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What Is Modern about "Modern" Mathematics? Hardy Grant In 1901 Bertrand Russell offered the opinion, subsequently much quoted, that in mathematics we never know what we are talking about nor whether what we are saying is true . I imagine that most mathematicians will have a sense of what lies behind these words, and will agree that the view they express is certainly modern. I should like, however, to put this last point a bit more strongly and suggest that Russell's stance is calculated to make a Euclid or even a Newton spin around in his grave. Now, beside Russell's pronouncement I want to set two others. The first will be recognized by readers of Hardy's Apology, where it is quoted with approval. It is not, however, about mathematics but about poetry; it is A. E. Housman's famous assertion (1933) that "poetry is not the thing said but a way of saying it" . Hardy did not add that this sentiment is calculated to make a Vergil or a Milton spin in his grave. And here is a third declaration, with the same power to upset distinguished shades, but this time bearing on the visual arts. In 1914 Clive Bell laid it down that "the representative element in a work of art may or may not be harmful; always it is irrelevant" . I venture to suggest that these three roughly contemporaneous quotations hint at certain parallels between mathematics and other spheres on the one hand, and certain distinctions between ancient and modern perspectives on the other, which may be worth exploring. sisted that for past generations an unvarying set of three assumptions underlay discourse in these realms, namely that, first, all genuine questions must have one true answer and one only, the rest being necessarily errors; secondly, there must be a dependable path towards the discovery of these truths; and, thirdly, The Classical Tradition To get some further anchorage in a large and amorphous subject, I call no less a witness than Sir Isaiah Berlin, the great historian of political and moral philo; sophy. Repeatedly, almost obsessively, Sir Isaiah has in62 THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 3 (~ 1995 Springer-Verlag New York the true answers, when found, must necessarily be compatible with one another and form a single whole . entities were guiding presences in the divine mind at the origin of the universe. "God the creator of the massive structure of the world," wrote Boethius (6th century Again, the reference here is primarily to moral and A.D.), "considered [arithmetic] as the exemplar of his own political thought. But Isaiah Berlin prefaces this list of thought, and established all things in accord with it" . perennial beliefs with the phrase "as in the sciences"; Thus mathematics seemed uniquely to promise knowland indeed it would seem that, both logically and psyedge of a transcendent reality. And of course an actual chologically, these convictions must be underwritten by example was ready to hand, for most of the life of the analogous postulates about the nature and reliability of tradition, of absolute and certain truth already attained human knowledge of the world around us. Translated by right method, namely Euclidean geometry. into such terms, that would mean that our ancestors beMore to the present point, this Platonist epistemollieved (at a minimum) that ogy was what John Dewey called a "spectator" theory of - - the world presents to us an objective reality, which knowledge . The grasping of the eternal verities was likened to the everyday experience of looking and seein principle we can know; there is about the world a unique set of truths which i n g - with the "mind's eye," to be sure. This meant that the acquisition of knowledge was in an important sense are objectively valid and mutually consistent; - - b y the right choice of method these truths are a passive experience. Not, of course, that it lacked emotional commitment and i n t e n s i t y - - n o reader of Plato discoverable. would suppose that. But the facts are "out there," and Dissenters and skeptics notwithstanding, these beliefs thus are, so to say, beyond human control. The observer were indeed at the heart of the mainstream epistemol- hopes to apprehend them, but cannot and does not hope ogy in the West in antiquity and beyond. "It was the uni- to change them or shape them, still less to in any sense versal conviction among those who thought seriously," create them. This is true in particular of math6matics. The says the great medieval scholar David Knowles, "that Greeks took the view that "the mathematician cannot crethere was a single true rational account of man and the ate things at will, any more than the geographer can; he universe.., as valid in its degree as the revealed truths too can only discover what is there and give it a name." of Christianity" . For convenience, and with all due I shall identify later the writer of these words. reservations, I shall label this theory of knowledge "clasThis passive character of knowledge acquisition sical" or (equivalently) "Platonist." shows clearly in the various metaphors by which our Whence came this trust that human beings can attain ancestors sought to explain the workings of our minds. certain knowledge of the real? Note first that it is after all Thus Plato and Aristotle already liken the mind to wax very much the common sense of the common man, and receiving the impression of a seal . Of all such imso is entirely natural. But sophisticated thought added its ages, probably the most powerful and pervasive was of own elaborate rationalizations. I shall indicate in a mo- the mind as a mirror, reflecting (but not shaping!) exment the powerful role played in this respect by mathe- ternal reality. That was a very natural metaphor in a matics. Plato's influential Timaeus told how the physical context where, as I have said, the mental act of knowworld was actually modeled on divine archetypes, an ori- ing was so commonly analogized to the physical act of gin which seemed to guarantee its rationality . Chris- seeing. Moreover, the mirror metaphor offered a convetianity offered convincing reinforcement. St. Paul wrote nient explanation of the manifest failings of our quest for to the Romans, "The invisible things of him from the understanding: one could say that human sin and error creation of the world are clearly seen, being understood bend or cloud the mirror and so distort its images (some by the things that are made" . Indeed, is man himself lovely lines in Shakespeare's Measure for Measure say just not made in the image of God? And has he not been en- this) . The methodological reformers of the Renaisdowed by his creator with a "natural light," by which he sance, like Francis Bacon and Descartes, can usefully be may see eternal truths? Many postulated a fundamen- pictured as attempting to get truer reflections of reality tal congruence or "fit" between our minds and "reality"; by straightening and polishing our mental mirrors. perhaps the best known statement is Spinoza's dictum The Platonist epistemology that I have been sketching that "the order and connection of [our] ideas is the same entailed a corresponding aesthetic, a traditional philosoas the order and connection of things" . phy of artistic purpose, and here the contrast that I want No secular agency under~rote this optimistic vision to draw between ancient and m o d e m ways of thought of human knowledge more powerfully than did mathe- appears with particular vividness, being in fact close to matics. Several strands of thought contributed. Plato's a polar opposition . The artist's aim, on the old view, theory of Ideas, the fountainhead of the classic epistemol- was the imitation ("mimesis") of the nature of things, ogy, drew deeply on the perceived status of numbers and of the eternal verities, as his culture by general consengeometrical shapes-7--independent of, yet accessible to, sus intuited them. The beauty of his work was in direct human minds. From the Timaeus to Kepler and Galileo, proportion to his success in this endeavour. Beauty was the conviction persisted that these same mathematical thus objective, and was in fact identical with truth. The - - THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 3, 1995 63 artist worked consciously within a tradition whose timehonoured conventions he was at pains to obey. It was no part of his business to express his own personality. His moral imperative was rather that of the good reporter: to set out the truth of things without letting his own idiosyncracies or his biases intervene. There is a touching example of this in Dante, when the poet invokes the Muses to help him describe with utmost fidelity the horrors of Hell as his vision revealed them . I have been trying to outline a complex of ideas which, to summarize, has three facets: the "classical" or Platonist epistemology, a corresponding monism in political thought, and an attendant philosophy of the arts. The tradition that these jointly represent had a very long careen Beginning with the Renaissance, to be sure, there were a number of challenges to those old beliefs, hints of changes to come. Let me mention just three of these: - - I n the Renaissance and afterward a growing cult of individual, self-conscious genius weakened the old vision of the artist as the humble, anonymous expositor of communally shared truths. - - A small minority of thinkers--posterity salutes especially Giambattista Vico ( 1 6 6 8 - 1 7 4 4 ) mounted a drastic challenge to the traditional epistemology. On their view we know with certainty only those things which we ourselves make (including mathematics!). The opening here for subjectivist and relativist theories of knowledge is obvious. --Similarly, from the 17th century's doctrine of the "secondary" qualities of objects, qualities like colours and tastes, held to be present in our minds rather than in the objects themselves, it can have been no great leap to the idea that when we form pictures of the world we are not passive recorders of the given but active contributors. Yet despite all such anticipations, the older tradition remained largely intact until well into the 18th century. Consider: in that century Voltaire, Leibniz, and others still believe that eternal, necessary laws govern even such slippery realms as morals and metaphysics. Rameau in France can still hope to reduce musical theory to a few clear and distinct principles and mathematically formulated rules . Winckelmann in Germany preaches an ideal, absolute beauty in the visual arts . Relativism, pluralism, subjectivism are still in the future. Nor is the primary reason for all this far to seek. The unprecedented triumphs of Newtonian science seemed to have solved the riddles of the physical world and to offer a methodology for unveiling eternal truths in other realms. The Romantic Revolution And so when, in the decades around 1800, the great change finally c o m e s - - when the mirror finally breaks - the impulse comes in large measure as a reaction against the mathematizing spirit. A full account would be the 64 THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 3, 1995 full story of the genesis and course of Romanticism, obviously not to be undertaken here even had I the competence. I can only cite instead some of the trends that bear most directly on my present theme. Three seem especially germane: the reaction against the perceived dominance of scientific thinking, the rise of nationalism, and the rise of historical consciousness. Singly or together, these ensure that the Romantic spirit refuses to grant sole possession of absolute truth to any particular group of people, any particular point of view, any particular methodology. There is now an insistence in some quarters that each group, each nation, has so to say its unique window on the world, its particular experiences, visions, values, which may be simply incommensurable with those of others. There is a growing belief that the supposedly natural, universal "laws" of the commonwealth are in fact shaped by particular human beings and are rooted in place, time, and circumstance. ("There are no laws," says Vautrin to Rastignac in Le p~re Goriot (1833), "there are only circumstances" .) Thus Isaiah Berlin locates in just this period, the late 18th and early 19th centuries, the abandonment of those age-old axioms of political and social thought that I cited earlier. For him a key figure is Herder, with his celebration of the unique history and character of each people, and of the German people in particular . Meanwhile in epistemology the Romantic age sees the radical replacement of one canonical metaphor of mind by another. The mirror as passive reflector of external impressions gives way t o - - the lamp, conceived as actively lighting up the outer world from within . Coleridge, who of course is near the heart of the whole movement, speaks of "informing the senses from the mind, and not compounding a mind out of the senses" . Thus there emerges an epistemology in which the known is shaped, even created, by the knower, and the very idea of objective, absolute truth becomes deeply problematic or is abandoned altogether. Knowledge and truth increasingly seem subjective, relative, fragmented, pluralistic. Correspondingly, the traditional philosophy of the arts is by now essentially stood on its head. Artists now specifically reject its underlying Platonism, specifically deny (here I quote a contemporary witness) that "beauty could ever own any fixed, absolute form" - - there speaks the quintessential Romantic musician, Franz Liszt . The ancient imperative of mimesis is now widely mocked. The artist no longer draws the content of his work from a communal vision of the real, and no longer fashions its form according to communal tradition; content and form alike become intensely subjective. Often, the form of a work is seen as more significant than its c o n t e n t - - a heresy on the older view, but precisely the attitude expressed in those remarks by A.E. Housman and by Clive Bell that I quoted at the outset. Creative freedom is unlimited, but operates in the absence of socially agreed canons of meaning and of value. "Modern" art is born. I have tried to stress how deep and pervasive and longlasting was the old Platonist tradition, here supplanted. Arguably, then, the breakdown of that whole complex of ideas, the giving up of its assumptions, the casting off of its conventions, the lo~s of its certainties, the embrace of other possibilities, was the pivotal watershed in the history of western sensibility, and its aftermath is the defining condition-- some would say "predicament" - of modernity. The novelist and critic Gabriel Josipovici writes : What all the moderns have in common--perhaps the only thing they have in common-- is an insistence on the fact that what previous generations had taken for the world was only the world seen through the spectacles of habit .... The modems unanimously rejected the implication that the norms of the Renaissance corresponded with the necessary structure of reality.... Mathematics Too? Now it is well known that great changes come over mathematics too in just the years of the Romantic revolution. To me, several of these developments echo strikingly the upheavals that I have tried to describe in general culture. Thus 19th- and 20th-century mathematics lays increasing stress on form and structure as against content-hence Russell's famous quip. Absolutes and certainties crumble, until Morris Kline can lament that mathematics "contains no truths" . Axioms are no longer statements of self-evident fact but mere assumptions, which may be more or less arbitrary. Axiom systems multiply, and internal consistency replaces truth as the criterion of their validity and value. Diverse mathematical structures coexist side by side, like euclidean and hyperbolic geometry, individually valid though mutually irreconc i l a b l e - w h i c h is exactly how Herder viewed the differing world-views of national groups. Euclidean geometry, for centuries the stock example of objective truth and incontestable knowledge, now seems merely "the world seen through the spectacles of habit," and not at all "the necessary structure of reality." More generally, the contacts of mathematical structures with "reality" become ever more mysterious, or in some eyes simply irrelevant. Meanwhile the creative freedom of the individual mathematician is much cherished, but sometimes seems to issue in waywardness and sterility. "Much of the current picture of pure mathematics," says Ivor GrattanGuinness, "is a dispiriting accumulation of scholastic exercises in difficult manipulations of complex structures without regard to their intellectual or empirical implications" . Mutatis mutandis, the remark's tone echoes countless exasperated responses to modern experiments in literature, music, and art. Thus to the question raised by my title, "What is modern about 'modern' mathematics?", the answer from a cross-cultural perspective would be precisely this partici- pation of mathematics in many of the trends and characteristics of modernity as a whole. One can go on to ask whether in this context there may have been actual influence between mathematics and the rest of the culture; here I offer only the briefest of remarks. First, remember that well before the rise of Romanticism, mathematics began dealing with concepts and entit i e s - irrational and complex numbers, infinities and infinitesimals, dimensions beyond the t h i r d - - t h a t would seem to be pure creations of the human mind, not to be discovered in a mirror of the physical world. Could this internal trend have "spilled over," operating suggestively on other minds? I can only say that I know of no direct evidence, no declaration of such indebtedness. Indeed, may it be significant that the mathematicians' full acceptance of these exotic entities generally awaited the early 19th c e n t u r y - - t h e very decades when Romanticism so weakened the old epistemology? Are there other hints of this opposite direction of influence, the impact on mathematics o f the wider milieu? I.am for the most part a convinced "internalist," especially at 19thcentury levels of mathematical sophistication, and I have no doubt that there were good mathematical reasons for all of those developments, cited above, that so closely parallel modernist trends elsewhere. But is that absolutely the whole story? Let me dramatize the issue with a single example. In 1823 the younger Bolyai, exulting in his discovery of hyperbolic geometry, rejoiced that "from nothing I have created another wholly new world" - - something which Euclid never would have said . At very much the same time, Wordsworth, remembering his time at Cambridge, wrote  that I had a world about me; 'twas my own, I made it .... which Dante never would have said. Is the similarity a mere coincidence? Those two men never met; but they breathed the same air, and that air was full of revolutionary things. Yet ... all that said, I cannot resist giving the story one final twist. Platonism dies very h a r d - - and nowhere harder than among mathematicians. Kurt G6del's deep commitment to it is well known. Ren6 Thom declared fervently that "mathematical forms indeed have an existence that is independent of the mind considering them" . David Hilbert spoke late in his life of a preestablished harmony between nature and our thoughts, a harmony realized by us in and through mathematics . Earlier I quoted a writer who said that mathematical entities are "out there" to be observed and described as the geographer describes things, and I said that this attitude was that of the Greeks. In fact the writer of those words was Gottlob Frege, in 1884, and the position he there described was his own . Perhaps mathematics is actually in this sense the least "modern" of modern endeavours. -- THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 3,1995 65 References 1. Bertrand Russell, "Recent Work on the Principles of Mathematics," International Monthly 4 (1901), 84. 2. A.E. Housman, "The Name and Nature of Poetry," in Collected Poems and Selected Prose, London: Allen Lane, The Penguin Press (1988), p. 364. 3. Clive Bell, Art, London: Chatto & Windus (1949), p. 25. 4. Isaiah Berlin, The Crooked Timber of Humanity, New York: Random House (Vintage Books) (1990), pp. 5-6. 5. David Knowles, The Evolution of Medieval Thought, New York: Random House (Vintage Books) (1962), p. 55. 6. Plato, Timaeus, 29 ft. 7. Romans 1:20. 8. Spinoza, Ethics, Part II, prop. vii. 9. Boethius, De Institutione Arithmetica (translated by Michael Masi as Boethian Number Theory), Amsterdam: Editions Rodopi B.V., 1983, I, 1. 10. John Dewey, The Quest for Certainty. New York: Minton, Balch, 1929, p. 23. 11. Plato, Thaeatetus, 191c ff, 193b-196a, 200c; Aristotle, De anima 424a. 12. Shakespeare, Measure for Measure, II, ii, 117-122. 13. The most learned and fascinating expositor and defender of the traditional philosophy of art was Ananda K. Coomaraswamy; see, for example, TraditionalArt and Symbolism, Princeton NJ: Princeton University Press (1986). 14. Dante, Inferno, xxxii, 10-12. 15. Richard Paul, "Jean-Philippe Rameau (1683-1764), the Musician as Philosophe," Proc. Am. Phil. Soc. 114 (1970), 140-154. 16. Johann Winckelmann, "History of Ancient Art," in Winckelmann: Writings on Art, David Irwin (ed), London: Phaidon (1972), pp. 117ff. 17. Honor6 de Balzac, Le p~re Goriot, trans. M. A. Crawford, London: Penguin (1951), p. 134. 18. Isaiah Berlin, (ref. 4) passim. 19. M. L. Abrams, The Mirror and the Lamp, New York: Oxford University Press (1953). 20. S. T. Coleridge, Table Talk and Omniana of Samuel Taylor Coleridge, quoted in Ref. 19, p. 58. 21. Howard E. Hugo (ed.), The Portable Romantic Reader, New York: Viking Press (1957), p. 61. 22. Gabriel Josipovici, The Worldand the Book:A Study ofModern Fiction, London: Macmillan (1971), pp. xiii-xiv. 23. M. Kline, Mathematics in Western Culture, New York: Oxford University Press (1953), p. 9. 24. Ivor Grattan-Guinness, Br. J. History Sci. 7 (1974), 186. 25. J~nos Bolyai, letter of 3 November 1823, in Roberto Bonola, Non-Euclidean Geometry. New York: Dover (1955), "Translator's Introduction" to The Science of Absolute Space, p. xxviii. 26. William Wordsworth, The Prelude, III, 142. 27. On both G6del and Thom, cf. Philip J. Davis and Reuben Hersch, The Mathematical Experience, New York: Penguin Books (1983), pp. 318-319. 28. Constance Reid, Hilbert-Courant, New York: SpringerVerlag (1986), p. 194. 29. G. Frege, The Foundations of Arithmetic, 2nd rev. ed., trans. J.L. Austin, Evanston IL: Northwestern University Press (1980), p. 108. 539 Highland Avenue Ottawa, Ontario K2A 2J8 Canada continued from p. 4 8. 9. 10. 11. 12. 13. JETP 8 (1938), 89-95 (I), 1340-1349 (II), 1349-1359 (III). Shortened translation of I und Ih On the theory of plastic deformation and twinning. J. Phys. (Moscow) 1 (1939), 137-149; translation of h On the theory of plastic deformation and twinning. Physikalis. Zeitschr. Sowjetunion 13 (1938), 1 - 10. A. Seeger, H. Donth, and A. Kochend6rfer, Theorie der Versetzungen in eindimensionalen Atomreihen III: Versetzungen, Eigenbewegungen und ihre Wechselwirkungen. Z. Physik 134 (1953), 173-193. E C. Frank and J. H. v.d. Merwe, One dimensional dislocations IV. Proc. Roy. Soc. London A 201 (1950), 261-268. J. K. Perring and T. H. R. Skyrme, A mode unified field equation. Nucl. Phys. 31 (1962), 550. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Method for solving the sine-Gordon equation. Phys. Rev. Lett. 30 (1973), 1262-1264. S. Coleman, Classical lumps and quantum descents. Zichichi: New Phenomena in Subnuclear Physics. Plenum Press, New York (1977), pp. 297-407. J. Rubinstein, Sine-Gordon equation. J. Math. Phys. 11 (1970), 258 - 266. Markus Heyerhoff Marienstrasse 18 58455 Witten Germany 66 THE MATHEMATICAL INTELLIGENCER VOL. 17, NO. 3, 1995 Number Magic Unexplained. On p a g e 48 of The Intelligencer's Winter 1995 issue, the author m a d e an error s o m e w h a t d a m a g i n g to his credibility in the rest of the article. H e said that "the value of 288 is a particularly felicitous one [for erecting spurious stoichiometric uniformities], for 288 has the largest n u m ber of integer divisors of a n y n u m b e r from 0 to 576." But in fact 360 has six m o r e divisors t h a n 288 and three m o r e than 576. Such a mistake s e e m s especially regrettable in an article so d e p e n d e n t u p o n s h a r p criticism. A fun piece withal. Charles Mus~s Mathematics & Morphology Research Centre 45911 Silver Avenue Sardis, British Columbia V2R 1Y8 Canada Editor's note: The error was made by the sharp critic H. Bull in 1941 and went undetected by The Intelligencer's author and myself. It does not diminish by much our admiration for Dr. Bull's debunking.