Philosophy of mathematics

 

                                                                                                       https://www.durham.ac.uk/departments/academic/mathematical-sciences/

Philosophy of mathematics should be tested against five kinds of mathematical practice: research, application, teaching, history, computing.

**

The need to check philosophy of mathematics against mathematical research doesn’t require explication. Many important philosophers of mathematics were mathematical researchers: Pascal, Descartes, Leibniz, d’Alembert, Hilbert, Brouwer, Poincaré, Rényi, and Bishop come to mind. Applied mathematics isn’t illegitimate or marginal. Advances in mathematics for science and technology often are inseparable from advances in pure mathematics. Examples: Newton on universal gravitation and the infinitesimal calculus; Gauss on electromagnetism, astronomy, and geodesy (the last inspired that beautiful pure subject—differential geometry); Poincare on celestial mechanics; and von Neumann on quantum mechanics, fluid dynamics, computer design, numerical analysis, and nuclear explosions.

**

G. H. Hardy “famously” boasted: “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” Nevertheless, the Hardy-Weinberg law of genetics is better known than his profound contributions to analytic number theory. What’s worse, cryptology is making number theory applicable. Hardy’s contribution to that pure field may yet be useful. Twenty years after the war, mathematical purism was revived, influenced by the famous French group “Bourbaki.” That period is over. Today it’s difficult to find a mathematician who’ll say an unkind word about applied math.

**

On the basis of this reduction, philosophers of mathematics generally limit their attention to set theory, logic, and arithmetic. What does this assumption, that all mathematics is fundamentally set theory, do to Euclid, Archimedes, Newton, Leibniz, and Euler? No one dares to say they were thinking in terms of sets, hundreds of years before the set-theoretic reduction was invented. The only way out (implicit, never explicit) is that their own understanding of what they did must be ignored! We know better than they how to explicate their work! That claim obscures history, and obscures the present, which is rooted in history.

An adequate philosophy of mathematics must be compatible with the history of mathematics. It should be capable of shedding light on that history. Why did the Greeks fail to develop mechanics, along the lines that they developed geometry? Why did mathematics lapse in Italy after Galileo, to leap ahead in England, France, and Germany? Why was non-Euclidean geometry not conceived until the nineteenth century, and then independently rediscovered three times? The philosopher of mathematics who is historically conscious can offer such questions to the historian. But if his philosophy makes these questions invisible, then instead of stimulating the history of mathematics, he stultifies it. Computing is a major part of mathematical practice. The use of computing machines in mathematical proof is controversial. An adequate philosophy of mathematics should shed some light on this controversy.

 

Hardy's Lectures

In lectures, his enthusiasm and delight in the subject fairly spilled over. “One felt,” wrote one of his later students, E. C. Titchmarsh, “that nothing else in the world but the proof of these theorems really mattered.” Norbert Wiener, the American mathematical prodigy who would later create the field known as “cybernetics,” attended Hardy’s lectures. “In all my years of listening to lectures in mathematics,” he would write, “I have never heard the equal of Hardy for clarity, for interest, or for intellectual power.” Around this time, a pupil of E. W. Barnes, director of mathematical studies at Trinity, sought Barnes’s advice about what lectures to attend. Go to Hardy’s, he recommended. The pupil hesitated. “Well,” replied Barnes, “you need not go to Hardy’s lectures if you don’t want, but you will regret it—as indeed,” recalled the pupil many years later, “I have.” Others who missed his lectures may not, in retrospect, have felt such regret: so great was Hardy’s personal magnetism and enthusiasm, it was said, that he sometimes diverted to mathematics those without the necessary ability and temperament.

But as lucid as were his lectures, it was his writing that probably had more impact. Later, speculating about what career he might have chosen other than mathematics, Hardy noted that “Journalism is the only profession, outside academic life, in which I should have felt really confident of my chances.” Indeed, no field demanding literary craftsmanship could fail to have profited from his attention. “He wrote, in his own clear and unadorned fashion, some of the most perfect English of his time,” C. P. Snow once said of him. That Hardy’s impressions of Ramanujan would be so relentlessly quoted, and would go so far toward fixing Ramanujan’s place in history, owes not alone to his close relationship with Ramanujan but to the sheer grace with which he wrote about him.

**

Thought, Hardy used to say, was for him impossible without words. The very act of writing out his lecture notes and mathematical papers gave him pleasure, merged his aesthetic and purely intellectual sides. Why, if you didn’t know math was supposed to be dry and cold, and had only a page from one of his manuscripts to go on, you might think you’d stumbled on a specimen of some new art form beholden to Chinese calligraphy. Here were inequality symbols that slashed across the page, sweeping integral signs an inch and a quarter high, sigmas that resonated like the key signatures on a musical staff. There was a spaciousness about how he wrote out mathematics, a lightness, as if rejecting the cramped, ungenerous formalities of the printed notation. He was like a French impressionist, intimating worlds with a few splashes of color, not a maker of austere English miniatures.

British Math vs European Math

Since the seventeenth century, Britain had stood, mathematically, with its back toward Europe, scarcely deigning to glance over its shoulder at it. Back then, Isaac Newton and the German mathematician Gottfried Wilhelm von Leibniz had each, more or less independently, discovered calculus. Controversy over who deserved the credit erupted even while both men lived, then mushroomed after their deaths, with mathematicians in England and on the Continent each championing their compatriots. Newton was the premier genius of his age, the most fertile mind, with the possible exception of Shakespeare’s, ever to issue from English soil. And yet he would later be called “the greatest disaster that ever befell not merely Cambridge mathematics in particular but British mathematical science as a whole.” For to defend his intellectual honor, as it were, generations of English mathematicians boycotted Europe—steadfastly clung to Newton’s awkward notational system, ignored mathematical trails blazed abroad, professed disregard for the Continent’s achievements. “The Great Sulk,” one chronicler of these events would call it.

In calculus as in mathematics generally, the effects were felt all through the eighteenth and nineteenth centuries and on into the twentieth. Continental mathematics laid stress on what mathematicians call “rigor,” the kind to which Hardy had first been exposed through Jordan’s Cours d’analyse and which insisted on refining mathematical concepts intuitively “obvious” but often littered with hidden intellectual pitfalls. Perhaps reinforced by a strain in their national character that sniffed at Germanic theorizing and hairsplitting, the English had largely spurned this new rigor. Looking back on his Cambridge preparation, Bertrand Russell, who ranked as Seventh Wrangler in the Tripos of 1893, noted that “those who taught me the infinitesimal Calculus did not know the valid proofs of its fundamental theorems and tried to persuade me to accept the official sophistries as an act of faith. I realized that the Calculus works in practice but I was at a loss to understand why it should do so.” So, it is safe to say, were most other Cambridge undergraduates.

Calculus rests on a strategy of dividing quantities into smaller and smaller pieces that are said to “approach,” yet never quite reach, zero. Taking a “limit,” the process is called, and it’s fundamental to an understanding of calculus—but also, typically, alien and slippery territory to students raised on the firm ground of algebra and geometry. And yet, it is possible to blithely sail on past these intellectual perils, concentrate on the many practical applications that fairly erupt out of calculus, and never look back.

In textbooks even today you can see vestiges of the split—which neatly parallels that between Britain and the Continent in the nineteenth century: the author briefly introduces the limit, assumes a hazy intuitive understanding, then spends six chapters charging ahead with standard differentiation techniques, maxima-minima problems, and all the other mainstays of Calc 101 . . . until finally, come chapter 7 or so, he steps back and reintroduces the elusive concept, this time covering mine-strewn terrain previously sidestepped, tackling conceptual difficulties—and stretching the student’s mind beyond anything he’s used to.

Well, the first six chapters of this generic calculus text, it could be said, were English mathematics without the Continental influence. Chapter 7 was the new rigor supplied by French, German, and Swiss mathematicians. “Analysis” was the generic name for this precise, fine-grained approach. It was a world of Greek letters, of epsilons and deltas representing infinitesimally small quantities that nonetheless the mathematicians found a way to work with. It was a world in which mathematics, logic, and Talmudic hairsplitting merged.

First Gauss, Abel, and Cauchy had risen above the looser, intuitive nostrums of the past; later in the century, Weierstrass and Dedekind went further yet. None of them were English. And the English professed not to care. Why, before the turn of the century, Cauchy—the Cauchy, Augustin Louis Cauchy, the Cauchy who had launched the French school of analysis, the Cauchy of the Cauchy integral formula—was commonly referred to around Cambridge as “Corky.”

Since Newton’s time, British mathematics had diverged off on a decidedly applied road. Mathematical physics had become the British specialty, dominated by such names as Kelvin, Maxwell, Rayleigh, and J. J. Thomson. Pure math, though, had stultified, with the whole nineteenth century leaving England with few figures of note. “Rigor in argument,” J. E. Littlewood would recall, “was generally regarded—there were rare exceptions—with what it is no exaggeration to call contempt; niggling over trifles instead of getting on with the real job.” Newton had said it all; why resurrect these arcane fine points? Calculus, and the whole architecture of mathematical physics that emanated from it, worked.

And so, England slept in the dead calm of its Tripos system, where Newton was enshrined as God, his Principia Mathematica the Bible. “In my own Tripos in 1881, we were expected to know any lemma [a theorem needed to prove another theorem] in that great work by its number alone,” wrote one prominent mathematician later, “as if it were one of the commandments or the 100th Psalm. . . . 

Cambridge became a school that was self-satisfied, self-supporting, self-content, almost marooned in its limitations.” Replied a distinguished European mathematician when asked whether he had seen recent work by an Englishman: “Oh, we never read anything the English mathematicians do.”

The first winds of change came in the person of Andrew Russell Forsyth, whose Theory of Functions had begun, in 1893, to introduce some of the new thinking—though by this time it wasn’t so new anymore—from Paris, Göttingen, and Berlin. Written in a magisterial style, it burst on Cambridge, as E. H. Neville once wrote, “with the splendour of a revelation”; some would argue it had as great an influence on British mathematics as any work since Newton’s Principia. By the standards of the Continent, however, it was hopelessly sloppy and was soundly condemned there. “Forsyth was not very good at delta and epsilon,” Littlewood once said of him, referring to the Greek letters normally used for dealing with infinitesimally small quantities. Still, it helped redirect the gaze of English mathematicians toward the Continent. It charted a course to the future, but did not actually follow it.

That was left to Hardy.

Cambridge, A Society of Bachelors

That Hardy’s life was spent almost exclusively in the company of other men, that he scarcely ever saw a woman, was, in those days, not uncommon. After all, among Havelock Ellis’s thousand or so British “geniuses,” 26 percent never married. In the academic and intellectual circles of which Hardy was a part, such a monastic sort of life actually represented one pole of common practice.

Thus, at Cranleigh School, all the teachers, except for the House staff, were men, most of them bachelors; dormitory masters had to be bachelors. Winchester was the same way. So was Cambridge. “In my day we were a society of bachelors,” wrote Leslie Stephen in Some Early Impressions of his time at Cambridge during the early 1860s. “I do not remember during my career to have spoken to a single woman at Cambridge except my bed-maker and the wives of one or two heads of houses.”

Not much had changed by the time Hardy reached Cambridge a generation later. Among the twenty or so colleges, two—Girton and Newnham—had been established for women in the previous two decades. But though women, with the lecturer’s consent and chaperoned by a woman don, could attend university lectures, by 1913 they still kept mostly to themselves and played little part in undergraduate life. Until 1882, college fellows couldn’t marry, but even after that most fellows remained bachelors. In 1887, a proposal was made to offer degrees to women; it was soundly defeated. Ten years later, on a May day in 1897, a straw-hatted mob thronged outside the Senate House, where the matter was again being taken up, demonstrating against the measure. A woman was hanged in effigy. A large banner advised (after Act II, Scene I of Much Ado About Nothing) “Get you to Girton, Beatrice. Get you to Newnham. Here’s no place for you maids.”

It was an almost laughably artificial environment, with dons left woefully ignorant of domestic life. One time at St. John’s College, the story goes, an elderly bachelor at High Table congratulated someone on the birth of his son. “How old is the little man?” he asked.

“Six weeks,” came the reply.

“Ah,” said the bachelor don, “just beginning to string little sentences together, I suppose.”

About the only time Hardy and other fellows encountered women was among the bedmakers who tidied up college rooms—and they were said to be selected for their plainness, age, and safely married status, presumably so as to minimize the distraction they represented to students and fellows of the colleges.

**

There was a hauntedness to Hardy that you could see in his eyes. “I suspect,” remembered an Oxford economist, Lionel Charles Robbins, who knew him later, that “Hardy found many forms of contact with life very painful and that, from a very early stage, he had taken extensive measures to guard himself against them. Certainly in his friendlier moments—and he could be very friendly indeed—one was conscious of immense reserves.” Always, he kept the world at bay. The obsession with cricket, the bright conversation, the studied eccentricity, the fierce devotion to mathematics—all of these made for a beguiling public persona; but none encouraged real closeness. He was a friend of many in Cambridge, an intimate of few.

In the years after 1913, Hardy would befriend a poor Indian clerk. Their friendship, too, would never ripen into intimacy.

Hardy

Hardy was forever judging, weighing, comparing. He rated mathematicians, the work they did, the books and papers they wrote. He held firm opinions on everything, and expressed them. When a Cambridge club to which he’d belonged moved to change its official colors, Hardy took six pages to attack the plan. He faulted a sacrosanct academic tradition of almost two centuries’ standing, and condemned it, unrelentingly, for more than twenty years. All his enthusiasms, peeves, and idiosyncrasies were like that—sharp, unwavering, vehement. He hated war, politicians as a class, and the English climate. He loved the sun. He loved cats, hated dogs. He hated watches and fountain pens, loved The Times of London crossword puzzles

After Mirzakhani

Complicating the global image of a nation

Much of the reactions to the death of Mirzakhani was both normal and predictable. The Iranian President Hassan Rouhani and the Stanford University President Marc Tessier-Lavigne both issued solemn statements of condolences. The Iranian media competed in showering her with accolades. The Iranian oppositional venues in the United States and Europe began to use and abuse the occasion to denounce the Islamic Republic for its various policies, especially for the conditions conducive to "brain drain".

Maryam Mirzakhani left Iran to pursue her advanced mathematical studies in the US. She would have gone to Baghdad to do so if she were born one thousand years ago - and she would have probably gone to Beijing if she were born just a couple of decades from now. The question of "brain drain" is, of course, a serious malady in Iran and many other similar countries. But Maryam Mirzakhani was no "brain drain". Hers was a superior intelligence and she travelled where she could nourish it best - and that travelling did not suddenly turn her into this strange thing called "Iran-born", instead of just plain "Iranian".

But something else, something quite simple and significant, was also happening, just like Maryam Mirzakhani herself, gently and quietly.

Mirzakhani is comparable to Omar Khayyam not just because they were both Iranian mathematicians. But because like Khayyam, Mirzakhani too complicates the vision of their common homelands in the European and now American imagination in unpredictable ways.

Markedly brilliant minds and beautiful souls like Maryam Mirzakhani both upon their global recognitions and perforce upon their early and tragic passing become a symbol, a sign, a citation far beyond who they are and what they have achieved in their professional calling. From the time that she achieved her coveted Fields Medal, Maryam Mirzakhani had begun to complicate the global image of her homeland against the backdrop of the pervasive demonisation of Iran by one brand of warmongering or another.

Nations need to be simplified to be targeted for military strikes. Afghanistan was reduced to Mullah Omar. Iraq was reduced to Saddam Hussein. The more the image of a nation is complicated the more difficult it is for warmongers in Washington, Tel Aviv, or Riyadh targeting it for destruction.

Precisely in the quiet dignity of her work, her avoiding publicity like a plague, the tiny, cancer-ravaged body of Maryam Mirzakhani shined like a beautiful star on the dark planet of her earthly life. An Iranian, a Muslim, a woman of modest middle-class background, rising gloriously to put a big brilliant question mark in front of everything mobilised against her people!

Those exonerating Iranian people when targeting "Iran" for demonisation should take a look at what they have done to Afghanistan, Iraq, Syria, or Libya before they might believe their own delusions. Ordinary people, not governments, are the primary targets and the final victims of warmongering anywhere and everywhere.

To be sure, much to the chagrin of warmongers, the image of Iran has been complicated by other prominent Iranians, in particular, those who have put Iranian cinema on the global map. I recall vividly when another accomplished Iranian young woman, Samira Makhmalbaf first appeared in Cannes Film Festival at the age of seventeen to premier her film "Apple" in 1998. She too succeeded that year, seriously altering the image of Iran from that of a bearded angry man to a gifted young woman.
 In the realm of art, no one, of course, did more to complicate the image of Iran than the late master Abbas Kiarostami who was the principal engine bringing the rest of Iranian art to global spotlight.

But much of that complexity has been in the realm of arts, not sciences. In the realm of science, the only thing publicly related to Iran is, of course, the nuclear scientists who are the targets of assassinations presumably by a settler colony on planet Mars that is very concerned about the Iranian nuclear programme. These Martians are suspected to have occasionally sent their assassins to kill these Iranian nuclear scientists.

Mirzakhani was not a nuclear physicist. A breed apart, she was a world-renowned mathematician. Her accomplishments, as a result, assume entirely historical proportions comparable to other Iranian and Muslim scientists at the historical level of Muhammad ibn Musa al-Khwarizmi, Muhammad ibn Zakariyya al-Razi, and above all to Omar Khayyam, the towering astronomer and mathematician.




Mirzakhani's uplifting of her homeland to its historical memories alerts the world to a whole different register of consciousness about Iran as a Muslim a country in circumstances that, because of pervasive Islamophobia in the US, even Rumi is read as if he were a New Age guru from California.

Mirzakhani is comparable to Omar Khayyam not just because they were both Iranian mathematicians. But because like Khayyam, Mirzakhani too complicates the vision of their common homeland in the European and now American imagination in unpredictable ways.
The reputation of Omar Khayyam as a poet, however, widely outshines his fame as a mathematician. But the difference is only on the surface. The beauty of Khayyam's mathematical mind, it now seems, had to be translated, as it were, into poetic scepticism to be registered for mortal beings, whereas Maryam Mirzakhani's poetry was and remained in pure mathematics.

**
The mathematician, poet, painter

On another occasion, she said: "Of course, the most rewarding part is the 'Aha' moment, the excitement of discovery and enjoyment of understanding something new - the feeling of being on top of a hill and having a clear view. But most of the time, doing mathematics for me is like being on a long hike with no trail and no end in sight". That is the mind of a mathematician in the soul of a poet. That is Khayyam incarnate.

Mirzakhani's mathematical equations were her poetry - a poetry only a happy few can decipher to their delight. The world at large is baffled at the beauty of that poetry. Thus, the other more recent kindred soul of Mirzakhani was, of course, the Indian mathematician Srinivasa Ramanujan (1887-1920) who died even younger than her.

The famous story narrated by Ramanujan's English colleague, the prominent mathematician GH Hardy, is now known as "Hardy-Ramanujan number 1729". According to Hardy, he was once going to see Ramanujan when he was bedridden. Hardy had just ridden in a cab number 1729 and upon arrival, he remarked to his friend that the number seemed to be quite dull and that he hoped it was not an unfavourable omen. "No", Ramanujan replied: "It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways". The two different ways are:

1729 = 13+123 = 93+103.

Now, that is pure poetry - for what is poetry other than a truth so obviously beautiful that mortals cannot see it. The joyous eyes of Maryam Mirzakhani's little daughter could see what her mother's beautiful mind knew with mathematical precision: "At the family's home, near Stanford University," Roberts tells us in her essay on the Iranian mathematician, "Mirzakhani would spend hours on the floor with supersized canvases of paper, sketching out ideas, drawing diagrams and formulae, often leading Anahita, now six, to exclaim, 'Oh, Mommy is painting again!"



Hamid Dabashi
aljazeera 

Matematiksel Düşünceler

Biz arkadaşlarımızı, insanlığın bütün iyi niteliklerini taşıdıkları için değil, kendileri oldukları için seçeriz. Aynı şey matematikte de geçerlidir. Çok sayıda nesnede var olan bir özellik çok ilginç olamaz. Matematiksel düşünceler de, özellikten yoksun iseler çok ilginç de olamazlar.

Matematiksel Düşünceler

Biz arkadaşlarımızı, insanlığın bütün iyi niteliklerini taşıdıkları için değil, kendileri oldukları için seçeriz. Aynı şey matematikte de geçerlidir. Çok sayıda nesnede var olan bir özellik çok ilginç olamaz. Matematiksel düşünceler de, özellikten yoksun iseler çok ilginç de olamazlar.

İnsanları Araştırmaya Yönelten Nedenler

İnsanları araştırma yapmaya yönelten pek çok neden vardır; ancak bunlardan üçü diğerlerinden çok daha önemlidir. Birincisi (ki bu olmadan öbür nedenler işe yaramaz), entellektüel merak, gerçeği öğrenme arzusudur. İkincisi, profesyonel saygınlık, yaptıklarının kendini tatmin etmeme endişesidir; ortaya koyduğu eser, yeteneği ile orantılı olmadığı zaman her onurlu zanaatçının duyduğu utanma hissidir. Sonuncusu da başarma hırsı, mevkii ve üne kavuşma arzusu, hatta sağlanacak para ve onun getireceği güçtür, işinizi yaptığınızda, başkalarının mutluluklarının artmasına, ya da acılarının hafifletilmesine katkıda bulunmuş olmayı görmek hoş bir duygudur; ancak siz o işi bu nedenlerle yapmamışsınızdır. O halde bir matematikçi, bir kimyacı, hatta bir fizyolog, bana çalışmasındaki güdünün insanlığa yararlı olmak olduğunu söylerse ona inanmam (inansam bile bundan dolayı onu daha saygıdeğer bulmam). Ona etken olan nedenler bu bahsettiklerimdir; bunda da dürüst bir insanın utanması gereken bir şey yoktur.

Araştırma için gerekli başlıca güdüler entellektüel merak, profesyonel saygınlık ve başarı ise, bunları karşılamakta hiç kimse bir matematikçiden daha şanslı değildir. Onun konusu diğerlerin-kinden çok daha merak uyandırıcıdır. Başka hiç bir alanda gerçekler aynı ölçüde şaşırtıcı oyunlar oynamaz. En incelikli, en büyüleyici teknikler ondadır; özgün mesleki hünerin sergilenmesi yönünden, matematik rakipsizdir. Nihayet, tarihin sağladığı pek çok örnekle de biliyoruz ki, matematiksel sonuçlar, içerdikleri gerçek değerler ne olursa olsun, diğerlerinin içinde en kalıcı olanlarıdır.

Bu gerçeği en eski uygarlıklarda bile görebiliriz. Babil ve Asur uygarlıkları yok oldu; Hammurabi, Sargon, Nabuchadnezzar artık anlamsız isimler. Fakat Babil matematiği hala ilgi çekicidir; 60 ölçekli Babil cetveli astronomide hala kullanılmaktadır. Ancak en önemli örnek, kuşkusuz Yunanlılarınkidir.

Eski Yunanlılar, bugün bile 'gerçek' olarak nitelendirdiğimiz ilk matematikçilerdir. Doğu matematiği ilginç bir merak konusu olabilir; fakat Yunanlılarınki gerçek matematiktir. Modern matematikçinin anlayabileceği dili ilk kez Yunanlılar geliştirmiştir. Littlewood'un bir keresinde bana dediği gibi, onlar zeki öğrenciler ya da "burs adayları" değil, "başka bir fakültenin araştırmacıları" idiler. Yunan matematiği kalıcı, hatta Yunan edebiyatından bile daha kalıcıdır. Aeschylus unutulsa bile Archimedes hatırlanacaktır; çünkü konuşma dilleri ölür, ama matematiksel düşünceler kalıcıdır. 'Ölümsüzlük' saçma bir sözcük olabilir; ancak, anlamı ne olursa olsun, ona erişmek için en şanslı olanlar matematikçilerdir.