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The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Sheldon Axler,
The Mandelbrot Set It is with some trepidation that I enter into a controversy (Opinion column, Mathematical Intelligencer, vol. 11, no. 4) which is clearly completely overheated and more than a little pointless. I only hope that I can add a little bit of flesh air and cold water to the subject. Regarding the issue of who invented what is now k n o w n as the Mandelbrot set, I think it is worth keeping in mind that Fatou had a pretty complete picture of the whole state of affairs by about 1920. There is absolutely no doubt in my mind that if Fatou had had access to modern computing facilities, he could have and would have drawn pretty much the same pictures that Matelski, Mandelbrot, and I did about sixty years later. The same is certainly true of a large n u m b e r of workers in the field in the intervening years. Frankly, I w o u l d be hard put to claim too much credit for what amounts to having lived during the computer revolution. But even without the aid of the computer, Fatou was led to a rather penetrating insight into the dynamical behavior of rational functions--some of which he could prove, some of which he only c o n j e c t u r e d - which stands up very well in comparison with all that has come since. I would recommend that anyone interested in the subject read Fatou's paper (P. Fatou, M6moire sur les 6quations fonctionnelles, Bull. Soc. Math. France, I: vol. 47 (1919), pp. 161-271; Ih vol. 47 (1920), pp. 33-94, III: vol. 47 (1920), pp. 208-314). It is a masterpiece of what might be called "descriptive mathematics." I feel I must comment on Mandelbrot's remark that Matelski and I were "close to something that was to prove special, but they gave no thought to the picture." I don't know h o w he can be so sure of what we gave thought to and what we d i d n ' t - - w e are not particularly fond of the idea of publishing our philosophical musings or our half-baked s c h e m e s - - b u t the bottom line is really in the first part of his phrase. If, after all this public bellowing, Mandelbrot can really believe
that this set is something special, then he is really and truly not listening to himself. Just as the coast of England is not really very different from all the other coastlines of the world, the Mandelbrot set is just one (and certainly neither the first nor the last) of a whole universe of mathematical curiosities, which reflect various types of beauty and subtlety in nature and in the world of the mind. For me to have spent too much time on this one curio w o u l d have meant spending less time elsewhere, and to my taste would have reflected a rather infantile and somewhat dull mathematical sensibility. I may have been mistaken on this point, but I would certainly invite Mandelbrot to peruse some of m y other papers to decide if I made the right choice. My only request is that he not announce his verdict in print.
Robert Brooks Department of Mathematics UCLA Los Angeles, CA 90024-1555 USA .FractalsThis letter is a response to the Opinion Column in the last issue of the Mathematical Intelligencer, which contained Steven Krantz's book review and Benoit Mandelbrot's rejoinder. As your readers might recall, the books in question were a collection edited by H.-O. Peitgen and D. Saupe, and also The Beauty of Fractals: Images of Complex Dynamical Systems by Peitgen and P. Richter. I know the latter book very well indeed since it is r e c o m m e n d e d reading in m y undergraduate course on computer studies of chaos. Somehow, in all the discussion, the qualities of this book did not get very much attention. The Beauty of Fractals is simply a lovely book, and it is in itself a very solid contribution to intellectual discourse. It is aimed at the intelligent and well-educated general reader w h o has a good mathematical background. It does contain some real mathematics. A1-
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though the book does not give proofs, Peitgen and Richter do give careful statements of elegant theorems by Adrien Douady, John Hubbard, and Dennis Sullivan, among others. And, in addition, they give some very beautiful pictures of the mathematical objects under discussion, which mostly are sets bearing the names of Julia and Mandelbrot. Furthermore, The Beauty of Fractals tells how the pictures are related to the theorems and in what ways an examination of one will tell you more about the other. This book suggests and illustrates the proposition that a combined process of looking, thinking, and proving is one possible road to truth (and to beauty). I think that The Beauty of Fractals is likely to be a classic.
Leo P. Kadanoff The ResearchInstitutes The University of Chicago 5640 S. Ellis Avenue Chicago, IL 60637 USA Proof as Explanation. James Gleick in his rejoinder (Mathematical Intelligencer, vol. 11, no. 3) to the article of Morris Hirsch has, it seems to me, missed an important point. He writes "there are times w h e n mathematical proof (essential t h o u g h it is!) c o m e s , historically, as an afterthought . . . . Lanford's proof of Feigenbaum was ingenious and admirable, but it did little, really, to validate Feigenbaum's b r e a k t h o u g h - - e x p e r i m e n t s accomplished that." The point is that Lanford and other mathematicians were not trying to validate Feigenbaum's results any more than, say, N e w t o n was trying to validate the discoveries of Kepler on the planetary orbits. In both cases the validity of the results was never in question. What was missing was the explanation. Why were the orbits ellipses? Why did they satisfy these particular algebraic relations? In his celebrated "afterthought" Newton provided the explanation while, incidentally, adhering meticulously to the "theorem-proof methodology" which, according to Gleick and Keith Devlin, "most scientists find peculiar." I believe many mathematicians share Gleick's excitement over the fascinating experimental discoveries of Feigenbaum as well as the beautiful self-symmetries of the Julia sets, the omnipresence of the Mandelbrot set, and so on. Indeed, these things have been a wonderful shot in the arm for mathematics, providing a whole new range of mathematical phenomena, phenomena which, to a mathematician, fairly cry out for explanation. T h a n k s to the work of Lanford and others one can now account for a great m a n y of these experimental results, though I'm told there still remain some challenging mysteries. But, to return to m y point, there's a world of difference b e t w e e n validating and explaining. Physics wouldn't be much of a science if physicists simply 4
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measured and recorded the light spectra of the various elements and let it go at that. The main goal of all science is first to observe and then to explain phenomena. In mathematics the explanation is the proof. It's as simple as that, and I doubt that any scientist who understands this can find our methodology peculiar. I can well imagine that some scientists might find it peculiar that a person is willing to devote a lifetime to, say, trying to find out if the 3-sphere is the only simply connected 3-manifold, but given that that's what one wants to do there is nothing peculiar about using the theorem-proof methodology, because in fact it's the only methodology there is.
David Gale Department of Mathematics University of California Berkeley, CA 94720 USA -Hausdorff I am pleased to see that recently the "Years Ago" column in the Mathematical Intelligencer (vol. 11, no. 1, 1989) was devoted to Felix Hausdorff and his epochmaking work. I would like to add a few comments designed to accentuate certain aspects of Hausdorff's personality and of his Grundziige der Mengenlehre. On page 62 of the Grundziige one finds unmistakable evidence of the author's musical bent of mind. Indeed, Hausdorff's love and talent for music were evident quite early; it was on his father's insistence that he abandoned his plan to study music. Hausdorff nevertheless retained his interest in music, literature, philosophy, and fine arts; and during his early professional career, he devoted at least as much time to these pursuits as he did to mathematics. Under the pseudonym Paul Mongr6, he published as many as four literary works of no mean accomplishment (these are discussed at some length in [Krull]). But it is his mathematical development that deserves to be studied more closely. The astonishing fact remains that prior to the appearance of the Grundz~ige, Hausdorff had not published anything at all on what must be regarded as the most original and influential part of the w o r k - - t h e theory of topological and metric spaces (chapters 7, 8, 9). As Katetov says [article on Hausdorff, Dictionary of Scientific Biography, vol. 6 (1981), p. 176], "The Grundziige is a very rare case in mathematical literature; the foundations of a new discipline are laid without the support of any previously published comprehensive work." Interestingly enough, the preface of the Grundziige reveals Hausdorff's own views on his labors. Expressing the hope that while the book might profitably be read by a student of the middle semesters (i.e., an advanced undergraduate), he declares that it would have failed its purpose if it did not offer the profes-
sional colleagues something new, at least in methodological and formal aspects. Saying that one should logically and systematically organize scattered facts, remove unnecessarily special or complicating assumptions from previous results, that one should strive for attaining simplicity and generality--surely the minimum that can be demanded of an author dealing with material already treated by o t h e r s - - H a u s d o r f f expresses his belief of having reasonably fulfilled this demand and of having opened some new lines of inquiry. He points out, especially, that by axiomatizing point-set theory, many theorems on point-sets on the real line have been so transformed, generalized, decomposed, and tied up in another context that a mere reference to the existing literature would give no correct picture. Thus, despite these rather modest and unassuming words, there is no question that Hausdorff was aware of what he had accomplished with this book. So, while he was no doubt pleased with the impact created by the Grundz~ige on the subsequent development, he was hardly surprised. The definition of a topological space by means of a set of axioms on neighborhoods (p. 213) is, beyond question, Hausdorff's greatest contribution. The concept of a n e i g h b o r h o o d as such was, of course, nothing new; Hausdorff's great credit was to make it the point of departure for an axiomatic development --abstract in form, but as Bourbaki has observed [General Topology, Part I, Addison-Wesley, 1966, p. 166 (Historical Note to Chapter I)], adapted in advance to a p p l i c a t i o n s - - a n d to have found the right set of axioms. That, in analogy with metric spaces, he incorporated the separation axiom since named after him, detracts nothing from this credit. The Grundz~ige was the source to which the spectacular rise of point-set topology in the 1920s and 1930s is due. Many of the basic notions and concepts of the subject that are to be found h e r e - - e v e n the name metric space (the concept itself is due to Fr6chet)--have come down to us unaltered; one has to e n v y Hausdorff for his singular acumen and extraordinary foresight. On the personal side, it is interesting to note that within a year of Hausdorff's appointment as Extraordinarius (ausserordentlicher Professor) at Leipzig, he received a similar call to GOttingen, but it was turned down. Considering the reputation of GOttingen in those days, this appears mystifying. According to Wolgang Krull, the eminent algebraist w h o was Hausdorff's colleague at Bonn from 1928, the refusal definitely harmed his academic career ([Krull], p. 54). As early as 1932, Hausdorff could sense the oncoming calamity of Nazism; but he made no serious attempt to emigrate while this was still possible. A memorial plaque to Hausdorff was unveiled at the entrance of the Mathematical Institute of the University of Bonn (Wegelerstrasse 10) on 25 January 1980. It bears the inscription:
AN DIESER UNIVERSITAT WIRKTE, 1921-1935 DER MATHEMATIKER FELIX HAUSDORFF 8.11.1868- 26.1.1942 ER WURDE VON DEN NATIONALSOZIALISTEN IN DEN TOD GETRIEBEN, WEIL ER JUDE WAR. MIT IHM E H I ~ N WIR ALLE OPFER DER TYRANNEI. NIE WIEDER GEWALTHERRSCHAFT UND KRIEG! (At this University worked, 1921-1935, the mathematician Felix Hausdorff: 8.11.1868-26.1.1942. He was driven to death by the national-socialists, because he was a Jew. With him we honor all victims of the tyranny. Never again dictatorship and war!)
Bibliography W. Krull, Felix Hausdorff 1868-1942, Bonner Gelehrte (Beitr~igezur Geschichte der Wissenschaflen in Bonn, Mathematik und Naturwissenschaften), Bonn: Universit/it Bonn (1970), 54-69. H. Mehrtens, Felix Hausdorff--Ein Mathematiker seiner Zeit (40 pp.), Bonn: Mathematisches Institut der Universit/it Bonn (1980).
M. R. Chowdhury Department of Mathematics University of Dhaka Dhaka-lO00, Bangladesh
DirichletVol. 10, no. 2, 1988, of the Mathematical Intelligencer contains a fascinating article by David Rowe on Dirichlet's early mathematical achievements. Unique evidence on his working procedure is provided in a letter written by Dirichlet in October 1827 to his mother. The letter is published in the article together with an English translation according to the transcription produced by Dirichlet's great-grandson Leonard Nelson and p r e s e r v e d a m o n g Felix Klein's p o s t h u m o u s papers. Rowe left the question open if the original of Dirichlet's letter is still extant. The question can be answered in the affirmative: The original has been preserved in the voluminous series of Dirichlet's letters to his mother, which constitutes one of the parts of Dirichlet's NachlaJJ now located in the Murhard Library/University Library in Kassel (Federal Republic of Germany). For information on the "trifurcation" of Dirichlet's NachlaJJ, see my article in Historia Mathematica 13 (1986), "The Three Parts of the Dirichlet Nachlass.'" A comparison of Nelson's transcript with the original reveals that the copy is not entirely correct--although Nelson succeeded in deciphering highly difficult passages. The major differences are the following: 9 the date of the letter is not "29. October 1827" but instead "19. October 1827"; 9 Dirichlet does not speak of an "'unknown" but of an " u n p u b l i s h e d " dissertation of Gauss: "In diesem THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
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Aufsatze (gelehrte Anzn. 11 April 1825) wird eine yon unserem Gauss der dortigen Societat (iberreichte, bisher ungedruckte Abhandlung iiber die biquadratischen Reste angekiindigt u n d . . . " ; 9 in the middle of the last paragraph, no additional word "Satzes" has to be inserted since the supposed adjective "searched" is in fact a noun: "Eines Abends ( . . . ) hatte ich einige Ideen, welche mich in den Besitz des so lange und so eifrig Gesuchten setzen zu miissen schienen"; 9 the last quoted sentence reads entirely different (the English translation given by Rowe corresponds already to the correct text): "Ich werde gewit] nicht ermangeln, das was er (iber die Schwierigkeiten, womit die Beweise dieser S/itze verbunden sind, sagt, in meiner Abhandlung auf geschickte Weise anzufiihren." It should be mentioned that the original bears a remark by Dirichlet on the margin of the first page that documents in a characteristic manner his energetic and strategic search for a mathematical career in Prussia's capital. Wenn meine Abhandlung in Berlin die Aufnahme finder, welche sie verdient, so werde ich wahrscheinlich nicht lange mehr hier bleiben sondern bald nach Berlin versetzt werden, w o e s so sehr an ordentlichen Mathematikern fehlt, daf~ Leute wie Crelle zu Mitgliedern der Akademie ernannt werden. (If my dissertation finds in Berlin the appreciation it merits, then it is probable that I shall not stay here much longer but will be transferred to Berlin. Qualified mathematicians are so scarce there that people like Crelle have been appointed members of the Academy.) Dirichlet's last remark shows him in opposition to his protector Alexander yon Humboldt: It had been Humboldt's intention to have A. L. Crelle act within the Academy as his organizational ally. The photo reproduced in Rowe's article as well as on the cover of issue 10/2 of the Mathematical Intelligencer warrants some critical commentary. This photo, from the Portr~tsammlung of the University Library G6ttingen is claimed to show "P. G. Lejeune Dirichlet as a young man" (p. 14). Reading this attribution, two objections came immediately to my mind: 9 the young man is shown without spectacles. Dirichlet's myopia dated from his very youth and was so excessive that it saved him from serving in the army; 9 the age of the young man seems to be thirty at most: if it was Dirichlet, the photo should have been produced in 1835 or even earlier. The quality of the photo seems to contradict the level of development of photography in those years. Therefore I asked Professor Kurt-Reinhard Biermann (Berlin) who first published this picture in his important biographical study of 1959 (J. P. G. Lejeune 6
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Dirichlet. Dokumente fiir sein Leben und Wirken, Berlin: Akademie-Verlag), providing it with the same legend. In his answer, Professor Biermann avowed not to have imagined at that time how erroneous the attributions in portrait collections can be. From some technical details, he explained, it is evident that the picture is a real photograph and not the earlier technique of daguerrotype. Hence, it is clear that the young man in that picture is not Dirichlet! (It might be one of his sons.) Gert Schubring Institut fiir Didaktik der Mathematik Universitdt Bielefeld 4800 Bielefeld, Federal Republic of Germany Hamilton's N o t e b o o k I would like to make a small correction to our paper R. Dimitri4, B. Goldsmith: "Sir William Rowan Hamilton," Mathematical Intelligencer, vol. 11, no. 2 (1989), pp. 29-30. The notebook where Hamilton wrote his quaternion formulae is not in the Royal Irish Academy as we stated but at Trinity College, Dublin, in Trinity Manuscript Collection, reference number TCDMS 1492. Radoslav Dimitrid Department of Mathematics University of Exeter Exeter EX4 4QE Great Britain
The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreements and controversy are welcome. An Opinion should be submitted to the editor-in-chief, Sheldon Axler.
H o w Reliable Is a Computer-Based Proof? C. W. H. Lam "Is a Math Proof a Proof If No One Can Check It?" asked the headline of a recent New York Times article [2]. The article reported the results of our just finished computer search for a finite projective plane of order 10. The computation was enormous and over 1014 cases were investigated, obviously too much for any h u m a n being to check. I discussed this question briefly with the reporter, M. W. Browne, when he was preparing that article. I did not tell him that the CRAY-1A, the supercomputer that did most of our work, was reported to have undetected errors at the rate of approximately one per thousand hours. Since we used several thousand hours of CRAY-1A computer time, we should expect a few errors. Imagine an expanded headline, "Is a Math Proof a Proof If No One Can Check It and It Contains Several Errors?" I try to avoid using the word "proof" and prefer to use the phrase "computed result" instead. However many mathematicians are willing to accept the result as a proof! For everyone's peace of mind, in the paper reporting the result [6], I included a section that speculated on the correctness of the computer search. It concluded that the probability of a few random hardware errors causing the search to miss a plane is extremely small. Notice that the assertion of correctness is not absolute, but only nearly certain, which is a special characteristic of a computer-based result. On second thought, perhaps I should not avoid the issue and should actually use the word "proof." This is not the first time that a computer has played an important role in "proving" a theorem. A notable earlier example is the four-color theorem [1]. As more and more mathematicians are realizing the power of a computer, maybe we should also consider its limitations. Above all, the computer is a new tool to be used with care. The use of a computer also has serious implications for what constitutes an acceptable proof. In this article, I shall outline some of my thoughts on this subject. 8
You might think that it is not your problem and that it affects only researchers like me who work in enumerative searches. "Occupational hazard," you might say. But, consider symbolic manipulation programs such as MACSYMA, REDUCE, or MAPLE. They are great time-savers and enable us to solve problems otherwise too large to do by hand. Suppose one of the steps in a proof is based on using one of these programs. Then we have to also consider whether the computer is giving the correct result. In [3], there is an example where one such program missed one of the two possible solutions to a system of equations. A colleague here at Concordia once asked MACSYMA to factorize a polynomial and was told that it was irreducible. Not believing the result, he did it again. This time MACSYMA returned the correct factors. Did MACSYMA make a mistake or did he input the wrong
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polynomial initially? The answer is not important to us. The important lesson is that a computed result may be wrong for whatever reasons. Hence, even if the computer played only a minor role in a proof, the result may not be absolute. In Section 2, we shall look into some of the common computing errors. We have to understand the enemy to prescribe counter-measures. In Section 3, I shall describe some of the methods I used to increase the confidence of a computed result. Section 4 outlines my thoughts on h o w computer-based proofs should be evaluated. Section 5 addresses the role of an independent verification of a computer-based proof. Section 6 contains a few brief concluding remarks.
Characteristics of Computing Errors Computing errors can be broadly classified into two categories: h u m a n errors and h a r d w a r e errors. Human errors are the most common. H u m a n errors can further be subdivided into user errors that are under the user's control or system errors that are outside the user's control. One type of user error is the input. If we give the computer wrong data, then we cannot expect correct
The CRAY-1A, the supercomputer that did most of our work, was reported to have undetected errors at the rate of approximately one per thousand hours. Since we used several thousand hours of CRAY-1A computer time, we should expect a few errors. answers back. There is a saying, "Garbage in, garbage out." It is always worthwhile double checking the input data. Another type of user error is the programming error, if the user is actually doing the programming. Programming is a process of giving the computer a sequence of instructions to follow. The computer will follows these instructions, with no regard as to whether they make sense. If we give a computer the wrong or incomplete instructions, the results may be correct some of the time and wrong some other time. As most programmers are painfully aware, computer programs have to be tested. Testing involves running the p r o g r a m w i t h s o m e test data a n d checking whether the expected results are obtained. A programming error is commonly called a bug. The process of catching and then correcting bugs is called debugging. Debugging has its limitations. The common wisdom is, "Testing can only show the presence of errors and cannot prove their absence." An important characteristic of programming errors
is that most of them are reproducible, meaning that given the same input, the same erroneous output will result. The possible exception is the programming of parallel computers, where timing-dependent errors are usually irreproducible. For reproducible errors, it
Although system programs should have undergone thorough debugging before reaching the hands of users, they are seldom error-free. is useless to rerun a program, because whether it is correct or not, the answer produced will be the same. We group all the other programming errors not made directly by the user as system errors. Whatever c o m p u t e r we use, we u s u a l l y need an operating system such as MSDOS or UNIX, which is a package of programs written by others to make it easier to use the computer. There are also compilers, which are translating programs that convert user programs written in high-level languages (such as Pascal) into the equivalent machine-level instructions. Some compilers are called optimising compilers because they can generate more efficient machine-level instructions. Although system programs should have u n d e r g o n e thorough debugging before reaching the hands of users, they are seldom error-free. I have encountered problems with several optimising compilers. One example is the early versions of the Pascal compiler on the VAX running under the VMS operating system. When used with the optimize option, it gave a wrong translation of some of the operations related to packed sets. Packed sets are used in many of our programs on projective planes to save both m e m o r y space and computer time. So we had to systematically avoid using the optimize option, even though the programs would run slower. We had enough worries about the correctness of our own programs without having to worry about the correctness of the compiler. Can programming errors be totally eliminated? Programmers have long resigned themselves to accepting a small number of undetected bugs in long and complicated programs. In fact, there is a saying, "The only programs with no bugs are the trivial ones." Whether the correctness of a program is provable is still very much a research topic. Hardware errors, on the other hand, are not under the control of humans. They occur randomly and u s u a l l y lead to catastrophic results, for example halting the computer or destroying the data on the disk. They seldom go unnoticed and rarely pose a problem regarding the correctness of a computed resuit, because either there is no result or the result is obviously wrong. However, the unnoticed errors that change the results in some subtle and non-obvious ways are the most dangerous. For example, a common THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990 9
error is the r a n d o m changing of bits in a computer memory. Most large computers use error-correcting memory to minimize this problem. Yet, there are error patterns that look correct to the error-correction circuits. Fortunately, such errors are very rare and the probability of being affected by them is small, though not always negligible, as indicated by the error rate of the CRAY-1A. Let us summarize the characteristics of the two types of computer errors. Human errors are the most common; they are usually reproducible, and they are almost unavoidable. Hardware errors are random and rarely unnoticed. The unnoticed ones do occur, but they are extremely rare. Looking at this summary, it is not surprising that we should be more concerned with h u m a n errors t h a n hardware errors. Fortunately, human errors are more under our control.
M e t h o d s to Increase Confidence in a C o m p u t e d Result Eliminating all human errors should be the ultimate goal, but it may be too ambitious. A more realistic goal is to aim for a nearly certain status in a computed result. We shall discuss four methods that can be used to increase the confidence of computer-based results.
Prove the Results by Hand. Suppose we used a symbolic manipulation program. Should we just accept the results unquestioningly? I suspect most of us will. However, we should at least try to prove it by hand. If we can prove a result by hand, then all the doubts about computing errors are eliminated. As Ronald L. Graham observed in [7], "Knowing what to prove is more than half the battle." If our problem is to factorize a composite number, then it is not difficult to multiply the factors by h a n d and check the result. Clearly, this is the best solution. The same approach could be used on most computer-based results. However, this is not always as easy or straightforward as in the factorization example. I would dearly like to see a computer-free proof of the non-existence of a finite projective plane of order 10. Even a proof of a major subcase would be illuminating. What if we fail to prove the result by hand? Certainly, we should not just sit on it and hope that someday we can find a computer-free proof. We may want to ask the computer to print out the intermediate steps. Maybe these can provide some hints for a proof. In this respect, I find most symbolic manipulation programs inadequate, because it is difficult or impossible to print out the intermediate steps. We might also try to run the program a second time. Unless we suspect that a rare, undetected hardware error has occurred, it is useless to merely rerun a program. Because programming errors are reproducible, we would get the same answer, whether it is fight or wrong. Instead, we should try for other means of double checking the result. Double Checking the Result. Consider again the example of solving mathematical problems using symbolic manipulation programs. There are several such programs with similar capabilities. It is worthwhile solving the same problem using different programs. If they agree, then the probability of the answer being wrong is very small. It is still not possible to claim that the answer is absolutely correct. We can definitely imagine a scenario where the answer is wrong and yet both programs agree. Maybe we gave them the same wrong input or maybe they use the same erroneous method! Even w h e n we develop our own programs, we can still do some double checking. We can develop two
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different programs, preferably using different methods and written by different individuals, and compare the results. You may think that the idea is unworkable, because developing one working program is already difficult enough, let alone two. However, the difficulty of developing a working program really arises in requiring the one program to be both efficient and general purpose, two often conflicting goals leading to complicated programs. It may be much easier to develop a second simpler program, where issues of efficiency or generality are not the main objective. For a long-running program, it is often necessary to develop two programs: one simple but slow, for exploration purposes, and another complicated but fast, for the actual run. In our search for the plane of order 10, we actually used many programs. The search is divided into cases, and we have fast programs tailor-made for each case. We also have another program that is slower by about four orders of magnitude, but which can work on any case. The results of these two programs are compared. Because it would have taken too long to run each case completely using the slower program, the comparison of results is restricted to a few selected small subcases. So, this is not a complete double checking of results, but the agreement of the results of selective subcases is still comforting. Even with only one program, it is possible to increase the confidence in its correctness by using internal consistency checking.
Consistency Checking. I shall start this section by describing one of my many programming errors. In 1975, as part of a study of the 1-width of a (0,1)-matrix [4], I had to generate all 6 x 6 (0,1)-matrices. A (0,1)matrix is a matrix all entries of which are either 0 or 1. The 1-width of a (0,1)-matrix is the minimum number of columns such that there is at least one 1 in each row. Since the matrix has 36 entries and each entry is either a 0 or a 1, there are 236 ~ 7 x 10TM such matrices. There are too many matrices to be considered directly, so I exploited the fact that the 1-width of a (0,1)-matrix is not changed by permuting rows or columns. These permutations divide the matrices into equivalence classes and I generated only one matrix from each equivalence class. The number of matrices to be considered is reduced by roughly a factor of 6! x 6!, to approximately 105. After running the program, I checked whether the answers made sense. By using P61ya's theory of counting or Burnside's Lemma, it is possible to calculate the size of each equivalence class. Adding up all these sizes, I obtained a number slightly smaller than 236. I spent the next weekend agonizing over what could be wrong without knowing where to start. I forget what the actual error was, except that it was very minor. I remember this story because it was the beginning of my conviction that an enumerative
search program should contain some internal consistency checking. What is consistency checking? It is the verification that a computed result does not contain contradictions. Many computational results contain internal relationships. For example, the sum of the degrees of the factors of a polynomial must equal the degree of the original polynomial, or the number of 6 x 6 (0,1)matrices is 236. To implement consistency checking, start by identifying some easily verifiable relationships among the expected results. Preferably, the relations should involve the whole computation rather than
A paper containing a computer-based proof creates a new challenge for our refereeing system. only a small part of it. Then, these relations should be tested and a failure would demonstrate the existence of errors. Where should we look for these relations? There is no simple answer and it depends on the nature of the problem being solved. In combinatorial computing, the counting method described above for the 1-width problem is typical. The permutation of rows and columns are generalized to the property-preserving operations. The idea of having to investigate only one representative from each equivalence class is the basis of isomorph rejection. This method can speed up a combinatorial program enormously, but it is also the source of many programming errors. Fortunately, the idea of counting objects in two different ways provides powerful consistency checking [5] and is responsible for catching many of my errors. I feel more comfortable with my computed results if they pass this check.
Use Well-tested Programs. The suggestion of using only well-tested programs seems a tautology. Yet, one can deduce different conclusions from this principle. For example, avoid writing unnecessary programs. In fact, very few mathematicians need to write complicated programs. There exist many excellent packages that are easy to use and can do most of the calculations we want to do. Since these packages are also used by many others, they have undergone extensive testing and hence are more reliable. A related conclusion is that we should use popular software packages, the more popular the better, because a popular package is more heavily used and less likely to have bugs. Mathematicians tend to be concerned about the cost of software packages. If there is a well-tested and free software package, by all m e a n s we should use it. However, there is cost involved in designing and deTHE M A T H E M A T I C A L INTELLIGENCER VOL. 12, NO. 1, 1990
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veloping a good package and someone has to pay for it. So do not let the cost of a package be the overriding factor. Investigate carefully its capability and its reliability. Consider part of the cost as an insurance premium for the correctness of the result. Perhaps the extra cost is worth the peace of mind. As mathematicians, we are using the computer as a tool, and we need a dependable and reliable tool. It is advisable to be conservative and follow the wellbeaten tracks rather than breaking n e w frontiers in computing technology. Leave the latter task to computer scientists or mathematicians with a strong background in computer science.
How Should Computer-Based Proofs be Evaluated? A paper containing a computer-based proof creates a new challenge for our refereeing system. Normally, a referee is expected to check the correctness of the proofs. H o w can a referee check a proof that "no one can check?" Is he or she expected to write a program to check the results? What if the program requires a lot of computer time, as the search for a finite projective plane of order 10 does. This is too much to expect from our referees. I believe it is best to treat computer-based proofs as scientists in other disciplines treat experimental results. In addition to the results, the author of the paper should include a description of the methodology, an analysis of the results, and a discussion of how the results fit into the known theory. The referee can check whether the methodology is correct and whether the author has taken care to ensure the correctness of the computed result. Ultimately, the referee has to make a judgement on whether the results are interesting and believable. The consequence of using the "believability" criterion is that, sometimes, wrong results are published. Scientists in other disciplines have a standard technique for handling this problem. They emphasize the importance of i n d e p e n d e n t verification of experimental results.
Independent Verification It is important to encourage the independent verification by a second party of computer-based proofs. The necessity of an independent verification may require a rethinking of what is publishable. Normally, in mathematics, w e s e l d o m p u b l i s h a s e c o n d p r o o f of a theorem, unless it contains new ideas. An independent verification may not contain any n e w ideas and hence not publishable by this standard. In this "publish or perish" world, few people are interested in redoing someone else's work. The consequence of this 12
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In this "'publish or perish" world, f e w people are interested in redoing someone else's work. The consequence of this attitude is that some wrong computer-based proofs m a y s t a y unchallenged for a long time.
attitude is that some w r o n g computer-based proofs may stay unchallenged for a long time. For the health of the discipline, I hope there will be a change of attitude. Personally, I believe that a journal that publishes a computer-based proof has a moral obligation to publish, albeit in a shorter form, an independent verification of the result. In a sense, the verification completes the proof.
Conclusion The advent of computers has provided mathematicians with a very powerful tool. It enables us to tackle complex problems. The slight uncertainty inherent in a c o m p u t e r - b a s e d proof should not deter us from using it. As Richard Feynman, a Nobel Laureate in Physics, said in his 1955 address to the National Academy of Sciences, "Scientific knowledge is a body of statements of varying degrees of certainty--some most unsure, some nearly sure, but none absolutely certain." As physicists learned to live with uncertainty, so we should learn to live with an "uncertain" proof. Maybe we can borrow a further leaf from the experimental physicist's b o o k - - t h e y have their particle accelerators, so why shouldn't we dream of mathematical supercomputers?
References 1. K. Appel and W. Haken, Every planar map is fourcolorable, Bull. Amer. Math. Soc. 82 (1976), 711-712. 2. M. W. Browne, Is a math proof a proof if no one can check it?, The New York Times (20 December 1988). 3. K. R. Foster and H. H. Bau, Symbolic manipulation programs for the personal computer, Science243 (3 February 1989), 679-684. 4. C. W. H. Lam, The distribution of 1-width of (0,1)-matrices, DiscreteMathematics20 (1977), 109-122. 5. C. W. H. Lam and L. H. Thiel, Backtrack search with isomorph rejection and consistency check, J. of Symbolic Computation 7 (1989), 473-485. 6. C. W. H. Lam, L. H. Thiel, and S. Swiercz, The nonexistence of finite projective planes of order 10, Can. Journal of Math. (to appear). 7. P. Wallich, Beyond understanding? Computers are changing the spirit of mathematics, ScientificAmerican (March 1989), 24.
Computer Science Department Concordia University Montrdal, QuebecH3G 1M8 Canada
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Mathematics and Poetry: How Wide the Gap? W. M. Priestley
A little learning is a dangerous thing; Drink deep, or taste not the Pierian spring: There shallow draughts intoxicate the brain, And drinking largely sobers us again. --Alexander Pope Until I was in m y late twenties and agonizing over the writing of a thesis in mathematics, I was never seriously engaged by a poem. Poetry had usually been thrust upon me in English classes in the same spirit that mathematics was thrust upon English majors. It was a passive experience. You could be active in mathematics. You could get inside a good problem and enjoy the discovery of unexpected connections with familiar things. Sometimes, as a result, you had the pleasure of seeing everything anew. You just needed to muster the self-discipline to keep the subject constantly before you. Then, as Newton said, "wait till the first dawnings open little by little into the full light." No one ever suggested to me that with the same approach, you might sometimes encounter the same pleasure inside a good poem. Then came a trying time when the prospect of my failing to complete a thesis in mathematics seemed imminent. ! was not prepared to face that prospect and, to m y surprise, I found myself recalling a poet's lament: "Thou art indeed just, Lord, if I contend with thee, b u t . . , send m y roots rain!" When I had first read these lines in college I thought them to express the kind of desperate state attainable only by an Old Testament prophet or an overly emotional poet. In fact, I thought of the whole poem as extravagantly exaggerated, a virtuosic piece written by Gerard Manley Hopkins just to show how free and 14
flamboyant he could be within the presumably confining r h y m e s c h e m e - - a b b a abba cdc c d c - - o f a sonnet. But m y problems with writing mathematics and Hopkins's problems with writing poetry had led us to essentially identical plights: You know what you want to write about, and you keep the subject before you. Yet inspiration descends grudgingly and fleetingly. You strain for months, but your best efforts produce only disappointments. To make things worse, others seem to get lively ideas without apparent effort. It isn't fair:
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Thou art indeed just, Lord, if I contend With thee; but sir, so what I plead is just. Why do sinners" ways prosper? and why must Disappointment all I endeavour end? Wert thou my enemy, 0 thou my friend, How wouldst thou worse, I wonder, than thou dost Defeat, thwart me? Oh, the sots and thralls of lust Do in spare hours more thrive than I that spend, Sir, life upon thy cause. See, banks and brakes Now, leaved how thick! laced they are again With fretty chervil, look, and fresh wind shakes Them; birds build--but not I build; no, but strain, Time's eunuch, and not breed one work that wakes. Mine, 0 thou lord of life, send my roots rain. Hopkins's poem now seemed to represent my own agony. However, the real revelation came in seeing that the poem's very existence gives assurance that such agony can be sublimated into flamboyant flight - - e v e n when that flight is held to the rigid demands of sonnet form. I had been absorbed by the daring flights of mathematics in the face of rigid formal demands, and I w a n t e d to make my own daredevil flight. How could I not become absorbed in this poem, which did, in essence, what I wished to do? Others have known emotional perils of mathematics [17], and other students of mathematics have been inspired by this poem [5, p. 31]. My encounter led me to try to learn a little about the world of serious poetry and to keep an eye out for similarities to the world of serious mathematics. (It pleased me no end to find out, incidentally, that Hopkins had made a brilliant record in mathematics at Balliol College, Oxford.) One similarity between mathematics and poetry paradoxically has the effect of preventing other shared characteristics from becoming more widely known. Each of the two callings seems--particularly from the other's point of v i e w - - t o aspire to create a world that "'the Philistine cannot enter without ceasing to be a Philistine." An apprenticeship does seem necessary to gain full entrance to either world. The longer I serve as an apprentice, the more I find that the gap between mathematics and poetry is not nearly so wide as it formerly seemed to be.
A Little Learning To probe the gap between mathematics and poetry, we might best begin by taking note of a kindred relationship that is less problematical and more familiar: the bond between mathematics and the basic elements of language. This bond was institutionalized in medieval times with the establishment of the "seven liberal
arts" as an educational unit. The trivium--grammar, rhetoric, and logic--could not have been wed to the quadrivium--arithmetic, geometry, astronomy, and m u s i c - - w i t h o u t general acknowledgment of a natural affinity between these two groups of disciplines. This affinity has persisted through modern times, w h e n a number of mathematicians have displayed more than a casual interest in the trivium. Gauss in his y o u t h was equally torn b e t w e e n mathematics a n d classical philology, and more recently we have seen mathematical treatments that have revolutionized aspects of logic and linguistics. As academic disciplines, mathematics and language are today intertwined at language's lower levels. The question to be addressed here may be put as follows: How wide is the gap between mathematics and language at language's highest level? Salomon Bochner, a distinguished mathematician and an avid student of linguistics, remarked that the Greek word for mathematics originally had the very general meaning of "something that has been learned or understood." The contraction of meaning around the time of P l a t o - - w h i c h r e s u l t e d in the w o r d ' s present denotation--Bochner [3, p. 25] thought to be noteworthy: This contraction is remarkable not only because it was so extensive, but because there was only one other contraction, in Greek, which was equally extensive, the contraction of the word "Poetics." By origin, poetikd means " s o m e t h i n g that has been done, manufactured, achieved." It thus appears to have been possible in 400 B.C. to use the words "poetry" and "mathematics" to refer on occasion to the same thing. Nothing, of course, could be further from the case today, when many students and professors do not even view mathematics as belonging to the liberal arts. The shortsightedness of this current view induced me to put the following words in the middle of a calculus book: It is hard to overestimate the value of appropriate symbolism. Of all creatures, only human beings have much ability to name things and to coin phrases. Poets do this best of all.
9 . . as imagination bodies forth The forms of things unknown, the poet's pen Turns them to shapes and gives to airy nothing A local habitation and a name9 A Midsummer-Night's Dream, Act V It can be contended that Leibniz's way of writing the calculus approaches the poetic. One can be borne up and carried along purely by his symbolism, while his symbols themselves may appear to take on a life all their own. Mathematics and poetry are different, but they are not so far apart as one might think [13, p. 110]. Though mindful that a little learning is a dangerous thing, I wish to expand upon the thesis expressed in THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
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the last sentence above. Let us take note of some ways in which mathematics and imaginative writing have touched each other.
Shallow Draughts Consider the following familiar lines:
Nature and Nature's Laws lay hid in night: God said, 'Let Newton be!"and all was light. Pope's "Epitaph Intended for Sir Isaac N e w t o n "
was intended to be taken as playfully composed sarcasm. When read that way in a poetry class it can elicit delighted chuckles from students. Read it aloud yourself, and you may feel an undercurrent of cynicism. Alexander Pope was not endorsing the Age of Reason. On the contrary, he could see nothing humanistic in Newton's legacy, and was reacting, perhaps partially out of envy, against it: Pope was, of course, one who ascribed to his own admonition.., that the proper study of mankind is man, [unlike] Sir Isaac Newton . . . . [who was] a student of the physical world, not [of] man. [This epitaph is] half-satiric, half-admiring . . . . The paraphrase of the line from Genesis is obvious--and cynical [12, p. 302]. Pope is easily misread by historians of mathematics. Eves [6, p. 307] and Kline [11, p. 281] q u o t e this epitaph for the purpose of epitomizing what they suppose, falsely, to be the unbounded admiration of an entire age for N e w t o n ' s accomplishments. Pope's words might better be taken in exactly the opposite way: to h e r a l d the e v e n t u a l b r e a k b e t w e e n the sciences and the humanities that led to the notorious "two cultures" of C. P. Snow.
P o e t r y and m a t h e m a t i c s do n o t belong to distinct cultures.
and mathematical ideas do not." This point is well taken, so long as we note that longevity does not imply superiority. If it did, Homo sapiens would not fare well in comparison with other species. A more contentious issue is raised with Hardy's analysis of "good lines" of poetry that are devoid of significant meaning [8, p. 84]. Hardy evidently thinks that the conventional w i s d o m of the 1930s w o u l d judge good lines to be good poetry even w h e n the lines are incoherent. It is curious that he should think so, for Hardy is at pains to contrast examples of beautiful mathematics (the Greek proofs of the irrationality of the square root of two, and of the infinity of primes) with examples of ugly mathematics (there are just four numbers which are the sums of the cubes of their digits). His point is that, in mathematics, truth is not a sufficient criterion for beauty. Seriousness and depth must be considered as well.
One hears of college courses labeled Mathematics for Poets. There is an equal need for courses in Poetry for Mathematicians. Hardy has only a vague notion [8, pp. 90-91] that a poet might have similar feelings about poetry. Yet poets do, of course, feel the importance of seriousness and coherence in judging a poem [4, pp. 340-343]. Perhaps the canonical textbook example of the good lines/bad poem syndrome is given by Joyce Kilmer's "Poems are made by fools like me, But only God can m a k e a t r e e . " Cleanth Brooks a n d Robert P e n n Warren [4, pp. 289-290] give a debunking of Kilmer's poem Trees. "Is there any basis," they ask, "for saying that God makes trees and fools make poems?" One hears of college courses labeled Mathematics for Poets. There is an equal need for courses in Poetry for
Mathematicians.
Poetical Power Poetry and mathematics do not belong to distinct cultures, however, despite the small number of people who would profess to truly appreciate them both. We shall see that Pope and Newton had something in common after all, that mathematics has within it humanistic elements of the same nature as poetry. First, however, we need to eliminate misconceptions about poetry that are likely to burden mathematicians. Even a mathematician like G. H. Hardy, whose graceful writing style could have come only from a deep love of letters, has made remarks about poetry that do not quite ring true. In his book A mathematician's apology he says [8, p. 81]: "Greek mathematics is 'permanent', more permanent even than Greek literature. Archimedes will be r e m e m b e r e d when Aeschylus is forgotten, because languages die 16
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Some mathematicians are not bad as poets. Here is the last of four stanzas written by Errett Bishop [2, p. 14] to w a r n mathematicians against the seduction of formal systems, and to implore them to turn instead to the bright sun of constructivism:
If Aphrodite spends the night, Let Pallas spend the day. When the sun dispels the stars, Put your dreams away. These are, in my opinion, good lines--both to the ear and to the mind. The last two rephrase the content of the first two. If you read the stanza aloud, however, you will notice the effect of a subtle change in meter.
The stanza exhibits the familiar iambic pattern of unaccented syllable followed by accented syllable, but in each of the last two lines the initial unaccented syllable is truncated. This change, coupled with the fact that these lines each begin with three monosyllables, forces us at mid-stanza to read more deliberately, to adopt a graver tone. This marked shift in tone, from fanciful to serious, has the effect of rendering to us the depth of Bishop's profound concern. Mathematics, of course, cannot render a feeling or an experience like poetry. To find similarities between poetry and mathematics, we must look elsewhere. Toward this end, consider Bishop's choice of the word dispels, which he may well have picked just because it "felt right." However, he may have considered and rejected other w o r d s or phrases: drives out, repels,
wards off, rebukes, rebuffs, etc. None of these other possibilities carries the overtones produced by the word Bishop picked. Literally, the word means "drives away," but in context dispels suggests "breaking the spell" of the magic of Aphrodite's stars. While there may be no logical or etymological connection between dispel and spell, their association by the ear can produce an association in the mind. If this happens, the word does double duty upon the reader. It is denoting and connoting at the same time in such a w a y that both actions carry forward the argument. To attain this poetic effect the association need not necessarily be produced by aural means; etymological or conceptual means work just as well. Regardless of h o w it is done, this imaginative way of mobilizing a w o r d - - o f making it speak in harmony on several levels at once--is a main weapon of poetry. Can mathematics do anything like this? It can. In fact, this is a key to the power of mathematics. Mathematicians engage in a kind of collective poetical undertaking by virtue of what has evolved as their common language. To take an example, consider the concept of a function. You may, if you choose, define it in a most unpoetic way as a set of ordered pairs with no first component repeated. Whatever you take the word to denote, the fact is that the concept carries with it a large baggage [3, p. 217], much as a single word can carry far-flung connotations. Mathematicians make a function do triple duty. As every student learns early on, a function can be thought of statically, kinematically, or geometrically; that is, as a pair of columns of elements, as a rule of correspondence between moving points, or as a curve. Through this word you may move at will, conceptually, in three vastly different realms at once, or shift back and forth until you get the best view. Philosophers of mathematics seem to pay little attention to this point, perhaps because the practice of couching much of mathematics within the language of functions is essentially poetic.
Of course, most of us like to speak in terms of functions just because it feels right. There are no critics of m a t h e m a t i c s a r o u n d to wax e l o q u e n t about w h y things sometimes work out so happily when functions are mobilized. Or when the deep analogies of isomorphisms are brought to bear. Or when algebra, analysis, and geometry train three points of view upon the same target at the same time.
A Final Epigraph That well-crafted mathematics has an aesthetic appeal is argued in different w a y s b y Halmos [7] and b y Hardy [8]. Bertrand Russell put this claim in the following way: Mathematics, rightly viewed, possesses not only truth, but supreme beauty--a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stem perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. In later life, when Russell no longer thought well of the essay that begins with these oft-quoted words, he still maintained [15, p. 212] that "the aesthetic pleasure to be derived from an elegant piece of mathematical reasoning remains." An elegant argument in mathematics may not give the same kind of aesthetic pleasure as an elegant argument in a poem, but the immediate response of a serious reader is often the same in either case. "Silence is the appropriate response to revelation," as an old professor of mine once wrote [9], putting it much better than I ever could. Of course, if revelation is taken to mean something that shakes us to our foundations and makes us adopt, or at least appreciate, a radically new point of view, then it is rarely encountered except in the greatest of works. G6del's theorem may leap to our minds as the purest example of revelation in mathematics. (My old professor, incidentally, was writing about King Lear.) However, revelatory works need not be vast and sublime. We are jarred by Pope when we hear that drinking a little will make you drunk, but drinking a lot will sober you up; by Shakespeare when we hear that airy nothing can be given a body and shape and name; and by Bishop w h e n we hear that we must choose the goddess of wisdom over the goddess of love in mathematics! We are jarred by Pythagoras when we hear about the square root of two, and by Euclid w h e n we hear of the long march of primes. A common function of poets and mathematicians is to jar us into place, to refocus our vision so as to order the chaos around us and to connect what is familiar with what is beyond. This function is objectified in THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
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Wallace Stevens's Anecdote of the Jar, where a jar upon a hill tames the wilderness in which it is placed:
The wilderness rose up to it, And sprawled around, no longer wild. The jar was round upon the ground And tall and of a port in air. It took dominion everywhere... Here, a bare and gray container is made to symbolize the effect of the adoption of a n e w point of view. Or, more briefly, Stevens has simply (!) made a jar symbolize a jar, giving to an abstract idea a local habitation that might have surprised even Shakespeare. Just as a piece of mathematics can model different situations, this poem is subject to different interpretations. The jar could be a poem, which may have been what Stevens had in mind, but it could be any artifact of the imagination made for the purpose of instituting order, control, understanding, or unity. Artifacts giving form to abstractions are as essential to mathematics as to poetry. Should prizes be offered for bodying forth the forms of things unknown, t h e n - - t o mention just one nomination from mathematics--the notion of a limit would have to be considered. What was a limit, before it was given a name? Before Cauchy gave a precise significance to the notion, the answer might have been an oracular utterance alluding to the Greek method of exhaustion, like "that which remains when everything to which it is not equal is eliminated." If anything was ever airy nothing before being given a name, this is it! What did Cauchy's jar do? The wilderness explored by Newton, Leibniz, and Euler rose up to it. The infinitesimal calculus sprawled around, no longer wild. It was perfectly grounded, this jar, yet tall enough to serve as port for the high-flown fancies of modern-day analysts. It took dominion everywhere. Stevens's poem once seemed to me the ideal epigraph to set forth the theme of a calculus text, and I used it that way in [13]. Now I see that my jar was too small. The poem could well be taken as an epigraph for all of mathematics.*
The Necessary Gap
to be a fine poet. Mathematics relies upon the eye, but not so much. Greek mathematicians, like Greek artists, may have relied primarily upon the sense of touch [10]. Ugly mathematics--such as a correct, but inelegant brute-force method hidden away in some computer p r o g r a m - - c a n be of great value. Ugly poetry is of no value except to illustrate bad form to be avoided. Society will let you copyright a poem, but not a theorem. Thus there is a legal distinction between poetical truth and mathematical truth. Society produces critics of poetry who are not poets, while mathematicians must serve as their own reviewers. In fact, the differences between mathematics and literature in society are an obvious subject for satire. What would the land be like if the roles of mathematics and literature
Society will let you copyright a poem, but not a theorem. were reversed in America? "'Acirema'" is the land to which Kenneth May and Poul Anderson [1] take us, where "mentioning the latest theorem is a standard ploy of social one-upmanship, whether or not one has actually studied it." It is argued in [14] and elsewhere that the spirit animating the study of mathematics is humanistic. H o w closely that spirit resembles the spirit of poetry is an inviting subject for a book. The only happy outcome of an expansion of this theme to booklength size, however, might be to prompt the reprinting of lines from Schwarzenberger's famous review [16]:
By and large any human activity Displays mathematical structures, But an author with some sensitivity Won't let them expand to six chapters. The material was once a nice lecture But expansion's a dangerous trap; My opinion as humble reviewer Is: the book fills a much needed gap.* Mathematics and poetry are different. Let that be conceded.
We have discussed some less obvious similarities between poetry and mathematics. Let us end on a lighter note about the less obvious differences. Poetry relies strongly upon the ear. A person deaf since birth could be a fine mathematician, but would have little chance
Despite the differences between mathematics and poetry, there are common threads enough to show that essential elements of mathematics are humanistic. In
* Editor's note: For a different analysis of t h e s a m e p o e m , see the review b y J o n a t h a n H o l d e n in this issue of t h e Mathematical Intelligencer.
* Professor S c h w a r z e n b e r g e r h a s v o l u n t e e r e d t h e information that the p h r a s e "'this book fills a m u c h - n e e d e d g a p " w a s b o r r o w e d f r o m a b o o k review written by Paul H a l m o s .
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Conclusion
fact, the only w a y to avoid seeing a n y t h i n g poetic in mathematics is to read its language in a p u r e l y formalistic fashion, p a y i n g attention to d e n o t a t i o n and the laws of logic, and to nothing else.
Acknowledgment: I wish to thank Henry F. Arnold, Jr., Professor of English at the University of the South, for help in making coherent a first draft of this paper.
References 1. Poul Anderson and Kenneth O. May, An interesting isomorphism, American Mathematical Monthly 70 (1963), 319-322. 2. Errett Bishop, Schizophrenia in contemporary mathematics, Contemporary Mathematics 39 (1985), 1-32. 3. Salomon Bochner, The role of mathematics in the rise of science, Princeton: Princeton University Press (1966). 4. Cleanth Brooks and Robert Penn Warren, Understanding poetry, New York: Holt, Rinehart and Winston (1960). 5. Freeman J. Dyson, Disturbing the universe, New York: Basic Books, Inc. (1979). 6. Howard Eves, An introduction to the history of mathematics, New York: Saunders College Publishing (1983). 7. P. R. Halmos, Mathematics as a creative art, The American Scientist 56 (1968), 375-389. 8. G. H. Hardy, A mathematician's apology, Cambridge: Cambridge University Press (1967). 9. Charles T. Harrison, The Everest of poems, The Sewanee Review 75 (1967), 662-671. 10. William M. Ivins, Jr., Art and geometry, New York: Dover (1946).
11. Morris Kline, Mathematics in western culture, New York: Oxford University Press (1953). 12. Frank N. Magill, Magill's quotations in context, New York: Harper and Row (1965). 13. W. M. Priestley, Calculus: an historical approach, New York: Springer-Verlag (1979). 14. , Review of The Mathematical Experience, by Philip J. Davis and Reuben Hersh, American Mathematical Monthly 89 (1982), 515-518. 15. Bertrand Russell, My philosophical development, New York: Simon & Schuster (1961). 16. R. L. E. Schwarzenberger, Review of Mathematics and Humor, by J. A. Paulos, Bulletin of the London Mathematical Society 42 (1981), 248. 17. Donald R. Weidman, Emotional perils of mathematics, Science 149 (1965), 1048.
Department of Mathematics and Computer Science The University of the South Sewanee, TN 37375 USA
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A Curious New Result in Switching Theory Lee C. F. Sallows
G6del turned out to be an unadulterated Platonist, and apparently believed that an eternal "'not" was laid up in heaven, where virtuous logicians might hope to meet it hereafter. --Bertrand Russell [1].
The following account relates how a puzzle brought to light a remarkably simple, highly intriguing, probably useless, but undeniably basic new result in switching theory. Spice is added to the story through the role played by construction of a wildly improbable electronic device in helping to establish the new finding. "Switching theory" has a slightly old-fashioned ring to it; what exactly does it signify? A brief remark on this and a couple of related matters will set our subject in perspective and prepare the way for issues arising later. Computer science emerges into view as a separate discipline from a fluster of related topics, chief among them symbolic logic, Boolean algebra, switching and automata theory. Logic, originating with Aristotle, concerns the study of deductive inference, of the conditions of truth-preservation in deriving one statement from another. More than two millenia following Aristotle, George Boole was to design his algebra to model logic, a step largely intended to replace reasoning with calculation, with the rule-governed manipulation of symbols. Boolean algebra, we remind ourselves, comprises a so-called formal system: a well-defined set of signs and conventions by means of which, starting with certain symbol strings, certain others may be legally substituted, the latter being deemed equivalent to the former. No specific meaning is attached to these transformations, except in the loose identification of signs with the things they usually signify when applying such formalisms to external systems (such as
logic). Whether the algebra applied really is an accurate model of the system in question is of course a problem not resolvable within the algebra itself. A notable success in the practical application of Boolean algebra occurred with the appearance of C. E. S h a n n o n ' s paper "Symbolic Analysis of Relay and Switching Circuits" in 1938 [2]. Ever since, the analogy of "0" and "'1" with open and closed switch contacts, and of series/parallel switch connections with AND/ OR B o o l e a n o p e r a t o r s , has b e e n a s t e r e o t y p i cal textbook example. Then, as today, a relay was an electromagnetically operated switch, a device opening
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up new realms of complexity in the possibilities it offered of switches controlling still other switches in endlessly convoluted networks. The problems thrown up in this new domain soon became the concern of "switching theory.'" Ten years following Shannon, switching theory advanced to a new level of maturity with G. A. Montgomerie's paper "Sketch for an Algebra of Relay and Contactor Circuits" [3]. By now a vital distinction had been recognized in the division of networks into combinational and sequential types. Combinational circuits were those in which the open or dosed states of every switch depended purely upon current input values (0, 1) to the network. A Boolean formula, simple or complicated, would always describe this relation satisfactorily. Sequential circuits, on the other hand, were those whose response to input patterns also depended in part on their past history: on the foregoing sequence of values presented. The behaviour of the circuit might thus change significantly after receipt of some critical input, the latter event thereby being in some sense "remembered.'" In fact memory--introduced via feedback effects--was the key property of such networks. Flow tables and state transition diagrams now displaced static formulas in the need to capture this temporal context-dependent behaviour. Thus was launched the study of what came to be called sequential or finite state machines, a field later to be known as automata theory. The progress of developments in automata theory is b e y o n d our p u r p o s e here: advances w e r e rapid, leading to theoretical results of great m o m e n t in connection with Turing machines and mathematical linguistics, the subject soon shading seamlessly into
The possibility of simulating three inverters by means of t w o had never so much as crossed my mind before; the bare contingency hinted indefinably at something wonderful. theoretical computer science proper. Back in the mainstream of switching theory however, by the 1960s advances in technology had shifted emphasis away from relays and onto "electronic digital logic" realized in micro-packaged integrated circuits or "'chips." Mechanically-actuated contacts gave way before "ANDgates" and "OR-gates" etc., the binary states (0, 1) whose input and output lines were represented by two discrete voltage levels. Soon Boolean algebra was a standard item on the training syllabus of electronics engineers; formerly recondite chapters of the now slightly outmoded-sounding "switching theory" became the stock-in-trade of every technician. Hence, overtaken by studies into the more chal22
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lenging finite state machines, as a subject of research, switching theory dropped into the background, furnishing instead a well-knit body of established results that found daily application in electronic logic design. This is not to say that all the theoretical questions raised had been successfully answered. Many problems, especially in the area of minimization, remained unsolved; later these would provide a point of departure for the currently vigorous theory of circuit complexity (see [4]), a field closely related to, yet historically distinct from the old switching theory. In any case, mass-production techniques had extinguished any practical need for minimal solutions. Gone forever was the pioneering impetus of the early days. Who then would have expected to stumble across an undiscovered nugget still reposing amid the slagheaps of this abandoned mine?
A Knotty Problem Recently browsing through A Computer ScienceReader (Selections from Abacus, Springer-Verlag, 1988), m y eye was caught by an article on Automated Reasoning by Larry Wos [5]. Wos illustrated the working of his r e a s o n i n g p r o g r a m by m e a n s of a few example problems, one of which immediately captured my attention. It was this:
z
.~
x /
.~
zI
The black box above receives binary inputs (0, 1) at x, y, and z. Each output line yields the complement of the corresponding input; that is, if x is 0, x' is 1, and so on. Each of the eight possible 3-bit input words thus gives rise to its complementary word at the outputs. Normally such a transfer function would be achieved by using three inverters or NOT-gates connected between each input and output.
Problem: Design a network using any number of AND and OR gates, but not more than two (2) NOT's to achieve exactly the same input-output function. (The AND's and OR's may have as m a n y inputs as required.) Now relays, gates, switchery, and logic hold powerful fascination for some. The possibility of simulating three inverters by means of two had never so much as crossed m y mind before; the bare contingency hinted indefinably at something wonderful. It seemed to call for ingenious circuitry. The puzzle had me hooked in no time.
Figure 1. Moore's circuit. It turned out to be a far tougher conundrum than first imagined. So much so, in its elusiveness it became hypnotic. In fact, on and off I took almost a fortnight to solve it, succeeding even then only through reasoning aided by trial and error. But the solution was worth waiting for: an intricate network of true Platonic elegance and inevitability. It is a logical constellation that was always there, sooner or later someone was bound to find it; a sheer poem for the switching theorist. As the sequel shows, the name of the man who did find it first turned out to be Edward F. Moore, a distinguished pioneer in the field of automata theory. From now on I shall refer to the basic arrangement as Moore's circuit. Incidentally, Wos's a u t o m a t e d r e a s o n i n g p r o g r a m was successful in solving the problem; his method being too complex to outline here, a detailed account can be found in [6]. One version of Moore's circuit is shown in Figure 1. The network admits of a number of (essentially minor) variations, some more economical in gates than others; our example is picked for its functional clarity. Interested readers might like to seek a more parsimonious circuit u s i n g one gate fewer t h a n Figure 1 (multi-input gates then being counted as if built up from 2-input equivalents; Figure 1 thus containing 11 AND's and 14 OR's). In view of its importance to what follows, a few comments on Moore's circuit will be worthwhile. Central to every variant of Moore's solution is circuitry leading to a b i n a r y r e p r e s e n t a t i o n of the number of zeros present in the input word (xyz) by the four possible states of the two inverter outputs: 00 = none, 01 = one, 10 = two, 11 = three zeros. Simple as this may seem, there is but a single way to achieve
it. In effect, each inverter's output state (0 or 1) must represent a classification of xyz according to whether the n u m b e r of ones it contains falls in the top or bottom row (first inverter, A), and in the left or right column (second inverter, B) of the table: B 1
0
In this way the intersection of A's row and B's column choice pinpoints the number of ones (and thus, zeros) in the input word. In our circuit, use of AND gates to combine this information with the specific input pattern enables a complete decoding of the input word. See how each of the seven lines feeding the three OR gates at the right is uniquely activated by a different input word (indicated). The circuitry to the left is thus a "3-bit to parallel d e c o d e r . " Note that a l t h o u g h available, the eighth line (23 = 8) is unused since, when active (i.e., when x = y = z = 1), all outputs are to remain 0. Similarly, the three interconnected output OR's comprise a "parallel to 3-bit re-coder," the coding in this case ensuring that xyz inputs that are 0 result in corresponding outputs that are 1, and vice versa: an active " x ~ " line turns on outputs y and z, for instance. A point to observe is that alternative OR combinations could replace (or supplement) this one to produce any desired input-output functions: An output word may have as many bits as we please, and distinct recoders working in parallel could realise unlimited simultaTHE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
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neous output words, if required. Already one senses a surprising latent potency here, although, as we shall see, most of the magic in Moore's circuit lies exactly in the mischievous recoding he did choose. Speaking of coding and recoding serves to recall that a circuit diagram is a kind of coded representation and thus itself capable of translation into different symbol systems. A change of m e d i u m often brings new aspects into view. An obvious alternative in this connection is Boolean algebra. Re-expressing Moore's circuit in these terms is a mere mechanical exercise. Designating the output of inverter A as A, for instance, we can work backwards through the circuitry towards the inputs, transcribing directly as we go:
A = Not[x & y) Or (x & z) Or (y & z)l. Comparing formula with circuit we find the inverter is replaced by Not, the 3-termed, square-backeted Or expression deputizes for the OR-gate wired to its input, and the three parenthesized terms stand in for the AND-gates communicating between the input pairs xy, xz, and yz and the OR inputs. Note how the nesting of expressions r e p r o d u c e s the pattern of outputs feeding into inputs in the circuit. We are looking at a fragment of Moore's circuit written in a different language. Analogously, and taking advantage of the above, a compact expression representing the output B of inverter B can also be written: B = Not[(x & A) Or (y & A) Or (z & A) Or (x & y & z)]. Note how the presence of A as an argument in the function describing B is more than a convenient abbreviation, it reflects A's antecedence in the signal processing path: the value of A must already be available in determining that of B, a point that will prove of importance later. However, the real convenience of these partial descriptions becomes clear in the crisp encap-
sulation of the complete Moore circuit they now facilitate: x' = [ ( y & A ) Or (z & A) Or (B & y & z) Or (A & B)] y' = [(x&A) Or (z & A) Or (B & x & z) Or (A & B)] z' = [(x&A) Or (y & A) Or (B & x & y) Or (A & B)]. See how the equations expose a (predictable) threefold functional symmetry hinted at, but less successfully conveyed, by their equivalent circuit diagram, an obfuscation resulting from the latter's confinement to two dimensions. (An amusing exercise is to design a 3-D version of the circuit recapturing the trilateral balance.) Still later we shall have occasion to recall these formulas. So much then for a preliminary look at Moore's circuit.
Networks and Notworks This was all fine as far as it went: an intriguing puzzle with a beautiful solution, if lacking in practical application. During an early stage in reaching that solution, however, a rather astounding thought hit me. As an electronics engineer the idea occurred to mind quite easily and, although perhaps ingenious in small degree, is certainly no creative tour de force. Nevertheless, the implications struck me as luminous and compelling. The idea was simply this: If it is possible to simulate three independent NOTfunctions using only two primary NOT's (or real inverters) then couldn't we use two of those three in order to simulate a second set of three NOT-functions? At this stage, having used only two of the first set of three, there would still be one left over. That means that a total of FOUR i n d e p e n d e n t NOT-functions would have been simulated while still using only two real inverters. Figure 2 makes the proposal explicit. C o n s i d e r the circuit s h o w n . N e t w o r k 2 is the
Figure 2. Four NOT-functions from two inverters. 24
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Figure 3. Recursive nesting to produce N negations from two inverters. straightforward Moore circuit; as such its behaviour is functionally equivalent to an outwardly similar box containing three separate NOT's or inverters connected between each of its three inputs and outputs. Network 1 is identical to Network 2 except that its two inverters have been removed. The internal inputoutput connections normally made to the missing inverters have been brought out and connected instead to two channels of Network 2. Network 2 thus furnishes the two NOT functions required for normal
working of Network 1 (channels a, b, c) while still leaving a fourth independent complement function over (channel d). That is all. The ramifications of this stratagem ripple swiftly outward. For clearly the four newly created NOTfunctions can again be nested in an endlessly expandable recursive hierarchy to produce an unlimited number of independent negation functions; see Figure 3. In other words (and striving for the infinite in the name of logic): THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990 2 5
THEOREM I. In any universe, exactly two fundamental negators suffice for concomitant synthesis of all others. But this soon leads us to a couple of other interesting consequences: THEOREM II. A device whose input-output relations are described by some system of Boolean functions is always constructible using a network comprising some number of AND- and OR-gates but no more than two inverters. Or, still more ambitiously (and relying on other well-known results in the field): THEOREM III. Every possible finite state machine (automaton) is realizable using no more than two primary complement functions. Am I alone in continuing to feel a sense of wonder in this simple discovery? As I say, the idea for the above configuration occurred to me at the time of reading Wos's article, even before solving his problem. Taking it to be a merely personal rediscovery of a presumably well-established result in logic, thought of any further development never arose. Being satisfied that the idea was sound, as an engineer I felt only sheer surprise that, in principle, all the millions of inverters in use throughout the world could be " s e e d e d " from a single pair. Having a romantic turn of mind, it conjured an imaginative vision of a sort of Yin-Yang dyad of inverters occupying a dusty, temperature-controlled glass case at the National Bureau of Standards. Wires leading away from the four old-fashioned knurled brass input and output terminals lead off for distribution to other boxes scattered about the nation. (I should say five terminals: a " g r o u n d " or zero voltage reference line would also be required.) Well-established result or no, the self-duplicating inverter circuit was a revelation to me and continued to exercise fascination. Having nothing better to do, for fun I typed out a devilish new version of Wos's problem, sending it around to tease friends and colleagues at computer science and mathematics departments at the University of Nijmegen. In the new version, otherwise identical to the old, four complement functions are to be realized instead of three. As before, of course, only two inverters are allowed. (As a matter of fact, by Theorem II above, the input-output functions demanded by any severer version of the problem could be made as complicated as one wished. Asking for four NOT-functions is the obvious choice, this representing the least jump in difficulty at the new level of complexity.) In a few cases the response to this teasing was sharper than anticipated. I suppose the problem is so clear-cut and inescapable it poses a provocative chap 26
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lenge to one's self-esteem as an engineer, mathematician, logician, or whatever. Prevarication in the face of this kind of simplicity is difficult; admitting one cannot solve such an apparently elementary problem, even more so. The trouble is, unless you happen to be aware of Moore's circuit (as most people are not), the two separate insights needed for reaching the solution put it well beyond all reasonable ingenuity. It would be a creative act to re-invent Moore's circuit from scratch; penetrating to the fact that such a circuit is necessary as a component in the 4-complement configuration asks too much of human imagination. In light of this, some of the scepticism poured on my assurances that the solution was complex but straightforward, involving absolutely no hanky-panky, becomes explicable. At this stage Hans Cornet, a mathematical friend at The Hague, ran across what proved to be the original source of the 3 - c o m p l e m e n t problem. In Marvin Minsky's book Computation: Finite and Infinite Machines (Prentice-Hall Inc., 1967, page 65) was a confirmation of m y assumption that Wos had merely borrowed r a t h e r t h a n i n v e n t e d the p r o b l e m . Looking u p Minsky's book in Nijmegen I learned the problem had first been "'suggested by E. F. Moore." Admittedly the solution circuit (shown only in skeletal form) is not overtly attributed to Moore but surely no one could pose such a riddle without first having unravelled it? A comment by Minsky following the problem statement drew from me an appreciative smile: "The solution n e t . . , is quite hard to find, but it is an extremely instructive problem to work on, so keep trying! Do not look at the solution unless desperate." It was a final remark of his, however, that brought me up with a jolt. With deepening puzzlement I ran my eye again and again over his two terminal sentences: To what extent can this result be applied to itself--that is, how many NOTs are needed to obtain K simultaneous complements? This leads to a whole theory in itself; see Gilbert [19541 and Markov [1958]. Clearly the sufficiency of two NOT's in obtaining K (an arbitrary number of) complements was unknown to Minsky. Yet to speak of "'applying the result to itself" was a pretty reasonable description of exactly the trick used in my 4-complement circuit. H o w could it be that he h a d envisioned the self-same possibility without ending up at the same configuration? Why on earth should a "whole theory" be required? My thoughts sped back to those sceptical friends who could "almost prove your 4-complement problem is insoluble." Previously I could afford to be smug, now it was me against Minsky, Gilbert, and M a r k o v - the latter a name of intimidating authority in the world of mathematics. Was it likely his theory would turn out to be wrong? Hadn't I after all overlooked some inherent logical flaw that rendered reflexive re-application of the circuit to itself in fact unworkable? I lost
no time in hunting up the papers from Gilbert and Markov. Alas, the journals were not available in Nijmegen; there was nothing for it but to order copies. That would take a week or so. In the meantime I returned to the 4-complement circuit, re-examining it from every angle. Later that evening I banged a defiant fist on the table. It was no good: Markov or no Markov, theory or no theory, there was nothing wrong with that circuit: it had to w o r k ! - - A n d w h y not demonstrate my reasoning agreed with reality by building it? The very next day saw me launched on construction.
The 4-Complement Simulator Physical realisation of the circuit followed conventional electronic practice. Taking standard ttl integrated circuits lying at hand (six SN74LS08's and eight SN74LS32's: 14 pin packages containing four 2-input AND's and OR's, respectively) and a prototype-development printed circuit card fitted out with 14-pin chip holders, using a wire-wrap pistol to make interconnections, assembly was completed within a matter of hours. An obvious approach in implementing the device was dictated by the very principle of operation: first build and test two quite independent Moore circuits, afterwards remove the inverters from one (a single SN74LS04 chip) and replace with connections to two inputs and outputs on the other. This is exactly what I did. In the cover photo, the twin Moore circuits are formed by the two groups of eight chips furthest from the multipin connector. In one circuit, four wires leading from the underside of the card to a small plug that replaces the discarded inverter chip are plain to see. Finally, to facilitate testing, a push-button controlled 4-bit binary counter and a sprinkling of light-emitting diodes (LEDs) were added. Successive presses on the button (see adjacent to the main connector) run the counter through 0000, 0001, 0010 . . . . . 1111, the seq u e n c e of sixteen possible 4-bit w o r d s . C o u n t e r outputs are wired to the four NOT-simulator inputs, the presently activated word being indicated by a line of four adjacent LEDs situated close by (on = 0, off = 1). Six remaining LEDs dotted about the board report on the high/low status of the 2 x 3 Moore circuit outputs. For ease of comparability one of these is duplicated so as to form a single line of four evenly spaced LEDs monitoring the four main outputs. These additions account for two of the three extra chips at one end of the board: an SN74LS93 binary counter and an SN74LS00 (four 2-input NAND's) used as a so-called set-reset flip-flop to eliminate pushbutton contact bounce problems. The need for still a further chip made itself felt when, having completed and tested the two separate Moore circuits, the final
4-complement simulator produced by combining them failed to work as anticipated! At first this was unnerving. Using an oscilloscope, however, the source of the trouble was soon tracked down: under certain input transitions Moore's circuit exhibits race-conditions. Race conditions arise w h e n delays introduced by hardware inertia result in unint e n d e d overlaps b e t w e e n logical state durations, leading to transitory "spikes" or pulses of very short duration (the antecedence of inverter A in the signal processing path now shows its significance; see Figure 4). Such spikes can be innocuous enough in many applications, but not so in the 4-complement simulator. Here, a spike emerging from output x' of the nested circuit becomes gated through the outer circuit to the input of its second inverter (B, see Figures 1 and 2 ) - itself, however, now simulated by channel y of the nested circuit. Our spike, in other words, traverses a sneaky feedback loop and now finds itself re-entering an input of the inner Moore circuit! A vicious circle has been established: regenerative oscillation sets in.
Figure 4. Race condition: Input word xyz changes from 101 to 100. Delayed reaction of second inverter (B) to first (A) causes brief pulse at the output of the AND to which they are connected. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
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Notice that the culprit here is not the feedback loop - - a n intrinsic feature of the nested scheme (to which we shall r e t u r n ) - - b u t the pulse generated by the race condition. Happily, a cure is easily effected through interposing a delay in the appropriate line (connecting the output of inverter A to the AND-gate input so as to ensure the latter cannot receive a 1 from the former until after the output of inverter B has changed to 0). This accounts for the last remaining chip in the photo (another SN74LS08: four AND's connected together head to tail, each contributing its own share to the aggregate delay thus created). With this modification completed, turning on the power once again, I finally had the satisfaction of verifying a perfectly functioning 4-complement simulator. It was a happy moment of vindication and triumph. The 4-complement circuit thus stood acquitted-though in hindsight it is amusing to recall exultation on completion of one of the most futile or, at least, r e d u n d a n t items of electronic apparatus ever constructed! The Great Unanswered Question n o w remaining, however, was how could this success ever be reconciled with the apparently contradictory theory of Gilbert and Markov? The working device was an unshakeable fact, yet a theorem in logic cannot be validated via any empirical demonstration, however suggestive. Could some sort of a disillusionment still lurk in the publications awaited?
A Gordian Not Unravelled Following eventual receipt of the anxiously awaited material, a rapid glance at Markov's and Gilbert's conclusions confirmed Minsky's original remark: blatant contradiction of the two-inverters-always-suffice idea. Steeling myself to the mathematics, I settled down to read. Gilbert's is the earlier, exploratory paper, his partial result later subsumed by Markov's more embracing work, " O n the Inversion Complexity of a S y s t e m of F u n c t i o n s " (translated b y Morris D. Friedman). We confine ourselves to the latter. Markov begins his monograph with a series of careful definitions. A small alphabet of the signs familiar from Boolean algebra is introduced, constants and variables included: {0, 1, xl . . . . . xn, &, Or, Not, (, )}. Certain words or strings of these are specified as formulas and sub-formulas, negative sub-formulas being characterized as those prefixed by " N o t . " The socalled inversion complexity of a system of sub-formulas is now identified with the number of distinct negative sub-formulas occurring in it. My pr6cis lacks his precision but the outline of what is going on here is already dear: substituting concatenations of discrete symbols for the tangled Celtic knotwork language of the switching engineer, Boolean formulas replace circuit diagrams: " N o t " s are to be counted instead of inverters. 28
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So far so good. Moore's problem itself might well have been so reformulated as to ask for a system of Boolean functions equivalent to x' = Not(x), y' = Not(y), z' = Not(z), but in which "Not" (preceding a distinct sub-formula) would occur no more than twice. O u r previously derived set of three formulas describing Moore's circuit is just such a solution. As before, we are merely talking about the same thing in a different language. Markov's list of definitions ends abruptly with a bold statement of his result, followed by a one-page l e m m a r u n n i n g into s u b - s u b s c r i p t e d , s u b - s u p e r scripted variables that "plays an essential role in the proof." The full proof is spared u s - - t h e author doubtless feeling a recapitulation of the obvious would be too t e d i o u s - - a n d so ends his paper. It is just as well: that lemma might have been written in Celtic for all I could make of it (printing errors abound, too). Not that I felt especially over-confident in challenging his mathematics. This was a g o o d instance of w h a t Richard Guy calls proof by intimidation. But what was that result? Following Markov we must be quite precise here. Consider a system of m Boolean functions of n arguments. It can be defined by different systems of m formulas in n variables. Take n o w a worst case instance of such a system of functions in which the number of distinct negative sub-formulas necessary to their definition is at its greatest. Then, says Markov, the least number of negative sub-formulas that will have to appear in the formulas defining the functions will be [log2n [ + 1 = the number of digits in the binary representation of n. (The vertical strokes indicate the truncated value of log2n. ) Thus Ilog2nl + 1, otherwise known as I or the inversion complexity of the system of functions, is indeed the Markovian equivalent to the minimum number of separate inverters that would be required in any network implementation of its formulas. Note that m, the number of functions (or formulas) does not actually enter into it. (Think of all the recoders that can be connected to Moore's 3-bit to parallel decoder, each yielding a n e w output function, none d e m a n d i n g extra negations). We can examine this further by taking Moore's problem as a example. Translating into Boolean terms, our question concerns a system of three functions (x' = Not(x), y' = Not(y), z' = Not(z)), of three arguments (x, y, z). Applying Markov's result we find n = 3, log23 = 1.5849 . . . . h e n c e ! = 1 + 1 = 2. That agrees with our conclusion: two inverters are sufficient. But what about the 4-complement problem? N o w n = 4, log24 = 2, I = 2 + 1 = 3. Three inverters are required. That disagrees with our conclusion. In effect, Theorem I above would assert that I = 2, irrespective of m and n. Here we have a contradiction. The collision here is so acute that something will
have to give way. And so it proves. Forcing the issue to a head, an o b v i o u s step n o w is to p r o d u c e a counter-example to Markov's result by writing out the 4-complement circuit as a system of Boolean formulas, thus demonstrating that only two distinct negative sub-formulas need appear. And indeed, with this comes a breakthrough and the resolution to this whole curious dilemma. The scales, so to speak, are about to fall from our I's. For with an attempt to write out a set of formulas depicting the 4-complement circuit comes the discovery that no Boolean representation of it exists. The barrier to deriving a Boolean representation is revealing. Looking back at the 4-complement block diagram (Figure 2), recall that Network 2, the nested box, is a pure Moore circuit for which we already have a system of formulas. To represent the complete 4complement box, however, we first need to respecify x, y, and z - - t h e inputs to the nested b o x - - i n terms of the n e w set of arguments: a, b, c, and d, the new main inputs. The obstacle to achieving this appears in finding that no expression for y can be derived without y occurring as one of its own arguments! H o w does this come about? It is our old friend the sneaky feedback loop, reminding us that this is no longer a simple combinational circuit like Moore's. Through the nesting of one box in another a primitive form of memory has been introduced whereby it has become a sequential switching circuit w h o s e subsequent internal state depends both upon present inputs and current state: the present value of y plays a part in determining y's n e w value. In short, the 4-complement circuit is really a finite state machine, a device w h o s e context-sensitive action lies b e y o n d the descriptive scope of Boolean formulas. The facile notion that everything can always be "talked about in a different language" is thus not without its pitfalls. Still sneakier (and this really is rather subtle), the 4-complement circuit is a finite state machine mimicking the behaviour of a non-sequential machine, the latter comprising a humble combinational circuit of just four inverters: a' = Not(a), b' = Not(b), c' = Not(c), d' = Not(d). Here we have the peculiar case of a higher or meta-Boolean form of life disguised as a lower, Boolean form. The camouflage is truly effective too, since no experiment conducted on the terminals of the 4-complement (black) box could determine whether it contained Boolean or non-Boolean-representable entrails. (Although c u r i o u s l y - - a n d here is another tricky t w i s t - - t h e non-Boolean circuit is actually composed entirely of Boolean components--AND's, OR's and N O T ' s - - a n indication both of the import of their interconnection pattern and of the source of weakness in the algebra that cannot describe it.) A fine distinction is involved in all this that it is worth being clear about. The familiar (two-element) Boolean function describes a relation or mapping be-
tween one two-valued variable (the value of the function) and others (its arguments). As such it may be expressed or specified in different ways; in a tabulation of corresponding values, for instance. Often we represent it as a Boolean formula, that is to say, as a legal expression in the formalism called Boolean algebra. In that case the dependence of the formula's value on that of its variables will strictly mirror that of the function on its arguments. Moreover, any Boolean function can always be described by a Boolean formula. But that is not to say that it has to be so represented or that a specification or implementation of the function must d e p e n d on some analogous structure or mechanism. The 4-complement finite state machine is an example of an alternative implementation, its effect representable by a' = Not(a), etc., but its internal operation (as embodied in its circuit diagram) having no counterpart in Boolean algebra. The importance of all this is that generalizations about devices that perform Boolean functions are not to be reliably based solely on inferences about Boolean algebra representations of those functions. So it is that the s u p p o s e d discrepancy b e t w e e n Markov's condusion and Theorem I turns out to be illusory. The meticulous definitions at the beginning of his paper are not to be overlooked. As a careful reexamination of the account above will show, the result he proves is explicitly restricted to Boolean functions realized in Boolean formulas. Our concern, on the other hand (if only lately appreciated), has been with Boolean functions realized otherwise. Minsky's implication notwithstanding, Markov's work is simply inapplicable to the case in hand. Like the 4-complement box, the K-complement box need employ no more than two inverters. But at least Ilog2KI + I distinct negative subformulas will be required in any Boolean formulas describing the input-output functions of the latter. No contradiction is implied. E. N. Gilbert's paper, incidentally, which also addresses the minimum inverter requirement question is equivalently restricted, his analysis being confined to loop-free networks. Even so, doesn't a.suspicion linger that the K-complement simulator is in some w a y yielding something for nothing? After all, inverting binary signals is a concrete if trivial operation, analogous to flipping over coins so as to make heads from tails or tails from heads. In the end, just h o w is it that K such reversals can be effected given only two reversing-machines? The answer is simple. It is done by using those machines more than once. Through reiterated application we can achieve serially the same result as K single-action machines working in parallel. But at a price, to be sure. Here is how John E. Savage puts it in The Complexity of Computing [4]: "Sequential machines compute logic functions, just as do logic circuits. However, since sequential machines use their memories to reuse THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
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their logic circuitry, they can realize functions with less circuitry than a no-memory machine but at the expense of time" [my italics]. As we saw earlier, hardw a r e - i m p l e m e n t e d logic introduces lag. As K increases, so will the number of passes through feedback paths in the nested circuitry, and the longer final outputs will take in responding to changing input patterns. In practice this would be a serious factor to consider.
I t is amusing to recall exultation on completion of o n e of the most futile or, at least, red u n d a n t items of electronic apparatus ever constructed! Lastly, note h o w Savage casts incidental light on the reason w h y a single i n v e r t e r - - h o w e v e r combined with AND's and O R ' s - - i s inadequate for simulating further negators. Negation of externally presented bits on one channel will always require one inverter. But at least a second will be d e m a n d e d in creating the memory needed in re-utilizing that first. In fact, as we have seen, two inverters are both necessary and sufficient. Simple but hard-won insights are compressed into the foregoing paragraphs. Having gained clearer understanding, a letter to the author whose casual remarks unwittingly triggered this improbable detective story seemed not inapposite. I was gratified thus w h e n , in a s u b s e q u e n t c o m m u n i c a t i o n , Marvin Minsky warmly concurred in the above analysis, graciously conceding a too hasty perusal of Gilbert's and Markov's articles. Likewise, his "To what extent can this result be applied to itself?" turned out to be a mere chance form of words, no reference to recursion i n t e n d e d , b u t r e s o n a n t to me u n d e r the circumstances. Thus were the K-nots disentangled from a Markov chain of deduction, and the sufficiency of two negators in producing moore inverters ad libitum confirmed.
Conclusion The inception of this narrative was a puzzle appearing in Abacus. As a matter of fact, the question there posed came in two, supposedly equivalent versions: Moore's original problem in circuit design and a seemingly analogous problem in computer programming. In the latter form we are asked to "write an [assembly language] program that will store in locations U, V, W the l's complement of locations x, y, and z. You can use as many COPY, OR and AND instructions as you like, but you cannot use more than two COMP (l's complement) instructions." This second version is absent from Wos et al's Automated Reasoning [6], ap30
THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
Figure 5. The 4-complement circuit in Pascal.
pearing only subsequently in his synoptic Abacus article. The trouble is, although purportedly isomorphic with the network puzzle, the conditions now imposed are actually more restrictive than Moore's, a whole class of solutions becoming inadvertently excluded. What is it that makes the program version different? In effect it is a silent prohibition against certain kinds of perfectly valid circuit configurations: a ruling out of the use of feedback loops implicit in the preclusion of a JUMP instruction. Self-modifying functions would be excluded from representation in software. That is, for every program solution there would be an equivalent circuit, but not vice versa. Just as sequential networks defy description in the notation of Boolean algebra, so loops in any circuit solution will defeat implementation in such a program. The slip is an easy one to make, and especially so w h e n Moore's o w n published circuit uses no feedback. Perhaps it was familiarity with this that unconciously acted to restrict Wos's contemplation to combinational type solutions only. Let us make no mistake however: discarding one channel from the 4-complement simulator would leave a three-channel device answering all the demands of Moore's problem. Here we have a finite state machine solution (one of an infinity) that cannot be represented in the above reduced instruction code. I suspect that in the urge to translate Moore's problem into terms suited to his automated reasoning program, Larry Wos temporarily underestimates and thus misrepresents the complexity and potential of networks using AND's, OR's and NOT's. [In p a s s i n g - - a n d without any reference to the aforementioned a u t h o r - - t h e tendency to see circuit diagrams as engineers' easy-to-read-picture-book-explications of "real mathematics" envisioned in putative formulas is not uncommon among mathematicians. Engineers, I may add, humble as their mental endowment may be, will be more impressed when condescension can be matched with insight into the advantages of a two-dimensional language.] Whether or not the automated reasoning technique could be successfully applied to the 4-complement problem is a further interesting question. Following the lead suggested here, the 4-complement circuit is elegantly modelled in a simple computer program using iteration to imitate the feedback loop. A series of assignment statements based on the earlier derived formulas describing Moore's circuit make up the body of the program. Figure 5 shows a version written in Turbo Pascal. Read in conjunction with Moore's circuit and Figure 2, the program is selfexplanatory: more eloquent in fact than any verbal commentary on circuit operation. Interested readers
THE MATHEMATICAL INTELL1GENCER VOL. 12, NO. 1, 1990
31
may like to try the effect of including a Write statement in the Repeat loop so as to expose the behaviour of y under different input sequences. A final observation on Markov's result must bring this account to a close. Figure 3 depicted the endlessly expandable system of recursively nested Moore circuits for producing an arbitrary number of NOT's. Winning three NOT's from two, every level of nesting yields a spare inverting channel. In practice, however, the mass-production of NOT-functions can be enormously accelerated. How? Notice that Markov's I is still only 3 for n as high as 7. But this is another way of saying that a simple combinational circuit exists that can simulate seven inverters directly from three. Similarly, from these seven a further 127 can be produced at only the third level of nesting: 2 --~ 3 --* 7 --~ 127 -~ . . . . Readers may like to test their grasp of the foregoing by writing a program that implements seven inversions while using only two Not operators. In conclusion, and before any false hopes are raised though, I ought to say that the above suggestion is intended merely as an exercise. Patents, it must be explained, have already been granted and the Sal-Mar International Inverter Hire Company, Inc., is due for launching at an early date. Prompt negations of the highest quality will be available to customers via standard phone lines. Charges are expected to be modest. In the meantime--call me an adulterated Platonist if you w i l l - - I presume that up in heaven two eternal NOT's await the arrival of virtuous logicians (and the occasional virtuous engineer). I look forward to rubbing that in with G6del and Russell hereafter.
As an engineer I felt only sheer surprise that, in principle, all the millions of inverters in use throughout the world could be "'seeded" from a single pair.
[Grateful thanks are due to Jim Propp, formerly of the Department of Mathematics, University of Maryland, whose searching criticisms brought to light various errors and made for substantial improvements to an earlier draft of this paper.]
References 1. B. Russell, The Autobiography of Bertrand Russell, Unwin (1978), p. 466. 2. C. E. Shannon, A symbolic analysis of relay and switching circuits, Trans. AIEE 57 (1938), 713-723. 3. G. A. Montgomerie, Sketch for an algebra of relay and contractor circuits, Jour. IEE 95, Part III (1948), 303-312. 4. J. E. Savage, The Complexity of Computing, Wiley (1976). 5. L. Wos, A Computer Science Reader, Ed. E. A. Weiss, Springer-Verlag, (1988), pp. 110-137. Orig. pub. Abacus, vol. 2, no. 3 (Spring 1985), pp. 6-21. 6. L. Wos, R. Overbeek, E. Lusk & J. Boyle, Automated Reasoning, Introduction and Applications, Prentice-Hall (1984). 7. M. Minsky, Computation: Finite and Infinite Machines, Prentice-Hall (1967), p. 65. 8. E. N. Gilbert, Lattice theoretic properties of frontal switching functions, Jour. Math. & Physics 33 (April 1954), 57-67. 9. A. A. Markov, "On the inversion complexity of a system of functions" (translated by M. D. Friedman), JACM 5 (1958), 331-334. Orig. pub. Doklady Akad. Nauk SSSR 116 (1957), 917-919. Buurmansweg 30 6525 RW Nijmegen The Netherlands 32
THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
Karen V. H. Parshall*
With this issue, the Years Ago department changes its editor but not its basic editorial policy. It will continue to chronicle major anniversaries (25th, 50th, 75th, lOOth, etc.) of important and interesting mathematical events. In a departure from the past, however, the editor will solicit a number of contributions each year for publication in the column. Readers are welcomed and encouraged to send proposals, suggestions, comments, and criticisms to me.
It is with great sadness that I report the death of my predecessor in this job, Allen Shields. I had hoped to take this opportunity to thank him for his efforts in bringing the history of mathematics before a wider audience on the pages of the Mathematical Intelligencer. I shall try to uphold the example he set. Karen V. H. Parshall
A Voyage with Three Balloons R.-P. Holzapfel Consider three examples of plane curves y = p2(x), y2 = p3(x), y3 = p4(x), where pk(x) denotes a polynomial of degree k. These simple curves are closely connected with the creation of mathematical theories. I want to use them as balloons for trips over old and new mathematical landscapes.
in the second half of the last century. The first equation describes a parabola. Much stimulation for the s t u d y of q u a d r a t i c s came f r o m physics: optics, acoustics, gravitation, etc. We switch from real geometry to complex analytic function theory by considering the integral
f
dz - I dz p~z) (z - a)(z - b)
The First Voyage The first of our curves will not bring us to recent developments. It should be used for training in navigation. Starting from antique shores we will reach Berlin
along a path in the complex z-plane. The integral can be calculated from the antiderivative ln[(z - a)/(z - b)]/ (a - b) of 1/p2(z) (a ~ b). So the problem is reduced to the study of the elementary function w(z) = In(z).
* Column Editor's address: D e p a r t m e n t s of Mathematics and History, University of Virginia, Charlottesville, VA 22903 USA.
THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 1 9 1990Springer-VerlagNew York 33
Looking around, we have reached the 18th century, w h e n elementary complex analytic function theory was b o r n a n d m a t h e m a t i c s b e c a m e an organized science with its o w n inner laws of development. The new era was opened by Isaac Newton and Gottfried Leibniz; L e o n h a r d Euler initiated f u r t h e r major changes in viewpoint. Understanding In(z) as a complex analytic function presented some difficulties. J. Bernoulli believed that I n ( - r) = In(r) for real numbers r # 0, whereas Leibniz thought that I n ( - r ) was an imaginary number for r > 0. In a correspondence of 1712/1713, each of these mathematicians found good arguments against the opinion of the other. The situation was clarified by Euler: In(z) is a multivalued function. We will symbolize multivalued functions by means of several arrows. For example, we write In: C* --~ C, C* = C \ {0}, C the complex numbers. The periodic inverse function z = exp(w) = ew clarifies the situation completely. It is described in the following inversion diagram: In / C*
Z~ex p
(1)
.C
The symbol 27 means that exp is also understood as the quotient map C --* C/Z = C* C C with the Z-action w --*w + m.2~ri, m E Z, o n C . Our route leads us n o w to special values of the elementary function exp. The values at the torsion points of the additive group C/77 are the roots of unity ~ = e 2~rim/n, m,n E Z, n ~ O. The theory of cyclotomic fields Q(~) is a cradle of algebraic number theory. We remember that Karl Friedrich Gauss reduced constructions of regular n-gons to calculations in these fields called "Kreisteilungsk6rper." His successful synthesis of geometry, algebra, and number theory, especially the construction of the 17-gon, led him to study mathematics rather than languages. The important role of cyclotomic fields in the theory of algebraic numbers is expressed in the completeness theorem of KroneckerWeber: Each absolutely abelian number field (finite abelian Galois extension of Q) is a subfield of a cyclotomic field. N o w our first trip is finished. We will come back to Leopold Kronecker later. The
Second
To me, there is little interest in that aspect of integral calculus where we use substitutions, transformations, etc.-merely clever mechanical tricks--in order to reduce integrals to algebraic, logarithmic, or trigonometric forms, as compared with the deeper study of those transcendental functions that cannot be so reduced. We are as familiar with circular and logarithmic functions as with one times one, but the magnificent goldmine that contains the secrets of higher functions is still almost completely terra incognita. I have, formerly, done much work in this area and intend to devote a substantial treatise to it, of which I have given a glimpse in my Disquiss. Arithmeficae p. 593, Art. 335. One cannot help but be astounded at the great richness of the new and extremely interesting relations that these functions exhibit (the functions associated with rectification of the ellipse and hyperbola being included among them). G a u s s on his o w n could not carry out the programme mentioned in the letter. A central theme in the creation of the fundamentals of analytic function theory during the last century was the theory of hypergeometric functions. Gauss was the first to consider hypergeometric series as functions of a complex variable. In this connection he introduced and investigated the circle of convergence (1813). Instead of a general definition of hypergeometric functions, we present a simple example coming from a variation of elliptic curves:
Voyage
The second balloon will carry us to new heights. If p3(x) has three different zeros, then the curve E defined by y2 = p3(x) is an elliptic curve. Elliptic curves arise from the rotation of a solid body around a fixed point. Karl Jacobi proved in Crelle's Journal 39 (1850) that for important special cases already investigated by Euler, Louis Poinsot, Ren6 Francois Lagrange, and Sim6on Poisson, the rotation can be described by 34
means of elliptic integrals t(s) = f~o dz/PV'~)k(z), k = 3,4. Knowledge of the function s(t), t the time coordinate, is the key to an explicit description of the motion. Periodicity of the rotation forces t(s) to be a multivalued function and s(t) to be periodic. Indeed, Niels Abel was the first to discover the double periodicity of the inverse meromorphic function s, called an elliptic function. Independently Jacobi came to the same result. Later he presented elliptic functions as quotients of theta-functions. Special cases of these quasiperiodic holomorphic functions had already appeared in the work of J. Bernoulli, Euler, and Jean Baptiste Fourier. Fourier used them to solve the heat equation (1826). Karl Weierstra~ later erected his pyramid of the theory of abelian functions of one, two, and several variables. However, with our second balloon we will not fly to higher dimensions. We are more interested in non-euclidean arithmetic and function theory. Coming from the arithmetic and geometric side, Gauss was led to a new point of view. In an 1808 letter to Heinrich Schumacher he wrote:
THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
k2 - k y = F(t) = 1 + ~~ ( . 1 . 3 . 5 - . .2. ( 2 It!
1)) "t k
(2)
satisfies the special hypergeometric differential equation 1 - 2t y"+ - - y ' t(1 - t)
1 4t(1 -
y = 0. t)
(3)
value of x, and conversely many values of x correspond to a given value of y, then one can ask whether one cannot express both y and x as single valued functions of a third variable t. A case in point is the example of elliptic functions, in which we can express, not merely the algebraic functions on a Riemann surface, but also the integrals of the first and second kinds, as single-valued functions of a third ('uniformizing') variable, namely the integral of the first kind. We can say, therefore: It is a general problem of analysis, in the discussion of analytic relationships, always to find a 'uniformizing parameter'. We look for a description of the inverse map of 4. It turns out that the triangles are fundamental domains of a lattice subgroup F of the group of biholomorphic automorphisms of the disc D. By means of a projective coordinate change we can assume that D C pl is the upper half-plane H = {z E C: Imz > 0} C pl with the automorphism group SL2(~). Then F is the congruence subgroup of SL2(77) defined by the exact sequence 1 --* F ~ SL2(7/) ~ SL2(Z/27/) ---) 1. The inversion diagram for ~ = yl/Y2 is"
Figure 1. Tesselation of the disc.
Up to a constant factor the hypergeometric function F can also be presented as a special hypergeometric integral dz
(4)
X/z(z - 1)(tz - 1)' where b,c ~ {0,1,~} and t ~ {0,1,~}. Think of F as a multivalued function of t on C\{0,1} = P1\{0,1,oo} (here pl denotes the Riemann sphere C U {~}). Following an idea of Georg Friedrich Riemann, Herm a n n Amandus Schwarz gave a nice description of the multivalence in 1873. He considered two independent solutions Yl,Y2 of (3) and their quotient ~ = yl/y2, which maps C \ ~ + (~+ = {r ~ ~: r I> 0}) onto an open triangle in a disc. By analytic continuation across the real half-line R +, the values of ~ fill the disc. The disc is tesselated by infinitely m a n y triangles as shown in Figure 1. The triangles are congruent from a non-euclidean point of view, making evident an important connection between analysis and non-euclidean geometry. We continue with a quotation from Felix Klein's lect u r e s on h y p e r g e o m e t r i c f u n c t i o n s ( G 6 t t i n g e n 1893/94): Now we want to apply our triangle figures in another direction, by asking: When is the (generally multi-valued) function ~(x) uniquely invertible, that is, when is x a single-valued function of ~? This is a special case of a quite general problem in function theory, arising from the desire to avoid operations with multivalued functions as far as possible. In general, if one has a functional dependence y(x) in which many, perhaps infinitely many, values of y correspond to a given
~////i~
ENN= El/E2
(5)
P~\{O,I,o~}--~ P~\{3 points} C P~ where e is a F-automorphic function. By analogy with elliptic and theta-functions, ~ can be realized as a quotient of two holomorphic quasi-periodic functions, more precisely, as a quotient of two F-automorphic forms of weight 1. Up to a cusp condition, F-automorphic forms of weight m are defined as holomorphic functions f on H satisfying the functional equations f(~(z)) " j~m(z) = f(z), ~/ e r, Jr = d~//dz.
(6)
From the algebraic point of view the forms E1,E2 appear as generators of the ring of F-automorphic forms of arbitrary weight. H. A. Schwarz's article is not restricted to our example involving modular forms. He also determined all h y p e r g e o m e t r i c f u n c t i o n s p r o d u c i n g triangle groups. Later the theory was greatly extended by F. Klein and Henri Poincar& Poincar4 started from a class of second order differential equations containing (3) as a special case, called Fuchsian equations. Looking not only to G6ttingen but also to Berlin, he brought into being a non-euclidean analogue of Weierstrass's elliptic function theory. We now turn to Berlin to understand Kronecker's "liebsten Jugendtraum." In some mysterious fashion, Kronecker saw the possibility of proving an analogue of the completeness theorem for absolutely abelian n u m b e r fields: The abelian extensions of imaginary quadratic n u m b e r fields are contained in n u m b e r fields generated by special values of elliptic functions THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990 3 5
and of the modular function j (Hauptmodul in Dedekind's terminology). This idea is the cradle of class field theory, the heart of algebraic n u m b e r theory. The precise theorem was proved by Tagaki in 1920. The special values on the elliptic curves are torsion points. The special values for the modular function are "singular m o d u l e s " ("singu1/ire M o d u l n " ) . We w a n t to u n d e r s t a n d t h e m as a non-euclidean version of torsion points.
the soul of our science. A last side-glance indicates the intimate interaction of arithmetic and analysis: The Eisenstein series oo
cr3(n)e2~rin"
E 2 = 1 + 240 ~ n=l oo
E3 = 1 - 504 ~ , r
2~in"
r
= ~
n=1
a k (7)
din
are the normalized Fourier series of g2('r) or g3(~-), respectively.
In some mysterious fashion, Kronecker saw the possibility of proving an analogue of the completeness theorem for absolutely abelian The Third Voyage number fields. First we look at the moduli space of elliptic curves, which parametrizes all isomorphism classes of elliptic curves. In (4) the variable t appears as a parameter for elliptic curves. With a glance at the diagram (5), we define the analytic family {E~},dn of elliptic curves in Weierstrass's normal form: E,:
y3 = 4(x - ex(~))(x - e2(T))(x - E3(~)) = 4x3 - g2(T)x - g3(T).
We choose ~1,e2,E3, with e 3 ~- - - ~1 - - ~ 2 , i n a symmetric m a n n e r with r e s p e c t to the s y m m e t r i c g r o u p S3 SL2(77)/['. The moduli curve is H/SL2(7/) = ( H / [ ' ) / S 3 pl\{oo}. The values of the modular function j(~) = 1728 ~(T)/(g23(~) - 27g2(-r)) are in one-to-one corres p o n d e n c e with all i s o m o r p h i s m classes of elliptic Curves.
In order to explain "singular moduli" w e look back to the exponential map. The pre-images ~ on C of torsion points of C/2~ri77 are characterized b y the two equivalent conditions:
The third balloon will bring us to higher dimensions. We start at the end of the last century in Paris, w h e r e w e find an immediate and important continuation of Weierstrass's w o r k in the hands and heads of H. Poincar6, Charles Emile Picard, and Paul-Emile Appell. We discover the first fine analysis of the family of curves {y3 = p4(x)} in a paper by Picard in Acta Mathematica (1883). This family appears in the hyperfuchsian integrals I(tl,t2) =
3Vz(z -
1)(z
-
6)(z - h ) "
w h e r e b,c E {0, 1, oo}, (tl, t2) E T** C C 2, T** = {(tl, t2) C2:tl, t 2 ~ {0, 1), t~ ~a t2}. The integrals satisfy the Euler partial differential equation 32F
a +
c~tlc~t2
-
a
-
t 2 --
OF
+ t2
-tl
--
0, a
=
1/3.
t2 3t2
This equation can be completed to a system of three partial differential equations w h o s e solutions are the multivalued functions I(t 1, t2) and their linear combinations (see Euler-Picard system below). The solution space has three dimensions at each point P0 = ( t~ to) (i) q(r -----0 m o d 2~ri, T**. Three of the above integrals 11, I2, I3 form a basis of (ii) (q + 1)or -= cr mod 2~ri, q e Q* ~ GLI(Q ). this space in a neighbourhood of P0. Starting at P0 w e can develop (/1,/2,/3) along paths cr in T** by analytic The torsion condition (i) is translated to the fixed point continuation. Passing through a cycle a each Ij, j = 1, condition (ii). Fixed point conditions make sense also 2, 3, changes to a linear combination of 11,/2,/3. In this for spaces w i t h o u t additive structure. The fixed points w a y one obtains a m o n o d r o m y representation of the q" of elements of GL~(Q) = {g E GL2(Q): det g > 0} fundamental group ~1 (T**, P0) in C 3. acting non-trivially on H are the "singular moduli." It Using five generators Picard found a unitary monoturns out that k, = Q(r is an imaginary quadratic d r o m y r e p r e s e n t a t i o n ~rl(T** ) --~ U((2, 1), C). The n u m b e r field, k = k, a p p e a r s a g a i n as the field group U((2, 1), C) consists of all elements of GL3(C ) Q (~) End(E,) of "complex multiplication" of E,. These respecting the hermitean metric lul2 + Ivl2 - Iwl2 on curves go with the child called class field theory. The C 3. In Picard's notation the five generators are called algebraic n u m b e r j(~) generates the maximal unrami- fundamental substitutions. Some years later Poincar6 defled abelian field extension of k, which is the absolute fined general f u n d a m e n t a l g r o u p s in his w o r k on class field of k. Analysis situs, which was the foundation-stone of algeThe theory of elliptic curves is a huge field of clas- braic topology. sical and present-day research, but w e will stop our N o w U((2, 1), C) acts on the complex unit ball B = second trip here. We w a n t to stay on the intersection {(u, v):lu[2 + Ivl2 < 1} in C 2. Picard was aware of the line of several branches of mathematics because this is central role of his model for further investigations. 36
THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
Figure 2. The ball B is projected onto the punctured complex branched along the six lines with ramification index 3. The projective plane T* = P2\{P1, P2, Pa, P4}- The four omitted S4-quotient of the complement T** of the six lines in p2 is the points are images of four cusps k1, k2, k3, k4 (and their F- moduli space of smooth Picard curves. The values of the orbits). Six discs in B (and their F-orbits) are projected on multivalued integral map (I1:12:13)fill the complement of the the six lines through the points P1, P2, P3, P4. The mapping is F-orbits of the six discs in B. Looking for unitary modular functions he remarked: "'Our functions of u, v p l a y . . , the same role as the modular function in the theory of elliptic functions." In the monograph "Th60rie de courbes et de surfaces alg6briques" Picard later referred back to his old work: "We have sought, Mr. Alezais and I, to prepare the ground for such i n v e s t i g a t i o n s . . , by doing some preliminary arithmetic studies." The situation can be clarified by modern methods. The main points are concentrated in the following inversion diagram: (I1 : I 2:I3) _,, B.
(el: ~2: e3: %)
(8) T** ~ T *
L \ P 2 \ { 4 points}.
11, 12, 13 are three linearly independent solutions of our Euler-Picard system. F is the congruence subgroup defined by the exact sequence 1 -+ r -+ U((2,1),9 ~ U((2,1),O/(1 - o~)O) --+ 1,
where 00 = e2"u3 and 9 = Z + 77o~is the ring of Eisenstein integers, e 1, e2, e2, e 4 satisfy el + (~2 + E3 q- ~4 = 0 and are fundamental F-automorphic forms (of Nebentypus). The normal forms 4
C(u,v):v 3
=
1-I (x
-
=
i=1
x 4 + G2(u,v)x 2 + G3(u,v)x + G4(u,v),(u,v ) E •,
define an analytic version of Picard's curve family. The moduli space for Picard curves is B/U((2, 1), O) = (B/F)/S 4 = (P2/$4)\{1 point} (compare with Figure 2.) Singular moduli are defined as isolated fixed points on B of elements of the group U((2, 1), Q(o~)) respecting the hermitean (2, 1)-metric on C 3 up to a constant factor. N o w it again happens, as in Kronecker's case, that the generating U((2, 1), 9 functions take values in algebraic number fields at the singular moduli. The first proof working for a class of these points was presented by H. Shiga [6] in 1986. He lifted Helmut Hasse's analytic proof for Kronecker's THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 1, 1990
37
case to the second dimension. The modular forms ~i are p r e s e n t e d explicitly as theta-constants. Their Fourier series look like those of E2, E3 in (7) but with explicit theta-functions as coefficients instead of crk(n). The intimate interaction of arithmetic, complex analysis, and n o n - e u c l i d e a n g e o m e t r y is obvious. It sounds surprising, but all the results can be obtained by means of algebraic geometry. Moreover, algebraic geometry plays a double role. On the one hand it The power
of algebraic geometry lies in the combination of technique with a new kind of intuition.
compensates for the loss of immediate intuition in classical non-euclidean geometry (see for example Figure 1). On the other hand the fine techniques of algebraic geometry remove all obstructions to a proof. The power of algebraic geometry lies in the combination of technique with a new kind of intuition. David Hilbert proclaimed a very general continuation of K r o n e c k e r ' s " J u g e n d t r a u m . " In the 12th problem of his Paris lecture (1900) Hilbert designated this extension as "one of the deepest and most farreaching problems of number theory and function theory." A more concrete continuation can be found in Erich Hecke's work on Hilbert modular functions. From his thesis we quote: "The most general thetafunctions of this kind lead directly to the functions of Hilbert and Picard." Now, for Picard curves the "Jugendtraum" climbs up towers of surfaces constructed recently by Hirzebruch, which we could identify as towers of Picard modular surfaces ([2], [4]). Keeping in mind the non-euclidean nature of the moduli space of Picard curves coming from the ball B, we turn back for a moment to the classical origins of Euler's equations O2F a OF a OF -+ --.-+ . . . . OrOs s - r Or r - s as
0, F = F(r,s).
(9)
There are only a few hints at Euler's work on these equations in connection with acoustics. One can be found in the "Festschrift zur Feier des 200. Geburtstages Leonhard Eulers'" (Teubner, 1907). In his treatise "Ueber die Fortpflanzung ebener Luftwellen von endlicher S c h w i n g u n g s w e i t e " (Abh. d. Kgl. Ges. d. Wiss. G6ttingen, 8, 1860) Riemann proved, without any reference to Euler's work, that the equations (9) govern the propagation of gas waves. We quote from the introduction to his paper: "Although the mathematical treatment up to this point has proved to be sufficient to explain the experimental phenomena observed so far, it m a y well be j u s t i f i e d - - w h e n one looks at the great advances made recently by Helmholtz in the e x p e r i m e n t a l t r e a t m e n t of acoustic 38 THE MATHEMATICALINTELL1GENCERVOL. 12, NO. 1, 1990
problems--to hope that the results of this more precise study may, in the not too distant future, provide some starting points for this n e w experimental research. This hope, apart from the intrinsic theoretical interest of treating non-linear partial differential equations, may justify the present communication." Riemann constructed two auxiliary functions r, s, starting from the density and velocity of the gas or its particles, respectively. Then there follows a remarkable inversion step. He switched the roles of the pairs (r, s) and (6, ~') where 6, ~"are the variables of place and time. That means that 6, ,r are locally understood as functions of r, s. The solution F = F(r, s) of an Euler equation provides multivalued functions 6, 7, solving the problem up to inversion. Mathematically, the situation is similar to the rotation of a solid body when time is expressed by an elliptic integral. Picard's curve family involves special Euler partial differential equations together with solutions. The study of papers by Picard, Appell, and related authors (modem treatments are [3] and [8]) induces the feeling of a certain duality between the Euler equations and n-th root integrals J'~t dx/"V'x(x - 1)(x - tl) . . . (x - tr) taken along cycles on the cycloelliptic curves (Serge Lang called them "superelliptic") Ct: y" = Pr+2(X) = Pr+2(x;t), t = (t 1..... tr) ~ Y'** C
Pr(C).
The connecting objects are algebraic curve families "--> V**, ~ = ~ ( r , y/) = {Ct} t C Z * * . This duality can be made precise in the most obvious manner for n ~> 2, r /> 1, g.c.d.(n, r + 2) = l, 0 < l < n, completing systems of Euler equations to Euler-Picard systems: --+ OtiOtj
-
F=O,
n(tj
ti)
i O2F
I 1 R 2, parameterized by arclength, with
tion, so that it can be used to guide robots, and it seems reasonable to suspect that large scale applications of this sort will be some time in coming. Still, there is an enormous amount of interest in these questions in the research engineering community, and I believe that sensor-guided robot control will, at some point in the future, become commonplace in industrial applications. The analysis given in the last section was predicated on the assumption that the robot mechanism under consideration admits a forward kinematics map, and not all possible mechanisms satisfy this condition.
10(s)[ ~< K and 0(0) = 0. The curvature b o u n d is necessary in order to prevent the mechanism from breaking if it is folded too sharply onto itself. A lot of engineers have enjoyed playing with this idea, and proposals for building such manipulators (or approximations to them) continue to crop up at regular intervals. It is my understanding that one of the earlier attempts, called the " O r m , " was built around 1968 at Stanford University, but work on it was eventually abandoned because of the problems involved in controlling it. (The Orm was, in fact, three-dimensional.)
r
Jv I
Figure 11. The maximum radius r that can be reached by a planar rope mechanism while a tangent vector at a point on the rope is pointing in the direction opposite to the position vector. No inverse function can be defined on a disk of radius greater than r centered at the base of the mechanism. (Compare with Figure 10).
THEOREM 2.5: (see [10]) If a planar rope manipulator has total length L, a kinematic inverse function for it cannot be defined on the disk of radius r if r > L - 2"tr/K. Figure 11 shows the curlicue configuration that the manipulator will have to make at some point on the boundary of any disk on which an inverse function is defined. Note that the tangent vector at the apex of the loop points in the opposite direction of the vector from the base to the endpoint of the manipulator, in analogy to Theorem 2.4.
What Is the Significance of All This? It is natural to ask what impact these results will have on the robotics industry, and, without trying to give a definitive answer to this question, I would like to offer up a context in which it can be considered, and then indulge in some conjectures about what the future may hold. A l t h o u g h our list of solution m e t h o d s to local problems is by no means complete, it does leave the (correct) impression that this area has not yet reached the stage of a marketable technology. In fact, there are still many very difficult and unsolved problems involving the real-time processing of sensor informa-
Figure 12. A Stewart Platform, an example of a closed link mechanism. The platform is attached to a fixed base via six telescopic legs, which can be viewed as translational joints. By varying the lengths of the six legs, both the position and the orientation of the coordinate frame attached to the platform can be changed. Since, for any position and orientation of the platform, the leg-lengths are uniquely determined, an inverse kinematics map is well defined. It is not, however, one-to-one, because every position and orientation of the platform can be reflected through the plane of the base without changing leg-lengths. THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990 7 5
This condition will, however, be satisfied by all seriallink m e c h a n i s m s , i.e., ones w h o s e m e c h a n i c a l linkages contain no closed chains, and the large majority of robots being b u t t at this time are either seriallink mechanisms or can be modeled as such for the purposes of path-tracking problems. The design of robot mechanisms that are not serial-link is an area of active research, but it entails a number of problems b e y o n d the scope of this article, which must be resolved before such designs become commercially viable. Figure 12 shows an example of one such mechanism. It consists of a base and a movable platform, connected by six translational joints, or legs. As the lengths of these legs vary, the platform changes both its position and its orientation in R 3. The problem of motion control for such a mechanism takes on a very different character from the problems examined in the previous section. For example, as reported in [5], this nonredundant mechanism comes with an inverse kinematics map g : R 3 x S0(3) -* j6
verse functions pose hard implementation questions, and most engineers will probably shy away from them unless concrete help is available in dealing with these issues. It should, however, be noted that constructing inverse functions is basically an off-line job, so that we can afford to be generous in the computer resources we use. Only the later evaluation of the inverse function, as paths are being tracked, must be done on-line. I believe that the future of inverse functions in rob o t i c s d e p e n d s largely on t h e d e v e l o p m e n t of methods for constructing them. If such methods are not forthcoming, or if they turn out to be too painful in their implementation, it seems reasonable to assume that the engineering community will focus its attention on the d e v e l o p m e n t of non-cyclic tracking methods, ways to live with the non-cyclic behavior, or on just accepting singularities where they arise and learning h o w to work around them.
instead of a forward kinematics map. The function g also has singularities, but in singular configurations infinitesimal motions in the operational space are now possible, while the leg-lengths are held fixed. Note also that many paths in the operational space will result in leg collisions, so that one does not really have access to the entire operational space, even if the singularities are ignored. The reader is referred to [5] for further details. For the class of robot mechanisms admitting forward kinematics maps, the results that we have discussed seem to be mostly of a clarifying sort; they make some very precise statements about what cannot be accomplished, and what trade-offs must be dealt with. Specifically, it has been s h o w n that cyclic tracking algorithms are equivalent to ones determined by inverse functions, and we have seen a number of situtions in which inverse functions cannot be defined on the whole operational space. These statements serve only as a background for the real engineering work, which still remains. It would be useful to have a repertoire of methods for determining whether or not inverse functions can be defined on various subsets of the operational spaces of different robots, but, even more important, the engineer needs a collection of methods by which inverse functions can actually be constructed in specific situations. These inverse functions must ultimately be given in a form that makes them easy for a computer to evaluate in real time. Furthermore, the problem of optimizing the inverse function to obtain some sort of minimization of the ratios of joint speeds to speeds in operational space must be dealt with, and this problem is clearly no longer topological in nature. In contrast with many of the non-cyclic tracking methods which have been proposed, in-
1. John Baillieul, Kinematic programming alternatives for redundant manipulators, Proc. of IEEE International Conference on Robotics and Automation, March 25-28, St. Louis, MO (1985), 722-728. 2. Daniel R. Baker and Charles W. Wampler, On the inverse kinematics of redundant manipulators, International Journal of Robotics Research 7 (2) (1988), 3-21. 3. Daniel R. Baker and Charles W. Wampler, Some facts concerning the inverse kinematics of redundant manipulators, Proc. of IEEE International Conference on Robotics and Automation, March 31-April 3, Raleigh, NC (1987), Vol. 2, 604-609. 4. J. E. Bobrow, S. Dubowsky, and J. S. Gibson, On the optimal control of robotic manipulators with actuator constraints, Proc. of Automatic Control Conference, June 22-24, San Francisco, CA (1983), Vol. 3, 782-787. 5. E. F. Fichter, A Stewart platform-based manipulator: general theory and practical construction, International Journal of Robotics 5 (2) (1986), 157-182. 6. Daniel H. Gottlieb, Topology and robots, Proc. of IEEE International Conference on Robotics and Automation, April 7-10, San Francisco, CA (1986), Vol. 3, 1689-1691. 7. C. A. Klein and C.-H. Huang, Review of pseudoinverse control for use with kinematically redundant manipulators, IEEE Trans. on Sys., Man, and Cybernetics, Vol. SMC-13, No. 3 (1983), 245-250. 8. Ki C. Suh and John M. Hollerbach, Local versus global torque optimization of redundant manipulators, Proc. of IEEE International Conference on Robotics and Automation, March 31-April 3, Raleigh, NC (1987), Vol. 2, 619-624. 9. Charles W. Wampler, Inverse kinematic functions for redundant manipulators, Proc. of IEEE International Conference on Robotics and Automation, March 31-April 3, Raleigh, NC (1987), Vol. 2, 610-617. 10. Charles W. Wampler, Winding number analysis of invertible workspaces for redundant manipulators, Proc. of 26th IEEE Conference on Decision and Control, Dec. 9-11, Los Angeles, CA (1987), Vol. 1, 564-569.
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THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
References
Mathematics Department General Motors Research Laboratories Warren, MI 48090 USA
Chandler Davis*
The Poetry of Wallace Stevens (Excerpted from Holden's article "Poetry and Mathematics," The Georgia Review 34 (1985), 770-783.)
Comments by Jonathan Holden 9. . And how seriously can analogies between poems and "certain pages of algebra" be drawn? Are such analogies fun but trivial? To get at answers to these questions, let us consider some of the poetry of Wallace Stevens. Stevens was by far the most mathematically sophisticated of recent American poets. His poems regularly allude to mathematical ideas, affectionately imitate mathematical demonstrations, and apply language "'mathematically" to the world. The most obviously mathematical poem of Stevens is Anecdote of the Jar:
Here, the jar is the origin of a Cartesian coordinate system imposed upon the "wilderness" of a physical world u n m a p p e d in human terms. 2 Stevens is even careful to propose a vertical z-coordinate: the jar was "tall and of a port in air." And he is careful to remind us that the terms being imposed upon this "wilderness" are, like lines and points, wholly imaginary, wholly ideal, that this "jar" was "the only thing" in Tennessee which "did not give of bird or bush." Anecdote of the Jar doesn't actually set out to measure anything in particular. It is about the conditions for measurement of "wilderness." Measurement is done, Stevens tells us, by imposing u p o n the world constructions of the imagination, ideal structures, terms which can only be sustained through something akin to Coleridge's "willing suspension of disbelief that is poetic faith."
I placed a jar in Tennessee, And round it was, upon a hill. It made the slovenly wilderness Surround that hill.
Department of English Kansas State University Manhattan, KS 66506 USA
The wilderness rose up to it, And sprawled around, no longer wild. The jar was round upon the ground And tall and of a port in air.
Advanced Calculus of Murder by Erik Rosenthal New York: St. Martin's Press, 1988, 263 pp.
Reviewed by Mary W. Gray It took dominion everywhere. The jar was gray and bare. It did not give of bird or bush, Like nothing else in Tennessee. 1
If I w a n t conversation on a plane, I a n s w e r m y neighbor's inquiry of "What do you do?" with "I'm a lawyer.'" On the other hand, if I want peace and quiet
* Column editor's address: Mathematics Department, University of Toronto, Toronto, Ontario M5S 1A1 Canada Copyright 1923 and renewed 1951 by Wallace Stevens. Reprinted from Collected Poems of Wallace Stevens, by permission of Alfred A. Knopf, Inc.
2 IS it an accident or characteristic of Stevens' wit and attention to minutiae that the round mouth of the jar just happens to resemble the zero at the origin of a Cartesian coordinate system and the letter O of the word origin?
THEMATHEMATICALINTELLIGENCERVOL.12, NO. 1 9 1990Springer-VerlagNew York 77
Derek Jacobi as Alan Turing in
Breaking the Code
I reply, "I'm a mathematician." The union of those bored or intimidated by this response is virtually the universe. Some of us might seek a more interesting image for the mathematics professions, but we have had little in literature to which we may turn. A couple of seasons ago Breaking the Code, based on the life of Alan Turing, enjoyed London and New York stage successes surprising to many. This season Tom Stoppard followed up his m a t h e m a t i c i a n in Jumpers with a physicist in Hapgood who talks about the Koenigsberger bridge problem. Occasionally we have seen a mathematician as m u r d e r e r - - m o s t notably recently in Scott Turow's Presumed Innocent [1], although my favorite remains Michael Innes's Weight of the Evidence [2]. Usually, however, an academic setring for a thriller means that all the interesting people are in literature; similarly, film rarely brings us a mathematician, Jill Clayburgh providing an intriguing exception in It's My Turn. Now Erik Rosenthal brings us, in Advanced Calculus of Murder, a second installment of his mathematician-private investigator Dan 78
THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 1, 1990
Brodsky [3]. This time author, victim, murderer, and sleuth are all mathematicians. Lest you think this unfairly reveals the solution, let me say that the murder takes place at a math conference at Oxford with several hundred mathematicians as the only suspects. Rosenthal is good at recreating the atmosphere of such meetings and more generally that of the inbred world of mathematics research. Those who were at Berkeley in the 60s will recognize some of his fictional characters, and perhaps recall nostalgically Steve Smale's "flight to Moscow" to pick up a Fields medal and the resulting fallout from the story that his work was done on the beaches of Rio under an NSF grant [4]. At least the jet-set image is in some ways an improvement over the "nerd" image. A recent occupational survey, rating jobs on the basis of such things as working conditions, stress, pay, and security, put actuary, computer programer, computer analyst, statistician, and mathematician (in that order) at the top of the list; college professor fell in the ll4th place and migrant worker was the least desirable. I wonder how we would rate on glamour and excitement. A minor quibble with the author's setting the stage: trains from Oxford come into Paddington station, not Victoria. Keeping London mainline stations straight seems to be a problem that American mystery writers often have. Martha Grimes, otherwise a superb creator of English atmosphere, has had this difficulty. On the other hand, perhaps this is a device to see whether the readers are paying attention. As a mystery Advanced Calculus of Murder is not very successful--Rosenthal telegraphs the killer's identity rather early on. As an anthropological study of mathe-
matics, it is more successful. It is said that academics battle so viciously because so little is at stake; Rosenthal captures the sense of how trivialities can come to dominate the lives of mathematicians. Moreover, he reminds us all of the cyclic nature of mathematical employment. Again today there is a shortage of mathematicians, but Brodsky is a remnant of less happy days. His gypsy academic character is still found in great numbers in fields less in demand. Unable to get a "real job," he teaches part-time at Berkeley for a pittance, but unlike many of the exploited underclass of part-timers he keeps up his research and has found a
R o s e n t h a l c a p t u r e s the sense o f h o w t r i v i a l ities can come to d o m i n a t e the lives o f m a t h ematicians.
declarations by young girls and boys that they want to grow up to be mathematicians or to delight by mathematicians" seatmates at the prospect of vicarious glamour for the duration of the flight. John yon Neumann acquired a measure of public recognition, in which he is said to have taken naive delight, but he cannot be said to have generated an aura of excitement about mathematics as a profession. Sonia Kovaleskaia was, in the view of some of her contemporaries, a romantic figure--but in roles other than as a mathematician. We need a folk hero. Law school applications are said to be way up this year due to viewer interest in L.A. Law. Would that someone could make a similar splash with M.I.T. Math! The closest that mass culture has come recently is Stand and Deliver.
References creative means of supplementing his income. His work as a private investigator is generally not exciting, process serving and such being the mainstay. The subplot in Advanced Calculus of Murder of reuniting a young woman with the mother who gave her up for adoption takes Brodsky not very convincingly to the porno underworld and to Wales. He should stick to math conferences. Is there a way to make mathematicians more exciting, short of taking up murder? Allen Paulos's Innumeracy [5] has made the New York Times best-seller list, but unfortunately Paulos's work is unlikely to lead to
1. Scott Turow, PresumedInnocent, New York: Farrar Straus Giroux (1987). 2. J. I. M. Stewart (Michael Innes), Weight of the Evidence, London: Gollancz (1943). 3. Brodsky first appeared in Rosenthal's Calculusof Murder, New York: St. Martin's Press (1986). 4. The story was retold by Smale in "On the steps of Moscow University," Mathematical Intelligencer, vol. 6, no. 2 (1984), 21-27. 5. John Allen Paulos, Innumeracy, New York: Hill & Wang (1989).
Department of Mathematics and Statistics The American University Washington, DC 20016 USA
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* C o l u m n editor's a d d r e s s : Faculty of M a t h e m a t i c s , T h e O p e n University, Milton K e y n e s MK7 6AA E n g l a n d 80 THE MATHEMATICALINTELLIGENCERVOL. 12, NO. 1 9 1990Springer-VerlagNew York