceedings were published in French to make them accessible to an international audience, see Sørensen (2006). 9 In 1886 Mittag-Leffler was sufficiently dissatisfied with his position in Stockholm to give serious consideration to applying for a job at the new university of Stanford in Palo Alto, California, even proposing to take Acta Mathematica with him (Stubhaug 2007, 365-366). 10 Sylow’s lecture has been translated into English: http://www.abelprisen.no/nedlastning/litteratur/sylow_abels-arbeider_en.pdf (accessed 27 July 2010).
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Of the participants from Finland, Veblen met with the function-theorist Ernst Lindelo¨f and the mathematical astronomer Karl Sundman, both from the University of Helsingfors (Helsinki). Although the two Finns were of not dissimilar age—Lindelo¨f was 43 years old and Sundman was 40—in terms of international reputation their trajectories had been very different. Lindelo¨f, the university’s Professor of Mathematics since 1903, had been well known outside Finland since the 1890s. A regular traveller to the Continent, he published vigorously and was the author of an admired monograph on the calculus of residues (1905). In contrast, Sundman, the acting Professor of Astronomy,11 had remained virtually unknown to Continental mathematicians until 1912 when his paper containing an analytic solution to the three-body problem—a solution of a type that mathematicians of the calibre of Poincare´ had considered impossible to achieve—was published in French in Acta Mathematica (Sundman 1912). At this point he was immediately projected into the mathematical limelight. (In fact, he had published the essentials of his solution some three years earlier in two papers in a Finnish journal but these earlier papers, which were also in French, were almost completely overlooked.12) Since Veblen knew that Birkhoff had a particular interest in the three-body problem, he knew he would be keen to hear about Sundman. Some years later Birkhoff devoted a chapter of his seminal book on dynamical systems to a discussion of Sundman’s solution (Birkhoff 1927, Chapter IX). Several Danish mathematicians were mentioned by Veblen in his letter, including Niels Nørlund who in 1912 had been appointed to a professorship of mathematics in Lund (hence Veblen’s description of him as Swedish). Nørlund was eager to learn from Veblen news about Robert Carmichael, who had been Birkhoff’s first doctoral student at Princeton, and had obtained his Ph.D. in 1910 for a thesis on difference equations (Carmichael 1911). Nørlund himself had earlier obtained results similar to Carmichael but by using different methods (Morse 1946, 369), and, as noted earlier, Birkhoff too had published an important paper on the subject (Birkhoff 1911a). But Carmichael was no longer at Princeton; shortly after receiving his Ph.D. he had been appointed Assistant Professor of Mathematics at Indiana University. A versatile and prolific mathematician, Carmichael was one of the first to publish a book in English on relativity theory (Carmichael 1913), and he is primarily remembered today for the Carmichael numbers,13 the first and smallest of which he found in 1910. Another Danish mathematician mentioned by Veblen was the topologist Poul Heegaard who revealed to Veblen
11
his interest in the four-colour problem. Both Veblen and Birkhoff had published already on this topic, Birkhoff’s interest in the problem having been stimulated by Veblen’s seminar at Princeton on analysis situs. Birkhoff introduced his chromatic polynomials—P(x) equal to the number of ways in which a given map can be coloured in x colours— in 1912 (Birkhoff 1912b) and afterward he made important steps toward the solution throughout his career. He later admitted that it was the only problem that kept him awake at night (Vandiver 1963, 386), and to solve it always remained ‘‘one of his dearest aspirations’’ (Veblen 1946, 281). Birkhoff’s last paper on the topic (joint with his former student D. C. Lewis) appeared posthumously in 1946. Christiania,14 Sunday, 7 September 1913 My dear Birkhoff: We came down from the mountains a week ago and found that the 3rd Scandinavian Math. Congress was about to come off.15 So I have attended some of the meetings and met a number of men.16 I thought you would be interested in hearing of it and so I shall write a few lines, although we are just packing up to go to Copenhagen. There were about 80 at most of the meetings I should guess, though I did not actually count them. The most distinguished figure was undoubtedly Mittag-Leffler. He is a fairly tall, erect and slender old man with a white moustache and long grey hair. At the first session he wore a grey frock coat evidently designed to match the hair flowing over his collar. He spoke on several occasions and, so far as I could judge did it very gracefully, with some humor and a great deal of historical reference. His manner is very gracious – for example he invited me very cordially to visit him in Stockholm, though he probably did not know a thing about me – but one or two things make me think that some of the men find him too condescending. Sylow, at 81 [actually 80, J. B.-G.], is very vigorous and young looking, but says that his memory is poor for all recent matters. He professed a great regard for American Math. You will find on the other side of this sheet the two best American mathematicians according to Sylow – (1) Dickson, (2) Osgood. Sundman, from Helsingfors, about whose recent work on the 3 body problem you of course know more than I, is a man in the prime of life, very clean cut and businesslike in appearance and manner. I had seen only a newspaper clipping about his work but he explained the nature of it very clearly in a couple of minutes and estimated it very modestly. He did not seem
Sundman was holding Anders Donner’s position while Donner was serving as Rector of the University. For a discussion of the publication and reception of Sundman’s work, see Barrow-Green (2010). 13 A Carmichael number is a composite positive integer n that satisfies the congruence bn–1 : 1(mod n) for all integers b relatively prime to n. 14 Veblen mistakenly used the outdated spelling for Kristiania. The official change from Christiania to Kristiania took place in 1877, although the new spelling was not universally adopted by the city until 1897. The change from Kristiania to Oslo took place in 1925. 15 Veblen started his tour in Norway probably for family reasons—he had Norwegian grandparents—which would explain why he did not know about the Scandinavian Congress until he arrived in Kristiania. 16 Although the titles of all papers presented at the Congress are given in the Proceedings (Størmer 1915), not all the papers were published. A list of the presented papers (with titles in English) is given in the Bulletin of the American Mathematical Society 20 (1914), 214. 12
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to know about your work on stability. Lindelo¨f, also from Helsingfors, seemed very agreeable and thoroughly awake, as well as the Swedes von Koch and No¨rlund. The latter, when Princeton was mentioned, asked at once if that was not where Carmichael is. I explained the system of relations in which C. figures as clearly as I could. No¨rlund seemed enthusiastic about your work and asked me to greet you for him. M. Riesz was present as from Stockholm. He is a docent there and seems to be a great favorite with Mittag-Leffler. He is a Hungarian, a small man with a bright, quick, gentle manner. The president of the organization committee was Carl St€ ormer, a Norwegian who has done some work on the aurora. He was one of Lovett’s lecturers at the opening of the Rice Institute,17 though he only sent his lecture and did not read it in person. The geometers all seem to be concentrated at Copenhagen. The ones who interested me most were Heegaard, Hjelmslev, and Juel.18 Heegaard had seen your paper on the 4-color problem and said he had been looking unsuccessfully for mine for two months.19 I expect to have a talk with him about Analysis Situs when I get to Copenhagen a few days hence. Probably I have omitted some of the more interesting personalities in the remarks above, but as it is late, this will have to do. I got practically nothing out of the papers I listened to, because of the language, but it was rather interesting to see their way of doing things. The first thing that struck me was the formality of everything. They wore frock coats to all the meetings and the papers were all given apparently in full and almost never followed by a discussion. The opening session was attended by the King of Norway20 and consisted of an address of welcome by St€ ormer and a paper about Abel by Sylow. After the session everybody marched away, looking solemn. Elizabeth and I attended two dinners, one at St€ ormer’s house and the other at a hotel. The latter was the closing event of the Congress and will be forever memorable because we were served six kinds of wine, each in its properly shaped glass, and two liqueurs. They have a pleasant custom in drinking. Someone bows to you and holds up his glass, you raise your glass—which must contain the same kind of wine as his—and also bow; then you both say ‘‘skaal’’ and drink; then you both bow again. It is easy to see how it might become very confusing toward the end of a long evening.
Well, there is a call for me to go to bed, so goodnight, O. V.
Go¨ttingen and Berlin By the time Veblen wrote the second letter he had been in Germany for 14 weeks, having visited both Go¨ttingen and Berlin. Go¨ttingen, with Hilbert at the helm and with Klein still active (although recently retired), was the mathematical centre of Germany. It was a hothouse of activity, and there were many mathematicians there, such as Landau, Carathe´odory, and Bernstein,21 who Veblen had a chance to meet. Berlin, in decline since the early 1890s (Haubrich 1998, 84), was altogether different. After its 19th-century heyday with Kummer, Kronecker, and Weierstrass, it now had Frobenius and the 70-year-old Hermann Amandus Schwarz. About the former, Veblen said nothing, although about the latter, he reported quite extensively. In Go¨ttingen Veblen had had plenty of opportunity to see Hilbert—the mathematical ‘‘king’’ of Germany—at work, although he never made personal contact; he was nevertheless impressed by what he saw. However, he also heard stories about Hilbert’s behavior, which were not to Hilbert’s credit. The only detail Veblen gave about these stories was a cryptic reference to Poincare´’s visit to Go¨ttingen in 1909. Information about Poincare´’s visit would certainly have been of interest to Birkhoff who had favourably reviewed the published version of Poincare´’s six Go¨ttingen lectures (Poincare´ 1910), commenting that Poincare´ had ‘‘treated a wide range of interesting subjects in a masterful and illuminating way’’ (Birkhoff 1911b, 190). Much later, in 1964, Courant, who had been present at the lectures, provided his own perspective on the occasion, reporting that Poincare´ in his lecture on transfinite numbers had made ‘‘a violent attack against Cantorism and against the principle of choice’’ and had enraged Zermelo, who had given a lecture on set theory the same day,22 by dismissing his recent result on the well-ordering of sets. Courant—using words that did not appear in the published version of the lectures—quoted Poincare´ as saying ‘‘Even the almost ingenious proof of Mr. Zermelo has to be completely scotched and thrown out of the window’’ (Courant 1981, 162). However, since Poincare´’s views on foundational issues had been well known for some time and Courant’s remarks seem rather at odds with the pub-
17
The Rice Institute (now Rice University), Texas, was formally opened in October 1912 with Edgar Odell Lovett, the former head of the department of mathematics and astronomy at Princeton University, as President. 18 At the Congress, Poul Heegaard spoke on ‘‘Contributions to the theory of graphs,’’ Johannes Hjelmslev on ‘‘Geometry of reality’’ and ‘‘Geometric experience,’’ and Christian Juel on ‘‘Toroidal non-algebraic surfaces of the fourth order.’’ 19 The papers referred to are either Birkhoff (1912b), or Birkhoff (1913b), and Veblen (1912). 20 The King of Norway was Haakon VII, the first king of Norway after the dissolution from Sweden in 1905. 21 Edmund Landau had been in Go¨ttingen since 1909 when he was appointed to the chair made vacant by the untimely death of Hermann Minkowski; Constantin Carathe´odory had succeeded to Klein’s chair in March 1913 (Georgiadou 2004, 91–92); Felix Bernstein had been appointed as associate professor for statistics and actuarial mathematics in 1911 (Rowe 1986a, 438). 22 The program for the Poincare´-Woche, which featured other speakers from Go¨ttingen, can be found in Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 18, pp. 78–79.
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lished text,23 Courant’s memory—he was recalling events that had taken place more than fifty years earlier—might not have been entirely reliable. In addition, Courant’s lingering impression of the event may have been coloured by Poincare´’s rather awkward lecturing style. Rather than speaking in his native French, Poincare´ had agreed to lecture in German, a language in which he was known to be uncomfortable.24 Veblen found little evidence that the other mathematicians in Go¨ttingen were interested in any mathematics beyond that favoured by Hilbert, or indeed that they had much interest in developments outside Germany. It was no wonder then that they showed little knowledge of American mathematics. One exception was Edmund Landau who was impressed by the recent outpouring on analytic number theory by Thomas Gronwall, the Swedish-American mathematician who had succeeded Birkhoff at Princeton in 1912.25 However, Landau’s interest in Gronwall’s work seems to have been the exception. Even though Birkhoff’s proof of Poincare´’s last theorem had ‘‘created the impression’’ that Birkhoff ‘‘probably [had] to be reckoned with,’’ Veblen could find no one with a proper understanding of Birkhoff’s work. Felix Bernstein claimed to be familiar with Birkhoff’s recent paper on linear differential equations and the generalized Riemann problem (Birkhoff 1913c), but in reality he knew only that it was an extension of the theorem proved by Hilbert, something he could have learned by reading the paper’s introduction,26 whereas Carathe´odory, the newly engaged successor to Felix Klein, was apparently willing to judge Birkhoff’s ability by the mere quantity of his work, without being too keen on actually reading the publications. The visit to Go¨ttingen also confirmed Veblen in his (mistaken) belief that Hilbert was a Jew, his judgement being at least partly based on what he perceived as a likeness between Hilbert’s voice and that of Saul Epsteen, a Jewish mathematician who both he and Birkhoff knew from 23
the University of Chicago.27 That Veblen should have considered such a topic worthy of mention, let alone that he made such a mistake, is not so surprising when considered in the context of the time when such discussions were quite common. Perhaps he even had a deeper interest in the matter in view of the fact that his uncle, the well known economist and sociologist, Thorstein Veblen, had offered a sociological explanation for the strong representation of Jews in academia (Veblen 1919). The mistaken idea that Hilbert was a Jew resurfaced again in the Nazi era in the writings of the physicist Johannes Stark (Rowe 1986a, 423).
Berlin, 25 December 1913 My dear Birkhoff: After 10 weeks in Go¨ttingen and 4 weeks in Berlin I am beginning to have definite impressions of Germany. Mathematically, even more than politically, it is a monarchy. The mathematical situation is well illustrated by a remark of Landau’s when I asked him whether there was any interest in Abelian functions and the like (more than one variable): No-one in Germany is interested in anything Hilbert has not worked with. They are only mildly interested in what is going on elsewhere, unless it touches pretty directly on their own work. Gronwall has made an impression on Landau and his circle and your proof of Poincare´’s theorem [Birkhoff 1913a] has created the impression that you probably have to be reckoned with. I was amused however, to see how little the people in related fields really know about your work. I was present when Carathe´odory opened a batch of reprints from you; his comment was: ‘‘he seems to cover a great many subjects’’. I also heard a literature report by F. Bernstein which was supposed to deal with some of your papers. What he said related almost entirely to your recent one in the American Academy Proc [Birkhoff 1913c]. He made much of
In the published text, Poincare´ wrote: Was haben wir von dem beru¨hmten Kontinuumproblem zu halten? Kann man die Punkte des Raumes wohlordnen? Was meinen wir damit? Es sind hier zwei Fa¨lle mo¨glich: entweder behauptet man, daß das Gesetz der Wohlordnung endlich aussagbar ist, dann ist diese Behauptung nicht bewiesen; auch Herr Zermelo erhebt wohl nicht den Anspruch, eine solche Behauptung bewiesen zu haben. Oder aber wir lassen auch die Mo¨glichkeit zu, daß das Gesetz nicht endlich aussagbar ist. Dann kann ich mit dieser Aussage keinen Sinn mehr verbinden, das sind fu¨r mich nur leere Worte. Hier liegt die Schwierigkeit. Und das ist wohl auch die Ursache fu¨r den Streit u¨ber den fast genialen Satz Zermelos. Dieser Streit ist sehr merkwu¨rdig: die einen verwerfen das Auswahlpostulat, halten aber den Beweis fu¨r richtig, die anderen nehmen das Auswahlpostulat an, erkennen aber den Beweis nicht an (Poincare´ 1910, 40–41). What can we make of the famous continuum hypothesis? Can one ‘‘well-order’’ the points of the space? What do we mean by that? Two cases are possible: either one claims the law of well-ordering can be finitely expressed, then the assertion is not proved, even Herr Zermelo will probably not claim to have proven that. Or we admit the possibility as well, that the law cannot be expressed finitely. In this case I cannot connect any meaning to the assertion which is then for me only empty words. Here is the problem. And this is presumably the origin of the dispute about the almost ingenious theorem of Zermelo’s. This dispute is very curious: the one party discards the axiom of choice, but considers the proof correct, the other party assumes the axiom of choice but does not recognize the proof. 24 Poincare´ delivered five of the six lectures in German. The final lecture, and linguistically the most challenging of the six, he delivered in French (Poincare´ 1910, 51). Hilbert’s welcoming address for Poincare´, which refers to Poincare´’s ‘‘willingness to make use of the German language despite the discomfort involved,’’ can be found in English translation in Rowe (1986b). It has also been said that Poincare´ unwittingly annoyed his audience by choosing to give a lecture on integral equations, a subject, due to Hilbert’s groundbreaking work, the Go¨ttingen mathematicians regarded as their own (Reid 1986, 120). However, so far no documentary evidence to substantiate this has come to light. 25 For details of Gronwall’s work in pure mathematics, and for a bibliography, see Hille (1932); for details of Gronwall’s work in applications, see Gluchoff (2005). 26 In the introduction to the paper, Birkhoff neatly summarizes the connection between his work and that of Hilbert: ‘‘The problem of Riemann for linear differential equations in its classic form was first solved by Hilbert [1905]. His treatment and that of Plemelj’s elegant completion thereof [1908] reposed alike upon a certain theorem whose proof was made by means of the Fredholm theory. Owing to the deep-seated analogy between linear differential and difference and q-difference equations, I have been able to supply a convenient extension of the same theorem in all cases; my proof is based on a method of successive approximations.’’ (Birkhoff 1913c, 521–522). 27 Saul Epsteen was an Associate in Mathematics at the University of Chicago during 1903 through 1905. In 1905 he moved to the University of Colorado, where, by 1913, he had become professor of Engineering Mathematics, see American Mathematical Monthly 20 (1913), p. 201. Epsteen is included in a list of ‘‘Jews of Prominence in the United States’’ published in the American Jewish Yearbook 1922–1923 (Dobsevage 1922, 136).
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the reference to Hilbert, but when it came to explaining what your work was about, he showed that he had not the least idea. He was not able to answer the simplest questions. Hilbert came fully up to my expectations. He has a much better style of exposition than I had been led to expect and one could not fail to realize his extreme intelligence after watching him a few minutes. He also struck me as being both urbane and magnanimous, although the stories one hears do not bear this out—for example, the stories told from the German point of view about Poincare´’s visit to Go¨ttingen put Hilbert and the others in rather a bad light. He conducts the ‘‘Gesellschaft’’ (‘‘Society’’) admirably in my opinion (I attended 3 times). He opens the sessions with remarks about his own work or what he has heard of interest from the physicists, then he calls for voluntary reports from anyone who has a new theorem or result which may be stated briefly. Meanwhile the literature which has come in during the week is circulated about, and if Hilbert is specially interested he may make a few remarks. Then there is a literature report on definite papers usually made by a professor, then they have the principal paper. In this I was pleased to see that the hearers, especially Hilbert, constantly interrupted with questions and comments, much as you and I used to do in Princeton. About once in two weeks the meeting is followed by a supper in a beer restaurant. In his attitude in the club Hilbert reminded me a great deal of [E. H.] Moore – and it is quite possible that I saw only his best side just as I did in the case of Moore. For a great many tales are told of unreasonable and discourteous performances of his just as there are about Moore. There is also a suggestion of an infinitely refined Epsteen in his voice. I no longer doubt that Hilbert is a Jew. If you ask anyone in Go¨ttingen he will tell you: ‘‘Mann weisst nicht’’.28 Altogether, what I saw of Hilbert (I did not meet him personally) added greatly to my admiration for him. But the rest of the people in Go¨ttingen did not impress me much. Naturally the way they are ready to run in any direction that the king (the math. one) points does not add to my respect for them or make me feel that they have much which is essentially their own. But aside from that [Carl] Runge, Landau, Carathe´odory, Bernstein is a combination that would not badly disturb the equilibrium in America. Landau is the one who has the largest following, next to Hilbert, among the students. I also had a couple of interviews with Klein who impressed me as a very smooth specimen, but not at present very interesting. Perhaps this was because I am not much
taken up with pedagogical and semi-professional philosophical questions, and partly because he ventured on a mistaken criticism of the Proj. Geom. Book based on a half hour glancing through the pages. He had no difficulty in seeing his oversight when I pointed it out, but he should not have made it. I believe he is the only one of the professors at Go¨ttingen who has opened the book. The copy in the Lesezimmer [reading room], however, shows sign of wear and a student who used to be at Princeton told me that it is read a great deal. But so far as my observation reaches, it has been entirely ignored. I have not seen as much of the mathematicians in Berlin as those in Go¨ttingen. But two points have struck me: the pressure is not nearly as high; and there is not nearly as great a proportion of Jews.29 As a matter of fact, there was no one I particularly wanted to see at the University here and so I did not go round until just before the Christmas holidays. When I did, I dropped into one of Schwarz’s lectures and afterwards went up and introduced myself. He was extremely cordial, especially when he found that I was interested in Geometry, and he has added greatly to the pleasure of my visit here. Naturally I was surprised to hear that he had a live interest in Synthetic Geom. and later asked him about it. He cited the example of his predecessor, Weierstrass, who always lectured on Geom. with great pleasure,30 and he also boasted (i.e. S boasted) that he was almost a student of Steiner.31 Steiner was away when Schwarz was ready to take his lectures, but he got it from a student of Steiner’s. Schwarz also has lectured on Geom. and been interested in it all his life, as he says. But the research he is on now will not electrify the world, I fear. However, he is a very interesting and agreeable old man. He invited me to a festival of the students’ Math. Club of which he was a charter member 52 years ago, and to a session of the Berlin Math. Ges. The former, a formal drinking bee, was attended by about 200 old and young students, and the latter, a scientific session, by about 50 professional mathematicians. (The numbers here are impressions. In Go¨ttingen all but the most advanced students are excluded from the Gesellschaft because of lack of room.) Besides this we had two or three private sessions in which I heard a pile of scandal about the mathematicians ‘‘of the upper semesters,’’ as well as the aforementioned geometrical ideas. The last session was highly convivial. The health of pure mathematics was followed by a toast to Euclid ‘‘in the fourth dimension’’ 32 and another to Abel. He also favored me with a verse of a German song to the effect that one can drink too much but not enough. In the end he was ready for more but I wasn’t. Someday I will repeat some of the scandal for you—but not all.
28
It is likely that Veblen meant ‘‘Man weisst es nicht’’ meaning ‘‘one does not know.’’ For a discussion on the numbers of Jews holding positions in Prussian universities, see Rowe (1986a). 30 Weierstrass lectured on synthetic geometry six times between 1864 and 1873, fulfilling a promise he had made to Jakob Steiner before his death in 1863. However in 1873, in contradiction to Schwarz’s claim, Weierstrass wrote to Sonya Kovalevskaya to say that he had little enthusiasm for the lectures, and gave them only on Steiner’s behalf and to continue a tradition that otherwise would be neglected in Berlin (Biermann 1966, 16). 31 Jakob Steiner was a professor of geometry in Berlin from 1834 until his death in 1863. Schwarz studied in Berlin from (at the latest) 1861 until 1867, receiving his doctorate for a thesis written under Weierstrass in 1864. 32 This would seem to be a reference to Minkowski space-time. 29
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It is after midnight and I am ordered to bed, otherwise I should have entertained you with my views on German politics. However, you will have to do without them for the present. Let me hear from you soon. Yours sincerely, Oswald Veblen Thanks for the reprints which came two days ago.
Paris At the beginning of May the Veblens arrived in Paris, having spent some time in Bandol, a small coastal town on the Coˆte d’Azur. When Veblen wrote the letter, the couple had been in Paris for 10 days, and most of that time had been taken up with sightseeing. Although Veblen considered Paris ‘‘much the finest city’’ he had seen, he found it very expensive. Knowing Birkhoff’s financial situation, he knew that this information would be useful with respect to Birkhoff’s plans for 1916. But Veblen had not been completely occupied with sightseeing. He had found time to attend several university lectures, including Picard’s course on ‘‘multiple integrals with applications to the theory of analytic functions of several variables’’ and Appell’s course on ‘‘figures of equilibrium of rotating fluid bodies’’ at the University of Paris,33 as well as a course of Humbert’s on group theory.34 To give Birkhoff an idea of the quality of lecturing, he adopted Maxime Boˆcher, who had taught both of them when they were undergraduates at Harvard, as a benchmark. Birkhoff, as Veblen knew, had a high opinion of Boˆcher’s ability as an expositor. Indeed, in 1913, on learning that Boˆcher was to be an exchange professor in Paris, Birkhoff had written to Veblen to say that he thought Boˆcher would ‘‘give great satisfaction,’’ adding that Boˆcher’s ‘‘sense of balance and gift in clear explanation are so good that they will be appreciated even at Paris.’’ 35 A few days before he wrote the letter, Veblen had been at a meeting of the French Mathematical Society where he had heard the presentation of a paper by the Danish mathematician Harald Bohr (Bohr 1914). Bohr, who had spent the earlier part of the year in Oxford and Cambridge studying with Hardy and Littlewood, was in Paris studying measure theory with Lebesgue (Bochner 1952, 73). At the meeting, he spoke on his joint work with Edmund Landau on the Riemann zeta function; the paper containing their result on the distribution of zeros of the Riemann zeta
function (now known as the Bohr-Landau theorem) had been presented to the Acade´mie des Sciences the previous December (Bohr and Landau 1914). It is notable that Veblen, while lauding Harald, made no mention of his elder brother, Niels, whose model of the hydrogen atom had been published the previous year and whose work had been under discussion in Go¨ttingen.36 But perhaps Veblen was not particularly curious about new trends in physics. Veblen had also encountered a number of other travelers in Paris. These included two of his doctoral students from Princeton, Albert Bennett and James Waddell Alexander, who were about to leave Paris to visit the geometer Federigo Enriques in Bologna. Both students would be awarded their doctorates in 1915, Bennett for a thesis on projective geometry (Bennett, 1915), and Alexander—who, although nominally a student of Veblen’s, was in fact being advised by Thomas Gronwall (Gluchoff 2005, 323)—for a pioneering paper on univalent function theory, his only venture into complex analysis (Alexander 1915).37 Veblen also met the Hungarian topologist Julius Pa´l (1881–1946), who had been a student of Carathe´odory in Go¨ttingen, and the Hungarian analyst Alfre´d Haar. Pa´l later completed his studies under Frigyes Riesz (brother of Marcel Riesz, mentioned previously), one of the founders of functional analysis, in Kolozsva´r (now Romanian Cluj) obtaining a doctorate in 1916. Veblen concluded the letter by referring to correspondence he had had earlier in the year with Bertrand Russell. At the time, Veblen had been in Bonn where he had had ‘‘several delightful conversations with [Eduard] Study’’ whom he found ‘‘even more interesting than you would expect from his writings.’’38 The correspondence, which had been initiated by Veblen in order to seek Russell’s opinion on a form of mathematical definition that he, Veblen, had proposed, pertained mostly to aspects of Veblen’s reading of Russell’s and Whitehead’s Principles of Mathematics. Paris, 17 May 1914 My dear Birkhoff: If you happen to be in Cambridge at the right time, you may have a visit from us sooner than you expected. We have engaged passage on the Carmania sailing from Liverpool for Boston on the 8th September and probably arriving on the 15th or 16th: As you were in C by that time last year we hope you will be there again.39 If so, a
33 Emile Picard held the chair of analysis and algebra, and Paul Appell held the chair of rational mechanics. The courses are listed in The Bulletin of the American Mathematical Society 20 (1914), 388. 34 Georges Humbert held the chair of analysis at the E´cole Polytechnique and the chair of mathematics at the Colle`ge de France. 35 Letter from Birkhoff to Veblen, 16 March 1913. Veblen papers, Library of Congress. An account of Boˆcher as a teacher is given in Osgood (1919, 343–344). 36 In the autumn of 1913, Harald wrote a letter to Niels Bohr in which he reported that the latter’s work was being carefully studied by the mathematicians and physicists in Go¨ttingen. It seems that Born, Madelung, and Runge were all sceptical, whereas Hilbert was rather excited about Bohr’s new theory (Mehra and Rechenberg c.1982, part 1, 201). 37 For a discussion of Alexander’s thesis, see Gluchoff and Hartmann (2000). Alexander, one of the leading topologists of his generation, is well known today for the construction he produced in 1924 that is now named for him, the Alexander Horned Sphere. 38 Letter from Veblen to Russell, 11 January 1914. The correspondence, which is held at The Bertrand Russell Archives, McMaster University, also contains a letter (19 January 1914) and a postcard (3 February 1914) from Veblen to Russell. 39 The RMS Carmania when launched in 1905 was the largest turbine-driven steamship in existence. Remarkable for the luxury of its accommodation, it was also one of the fastest ships in the world. It is not clear why the Veblens had planned to arrive in Boston rather than New York but it may have been due simply to the convenience of the date of sailing.
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sufficiently pressing invitation might induce us to stop off a day or two. But don’t let us interfere with your plans in any way. The spirit does not move me particularly in the way of letter writing today; so I fear you will have no adequate return for the interesting letter you sent me while we were in Bandol. I regret to report that I am not nearly as advanced with my work as I should be. Part of the time I have worked very hard, but I continually find myself impinging on pedantic details. I never seem to be satisfied with any method till I have tried all the others which occur to me, and about half the time I ultimately adopt the first experiment. Naturally, I have not failed to think of some things which interest me – but I don’t recommend my method to the young man looking for a good job. The week and a half which we have spent in Paris have been more prolific in sightseeing and amusement than anything else. I visited the French Math. Soc. where I heard a paper by Bohr which is practically identical with a report I heard him make when I was in Go¨ttingen—it was on his and Landau’s work on the Riemann zeta function in which they ‘‘almost’’ prove the R. Hypothesis on the zeros. Bohr strikes me as one of the best young mathematicians on this side of the water. Speaking of y[oung] m[athematicians] have you heard that Bennett and Alexander are both to be instructors in Princeton next year? I don’t have much responsibility for this event as you might suppose, but it will still be agreeable for me. The day after we arrived in Paris they left for Bologna where Enriques is organizing a special seminar for their benefit. Alexander met E. here in April and was very much taken with him. It appears that E. knows nothing to speak of about the analysis situs of the algebraic manifolds, but he wants to; and, of course, he can put them in touch [with] the algebraic work as well as anyone. Neither A. nor B. seems to have turned in a new direction as a result of his European trip, though I urged them both to try to – Bennett particularly. As the twig is bent, …40 Hadamard has stopped all his courses till the 27th of May, after the agreeable manner which they have here; so I have not seen him. Borel has been to see me, but I was not at home, and as he is not giving any lecture courses I have not seen him either. I visited one course of Picard’s, one of Appell’s, and one of Humbert’s. Appell is the best lecturer of the three, perhaps a little better than Boˆcher (to adopt a convenient form of reference) in that he seems to presuppose less and yet to advance a little faster. Picard is distinctly poorer than B. He sacrifices precision without gaining excitement or interest. Humbert has a very good
style but was talking about some group theoretic matters which I thought I could have treated with more insight. Paris is much the finest city we have seen. Most of the things which one admires in the other places (e.g. Berlin) have been imitated from Paris. However, it has one disadvantage: money flows–away–like water. We shall probably stay till the end of the month and then settle down in some quieter place to recuperate. If I did not have my work to keep me occupied, I am sure we should have been unhappy on this trip for the lack of money. Wherever we travel and whenever we stay in a large place we find that we haven’t nearly enough. But when we settle down in a small place as we did in Bandol we live more cheaply than at home. This is intended as a helpful counsel toward your trip in 1916. If you want to travel very much you had better do it for a half year on a full year’s pay. I won’t write any more, for I see I am coming too close to my own prediction, but I hope you will write soon and return good for bad. I am curious to hear about Russell. I have exchanged some letters with him recently and hope to see him next month. In the last letter he seemed rather more fixed in his ideas than he used to. I should have liked to witness his meeting with Schweitzer.41 With best greetings for Mrs Birkhoff and Barbara and Garrett, Sincerely yours, Oswald Veblen P.S. I have just had a talk with a Hungarian J. Pa´l who says that he found a proof of the Poincare´ B[irkhoff] Theorem independently of you but did not publish it because someone told him in time of your work.42 However his method, so far as I could guess from what he said is different. I urged him to publish it, naturally. With the outbreak of the War—the United Kingdom officially declared War on Germany on the 4 August 1914— and the requisitioning of the Carmania for conversion into an armed merchant cruiser, the Veblens had to revise their travel plans. They booked a passage on the equally luxurious Laconia43 and sailed from Liverpool for New York on 8 August, a month earlier than originally planned. Once home, Veblen settled back into everyday life at Princeton. It was a routine mostly untroubled by the War until 1917 when America was drawn into the conflict and Veblen himself was commissioned into the army (Grier 2001). Veblen’s letters provide a rare glimpse of an outsider’s experience of the pre-War European mathematical community. It was a community soon to be shattered and one
40 ‘‘As the twig is bent, so grows the tree.’’ A proverb quoted in various forms, most notably by Alexander Pope in 1732 ‘‘Tis Education forms the common mind, Just as the Twig is bent the tree’s inclined.’’ (Epistle to Cobham, 149–150). Veblen’s words may not have fallen on completely stony ground. The following year, Bennett published papers on iteration in one or more variables, a topic quite distinct from his thesis work in projective geometry. 41 So far I have been unable to find any further information relating to Russell’s meeting with the theologian and physician Albert Schweitzer. 42 The list of Pa´l’s publications in Filep and Elkjaer (2000) contains no paper on Poincare´’s ‘‘last geometric theorem,’’ so it appears that Pa´l’s method of proof was either flawed or no different to that of Birkhoff’s. 43 For a glimpse of accommodation on board the Laconia, see the advertising brochure of 1912: http://www.gjenvick.com/CunardLine/VintageBrochures/ 1912-RMS-FranconiaAndLaconia.html (accessed 5 October 2010).
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that took many years to rebuild. In the aftermath of the War, the controversial exclusion of the Central Powers from the ICMs drove a wedge between former colleagues. Although Veblen himself made a visit to Europe in 1921 (and several more thereafter), it was not until the ICM in Bologna in 1928 that the exclusion order was finally broken and mathematicians of all nationalities were once again able to mix freely with one another. Veblen and Birkhoff were both at the ICM and each gave a plenary address. American mathematics had come of age and Veblen and Birkhoff were firmly established as leaders within it.
Carmichael, R. (1913). The Theory of Relativity, New York: J. Wiley & Sons. Courant, R. (1981). Reminiscences from Hilbert’s Go¨ttingen, The Mathematical Intelligencer 3, 154–164. Dobsevage, I. G. (1922–1923). Jews of Prominence in the United States, American Jewish Yearbook Vol. 24, Jewish Publication Society of America: Philadelphia, 109–218. Filep, L., Elkjaer, S. (2000). Pa´l Gyula—Julius Pal (1881–1946), the Hungarian-Danish Mathematician, Acta Mathematica Academiae Paedagogicae Nyı´regyha´ziensis 16, 89–94. Georgiadou, M. (2004). Constantin Carathe´odory. Mathematics and Politics in Turbulent Times, Berlin: Springer.
REFERENCES BCMP = George David Birkhoff, Collected Mathematical Papers, 3 vols.,
Gluchoff, A., Hartmann, F. (2000). On a ‘‘Much Underestimated’’ paper
Providence: American Mathematical Society (1950); reprinted New York: Dover
Gluchoff, A. (2005). Pure mathematics applied in early twentieth-
Publications (1968).
Alexander, J. W. (1915). Functions which map the interior of the unit disk upon simple regions, Annals of Mathematics 17(2), 12–22. Barrow-Green, J. E. (2010). The dramatic episode of Sundman, Historia Mathematica 37, 164–203. Beck, H. (1910). Review of O. Veblen and J. W. Young ‘‘Projective Geometry,’’ Archiv der Mathematik und Physik 18, 84–86. Bennett, A. A. (1915). An algebraic treatment of the theorem of
of Alexander, Archive for History of Exact Sciences 55, 1–41. century America: The case of T. H. Gronwall, consulting mathematician, Historia Mathematica 32, 312–357. Grier, D. A. (2001). Dr. Veblen takes a uniform: Mathematics in the First World War, American Mathematical Monthly 108, 922–931. Haubrich, R. (1998). Frobenius, Schur and the Berlin Algebraic Tradition, in Mathematics in Berlin, H. G. W. Begehr, et al. (eds.), Berlin: Birkha¨user, 83–96. Hille, E. (1932). Thomas Hakon Gronwall—In Memoriam, Bulletin of the
Biermann, K.-R. (1966). Karl Weierstrass, Journal fu¨r die reine und
American Mathematical Society 38, 775–786. Horva´th, J. (2008). Marcel Riesz, Complete Dictionary of Scien-
angewandte Mathematik 223, 191–220. Birkhoff, G. D. (1911a). General theory of linear difference equa-
tific Biography, Vol. 18, Detroit: Charles Scribner’s Sons, 743– 745.
tions, Transactions of the American Mathematical Society 12,
Mehra, J., Rechenberg, H. (c.1982). The quantum theory of Planck,
243–284; BCMP I, 476–517. Birkhoff, G. D. (1911b). Poincare´’s Go¨ttingen lectures, Bulletin of the
Einstein, Bohr, and Sommerfeld: Its foundation and the rise of its
American Mathematical Society 17, 190–194; BCMP III, 190–194. Birkhoff, G. D. (1912a). Quelques the´ore`mes sur le mouvement des
Morse, M. (1946). George David Birkhoff and his Mathematical Work,
closure, Annals of Mathematics 16(2), 97–118.
difficulties, 1900–1925, Vol. 1, Part 1, New York: Springer.
syste`mes dynamiques, Bulletin de la Socie´te´ Mathe´matique de
Bulletin of the American Mathematical Society 52, 357–391. Osgood, W. F. (1919). The Life and Services of Maxime Boˆcher,
France 40, 305–323; BCMP I, 654–672. Birkhoff, G. D. (1912b). A determinant formula for the number of ways of
Bulletin of the American Mathematical Society 25, 337–350. Parshall, K. H., Rowe, D. E. (1994). The Emergence of the American
coloring a map, Annals of Mathematics 14, 42–46; BCMP III, 1–5. Birkhoff, G. D. (1913a). Proof of Poincare´’s Geometric Theorem,
Mathematical Research Community, Providence: American Math-
Transactions of the American Mathematical Society 14, 14–22; BCMP I, 673–681. Birkhoff, G. D. (1913b). The reducibility of maps, American Journal of Mathematics 35, 115–128; BCMP III, 6–19. Birkhoff, G. D. (1913c). The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proceedings of the American Academy of Arts & Sciences 49, 521–568; BCMP I, 259–306. Birkhoff, G. D. (1927). Dynamical Systems, Providence: American Mathematical Society. Bochner, S., Harald Bohr, (1952). Bulletin of the American Mathematical Society 58, 72–75. Bohr, H. (1914). Confe´rence de M. Harald Bohr, Sur la fonction f(s) de Riemann, Bulletin de la Socie´te´ Mathe´matique de France 42, 52–66. Bohr, H., Landau, E. (1914). Sur les ze´ros de la fonction f(s) de Riemann, Comptes Rendus Hebdomadaires des se´ances de l’Acade´mie des Sciences 158, 106–110. Carmichael, R. (1911). Linear difference equations and their analytic solutions, Transactions of the American Mathematical Society 12, 99–134.
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ematical Society and London Mathematical Society. Poincare´, H. (1910). Sechs Vortra¨ge u¨ber ausgewa¨hlte Gegensta¨nde aus der reinen Mathematik und mathematischen Physik, Leipzig and Berlin: Teubner. Reid, C. (1986). Hilbert—Courant, New York: Springer. Rowe, D. (1986a). ‘‘Jewish Mathematics’’ at Go¨ttingen in the Era of Felix Klein, Isis 77, 422–449. Rowe, D. (1986b). David Hilbert on Poincare´, Klein, and the World of Mathematics, The Mathematical Intelligencer 8(1), 75–77. Siegmund-Schultze, R. (2001). Rockefeller and the internationalization of mathematics between the two world wars, Basel: Birkha¨user. Sørensen, H. K. (2006). En udstrakt broderlig ha˚nd: Skandinaviske matematikerkongresser indtil slutningen af 1. verdenskrig. Nordisk Matematisk Tidsskrift (NORMAT) 54, 1–17 and 49–60. Størmer, C. (ed.) (1915). Den tredje skandinaviske Matematikerkongres i Kristiania 1913. Beretning. Kristiania: H. Aschehoug & Co. (W. Nygaard). Stubhaug, A. (2007). Med Viten og Vilje: Go¨sta Mittag-Leffler (1846– 1927), Oslo: Aschehoug. Sundman, K. (1912). Me´moire sur le proble`me des trois corps, Acta Mathematica 36, 105–179.
Vandiver, H. S. (1963). Some of my recollections of George David
Veblen, O., Young, J. W. (1910). Projective Geometry, Boston: Ginn & Co.
Birkhoff, Journal of Mathematical Analysis and its Applications 7, 271–283.
Veblen, T. (1919). The intellectual pre-eminence of Jews in Modern Europe, Political Science Quarterly 34, 33–42.
Veblen, O. (1912). An application of modular equations in analysis
Whittaker, E. T. (1945). George David Birkhoff, Journal of the London
situs, Annals of Mathematics 14(2), 86–94. Veblen, O. (1946). George David Birkhoff, Yearbook of the American Philosophical Society, 279–285; BCMP I, xv–xxi.
Mathematical Society 20, 121–128. Wiener, N. (1953). Ex-Prodigy: My Childhood and Youth, New York: Simon and Schuster.
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The Mathematical Tourist
Dirk Huylebrouck, Editor
The Kepler Memorial in Graz MICHAEL LONGUET-HIGGINS
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe´ where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to
ohannes Kepler was a pivotal figure in the emergence of modern science from earlier superstitions and rigid doctrine. He was born in December 1571, in the small German town Weil der Stadt, of which his grandfather had formerly been mayor, and which is located 20 miles west of Stuttgart. He became the first to treat the planets and other celestial objects as governed by simple physical laws. He insisted that such laws can be discovered not by abstract thought alone, but rather by accurate and painstaking observation. Kepler’s outstanding mathematical abilities were recognized early in his life, but childhood smallpox left him with crippled hands and weak vision, and he was assumed to be destined for a religious life. In 1584 he was placed in the seminary of Adelburg, and after two years was transferred to that of Maulbronn. In 1588, however, he entered the University of Tu¨bingen where he studied astronomy privately under the guidance of Michael Maestlin, the professor of mathematics. Kepler became an adherent of the Copernican description of the solar system, in preference to the Ptolemaic. After graduating from Tu¨bingen, Kepler’s first appointment was as a teacher at the Protestant school in Graz, Austria. His home in Graz can no longer be pointed out, but in a sheltered square near the city centre one can find the plaque shown here.
J
this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
An English translation follows:
â
Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail:
[email protected] 48
THE MATHEMATICAL INTELLIGENCER Ó 2011 Springer Science+Business Media, LLC
DOI 10.1007/s00283-011-9240-5
JOHANNES KEPLER 1571-1630 Taught here at the former Protestant monastery school from 1594 to 1599 as Professor of Mathematics. While in Graz he wrote his first astronomical work ‘‘The Secret of the Structure of the Universe’’ which brought him fame throughout the Western world. In the year 1600 following the Counter-Reformation he had to leave Graz and was appointed at the court of Kaiser Rudolf II in Prague, first as a collaborator of Tycho Brahe, and then as his successor.
The Evangelical monastery school was closed and converted to a cloister for nuns of the order of St. Clare. Kepler was a prolific author. The three laws of planetary motion which bear his name were not all published together. The first two laws – that the planets move in plane elliptical orbits with the sun in one focus and that the radius from the sun sweeps out equal areas in equal times – were included in his Astronomia Nova, published (eventually) in 1609. The third law, relating the period and the distances of the planetary orbits, appears in Harmonices Mundi (On the Harmony of the Universe), which was published in 1619. Note that ‘‘Harmonices’’ is a genitive singular, meaning ‘‘of the Harmony.’’ As might be expected, Kepler was also an accomplished geometer. He was the first to describe the ‘‘rhombic triakontahedron,’’ the polyhedron obtained by drawing tangent planes to the sphere touching the 30 edges of a dodecahedron. This polyhedron, illustrated in Harmonices Mundi (1619) was formerly known as ‘‘Keplersche Ko¨rper’’ or ‘‘KK.’’ It is the outer surface of a ‘‘Kepler Ball,’’ a solid that can be formed out of 20 ‘‘golden rhombohedra,’’ 10 sharp and 10 flat, as building blocks. Moreover, one of the stellations of KK can be built up from 20 of the sharp rhombohedra, and has been named a ‘‘Kepler Star.’’ Kepler also made significant contributions to optics; in his ‘‘Dioptrice’’ (1611) he expounded the theory of refraction by lenses and suggested the principle of the inverted telescope. While in Graz and also in Prague, Kepler found that one of his duties was to compose astrological forecasts. With the help of a good sense of the current political situation, he carried this out with some success, but with increasing skepticism. Toward the end of his life he would not make astrological forecasts himself, although he allowed others to use his data for such purposes.
Kepler’s major work at the time was considered to be his completion and publication of the Rudolphine Tables (Ulm 1627) based on Tyche Brahe’s extensive astronomical observations. These gave the parameters for the orbits of the planets. Although not without errors, they were subsequently much used. Throughout his life Kepler had to deal with many personal problems, including the nonpayment of his salary as court astronomer and the defense of his own mother from charges of witchcraft. In the latter he was eventually successful. The plaque was photographed on a return journey from the Bridges Conference in Pe´cs, Hungary, in July 2010. It was quoted at the ISIS-Symmetry Congress held at Gmuend, Austria, in August of that year.
Sources [1] Article titled ‘‘Kepler, Johann’’ in Encyclopaedia Britannica, 14th ed., and the authorities quoted therein. Encyclopaedia Britannica, London (1929). [2] Article titled ‘‘Johannes Kepler,’’ in Wikipedia, http:// wikipedia.org/wiki/JohannesKepler.
Institute for Nonlinear Science University of California San Diego 9500 Gilman Drive, La Jolla, CA 92093-0402 USA e-mail:
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Pursuit of Genius: Flexner, Einstein, and the Early Faculty at the Institute for Advanced Study by Steve Batterson WELLESLEY, MASSACHUSETTS: A. K. PETERS, LTD, 2006, 302 PP., US $39.00, ISBN: 1568812590 REVIEWED BY MARIANNE FREIBERGER AND RACHEL THOMAS
hen the Institute for Advanced Study officially opened in 1933, it did so with a bang. Its School of Mathematics included Albert Einstein, Herman Weyl, and John von Neumann. The Institute offered unrivalled salaries for researchers, and working conditions conducive to research that were second to none. Ever since, the IAS has lived up to the expectations raised by its spectacular start. Mathematicians who have worked at the Institute famously include Kurt Go¨del, Paul Dirac, and Paul Erd} os, to name just a few from a truly impressive list. In this carefully researched book, Steve Batterson chronicles the early history of the Institute, focusing on its inception and the first generation of researchers. With meticulous attention to detail, Batterson describes the vision on which the Institute was founded, the protracted efforts in diplomacy, behind-the-scenes intrigue that accompanied its establishment, and the early developments that shaped its future. Batterson’s background as a mathematician is clear from the extensive footnotes providing evidence for his account with an almost mathematical level of rigour. For all those with a special interest in the IAS, because they have worked there or out of sheer admiration, this book makes a fascinating read. Batterson starts his narrative with the birth in 1866 of Abraham Flexner, the first, and rather unlikely, director of the Institute. Flexner was not a mathematician, not even an academic, but had gained considerable prestige as the author of an influential and scathing report on the teaching standards in US medical schools. This expertise, as well as Flexner’s philanthropic bent, attracted the interest of high-school dropout and department-store magnate Louis Bamberger and his sister Carrie Bamberger Fuld. In search of opportunities to build a lasting legacy from their vast fortune, the Bambergers were interested in endowing a medical school in their native Newark. Flexner skillfully redirected their ambition – it wasn’t a medical school that was needed in New Jersey, but a ‘‘modern university’’ after Flexner’s own taste: with high-calibre, highly paid researchers enjoying a maximum of academic freedom and teaching a small number of the brightest postgraduate students, all set to produce groundbreaking results. The first half of the book describes Flexner’s endeavours to make his vision a reality, backed by generous funds from the
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DOI 10.1007/s00283-011-9231-6
Bamberger family. Flexner soon realised that mathematics was an ideal subject with which to start off the Institute. Not only was there a niche in the market, as no institution in the US could boast much mathematical expertise, but also the subject was cheap, only requiring chalk, blackboards, and access to a good library. While Flexner travelled extensively around Europe and the US to seek support, guidance, and contacts in a subject he knew nothing about, Hitler did his bit in Germany, driving the intellectual elite to seek positions abroad. Thus the instant mathematical prestige of the Institute was a result not only of Flexner’s enlightened vision and the Bambergers’ money, but also of the political situation of the day – as Batterson puts it, it was the result of the diverse aspirations of two laypeople: Hitler and Flexner. Mathematicians will encounter many familiar names in Batterson’s narrative. His scholarly style is punctuated with glimpses into his protagonists’ lives, their work, and some juicy accounts of diplomacy and intrigue. Batterson carefully unravels the confusing developments in the run-up to the Institute’s official opening, which see Flexner coping with a neurotic Herman Weyl, poaching mathematical talent from the Institute’s neighbour, Princeton University, and even censoring Einstein’s correspondence. But despite some decidedly underhanded tactics, Flexner’s efforts met with enormous success: the first half of the book culminates in the official opening of the Institute with a star line-up. The book then moves on to Flexner’s efforts to establish other schools of ‘‘comparable distinction’’ to his triumphant start with mathematics. He set his sights on a school of Economics, and one of Humanistic Studies. However the path was not smooth, and the process of finding talented researchers and the money to fund them contained as much wheeling and dealing as you’d find in any corporate boardroom. The book ends with Flexner’s final efforts to put the Institute on a sound financial footing before his retirement. But in the midst of the Great Depression this was a profoundly difficult task. Batterson gives us a thorough account of the competing issues Flexner faced, which makes some of the compromises he had to make very understandable. And, although Batterson doesn’t hide Flexner’s flaws or doubtful decisions, his insight into his principles and his efforts to create the Institute paint a sympathetic picture, which is what makes the final chapter so poignant, as Flexner himself is manoeuvred out of the directorship by the faculty he brought together. We usually think of our august academic institutions as some sort of solid edifices that exist apart from the bodies and brains that fill them. Batterson’s book about the early days of the Institute for Advance Study is a good reminder that these institutions are very much the result of the people who found them, fund them, and work in them – both academics and administrators. Plus Magazine, Millennium Mathematics Project Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WA UK e-mail:
[email protected] e-mail:
[email protected] Measure of the Earth by Larrie D. Ferreiro NEW YORK: BASIC BOOKS, 2011, 320 PP., US $28.00, ISBN: 978-0-465-01723-2 REVIEWED BY ANDREW J. SIMOSON
hat is the shape of our earth? is an old question. Legends suggest that an early answer was that of a flat disk. The Greek philosophers, notably Aristotle and Archimedes, championed a stationary ball at the very center of the universe. Two millennia later, astronomers such as Copernicus, Galileo, and Kepler still mostly agreed, but insisted that the earth moved about the sun. And what kept the earth moving in such an orbit? Rene´ Descartes hypothesized celestial vortices of ether, somewhat like prevailing winds or Gulf Stream water currents, that kept the planets in their courses. In the late seventeenth and in much of the first half of the eighteenth centuries French savants, as illustriously led by the father and son astronomers Giovanni and Jacques Cassini, further reasoned that such vortices would constrict the planets so that their shape resembled a lemon. Not so! interjected Isaac Newton with his maverick ideas of gravity as elaborated in the Principia of 1687; a spinning earth should be flattened at its poles, its shape resembling a mandarin orange. He went on to calculate this degree of flattening, predicting that the difference Dr between earth’s equatorial radius q and its polar radius R was Dr ¼ q R 17:1 miles & 27.5 km for a flattening ratio of 1/230 [1, p. 824]. Which model—the lemon or the orange—was the more accurate? For decades the debate raged. Members of the Royal Society and the French Academy of Sciences more or less sided with their country’s protagonist, so much so that, as Larrie Ferreiro describes it, ‘‘each side glowered warily at the other across the Channel.’’ However, a small Newtonian party began to grow in France, represented by Pierre Louis Moreau de Maupertuis, Voltaire, and E´milie du Chaˆtelet; see J. B. Shank’s Newton Wars [2] for a full story of these developments. Finally, in 1735, the French Academy of Sciences decided to settle the question by sending a geodesic mission to measure a degree of arc along a line of longitude at the equator in the Andes Mountains in the region now called Ecuador. Ferreiro’s new book narrates the story of this very first international joint scientific expedition. Briefly, if their measurements were more or less than that of the degree of arc already measured near Paris, then the world’s shape was respectively oblique (like a lemon) or oblate (like a mandarin orange). The story of this expedition has been told before in English, albeit in less detail, most recently in The Mapmaker’s Wife [8], which focuses on the wife of a junior member of the scientific team. In short, the husband has been marooned for many a long year on his way back to France in what is now French Guiana, and she travels down the Amazon from the Andes, finally reuniting with her
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husband after having been lost alone in the jungle for many days. A terse account of the geodesic mission is J. R. Smith’s From Plane to Spheroid [6], who devotes 73 pages to it in his chronicle of earth measurers throughout time. Ferreiro’s approach to the expedition’s story is somewhere between Whitaker’s novel-like and Smith’s journallike approaches. He relates the story of each member of the team, sometimes giving us the names of the slaves bought and occasionally freed by the team members. All told, the team had five principal members: three French academicians and two Spanish naval officers. Although the Spaniards had been initially appointed to prevent blackmarket activities by the French savants in the Viceroyalty of Peru, they eventually became coequal researchers with the French. Louis Godin was the designated leader of the team but was soon largely discredited for poor leadership and mismanagement of funds. At the end of the expedition he took a professorship in Lima instead of returning to France, and he became the principal architect in rebuilding the city after a horrific terremoto in 1746, as its colonial part attests to this day. Pierre Bouguer, a recognized mathematical genius, who wrote a watershed treatise on shipbuilding techniques in between geodesic observations while atop various Andean mountains, ended up being the unofficial leader of the team. Charles Marie de La Condamine, a soldier of fortune, gambler, universal savant, and one whose exploits, Ferreiro claims, inspired his friend Voltaire to write the play Alzire, wound up underwriting a significant chunk of the total cost of the expedition, roughly two million dollars in today’s currency. Jorge Juan and Antonio de Ulloa were the Spaniards. They began the expedition as newly minted lieutenants. Toward the end of their careers, Juan rose to admiral in the Spanish navy whereas Ulloa rose to secretary of the navy. As Ferreiro weaves his story of the ten years needed for this expedition, he reminds us of the political and social tensions of the era: of Spain’s paranoid reluctance to allow foreign travel in its American colonies, of the often maniacal savant tendencies to focus on protocol and rank, and of the low value Europeans placed on Amerindian life. He describes the human failings as well as the human resiliency of the team members. We learn about the official funds Godin lavished on a harlot in the island of Haiti while waiting months for a boat to ferry the team to the Panamanian coast. We learn that halfway through the mission, La Condamine served as lawyer in an extensive wrongful death suit on behalf of the team’s surgeon who had been fatally wounded in a love triangle gone wrong. We learn about the team’s medical doctor, Joseph de Jussieu, who took his Hippocratic oath seriously. He remained in Peru for 36 years battling yellow fever and smallpox epidemics. He also doubled as a botanist. In 1747, thanks to his samples and descriptions of South America’s flora sent back to Paris, he was elected associate botanist in the Academy. While collecting yet more specimens in what is now Bolivia, he inspected the silver mines and ‘‘was haunted by the appalling conditions he saw at the mines [the workers handled liquid mercury to extract silver from the ore] and could not bring himself to walk away.‘‘ He stayed there for many years tending the sick and engineering an improved Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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DOI 10.1007/s00283-011-9245-0
infrastructure for water management. Later when he was struck down by mercury poisoning himself and no longer able to care for his basic needs, his friends in Lima arranged passage to France. There for his last eight years of life amid the ‘‘silence and torpor’’ of advanced mercury poisoning, he occasionally hallucinated that ‘‘he was back in the Caribbean botanizing the new world.’’ To measure the earth, the team used a triangulation scheme pioneered in 1617 by Willebrord Snell. That is, define a geodesic family of triangles in the plane as one that can be assembled by starting with a single triangle, and thereafter appending triangles successively so that each appended triangle shares at least one of its sides, vertex to vertex, with a current family triangle. If any one side, the baseline, of one of the triangles in such a family is known, and the angles from every triangle are known, then it is an easy although tedious matter to determine the distance between any two vertices in the family of triangles. For their baseline, the team chose a flat plain called Yaraqui, now the site of Quito’s airport, seven miles long. The team split into two squads, each independently measuring the baseline starting from its opposite endpoints using poles of twenty feet. When they finished a month later, their baseline measures differed by less than eight centimeters! The remaining vertices in their triangular family were atop mountains, some of them active volcanoes. In fact, one of these erupted during their tenure, albeit two years after they had camped thereon. One of the reasons the mission took so long was that for much of the time fog shrouded their observation points. At long last, through wars, riots, murder, lack of funds, romantic attachments, and disease, the average of the computations independently deduced by Bouguer, La Condamine, Juan, and Ulloa, the arclength of one degree at the equator along a line of longitude at sea level was B & 56760 toises (B for Bouguer) [6, p. 248], where one toise is approximately 1.949 meters. When compared with modern measurements, this value differs by less than half the length of a standard soccer field. Bouguer went on to calculate that this measurement corresponded to a flattening factor of 1/179. Meanwhile, one year into this equatorial expedition, the French Academy of Sciences launched a second geodesic mission. This one was led by Maupertuis who, in trying to go as near to the North Pole as possible, traveled to the Arctic Circle to measure one degree of arc in what is now Finnish Lapland. Although Ferreiro writes that news of these developments blindsided the equatorial team when they learned about it much after the fact, even depressing them, plans for twin missions to the extremes of the earth had been coordinated from the beginning. The two expeditions were not meant to be rival enterprises. Instead their purpose was to measure the earth at its extremes and compare those measurements. Within a year of its commencement, this second expedition returned to Paris with a measurement of M & 57438 toises (M for Maupertuis) [6, p.184]. They too had adventures: shipwrecks, continual nights of freezing cold and snow, and biting insect hordes during the midnight suns, as is well chronicled in [7], a book also reviewed in this journal [3]. Whereas La Condamine and Maupertuis were always close friends, 52
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Ferreiro mentions multiple times that Bouguer considered Maupertuis a rival, even an enemy. Yet nowhere in Mary Terrall’s exhaustive biography does she mention that Maupertuis had reciprocal thoughts about Bouguer. Indeed, before their geodesic expeditions they often collaborated. For example, Bouguer solved the classic pursuit problem of a privateer overtaking a merchant ship on the high seas where the merchant ship flees along a straight path and the privateer ever sails toward its prey; Maupertuis generalized the problem so that the pursued can follow any curve; for details, see [5]. Of course, Maupertuis had many naysayers, and perhaps Bouguer was one of the least of these. Or perhaps Bouguer used Maupertuis as a quasi-imagined antagonist to help spur himself on to greater mathematical achievements much as a sports star might use the trash talk of a rival to impel himself to greater feats. It is not difficult to calculate the least squares error estimate for Dr given B and M. For simplicity, we center these arclength measurements at the Equator and at the Arctic Circle, respectively. As with the Cassinis and Newton, assume that the profile of the earth is an ellipse whose parametric equations are ðx; yÞ ¼ ðq cos /; R sin /Þ; where / is a parameter. Since dy dy=d/ R cos / R ¼ ¼ ¼ cot / dx dx=d/ q sin / q then the latitude h for a given parameter / is q hð/Þ ¼ tan1 tan / R or, equivalently, the parameter / corresponding to the latitude h in radians is R /ðhÞ ¼ tan1 tan h : q By this relationship, let /1, /2, /3 be the parameter values associated with the latitudes 0.5°, 66°, and 67°, respectively. Then the arclengths along one degree of arc along a line of longitude in terms of q and R at the Equator is Z /1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 sin2 / þ R 2 cos2 /d/ Qe ðq; RÞ ¼ 2 0
and at the Arctic Circle is Z /3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Qa ðq; RÞ ¼ q2 sin2 / þ R2 cos2 /d/; /2
which means that the distance D(q, R) between (Qe, Qa) and (B, M) is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dðq; RÞ ¼ ðQe BÞ2 þ ðQa MÞ2 : D experiences its minimum at a unique (q,R), and at this point Dr ¼ q R 18:7 miles & 30.1 km, for a flattening ratio of 1/213. However, in the early nineteenth century, Maupertuis’s arclength was remeasured at M & 57188 toises [6, p. 189]. With this new measurement, the minimum value of D occurs when Dr 11:8 miles & 19.0 km for a flattening
ratio of 1/336. The actual values are Dr 13:3 miles 21:4 km for a flattening ratio of 1/298. For further details on these calculations see [4]. All in all, these results are truly remarkable achievements of the Enlightenment. Finally, it is an irony of this entire story of a grand intellectual quest that, as Ferreiro points out, no matter how often the French and Spanish savants patiently explained their mission to skeptical onlookers, the listeners almost always left ever more convinced that these explorers were but prospecting for gold. Yet what these explorers sought was better than gold: the resolution of the earth’s size and shape. Ferreiro himself felt this veritable better-than-gold fever. As he explains it, ‘‘[when first hearing of the expedition] the epic saga gripped me almost immediately, and for two decades now it has refused to let me go.’’ Readers of his resultant book will be wafted by this very fever.
REFERENCES
[1] Isaac Newton, The Principia, translated by I. B. Cohen and A. Whitman, University of California Press, Berkeley, 1999.
[2] J. B. Shank, Newton Wars, Chicago University Press, Chicago, 2008. [3] Andrew Simoson, A review of Mary Terrall’s The Man Who Flattened the Earth, The Mathematical Intelligencer 32:4 (2010) 69–72. [4] Andrew Simoson, Newton’s radii, Maupertuis’ arclength, and Voltaire’s giant, College Mathematics Journal, 42 (2011) 183– 190. [5] Andrew Simoson, Pursuit curves for The Man in the Moone, College Mathematics Journal, 38 (2007) 330–338. [6] James R. Smith, From Plane to Spheroid, Landmark Press, 1986. [7] Mary Terrall, The Man Who Flattened the Earth, University of Chicago Press, Chicago, 2002. [8] Robert Whitaker, The Mapmaker’s Wife, Basic Books, New York, 2005.
King College Bristol, TN 37620 USA e-mail:
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The Tourist by Florian Henckel von Donnersmarck, Christopher McQuarrie, and Julian Fellowes GK FILMS, COLUMBIA PICTURES, IN ASSOCIATION WITH SPYGLASS ENTERTAINMENT, 2010 REVIEWED BY MARY W. GRAY
rilliant, but remote, obsessed, at least a little crazy. That’s the usual film characterization of a mathematician. Mathematics is the occupation of choice to convey alienation and inexplicable genius, even when actually doing mathematics has nothing to do with the plot. What a surprise then to see ‘‘mathematician’’ adopted to project the ultimate in blandness in The Tourist. Although portrayed by Johnny Depp, most noted in Hollywood for his roles as a pirate, and sexy enough to attract the attention of the Angelina Jolie character, he is intended to be the ‘‘ordinary’’ American tourist, an innocuous math teacher from Wisconsin, not the stereotypical mathematician. In fact, the character is so unlike the way a mathematician is usually depicted on screen that mention of the character’s occupation is made in few if any reviews of the film. Brilliant but homicidal is the most extreme model of a mathematician—The Girl Who Played with Fire [1] or Dustin Hoffman in Straw Dogs [2]. The theme of these two movies might be ‘‘don’t mess with mathematicians’’ [3]. Lisbeth Salander’s attempts to prove Fermat’s Last Theorem do not survive the transition from book to film, but math ability is part of her image as an antisocial, near-psychotic hacker. In the novel on which Straw Dogs was based, the isolated professor defending his ‘‘castle’’ was an English professor. The only apparent reason for the change is that equations on a blackboard, over which the protagonist might obsess, work better on screen than something from the biography of a late 18th-century diarist; the equations give the wife who is feeling neglected the opportunity for clandestine sabotage by changing a plus to a minus in an equation. In a recently released 2011 remake, the mathematician is transformed into a Hollywood screenwriter; no indication of why—perhaps because public perception of mathematicians has changed? For better, or for worse? Or maybe because the scene has shifted from England to the American South, considered a less likely choice for an academic’s retreat? In a later film [4], Dustin Hoffman plays an autistic savant who has great skill with numbers, a trait often mischaracterized as mathematical talent. There are two other characterizations mistakenly perceived in films and in public perception as representing mathematical talent. One is the numerologist, sometimes incorrectly called a number theorist, such as is encountered in Pi [5], in which the protagonist’s obsession involves him with Gematria, with Hebrew numerology, and with WallStreet types who believe his mysterious 216-digit numeral is the key to making a fortune. The other is the hacker, sometimes with a nod to real mathematics as in the case of
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Salander and the mathematician among hackers in Sneakers [6] and sometimes not. Some screen portrayals of real-life mathematicians—Alan Turing in Breaking the Code [7] and John Nash in A Beautiful Mind [8]—certainly contribute to the stereotypical image. Although it is true that Russell Crowe’s visual hallucinations as Nash are fictional and the emphasis in the film version of Breaking the Code is more on Turings’s sexuality than on his mathematics, the sense of isolation and fixation pervades the films. An American Public Broadcast System docudrama Newton’s Dark Secrets [9] does the image of mathematicians little good; although it credits Newton with great intellectual advances, it depicts an obsessive, secretive, and vindictive personality, and focuses on his explorations in alchemy and numerology, obviously thought to make better TV. A reviewer did praise it for ‘‘humanizing’’ Newton, which may or may not be for the better. Enigma [10], a film based very loosely on the activities at Bletchley Park, was heavily criticized for its misrepresentation of the Turing character as well as for its complete neglect of the contributions of Polish cryptologists. The depiction of the mathematician in Enigma is spiced with spy-catching intrigue and romance but relies on the all-too-usual image of a slightly bizarre genius. Another film purported to be based on the life of a mathematician is The Professor’s Beloved Equation [11], but if the Professor, who can retain only 80 minutes of memory, has anything other than number theory and unworldliness in common with Erd} os on whom the character is said to be based, it is difficult to see. At least the kind and gentle professor is a welcome counterbalance to the arrogance all too often seen in film mathematicians. Also off the beaten track is a short erotic film created by and starring mathematician Edward Frenkel. Said to be about the beauty of mathematics, Rites of Love and Math [12] was inspired by the recently revived 1966 Japanese Rite of Love and Death [13]. If more widely released, the film might do much to alter the public image of mathematicians. There has been a TV docudrama of the Unabomber [14], but perhaps fortunately for the image of mathematicians it focuses on his brother and on the pursuit by a mail inspector. There may be more about Ted Kacyznski to come. A film based on the Perelman biography [15] could also once again show the public an extreme in otherworldly genius. One real-life mathematician fares somewhat better in film, perhaps because we know so little about her. In Agora [16], Hypatia appears to uncover the concept of elliptical orbits before being torn apart by a Christian mob; although the discovery may be purely speculative, it is a nice touch and her devotion to learning is probably accurately rendered. On the other hand, the real mathematical contributions of Sonya Kovelevskaya are given short shrift in A Hill on the Dark Side of the Moon [17]. And who could believe Walter Matthau as a kindly Uncle Einstein cavorting with his fun-loving friend G} odel in I.Q. [18]? Meg Ryan as a Ph.D. candidate in mathematics in I.Q. is also a bit off-key, although there is a nice scene involving Zeno’s paradox. But in any case can anyone take the characters seriously? Women have had their share of less than positive stereotypes as well. In addition to Lisbeth Salander, we have Gwyneth Palmer’s mathematician (or maybe not) in Proof
[19]. None of the three mathematicians in that film displays many positive characteristics: the father was definitely mentally disturbed, the daughter probably so, and the young man is attracted not to her but to the possibility of stealing another’s results. Coincidently, before his turn as Nash, Crowe was in an Australian film called Proof [20], but the proof was of a photo, not a theorem. In another version of father–daughter shared mathematics, An Invisible Sign [21] asks whether a mathematics-obsessed dysfunctional daughter can become an effective math teacher after bringing an ax to her elementary school. But love triumphs in the end. In Incendies [22] mathematics as the subject of study of one of the protagonists is apparently used to suggest the single-minded obsession also demonstrated in her quest to find her father. It would be interesting to see how the discovery of chaos theory by Thomasina, the precocious 13-year-old said to be modeled on Ada Lovelace, would be portrayed, were Tom Stoppard’s Arcadia [23] to be filmed. The more mature Jill Clayburgh proves the Snake Lemma in the opening of It’s My Turn Now [24], and there is a positive conclusion in her exchange with a male graduate student, but the portrayal appears to be another instance of mathematics being used to characterize someone as smart but emotionally repressed (until she meets a baseball player). In the strongly feminist Antonia’s Line [25] the choice of mathematics as a profession reinforces the matriarchal society’s breaking down of stereotypes, especially when the mathematician neglects her baby in favor of differential geometry. Another illustration of the advantage of blackboards in Good Will Hunting [26] has young Matt Damon demonstrating another stereotype—the sudden solution, in his case of a problem a Fields medalist has left on a blackboard at MIT where Damon is a janitor. The mathematicians in the film are a fairly arrogant bunch, making mathematics look very ego-driven, not Will’s thing at all. The plan to channel his talent into a productive career is foiled as the incipient mathematician finds that a job in mathematics is not as enticing as riding off to join his girlfriend in California. Then we have mathematicians as observer and suspect in The Oxford Murders [27]. As the film is based on the novel of the same name by a real-life mathematician, one would expect believable characterizations. The introduction we see is a lecture by the mentor figure (played by John Hurt) on Wittgenstein—okay, this is Oxford, and maybe the title alerts moviegoers, but it is neither your usual film fare nor a usual math lecture. What mathematicians may really find hard to believe is that the graduate math student passes up a chance to travel to Cambridge to hear Wiles’s climactic lecture revealing his initial proof of Fermat’s Last Theorem; the viewers do get to go along, but the lecture-room scene will seem all too brief. For a mixture of the real and imaginary, we have Fermat’s Room [28]. The participants, who have some connection with mathematics, are lured to a remote location and presented with a series of Martin-Gardner-type problems that they must solve to stay alive. They are given labels as mathematicians (except for the woman for whom the best they could do is a 16th-century Spanish philosopher Oliva Sabuco), the
significance of which becomes apparent as the mastermind pulls the string and the room closes in on them. None of the eponymous mathematicians were in real life deranged— obsessed maybe, but not crazy—but one of the namesakes clearly is. The key is that ‘‘Galois’’ faked a proof of the Goldbach conjecture, which ‘‘Hilbert’’ actually proved, but ‘‘Pascal’’ throws the solution into the river. No doubt, an interesting manifestation of the madness theme. There are a number of films in which a major character is a mathematician, but his work has no discernible connection with the plot. Sean Penn in 21 Grams [29] is such an example, as are Jeff Bridges in The Mirror has Two Faces [30] and Jeff Goldblum in Jurassic Park [31]—as a mathematician he may work on chaos theory, but not the chaos caused by escaping dinosaurs. The message, if any, is that mathematics is something done by smart people, but it is not all that important to them nor to the story. In at least one film, mathematics—in this case chaos theory and fractals, making for some great graphics of the Mandelbrot set—actually is essential to the plot. The main character in The Bank [32] is an unscrupulous, narcissistic executive to whom a mathematician has sold his program BTSE (Bank Training Simulation Experiment), the goal of which is to predict market downturns. The mathematician’s precocity is nicely illustrated in the opening middle-school classroom scene where he is the only one who understands the concept of compound interest, but what might be his adult motivation is murky. Are we to believe that he too is concerned only with profit, or is he meant to be the usual unworldly mathematician not well-connected with reality and susceptible to manipulation by others? A TV series presents the chance for character development that a single film lacks. In its six seasons on U.S. television, Numb3rs [33] has shown Charlie Eppes, mathematician at fictional CalSci, assisting his FBI-agent brother by using mathematics to solve crimes. The problems are sometimes ridiculous but in at least a few cases there is an opportunity to see how mathematics works in real life, although the speed at which this occurs in an hour segment is not very realistic. The major development in Charlie’s character lies in his relationship with another mathematician, Amita Ramanujan (!). Complaints from viewers about his potentially exploitive relationship with her as his graduate student were answered by having her complete her Ph.D. and having her appointed as an assistant professor in the department, with the obvious attendant problems. The conflicts arising from her efforts to establish an independent career are realistic, but their resolution seems unduly fortuitous. It has been reported that David Krumholtz, who played Charlie, spent time with mathematicians to make his character more convincing, and many aspects of the story line will resonate with mathematicians. In spite of some problems, Numb3rs presents the most all-around positive and believable image on film, although of necessity it does not show what doing mathematics is really like. That is left to an American Public Broadcasting System documentary The Proof [34]. In this documentary Andrew Wiles talking about his result is deeply moving; whereas most cannot claim such success, many can identify with his commitment to and pleasure in mathematics. Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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But what of our mild-mannered, innocuous mathematician, in The Tourist? In common with most other screen characterizations of mathematicians, there is not much mathematics to be seen. On the other hand, there is also little indication of madness. Great skill at evading his pursuers, agility at cavorting over the canals and rooftops of Venice, dedication to the woman in the story—indeed a new image for a profession meant in this film to indicate an unremarkable, ordinary sort of person.
[16] A. Amena´bar and M. Gil, Agora, Barcelona: Mod Producciones, 2009. [17] S. Pleijel, A Hill on the Dark Side of the Moon, Stockholm: Moviemakers Sweden, 1983. [18] A. Breckman and M. Leeson, I.Q., Hollywood: Paramount Pictures, 1994. [19] D. Auburn and R. Miller, Proof, Los Angeles: Miramax Films, 2005, from the play of the same name by David Auburn, New York: Faber & Faber, 2002. [20] J. Moorhouse, Proof, Richmond, Victoria, Australia: House and Moorhouse Films, 1991.
REFERENCES
[1] J. Frykberg and Stieg Larsson, The Girl Who Played with Fire (Flickan som lekte med elden), Stockholm: Yellow Bird Films, 2009, based on the novel of the same name by Stieg Larsson, New York: Knopf, 2009. [2] D. Z. Goodman and S. Peckinpah, Straw Dogs, New York: ABC Pictures, 1972, based on the novel by Gordon M. Williams, The Siege of Trencher’s Farm, London: Socker and Warburg, 1969, reprinted as Straw Dogs, London: Bloomsbury Publishing, 2003. [3] http://world.std.com/*reinhold/mathmovies.html. [4] B. Morrow and R. Bass, Rain Man, Los Angeles: MGM, 1988. [5] D. Aronofsky, S. Gullette, and E. Watson, Pi, Santa Monica, CA: Artisan Entertainment, 1998. [6] W. F. Parkes and P. A. Robinson, Sneakers, Los Angeles: Universal Studios, 1992. [7] A. Hodges and H. Whitemore, Breaking the Code, London: BBC, 1996, from the play of the same name by Hugh Whitemore, Charlbury, UK: Amber Lane Press Ltd., 1987. [8] A. Goldsman and S. Nasar, A Beautiful Mind, Universal City: Universal Pictures, 2001, based on the biography of the same name by S. Nasar, New York: Simon & Schuster, 1998. [9] Nova, Newton’s Dark Secrets, Boston: WGBH, 2005. [10] T. Stoppard, Enigma, Burbank, CA: Buena Vista Films (Walt Disney), 2001, based on the novel of the same name by R. Harris, New York: Random House, 1995. [11] T. Koizumi, The Professor’s Beloved Equation, Tokyo: Ace Entertainment, 2006, based on the novel by Yoko Ogawa, The Housekeeper and the Professor (English translation), New York: Picador, 2009. [12] E. Frenkel and R. Graves, Rites of Love and Math, Tunbridge
[21] P. Falk and M. Ellis, An Invisible Sign, Los Angeles: J2 Pictures, 2010, based on the book by Aimee Bender, An Invisible Sign of My Own, New York: Doubleday Books, 2000. [22] W. Mouawad and D. Villeneuve, Incendies, Montreal: micro_scope films, 2010, from the play of the same name by Wadji Mouawad, Montreal: Lemeac Editeur, 2009. [23] T. Stoppard, Arcadia, New York: Doubleday Books, 1995. [24] E. Bergstein, It’s My Turn Now, Culver City, CA: Columbia Pictures, 1980. [25] M. Gorris, Antonia’s Line, Netherlands: Bergen Film Company, 1995. [26] M. Damon and B. Affleck, Good Will Hunting, Los Angeles: Miramax Films, 1992. [27] J. Guerricaechevaria, A. de la Iglesia, and G. Martinez, The Oxford Murders, Strasbourg: Eurimages, 2008, based on the book by Guillermo Martı´nez, The Oxford Murders, London: Abacus, 2005. [28] L. Piedrahita and R. Open˜a, Fermat’s Room (La Habitacio´n de Fermat), Barcelona: Notro Films, 2007. [29] G. Arriaga, 21 Grams, Los Angeles: Universal Studios, 2003. [30] R. La Gravenese, The Mirror Has Two Faces, Los Angeles: TriStar Films, 1996. [31] D. Koepp, The Lost World: Jurassic Park, Los Angeles: Universal Studios, 1997, based on the novel by Michael Crichton, The Lost World, New York: Knopf, 1995. [32] M. Betar, R. Connolly, and B. Price, The Bank, Melbourne: Arenafilm, 2001. [33] N. Falacci and C. Heuton, Numb3rs, produced by Scott Free Productions, CBS Television, 2005-2010. [34] Nova, The Proof, Boston: WGBH, 1997.
Wells, UK: Sycomore, 2010. [13] Y. Mishima, Rite of Love and Death (Patriotism) (Yuˆkoku), Tokyo: Toho Company, 1966. [14] J. McGreevey, Unabomber: The True Story, Toronto: Atlantis Releasing, 1996. [15] M. Gessen, Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century, Boston: Houghton Mifflin Harcourt, 2009.
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American University 211 Gray Hall Washington, DC 20016 USA e-mail:
[email protected] Oxford Handbook of the History of Mathematics by Eleanor Robson and Jacqueline Stedall (eds.) OXFORD: OXFORD UNIVERSITY PRESS, 2009, 800 PP., US $150.00, ISBN-13: 9780199213122, ISBN-10: 0199213127 REVIEWED BY STAFFAN RODHE
e hope that this book will not be what you expect. It is not a textbook, an encyclopaedia, or a manual. If you are looking for a comprehensive account of the history of mathematics, divided in its usual way into periods and cultures, you will not find it here.’’ This uncommon beginning of the Handbook fills me with expectation. Will this book reveal hidden parts of our subject, describe little-mentioned mathematical cultures, and give results of recent research, too? Yes, it does. Many of the 36 authors are not historians of mathematics, but are specialists in, for example, anthropology, archaeology, art history, philosophy, and literature. They show that the history of mathematics is not just the history of counting, number-systems, or the evolution of mathematical tools, but is full of nuances. The idea of mathematics itself is scrutinized: ‘‘What has it been, and what has it meant to individuals and communities? How is it demarcated from other intellectual endeavours and practical activities?’’ The first chapter ‘‘What was mathematics in the ancient world? Greek and Chinese perspectives’’ by G. E. R. Lloyd gives some clue of what will come. Lloyd surveys the Greek axiomatic-deductive methods based on Plato’s, Aristotle’s, and Euclid’s works among others, objecting to the widespread idea that these methods dominated Greek ancient mathematics. His first point is worth quoting for its relevance today: ‘‘…axiomatics was quite unnecessary. Not just in practical contexts, but in many more theoretical ones, mathematicians and others got on with the business of calculation and measurement without wondering whether their reasoning needed to be given ultimate axiomatic foundations.’’ Lloyd’s survey of Chinese mathematics is shorter. It is based on some of the Chinese mathematical masterpieces, such as the Nine Chapters, and the mathematical commentator Liu Hui’s texts. Lloyd sees the differences with the Greek mathematicians: ‘‘The Chinese, by contrast, were far more concerned to explore the connections and the unity between different studies, including between those we consider to be mathematics and others we class as physics or cosmology. Their aim was not to establish the subject on a self-evident axiomatic basis…’’ He concludes: ‘‘It is apparent that there is no one route that the development of mathematics had to take, or should have taken. … The value of asking the question … reveals so clearly … the fruitful heterogeneity in the answers that were given.’’ This is good to hear. In our global world we need diversity to build our route.
‘‘
W
The chapters are grouped into three main sections: ‘‘Geographies and cultures’’, ‘‘People and practises’’, ‘‘Interactions and interpretations’’. Each of these is divided into three subsections of four chapters each. Lloyd’s chapter appears in ‘‘Global’’, the first subsection of ‘‘Geographies and cultures’’. An appropriate name, as we have seen. As the editors say: ‘‘The dissemination and cross-fertilization of mathematical ideas and practices between world cultures is a recurring theme throughout the book.’’ There is diversity in time periods, cultures, and people. In ‘‘Global’’ you will find Urton’s interesting study of Incan and European accounting and Jami’s chapter on the transmission of Jesuit mathematics from Europe to China. Parshall notes: ‘‘By 1962 and the occasion of the fourteenth ICM in Stockholm, it was a well established fact that a community of mathematicians existed not just in individual national settings but internationally as well.’’ And, she continues, ‘‘Unlike the world’s nations, however, mathematics and its practitioners were naturally united in a common goal, the development—increasingly expressed in a common, transcendent language—of a fundamental body of scientific knowledge.’’ The editors are authors of chapters in the first main section. Stedall gives us a thrilling and very pedagogic chapter on relations between minor mathematicians in seventeencentury England. A recently found copy of an unpublished treatise by Nicolaus Mercator starts a journey backward to Thomas Harriot. The diagram, which gives the known circulation of Harriot’s method of differences and interpolation passing John Pell, Mercator, John Collins, John Wallis, William Jones, and at last the purchase of the Macclesfield Collection by Cambridge University Library in 2000, is an instructive example for historians looking for ways to present networks they are studying. Robson’s chapter is of course on Babylonian mathematics. The clay tablets she has studied are from an excavation of a scribal school in Nippur; most teach elementary measurement and arithmetic. This is not a chapter to read leaning lazily back in a sofa. It would be a valuable study for a student project at the university level. In two chapters we meet a study of the planning of the Roman city of Corinth and the controversy between engineers and their critics about the modernization of Naples in 1808. Aubin’s ‘‘Observatory mathematics in the nineteenth century’’ takes us to several observatories, in Greenwich, Paris, Go¨ttingen, and Kazan, to ‘‘examine the specific spatial arrangements of mathematical work within observatories’’. In presenting the main section, ‘‘People and practices’’, the editors ask: ‘‘Who creates mathematics? Who uses it and how? … Further, the ways in which people have chosen to express themselves—whether with words, numerals, or symbols, whether in learned languages or vernaculars—are as historically meaningful as the mathematical content itself ’’. There are two chapters on Islamic mathematics. In ‘‘Patronage of the mathematical sciences in Islamic societies’’, Brentjes explains that the Islamic mathematicians from the eighth to the twelfth centuries often worked in courts and were expected to offer the patron their ‘‘expertise in areas such as healing, observing the planets and stars, casting horoscopes, constructing instruments, 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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DOI 10.1007/s00283-011-9221-8
writing books, making automata, and repairing clocks, water wheels, channels, and other infrastructural components’’. You can find a similar mathematical diversity in Europe more than half a millennium later, in, for example, Milliet Dechales’s 4000-page university textbook Cursus seu Mundus Mathematicus (1674). Spooner’s and Hanaway’s chapter on ‘‘Siyaq: numerical notation and numeracy in the Persianate world’’ gives another perspective to ‘‘our’’ Arabic numerals. Siyaq numerals were used in the northern and eastern areas of the Islamic world. Even in the late medieval period siyaq was used in government, land management, and trade alongside the Arabic numerals to make falsification more difficult. This is a fascinating chapter, and it is impossible not to draw parallels to the development of the numeral symbolism in Europe. In John Denniss’s brilliant account of English textbooks on arithmetic, 1500–1900, we learn that pupils learnt arithmetic through rote learning of rules and methods. Robert Recorde’s ‘‘The ground(e) of artes’’ (1543) was published in many editions over the next century. Denniss suggests that its informal style – a sometimes humorous dialogue between Master and Scholar – contributed to its popularity. Recorde0 s mnemonic verses for root learning are superb. The ‘‘Rule of False’’ is described in twelve lines, of which the last five are: From too few take too few also: With too much joyn too few again; To too few add too many plain; In cross-wise multiply contrary kinde, And all truth by falshood for to find. Plofker tells us that the Indians wrote mathematical verses, too, in Sanskrit. Even the numbers are written poetically. I like the example where the number 375 is written in words as ‘‘arrow-mountain-Rama’’ (arrow = 5, mountain = 7, Rama = 3). The third main section is called ‘‘Interactions and interpretations’’, by which the editors mean: Interactions: ‘‘Mathematics is not a fixed and unchanging entity. New questions, contexts, and applications all influence what count as productive ways of thinking or important areas of investigation.’’ Interpretations: ‘‘The history of mathematics does not stand still either. New methodologies and sources bring new interpretations and perspectives, so that even the oldest mathematics can be freshly understood.’’ In ‘‘Modernism in mathematics’’, Jeremy Gray proposes that the work of Noether, Dedekind, Lebesgue, Poincare´,
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and Hilbert ‘‘can be regarded as a ‘modernism’ like modernism in painting, music and literature.’’ Cullen’s ‘‘People and numbers in early imperial China’’ brings us back to the question we meet in the first chapter: What is mathematics in the ancient world? Cullen sees this differently from Lloyd; his main interest is ‘‘the people who wrote and used texts rather than the texts themselves’’. The ‘‘Mathematical’’ section includes material new to this historian of mathematics: I did not know about Rolle’s cascades! June Barrow-Green’s history of Rolle’s theorem is a ‘‘must’’ for all teachers of calculus. In the ‘‘Historical’’ section we revisit four Egyptian mathematical myths: Egyptian p, the Horus eye fractions, right-angled triangles, and Egyptian unit fractions. Annette Imhausen shows that the Egyptians did not find an approximate value for p; they used the unit fraction 1/9 to calculate the area of a circle. The myth of the Horus eye is shown to have originated in 1911! The last chapter is also the last chronologically. Reinhard Siegmund-Schultze, in ‘‘The historiography and history of mathematics in the Third Reich’’ gives an account of recent research on mathematics under Nazi rule and a brief outline on the mass emigration of mathematicians from the Third Reich. The book ends with short biographies of its thirty-six authors with references to their writings. I recommend this book as a source of inspiration. Don’t try to read it straight through. Find your own thread; I assure you that you will be inspired. Leo Corry points out that ‘‘Writing a textbook involves much more than simply putting together previously dispersed results. Rather, it requires selecting topics and problems, and organizing them in a coherent and systematic way…’’ In his view, Bourbaki’s E´le´ments de mathe´matique is this ultimate textbook. Writing a handbook is similar. Robson and Stedall have organized the chapters in a coherent and systematic way. Though it is not the common way, it is inspiring. If this is not the ultimate handbook of history of mathematics, it is close to it.
Department of Mathematics University of Uppsala Box 480, S-751 06 Uppsala Sweden e-mail:
[email protected] Biscuits of Number Theory
have two distinct prime factors. Now, since n(n + 1) and n(n + 1) + 1 are relatively prime, the integer ðnðn þ 1ÞÞðnðn þ 1Þ þ 1Þ
by Arthur T. Benjamin and Ezra Brown (eds.) WASHINGTON, DC, THE MATHEMATICAL ASSOCIATION OF AMERICA, 2009, XIII + 311 PP., US $62.50, ISBN: 978-0-88385-340-5 REVIEWED BY JOHN J. WATKINS
he American biscuit is a glorious and wonderfully versatile culinary creation. My mother’s biscuits, hot out of the oven, were light and fluffy; we broke them in half to eat with butter and honey, a memory that still makes my mouth water. Our Uncle Dan made batches of biscuits in a cast iron skillet on family camping trips in the mountains of New Mexico; he served them piping hot as we crawled out of our sleeping bags on frosty mornings. Even now, whenever I head out for a day of skiing, my favorite breakfast is a huge plate of biscuits and gravy at a roadside cafe. Garrison Keillor talks about the virtues of biscuits every Saturday on his radio show A Prairie Home Companion: ‘‘Powdermilk Biscuits: Heavens, they’re tasty and expeditious! They’re made from whole wheat, to give shy persons the strength to get up and do what needs to be done.’’ It is the expeditious quality of biscuits that made them so popular with early-day pioneers; they are quick and easy to make. Biscuits of Number Theory is a collection of 40 exceptionally well-written articles and notes on number theory. As editors Arthur Benjamin and Ezra Brown say, ‘‘each item is not too big, easily digested, and makes you feel all warm and fuzzy when you are through.’’ Each item in this marvelous collection is a tasty and mathematically nourishing biscuit. Benjamin and Brown have performed an extraordinary service for mathematicians, teachers, and students by compiling this superb collection of articles. The idea sprang from a session called ‘‘Gems of Number Theory’’ at MathFest 2005 in Albuquerque (biscuits would come later), at which Mark Chamberland spoke on the Collatz 3x + 1 Problem, Ed Burger on Diophantine Approximation, Jennifer Beineke on Great Moments of the Riemann Zeta Function, and Roger Nelson on Visual Gems of Number Theory. Many of the articles in this book have been awarded prizes and, not surprisingly, many of the authors have won distinguished teaching awards. It is also a joy to find in this book articles by Leonard Euler (translated by George Po´lya), Paul Erdo¨s, Ivan Niven, Carl Pomerance, and Martin Gardner. One article—on the Riemann zeta function by Jennifer Beineke and Chris Hughes—was written especially for this book. Perhaps the most delightful paper in the book is one by Filip Saidak that presents a proof of Euclid’s theorem on the infinitude of the primes. His proof is not only new but is even simpler than Euclid’s. Saidak’s proof depends only upon two fundamental facts about primes: any integer greater than 1 has a prime factor, and two consecutive positive integers are always relatively prime. Let n [ 1 be an integer. Then, since n and n + 1 are relatively prime, the integer n(n + 1) must
T
must have three distinct prime factors. We can continue in this fashion to construct numbers having an ever larger and larger number of prime factors. For example, if we begin by letting n = 2, then 2 3 = 6 has two distinct prime factors; and then 6 7 = 42 has three distinct prime factors, and 42 43 = 1806 has four distinct prime factors. Next, observe that although we know that 1806 1807 = 3 263 442 must have five distinct prime factors, it in fact has six because 1807 factors into primes as 13 139. The next two iterations of this process produce numbers having seven and eleven distinct prime factors, respectively. It is truly astonishing that this proof remained undiscovered all these years. Another proof I enjoyed pffiffiffi enormously was a clever new proof by Tom Apostol that 2 is irrational. His proof is geometric and in effect uses (without mentioning) Fermat’s ‘‘method of infinite descent.’’ Begin with an isoscelespright ffiffiffi triangle with integer sides (such a triangle exists if 2 is rational), and let A be the vertex opposite the hypotenuse. From A draw a circular arc with center at either of the other two vertices, and let B be the point where this arc intersects the hypotenuse perpendicularly. Then, from B draw a perpendicular to the hypotenuse (that is, tangent to the arc) to intersect a leg of the triangle. This construction creates a smaller isosceles right triangle with integer sides. Since this process cannot be repeated indefinitely (Fermat’s infinite pffiffiffi descent), 2 cannot be rational. In 1638, in a letter to Mersenne, Fermat made one of his most remarkable conjectures, claiming that every nonnegative integer is ‘‘the sum of three triangular numbers, of four squares, of five pentagonal numbers, and so on.’’ Gauss proved this conjecture for triangular numbers and Lagrange proved it for squares; and then, Cauchy proved Fermat’s statement in general. Melvyn Nathanson provides a short, extremely accessible, two-page proof of Cauchy’s theorem that every nonnegative integer is the sum of m + 2 polygonal numbers of order m + 2. Euler discovered—first empirically, and then providing a proof many years later—a beautiful formula we now call Euler’s pentagonal number theorem: 1 Y
ð1 x n Þ ¼ 1 x x 2 þ x 5 þ x 7 x 12 x 15 þ x 22
n¼1
þ x 26 where on the right-hand side we have the Euler pentagonal numbers (3n2 ± n)/2. Euler used this theorem, and generating functions, to find the following recurrence for p(n), the number of partitions of an integer n: pðnÞ ¼ pðn 1Þ þ pðn 2Þ pðn 5Þ pðn 7Þ þ : Even today, there is no better algorithm for computing values of p(n). For example, given that p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 5, p(5) = 7, and assuming that p(0) = 1, we can compute p(6) = p(5) + p(4) - p(1) = 11 and p(7) = p(6) + p(5) - p(2) - p(0) = 15. David Bressoud and
Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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DOI 10.1007/s00283-011-9235-2
Doron Zeilberger provide a one-page combinatorial proof of Euler’s recurrence in their paper ‘‘Bijecting Euler’s partition recurrence.’’ The shortest paper in this book is Don Zagier’s elegant one sentence proof of Fermat’s famous theorem that every prime p 1 ðmod 4Þ is a sum of two squares. Of course the reason the proof is so short is that Zagier leaves out all the details. As the editors suggest, it is enlightening for students to work out these details for themselves. Several authors provided updates to their original articles for this collection—the editors call these ‘‘second helpings.’’ One such second helping by Stan Benkoski tells the charming story of how, as a student just beginning his graduate studies at Penn State, he came to write a paper on weird and pseudoperfect numbers with Paul Erdo¨s. The positive integer 20 is pseudoperfect because it is equal to the sum of some of its proper divisors (20 = 1 + 4 + 5 + 10); 20 is also an abundant number because the sum of all its proper divisors is at least 20 (1 + 2 + 4 + 5 + 10 = 22 C 20). On the other hand, the number 70 is abundant (1 + 2 + 5 + 7 + 10 + 14 + 35 = 74 C 70) without being pseudoperfect (it is clearly impossible to reduce this sum of divisors by exactly 4). Benkoski coined the term weird for positive integers such as 70 that are abundant but not pseudoperfect, and he raised a question that caught the attention of Erdo¨s: are there any odd weird numbers? When Benkoski first met Erdo¨s one morning at 8:00 a.m., Paul was still in his pajamas, and Stan helped him get dressed so they could go have breakfast and spend an hour talking mathematics prior to Paul’s scheduled lecture later that morning. The Benkowski/Erdo¨s paper contains a table of all weird numbers less than one million, beginning with the smallest weird number 70. This table appears on page 70 (merely a ‘‘weird coincidence’’ claim the editors). Although numbers have been studied in a serious way for more than three dozen centuries, there is always something new to discover. We have known how to generate all Pythagorean triples since the time of Euclid, but in 1970 A. Hall produced a gorgeous family tree of all Pythagorean triples with, of course, (3, 4, 5) as the root, such that each member of the family has exactly three offspring. I can’t help but wonder what Euclid would have thought of this extraordinary construction! By the way, the three offspring of (3, 4, 5) are (5, 12, 13), (21, 20, 29), and (15, 8, 17); but, to see the the entire tree, you’ll need to buy the book. This book achieves a nice balance among short proofs of famous results, attractive visual demonstrations of familiar formulas, and longer expository articles. One of the best expository articles is Jennifer Beineke and Chris Hughes’s ‘‘Great moments of the Riemann zeta function.’’ This sparkling paper traces this story from Euler’s proof of the infinitude of the primes, to the celebrated Riemann hypothesis, through to contributions by Hardy and Littlewood and others, and finally to a detailed discussion of recent advances on what is now one of the seven Millennium Problems. This delightful paper even includes a song by Tom Apostol, ‘‘Where are the zeros of zeta of s?,’’ sung to the tune of ‘‘Sweet Betsy from Pike,’’ that begins Where are the zeros of zeta of s? G. F. B. Riemann has made a good guess: ‘‘They’re all on the critical line,’’ stated he, ‘‘And their density’s one over two pi log T.’’ 60
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and ends with the advice There’s a moral to draw from this long tale of woe That every young genius among you must know If you tackle a problem and seem to get stuck, Just take it mod p and you’ll have better luck. A fabulous paper by Dan Kalman and Robert Mena makes the case that many of the properties we find so appealing in the Fibonacci sequence also occur more generally in any sequence that satisfies a second-order recurrence relation. This is why the Lucas sequence shares many properties with the Fibonacci sequence. This is one of many excellent papers in this book that would serve as an ideal source for a student project in a standard number theory course. Selecting the forty articles to include in this book was not an easy task. The editors selected a visual proof by J. Barry Love of the ancient identity involving sums of cubes and triangular numbers: 13 þ 23 þ 33 þ þ n3 ¼ ð1 þ 2 þ 3 þ þ nÞ2 : Love’s proof is indeed lovely, but I think the following proof by Solomon W. Golomb (Mathematical Gazette, 49, May 1965, p. 199) is equally good: 1
2
3
4
5
1 2 3
4
5
Each cube is visible in this figure; for example, 53 can be seen as the sum of its five 5 9 5 square layers. Note how Golomb cleverly handles the even cubes with the use of positive and negative space; Love similarly handles the even cubes by splitting one of the n 9 n squares into two n n2 rectangles. I mention this to suggest that there is ample excellent material for yet another volume of articles on number theory. I urge the editors and the Mathematical Association of America seriously to consider producing a second volume: More Biscuits of Number Theory. You can’t ever have too many! I hope I have managed to whet your appetite for the many tasty morsels this fine book has to offer, but all this mathematics has made me quite hungry. I think I’ll go whip myself up a batch of biscuits. Department of Mathematics and Computer Science Colorado College Colorado Springs, CO 80903 USA e-mail:
[email protected] Mathematics and Reality by Mary Leng OXFORD: OXFORD UNIVERSITY PRESS, 2010, X + 278 PP., £37.50, US $65.00, ISBN 978-0-19-928079-7 REVIEWED BY ROBERT THOMAS
his is a book written by a professional academic philosopher for other academic philosophers, not primarily for mathematicians. Because the author is a young academic, I conjecture that her conclusions are roughly those she came to in her thesis research and attending a high-level research seminar in mathematics. On the other hand, her mode of presentation is entirely within the framework set out by the American philosopher W. V. O. Quine in the latter half of the twentieth century. Quine has replaced Plato as the author of the texts to which philosophy is footnote. Most of the book is framed by Quine’s ‘‘indispensability’’ argument. In her version it goes like this: P1 (Naturalism): We should look to science, and in particular to the statements that are considered best confirmed according to our ordinary scientific standards, to discover what we ought to believe. P2 (Confirmational Holism): The confirmation our theories receive extends to all their statements equally. P3 (Indispensability): Statements whose truth would require the existence of mathematical objects are indispensable in formulating our best confirmed scientific theories. C (Mathematical Realism): We ought to believe that there are mathematical objects. A mathematician might very well attack P3, but Leng attacks P2: since scientists use ideal objects that no one thinks exist in their best theories, confirmational holism is plainly false. This is a fresh approach to a matter that has been discussed to death for years. With rare exceptions [1, 2], philosophers call a statement true only if it refers to something real. (This appears to be a twentieth-century aberration.) Mathematicians, on the other hand, present statements that are justified on the basis of axioms as true theorems. This is the kind of fiction that the philosopher Hartry Field [3] attributes to mathematicians, and he has tried to show that it is possible to do mathematics—in particular applied mathematics, having in mind the indispensability argument—without existent numbers. Mary Leng does not follow him in attacking P3, but she does associate the idea of fiction with what does not exist. After introducing the indispensability argument and the approach of her book, Leng begins by defending Quine’s criterion for existence in the context of twentieth-century philosophy. As it provides evidence for existence but only negative evidence for nonexistence, one has to fall back on Ockham’s razor to cut away things that Quine says we don’t need. I agree that it is good advice not to bring more entities
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into an explanation than are needed, but this has no metaphysical force [4]; all it tells me is that we do not need to consider the existence of mathematical objects seriously. Philosophy does not have a good answer to this question [5]. Hartry Field’s attack on P3 is discussed in Chapter 3. Leng concludes this chapter by assuming P3 for the sake of argument and beginning her attack on P2, impugning the physical assumption of the continuity of space and time. Chapter 4 is on pure mathematics, which she calls, following Quine, ‘‘recreational’’ because it is not applied and so is ontologically neutral. This is unfortunate: dismissive names do us no good. Names aside, she writes sensibly enough, The picture of pure mathematical practice I am suggesting views mathematicians as engaged in (a) characterizing mathematical concepts, and (b) enquiring into the consequences of the assumption that these concepts are satisfied [by things that are real]. The practice of choosing mathematical axioms falls under the first of these activities. What I would like to argue, then, is that reasons offered in favour of adopting a given collection of axioms can be viewed as justifying their value as characterizations of interesting mathematical concepts, implying nothing about whether these concepts are true of any objects. The chapter ends with a detailed consideration of three arguments against the possibility of ontologically neutral mathematics. She makes the point that there is no more reason to demand ontological sophistication of mathematicians than to demand mathematical creativity of philosophers. Next (Chapter 5) she establishes that mathematical posits are treated differently from physical posits in the formulation and testing of physical theories. As mathematical relations are sometimes explanatory of empirical phenomena, we must initially allow for the possibility of evidence for the existence of mathematical objects. In Chapter 6 Leng follows Steve Yablo and Hilary Putnam part way down their paths through naturalized ontology to argue that scientists ought in some cases to work without ontological baggage even if in fact they do not. I find Chapter 7 the most interesting. Leng uses the term ‘‘fiction’’ and Kendall Walton’s theory of fictions [6]. It is delightful to see someone taking seriously the notion that mathematics is a representational art, even if not in quite the way Walton intends. Walton is clear that representation does not need to represent something beyond itself, still less something real. And we can pretend in a wonderful variety of ways. He is interested mainly in shared pretences that are made objective and so sharable because they have a real basis in what he calls props. In the appreciation of a work of art, the prop is the work of art; it can be the text of a novel. Leng uses the fact that standard props, whether dolls or artworks, have standard games. When playing with a doll one pretends that the doll is a child; when looking at a painting of a rose, one imagines a rose that looks like the rose ‘‘in’’ the picture. Even to say ‘‘in’’ the picture is to pretend, since the picture is oil on canvas, say. Imagining the prop Italy as a boot allows us to locate Crotone ‘‘in the arch of the boot’’. Leng wants real props, not just ‘‘prescriptions to imagine that [physical] laws are true’’, because we are concerned with the actual truth of statements about the real. To connect between Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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the real things that our theories are about and the theorizing of mathematics, she proposes to connect statements about mathematical objects with statements about nonmathematical objects by set theory. Axioms of this set theory are not vacuous; ‘‘we can hypothesize that set theory with urelements is all we need in order to view the assumptions of our ordinary scientific theories as hypotheses concerning what is fictional in this game’’. The link between correct mathematical statements and the real facts that make them fictional is not the fiction generated by pure-mathematical statements but the statements about urelements in the set-theoretical axiomatization of these statements. ‘‘Then how things are with the non-mathematical props will make fictional some utterances in the context of this make-believe.’’ Or not. Hypotheses about the urelements—what we think do exist— automatically generate imaginings in our mathematics that express physical content. The similarity of this view to van Fraassen’s constructive empiricism leads to a chapter (8) on that topic. In Chapter 9 Leng explains how mathematics, viewed as fiction in her sense, can be so successful. Not surprisingly, she concludes that her mathematical fictionalism is preferable to van Fraassen’s discussion of science for both mathematics and science. Chapter 10 ties up loose ends. The book is a sustained argument that the metaphysical existence of mathematical objects is not needed for anything. Mathematicians innocently assume it for the sake of argument. The book will be a great success, not by changing the minds of philosophers, but by being discussed for years to come. I want to turn now to a criticism, not of approach or of conclusion but of a detail, the use made of ‘‘fiction’’ and of Walton’s theory of fictions. Our pretence that what we are talking about is real is just the grammatical pretence involved in making sentences. But for Field and other antirealists, ‘‘fiction’’ is a polite way of characterizing the nonexistent. Pretence is not involved. Walton’s theory of fiction is not a theory of a kind of thing but rather of a function that a variety of kinds of real things can serve. Something is a fiction for Walton if it is used for the purpose of acting as a prop in games. In a game certain propositions are ‘‘fictional’’ because they are ‘‘true in the game’’. This functional definition of fiction has no ontological implications. Imaginings are real imaginings, and what is imagined may or may not be real. Philosophy of science, if written as a dialogue, is explicitly fiction in this sense despite its conveying nonfiction (in a librarian’s sense). Walton is quite clear that fiction in his sense overlaps nonfiction. What it talks about need not be made up; it’s the way it is talked about, the prompting of imaginative engagement. Apparently mathematical writing prompts imaginative engagement of a strikingly objective sort. Most nonfiction does not prompt— still less authorize—games of make-believe. It tries to convince us of propositions about the world as it is—not the world of our imaginations. Mathematics can be interpreted in that way too. For Walton, being a fiction is a matter of how something is interpreted or used.
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Mary Leng’s complex and ingenious use of Walton’s theory, however, is entirely at the service of the unreality of mathematical objects. A historical novel, according to Walton, prescribes certain imaginings and seeks to elicit belief in some of them. In such a document certain propositions are fictional in his sense and certain are historical. They can overlap. Such a document can be indistinguishable from history (a problem for historians). He explicitly allows that something can be a fiction in his sense and at the same time be ‘‘a means to the end of conveying information and insight about actual historical events’’, ‘‘especially in communicating understanding in a sense that goes beyond acquiring factual information’’. This seems to me to be too valuable a way of looking at mathematics to be downplayed as Leng does. Here is a simpler way of deploying Walton’s theory of fictions directly on pure mathematics. Definitions and proofs can be taken to be props that ground the objectivity of our theorems. The game that they authorize is that of imaginative understanding of the theorems. Fictionality in this use of the theory is having been proved. What makes a statement, an equation for example, fictional is its proof—a prop. Other props could ground the fictionality of the same equation. The 2009 Man Booker Prize was awarded to the author of the historical novel Wolf Hall, and the genre is attracting critical attention [7]. Being factual in applications is no more necessary to correct mathematics than being historical is necessary to being a well-written novel. Being a historical novel and being applied are both options that a novel or a piece of mathematical theory may have. We have much more scope for applying mathematics than for recognizing that a novel is historical, because typically a novel has only one possible application to the actual course of events in the past, but both sorts of application are similarly optional.
REFERENCES
[1] Priest, Graham. Towards Non-Being: The Logic and Metaphysics of Intentionality. Oxford University Press, 2007. [2] Azzouni, Jody. Talking about Nothing: Numbers, Hallucinations, and Fiction. Oxford University Press, 2010. [3] Field, Hartry. Science without Numbers: A Defence of Nominalism. Princeton University Press, 1980. [4] Coleman, Edwin. ‘‘Pernicious logical metaphors’’, Logique et Analyse 53 (2010), 185–210. [5] Hellman, Geoffrey. Mathematics without Numbers. Oxford University Press, 1989. [6] Walton, Kendall. Mimesis as Make-believe: On the Foundations of the Representational Arts. Harvard University Press, 1993. [7] de Groot, Jerome. The Historical Novel. Routledge, 2009.
St John’s College and Department of Mathematics University of Manitoba Winnipeg, Manitoba R3T 2N2 Canada e-mail:
[email protected] The chapters are:
Loving and Hating Mathematics by Reuben Hersh and Vera John-Steiner PRINCETON AND OXFORD: PRINCETON UNIVERSITY PRESS, 2010, 428 PP., US $29.95. ISBN-10: 0691142475, ISBN-13: 978-0691142470 REVIEWED BY JONATHAN M. BORWEIN AND JUDY-ANNE OSBORN
oving and Hating Mathematics (hereafter referred to as Loving and Hating) is the child of two passionate scholars, a mathematician and a social scientist. Reuben Hersh is known to some readers for his many articles in the Intelligencer and earlier books, such as The Mathematical Experience coauthored with Davis and Marchisotto, and What is Mathematics Really?. The latter had a substantial effect upon the older of us; it was, at the time of publication, a welcome blast of mathematical humanism. Loving and Hating is written in the same clear, gentle style, and it aims to vanquish four myths:
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1. Mathematicians are different from other people, lacking emotional complexity. 2. Mathematics is a solitary pursuit. 3. Mathematics is a young man’s game. 4. Mathematics is an effective filter for higher education.
Outline of Loving and Hating Loving and Hating addresses mathematical beginnings, culture, solace, addictive potential, communities, gender and age related-issues, philosophies of teaching of mathematics in universities, and the teaching of mathematics in schools. At 416 pages, it is as compact as it could be, given the ambitious breadth of its scope. Only the last and first chapters of the book deal directly with school mathematics. This makes the cover design (credited to Lorraine Betz Doneker) particularly worthy of mention. The image is emblematic of a key message, which the authors state as they conclude their last chapter: Because we love mathematics, we want to minimize the number of those who hate it. Two adorable little boys in short-trousered school uniforms sit in front of a blackboard, with a silver trophy between them. One is forward facing and looks quite content, the other is notably unhappy and gazes to the side. They personify the Loving and Hating of the title, as we realized when a young person who glanced at the book instantly associated himself with the sad-looking child. Loving and Hating is not a recipe-book for addressing problems with school experiences of mathematics, however. Rather, it is a tour of mathematical life in the large, carrying with it a recommendation that issues relating to the schoollevel experience of mathematics should be addressed in terms of mathematics in its entirety, and in particular the joy that its practitioners take in the endeavor.
Chapter 1: Mathematical Beginnings Here the authors address how a child becomes engaged in mathematics. The trophy on the front cover points to the section on mathematics competitions. We learn of the childhood mathematical experiences of famous mathematicians, such as Terence Tao, Carl Friedrich Gauss, Sonia Kovalevskaya, and many others, and observations of personality and psychological traits recurrent in children’s enjoyment of mathematics.
Chapter 2: Mathematical Culture Mathematics has a culture reaching back over a long, long time. The authors’ description encompasses thoughtful forays into four main ideas: abstraction, aesthetics, belongingness, and the tension between collaboration and competition.
Chapter 3: Mathematics as Solace The authors ask: ‘‘Is mathematics a safe hiding place from the miseries of the world?’’ and answer that it can be. For instance, absorption in mathematics can temporarily keep the worries of the world at arm’s length. It can even be a means of coping with situations as extreme as imprisonment.
Chapter 4: Mathematics as an Addiction: Following Logic to the End What does ‘‘mathematics as addiction’’ mean? The authors give us a sample of some extremes: after a mention of John Nash, whose life and schizophrenia were the subject of the book and movie A Beautiful Mind, they paint a detailed picture of the extraordinarily creative and intense life of Alexander Grothendieck. Then follow ‘‘five cases of actual criminal or suicidal insanity in other mathematicians,’’ including famous cases such as the tragic later life and death of the renowned logician Kurt Go¨del.
Chapter 5: Friendships and Partnerships This chapter describes some famous friendships between mathematicians, including: Karl Weierstrass and Sonia Kovalevskaya, the trio of Hardy, Littlewood, and Ramanujan, and the friendship between Go¨del and the physicist Albert Einstein. Mathematical marriages such as the Robinsons (Julia and Raphael) are also described – see also the book Julia and film Julia Robinson and Hilbert’s Tenth Problem. The importance of friendships and partnerships in sustaining the individuals involved is described both in particular and in general.
Chapter 6: Mathematical Communities Communities of mathematicians have formed spontaneously or in organized ways to meet the needs of the groups that comprise them. Examples given here range from the faculty at the University at Go¨ttingen in Germany (1890s-1930s), the famous French group Bourbaki, which began in the 1930s, and the short-lived Jewish People’s University (1978-1983) in Moscow, to contemporary examples such as the Association for Women in Mathematics (AWM) and the web-supported 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
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Polymath Project. The authors describe ways in which communities support their members, how communities themselves die or flourish, and how they relate to larger mathematical communities.
Chapter 7: Gender and Age in Mathematics In this chapter the authors examine mathematical life through the lenses of gender and aging. The experiences of many famous women mathematicians are described, including historical examples in which being female was a considerable impediment to mathematical life, such as experienced by Sonia Kovalevskaya and Emmy Noether. Notably absent is Lady Ada Lovelace, famous for her early work on algorithms and information in relation to Babbage’s analytical machine. The varied experiences of contemporary women mathematicians, such as Karen Uhlenbeck, Joan Birman, and Fan Chung, are described. Again, this is just a selection—pleasingly, there are now too many accomplished women in the profession to be comprehensive; the selection does not include other equally notable women such as Cathleen Morawetz, nor any non-Americans. The experience of being a mathematician and growing older is also described in some of its varied detail, reprising results from an earlier published survey conducted by Hersh as well as comments from other surveys.
Chapter 8: The Teaching of Mathematics: Fierce or Friendly? The focus here is on two University-level methods: those of Robert Lee Moore (Moore Method) and Clarence Francis Stephens (Potsdam Model), both from the USA. They are extreme points rather than the sort of barycentric averages that may be common practices today in the USA and other countries. These models, the authors explain, ‘‘embody two different, opposed strains in American Education: the egalitarian versus the elitist; the cooperative versus the competitive; the heritage of the Declaration of Independence versus the heritage of the Confederate States of America.’’
Chapter 9: Loving and Hating School Mathematics The final chapter begins with observations on how school mathematics affects the feelings of adults toward mathematics, including the ‘‘hating’’ of the book’s title. The authors provide very little description of mathematics in the classroom as experienced by school students; they refer to mathematical learning in a variety of contexts, such as the shopping contexts investigated by anthopologist Jean Lave. The chapter includes many suggestions for reform in the teaching of mathematics, with reference to various trial programs. That people have multiple different kinds of intelligences, and that teaching generally should not privilege mathematical thinking or even specific kinds of mathematical thinking, is a thread underlying many of these suggestions.
End Matter The book ends with five pages of conclusions, nine pages of ‘‘literature review’’ listing other popular books on mathematics, and thirty-four pages of paragraph-long biographies 64
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of mathematicians. This last compendium was of particular interest to the (mathematician) husband of one of us, who picked up the book upon its arrival in the household and, on discovering the biographies, sequestered it until he had read them (and the rest of the book) through. This biographical ‘‘digestif,’’ with its overview of the lives of well known and less known mathematicians, may be one of the highlights for readers in the mathematical community. To summarize, Loving and Hating is a sweeping survey of mathematical life, into which the four myths and the antidotes the authors provide are woven. We find ourselves largely in agreement on its strengths and weaknesses; the remainder of this review is a snapshot of our discussions. We write in explicit dialogue (JB for Jon Borwein and JO for Judy-anne Osborn), to clarify our different perspectives and occasional disagreements. Where we write in one voice, we agree with each other. We structure our remarks around the following two sets of questions, which arose for us in the reading.
About the Myths 1a. Are the four claims actually myths? 1b. Who believes them? 1c. Are these myths about mathematicians, or about broader groups?
About the Audience 2a. To whom is the book addressed by the authors? 2b. To whom would the book be useful? 2c. Will the book Loving and Hating find its audience?
1 The Myths One by One Myth 1 Mathematicians are different from other people, lacking emotional complexity Is this a widely held belief? Does it have a basis in fact? Hersh and John-Steiner have run two claims together here. Is obfuscation the result? JB:
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If Myth 1 is meant to say that mathematicians are ‘‘a bit odd,’’ then it is widely believed. It is sometimes, but not always, true. I grew up around many mathematicians, who ranged from the urbane and articulate to the seemingly mute. Quite a few anecdotes in Loving and Hating reinforce a perspective of eccentricity. For instance the one where R. H. Bing drives colleagues to a conference, and when the windscreen fogs up, uses it to draw mathematical diagrams on, instead of wiping it clean. Films such as A Beautiful Mind pick up on and emphasize the idea of the eccentric or insane mathematician. It is a myth that being crazy helps a person do good mathematics (or much of anything else): it doesn’t. As Michael Crichton had said, however, ‘‘All professions look bad in the movies - why should scientists expect to be treated differently?’’
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Evidence against the second part of the claim - that mathematicians lack emotional complexity - is overwhelming. Hersh and John-Steiner show this in story after story. Attachment, affection, joy, courage, fear, empathy, anxiety, sorrow, indignation, depression, and wonder feature in their accounts of discoveries, friendships, prison terms, politics, competition, collaboration, and everyday life. I recognized the joy that Jenny Harrison finds exploring paths through the woods and looking at mathematical landscapes with something of the same feeling. Yes, the sense of wonder is palpable in Grothendieck’s description of his feelings when he switched mathematical fields, from analysis to geometry: It was as if I had fled the harsh arid steppes to find myself suddenly transported to a kind of ‘promised land’ of superabundant richness, multiplying out to infinity wherever I placed my hand on it, either to search or to gather … I was also taken by the description of Chandler Davis’s response to his six months in prison for his refusal to cooperate in McCarthy-era questionings by the Committee on Anti-American Activities. Davis’s sense of humor, and courage, comes through in a footnote to one of his subsequent papers: Research supported in part by the Federal Prison System. Opinions expressed in this paper are not necessarily those of the Bureau of Prisons. This myth makes me think the book is aimed at the general public, for surely the belief that ‘‘mathematicians lack emotional complexity’’ could only be held by members of the public who don’t know any mathematicians! But the part on Grothendieck’s work, for instance, is far too technical for the lay reader. I agree that is very technical. I’m a mathematician and it required more concentration than I was willing to give at the time. Yet surely it gives a flavor of a kind of mathematics in a way that would be impossible otherwise, and gives people who have never been part of a university mathematics department a sense of what it is like. I think the book is aimed at lay readers and mathematicians alike. Much of what Loving and Hating suggests is ‘‘unique to mathematicians’’ applies to any group of people who pursue a life of the mind. As for madness or emotional range, I see no difference with physicists or English scholars. Just as mathematicians are portrayed as mad in the movies, the idea of the ‘‘mad poet’’ is a romantic concept. There is no reason to think that fewer physicists and writers go off the deep end. Famous physicist Ludwig Boltzmann and famous author Virginia Woolf? Yes, those instances and more. Surely there’s no harm in just focusing on mathematicians in a book about mathematical life? Loving and Hating would be better if it were situated better. It is misleading, for instance, to write about the Unabomber as a mathematician without talking about
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other scientists who also did crazy violent things. I searched the Unabomber’s massive manifesto in Altavista at the time it was published. The word mathematics occurred in only the phrase ‘‘science and mathematics,’’ and that only three times. I asked my sister, as a person outside of academia, whether she thinks that Mathematicians are different from other people, lacking emotional complexity. She replied that most people don’t think mathematicians lack emotional complexity. She thinks that the general public thinks of mathematicians as people who spend time writing strange complicated things on bits of paper. That’s an interesting reply. Yes, so was Myth 1 ever a myth at all? It has the weakest claim of the four to being a myth. What is true is that many mathematicians have interesting life stories. Any reader of Loving and Hating who begins the book thinking mathematicians are uniquely and exclusively passionate about their particular field, and idiot savant in everything else, will come away knowing that is not the case.
Myth 2 Mathematics is a solitary pursuit. JO:
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It is not very surprising that people outside of mathematical research would think of it as a solitary pursuit, since it is often depicted in the movies as being the work of lone geniuses in their own private ivory towers replete with chalkboards. The movies aren’t the only source for this view. It is a default perception of creative work. We also think of novelists as working in solitude. Yet in mathematics, the idea of solitary creation is only a half-truth. Excellent mathematical work can be done in solitude (as the examples of mathematicians in prison testify), but this is not the norm. Interaction between mathematicians, both planned and accidental, sharing of ideas, and mutual inspiration, is typically important or even necessary in mathematical creativity. I think that we’re agreed that Myth 2 is a myth: a belief both widely held, at least outside of mathematical circles, and in the large not true. That said, it is still true that we have no technology for telepathy, so that the creative thinking that is essential to mathematical progress is still necessarily an individual part of the activity. Yes, indeed, but Hersh and John-Steiner do a good job of rebutting the myth in the large. I was fascinated to read of the friendship of Hilbert, Minkowski, and Hurwitz, and how the latter two helped Hilbert plan the famous ‘‘23 important open problems’’ talk he gave in Paris in 1900, which shaped much of mathematics for the next hundred years. My colleague at Carnegie Mellon, Massera’s student Shaeffer, was heavily involved in the campaign to free Massera when he was interned in prison in Uruguay for 9 1/2 years. Massera was a collaborative and generous
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person both in and out of prison. As a prisoner he was involved in circulating forbidden papers on dialectic, logic, and mathematics that helped the prisoners keep their spirits up. It is a fine example of humans supporting ideals and each other in hard times. Another interesting but entirely different instance of a cooperative spirit is Timothy Gower’s 2009 PolyMath project, in which a major open mathematics problem was posed on the web and the problem was solved in the astonishingly short time of a few weeks by ‘‘the shared effort of over two dozen contributors from several countries.’’ The internet has made multi-location collaboration practical. Hersh and John-Steiner comment that students receive guidance and inspiration not only face to face but also at a distance. I found it curious that they chose to illustrate this with a century-old example of Birkhoff in the USA being inspired and learning from Poincare´ in Paris. Yes, collaboration and mentorship has been important in mathematics throughout the ages, but there are many more contemporary examples—such as arXiv—that they could have drawn upon. For instance, in 2003, when I was still at Simon Fraser University in Vancouver, one of the biggest power blackouts in Northern America in a generation occurred. In our research centre we knew instantly that there was a problem when none of our contacts east of the Mississippi would come up. We were more quickly aware of a problem affecting our distant colleagues than if there had been major trouble in part of our own physical campus. The support that mathematicians can give each other is beautifully described in a quote about Chern who told his thesis students ... the philosophy that making mistakes was normal and that passing from mistake to mistake to truth was the doing of mathematics. And somehow he also conveyed the understanding that once one began doing mathematics it would naturally flow on and on. About ten years after my thesis I started collaborating seriously with other scientists and gained inspiration from them, too. I came to realize that how good you are at formalizing 20th-century rigorous expressions is not a complete measure of your mathematical worth. I would often rather collaborate with a physicist on a poorly posed problem than with a mathematician close to my own field, because there may be more undiscovered nuggets in the bringing together of two more different perspectives. That’s something that doesn’t shine through much in Loving and Hating - the rich mathematical opportunities for mathematicians in collaboration with other scientists. I’d like to see a book that does for the sciences what Loving and Hating attempts to do for mathematics. What would such a book look like? The closest that I can think
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of at the moment is A Passion for Science, by Alison Richards and Lewis Wolpert. I am coming to think that there may be more differences within mathematics than there are across the sciences in their entirety. Wolpert interviewed Christopher Zeeman who told a story about one of his (non-mathematician) administrators who helped run an annual summer conference series in Warwick, which rotated every three years between the three mathematical areas of algebra, topology/geometry, and analysis. She said that after a while she could tell which was the year’s theme without looking at the program, but just by observing the behaviour of the participants. The algebraists were punctual, organized, and thrifty. They wanted single cheap rooms and arrived by train when they said they would. The topologists wanted big houses, they brought their families, and wanted to stay the whole week. The analysts were predictably unpredictable, promising to turn up Tuesday with their partners and arriving on a different day with somebody else. Maybe the main cleavages are within our discipline. There are differences in levels of rigour; perhaps there are differences in levels of embodiment. Loving and Hating would be better if it were more contextualized. Hersh and John-Steiner describe characteristics of the whole tribe of mathematicians: curiousity, determination, willingness to spend time doing mathematics, not minding being alone so much as others might, cherishing independence, and having a love for symmetry or logic or pictures or, sometimes, how things work. I think willingness to be alone is the grain of truth in the myth that gives it some traction. If Hersh and John-Steiner had included the context you want them to, it would be a much bigger book. Currently it is just a comfortable size for carrying and reading on the train. But you can also get it on Kindle. Being willing to spend time alone thinking isn’t special to mathematicians or even scientists. It is a property of leading a life of the mind, and that is a point worth making in a book that endeavors, at least in part, to introduce the wide public to the mathematical life.
Myth 3 Mathematics is a young man’s game. JO:
Hardy wrote that ‘‘Mathematics is a young man’s game’’ when he was in his sixties and in a self-confessed melancholy mood, writing about mathematics instead of doing mathematics, which he would have preferred. Hersh and John-Steiner address this myth in their Chapter 7: Gender and Age in Mathematics, where they make it rightly clear that Hardy would have meant ‘‘person’’ by ‘‘man,’’ and that he was making a claim about age, not women. Their chapter deals with issues of both gender and age; here we focus on ‘‘age.’’ Is it widely believed that mathematics is ‘‘a young person’s game’’? And is it true?
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This is widely believed, both within and outside of the mathematical community. I agree it is prevalent inside the mathematical community. I disagree about outside of it. For many of the general public, the iconic mathematician is Einstein, a brilliant old man with a shock of white hair. I think the public myth of ‘‘genius’’ is of ‘‘Age and Wisdom.’’ If that’s the case, then the public has forgotten what a media star Einstein was in his youth. Is it true that mathematics is ‘‘a young person’s game’’? There’s an aspect of truth to this in many fields, not just mathematics. In art there are young geniuses and old masters, as David Bailey and I relate in my book about experimental mathematics. The difference is like that between an early Picasso cubist work and a Renaissance masterpiece. Breakthroughs tend to be made by the young. Hardy’s own long-time collaborator Littlewood became a counter-example to Hardy’s claim when he was in his seventies, publishing a ‘‘monster’’ paper with Mary Cartwright that ‘‘was recognized as an early breakthrough in the discovery of chaos.’’ Yes, if you maintain a passion for your field, as I do, and reasonable health, then you can continue to make fine contributions as you age. I can still do good mathematics in my sixties. My father enjoys doing mathematics as much at 87 as he did at 18. The myth with the strongest credos within the mathematics community is really: Doing first-rate research mathematics is something that you had better hurry up and do before you’re 35 or 40. It is very unusual to find someone who has been a toiler in the ‘‘mathematical vineyards’’ who suddenly has a huge result at age 50. The mathematician Mary Ellen Rudin is quoted in the book as stating in an interview I don’t think most people’s best work will be done by the time they’re thirty, and certainly my best work wasn’t done until I was fifty-five years old. Might the myth you state be self-perpetuating? For instance only mathematicians not over 40 years of age are eligible for the Fields Medal, which is the mathematical equivalent of a Nobel Prize. In fact the Fields medal is awarded for the contribution that the person has made and is expected to continue to make! That fits with the responses to Hersh’s survey of his fellow mathematicians. Many said they continue to make good progress, in their later years, though their mix of skills and strategies tends to change. Yes, some mathematicians make more and more impact as they get older, whereas others burn out. Again, look at other creative fields - the synthetic rather than the accretive ones. Look at ‘‘geriatric rock.’’ The Rolling Stones are still energetic, creative musicians. The book also tackles the claim that if you’re any good at all as a mathematician it’s going to show up before
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you turn 40, and intriguingly, much of the data refer to women. For instance, Joan Birman, a topologist at Columbia UniversityBarnard College, did not get her Ph.D. until she was 40 years old. Birman focused better on math after the issues of marriage were sorted, her children older, etc. ‘‘I think doing mathematics when you’re enthusiastic is important – not your age.’’ If it is true that women often make their first good achievements later on, it is consistent with what my (male) thesis advisor used to tell me. He didn’t believe that mathematicians become notably worse as they age, but rather that there are more other responsibilities and distractions that tend to get in the way of creative achievement. He used to say that if you can keep yourself clear of the ‘‘crap,’’ you can still create. Yes and yes. I learned to keep the ‘‘crap’’ under control from my father who was a research-active mathematics department head for a very long time. My mother, who got an anatomy Ph.D. the year before I got mine, was a wonderful example of deferred female achievement.
Myth 4 Mathematics is an effective filter for higher education. JO:
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It is said that the door to Plato’s Academy was engraved with the phrase: Let none ignorant of geometry enter here. Whether or not this fable is true, the spirit of the idea is consistent with Plato’s conception of mathematics as a means of training the mind, an idea that has been embedded in western thought ever since. Yes, historically in Cambridge, the earliest, and for a long time the only, examined degree awarded was in mathematics. It was called the Tripos. Keynes did the Tripos, as did Airy, Herschel, and many other famous people in and out of the profession of mathematics. Such as the economist Thomas Malthus; astrophysicist Arthur Eddington; discoverer of argon Lord Rayleigh; founder of the theory of electromagnetism James Clerk Maxwell; philosopher, logician, mathematician, historian, and social critic Bertrand Russell? At much the same time, the degree of equivalent standing at Oxford University was called Greats (or Classics), an archetypal humanities degree, emphasizing literature, language, philosophy, history, and art, and was the course taken by aspiring ministers of the church, social thinkers, writers, politicians, and civil servants. So the special status of the subjects Mathematics and Classics in schools throughout the English-speaking world, is inherited from the two great English Universities of the Middle Ages? In many ways, yes. Even now, although individuals claim to ‘‘hate math,’’ or not be able to do it, nobody doubts its importance. I am not so sure.
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Every serious business at every serious level relies these days on an enormous amount of mathematics. The trend is reported by publications such as Business Week, with cover stories such as ‘‘Top Mathematicians Are Becoming a New Global Elite’’ (January 2006). Successful tech companies are employing mathematically trained people. At the height of Nortel’s success, that was the only factor that the company could identify that distinguished successful from less successful research groups. That understanding may not be as broad as you think, particularly in schools. Students are asking, What is the point? and, When am I ever going to use this? Yes, the mathematician Underwood Dudley is quoted as arguing that algebra and calculus are seldom used or needed by most people. Hardy made the same point with more eloquence in his famous ‘‘Apology’’: ... some mathematics is certainly useful in this way; the engineers could not do their job without a fair working knowledge of mathematics, and mathematics is beginning to find applications even in physiology. ... It is useful to have an adequate supply of physiologists and engineers; but physiology and engineering are not useful studies for ordinary men. Hardy was born in 1877 and died shortly after the end of the second world war. Since then, the professions that rely on mathematical thinking have multiplied. I think he would have been greatly amused at the number of number-theorists who work for the security agencies around the world, such as CSIS or NSA. That said, I do not believe that we teach the right mathematics to the right students. As an eleven-year-old in Britain, I experienced the ‘‘11 plus,’’ an examination taken by all British children. It determined, at that young age, the rest of my academic future by specifying what kind of secondary school I was admitted to. The pressure on my eleven-year-old self was horrendous, even though I was capable of doing well on that exam and did so. I do not advocate the ‘‘11 plus’’ or anything like it; yet in its absence, I challenge anyone to find a diagnostic for those who will need mathematics beyond arithmetic better than letting everyone try it. You’re saying that we need engineers and so forth, and that they need mathematics, and - I’m inferring here, since mathematics is cumulative and takes a long time to learn - we had better ensure everyone does mathematics up to a certain level. Yes. There is an analogy with learning a foreign language. One of the few things that linguists agree about is that learning a language before age eight is different from learning it afterwards. In the same way, it is very hard to lay down high-end mathematical skills after school; to say, ‘‘I’m going to
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catch up at University.’’ There are some counterexamples but they are rare. What about the claim that mathematics should be used as a filter to enter professions that don’t need it, as described in Loving and Hating? That is very different. However, a belief in mathematics as a general filter has had considerable staying power in universities. For instance, by and large, a student may not fail much else in a Business Degree, but he or she will need to get through Calculus 100 and something Actuarial. I am puzzled by what is meant by the phrase, an effective filter? Does it mean that a moderate proportion of students tend to fail first-year calculus, so that this can be used as a sieve independently of whether it is sieving on some sensible criteria? Or does effective filter mean that an ability to do mathematics courses is being used as a proxy for general intelligence? People sometimes say to me, ‘‘Oh you do maths; you must be so smart.’’ It strikes me as an odd comment, coming as it often does from people who do work that seems to me to be much more subtle and difficult, like the clinical practice of medicine. Perhaps the belief that mathematical capacity implies intelligence is widely held? I don’t know about the other way around. The authors of Loving and Hating counter the myth by quoting Gardner’s theory of multiple intelligences, describing mathematical abilities as distinct from and not necessarily related to other equally important intelligences, such as linguistic, musical, and interpersonal. We filter inappropriately. We conflate the idea of the ability to do mathematics well with the ability to appreciate it. We don’t make the same mistake with Shakespeare, or sports. What I would like to be able to do, when teaching mathematics, is to ask students in the class who is there for appreciation, and who needs mastery for subsequent professional use. Those in the first group could take the course as Pass/Fail, and those in the second for a numerical grade. I don’t mind which group individuals are in. Both are worthwhile. With modern technology there is more and more capacity to meet both kinds of needs in the same classroom. That’s an arrestingly interesting idea. Hersh and JohnSteiner suggest that There is a wrenching strain between opposing pressures: a continuing demand for enough sophisticated math specialists, with a shrinking need for traditional math skills in the general population. ... Studying math(s) should continue to be required, but not in such a manner that students remember it with antagonism and loathing. Many of the points they make at the end are about what could or should be, as opposed to what is. This is a shift in style from the earlier chapters, which were more driven by personal experience and anecdote.
JO:
Yes, there is an enormous idealism in the final chapter, which collects together lots of stories of ideas that individuals have for trying to begin to make a difference in the experience that school-children have of mathematics classes.
JB:
Conclusions Loving and Hating is a book filled with gems. We could open it on any page and find something interesting. It is imbued with the authors’ love of mathematics and respect for people. The message that mathematics is a fundamentally human activity, in which people can find meaning and joy, is clearly conveyed. The book has flaws. We liked the parts each in turn more than the whole. Although all mathematicians of a generally philosophical nature are likely to enjoy browsing in Loving and Hating, we are less sure that the order in which the material is presented is of service to the rest of its potential readership, including teachers, policy makers, and the general public. For readers who are not professional mathematicians but who are interested in mathematics, the chapter on school mathematics may be placed rather late on the scene: one of the main touchstones of mathematical experience for these readers may have been in school. That said, the immersion in mathematical life with its joys of discovery, which the earlier chapters provide, provides a fresh mindset for thinking about improving the experiences in the school classroom. This book may be of special interest to graduate students in mathematics as part of an introduction to the stories and culture of the community that they are joining. JB usually recommends that his new graduate students read Lakatos’s Proofs and Refutations, Medawar’s Advice to a Young Scientist, Davis and Hersh’s The Mathematical Experience and Yandell’s The Honors Class. It is tempting to add Loving and Hating Mathematics to that list. JB: JO:
Will the book find its audience? It is readable, informative, interesting: I think so.
JO:
A school-girl asked to review a book on penguins wrote: This book told me a great deal more about penguins than I really wanted to know. If mathematicians are penguins and the readers are all marine ornithologists, the book is great. Sadly, I am unconvinced that a broader audience will take the time to read this book. As a professional penguin, I think that this is a pity. Your point is so beautifully made that it is an exquisite pain for me to differ, but I do. I don’t think that you have to be a bird specialist to love penguins, or, for instance, to enjoy a well-made David Attenborough special on them. Maybe Hersh and John-Steiner are not quite David Attenborough, but they’re close. The descriptions in Loving and Hating are sympathetic and understandable. The lives that Hersh and John-Steiner have led have allowed them to get up-close and personal with a species (mathematicians, and more generally people whose work is creative thinking) whose world many people don’t ordinarily get to see, and may welcome a window into.
ACKNOWLEDGMENTS
JO thanks her sister and husband for helpful discussions and her former Ph.D. supervisor for allowing his advice to be quoted. Centre for Computer Assisted Research Mathematics and Its Applications The University of Newcastle Australia.
University of Newcastle Newcastle Australia e-mail:
[email protected] 2011 Springer Science+Business Media, LLC, Volume 33, Number 4, 2011
69
Stamp Corner
Robin Wilson
Fields Medallists Anil Nawlakhe, Ujwala Nawlakhe, and Robin Wilson
he Fields Medal is awarded to up to four mathematicians at each International Congress of Mathematicians, held every four years. It gives recognition and support to mathematical researchers under the age of 40 who have made outstanding contributions to their subject. John Charles Fields was a mathematics professor at the University of Toronto, and President of the Toronto Congress in 1924. The idea for the medals came from Fields after the Toronto meeting ended with a financial surplus. This profit, together with later money from his estate, provided the funding for these medals. The first Fields Medals were awarded in 1936 at the International Congress in Oslo, to Lars Ahlfors and Jesse Douglas. Made of gold, they feature Archimedes on one side and an inscription on the other. Over fifty mathematicians have received Fields medals, the youngest being Jean-Pierre Serre who received it in Amsterdam in 1954 at the age of 27. In 2008 the Republic of Guinea issued two sheets of stamps depicting Fields Medallists. One sheet features David
Mumford (Vancouver, 1974) and a stamp showing Laurent Schwarz (Cambridge, 1950) and one side of the medal. The other sheet contains illustrations of Efim Zelmanov (Zu¨rich, 1994) and Pierre Deligne (Helsinki, 1978), a stamp showing J. C. Fields and the other side of his medal, and five other stamps featuring Andre´ Weil and Fields medallists Alain Connes (Warsaw, 1983), Alan Baker (Nice, 1970), Vladimir Drinfel’d (Kyoto, 1990), Simon Donaldson (Berkeley, 1986), and Andrei Okounkov (Madrid, 2006).
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J. M. Patel College Bhandara India
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Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, Computing and Technology The Open University, Milton Keynes, MK7 6AA, England e-mail:
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THE MATHEMATICAL INTELLIGENCER Ó 2011 Springer Science+Business Media, LLC
DOI 10.1007/s00283-011-9244-1
Sarwajanik Wachanalaya Bhandara India
