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Non-Euc l i dean Pythagorean Tripl es, A Prob l e m of Eu l er, and Rational Poi nts on 1 3, n - 1 and n + 1 have to be even and greater than 2. So they can not be primes. So they cannot be sub primes. Problem solved! We call these neighboring numbers the housing for the given subprime. They play a similar role to that played by the housing for an electrical conduit, serving as a protective layer that keeps out weather and rodents. Often, when one leases a subprime, one also leases the housing to go with it.
And What About the "Mortgage"? The term "mortgage" traditionally refers to a contractual agreement to borrow money for the purchase of a domicile or other piece of real estate. The so called collateral is the building or prop erty itself. In the case of the subprime market, there is no real estate. There is simply a number, together with its "house." But by abuse of terminology, the act of bor rowing the money to purchase a lease on a subprime and its house has be come known as a mortgage. These
mortgages are a means to dramatically increase your investment potential. In stead of being limited to the funds you have on hand, you can "mortgage your future" and invest funds that actually
belong to someone else.
And "Mess"? As you know, if you still own a house, it doesn't take long for it to become a mess. The same holds true for these number houses. A lease not only pro vides the rights to n 1 , n, and n + 1 , it also provides the rights to a ll the real numbers in between. That is an un countable collection of numbers. Many of them are given by nonrepeating dec imals that GO ON FOREVER. If this col lection of numbers gets just a little bit out of order, you can imagine the mess that ensues. But don't worry. Things are not as bad as they first appear. For it turns out that the real numbers are well ordered. This means that there is a choice of or dering on the numbers such that a ny subset has a least element. It's not the usual ordering, but so what? If your real numbers get mixed up, just apply this ordering, and find the least element in the entire set. Then remove this element, and repeat the process. In no time at all, you will have your house in order. -
Is There Any Risk? Getting out of bed every morning is a risk. But if you stay in bed, you end up
covered in bed sores. Not to mention a meteor smashing through your bed room ceiling, and off you go to join the dinosaurs. So yes, there is some risk. But keep in mind that these instruments are trusted by brokerage houses that are the bedrock of the entire financial commu nity. If they feel safe and protected, why shouldn't you?
I'm Still a Little Confused About What I Do with a Subprime Once I Lease lt. That's not a question.
What Do I Do with a Subprime Once I Lease It? That's entirely up to you. It's your oys ter. You get to decide. There are es sentially no restrictions. So go ahead, cut loose. Have some fun!
How Do I Sign Up? Legally, you cannot just send us your credit card information. We are sup posed to send you a prospectus which you are then supposed to read. But the truth is that hardly anyone ever reads prospectuses, let alone using the plural of that word. Given the need for fast action, and the fact you look smart, we can forgo the "information" stage. So don't wait. It's not so clear how long this opportunity will remain available.
© 2008 Spnnger Sc1ence+ Bus1ness Media, I n c , Volume 30, Number 4, 2008
13
Vie\N p o int
The G eometry of Paradise MARK A. PETERSON
The Viewpoint column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. Viewpoint should be submitted to the editor-in chief, Chandler Davis.
14
�
�
1 '/
athematics before 1700 presents a peculiar picture. It is difficult to avoid the impres sion that there was a golden age of mathematics in the Hellenistic period of Euclid and Archimedes, that a new mathematical golden age began in the 17th century, the golden age in which we are now living, and that in the long period in between, mathematics for some reason languished. Medieval Arab and European cultures inherited classi cal mathematics, and good mathemati cal minds were undoubtedly at work, but the circumstances put them some how at a disadvantage. Lucio Russo's remarkable book Tbe Forgotten Revolution (1) argues that the damage to mathematics as a collective enterprise was done already in the Ro man period. Thus what later civiliza tions inherited was already somehow maimed, cut off from the problems that gave rise to it. Medieval cultures were in the peculiar condition of being un mathematical cultures in possession of sophisticated mathematics. They pos sessed it in the sense of having the books, studying them and translating them, and even doing some mathe matics, but they had no clear indica tion where this rich subject had come from or what it would be good for. They did not know, in our terms at least, what it was. The story is complicated by excep tions to this sweeping characterization. Certain constructions called geom etria practica, useful in building and com merce, had a continuous existence right through the period, changing hardly at all, as if they were already adequate to their problems, with no need for inno vation. Methods and notations for do ing arithmetic certainly changed, but ancient civilizations had already been good at arithmetic, so the main inno vation here was perhaps the diffusion of arithmetical competence to a large commercial and professional class. Above all, algebra developed, with al gorithms for the solution of polynomial
THE MATHEMATICAL INTELLIGENCER © 2008 Spnnger Sc1ence +Busmess Med1a, Inc
equations and systems of equations, in response to problems that arose first in commerce, and then took on an ab stract life of their own. This develop ment looks like normal mathematics, and it is noteworthy that algebra did not have any essential connection to the Greek mathematics of the classical past. Geometry, on the other hand, was moribund. It is as if its high sophisti cation, precisely its roots in the classi cal past, somehow disadvantaged it, and made it almost a dead subject, in spite of its high status. I have stated this view of medieval geometry more starkly than I could jus tify. It is not given as the topic of this article (I am not so ambitious as to un dertake a proper evaluation of it), but as a background against which one ex ample stands out dramatically: Dante's geometry in The Divine Comedy, specif ically in Paradise. Dante seems to have an unusual mathematical gift, but in an unmathematical age this gift finds a peculiar outlet. What Dante does with mathematics may bear out the previous suggestion that late medieval European culture possessed mathematics, but without knowing what it was. Where we expect mathematics to find appli cation in practical, earthly problems, medieval mathematics, apart from geometria practica and commercial arithmetic, did not come with that ex pectation at all. If anything, its associ ations were with astronomy and ce lestial things (2]. It must have seemed natural to Dante to find, as we will see, applications of mathematics in theology! And perhaps it is we who are not sufficiently imaginative. Math ematics, as the ultimate in abstraction, does not come with any prescription for what it might mean, so-theology, why not?
Dante's Universe is S3 It has been noticed by many readers that Dante's universe is topologically S3 [3, 4, 5. 6) . Still, the occurrence of a compact 3-manifold without boundary
in a late medieval poem is so unex pected, and the suggestion seems so implausible, that it might be good to go over the evidence here, lines of the poem that cumulatively leave little doubt. Dante invented concepts that were reinvented long afterward. Paradise represents the ascent of Dante and Beatrice through the spheres of the Aristotelian heaven, concentric with the Earth, beginning with the sphere of the Moon, then Mercury, Venus, the Sun, etc. [7]. In Canto 27, Dante and Beatrice make the ascent from the sphere of the fixed stars to the Primum Mobile, the outermost sphere of the universe, the one that turns all the others. Beyond that is the Empyrean, which has no conventional geometric description, but which Dante must now describe. The exposition begins as early as the end of Canto 22, in lines that are ad dressed to Dante, but are also prepar ing the reader, "Tu se' si presso a l'ultima salute," comincio Beatrice, ''che tu dei aver le luci tue chiare e acute; e pero, prima che tu piu t'inlei, rimira in giu, e vedi quanta mondo sotto li piedi gia esser ti fei . . . ' '
Beatrice began: "Before long thou wilt raise Thine eyes and the Supreme Good thou wilt see; Hence thou must sharpen and make clear thy gaze, Before thou nearer to that Presence be, Cast thy look downward and con sider there
How vast a world I have set under thee . . . " [8] Par. 22: 1 24-129 Dante does look down, seeing "this little threshing floor" the Earth below, surrounded by the heavenly spheres through which he has ascended, e tutti e sette mi si dimostraro quanta son grandi e quanta son ve loci e come sono in distante riparo. All seven being displayed, I could admire How vast they are, how swiftly they are spun, And how remote they dwell . . . Par. 22:148-150 This calling attention in Canto 22 to the sizes and velocities of the heavenly spheres before ascending to the sphere of the fixed stars is a kind of fore shadowing, to be recalled in Canto 28. In Canto 27 Beatrice asks Dante once again to look down, admiring the spheres below, as they ascend from the sphere of the fixed stars to the Primum Mobile, the ninth sphere. Dante is care ful to say of the Primum Mobile Le parti sue vivissime ed eccelse si uniforme son, ch'i' non so dire qual Beatrice per loco mi scelse. This heaven, the liveliest and loftiest, So equal is, which part I cannot say My Lady for my sojourn there deemed best. Par. 27:100-102
That is, this sphere has full rotational symmetry, and the ascent they have chosen is in no way distinguished from any other way they might have come. That S0(3) symmetry is a crucial in gredient of Dante's image, and he does not want it to be missed. At the beginning of Canto 28, Dante sees reflected in Beatrice's eyes a bright Point. Turning, he sees the Point in re ality, surrounded by nine whirling cir cles, moving the more slowly as they are larger. The seventh of these is al ready larger than the rainbow's circle. It emerges in the next lines, where this whole structure is discussed, that these angelic circles are a kind of mirror im age of the heavenly circles below. Mir ror symmetry is already suggested in the way that Dante first sees them, as a reflection. The Point, representing God, the center of the angelic circles, is the mirror image of the center of the material universe, the center of the Earth down below, where Satan is fixed in ice. La donna mia, che mi vedea in cura forte sospeso, disse: "Da quel punta depende il cielo e tutta la natura. " Observing wonder i n m y every feature, My Lady told me what I set below: "From this Point hang the heavens and all nature. '' Par. 28:40-42 The word "depende" used in this way expresses essentially what we mean when we say S3 is IS2, the suspension of the 2-sphere. Dante even describes himself as "sospeso," suspended, per-
MARK A. PETERSON is Professor of Physics and Mathematics on
the Alumnae Foundation at Mount Holyoke College. His interest in Renaissance mathematics goes back to h1s graduate student days in physics at Stanford. He has just completed a book, from which the material of this article is taken, Ga/J!eo's Muse: The Renaissance
Re-invention of Science by the Arts. He also works on geometrical methods for continuum mechanics. Department of Mathematics and Statistics Mount Holyoke College South Hadley, MA 0 I 075-6420 USA
e-mail:
[email protected] © 2008 Spnnger Sclence+BusJness Media, lnc , Volume 30, Number 4, 2008
15
haps suggesting the precariousness of the soul between these theological poles, in addition to the more literal meaning of being "in suspense," al though what is literal and what is fig urative here is hard to pin down. In any case, the picture is the suspension construction, with a sense of "higher" and "lower" imposed on it. Dante is bothered by something that seems wrong to him. The heavenly spheres tum faster the larger they are, but the angelic ones tum slower the larger they are. We have already been alerted to these velocities, but now the matter is brought up explicitly. As Dante puts it, he would like to know why "l'essemplo e l'essemplare non vanno d'un modo," that is, why the pat tern and the copy don't move in the same way. Beatrice laughs "Se li tuoi diti non sono a tal nodo sufficienti, non e maraviglia: tanto, per non tentare, e fatto sodo!" "There's naught to marvel at, if to untie This tangled knot thy fingers are un fit, So tight 'tis grown for lack of will to try. Par. 28:58-60 This is an assertion that we are look ing at new mathematics! Beatrice ex plains that the spheres are ordered by velocity, and that they tum faster the higher they are (in the sense of higher knowledge, higher love, that is, prox imity to God), not the larger they are, another nice way to think of S3. The Primum Mobile may be the largest, but it is only the equator of the universe as a whole, being only midway in the or dering. The smallest circles, closest to the Point, tum really fast, as Beatrice points out. It is clear that Dante invents the no tion of manifold here, in building the universe out of two balls, glued along their common boundary. That is the meaning of "l'essemplo e l'essemplare. " The reader may b e bothered by the frequent use of the word "circle" where the right word would seem to be "sphere," but Dante explicitly says, in another place that we will see later, that he uses the word circle for both, and in general for anything round. In any case he makes clear more than once that
16
THE MATHEMATICAL INTELUGENCER
the circles about the Point are actually spheres. For example, "l'essemplo" is the spheres of the conventional uni verse, and it is the model for "l'essem plare," the angelic "circles. " Also, the ho mogeneous uniformity of the Primum Mobile, in which Dante can't distinguish any location, implies that the angelic "circles" are not just off to one side of the Primum Mobile but surround it. A skeptic might reluctantly agree that this looks very much like S3, de scribed in several different ways, in fact, but that it is simply impossible: Dante could not so easily have over come the normal tendency of the hu man mind to regard space as infinite (and Euclidean). I wonder, though. It might have been easier for Dante to in vent S3 than for, say, Immanuel Kant. Aristotle's universe, which was also Dante's universe, was explicitly de clared to be finite, although in Aristo tle's version it went only up to the Pri mum Mobile and no resolution was offered for the puzzle of what lay be yond. From the beginning, therefore, Dante was describing a finite universe. Going on, it appears that he did not have the Euclidean prejudice in favor of infinite straight lines. This objection might not even have occurred to him. His preferred geometrical object was the circle, and a space built out of cir cles might well be S3. The internal evidence of Dante's writing suggests that although he knew Euclid's geometry, and made casual, easy use of it, he does not necessarily regard it as a model for space, espe cially globally. Rather he regards it as a branch of philosophy whose propo sitions are true with peculiar certainty (as Aristotle also regards geometry). There are two Euclidean theorems in Paradise, but neither of them carries a meaning that has anything to do with space. The theorem in Par. 1 3 : 1 0 1-102 is Elements III.31, a triangle inscribed in a semicircle is a right triangle. This the orem occurs as just one of several learned propositions in a list, the other propositions not being from geometry. The list consists, tellingly perhaps, of things that King Solomon did not ask to know when he was granted wisdom. This rather backhanded reference might even be read as slightly dismis sive of geometry.
In Par. 17: 1 3-18 Dante says, ad dressing Beatrice, that she sees the fu ture as clearly as men see that no tri angle can have two obtuse angles. Neither this occurrence nor the previ ous one uses Euclid to describe some thing in space. When Dante asks about his future, he does not mean triangles. Rather, these theorems are cited as ex amples of things that are known with certainty to be true. In short, geometry seems to be less geometrical for Dante than it is for us.
Dante's Geometer Dante discusses the seven liberal arts of the trivium and quadrivium, geom etry being one of them, in his earlier unfinished book of classical learning, Tbe Banquet. From Tbe Banquet II. l3, Geometry moves between two things antithetical to it, namely the point and the circle-and I mean "circle" in the broad sense of any thing round, whether a solid body or a surface ; for, as Euclid says, the point is its beginning, and as he says, the circle is its most perfect fig ure, which must therefore be con ceived as its end.Therefore Geome try moves between the point and the circle as between its beginning and end, and these two are antithetical to its certainty; for the point cannot be measured because of its indivis ibility, and it is impossible to square the circle perfectly because of its arc, and so it cannot be measured ex actly. Geometry is furthermore most white insofar as it is without taint of error and most certain both in itself and in its handmaid, which is called Perspective. [9] This passage cites Euclid, but the sen timents attributed to Euclid are virtually unrecognizable. The implied meaning of geometry in this passage is precise measurement, and the point and the circle are "antithetical" to the certainty of geometry because they can't be mea sured, not at all a Euclidean idea. Nor does Euclid call the circle "most per fect. " The enthusiasm for the circle ex pressed here must be Dante's own. The problem of measuring the circle, given such prominence here, is of course not a problem of Euclid. One is left with the impression, con-
sistent with the two theorems in Par adise, that although Dante knows and respects Euclid, he does not find him very interesting. The passage in The Banquet summarizing geometry essen tially ignores Euclid, even as it cites him. The certainty of geometry seems less interesting to Dante than its op posite, the antithetical point and circle, for he devotes most of this little state ment to them. The unmeasurability of the circle definitely interests him.
Dante and Archimedes D ante returned to the problem of measuring the circle in one of the most astonishing passages he ever wrote, the final image of Paradiso. He is looking at an image of the Trinity, as three cir cles, and staring especially at the sec ond of these, representing the Son: dentro da se, del suo colore stesso mi parve pinta de Ia nostra effige: per che 'l mio viso in lei tutto era messo. Qual e 'l geometra che tutto s'affige per misurar lo cerchio, e non ritrova, pensando, quel principia ond' elli indige, tal era io a quella vista nova: veder voleva come si convenne !'imago al cerchio e come vi s'indova; rna non eran da cia le proprie penne: se non che Ia mia mente fu percossa da un fulgore in che sua voglia venne. A !'alta fantasia qui manco possa; rna gia volgeva il mio disio e 'l velle, si come rota ch'igualmente e mossa, l'amor che move il sole e l ' altre stelle. Par. 33: 1 30-145 Within itself, of its own coloration I saw it painted with our own hu man form: So that I gave it all my attention. Like the geometer, who exerts him self completely To measure the circle, and doesn ' t succeed, Thinking what principle he needs for it, Just so was I, at this new sight. I wanted to see how the human im age
Conforms itself to the circle, and finds its place there; But there were not the means for that, Except that my mind was struck By a flash of lightning, by which its will was accomplished. Here strength for the high imagin ing failed me, But already the love that moves the Sun and the other stars Turned my desire and my will Like a wheel that is turned evenly. I have preferred the unpoetic transla tion here in order to be as literal as possible, for the purpose of a close reading. Notes to this passage always point out the futility of trying to square the circle. They suggest that squaring the circle functions here as a metaphor for the impossibility of understanding the mystery of salvation by Christ's cruci fixion. In the last century or so, notes on this passage even cite Lindemann's 1882 proof that 7T is transcendental! That result cannot be relevant to Dante's intention in this image, but we already have Dante's own opinion in The Banquet that the circle cannot be squared. The message of futility might appear to be unavoidable, in view of the gen eral agreement that what the geometer is trying to do is impossible, but there are subtle problems with this reading. In the first place, it just doesn't sound like Dante to give up. Why would he come to the end of his amazing epic poem and then admit defeat by intro ducing an impossible problem in the very last lines? It isn ' t even so clear that he is defeated. That flash of light ning might indicate the opposite. Typ ical notes suggest that the flash is a metaphor for the acceptance of God's grace, as if the struggle with geome try were over, but the geometrical metaphor seems to continue even af ter the lightning flash, in the image of the turning wheel. Scholarship has re turned to this enigmatic passage again and again, without a wholly satisfac tory conclusion. Dante never mentions the name Archimedes, but Archimedes's little treatise On the Measure of the Circle had been translated several times be fore Dante wrote, from Arabic by both
Plato of Tivoli and Gerard of Cremona, and from Greek around the time of Dante's birth by William Moerbeke. Given Dante's interest in the question of measuring the circle, he would nat urally have sought out this treatise and studied it. It seems impossible that he would not have. It is short and easily copied, especially its Proposition I, which is the relevant one. According to Marshall Clagett [10] , versions of the Gerard translation were widely circu lated, and we will notice evidence be low that this is the version that Dante knew. I believe that Dante used the Archimedes proof as an extended metaphor in the last lines of Paradise for the drama of salvation, as I will now explain. It will follow that Dante un derstood the proof perfectly, and used it with precision. Let me recall the familiar magiste rial argument of Archimedes in On the Measure of the Circle. Archimedes shows by a method of exhaustion that the circle of radius R and circumference C is, in area, neither larger nor smaller than 1RC. Thus it is exactly 1RC. For as sume that the circle is larger than �RC. We construct a sequence of regular polygons inside the circle, beginning with the inscribed square, and doubling the number of sides at each step, as in Figure 1 . More than half the remaining area outside the polygon is incorpo rated at each step into the next poly gon in the sequence. Hence, by our as sumption that the circle is larger than iRC, some polygon in the sequence will also have area larger than 1RC. But this is impossible, because the area of the polygon is the sum of the triangular wedges in Figure 2, namely !z hNb in the notation defined there, and b < R, and Nb < C. Thus the circle is not greater than !zRC. A similar argument using a sequence of polygons outside the cir cle shows that the circle is also not less than iRC. Figure 1 shows the sequence of drawings that anyone would make who actually carried out the constructions of the proof. Only the rightmost figure of those four is found in the manuscripts, and only the Gerard translation manu scripts show the whole polygon [10] . The Moerbeke translations confine the construction lines to the upper left quadrant, so that the visual impression is quite different, although someone re-
© 2008 Spnnger Se>ence+Bus>ness Med>a. Inc . Volume 30. Number 4. 2008
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Figure I. A sequence of regular polygons is constructed in the circle.
capitulating the process might still draw it as in Figure 1 . I believe that Dante worked with the Gerard translation, and made the figures in the sequence shown in Figure 1 , because the second figure in that sequence, the cross in the circle, must have struck him as crucially significant (pun intended). The ap pearance of the cross in an argument that already seemed to have a tran scendent meaning must have been ir resistible to him. If we examine again the last lines of Tbe Divine Comedy, we see that they follow the Archimedes proof thought for thought. The second circle, painted with man's image, is the Son, and "Ia nostra effige, " our own human form, is the cross, that is, a man stretched out (crucified), exactly the second figure in Figure 1 . The strange word "painted" depicts the geometer's literally adding the lines of the cross to the circle with a drawing instrument. The cross gives rise to the square, then to the octagon, and so forth. The geometer wants to know "how the human image/Con forms itself to the circle, and finds its place there. " That is just the question, geometrically, how the sequence of polygons approaches the circle, and theologically how the human and mea surable becomes the divine and im measurable. The geometer knows that no matter how many sides it has, a polygon still has straight sides and so cannot become the circle, which is curved. It is just Dante's point in Tbe Banquet, that the circle cannot be mea sured "because of its arc . " He vainly seeks the principle, until suddenly there is "a flash of lightning, " which re solves the problem. This is the argu ment of Archimedes that shows how the polygons do become the circle in the limit. That result is finally asserted in Dante's saying that the wheel turns evenly, as only a circular wheel can do, not a polygon.
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THE MATHEMATICAL INTELLIGENCER
The Archimedes proof makes pre cise sense of so many odd details in these last lines of the poem that it seems quite believable that Dante had this proof in mind as an extended metaphor for the union of human (straight) and divine (curved) . If so, he understood that the limit of a sequence can have a property that no member of the sequence had, in this case the property of being curved, since that is the point of the metaphor and the mys tery to which he is leading us. Like the construction of S3 in Canto 28, it is possible that this final image was understood in mathematical terms by no contemporary of Dante, but there is an odd hint in this latter case that some people did. The first commentary that actually mentions the geometer is Benvenuto da Imola 0 375), who says [ 1 1] :
Et explicat summum conatum suum per unam comparationem elegantis simam de geometra, qui volens men surare circulum colit se tatum sibi; et quamvis autor videatur loqui com-
Area of polygon
=
muniter de geometria, tamen iste ac tus et casus quem ponit maxime ver ificatur de Archimede philosopho; ad quod est praenotandum quod sicut scribit Titus Livius etc. And he explains his highest effort by a most elegant comparison with a geometer, who, wanting to measure the circle, gives himself completely to it; and however much the author is seen to speak generally about geometry, this particular case, which he places most highly, is proved by the philosopher Archimedes, about whom it is well known, as Livy writes, etc. Benvenuto seems to know that there is a proof of Archimedes behind this im ager He does not, however, refer to Archimedes's treatise, but to Roman his tories that tell the story of Archimedes. That suggests that he is not one of the people who has actually seen or un derstood the proof. The next commentators seem to have misunderstood this idea in a
( 1/2) h·circumference
Figure 2. The area of the regular polygon is
�hNb,
where
N is the number of sides.
rather hilarious way. Chiose Vernon ( 1390) [1 1], probably from reading Ben venuto, thinks that the geometer star ing at the circles is Archimedes at the moment of his death at the siege of Syracuse, and inserts that whole story into his commentary. Since this makes a ridiculous ending for The Divine Comedy, and is clearly impossible, the idea was dropped in the next genera tion of commentaries, and all connec tion to the Archimedes proof seems thereafter to have been forgotten. If we restore the idea of the Archimedes proof as metaphor, we see that Dante's image might well represent not the failure of human intellect to com prehend the divine, as it is usually un derstood, but rather something more positive, more like a triumph of the hu man intellect, and more characteristic of Dante himself. Understanding the math ematics behind the image potentially changes its meaning.
Geometry as Philosophy It is noteworthy that although Dante refers to geometry, and even does geometry, in ways that we can recog nize (with some difficulty), the mean ing of mathematics for him is philo sophical. Euclid and Archimedes are philosophers. What we call mathemat ics is, for him, and presumably for his
contemporaries and for his culture, a corner of philosophy having to do with the celestial part of creation, exempli fying a particular kind of truth. In par ticular, mathematics does not deal with messy, earthly problems, or with ter restrial space. Is it credible that a philosophical stance of this kind, even if it is accepted by a whole culture, could change the nature of mathematics so drastically in practice? Restricting the problems that mathematics could address appears to have restricted mathematics itself. In principle, mathematics could be an ab straction that feeds on its own abstract problems. The Romans believed, and therefore later civilizations also be lieved, that the origin of Greek mathe matics was a love of abstraction. If the long medieval period attempted to see whether mathematics could flourish without earthly applications, the an swer seems to be a resounding no. The 1 7th-century revolution in mathematics came when it began addressing ques tions that we now call physics, concrete problems with experiments and data. Even if we formally share the math ematics inherited from the ancients, what we make of it depends on our culture, not simply on the contents of mathematics books. To a surprising de gree, the meaning of mathematics is
� Springer
the language of science
what we think it is, and what we want it to be. REFERENCES
[1] Russo, L., The Forgotten Revolution, Springer-Verlag, Berlin, Heidelberg, New York, 2003. [2] Ptolemy, Almagest, Book I. [3] Speiser, Andreas, Klassische StUcke der Mathematik,
Verlag Orell Fuselli, Zurich,
1 925. [4] Callahan, James, "The curvature of space in a finite universe," Scientific American 235, August, 1 976, 9Q-1 00.
[5] Peterson, Mark, "Dante and the 3-sphere," American Journal o f Physics,
47, 1 979,
1 03 1 -1 035.
[6] Osserman, Robert, Poetry of the Universe, Garden City, NY: Doubleday, 1 995. [7] An amusing riff on these spheres is Osmo Pekonen, "The Heavenly Spheres Re gained," The Mathematical lntelligencer 1 5, No. 4, 1 992, 22-26. [8] Verse translations are those of Barbara Reynolds from Dante's Paradise, Penguin Books, 1 962. [9] Dante's II Convivio (The Banquet), tr. R. H. Lansing, Garland Publishing, New York, 1 990, 72. [1 0] Clagett, Marshall, Archimedes in the Mid dle Ages, Vol. 1 , University of Wisconsin Press, Madison, 1 964. [1 1 ] Over 70 Dante commentanes can be searched online at dante.dartmouth.edu.
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© 2008 Spnnger Sctence+Bustness Media, Inc , Volume 30, Number 4, 2008
19
An I ntrod uction to I nfi n ite H at P rob l e ms CHRISTOPHER 5. HARDIN AND ALAN 0. TAYLOR
l_l
at-coloring puzzles (or hat problems) have been around at least since 1961 (Gardner 196 1 ) , and probably longer. They gained wider public attention with a question posed and answered by Todd Ebert in his 1998 Ph.D. dissertation (Ebert 1 998). The problem was presented by Sara Robinson in the April 10, 200 1 , Science section of Tbe New York Times as follows:
1
Three players enter a room and a red or blue hat is placed on each person's head. The color of each hat is determined by a coin toss, with the outcome of one coin toss having no effect on the others. Each person can see the other players' hats but not his own. No communication of any sort is allowed except for an initial strategy session before the game begins. Once they have had a chance to look at the other hats, the play ers must simultaneously guess the color of their own hats or pass. The group shares a hypothetical $3 million prize if at least one player guesses correctly and no play ers guess incorrectly. The same game can be played with any number of play ers. The general problem is to find a strategy for the group that maximizes its chances of winning the prize. If one player guesses randomly and the others pass, the probability of a win is 1/2. But Ebert's three-player solu-
20
THE MATHEMATICAL INTELLIGENCER © 2008 Spnnger Sc1ence+Bus1ness MeCia, Inc
tion is better: pass if the two visible hats are different col ors, and guess the missing color if they are the same. This strategy yields a win, on average, 3/4 of the time: of the eight possible hat assignments, it fails only on the two in which all three hats are the same color. Elwyn Berlekamp generalized this to n = 2 k - 1 players, using Hamming codes to show the existence of a strategy that yields a win with probability n/( n + 1 ) . Joe Buhler gives an account of this, and further variations, in Buhler (2002). In Spring 2004, Yuval Gabay and Michael O'Connor, then graduate students at Cornell University, produced a num ber of hat problems involving infinitely many players, one of which was (an equivalent of) what we will call the Gabay O'Connor hat problem: Infinitely many players enter a room and a red or blue hat is placed on each player's head as before. Each player can see the other players' hats but not his own. Again, no communication of any sort is allowed except for an initial strategy session before the game begins. But this time, passing is not allowed and each player receives $ 1 million if all but finitely many players guess correctly. There are simple strategies ensuring that infinitely many players will guess correctly. For example, let a player guess red if he sees infinitely many red hats, and guess blue oth erwise. If there are infinitely many.red hats, everyone will guess red, and the players with red hats will be correct; if
there are finitely many red hats, everyone will guess blue, and the infinitely many players with blue hats will be cor rect. The problem, however, seeks a strategy ensuring that all but finitely many-not just infinitely many-are correct, and this is what Gabay and O'Connor obtained using the ax iom of choice. The special case in which the set of play ers is countable follows from a 1964 result of Fred Galvin ( 1965); see also Thorp ( 1 967). Although Galvin's argument and the Gabay-O'Connor argument are similar, they are dif ferent enough that neither subsumes the other; a compar ison will appear elsewhere. As the title suggests, this paper is meant to be only an introduction to infinite hat problems, and as such proceeds in a somewhat expository manner. We have made no at tempt here to say anything of the relevance of hat prob lems to other areas of mathematics, but the reader wishing to see some of this can begin with Galvin and Prikry 0976), George (2007), and Hardin and Taylor (2008). The rest of the paper is organized as follows. In "The Formalism and the Finite, " we set up a general framework for hat problems of the Gabay-O'Connor type, and present a few results in the finite case. "Theorems of Gabay O'Conner and Lenstra" concerns the infinite case, which is our primary interest; we present the Gabay-O'Connor The orem, and a theorem of Lenstra involving strategies that ei ther make every player correct or every player incorrect. In "The Necessity of the Axiom of Choice, " we discuss the ne cessity of the axiom of choice in the Gabay-O'Connor The orem and Lenstra's Theorem; this section requires some ba sic facts about the property of Baire, so a short appendix on the property of Baire appears afterward. Our set-theoretic notation and terminology are standard. If is a set, then is the cardinality of and is the complement of If f is a function, then f i is the restric tion of f to A, and Pc is the set of functions mapping the set P into the set C. If x is a real number, then Lx J is the greatest integer that is less than or equal to x. We let N = (0, 1 , 2, . . . }. The authors thank James Guilford, John Guilford, Hen drik Lenstra, Michael O'Connor, and Stan Wagon for al-
A
A.
IAI
A A
Ac
CHRISTOPHER S. HARDIN, B.A. Amherst
1 998 and Ph.D. Cornell 2005, 1s currently a visiting professor at Wabash College. His pub lications are in the area of mathematical logic,
lowing us to include unpublished proofs that are in whole or in part due to them. Their specific contributions will be noted at the appropriate places. We also thank Andreas Blass for bringing Galvin's work to our attention, and thank the referee for many helpful suggestions.
The Formalism and the Finite The problems we consider will resemble the Gabay-0' Connor hat problem, but we allow more generality: the set of players can be any set, there can be any number of hat colors, players do not necessarily see all other hats, and the criterion for winning is not necessarily that all but fi nitely many players guess correctly. So, a particular hat problem will be described by (i) the set of players, (ii) the set of possible hat colors, (iii) which hats each player can see, and (iv) a rule that indicates, given the set of players who guess correctly, whether or not they win the game. We formally define a hat problem to be a tuple (P, C, V,"W') with the following properties. (i) The set ofplayers P is any set. (ii) The set of colors C is any set. (iii) The visibility graph V is a directed graph with P as the set of vertices. When there is an edge from a to b (which we denote by a Vb or b E Va), we interpret this as meaning that a can see (the hat worn by) b. In par ticular, Va is the set of players visible to a. We are only interested in cases where players cannot see their own hats, so we require that V has no edges from vertices to themselves. (iv) The winning family "W is a family of subsets of P. The players win iff the set of players who guess their own hat color correctly is in "W. A function g E Pc assigns a hat color to each player; we call g a coloring Given a hat problem (P, C, V,"W) , a strat egy is a function S : (P X PC) � C such that for any a E P and colorings g, h E Pc, g l Va =
bj Va
=>
S( a, g)
=
S(a, h).
(1)
We think of S(a, fi) as the color guessed by player a under coloring g. Condition (1) ensures that this guess only depends
ALAN D. TAYLOR received his Ph.D. from
Dartmouth in 1 975, and has been at Union College ever since. He has published six books on set theory, combinatorics, fair division, and
and include (wiTh Taylor) "A peculiar connec
the theory of voting, including Soda/ ChoiCe and
tion between the axiom of choice and pre
the Mathematics ofManipulation (Cambridge
dicting the future," Amer: Moth. Monthly
2005). He still keeps trying to run track
(2008). He enjoys climbing and music. He
meters).
(400
does not own blue jeans. Department of Mathematics Department of Mathematics and
Union College
Computer Science
Schenectady, NY 1 2308
Wabash College Crawfordsville, IN 47933
USA
e-mail:
[email protected] USA '
e-mail:
[email protected] © 2008 Spnnger ScM3nce+Busmess Med"', Inc , Volume 30. Number 4, 2008
21
l
i
on the hats that a can see, since g Va h Va means that the colorings g and h are indistinguishable to player a. We will frequently consider strategies player by player; for a E P and a strategy S, we define Sa : Pc� C by Sa(i) S( a, gy, and call Sa a strategy for player a. We say that player a guesses correctly if Sa(i) g(a). We call S a winning strategy if it ensures that the set of players who guess correctly is in the winning family, re gardless of the coloring; that is, {a E P : Sa(i) = g(a)} E W for any coloring g. To illustrate the kinds of questions and answers that arise within this framework, we present two results in the con text of finitely many players. For the first, say that a hat problem is a minimal hat problem if it asks for a strategy ensuring that at least one player guesses correctly, and call such a strategy a minimal solution. Our first result (the sec ond half of which is due, in part, to James Guilford and John Guilford) answers the following question. =
=
=
With k players and 2 colors, how much visibility is needed to guarantee the existence of a minimal solu tion? What if there are k players and k colors?
THEOREM 1 A k--player, 2--color hatproblem has a minimal solution iff the visibility graph bas a cycle. A k--player, k-color hat problem has a minimal solution iff the visibility graph is complete. To prove Theorem 1 , it will help to have a lemma that confirms an intuition about how many players guess cor rectly on average.
LEMMA 2 In a k--player, n-color hat problem, for any par ticular strategy, the average number ofplayers who guess cor rectly is k/n. (/be average is taken over all colorings.)
this, we first note that because V has no cycles, we can as sign each player a rank as follows: a has rank k if there is a directed path of length k beginning at a, but none of length k + 1 . Now, if there is a directed edge from vertex a to vertex b, then the rank of player a is strictly greater than the rank of player b. Thus, a player can only see hats of players of strictly smaller rank. Hence, given any strat egy, we can assign hat colors in order of rank to make everyone wrong: once we have colored the hats of players of rank < k, the guesses of players of rank k are determined , and w e can then color their hats t o make them wrong. Now suppose there are k colors. For the right-to-left di rection, assuming the visibility graph is complete, the strat egy is as follows. Number the players 0, 1 , . . . , k - 1 , and the colors 0, 1, . . . , k - 1 , and for each i, let si be the mod k sum of the hats seen by player i. The plan is to have player i guess i - s, (mod k) as the color of his hat. If the colors of all the hats add to i (mod k), then player i will be the one who guesses correctly. That is, if Co + + ck- J = i (mod k) then c, i - s, (mod k). For the other direction, assume that there are k players and k colors, and assume the visibility graph is not com plete. Let S be any strategy. We must show that there is a coloring in which every player guesses incorrectly. Suppose player a does not see player lis hat (with a =f:. b), and pick a coloring in which player a guesses correctly. If we change the color of player lis hat to match player b's guess, player a will not change his guess, and we will have a coloring in which a and b guess correctly. By Lemma 2, the aver age number of players who guess correctly is k/k 1 ; be cause we have a coloring with at least 2 players guessing correctly, there must be another coloring in which fewer than 1 (namely, zero) players guess correctly. 0 ·
·
·
=
=
Our second result along these same lines is also due, in part, to James Guilford and John Guilford (the n 2 case appears in Winkler 2001). It answers the following ques tion. =
Proof Suppose there are k players and n colors. Let S be any strategy. It suffices to show that any particular player a is correct in 1 out of n colorings. Given any assignment of hat colors to all players other than a, player a's guess will be determined; of the n ways to extend this hat as signment to a, exactly one will agree with a's guess. 0
Proof of Tbeorem 1 . Suppose first that there are 2 colors. The right-to-left direction is easy; assuming that the visibil ity graph has a cycle, the strategy is for a designated player on the cycle to guess that his hat is the same color as that of the player immediately ahead of him on the cycle, while all the others on the cycle guess that their hat color is the opposite of the player immediately ahead of them. To see that this works, assume that the first player on the cycle has a red hat and that everyone on the cycle guesses in correctly using this strategy. Then the second player on the cycle has a blue hat, the third player on the cycle has a blue hat, and so on until we're forced to conclude that the first player on the cycle has a blue hat, which we assumed not to be the case. For the other direction, we show that if there is no cy cle in the visibility graph V, then for every strategy there is a coloring for which everyone guesses incorrectly. To do
22
THE MATHEMATICAL INTELLIGENCER
With k players and n colors, how many correct guesses can a strategy guarantee, assuming the visibility graph is complete? Lemma 2 shows us that, regardless of strategy, the num ber who guess correctly will on average be k/ n. But this is very different from ensuring that a certain fraction will guess correctly regardless of luck or the particular coloring at hand. Nevertheless, the fraction k/ n is essentially the cor rect answer.
THEOREM 3 Consider the hatproblem with IPI = k, le i = n, and a complete visibility graph V Tben there exists a strat egy ensuring that L k!nJ players guess correctly, but there is no strategy ensuring that L k!nJ + 1 players guess correctly. Proof The strategy ensuring that L k/nJ players guess cor rectly is obtained as follows. Choose n X L k!nJ of the play ers (ignoring the rest) and divide them into L k/nJ pairwise disjoint groups of size n. Regarding each of the groups as an n-player, n-color hat problem, we can apply the previ-
ous theorem to obtain a strategy for each group ensuring that (precisely) one in each group guesses correctly. This yields L k!nJ correct guesses altogether, as desired. For the second part, we use Lemma 2. For any strategy, the average number of players who guess correctly will be L k!nJ , and L k/nJ < L k!nJ + 1 , so no strategy can guarantee at least L k/nJ + 1 players guess correctly for each coloring. 0 Theorem 3, and most of Theorem 1 , were obtained in dependently by Butler, Hajiaghay, Kleinberg, and Leighton (2008; see this for a considerably more detailed investiga tion of the finite context). With two colors and an even number of players, Theo rem 3 says that-with collective strategizing-the on-aver age result of 50% guessing correctly can, in fact, be achieved with each and every coloring. But it also says that this is the best that can be done by collective strategizing. In the finite case, this latter observation does little more than pro vide proof for what our intuition suggests: collective strate gizing notwithstanding, the on-average result of 50% can not be improved in a context wherein guesses are simultaneous. The infinite, however, is very different, and it is to this that we next turn.
Theorems of Gabay-O'Connor and Lenstra We begin with a statement and proof of what we will call the Gabay-O'Connor Theorem. As stated, this result is strong enough to solve the Gabay-O'Connor hat problem and to al low us to derive Lenstra's Theorem (below) from it. (One can use an arbitrary filter in place of the collection of cofinite sets, with essentially the same proof, to generalize the result.)
THEOREM 4 (GABAY-O' CONNOR) Consider the situa tion in which the set P ofplayers is arbitrary, the set C of col ors is arbitrary, and every player sees all butfinitely many of the other hats. Tben there exists a strategy under which all but finitely many players guess correctly. Moreover, the strategy is robust in the sense that each players guess is unchanged ifthe colors offinitely many hats are changed. Proof For h, g E Pc, say h g if (a E P : h(a) =f. g(a)} is finite; this is an equivalence relation on PC. By the axiom of choice, there exists a function : Pc � Pc such that h, and if h g, then ( h) = (g). Thus, is choos (h) ing a representative from each equivalence class. Notice that for each coloring h, each player a knows the equivalence class [h), and thus (h), because the player can see all but finitely many hats. The strategy is then to have the players guess their hat colors according to the chosen representa tive of the equivalence class of the coloring; more formally, we are letting SJ_ h) = (h)(a). For any coloring h, since this representative a are red-even in the sense that changing player a's hat color causes the set of red-even numbers greater than a to be complemented . The strategy is for player a to make his choice so that, if this choice is correct, then the set of red-even numbers is in the ultrafilter OU. The strategy works because either the set of red-even numbers is in au (in which case everyone is right) or the set of red-even num bers is not in au (in which case everyone is wrong). .
The Necessity of the Axiom of Choice Some nontrivial version of the axiom of choice is needed to prove Lenstra's Theorem or the Gabay-O'Connor Theo rem. Specifically, if we take the standard axioms of set the ory (ZFC) and replace the axiom of choice with a weaker principle known as dependent choice, the resulting system ZF + DC is not strong enough to prove Lenstra's Theorem or the Gabay-O'Connor Theorem, even when restricted to the case of two colors and countably many players. His torically, the precursor to our results here is a slightly weaker observation (in a different but related context) of Roy 0. Davies that was announced in Silverman (1966). The reader does not need any familiarity with ZF + DC; all that must be understood is that, as an axiom system, ZF + DC is weaker than ZFC, and somewhat stronger than ZF (set theory with the axiom of choice removed altogether). To follow our argument, some basic facts about the prop erty of Baire are needed; to this end, the appendix gives a short introduction to the property of Baire. As an aid to in tuition, having the property of Baire is somewhat analo gous to being measurable, whereas being meager (see ap pendix) is somewhat analogous to having measure 0. (The two notions should not be conflated too much, however: the real numbers can be written as the disjoint union of a measure 0 set and a meager set.) Let BP be the assertion that every set of reals has the property of Baire. It is known (assuming ZF is consistent) that ZF + DC cannot disprove BP (Judah and Shelah 1993). (This was established earlier, assuming the existence of a large cardinal, in [Solovay 1970]. ) It follows that ZF + DC cannot prove any theorem that contradicts BP, as any such proof could be turned into a proof that BP fails. We will show that Lenstra's Theorem and the Gabay-O'Connor The orem contradict BP, and thus ZF + DC cannot prove Lenstra's Theorem or the Gabay-O'Connor Theorem. AI-
24
THE MATHEMATICAL INTELLIGENCER
though BP is useful for establishing results such as these, one should note that BP is false in ZFC (for instance, ZFC can prove Lenstra's Theorem, which contradicts BP). Throughout this section, we take the set P of players to be the set N of natural numbers, and we take the two col ors to be 0 and 1 . The topology and measure on N{o, 1 } are the usual ones. That is, if s is a finite sequence of Os and 1s, then the set [s] of all infinite sequences of Os and ls that extend s is a basic open set whose measure is z - n, where n is the length of s. Identifying N{O, 1 } with the bi nary expansions of reals in [0, 1], this is Lebesgue measure. The topology is that of the Cantor set. Let Tk be the measure-preserving homeomorphism from N{o, 1 } to itself that toggles the kth bit in a sequence of Os and 1s. Call a set D � N{o, 1} a toggle set if there are infi nitely many values of k for which T,j_D) n D 0. The next lemma is key to the results in this section; its proof makes use of the following observation. If a set D has the property of Baire but is not meager, then there ex ists a nonempty open set V such that the symmetric dif ference of D and V is meager. Hence, if we take any ba sic open set [s] � V, it then follows that [s] - D is meager. =
LEMMA 6 Every toggle set with theproperty ofBaire is mea ger. Proof Assume for contradiction that D is a nonmeager tog gle set with the property of Baire, and choose a basic open set [s] such that [s] - D is meager. Because D is a toggle set, we can choose k greater than the length of s such that T,j_D) n D = 0. It now follows that [s] n D � [s] - T,j_D). But T,j_[s]) [s] , because k is greater than the length of s. Hence, T,j(s] - D). Thus, [s] n D � [s] - T,jD) = T,j[s]) - T,jD) [s] n D is meager, as was [s]- D. This means that [s] itself is meager, a contradiction. D =
=
With these preliminaries, the following theorem (of ZF + DC) shows that Lenstra's Theorem contradicts BP, and hence it cannot be proven without some nontrivial version of the axiom of choice.
THEOREM 7 Consider the situation in which the set P of players is countably infinite, there are two colors, and each player sees all of the other hats. Assume BP . Tben for every strategy there exists a coloring under which someone guesses correctly and someone guesses incorrectly. Proof Assume that S is a strategy and let D denote the set of colorings for which S yields all correct guesses, and let I denote the set of colorings for which S yields all incor rect guesses. Notice that both D and I are toggle sets, since changing the hat on one player causes his (unchanged) guess to switch from right to wrong or vice versa. If D and I both have the property of Baire, then both are meager. Choose h E N{o, 1 } (D U I). Under h, someone guesses correctly and someone guesses incorrectly. D -
In ZFC, nonmeager toggle sets do exist: as seen in the previous proof, if all toggle sets are meager, then Lenstra's Theorem fails, but Lenstra's Theorem is valid in ZFC.
We derived Lenstra's Theorem from the Gabay-O 'Con nor Theorem, so Theorem 7 also shows that some non trivial version of the axiom of choice is needed to prove the Gabay-O'Connor Theorem. However, the Gabay O'Connor Theorem, even in the case of two colors and countably many players, is stronger than the assertion that the Gabay-O'Connor hat problem has a solution: the theo rem does not require that players can see all other hats, and it produces not just a strategy, but a robust strategy. The following theorem (of ZF + DC) shows that any solu tion to the Gabay-O'Connor hat problem, even in the count able case, contradicts BP and hence requires some non trivial version of the axiom of choice.
THEOREM 8 Consider the case of the Gabay-O'Connor hat problem in which the set ofplayers is countably infinite. As sume BP . Then for every strategy there exists a coloring under which the number ofplayers guessing incorrectly is infinite. Proof Assume that S is a strategy and, for each k, let Dk denote the set of colorings for which S yields all correct guesses from players numbered k and higher. Notice that each Dk is a toggle set, since changing the hat on a player higher than k causes his (unchanged) guess to switch from right to wrong. If all the DkS have the property of Baire, then all are meager. Let D be the union of the DkS, and choose h E N{O, 1} D. Under h, the number of people guessing incorrectly is infinite. D -
A set B has the property of Baire if it differs from an open set by a meager set; that is, there is an open set V and a meager set M such that BllV = M (equivalently, B = VJlM), where Jl denotes symmetric difference. A topological space is a Baire space if its nonempty open sets are nonmeager.
THEOREM 10 (BAIRE CATEGORY THEOREM) Every nonempty complete metric space is a Baire space. We do not show the proof here, but it can be carried out in ZF + DC. For the special cases of the reals and Can tor space, the proof can be carried out in ZF. REFERENCES
Steven Butler, Mohammad T. Hajiaghayi, Robert D. Kleinberg, and Tom Leighton. Hat guessing games. SIAM Journal of Discrete Mathe matics
22:592-605, 2008.
Joe P. Buhler. Hat tricks. Mathernatlcal lntelligencer 24:44-49, 2002. Todd Ebert. Applications of recursive operators to randomness and complexity.
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1 998. Fred Galvin. Problem 5348. American Mathematical Monthly 72: 1 1 36, 1 965. Mart1n Gardner. The 2nd Scientific Amencan Book of Mathematical Puz zles & Diversions.
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Alexander George. A proof of Induction? Philosophers ' Imprint 7 : 1 -5, March 2007.
Theorems 7 and 8 can be recast in the context of Lebesgue measurability to show that both Lenstra's Theo rem and the Gabay-O'Connor Theorem imply the existence of nonmeasurable sets of reals. However, to show that ZF + DC cannot prove the existence of nonmeasurable sets of reals, one must assume the consistency of ZFC plus the ex istence of a large cardinal (Solovay 1970, Shelah 1 984). Al though this is not a particularly onerous assumption, it is why we favored the presentation in terms of the property of Baire. It turns out that with infinitely many colors, some non trivial version of the axiom of choice is needed to obtain a strategy ensuring even one correct guess; this will appear elsewhere.
Appendix: The Property of Baire 9 A subset
N of a
topological space is nowhere dense if the interior of its closure is empty. A set is meager if it is the union of countably many nowhere dense sets.
DEFINITION
Fred Galvin and Karel Prikry. lnfinitary Jonsson algebras and partition relations. Algebra Universal1s 6:367-376, 1 976. Chnstopher S. Hardin and Alan D. Taylor. A peculiar connection be tween the ax1om of choice and predicting the future. American Math ematical Monthly
1 1 5:91-96, February 2008.
Haim Judah and Saharan Shelah. Baire property and axiom of choice. Israel Journal of Mathematics
84:435-450, 1 993.
Saharan Shelah. Can you take Solovay's inaccessible away? Israel Journal of Mathematics
48: 1 -47, 1 984.
D. L. Silverman. Solution of problem 5348. American Mathematical Monthly
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Robert Solovay. A model of set theory in which every set of reals is Lebesgue measurable. Annals of Mathematics 92: 1 -56, 1 970. B. L. D. Thorp. Solution of problem 5348. American Mathematical Monthly
74:730-731 , 1 967.
Peter Winkler. Games people don't play. In David Wolfe and Tom Rodgers, editors, Puzzlers' Tribute, pages 301-31 3. A. K. Peters, Ltd. , 2001 .
© 2008 Spnnger Sc1ence+Bus1ness Med1a, Inc , Volume 30, Number 4, 2008
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l$@u:i§u@iil¥11@1§4fii,iui§,IN
Sub S h oot ! MICHAEL KLEBER
This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one bas an urge to pass them on. Contributions are most welcome.
Michael Kleber and Ravi Vakil , Editors
� y day, Leonidas Kontothanassis works as my colleague at Google. LJ By night, he runs a gaudy carni val booth on the boardwalk outside of town. "Step right up and try your luck! Shoot the sub and win a prize," Leonidas was calling out one fine Fall evening. He was standing in the middle of the booth, surrounded by a moat packed with plastic toy submarines. The subs were of every shape and size, and they circled the moat at all different speeds, propelled by a complex system of cur rents. "On the bottom of one of these sub marines is an X," he assured me. "Find the right sub and the prize is yours. " I eyed him suspiciously-he wasn't the most trustworthy character. "All right, how many chances do I get?" "Let me explain the rules," he an swered, "and then you tell me how many guesses you think you'll need." Leonidas's moat, he pointed out, was divided into n regions, numbered in or der from 1 through n as you walked around the booth. On each shot, I would get to pick not just one subma rine, but one whole region, and we would scoop out all of the subs in that region. If any of them was the one marked with the X, I would win. If not, I could take another shot at the re maining subs-which would all have had time to move to new regions by then, of course. "But how fast do the subs move?" The swirl of motion was dizzying, with subs passing each other right and left. "It's actually very orderly, " he as sured me. "At the beginning, there are exactly n2 submarines, with n in each of the n regions of the moat. Moreover, the subs in a region each move at a dif ferent constant velocity: Every velocity 1 is represented from 0 through n once. Velocity is measured in regions per shot. So you get one shot at each . , and there is a time t 0, 1 , 2, 3, sub with position a + bt for every a and b (mod n)." Now that I knew what I was looking at, I had to admit it was very orderly, -
Please send all submiSSions to the Mathematical Entertainments Co-editor, Ravl Vakil, Stanford Un iversity , Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-mail:
[email protected] 26
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THE MATHEMATICAL INTELLIGENCER © 2008 Spnnger Sc1ence+Bus1ness Med1a. Inc
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right down to the one b 0 submarine anchored, unmoving, in each region. "Do I have to take a shot at each time t, or can I take a small number of shots but space them out?" "I don't have all night," Leonidas replied. "The question is how quickly you can find the X, not in how few shots. Anyway, if I let you have arbi trary breaks, then you could shoot only at times 0, n, 2 n and so on, when all the subs got back to their original lo cations. Too easy!" "You don't sound like you've left anything to chance, you old huckster. Do you really expect me to believe I might get lucky and hit the winning sub on my first shot?" "Of course not," he answered with a twinkle in his eye. "I guarantee, the X will be in the last place you look." =
Greed In how few shots can we hit all of Leonidas's submarines? Certainly we can manage in n2 shots, one per submarine, but, of course, we can do much better. On the other hand, we can't hope to do it in fewer than n shots: There are n2 subs, and we can only hit n of them at a time. In fact our 1 subs, so second shot will only hit n even n shots is unattainable. Also, the n stationary submarines--or, indeed, the n subs with velocity b, for any b will each require its own shot. So n is a lower bound that's too low, n2 is an upper bound that's too high, and the truth lies somewhere in be tween. The greedy algorithm is the first refuge of the lazy: On each shot, we could target the most heavily-occupied region. Ignoring the precisely choreo graphed submarine movement entirely, we know that there will always be a re gion holding at least 1/ n of the re maining subs, by the pigeonhole prin ciple. We can get a bound on the efficiency of this approach by working backwards. The last n submarines will, at worst, take one shot each, or n shots total. Before that stage, when from n + 1 to -
2 n subs remain, we can hit at least two at a time, so we need at most n12 shots to make it through. Likewise, we need at most r n/31 shots to get from 3 n subs down to 2 n, and s o o n . We can be sloppy about rounding and still get within n shots of the true answer. The total number of shots to hit all n2 subs in the worst case is within n of + .!� = n ( 1 + _1_ + n + .!� + .!� +
r
..!. + 3
2
·
·
·
3
·
·
·
n
J
2
+ 1._), or around n In n. In fact, II
it's around n/2 + n In n: Experimen tally, this is how many times you need 1 to perform "multiply by 11 - and round down" to get from n2 to �ero. So we know about how long the greedy algorithm would take against submarines that deliberately spread themselves out to avoid our shots. But Leonidas's aren't doing that. If subs conveniently clump together, the greedy algorithm will find more ap pealing targets, and will finish sooner. My friend Pablo Alvarez wrote a simu lation, and his results are plotted in Fig ure 1 . (In case of ties, he shot at the lowest-numbered region which attained the maximal sub count.)
The greedy algorithm appears to do much better than the worst-case analysis would suggest. Empirically, it seems to take 2 n 1 shots when n is prime or a power of two, and n 3 shots when n is twice a prime. But at n 575, the greedy algorithm takes more than 4 n shots for the first time, and the trend for the worst n does seem to keep growing faster than linearly . From a practical point of view, the greedy algorithm's implementation is not as easy as its description. Suppose n were a million. A computer program calling the shots would need to track which of the trillion submarines in the fleet were still afloat and how they were allocated among the regions. This seems like a lot of work, particularly for the approach that was supposed to be the lazy way out. -
f
-
=
Divisiveness If we make use of the precise move ment of the subs, there is a much simpler algorithm that does better than n In n.
Consider a submarine whose veloc ity b is relatively prime to the number of regions n. In n moves, that sub will visit all the regions, in some order. In particular, it will visit region 1 at some point. So if we shoot at region 1 for n shots in a row, we will hit every sub with gcd( b, n) = 1 . Supposing for a moment that n is even, look at the subs with gcd(b, n) = 2. These subs will eventually visit half the regions, either all the even- or all the odd-numbered ones, completing their route with period n/2. We can therefore hit all of them in n shots as well: Shoot n/2 times at region 1, and then n/2 times at region 2. More gen erally, for any d dividing n, we can hit all the subs with gcd(b, n) = d in only n shots, by taking n/ d shots in a row at each region 1 , 2, . . . , d. To hit all the subs, we can pile the strategies for each divisor d on top of each other. First take n shots at region 1 : We've now hit the relatively prime velocities, but at the same time we've hit half the d = 2 subs as well. Next, fire at region 2 for n/2 shots, wiping
Figure I. Number of shots needed by the greedy algorithm for 1 :s n :s 1000. The lines 2n, 3n, and 4n are shown for refer ence. In practice, greedy performs substantially better on Leonidas's submarines than the order n In n upper bound.
© 2008 Spn�er Sc1ence+Busmess Med>a, Inc , Volume 30. Number 4, 2008
27
out the rest of the d = 2 part of the fleet. If n is a multiple of 3 as well, we've now fired more than enough times at regions 1 and 2, and firing n/3 times at region 3 will wipe out the re maining subs with d = 3. If we ignored the question of which d divide n-for example, if Leonidas had only told us n approximately, not exactly-this shows that we could hit all the submarines by firing a volley of nlk consecutive shots at region k, for k = 1 , 2, . . . , n. This may be less ef ficient than the greedy algorithm, where n In n is the upper bound, not the true performance, but at least there's no hard work involved. But if we know n's divisors 1 = d1 < d2 < < dr = n, we can shave off some guesses. Instead of bombarding region k with n/k shots, we can "round down to the nearest divisor," firing only n/d; times, where d;- 1 < k � d;. For ex ample, if n is a multiple of 3 but is not even, we can follow the n shots at re gion 1 with only n/3 shots at region 2, not n/2. When n has many small divisors, the savings over the n In n bound is real if modest. When n is one million and the greedy algorithm's worst-case bound is 14,392,720 shots, the divisor-aware ver sion requires only 1 2,687,500, a 10% savings. The advantage is most telling when n is prime, though: The divisor-aware version requires only 2 n - 1 shots. First, take n shots at region 1, hitting all the submarines whose velocity is nonzero, and then take one shot at each other region, to clean up the stationary subs. This is exactly how the greedy al gorithm behaves for prime n. ·
·
·
Exhaustion I visited Leonidas again the following night. The number of regions at his booth is not prime. "So for a million regions," he summed up, "you have one algorithm that you think will do well, but you don't know how well and it will require a trillion pieces of bookkeeping. And you have another algorithm that's easy to use, but will take more than twelve million tries." Under the flashing board walk lights, neither option seemed par ticularly appealing. Leonidas sent me home to think some more. In conversations with friends and
28
THE MATHEMATICAL INTELLJGENCER
colleagues, we produced a few refor mulations. Ignore the n2 - 1 sub marines without the X, and say that there is one sub at position a + bt with unknown a and b. You can think geo metrically: The sub at a + bt is a line in the space 7L. X 7L./n7L., and you're try ing to pick a set of points (So, 0), (s1 , 1), etc. that hit each line. Dually, you can view the a + bt sub at the point (a, b) in 7L./ n7L. X 7L./ n7L., and your goal is to 2 cover the n points with one line of slope 0, one line of slope 1 , one line of slope 2, and so on. The slope of the lines must be considered mod n, which highlights the fact that the shots at time t and t + n are equivalent. But none of these perspectives led to a break through. I managed to enlist some coconspir ators: Rich Schroeppel, Steve Witham, Scott Huddleston, and Edwin Clark all took up the computational hunt. They observed empirically that the n-region problem can be done in 2 n - 1 shots for primes and for prime powers as well-and for powers of odd primes, this turns out better than either the di visor strategy or the greedy algorithm. Rich and Steve ran exhaustive searches for small n. To make this fea sible, they needed to use some sym metry considerations that are not im mediately obvious. Without loss of generality, you may assume that your first shot is at the region numbered 1 : I f it isn 't, just add a constant t o each re gion's number, mod n. More surprising, they observed, you may assume your second shot is at region 1 also: Renum bering region r to r + st at time t is an automorphism of the original problem, transforming the submarines moving at velocity b into ones moving at velocity b - s (mod n). You do get to make a nontrivial choice for your third shot, but without loss of generality, the difference between shots two and three might as well be a divisor of n, thanks to multi plication by a unit mod n. Finally, the subs all have paths which repeat in time n, so the shots at times t, t + n, t + 2 n, etc., are all equivalent, as we mentioned before. Using such considerations and well pruned searches, they showed that 2 n 1 is minimal for prime powers up to n = 9. The sequence of 2 n - 1 shots for n = p k is a version of the one we described for primes above: First shoot n times at
region 1 , then shoot once each at re gions 2, 3, . . . n, in that order. Order didn't matter for primes, but it certainly does for prime powers. I'll present a proof of this construction below. For n = 6, the exhaustive searchers report that the minimum to hit all subs is 10 shots, for example 1-1-1-2-1-3-45-4-6, giving us for the first time a so lution under 2 n - 1 ; for n = 10 the minimum is 17 shots, attained by 1 - 1 1-1-1-2-2-7-2-5-9-6-4-10-3-8-8. Scott and Edwin generalized empir ically observed patterns and figured out how to get the job done in time 2 n 1 for n = 2pk, for odd primes p. That construction and proof is also forth coming, but we need a new point of view first.
Shifty Sweeps To make things easier to discuss, let's define a b-sweep to be a sequence of n shots which hit every submarine with velocity b. For the moment we'll restrict sweeps to be n consecutive shots, though we'll relax that requirement soon. We already mentioned that you can't accomplish this with fewer than n shots, since the n subs with velocity b are par allel-that is, their paths never intersect, so no single shot can hit two of them. There are exactly n! b-sweeps (for a given n and b) : You can hit the n subs in any order. Example 1: The permutations are ex actly the 0-sweeps. To hit the station ary subs, shoot at every site (in any or der). Example 2: The sequence of n shots at region 1 is a b-sweep for every ve locity b such that gcd(b, n) = 1 , one of our first observations. These shots hit many other subs as well, but for no other velocities do they hit all the subs. So one (very limiting) strategy for hit ting all the subs is to partition the set of possible velocities B into, say, B1 U B2, and then find one sequence of shots that is a b-sweep for all b in B1 and an other sequence that works for all b in Bz. If n is a prime, for example, then the two examples above go together to reproduce our old 2 n - 1 solution. (Why 2 n - 1 , not 2 n? Well, sweeps are translation-invariant-that is, you can freely add a constant to each region number: They only care about the ve locities of the subs, not their initial po-
sitions. So if we're going to shoot one sweep and then another, we can trans late one so that the final shot of the first sweep also serves as the initial shot of the second sweep.) Here's a useful lemma: If shooting at the sequence of regions r1 , r2, r3, . . . , rn is a b-sweep, then shooting at r1 + w, r2 + 2 w, r3 + 3w, . . . , rn + nw (all mod n) is a (b + w)-sweep. This is another invocation of the automorph ism used by the exhaustive searchers. It's easy to justify with some mod n arithmetic: The sub with position func tion a + bt gets hit by sweep shot i if a + bi == r, which is the same as a + (b + w) i == r, + iw. Now I can give a simple proof that we can hit all subs in 2 n - 1 shots whenever n is a prime power, say pk. First, fire n shots at region 1 , hitting all subs with velocity prime to n, i.e., not divisible by p. This is the sweep 1, 1 , 1 , . . . , 1 . Now transform it to the sweep 1 , 2 , 3, . . . , n, and, by the lemma, it will hit every sub with velocity b as long as b - 1 is not divisible by p. Consec utive numbers can't both be multiples of p, so every sub will be hit by one of the two sweeps. We can generalize this to n with more complicated prime factorizations. This solution schema just depends on the shifts by which we transform the sweeps. Round 1 : Take n shots at re gion 1 . Rounds 2, 3, . . . , k: Take n 1 shots, adding c2, c3, . . . , ck to the region number before each shot. The k rounds take kn - k + 1 shots, and will hit every submarine with a velocity b such that any of b, b - Cz , b - c3, . . . , b - ck is prime to n. (We assumed c1 = 0 by the symmetry arguments made earlier.) For example, take n to be one mil lion again. If we set c2, c3, c4 to be 1 , 2, 3, this method will let u s clear all the subs in four million shots (well, 3,999,997). These shifts work because in any four consecutive numbers, there is one which is relatively prime to a mil lion, i.e., divisible by neither 2 nor 5. We can't do it using this technique in only three rounds. Two of the shifts, say Cx and c would have the same parity, .Y' and we could always find a b that made Cx + b and Cy + b even and Cz + b a mul tiple of 5, if odd. Indeed, the Chinese Remainder Theorem tells us that this is the general case: No choice of shifts will
do any better than taking c1 = 0, c2 = 1 , . . . , ck = k - 1 for some k. The number of rounds we need, k, is therefore the maximal distance be tween consecutive integers relatively prime to n. Courtesy of Neil Sloane and his Online Encyclopedia of Integer Se quences (http://www .research.att.com/ -njas/sequences/), I learned that this maximal distance is known as the jacobsthal function of n, sequence A04-8669, hereafter j(n). So we now have a very simple al gorithm which will succeed in just un der nj( n) shots. The value of j(n) de pends only on the set of primes dividing n-the powers in n's prime factoriza tion are irrelevant to questions of what's relatively prime-so this approach pro duces solutions that scale up well to large n with only a few prime factors. When n is a prime power, j(n) = 2, and this reduces to our old 2 n - 1 so lution. When n p kq C for odd primes p and q, j(n) = 3: certainly if p divides b, and q divides b + 1 , then neither can divide b + 2 . If, instead, n = 2 kq C , we have j(n) = 4, just as we saw for n of one million. Not bad. But not always great, either. For n = 210 = 2 3 5 7, for instance, this con struction does very poorly. j(n) 10, since no number between 1 and 1 1 is prime to 2 10. So this scheme requires IOn - 9 = 2091 shots, while the divisor aware algorithm uses 1 1 18 and the greedy algorithm only 648. =
·
·
·
=
Time Warp Thej(n) bound is great for prime pow ers, but when n = 2pk, it degrades to a length 4 n - 3 solution. Scott Huddle ston and Edwin Clark figured out how to get the job done in time 2 n - 1, but their solution isn't a concatenation of two sweeps. For n = 10, for example, the sequence of shots is 10-2-10-4-106-10-8-10- X -5-7-5-9-5-1-5-3-5. (The X means you don't fire at all.) Scott figured out the general picture of which this is one example. The key is removing the limitation that sweeps consist of consecutive shots. Useful lemma number two: If shoot ing at regions r1 , r2, . . . , rn at times t1 , t2 . . . , tn is a b -sweep, then shoot ing at those same regions at times wt1 , wt2, . . . , wtn is a b' -sweep for any ve locity b' with wb' = b. When w divides n, multiplication by w mod n is a many-
to-one function, so spreading out the timing of our shots can greatly increase the number of velocities swept. The simplest use of this lemma is transforming our old friend n-shots-at the same-region. Applying the earlier lemma as well, we get Scott's key result, describ ing the effect of a sequence of shots that are linear in both time and space.
THEOREM 1 In the n-region subma rine problem, shooting at region ro + sdr (mod n) at time t0 + sd1, for 1 ::::; s ::::; n, will hit all submarines with all velocities b such that gcd(n,dr - bd 1) = 1 . The rules o f the problem don't look kindly on widely-spaced shots. Indeed, the most extreme application of this the orem, with .:11 = n and dr 1, reminds us that we could hit all the submarines in only n shots total, if only we were al lowed to wait for time n between shots. But now we can improve on this: It's suf ficient to let .:11 be the square-free part of n---let's call it �n) = II� n p. It's still in efficient time-wise, but it's good to know that we can finish the million-region sub marine problem in a million shots and time ten million, by shooting at regions 1, 2, 3, . . . at times 10, 20, 30, . . . These temporally-dilated sweeps can be used efficiently, also: If we want to shoot several of them, we can interleave them and not skip shots. Our old j(n) based approach filled time 2 n by two concatenated sweeps; in Scott's ap proach, you could instead fill time 2 n with two sweeps with .:11 = 2, alternat ing between them. For n of one million, Scott can do just that, hitting all the subs in two mil lion shots: =
•
•
on even-numbered shot 2 t, fire at re gion t; on odd-numbered shot 2 t + 1, fire at region 3t.
Setting .:11 = 2 means every sub's effec tive velocity bL11 is always even; sub tracting either one or three is sure to leave you with a velocity relatively prime to a million. Any n of the form 2 kpC works the same way. Without going into any details, let me also mention that Scott has a way of whittling a few shots off the end of a solution, like the overlapping sweep trick that led to j(n) - n + 1, but more subtle. (It also lets him skip some shots entirely, though in ways that don't re-
© 2008 Spnnger Sctence+Bustness Medta, Inc., Volume 30, Number 4, 2008
29
Questions
duce the total time needed.) Choosing the two 11/s correctly, you could finish in only 1 ,999,999 shots. For a given (!n), the problem be comes one of efficiently arranging and packing sweeps with the same 11/s. In addition to the above time 2 n - 1 re sult for (X n) = 2p Scott reports these successes:
"Bravo! " applauded Leonidas when I described how to clean out a million regions in two million shots. But he was ruthless in his interrogation. "For primes and prime powers, you make it sound like 2 n - 1 is the mini mum possible number of shots. Have you proved that?" No, I admitted-not even when n is prime, which felt par ticularly galling. "And your algorithms have prime powers as their best performance. But didn't the exhaustive search data show other small n needed fewer than 2 n shots, not more?" H e was right again, of course. Should we expect a million regions to take substantially less than two million shots? And I have no rea son to think that products of small primes are inherently harder than other n, just because they are the Achilles' heel of one tactic. The greedy algorithm only needs 68 and 648 shots for n 30 and 2 1 0, respectively, far better than any organized scheme I know. "Do you think there is an algorithm which is at worst linear in n?" Inspect-
,
•
•
•
time 3 n - 2 for (! n) 3pq with primes q > p > 3; time 4n - 3 for (X.. n) = 2 3 5, and, at most, 3 n - 2 shots are needed; time 6n - 5 for (! n) of 2 3 5 7 2 3 5 7 11 and 2 3 5 7 1 1 . 13; time 8 n - 7 for (X.. n) = 2 3 5 7 1 1 13 17 =
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•
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·
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·
·
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·
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.
The first three use 111's of 3, 2, and 6, respectively. The fourth uses six sweeps with 111 = 6 and two with 111 = 2, and requires careful work to get them to all align correctly. As (X n) grows to the product of many initial primes, Scott says he would eventually use sweeps with time dilation factors of 30, 6, and 2 .
=
fl Springer
ing the data, it seems tantalizingly pos sible. As my friend and colleague Thomas Colthurst pointed out, a con stant-time bound would let us refor mulate the game: Suppose you are al lowed to fire at up to k regions at a time, for some fixed k. Now can you clean out all n2 subs in time n? And perhaps even k = 2 would do the trick! Leonidas smiled at me and walked away from his booth. "See you tomor row,'' he called back to me. ''We have work to do!"
Thanks I am much obliged to the real Leonidas Kontothanassis for inventing the sub marine-mod- n problem and for letting me put words in his mouth now. Doubtless he had no idea what he was getting into when he posed the ques tion to me. Thanks also to the collab orators mentioned by name for letting me report on their advances, particu larly Scott Huddleston for feeding me quickly when it counted, and to Jessica Polito, Will Brockman, and Thomas Colthurst for many discussions.
.
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30
THE MATHEMATICAL INTELLIGENCER
Stefan Banach Re me mbe red 1n l 0 arbitrary). Let A(x) be a continuous, strictly monotone increasing nonnega tive function on [ 0 oo) . Then for A C !Rq Hausdorff introduces ,
L�(A) = inf
�'
and
A(d(U.)), A C
�
u.
d(U,)
0: J!Pl(A) = oo) = inf{p > 0: J!P)(A) = 0),
where L(P) = LA and with A(x) = :xf'. Hausdorff's concept of dimension is a finely-tuned instrument for charac terizing and comparing sets that are "highly jagged." The concepts in Di mension und iiufleres Mafl have been
applied and further developed in nu merous areas, for example, in the the ory of dynamical systems, geometric measure theory, the theory of self-sim ilar sets and fractals, the theory of sto chastic processes, harmonic analysis, potential theory, and number theory.24 Unfortunately the boom of interest in "fractal theory" has often led to misun derstandings and m1smterpretations about Hausdorff's conceptions.25 The University of Greifswald was a small Prussian provincial university of merely local importance. Its mathemat ics institute was small, and in the sum mer semester of 1 9 1 6 and the follow ing winter semester, Hausdorff was the only mathematician teaching there! Thus his teaching activities were almost completely dominated by elementary courses. His situation improved markedly from a scientific standpoint when he went to Bonn in 1 9 2 1 . Here he had the opportunity to expand his teaching to a wide number of themes and to lecture over and again on his current research interests. Particularly noteworthy, for example, is the lecture course he offered in the summer se mester of 1923 on probability theory,26 in which he placed this theory on ax iomatic and measure-theoretic founda tions, ten years before the publication of A. N. Kolmogoroff's Grundbegri.ffe der Wah rscheinlichkeitsrechnung. In Bonn, Hausdorff found in Eduard Study and later Otto Toeplitz colleagues who were not only outstanding mathema ticians but who also became good friends. During this second period in Bonn, Hausdorff produced important work in analysis. In Hausdorff 0921), he de veloped an entire class of summation methods for divergent series which to day are known as Hausdorff methods. 27 The classical methods of Holder and Cesaro are special cases of these Haus dorff methods. Each such Hausdorff method is given by a sequence of mo ments; in this context Hausdorff pre sented an elegant solution of the prob-
lem of moments for a finite interval that bypasses the theory of continued frac tions. In Hausdorff ( 1 923b) he dealt with a special moment problem for a finite interval (subject to certain re strictions on the generating density cp (x), for example that cp (x) E LP[0, 1]). Hausdorff spent many years working on criteria for the solvability and de termination of moment problems, as evidenced by hundreds of pages left in his posthumous papers. 28 Hausdorff made a major contribu tion to the emergence of functional analysis in the 1920s with his extension of the Fischer-Riesz theorem to LP spaces in Hausdorff (1923a). There he also proved the inequalities named af ter him and W. H. Young29: If an are the Fourier coefficients of /E Lq(0, 2 7T), q � 2,
1..
+
.!.
= 1 , then
(� i �)h ( 00
p
an
q
�
l l
1 27T
)
21T � L lfl q dx . 0
If le 0 for which one can perform this subdivision for any z E S and any 0 < r < 1 , then the set is porous (or shal low). We call N the constant of porosity. (There is an equiv alent definition of porosity using disks rather than squares. A set is porous if we can find circular holes in our set at every scale, that is, within any disk a hole is guaranteed to be big ger than a constant multiple of the original disk size.) The definition can be modified in several different ways. The notions of mean porosity and E-mean porosity were developed as extensions of porosity by Koskela and Rohde (1997) and loosen the requirements on the uniformity of the size of the holes within certain limits. I introduced the notion of nonuniform porosity as a weakening of porosity (Roth 2006). This retains the holes at every scale while com pletely eliminating uniformity. Porosity and Dimension
Why do we care about porosity? Because porosity implies that upper box dimension is less than 2, which implies that the Lebesgue measure is zero. If the constant of porosity,
Figure 3. Box coverings for the first three rows of Table I. Note in the rightmost picture that the center box is not needed for the covering.
" 1 Most results in the field are stated in terms of Hausdorff dimension, which 1s less than or equal to lower box dimension However it is not necessary to state results 1n these terms for th1s paper
© 2008 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 30, Number 4, 2008
53
and so
side-length r
--
dimBS :S
log
Ndn
log Nn
= d < 2.
0
For a complete proof, see Martio and Vuorinen 0987). How Do We Show Porosity?
a
hole
Figure 4. Illustration of definition of porosity. N, is exactly calculated, which it often cannot be, a better bound for dimension can be provided. See Martio and Vuorinen (1987) for details. Similar bounds exist for mean porosity and e-mean porosity, although the bounds are slightly different (Koskela and Rohde 1 997). Unfortunately nonuniform porosity does not provide a dimension bound, although it does imply Lebesgue measure zero.
THEOREM 1 A porous set in the Riemann sphere has upper box dimension less than 2 .
PROOF
IDEA .
Let's call our porous set S. We want to estimate
N_S, r), the number of boxes needed to cover the set.
First, we cover S with squares of a side-length r, where r is a number greater than zero and less than one. We can assume we have used a finite number of squares, since S is compact by the definition of a porous set. We will call that number of squares k. Now, examine a specific square. Divide it into N2 subsquares, where N is the porosity con stant. By the definition, there must be at least one square that contains no points of S, as in Figure 4 . We do not need that square to cover S, so throw it out. Repeat this process on the rest of the k squares in our cover. Now, we r have k(N2 - 1) squares of side-length '
N
We repeat the process, subdividing our cover of squares of side-length ....!..._ After subdividing and throwing out Jv.
squares not needed to cover S, we now have squares of side-length
�· N
k(N2 - 1) 2
If we repeat this process n
. times, we have k(N 2 - l) n squares of side-length that N 2 - 1 = Nd for some d < 2 . S o the upper box dimension log N(S, r)
dimES = }!p0
54
log
(�)
THE MATHEMATICAL INTELLIGENCER
= lim
n -. oo
r
N
" . Note
Let's discuss the simplest example. The Julia set of j(z) = z2 is the unit circle centered at the origin. At any point z on the unit circle we take a box of side-length r centered at z. The circle takes up less than half the area of the box, because the tangent line to the circle at z lies outside the circle and divides the box into two parts with equal area. Using N = 3 guarantees a hole when you subdivide. Since this is true for all r at z, the Julia set is porous there. This is true for all points on the circle, and the constant of poros ity remains the same for each point. For a general Julia set, showing porosity is more com plicated. First we must find holes at some point in the Ju lia set. Once we have porosity for some z E ]fi we find a conformal mapping that allows us to duplicate these holes at every point. For rational functions expanding on a Julia set, the proof of porosity is easy. The idea was known for a long time. Just pull back large-scale holes to all small scales by iteration of inverse branches of f (Przytycki and Urbanski 2001). The main difficulty in this approach is keeping the holes reasonably close to the same size while pulling back. If we distort our holes too much under the pull-back we cannot replicate the porosity. To keep the distortion of our holes bounded, we use the Koebe One-Quarter Theorem and the Koebe Distortion Theorem (Carleson and Gamelin 1993). Of course, guaranteeing that we have one constant of poros ity for all of the holes is a much more complicated matter that I will not discuss here.
Julia Sets that are Full of Holes Several types of Julia sets have been shown to be porous. Most of these results have been published relatively re cently, and the Julia sets considered are subjected to re strictive hypotheses to make the mathematics easier. I will
consider a few related sets, although I will not discuss the conditions on the sets in detail. For other known results on porosity of Julia sets, see Geyer 0999), Jarvi 0995), Przy tycki and Rohde 0998), Przytycki and Urbanski ( 2001 ), Sul livan ( 1 983), and Youngcheng (2000). Our first Julia set that is full of holes is the rabbit shown in Figure 2 (Przytycki and Urbanski 2001). It belongs to a class of Julia sets for nonrecurrent and parabolic Collet-Eck mann rational functions. The second group of porous Julia sets are the polyno mials R.. z) = fi27T'8z + ZZ, which were shown to be porous for irrationals (} of bounded type (McMullen 1 998) . The Julia set fp for (} =
1 CVS - 1 ) is shown in Figure 5 .
The third group o f porous sets i s not quite a Julia set. (z - 3) ]e is a subset of the Julia set fR of %.. z) = fl-7rrr ;i2 , 1 - 3z where T is a number chosen so that the rotation number is (} (see Figs. 6 and 7). The set ]e is important because of its relationship to the previous class of Julia sets fp. ]e is porous for irrationals of bounded type (McMullen 1 998), and for other irrationals it is a close call: ]e for the function (z - 3 ) . . any 1rrauona . . ? "mr(O) z:1 , 1s nonun11orm , where (} 1s e·c 1y 1 - 3z porous (Roth 2006). Despite all that is known aboutj0, it is unknown whether ]R is porous. The problem is finding the initial point on the Julia set where the set is porous. If such a point could be found, the techniques from the previous results could be applied to find holes at every scale. Mathematically speak-
ing, Julia sets are part of a "young" field, and it is easy to find questions that are still not answered.
Conclusion Many of the most fundamental properties, such as measure and dimension, remain unknown for most Julia sets. Al though there are Julia sets that are the whole Riemann sphere and so have dimension two and positive measure, no other Julia sets of measure bigger than zero have been found. Shishikura's surprising result 0 998) shows that there are other Julia sets of dimension 2, which makes it appear possible that there are other Julia sets of positive measure. Proving that a Julia set is full of holes, or porous, pro vides a bound on the upper box dimension, but this has so far been possible only for special classes of Julia sets. Mean porosity and mean E-porosity, both found in Koskela and Rohde 0997), provide better dimension bounds; nonuniform porosity (Roth 2006) implies measure zero, but is not known to provide dimension bounds. These notions can be used in some cases when it is not possible to prove porosity. In the end, we do not know in general which Ju lia sets are porous and which are not. In fact, for ]R, little is known about its dimension or measure. There is much left to explore. ACKNOWLEDGMENTS
The figures in the paper were generated using several differ ent programs. Figures 1 and 2 were generated using Fractal Explorer. Figures 5 and 6 were generated using the software of Curt McMullen (2001). Figure 6 comes from Peterson (1996). The Sierpinski gasket was generated using BRAZIL Fractal Builder. REFERENCES
L. Carleson and T. Gamelin. Complex Dynamics. Universitext, Springer Verlag, New York, 1 993. R. Devaney, A First Course in Chaotic Dynamical Systems. Westview Press, Perseus Books Group, Boulder, 1 992. L. Geyer, Porosity of parabolic Julia sets. Complex Variables Theory Appl.
39 (1 999), 1 91 -1 98.
P. Jarvi, Not all Jul1a sets are quasi-self s1m1lar. Proc. Amer. Math. Soc.
Figure 6. The Julia set, }R, for R(z) T
6
=
e27T1" z2 1!..:-_ll, where
I - lz
1 25 (1 995), 835-837 . P. Koskela and S. Rohde, Hausdorff dimension and mean porosity.
is a number chosen so the rotation number is =
t (vS -
1 ).
Math. Ann.
309 (1 997), 593-609.
N. Lesmoir-Gordon, W. Rood, and R. Edney, lntroducing Fractal Geom etry.
Icon Books Limited, Cambridge, 2000.
C. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets. Acta Math. 1 80 (1 998), 247-292. C.
McMullen,
Julia Sets of Polynomials and
Rational
Maps.
http://www.math.harvard.edu/-ctm/programs.html, 2001 . 0. Martio and M. Vuorinen, Whitney cubes, p-capacity, and Minkowski
content. Expo. Math. 5 (1 987), 1 7-40.
Y. Pes1n, Dimension Theory in Dynamical Systems: Contemporary Views and Applications.
Chicago Lectures in Mathematics. University of
Chicago Press, 1 997. C. L. Peterson, Local connect1v1ty of some Julia sets containing a cir cle with an irrational rotation. Acta Math. 1 77 (1 996), 1 63-224.
Figure 7. A subset of the previous Julia set, we'll call it }e, for 6 (Vs I) (from Peterson 1 996) =
1
-
.
F Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets, Fund. Math .
1 55 (1 998), 1 89-199.
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F. Przytyck1 and M. Urban-ski, Porosity of Julia sets of non-recurrent
M. Sh1sh1kura, The Hausdorff dimension of the boundary of the
and parabolic Collet -Eckmann rat1onal functions, Ann. Acad. Sci. Fen
Mandelbrot set and Julia sets. Annals of Math. 1 47 (1 998), 225-
nicae
267.
26 (2001), 1 25-154.
K. Roth, Non-uniform porosity for a subset of some Julia sets, Com
D. Sullivan, Conformal Dynamical Systems, Geometric Dynamics (Rio
plex Dynamics: Twenty-Five Years After the Appearance of the Man
de Janeiro, 1 981), Lecture Notes in Math. , vol. 1 007, Springer-Ver
de/brat Set,
American Mathematical Society, Contemporary Math.
lag, Berlin, 1 983, 725-752. Y. Youngcheng, Geometry and dimension of Julia sets. The Mandel
396 (2006), 1 53-1 68. D. Schleicher, Hausdorff dimension, 1ts properties, and its surprises, Amer. Math. Monthly
� Springer
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brot set, theme and variations, ture Note Ser.
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THE MATHEMATICAL INTELUGENCER
pp. 281 -287, London Math. Soc. Lec
27 4, Cambridge Univ. Press, Cambridge, 2000.
la§'h§'.Jtj
Osmo Pekonen ,
Editor
I
The Architecture of M odern Mathematics by jose Ferreir6s andjeremy Gray (eds.) NEW YORK. OXFORD UNIVERSITY PRESS, 2006, 442 PP, US $69.50 ISBN 0198567936 REVIEWED BY ANDREW ARANA
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his collection of essays explores what makes modern mathematics "modern," where "modern mathematics" is understood as the mathe matics done in the West from roughly 1 800 to 1970. This is not the trivial mat ter of exploring what makes recent mathematics recent. The term "modern" (or "modernism") is used widely in the humanities to describe the era since about 1900, exemplified by Picasso or Kandinsky in the visual arts, Rilke or Pound in poetry, or Le Corbusier or Loos in architecture (a building by the latter graces the cover of this book's dust jacket). Though it is hard to say precisely what modernism is, or what distinguishes it from other eras, Gray at tempts a definition in his closing essay in this collection:
I
Modernism can be defined as an au tonomous body of ideas, pursued with little outward reference, main taining a complicated, rather than a naive, relationship with the day-to day world and drawn to the formal aspects of the discipline. (p. 374)
Column Editor: Osmo Pekonen, Agora Centre, 40014 University of Jyviiskylii, Fmland e-mail: osmo.
[email protected] This is a good start. Gray mentions modern algebra, topology, and logic as examples fitting this description, and explains why they fit. These characteristics, though, fit high-profile examples of mathematics before this era also. Ancient geometry as in Euclid's Elements seems to have been pursued as an autonomous body of ideas, at least as far as I understand what this means. D'Alembert's work on differential equations, for instance on
the vibrating-string problem, was criti cized for failing to model empirical re ality adequately, though d'Alembert dis puted this: hence d'Alembert's work had a complicated, rather than naive, relationship with the day-to-day world. Lastly, Euler and Lagrange, among many others in the eighteenth century, were drawn to the formal aspects of analysis. Gray's definition of modernism could be tightened to disqualify these examples (and to be fair, his essay in dicates some ways to do this, as I'll soon point out). But if we are going to make a case for the continuity of modern mathematics with modernism, we must look beyond the quoted definition for another account. We can see how the essays in this collection contribute toward answering what makes modern mathematics mod ern if we instead view modernism as a crisis concerning foundations. Let me explain. All of the artists and architects mentioned previously looked for a new way of practicing their art, because the old ways had been discredited or had ceased to speak to them. This loss opened up many new possibilities. Each experimented with form and content. The results were radically new, and many found them alien upon first con tact. Think of people asking if Kandin sky's spirals of color are really "art." What would it be to really be art? Mod ern artists realized many answers to this question that would not have seemed open in earlier times. This sense of openness, in contrast to a time in which some options would have seemed in escapably "correct," is what I want to call a foundational crisis. As has often been remarked, math ematics in the early twentieth century underwent a foundational crisis. This is usually said to be a result of the para doxes in set theory, which threw into question whether mathematics was con sistent. I agree that this was a crisis of sorts, but there was another crisis con temporaneous with this one that had wider reach. This wider foundational crisis mirrors the crisis in art just dis cussed. There had been consensus in
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the past on what makes a mathemati cal theory, such as arithmetic, true: it was true if it described the way things really were. By the turn of the twenti eth century, this view had lost much of its credibility. It now seemed open whether there were any true mathe matical theories, and if so, there were a variety of possible answers to this question. As with art, I want to call this sense of openness in mathematics a foundational crisis. Viewing modernism as a foundational crisis in the sense de scribed here provides a sharper answer to what makes modern mathematics modem. The modern turn in mathe matics happens in parallel with the modem turn in the arts, both trying to progress despite an awareness that old orders that used to underwrite their on tologies and values had been discred ited. The essays in this collection address this modernist foundational crisis in mathematics in a variety of ways. The essays concern the years following 1800, when non-Euclidean geometries were beginning to receive attention. These geometries gave new urgency to the problem of what it was for a math ematical theory to be true. Which is the real geometry of space? There was con sensus in the past: a geometry was true if it described the way things really were. Until the mid-eighteenth century, there were two main explanations of how this worked: either the description of reality was a result of abstraction from nature, or an expression of Pla tonic forms and their ordering. From ei ther standpoint, Euclidean geometry was thought to be a true description of space. During the eighteenth century, an other explanation gained currency. Gray describes this view as follows: mathematics is "what is presented by idealized common-sense." (p. 390) On this view, "every rational person can recognize a straight line when they see one. " The true geometry is thus the one acknowledged by all rational people; naturally this was thought to be Eu clidean geometry. Around the tum of the nineteenth century, Kant offered a more sophisticated version: roughly, the world appears to us the way it does (for instance, as having unified objects) be cause our minds structure it to appear that way. We can't help but experience
58
THE MATHEMATICAL INTELLIGENCER
the world as we do, but whether the world really is as we experience it is unanswerable. A theory of space is true, on this account, if it expresses the struc ture that our minds are constrained to experience space as having. (Kant seems to have thought that Euclidean geometry expressed this structure.) During the nineteenth century, as mathematicians became aware of non Euclidean geometries, it no longer seemed obvious that space really is Eu clidean, or that the mind structures spa tial experience as Euclidean. The foun dational crisis for mathematics begins here. One option is to conclude that no geometry is the "true" one. Instead, there are different geometries and none is any more true than the others. Some geometries suit our individual situations and present purposes better than oth ers, and our situations and purposes can change. This view is similar to the philosopher Friedrich Nietzsche's "per spectivalism" on moral and scientific matters: there is no single "true," "God's eye view" of morality and the world, only biased individual perspectives. To see things more clearly we should learn to view morality and the world from many different perspectives . As Moritz Epple explains in his fascinating essay in this volume, one mathematician who explicitly took up this Nietzschean ban ner in his work was Felix Hausdorff. Hausdorff lived a double life in print, publishing as a mathematician under the name Felix Hausdorff, and as a Ni etzschean philosopher under the name Paul Mongre. Hausdorffs view, which he called "considered empiricism, " was that mathematics is useful for construct ing axiomatic theories that "model" em pirical phenomena. Each theory repre sented a "perspective" on the empirical matter it concerned. Whether a theory is good is a practical question, to be evaluated based on how well the the ory describes, explains, and predicts data. Since the empirical data may be consistent with several different mathe matical theories, each theory should continue to be developed; Hausdorff thought this was the case with dimen sionality and with the ongoing devel opment of various axiomatic geometries. Relativism regarding the truth of mathematical theories is a radical de parture from the past. It constrains
mathematical activity to the construc tion of theories, without any preten sions to "getting it right." On this view, there is no ·'right" way to think of space, no single true analysis of the concepts of circle or line or continuity. Relativism can extend even into the allegedly "foundational" areas of arithmetic and set theory. There are many axiomatic theories of arithmetic that we can study freely for their mathematical structure, but we must not confuse any of them with the "real thing," for there is no "real" thing. Similarly, there are many different set theories; the relativist can claim that none is absolutely true. As Epple explains, Hausdorff became in terested in a set-theoretic approach to topology in seeking a continuous model of space and time, which led him to Cantor's point-set analysis of the con tinuum. Yet he thought this was just one perspective on the continuum. Alfred Tarski's work on logical con sequence, carefully discussed by Paolo Mancosu in this collection, gave more tools to the technically-minded rela tivist. Today we follow Tarski in saying that a sentence u is a logical conse quence of a set of sentences � if every interpretation or "model" of all the sen tences in � is also a model of u. But the "every" in this analysis of logical consequence raises a question. Suppose we are asking whether a sentence in the language of arithmetic is a logical consequence of the axioms of arith metic. Should we consider just models with the intended domain N, or do we consider models with other domains also? The more radical conception is to vary domains widely, perhaps out of skepticism that the notion of an "in tended domain" makes any sense. Man casu argues that in 1936 at least, Tarski avoided the more radical option. I find Mancosu's argument convincing, but more work is being done on this topic, and there may be new arguments worth considering. In any case, this article is an excellent starting point for under standing this active area of research. Gray notes that an even more radi cal relativism arose in the early twenti eth century: The Modernist foundations of math ematics ultimately dispensed with the idea that the subject matter of logic was the correct rules of rea-
son-those that would be followed by any undamaged mind. A part of logic does consider such rules, but it seemed ever more obvious that the logic needed to create genuine mathematics is not a candidate for even an idealized description of the way people think. Not only geome try, not only the conception of num ber, but eventually any simple minded association of logic with correct thinking was made anew. (p . 396) Is there a principled reason to reject relativism, and maintain the traditional view? Because this question is a live one, I call the ongoing situation a foun dational crisis. Epple and Gray's essays explore (without advocating) this crisis directly, whereas Mancosu's essay bears on this issue without addressing it explicitly. Other essays respond to this view, in one way or another. Two essays concern, in different ways, Hilbert's attempt to reconcile a Kantian approach with these new mathematical developments. Wilfried Sieg considers ongoing developments in the spirit of "Hilbert's program" in proof theory. Hilbert thought mathematical methods could be divided into two categories, the "real" and the "ideal. " In his early work, Hilbert echoed the traditional view that the natural and real numbers, and the points of Euclidean geometry, are real, whereas imaginary numbers and points at infinity in projective geometry are ideal; later, he identified the finitary mathematics of the natural numbers as real and the infinitary meth ods of higher mathematics as ideal. In drawing this real/ideal distinction, Hilbert was echoing Kant's distinction between constitutive and regulative principles, where the former are real ized in experience, and the latter are not, but instead are tools for organiz ing our thoughts concerning experi ence. Theorems proved by real meth ods were contentual, whereas theorems proved by ideal methods were useful tools for theorizing, but lacked content. Hilbert hoped to show that every the orem provable by ideal methods could be proved by real methods. Sieg agrees with the received view that Hilbert's program is dead, as a re sult of Godel's second Incompleteness theorem. But he thinks ongoing work
in proof theory might salvage some thing like it, if the base theory is "ac cessible"-that is, if it has "a unique build-up through basic operations from distinguished objects," so that it consists of "principles that are evident. " For then the base theory would be contentual and thus capable of yielding knowl edge. Sieg doesn't offer criteria for eval uating when an operation is basic, or for when a principle is evident. Instead, he raises this as a project for future work. Though there are important Kantian elements in Hilbert's thought, Hilbert re jected the details of Kant's views on in tuition as "anthropomorphic garbage. " Nevertheless, intuition and experience, and in particular visualization, played an important role in Hilbert's thought. Leo Corry explains that Hilbert believed that we are guided in our formulations of axiomatic theories of geometry by in tuition and experience, and that Hilbert continued to believe this even as he came to understand general relativity. Although in the past Hilbert had thought Euclidean geometry was the true geometry of space, he recognized that general relativity cast doubt on this. This was no problem for Hilbert's views about axioms and experience, because as our experience changes, so should our axiomatic theories. Such a view, though, seems to leave open the ques tion of whether mathematics can "get it right" when describing the world, or if instead it is just our way of describing things, which can only be judged prag matically by what those models can do for us. That is, Hilbert's view as de scribed by Corry leaves open the pos sibility of relativism for geometry. These matters troubled Hilbert's stu dent Hermann Weyl, whose fascinating views Erhard Scholz discusses. Like Hausdorff and Hilbert, Weyl thought mathematical activity was largely a mat ter of producing systems of symbols. But Weyl was no relativist. According to Scholz, Weyl thought that mathematics did more than offer mere tools for the formation of mathematical models of processes or structures, in a purely pragmatic sense. A good mathematical theory of nature . . . expressed, if well done, an aspect of transcendent reality in "symbolical form. " (p. 296)
That is, a good mathematical theory must include a " metaphysical belief in some transcendent world core , " so that the meanings of the symbols used in ordinary practice are not merely the stipulations of individual practitioners (as the relativist would have it), but in stead are rooted in a transcendent re ality; otherwise "no meaningful com municative scientific practice would be possible . " Weyl was inspired by post Kantian German philosophy, particu larly the work of Wilhelm von Hum boldt, Martin Heidegger, Karl Jaspers, and Ernst Cassirer. Following their writ ings, Weyl considered our use of sym bols analogous to the use of tools by carpenters and other craftsmen. The raw materials, the abilities of the car penter, and the item the carpenter wants to build, all put demands on what makes a tool for that task good. Similarly, Weyl thought mathematical language is a tool for the mathemati cian, and it is not entirely up to the mathematician to determine what makes that tool good. Scholz does not explain what Weyl thought were the analogues of the raw materials, etc., for the carpenter that would constrain the goodness of mathematical tools, or how this constraining would work. I agree with Scholz that Weyl's idea is more a plan for future work than a complete solution. But it is a fascinat ing idea, and one that is worth devel oping. (In his fine article, Jean-Pierre Marquis also takes up the idea of tool production and use in mathematics, ar guing that some mathematical theories, specifically homotopy theory, are worth knowing only because of their practical value for work in other sub ject-matters of intrinsic interest, even if these theories are not themselves of in trinsic interest.) Let us turn now to the essays on Gott lob Frege. Frege was a key instigator of the contemporary Anglo-American ap proach to philosophy, in which the log ical analysis of language is of central importance. Though Frege is mostly studied by philosophers nowadays, he was very much a mathematician: his doctorate was in mathematics; he was employed in the ]ena mathematics de partment; he regularly taught courses in complex analysis, elliptic functions, and potential theory. Within the philosophy of mathematics, he is best known for
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his "logicist" project, the goal of which was to show that all the truths of arith metic and analysis (though not geome try) were really truths of logic. Frege, too, was concerned about the problem of relativism, particularly for the concept of number. As he wrote in the introduction to his Foundations of
Arithmetic, Yet if everyone had to understand by this name ["the number one"] whatever he pleased, then the same proposition about one would mean different things for different peo ple,-such propositions would have no common content. (Frege, p. i) Like Hilbert and Weyl, Frege was con cerned that if there is no single correct answer to what a number is, then the intersubjectivity of arithmetic and analysis would fail. He accounted for his answer's correctness by appealing to his logicism, that is, to his view that the laws of arithmetic are reducible to the laws of logic, which are laws of thought and thus are common to every rational person. Nevertheless, Frege recognized that human practices played a role in mak ing explicit the sense of the number concept. Michael Beaney's essay in this volume carefully explores how Frege went about "elucidating" the basic con cepts of arithmetic and analysis, draw ing on "our common conceptual her itage" (p. 53) to make explicit what these concepts really are. (In her en gaging article on twentieth-century French philosophy of mathematics in this volume, Hourya Benis Sinaceur's description of Jean Cavailles' project of unwinding the historical ''dialectic" of mathematical concepts suggests paral lels with Frege's idea of elucidation, al though these parallels are not explored here.) In his essay, Jamie Tappenden care fully situates Frege within the nine teenth-century struggle in Germany over how best to think about complex analysis. This was (roughly) a struggle between two camps. Weierstrass and his followers thought complex analysis should be "arithmetized," meaning in particular that analytic functions should be defined as functions representable by power series. By contrast, Riemann and his followers favored representa-
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tion-independent definitions, which in practice meant defining analytic func tions as those satisfying the Cauchy-Rie mann equations. Riemann advanced the term "geometric" for this kind of ap proach to analysis, but this wasn't a stretch: he encouraged visualization in analysis, developing the notion of a Rie mann surface to help. He even en couraged physical reasoning in analy sis, using Dirichlet's principle freely even though his evidence for it was based on potential theory. (Riemann's own reflections on these matters were quite rich, provocatively engaging philosophical matters, as Ferreir6s doc uments in his delightful essay.) Tappenden thinks it is wrong to see Frege as a Weierstrassian. Instead, he argues, Frege's work should be seen within the Riemannian tradition, where a central aspiration was the identifica tion and clarification of concepts-in Frege's case, of the concept of number. Weierstrass holds that there can be no dispute about the kind of thing that counts as a basic operation or concept: the basic operations are the familiar arithmetic ones like plus and times. Nothing could be clearer or more elementary than explanation in those terms. Series representa tions count as acceptable basic rep resentations because they use only these terms. By contrast, the Rie mannian stance is that even what is to count as a characterization in terms of basic properties should be up for grabs. What is to count as fundamental in a given area of investigation has to be discovered. (p. 1 1 2) Tappenden argues that Frege's attempt to provide a logicist definition of num ber should be understood as an in stance of this general Riemannian quest. To understand better what this quest is all about, I want to pose and answer two questions about these Riemannian definitional "quests. " First, does the Rie mannian think there must always be a "right" definition, and if so, what makes it right? Second, how does the Rie mannian think we are to know when the "right" definition has been found? On the first question, I think Tappen den's discussion is inconclusive. When
he says that the Riemannian thinks what is fundamental in an area has to be dis covered, does he mean that the Rie mannian always thinks there is a fact about this to be discovered? If the an swer is "no , " then the Riemannian is a relativist. As to the second question, Tappenden's answer is that for the Rie mannian, definitions prove their cor rectness by their "fruitfulness," for in stance in organizing our practice well or in playing a role in important further research. Tappenden discusses this view in more detail in other work, but we can address it without leaving this volume by turning instead to the essays on the Riemannians Richard Dedekind and Emmy Noether, by Jeremy Avigad and Colin McLarty, respectively. Avigad writes about Dedekind's Rie mannian approach to developing ideal theory, which he took to mean in prac tice avoiding computation as much as possible. Dedekind instead adopted the axiomatic, set-theoretic approach famil iar to us from contemporary algebra. His work in turn influenced Noether and subsequently a central strand of twentieth-century algebra and algebraic geometry. McLarty presents an overview of how Noether brought this contemporary approach to topology. Continuing Dedekind's Riemannian quest to avoid computation in algebra, Noether took a "purely set-theoretic" approach that was, in her words, "in dependent of any operations" (p. 193). Instead of studying addition or multi plication of the elements of a ring, for instance, she proposed studying partic ular subsets and homomorphisms pre serving the structure of those subsets. In Riemann's terms, she saw these struc tural properties as the "internal charac teristic properties" of rings, rather than the computational properties that she thought were merely "external. " This approach gives special value to homo morphism theorems, as McLarty ably documents. Thus both Dedekind and Noether had views on what the "right" defini tions are in algebra and algebraic topol ogy. How did they think we were to know when we'd found those right de finitions? Avigad suggests some answers for the case of Dedekind, and I want to consider three of these. First, Avigad suggests that Dedekind thought the right definitions in algebra would avoid ·
elements "extraneous" to algebra. This suggestion just pushes the question back, into what it is to be extraneous to algebra. Second, Avigad suggests Dedekind thought the right definitions in algebra would unify the domain be ing defined; as he puts it, "A single uni form definition of the real numbers gives an account of what it is that par ticular expressions are supposed to rep resent" (p. 178). But why should we ex pect that the right definitions will be uniform, rather than having lots of case distinctions? It would be nice if that were so, but wishing doesn't make it so, unless what makes a definition right is that it's the one we want. Correct de finition as wish-fulfillment: if this were Dedekind's view, he would have been a relativist. Fortunately, there is a third possibil ity. Avigad suggests that Dedekind thought the right definitions for a do main would yield properties familiar from other domains. For instance, in ideal theory, Dedekind's "overall goal [was] to restore the property of unique factorization, which [had] proved to be important to the ordinary integers" (p. 171). Then many results following from unique factorization in the integers could be carried over to ideal theory. This is surely an important labor-saving technique. But why should we think that this technique leads to the right de finitions for a domain? There would have to be something "inevitable" about those properties if this technique were to avoid being another type of rela tivism. And indeed Dedekind seems to have thought certain properties were in evitable. Like Frege, Dedekind thought the familiar laws of arithmetic are laws of logic, and he seems to have believed that laws of logic are laws of thought; thus, we can't help but arrive at the properties we do in arithmetic because of the way our minds are constrained to think. Furthermore, he thought that this made inevitable properties in higher mathematics also: as he wrote in his 1 888 essay "Was sind und was sollen die Zahlen?," "every theorem of algebra and higher analysis, no matter how re mote, can be expressed as a theorem about natural numbers-a declaration I have heard repeatedly from the lips of Dirichlet." Thus Dedekind resorted to logicism to solve the fou'ndational cri sis.
I've addressed these matters at length because they help clarify the unity of the subject matter of this essay collec tion. Each essay documents a reaction to the problem of relativism, a problem I argue is central to understanding modernity, not just in mathematics, but in our culture generally. The essays are uniformly a joy to read, and the bibli ography is ample, giving interested readers an extensive springboard for further exploration. I recommend the book highly. [Thanks to my colleague Amy Lara for helpful comments on an earlier draft.] Department of Philosophy Kansas State University Manhattan, KS 66506-2602 USA e-mail:
[email protected] REFERENCES
Gottlob Frege, The FoundatiOns of Anthmetic, translated by J. L. Austin, second revised edi tion, Northwestern University Press, 1 994.
Leonhard Euler. Ein Mann1 mit dem man rechnen kann Leonard E uler. A Man to Be Reckoned With comic album by Andreas K. Heyne, Alice
K.
Heyne (text) and Elena S.
Pini (illustrations); English edition translated by Alice K. Heyne and Tabu Matheson BASEL: BIRKAUSER VERLAG, 2007, HARDCOVER 45 PP.,
18.60 ISBN: 978-3-7643-7779-3,
(GERMAN EDITION);
19.90 ISBN 978-3-7643-
8332-9 (ENGLISH EDITION) REVIEWED BY JOHAN STEN
here are not many mathemati cians, or intellectuals in general for that matter, to whom a comic
book has been devoted. If anyone de serves to be commemorated with such an honor, surely it is Leonhard Euler 0707-1783) on the occasion of his 300th anniversary. The challenge facing such a project is how to present the life and work of the principal character in an interesting and appealing fashion: Euler is more known for his immense scientific out put and his love for peaceful family life. But, the authors show, Euler's life was in fact quite eventful. Throughout his career, Euler was in the service of the rulers of Prussia or Russia, which were in a state of war and political turmoil. Euler had to cope with complicated per sonalities, not least his employer Fred erick, the king of Prussia. Moreover, Euler's interests ranged far beyond pure mathematics and theoretical physics. As many panels suggest, he was deeply in volved in solving engineering problems, such as designing naval vessels, study ing projectile motion, and designing telescopes (the "high-tech" of the time). On the other hand, the mathematical formulas and theories we all know and admire are completely absent. One wonders whether it would have not been possible to flash up some of the celebrated expressions. The first 12 pages of the comic fo cus on Euler's prodigious childhood, his studies of the Classics in Basel, his ac quaintance with the Bernoulli dynasty, and his private studies of mathematics under the auspices ofJohn Bernoulli the elder. Pages 13-19 .describe events in chaotic St. Petersburg during Euler's first stay there 1727-1741. During this pe riod of concentrated scientific investi gation in various fields, which could be called his formative years, Euler also suffered from a grave illness which would cost him the sight of his right eye. Pages 20-37 illuminate Euler's stay in Berlin in the service of the Prussian Academy of Sciences, 1741-1766, dur ing which he earned his reputation as the Princeps Mathematicorum of his time. Of this period, the book highlights the Silesian wars, Frederick's caprices and misbehavior against Euler, as well as Maupertuis's presidency of the Acad emy and his priority affair concerning the Principle of Least Action. Finally, pages 41-45 are devoted to Euler's second St. Petersburg period, 1 766-1783, during which he composed
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nearly half of his scientific oeuvre. The move to St. Petersburg was a result of an invitation by the empress Catherine the Great, who is pictured as a gracious protector of the arts and sciences, but who also displayed an unusual appetite for men. A special event captured in this section (and on the cover of the al bum) is the dramatic shipwreck suffered by Euler's family and household on their journey to St. Petersburg on the Baltic sea. Also illustrated is the great fire of St. Petersburg in 1771, which de stroyed a large part of that city, and from which Euler (then totally blind) and his manuscripts were neatly res cued. The last scene of the album shows a team of contemporary scientists, some easily identifiable, occupied with the editing of Euler's legacy of manuscripts and letters. In general, the quality of Elena Pini's drawings is good and her palette is qui etly colorful. Occasionally, however, the characters are not easy to identify. As for whom such a comic is suit able, I am in doubt: Is it for school children, for undergraduate university students, or for academic scholars? Of these alternatives, I would opt for the third one. I fear that a school child showing interest in the sciences and the accomplishments of the great heroes of mathematics would be disappointed, and I also doubt whether he or she would be very much amused by it. It is not that the album lacks humour, but the wit is so subtle that it is impossible to grasp without at least some familiar ity with the life, science, and politics of the Enlightenment era. Fortunately, the authors have recognized this by col lecting some helpful "appendices" list ing Euler's life and work, contemporary rulers, scientists, and artists, and by ap pending footnotes explaining some dis joint events and quotations in German (often in a Swiss dialect) or Russian. Ad ditionally, as a challenge, the authors have left several anachronisms (and er rors) for the readers to find (the right answers are promised to be announced on the Euler tercentenary website). I, for my part, take this opportunity to point out some less trivial inaccura cies on pages 34-35, where Euler is first pictured in the company of the mar grave Heinrich von Brandenburg Schwedt. As suggested by the picture, the two men were brought together by
62
THE MATHEMATICAL INTELLIGENCER
their love for music. In fact, in the scene, the margrave tells Euler that he very much admires his theory of music (alluding to Euler's Tentamen Novae Tbeoriae Musicae, 1739) and that he considers him to be worthy of teaching his "lazy" daughter something, not nec essarily mathematics. Of course, Euler is a man to be reckoned with, and in the next picture, the margrave's daugh ter, the princess Sophie Charlotte, is seen at her desk crowded with piles of Euler's letters on various philosophical subjects, looking rather desperate while the voice behind the door tells her not to go out before she has read every let ter. In the last picture of this episode, the princess cries out: "This will take years. I may as well go to a nunnery. " Now, what the reader may not know is that this is precisely what happened. The eldest daughter of margrave Friedrich Heinrich von Brandenburg Schwedt and princess Leopoldine of Anhalt-Dessau, whose name in reality was Friederike Charlotte Leopoldine Luise von Brandenburg-Schwedt ( 17451808), would indeed ascend the throne of the protestant nunnery of the city state of Herford (Germany) and rule as its last princess-abbess. Besides, nowhere have I seen evidence that the princess was not diligent. On the con trary, based on the engaging style of the Letters (and I mean of course the orig inal Lettres a une Princesse d'Allemagne
sur divers sujets de Physique et de Philosophie, written 1760-1762 and
published 1768-1772; not the numerous translations that have suffered from un fortunate alterations and omissions), I infer that Euler felt that his pupil was genuinely interested in science. More over, in the Letters there is a passage where Euler recalls an encounter with the margrave's family six years earlier and where Euler's scientific tutoring of the margrave's two daughters had been going on in Berlin already some time before 1760, when (following a move of the margrave's family to Magdeburg) the lectures were pursued in written form. To conclude, I would say that this intellectual comic album is a nicely il lustrated companion to the existing bi ographical literature on Euler. As a work of art it is charming, but unfortunately not quite easily accessible. However, if the book manages to create a serious
interest in the life and achievements of a great mathematician and a human spirit, then it has appropriately justified itself. Technical Research Centre of Finland P.O. Box 1 000 FIN-02044 VTT Espoo, Finland. e-mail:
[email protected] GraBmann (Vita mathematica) by Hansjoachim Petsche BASEL: BIRKHiiUSER, 2006,
58, ISBN 978·3·
7643·7257-6; ISBN 3-7643-7257-5 REVIEWED BY GERT SCHUBRING
� iographies
of mathematicians constitute a key dimension of L mathematical historiography and its literature. Active mathematicians en joy sharing the hopes and disillusions, the successes, sufferings, contests and rivalries of famous mathematicians. Such biographies need not necessarily contain essentially new material-as did, for instance, Laura Toti Rigatelli's biography of Evariste Galois [12] , with her findings on the political motivations for Galois's "sacrifice" of his life. A fresh presentation of knowledge already known that addresses complementary contexts or integrates hitherto separated traditions or research approaches will find interested readers, too. One will therefore open this biography of Her mann GrafSmann with the expectation of additional insights, even if it does not much exceed Friedrich Engel's classical biography of 1 9 1 1 . And even if one does not share the author's opinion that GrafSmann was "one of the most extra ordinary research personalities of the nineteenth century" [ 1 , p. xv], GraB mann is distinguished in fact by sev eral characteristics-autodidact, outsider, idiosyncratic terminology-which make him attractive historically. A dimension complementing traditional biographical literature is the detailed study of the political and social context of Stettin, the town in Pomerania which deter mined the largest part of GraBmann's life. Yet this analysis is based on rather raw-boned categories of political his-
tory (see, for instance, Figure 1 on p. xxi). The present book by Hans-Joachim Petsche, a philosopher at Potsdam Uni versity, is essentially his PhD thesis, which he defended in 1979 at the Piid agogiscbe Hocbschule in Potsdam, then in the German Democratic Republic. One is charmed to see that the author has refrained from deleting the quota tions of Karl Marx and Friedrich Engels in his original version of 1979 from his new version, published after the demise of the GDR. An important question that must be raised by republication after a long pe riod of time concerns the more recent literature on GraBmann published be tween 1979 and 2006. Except for some reordering, all chapters have been adopted from the original 1979 thesis in a more or less unmodified way. For in stance, the chapter analysing GraB mann's mathematical achievements has remained unchanged although there are, among other relevant studies, the book by Zaddach [13] and a number of contributions in the Proceedings of the International GraBmann Conference of 1994 [9]. Disregarding international lit erature, the author even claims that GraBmann "nowadays is a largely un known mathematician" [1, p. xiiil. Like wise, as regards GraBmann's biography and his Nachlass, there are-contrary to the author's assertion [ 1 , p. 1 05]-a number of new documents which have since been unravelled and which are important for the methodology issue (see below) and for Grassmann's ideas on the teaching of language [4]. Given the intense research on the history of nineteenth-century mathe matics since the 1970s, one reads with surprise that this history still waits to be investigated [1, p. xiii]. Given GraB mann's, in fact, extraordinarily innova tive mathematical conceptions, one of the key interests of the historiography of mathematics, and of many mathe maticians, has always been to find their sources. The major obstacle has been GraBmann's own discretion with regard to his sources. This silence is actually a common characteristic of the publica tions presented by many Prussian Gym nasium teachers of the first half of the nineteenth century: They preferred to proudly pretend to intellectual original ity. Any attempt towards progress on
this issue would have to start by ad mitting ignorance and by presenting reasonable tentative hypotheses in or der to attain some approximate result. What is Petsche's approach to this key methodological question? Instead, he starts from an alleged certitude. More strongly than in his original thesis of 1979, he claims to be able to "deter mine" the factors conducive to the emergence of GraBmann's novel scien tific conceptions [1, p. xvii]. Regarding the philosophical-method ological factor, the author is convinced that Schleiermacher's Dialektik was most decisive. Since Schleiermacher's Dialektik presents a recurrent topic in the literature about GraBmann, this needs some discussion. Engel was the first to ascribe to Schleiermacher a key influence on GraBmann, in his exten sive biography of 191 1 . Engel, who had intensely searched for sources, even in archives, had found two curricula vitae which GraBmann had composed for dif ferent exam purposes, and gave a num ber of quotes from both [3]. The sec ond CV of 1833-not of 1834, as Engel wrote erroneously-written for GraB mann's application to be admitted to the first exam for future pastors, referred ex plicitly to Schleiermacher, the then lead ing Protestant theologian and major au thority within the Prussian Protestant church. While the applicant claimed he understood nothing of Schleiermacher's lecture on Dialektik in his second term, he professed to have been able to ap preciate the latter's lecture in the last term (on psychology) as a means for heuristics, to find the "Positive" on one's own [3, p. 2 1 f.]. Without reflecting on the purpose and the occasion of this text, and a possible theological mean ing of the "Positive, " Engel understood this CV to be an objective source, de ducing from it his own general assess ment that it was Schleiermacher who had crucially influenced GraBmann's mathematical approaches [3, p. 28]. Albert C. Lewis, in his study of 1977, took up more systematically the ques tion of the conceptual basis of GraB mann's ideas. He dwelt upon Engel's assertion of the decisive character of Schleiermacher's influence, trying to show that Schleiermacher's lectures on Dialektik had indeed provided that con ceptual basis. Using Engel's quotes from the 1833 CV, Lewis no longer mentions
this text's specific nature, dissociating it from a concrete purpose and date [5, p. 109, n. 24]. Although admitting that the Dialektik contains but few references to mathematics, Lewis maintains that mathematics "is presumably the subject of the Dialektik the same as any other branch" of knowledge [5, p. 1 12] . Lewis tried to establish such a relation be tween the Dialektik and GraBmann's key opus, the Ausdebnungslehre of 1844, the so called A 1 . Strangely enough, the bulk of his study consists in a methodological analysis of the A1 itself, with few references to Schleier macher's Dialektik. Lewis's principal ap proach is to highlight opposed mean ings and dualities in GraBmann's approaches in constructing mathemati cal knowledge-with the tacit assump tion that dialectics is a method of pro ceeding by opposites [5, p. 1 1 2 and 160f.J. This seems to constitute a funda mental misunderstanding of Schleier macher's approach. Usually Schleier macher's key opus is interpreted along the very well known lines of Hegel's Dialektik. Instead, one must be aware that Schleiermacher exhibits an entirely independent approach and thinking. He understood dialectics according to the original Greek meaning of the term: as a dialogical way of establishing the cer tainty of knowledge. Settling disputes between different views of individuals by means of language, of dialogue, and by applying general rules, dialectics has the task of establishing an organism of knowledge, to act as a Wissenscbafts lehre. Starting from common knowledge and achieving certain knowledge by di alogue of the partners clarifies what "knowledge" means and how in it thinking and experience are related (see [7, p. 1 16ff.]). Petsche, in his PhD thesis of 1979, although apparently not aware of Lewis (1977), proceeded analogously. Con vinced of Schleiermacher's determina tive function and based on Engel, he collected the few hints that GraBmann had read Schleiermacher and trans formed them into proof of influence. In the present book version, Petsche has confirmed and elaborated this view of essential determination by Schleierma cher-without, however, demonstrating it concretely in an analysis of GraB mann's work. He restricts himself to as-
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serting a "masterfully implicit application [of Schleiermacher's Dialektik] in Her mann's Ausdehnungslehre' [ 1 , p. 173]. Even in Chapter 4, expressly devoted to the philosophical concepts of the A1, there are but few general references to Schleiermacher's ideas [1, p. 252f.; 270f.]. Actually, Victor Schlegel-GraBmann's first biographer and one who knew him personally-gives an entirely different assessment of Schleiermacher's impor tance for GraBmann. While attributing to Schleiermacher a considerable influ ence, he reports it as a deviation from mathematics. According to Schlegel, GraBmann-after having finished in 1 840 his thesis on the tides, the basis for his later Ausdehnungslehre--did not continue elaborating his new mathe matical conceptions, but diverted him self to philosophical studies of lan guage: The Dialektik of his venerated mas ter Schleiermacher published shortly before attracted him too mightily and tugged him temporarily into a new current, for which he worked jointly with his brother Robert. Due to this current, they elaborated the next year (184 1 ) a philosophical grammar; they used its achieve ments in putting into writing the 'Grundriss der deutschen Sprach lehre ' [outline of German grammar] and the 'Leitfaden fUr den deutschen Unterricht' [manual for German lan guage instruction], both published in 1842. Schlegel stated that "eventually," about Easter 1 842, GraBmann "returned to mathematics with all his forces" [8, p. 4f.]. In my own research for the GraB mann Nachlass [10], I succeeded in find ing the two CVs. The first one, of 1831 for the teacher examination, mentioned Schleiermacher just once and had Greek philology as its focus. It is evident, hence, that the CVs-just as any his torical text-must be interpreted in their proper context. Petsche dismisses this remark [ 1 1 , p. 60f.] by claiming that "the sincerity of his soul" would have ex cluded any "instrumental use" by GraB mann in writing a text [ 1 , p. 279, n. 148]. Actually, there are two texts by GraB mann with differing emphases, a fact
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THE MATHEMATICAL INTELLIGENCER
which makes the requirement of a con textualized interpretation imperative. I am preparing a publication of the two curricula vitae. One recent research study [6] the au thor does mention is the assessment of the arithmetic textbook that the broth ers Hermann and Robert GraBmann published in 186 1 . Contrary to a widely accepted view (see [2]), the author does not understand it as a moment in the movement of axiomatizing arithmetic, but as a constructivist approach [ 1 , p. 2 2 1 ff.] . He does not make explicit, however, his conception of construction and to what extent it excludes an ax iomatic approach. Here again, all de pends on whether the interpretation by an overarching Schleiermacher frame work is acceptable or not. This book underscores the dimen sion of the methodological challenge in understanding historical texts in mathe matics, even for a biography.
[8] Victor Schlegel,
Hermann Grassmann: sein
Leben und seine Werke
(Leipzig: Brock
haus, 1 878). [9] Gert Schubring {ed .}, Hermann Gunther GraBmann (1809- 1877): Visionary Mathe matician, from
Scientist and Scholar: Papers
a
Sesquicentennial
Conference
(Boston Studies in the Phtlosophy of Sci ence, Vol. 1 87). (Dordrecht: Kluwer Acad emic Publishers, 1 996). [ 1 0] Gert Schubring, "Remarks on the Fate of Grassmann's NachlaB", In: [9], 1 9-26. [1 1 ] Gert Schubring, "The cooperation be tween Hermann and Robert Grassmann on the foundations of mathematics," In: [9], 59-70. [1 2] Laura Toti Rigatelli, Matematica su/le bar ricate: vita di Evariste Galois
(Firenze: San
soni, 1 993). [1 3] Arno Zaddach, GraBmanns Algebra in der Geometrie:
mit Seitenblicken
wandte Strukturen
auf ver
(Mannhetm: BI-Wis
senschafts-Verlag, 1 994). Gert Schubring
REFERENCES
[1 ] Hans-Joachim Petsche, GraBmann (Basel: Birkhauser, 2006).
Fakultat fOr Mathematik Universitat Bielefeld Postfach 1 00 1 3 1
[2] L. G. Biryukova; B. V. Biryukov, "On the ax
D-33501 Bielefeld
iomatic sources of fundamental algebraic
Germany
structures: the achtevements of Hermann
e-mail:
[email protected] Grassmann
and
Robert
Grassmann,"
[Russian] Modern Logic, 7 (1 997), no. 2, 1 31 -1 59.
[3] Friedrich Engel,
Grassmanns Leben: Nebst
einem Verzeichnisse der von Grassmann ver6ffentllchten Schriften und einer Ober sicht
des
handschriftlichen
Nachlasses
(Leipzig: Teubner, 1 9 1 1 ). [4] Erika Hultenschmidt,
"Hermann
Grass
mann's contribution to the construction of a German 'Kulturnation' -Scientific school grammar between
Latin
tradition and
French conceptions," In: [9] , 87-1 1 3. [5] Albert Lewis, "Hermann Grassmann's 1 844
Arthur Cayley Mathematician Laureate of the Victorian Age by Tony Crilly THE JOHNS HOPKINS U NIVERSITY PRESS,
Ausdehnungslehre and Schleiermacher's
BALTIMORE, MARYLAND, 2006, HARDCOVER,
Dialektik," Annals of Science
US$69.95, XXII + 610 PP, ISBN 0.8018-8011-4
34 (1 977),
1 03-1 62. [6] Mircea Radu, Nineteenth century contnbu tions to the axiomatizat1on of arithmetic: a historical reconstruction and comparison of the mathematical and philosophical ideas of Justus GraBmann, Hermann and Robert GraBmann, and Otto Holder.
PhD thests,
Btelefeld Untverstty, 2000.
[7] Wolfgang Rod, Dialektische Philosophie der Neuzeit.
Vol.
1:
Von
(MOnchen: Beck, 1 97 4).
Kant
bis
Hegel
REVIEWED BY HENRY E. HEATHERLY
I
ony Crilly has written a definitive biography of Arthur Cayley. Not only is Cayley's mathematical and scientific work discussed thoroughly, but careful attention is given to his pro fessional career, to his personal life, and to his many colleagues and friends. First and foremost, Cayley was a mathematician. He also did important
work in astronomy and physics, as well as making significant contributions to a widely used legal manual. Cayley was one of the most prolific mathematicians of all time, publishing well over nine hundred papers [2]. Arthur Cayley was born on August 16, 1 82 1 , at Richmond upon Thames, a genteel town outside of London. His family was visiting England at the time, on home vacation from St. Petersburg, Russia, where his father, Henry Cayley, was a merchant. Arthur lived his early childhood in St. Petersburg, until his family returned to England when Arthur was 10. There Arthur first attended school, having been tutored at home while in Russia. At age 1 4 he was en rolled in the senior department of King's College in London. There he received a comprehensive mathematical educa tion. Cayley was well prepared for the next step in his formal education, Trin ity College, Cambridge University, which he entered in 1 838. There he was "A Cambridge Prodigy," as Crilly em phasizes with the title of his second chapter. Cayley's course of study lasted for ten terms, culminating in an honors degree after taking the six-day Tripos exams in January 1842. In this highly competitive exam, Cayley scored well above all the other competitors, earn ing the title of "Senior Wrangler." Soon afterward he became the First Smith's Prizeman. The Smith's Prize examina tion was another week-long schedule of competitive tests, 1 13 questions of fered over a five-day period. Even while preparing for these ar duous exams, Cayley was doing math ematical research. His first published paper appeared in May 1 84 1 . This pa per, "On a Theorem in the Geometry of Position, " related algebra to geome try and made use of determinants, though Cayley did not use that term. In terestingly, the young author made the contribution anonymously, "from a cor respondent." With this outstanding undergraduate record, Cayley was an obvious choice for a Cambridge fellowship. Unique among Cambridge colleges at the time, Trinity required fellowship aspirants to take further competitive exams in the classics as well as in mathematics. In September 1842, Cayley took these ex ams. Twenty-two candidates competed
for seven fellowships. Cayley did well on a diverse array of topics, from the philosophy of Plato, Aristotle, and Locke to finding the variation in the moon's orbit. Cayley was selected as a Trinity fellow in October 1842. After visiting Switzerland and Italy, the new Cambridge Fellow turned to re search with delight. He quickly pub lished papers on geometry, elliptic func tions, mechanics, and algebra. He began a long correspondence with George Boole on n-dimensional deter minants and other algebraic topics. Cay ley became familiar with the new sub ject of quaternions, introduced by Hamilton in 1843, and in December 1844 submitted his first paper on that topic, making connections between ro tations and quaternions. In January 1845 Cayley discovered the eight dimen sional, nonassociative, noncommutative algebra that became known as "the Cay ley numbers . " He published these re sults in a postscript to a paper on el liptic functions [1). Here Cayley's habit of publishing his results quickly served him well, for John Graves had discov ered this algebra in December 1843 and had communicated this to Hamilton. Hamilton (1844) noted that Graves's "octaves" are nonassociative, an at tribute that Hamilton felt detracted from their worth. Graves did not publish his results until after Cayley's paper had ap peared [3). Also in 1845, Cayley began publishing papers in what would even tually become a life-long work, invari ant theory. His early work on the sub ject received a tepid response in England. But Cayley's reputation as a mathematician was rising as his publi cation record grew in size and diver sity. Of the 500 or so students who en tered Cambridge with Cayley, more than half went into the Church of En gland after graduation. The next most popular career choice was the law. In April 1846, Cayley took the decisive step onto this second path: He entered Lincoln's Inn as pupil barrister. At that time the four legal inns in London also served an educational function. Each of these inns had a library, and the pupil barristers were supervised by working barristers. The young Cayley was a pupil of the leading conveyancy coon sui and barrister, ]. H. Christie. Cayley rapidly gained expertise in this area, but
he also continued working at mathe matics. Sometime in 1847, Cayley met another mathematician who was work ing as a barrister only a short walk away from Lincoln's Inn, James Joseph Sylvester. They became friends and mathematical confidants. It was a close relationship that lasted until the end of Cayley's life, but never resulted in a joint publication. Fortunately for historians, a considerable amount of the written cor respondence between the two has been preserved. Many of these letters can be found in [5). Cayley led a measured, tranquil life, developing in his legal career and grow ing in mathematical power and renown. A common bond of mathematics dom inated his immediate circle of corre spondents, which included Sylvester, Boole, Kirkman, Salmon, and Hermite. Cayley maintained contact with Cam bridge by taking the role of examiner for the regular Trinity College internal examinations. After three years as a le gal pupil he received full membership in the Society of Lincoln's Inn. He was a barrister. In the 1850s, Cayley's mathematical scientific reputation flourished. In 1852, he was elected a Fellow of the Royal Society of London. In 1857 he became a fellow of the Royal Astronomical So ciety, and in 1859, he was awarded the Royal Medal of the Royal Society of Lon don. His professional interest in astron omy had begun in 1 855, and his first paper on that subject, on lunar theory, appeared in 1857. The subject of lunar motion was one to which he would re turn. During the decade, he had made substantial contributions to the theory of permutation groups, further devel oped the notion of a matrix, and ex plored the algebra of matrices. In both group theory and matrices, Cayley made connections with Hamilton's quaternions. In one of his 1858 papers, he gave what became known as the Cayley-Hamilton Theorem for n-by- n matrices. However, he only gave a proof in the 2-by-2 case, writing that he had "not thought it necessary to un dertake the labor of a formal proof. " This hand-waving argument drew criti cism from Boole. A lack of rigor also appeared in Cayley's attempt to prove the fundamental theorem of algebra in 1859. There he showed the desired al gebraic result was equivalent to a geo-
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metrical statement which he said was "a matter of intuition. " Once Cayley had convinced himself of the truth of a mathematical statement, he would often verify it in some elementary cases and then just assert the general statement. By the beginning of 1859, Cayley's need for an academic position, and his suitability for one, was clear to Cayley and to his friends. Many of the latter al ready held suitable positions: Thomson, Stokes, and Boole held chaired profes sorships, while Salmon and Sylvester were lecturers. Cayley had unsuccess fully applied in 1856 for the chair of natural philosophy in Aberdeen, and tried to secure a position at the pro posed, but never formalized, "Western University of Great Britian," in Wales. In 1863, the Sadlerian Chair in pure mathematics was established at Cam bridge. Cayley applied for this chair and was elected to it in June of 1863. Perhaps the most important conse quence of his secure academic position was that Cayley could settle down. In September 1863, he married Susan Mo line. This was a lifelong and tranquil marriage, which produced two off spring, Henry in 1870, and Mary in 1872. Geometry dominated Cayley's early life as a professor. His inaugural lecture at Cambridge (November 3, 1863) was on analytical geometry. As early as 1846, he had published work on n-dimen sional analytic geometry. In his career he published a large number of papers on a wide variety of geometric subjects. In a short note published in 1865, Cay ley introduced the idea of non-Euclid ean geometry to English readers. In this paper, Cayley referred to Lobachevsky's 1837 paper in Grelle'sjournal, mention ing it as a "curious paper" with certain parts as "hard to be understood. " This serves to illustrate the attention Cayley paid to the mathematical literature, in cluding that in the major continental journals. During this period Cayley was influenced by the work of Chasles and Plucker. He had most in common with Plucker, sharing a belief in the primacy of projective geometry and a commit ment to the analytic method. Cayley's ever growing scientific rep utation, his distinguished university po sition, and his reputation as a man both wise and prudent led to his being elected to several leadership positions in scientific organizations: President of
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THE MATHEMATICAL INTELLIGENCER
the London Mathematical Society (1868-1870); President of the Cambridge Philosophical Society (1869-1870); Pres ident of the Royal Astronomical Society (1872-1874); and President of the British Association for the Advancement of Science (1883). He was awarded the (first) De Morgan Medal of the London Mathematical Society in 1884. Cayley's mathematical and scientific interests remained diverse. He contin ued his great work on invariant theory, which included his monumental 10 memoirs on quantics, the last appear ing in 1878. That year he also stated the representation theorem for groups which we know as "Cayley's Theorem." He addressed most of the then-current topics in algebra, did work in differen tial equations, and continued his long standing work on elliptic functions. The latter led to the only full-length book Cayley published: An Elementary Trea tise on Elliptic Functions, 1876. At a meeting of the London Mathe matical Society in June 1878, Cayley raised a question that puzzled mathe maticians thereafter for almost a cen tury: "Has a solution been given of the statement that in coloring a map of a country, divided into counties, only four distinct colors are required, so that no two adjacent counties should be painted in the same color?" He followed up this query in 1879 with a paper on this "Four Color Problem" in which he ascribes the problem to De Morgan. There Cayley stated, "I have not suc ceeded in obtaining a general proof. " Cayley's interest in graph theory goes back to the 1850s. It was originally mo tivated by questions from organic chem istry. This Jed him in 1857 to introduce the idea of a "tree" in graph theory. In the 1870s, Cayley worked on the prob lem of enumerating "unrooted trees. " These mathematical problems were closely connected to chemistry, and Cayley's theory successfully predicted the existence of some alcohols before they were known to exist in nature. Commentary from some of Cayley's Cambridge students who later became mathematicians of note and the official Cambridge records give one a good idea of what Cayley did in his professional lectures. Until 1886 he was required to give only a single course of lectures for one term each year. The topic for his course varied from year to year and re-
fleeted his current research interests. A five-year sample is as follows: 1877, algebra; 1878, solid geometry; 1879, differential equations; 1880, theory of equations; and 1881, Abel's theory of theta functions. From George Andrew Forysyth, who attended the 1879 lec tures, we have that "old notes were never used a second time," and that the lectures were on Cayley's latest research. Cayley played an important role in the development of many mathemati cians and scientists. Less well known, perhaps, is that he "was a significant fig ure in the movement towards women's education. " He was the first president and chair of the college of what is now Newnham College, one of the first two colleges for women at Cambridge. His best-known female student was Char lotte Scott. She regularly attended Cay ley's lectures, and he helped her later in her mathematicaVprofessional career. (For more on the career of Charlotte Scott, see [4].) In 1892, Cayley's general health de teriorated, very likely due to the long term effect of cancer. Even in this de bilitated state, during the last three years of his life Cayley published 40 papers embracing almost the whole field of pure mathematics. During his last sum mer, he wrote a short monograph on the Principles of Book-Keeping by Dou ble Entry and a paper on quaternions. Until the end, Cayley lived the life of the mind, wholly devoted to intellectual endeavors. Arthur Cayley died at 6 p.m. on Saturday, January 26, 1895. The work under review contains much useful and interesting supple mentary material. This includes a lengthy list of Cayley's community of scholars and friends, each given with their link to Cayley. A nine-page glos sary of mathematical terms is provided, together with an extensive section of supplementary notes and a substantial bibliography. There are 24 pages of photographs, portraits, and illustrations. These include a portrait of Cayley as Se nior Wrangler and photographs of him at ages 35 and 69. The book is remarkably free of er rors. The mathematical typo in the ex ponent of e in note 54 on page 528 is a rare exception. This is a scholarly work of the highest quality. It should be in every university library, and I rec ommend it to all who wish to delve
deeply into the life of Arthur Cayley and mathematical life in nineteenth century Britain. REFERENCES
(1] Arthur Cayley, "On Jacobi's elliptic func tions and quatemions," Philosophical Mag azine
26 (1 845), 208-21 1 .
(2] Arthur Cayley and A. R. Forsyth, (eds.), The Collected Mathematical Papers of Arthur Cayley,
1 4 vols., Cambridge Univ. Press,
Cambridge, 1 889-1 898. [3] John Graves, "On the theory of couples," Philosophical Magazine
26 (1 845), 3 1 5-
320. (4] Patricia Clark Kenschaft, "Charlotte Angas Scott (1 858-1 931 )," 1n Women of Mathe matics ,
Louise S. Grinstein and Paul J.
Campbell, (eds.), Greenwood Press, West port, CT, 1 987, 1 93-203. [5] Karen Hunger Parshall, James Joseph Sylvester: Life and Work m Letters,
Claren
don Press, Oxford, 1 998. Mathematics Department University of Louisiana, Lafayette Lafayette, LA 70504-1 0 1 0 e-mail:
[email protected] H e l mut H asse u nd Emmy Noether. Die l
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