"Our country is full of Oblomovs"

 

“We’re dreamers, of course. Our souls strain and suffer, but not much gets done—there’s no strength left over after all that ardor. Nothing ever gets done. The mysterious Russian soul…Everyone wants to understand it. They read Dostoevsky: What’s behind that soul of theirs? Well, behind our soul there’s just more soul. We like to have a chat in the kitchen, read a book. “Reader” is our primary occupation. “Viewer.” All the while, we consider ourselves a special, exceptional people even though there are no grounds for this besides our oil and natural gas. On one hand, this is what stands in the way of progress; on the other hand, it provides something like meaning. Russia always seems to be on the verge of giving rise to something important, demonstrating something completely extraordinary to the world. The chosen people. The special Russian path. Our country is full of Oblomovs, lying around on their couches, awaiting miracles. There are no Stoltzes. The industrious, awaiting miracles. There are no Stoltzes. The industrious, savvy Stoltzes are despised for chopping down the beloved birch grove, the cherry orchard. They build their factories, make money…They’re foreign to us…”

**

“The Russian kitchen…The pitiful Khrushchyovka kitchenette, nine to twelve square meters (if you’re lucky!), and on the other side of a flimsy wall, the toilet. Your typical Soviet floorplan. Onions sprouting in old mayonnaise jars on the windowsill and a potted aloe for fighting colds. For us, the kitchen is not just where we cook, it’s a dining room, a guest room, an office, a soapbox. A space for group therapy sessions. In the nineteenth century, all of Russian culture was concentrated on aristocratic estates; in the twentieth century, it lived on in our kitchens. That’s where perestroika really took place. 1960s dissident life is the kitchen life. Thanks, Khrushchev! He’s the one who led us out of the communal apartments; under his rule, we got our own private kitchens where we could criticize the government and, most importantly, not be afraid, because in the kitchen you were always among friends. It’s where ideas were whipped up from scratch, fantastical projects concocted. We made jokes—it was a golden age for jokes! “A communist is someone who’s read Marx, an anticommunist is someone who’s understood him.” We grew up in kitchens, and our children did, too; they listened to Galich and Okudzhava along with us. We played Vysotsky, tuned in to illegal BBC broadcasts. We talked about everything: how shitty things were, the meaning of life, whether everyone could all be happy. I remember a funny story…We’d stayed up past midnight, and our daughter, she was twelve, had fallen asleep on the kitchen couch. We’d gotten into some heated argument, and suddenly she started yelling at us in her sleep: “Enough about politics! Again with your Sakharov, Solzhenitsyn, and Stalin!”

"Only a Soviet can understand anoter Soviet"

 

“Why does this book contain so many stories of suicides instead of more typical Soviets with typically Soviet life stories? When it comes down to it, people end their lives for love, from fear of old age, or just out of curiosity, from a desire to come face to face with the mystery of death. I sought out people who had been permanently bound to the Soviet idea, letting it penetrate them so deeply that there was no separating them: The state had become their entire cosmos, blocking out everything else, even their own lives. They couldn’t just walk away from History, leaving it all behind and learning to live without it—diving headfirst into the new way of life and dissolving into private existence, like so many others who now allowed what used to be minor details to become their big picture. Today, people just want to live their lives, they don’t need some great Idea. This is entirely new for Russia; it’s unprecedented in Russian literature. At heart, we’re built for war. We were always either fighting or preparing to fight. We’ve never known anything else—hence our wartime psychology. Even in civilian life, everything was always militarized. The drums were beating, the banners flying, our hearts leaping out of our chests. People didn’t recognize their own slavery—they even liked being slaves. I remember it well: After we finished school, we’d volunteer to go on class trips to the Virgin Lands, and we’d look down on the students who didn’t want to come. We were bitterly disappointed that the Revolution and Civil War had all happened before our time. Now you wonder: Was that really us? Was that me? I reminisced alongside my protagonists. One of them said, “Only a Soviet can understand another Soviet.” We share a communist collective memory. We’re neighbours in memory.”

İslam Dünyası'nda Fetva Sorunu

 

Bugün yeryüzünde yaşayan 2 milyara yakın Müslüman nüfusunun evet iman problemi vardır, ahlak problemi zirvelerdedir, Batı’lıların “fixed mindset” dedikleri donmuş bir zihniyet problemi had safhadadır ve tabii ki kurucu metinlere yaklaşım keyfiyeti açısından bilgi üretme ve düşünce sistematiğini oluşturan metodoloji problemi inkar edilemez bir gerçeklik olarak ortada durmaktadır. Bütün bunların üzerine koyabileceğiniz adeta yemeğe ekilen tuz biber misali bir fetva problemimiz de vardır.

**

Türkiye’nin iflah olmaz ve olması da yakın bir gelecekte mümkün görünmeyen siyasi hayatta yalancının yamacısı, hırsızın yandaşcısı durumunda diyebileceğim devlet şeyhülislamlarının, parti müftülerinin her türlü yanlışı din ile meşrulaştırmak için verdikleri fetvalar. Belki şöyle demek daha doğru, yapılan yanlışlıklara dini kılıf bulup meşrulaştırma değil aksine dinin bir emri gibi sunma. Müstakil bir yazı ister bu son cümlem ama ben devam edeyim.

İsim vermeye gerek yok sanıyorum. Alimlerin sultanı olma yerine sultanların alimi olmayı bile isteye tercih etmiş öyle insanlar türedi ki insan “Nereden çıktı bunlar Allah aşkına?” demekten kendini alamıyor. Eskiden ruhu kara, vicdanı kara, muhakemesi kara ve belki cüzdanı kara birkaç insan örnek olarak gösterilirken şimdilerde bunlara adı sanı sözünü ettiğim fetvalarla duyulan birçok insan eklendi maalesef. Eskiden “Hırsızlık yolsuzluk değildir, devletin âli menfaatleri için şahsın, grubun hukuku feda edilebilir, iktidara zarar verecekse doğruları söylemek caiz değildir, dövize endeksli faiz mevduatları faiz değil hibedir,” gibi ne dinin ne usul ve füruu, ne fıkhın kaide ve hükümleri ne İlahi iradenin maksadı ne de insanların maslahatları ile örtüşen fetvalar akla gelirken şimdilerde bunlara da evet deyip nice ilavelerde bulunan bir çok yeni isimler katıldı. İsterseniz, “Ziraat Bakası Kur Korumalı TL Katılma Hesabı İcazet Belgesi” diye piyasaya sunulan ve içeriği itibariyle modern fetva şekli denilen evrakın altındaki imzalara bakın, ne demek istediğimi daha net anlayacaksınız.

“Sadece Türkiye mi?” diye aklınıza bir soru gelebilir burada. Elbette sadece Türkiye değil, İslam dünyasının her yeri için geçerli bu dediklerim. Ele alınan meseleler farklı olabilir ama değişmeyen gerçek dinin ruhu, insanların maslahatı ile örtüşmeyen fetva gerçeğidir. Hatta bunlar arasında öyleleri vardır ki bütün Müslümanlara dünyayı dar edecek boyutlara ulaşmıştır. Ayetullah Humeyni’nin Selman Rüşdi’nin katli, Yusuf Karadavi’nin intihar bombacılarına verdiği fetvaları hatırlayın. Taliban’ın yıktığı Buda heykelleri, 11 Eylül Amerika, 7 Temmuz Londra, 11 Mart Madrid saldırıları, IŞİD’in masum sivil insanları boğazlayarak öldürmesi ve alt alta sıralayabileceğimiz onlarca-yüzlerce vakıanın altında hep bu tür fetvalar vardır.

 Ahmet Kurucan, tr724 , Ocak 2022

Descartes (3)

 

The two aspects of Descartes split apart in his intellectual heritage. His Geometry instructed Newton and Leibniz, and is forever integrated into the living body of mathematics. Mathematicians remember him with the name “Cartesian product” for ordered pairs in set theory and geometry.

In theology, the story is more complex. It would be unfair to suppose Descartes’ bows to the Church were insincere. He really was a devout Catholic. His commitment to philosophy and science originated in a dream-vision of the Blessed Virgin.

Descartes was very considerate of the Church’s worries. Still some denounced him as a skeptic or crypto-skeptic. His First Meditation raises profound doubt. Does the Third Meditation really dispel it? Will his “clear and distinct idea” really revive faith, once his doubt has shaken it? In 1663 he was put on the Index (Vrooman, p. 252). In 1679 Leibniz wrote of Descartes’s philosophy, “I do not hesitate to say absolutely that it leads to atheism” (Leibniz, p. 1).

In the following century, nevertheless, Cartesianism became popular among Church apologists. But Descartes’s follower, the “God-intoxicated” Spinoza, was denounced as an atheist, and “Spinozism” became a synonym for atheism. Cartesians were prominent denouncers of Spinoza (Balz, pp. 218–41).

Descartes (2)

 

In Rules for the Direction of the Mind, Descartes wrote: “The first principles themselves are given by intuition alone, while, on the contrary, the remote conclusions are furnished only by deduction. . . . These two methods are the most certain routes to knowledge, and the mind should admit no others. All the rest should be rejected as suspect of error and dangerous.” 

Descartes was embracing the Euclidean ideal: Start from self-evident axioms, proceed by infallible deductions. But in his own research, Descartes forgot the Euclidean ideal. Nowhere in the Geometry do we find the label Axiom, Theorem, or Proof. 

In classical Greece, and again in the Renaissance and after, mathematicians distinguished two ways of proceeding—the “synthetic” and the “analytic.” The synthetic way was Euclid’s: from axioms through deductions to theorems. In the analytic mode, you start with a problem and “analyze” it to find a solution. Today we might call this a “heuristic” or “problem-solving” approach. 

In formal presentation of academic mathematics, the synthetic was and still is the norm. Foundationist schools of the nineteenth and twentieth centuries identify mathematics with its synthetic mode—true axioms followed by correct deductions to yield guaranteed true conclusions. 

In his Rules for the Direction of the Mind, Descartes insists on the synthetic method. But his own research, in the Geometry, uses only the analytic mode. He solves problems. He finds efficient methods for solving problems. Never does he bother with axioms. 

Descartes’s conviction of the certainty of mathematics might lead readers to expect that at least Descartes’s own mathematics is error-free. But of course, as we will see, the Geometry, like every other math book, has mistakes. Certitude is only a goal.

**

You won’t find in the Geometry the method we teach nowadays as Cartesian or “analytic” geometry. Our analytic geometry is based on rectangular coordinates (which we call “Cartesian”). To every point in the plane we associate a pair of real numbers, the “x” and “y” coordinates of the point. To an equation relating x and y corresponds a “graph”—the set of points whose x and y coordinates satisfy the equation. For an equation of first degree, the graph is a straight line. For an equation of second degree, it’s a circle or other conic section. Our idea is to solve geometric problems by reducing them to algebra. Nowhere in Descartes’s book do we see these familiar horizontal and vertical axes! Boyer says it was Newton who first used orthogonal coordinate axes in analytic geometry.

**
The conceptual essence of analytic geometry, the “isomorphism” or exact translation between algebra and geometry, was understood more clearly by Fermat than by Descartes. Fermat’s analytic geometry predated Descartes’s, but it wasn’t published until 1679. The modern formulation comes from a long development. Fermat and Descartes were the first steps. Instead of systematically developing the technique of orthogonal coordinate axes, the Geometry studies a group of problems centering around a problem of Pappus of Alexandria (third century A.D.). To solve Pappus’s problem Descartes develops an algebraic-geometric procedure. First he derives an algebraic equation relating known and unknown lengths in the problem. But he doesn’t then look for an algebraic or numerical solution, as we would do. He is faithful to the Greek conception, that by a solution to a geometric problem is meant a construction with specified instruments. When possible Descartes uses the Euclidean straight edge and compass. When necessary, he brings in his own instrument, an apparatus of hinged rulers. Algebra is an intermediate device, in going from geometric problem to geometric solution. Its role is to reduce a complicated curve to a simpler one whose construction is known. He solves third and fourth-degree equations by reducing them to second degree—to conic sections. He solves certain fifth-and sixth-degree equations by reducing them to third degree. A modern reader knows that the general equation of fifth degree can’t be solved by extraction of roots. So he’s skeptical about Descartes’s claim that his hinged rulers can solve equations of degree six and higher. Descartes was mistaken on several points. In themselves, these are of little interest today. But they discredit his claim of absolute certainty. Descartes’s mathematics refutes his epistemology. 

Emily Grosholz and Carl Boyer point out errors in the Geometry. “When he turns his attention to the locus of five lines, he considers only a few cases, not bothering to complete the task, because, as he says, his method furnishes a way to describe them. But Descartes could not have completed the task, which amounted to giving a catalogue of the cubics. . . . Newton, because he was able to move with confidence between graph and equation, first attempted a catalogue of the cubics; he distinguished seventy-two species of cubics, and even then omitted six” (Grosholz, referring to Whiteside).

**

Descartes claimed his Method was infallible in science and mathematics. He was more cautious with religion. He didn’t derive Holy Scripture or divine revelation by self-evident axioms and infallible deductions. When he heard that Galileo’s Dialogue on the Two Chief Systems was condemned by the Holy Church, he suppressed his first book, Le Monde, even though he was living in Holland, safe from the Church. (Galileo was kept under house arrest at first. For three years he had to recite the seven penitential psalms every week.) Descartes wrote to Father Mersenne, “I would not want for anything in the world to be the author of a work where there was the slightest word of which the Church might disapprove.”

**

Like Pascal, Newton, and Leibniz, Descartes may have valued his contributions to theology above his mathematics. His struggle against skeptics and heretics is the major half of his philosophy, more explicit than his battles with scholastics. “In Descartes’ reply to the objections of Father Bourdin, he announced that he was the first of all men to overthrow the doubts of the Sceptics . . . he discovered how the best minds of the day either spent their time advocating scepticism, or accepted only probable and possibly uncertain views, instead of seeking absolute truth. . . . It was in the light of this awakening to the sceptical menace, that when he was in Paris Descartes set in motion his philosophical revolution by discovering something so certain and so assured that all the most extravagant suppositions brought forward by the sceptics were incapable of shaking . . . in the tradition of the greatest medieval minds, (he) sought to secure man’s natural knowledge to the strongest possible foundation, the all-powerful eternal God” (Popkin, p. 72). The essence of the Meditations is a proof that the world exists by first proving Descartes exists, and then, by contemplating Descartes’s thoughts, proving that God exists and is not a deceiver. Once a non-deceiving God exists, everything else is easy.

 

Descartes

 

D’Alembert wrote that it was Descartes who first “dared . . . to show intelligent minds how to throw off the yoke of scholasticism, of opinion, of authority—in a word, of prejudices and barbarism. . . . He can be thought of as a leader of conspirators who, before anyone else, had the courage to arise against a despotic and arbitrary power, and who, in preparing a resounding revolution, laid the foundations of a more just and happier government which he himself was not able to see established.” 

Philosophers of the scholastic persuasion pointed to the dangerous parallel between Descartes’s scientific individualism and the outlawed Protestant heresy. Descartes said individual thinkers could find scientific truth; Protestants said individual souls could find direct communion with the Almighty. But the Holy Roman Catholic Church knew that individual souls and thinkers could be deceived. It took the experience and wisdom of the Church to prevent the seeker from wandering astray. Despite such scholastic criticism, Descartes quickly came to dominate West European intellectual life.

**

Like Galileo, Descartes recognized mathematics as the principal tool for revealing truths of nature. He was more explicit than Galileo about how to do it. In every scientific problem, said Descartes, find an algebraic equation relating an unknown variable to a known one. Then solve the algebraic equation! With the development of calculus, Descartes’s doctrine was essentially justified. Today we don’t say “find an algebraic equation.” We say “construct a mathematical model.” This is only a technical generalization of Descartes’s idea. Our scientific technology is an inheritance from Descartes.


 

Nicholas of Cusa and Theology in Math

 

The son of a fisherman, Nicholas rose to become a diplomat and counselor for the Church. “He was a member of the commission sent to Constantinople to negotiate with the Eastern church for reunion with Rome, which was temporarily effected at the Council of Florence (1439).” In 1448 he became cardinal and governor of Rome. 

Cusa was not a philosopher of mathematics. He was a philosopher whose thinking was imbued with mathematical images, so that he used mathematics to teach theology. He knew that there are different degrees of infinity. He said, amazingly, that the physical universe is finite but unbounded. He showed that a geometric figure can be both a maximum and a minimum, depending on how it’s parametrized. 

Again from the Encyclopedia, “According to Cusa, a man is wise only if he is aware of the limits of the mind in knowing the truth. . . . Knowledge is learned ignorance (docta ignorantia). Endowed with a natural desire for truth, man seeks it through rational inquiry, which is a movement of the reason from something presupposed as certain to a conclusion that is still in doubt. . . . As a polygon inscribed in a circle increases in number of sides but never becomes a circle, so the mind approximates to truth but never coincides with it. . . . Thus knowledge at best is conjecture (coniectura).” 

Cusa was a Platonist at a time when Aristotelians were dominant. “He constantly criticized the Aristotelians for insisting on the principle of noncontradiction and stubbornly refusing to admit the compatibility of contradictories in reality. It takes almost a miracle, he complained, to get them to admit this; and yet without this admission the ascent of mystical theology is impossible. . . . He constantly strove to see unity and simplicity where the Aristotelians could see only plurality and contradiction. 

“Cusa was most concerned with showing the coincidence of opposites in God. God is the absolute maximum or infinite being, in the sense that he has the fullness of perfection. There is nothing outside him to oppose him or to limit him. He is the all. He is also the maximum, but not in the sense of the supreme degree in a series. As infinite being he does not enter into relation or proportion with finite beings. As the absolute, he excludes all degrees. If we say he is the maximum, we can also say he is the minimum. He is at once all extremes. . . . The coincidence of the maximum and minimum in infinity is illustrated by mathematical figures. For example, imagine a circle with a finite diameter. As the size of the circle is increased, the curvature of the circumference decreases. When the diameter is infinite, the circumference is an absolutely straight line. Thus, in infinity the maximum of straightness is identical with the minimum of curvature. . . . 

Cusa denied that the universe is positively infinite; only God, in his view, could be described in these terms. But he asserted that the universe has no circumference, and consequently that it is boundless or undetermined—a revolutionary notion in cosmology. . . . Just as the universe has no circumference, said Cusa, so it has no fixed center. The earth is not at the center of the universe, nor is it absolutely at rest. Like everything else, it moves in space with a motion that is not absolute but is relative to the observer. . . . 

“Beneath the oppositions and contradictions of Christianity and other religions, he believed there is a fundamental unity and harmony, which, when it is recognized by all men, will be the basis of universal peace.”

Plato's mathematics

  

Plato didn’t have a “philosophy of mathematics” as we understand that phrase today. Mathematics is central in his philosophy. His believes the physical world of visible, changeable entities is illusion. What’s real is invisible, immaterial, eternal. Mathematics is real because it’s immaterial and eternal. It’s tied to religion, as a stepping stone in one’s ascent toward “the good,” the loftiest aspect of invisible reality. A challenge to his notion of mathematics would be a challenge to his religion.

Rules for educating "guardians"

  

My dear Glaucon, what study could draw the soul from the world of becoming to the world of being? . . . this, which they all have in common, which is used in addition by all arts and all sciences and ways of thinking, which is one of the first things every man must learn of necessity.” 

“What’s that?” he asked again. 

“Just this trifle, I said—to distinguish between one and two and three: I mean, in short, number and calculation . . .” 

“Number, then, appears to lead towards the truth?” 

“That is abundantly clear.” 

“Then, as it seems, this would be one of the studies we seek; for this is necessary for the soldier to learn because of arranging his troops, and for the philosopher, because he must rise up out of the world of becoming and lay hold of real being or he will never become a reckoner.” 

“That is true,” said he. 

“Again, our guardian is really both soldier and philosopher.” 

“Certainly.”

“Then, my dear Glaucon, it is proper to lay down that study by law, and to persuade those who are to share in the highest things in the city to go for and tackle the art of calculation, and not as amateurs; they must keep hold of it until they are led to contemplate the very nature of numbers by thought alone, practicing it not for the purpose of buying and selling like merchants or hucksters, but for war, and for the soul itself, to make easier the change from the world of becoming to real being and truth.” 

“Excellently said,” he answered. 

“And besides,” I said, “it comes into my mind, now the study of calculations has been mentioned, how refined that is and useful to us in many ways for what we want, if it is followed for the sake of knowledge and not for chaffering.” 

“How so?” he asked. 

“In this way, as we said just now; how it leads the soul forcibly into some upper region and compels it to debate about numbers in themselves; it nowhere accepts any account of numbers as having tacked onto them bodies which can be seen or touched. . . . I think they are speaking of what can only be conceived in the mind, which it is impossible to deal with in any other way.” 

“You see then, my friend, said I, that really this seems to be the study we need, since it clearly compels the soul to use pure reason in order to find out the truth.” 

“So it most certainly does. . . . 

“For all these reasons, the best natures must be trained in it.” 

After arithmetic, Glaucon and Socrates consider geometry. 

Says Socrates, “The knowledge the geometricians seek is not knowledge of something which comes into being and passes, but knowledge of what always is.”

“Agreed with all my heart, said he, for geometrical knowledge is of that which always is.” 

“A generous admission! Then it would attract the soul toward truth, and work out the philosopher’s mind so as to direct upwards what we now improperly keep downwards.” 

After arithmetic and plane geometry, Glaucon proposes astronomy as the third subject in the curriculum of the Guardians. Socrates objects; solid geometry is more appropriate, he says. 

“Quite so,” says Glaucon, “but it seems that those problems have not yet been solved.” 

“For two reasons,” I said, “because no city holds them in honour, they are weakly pursued, being difficult. Again, the seekers lack a guide, without whom they could not discover; it is hard to find one in the first place, and if they could, as things now are, the seekers in these matters would be too conceited to obey him. But if any whole city should hold these things honourable and take a united lead and supervise, they would obey, and solutions sought constantly and earnestly would become clear. Indeed even now, although dishonoured by the multitude, and held back by the seekers themselves having no conception of the objects for which they are useful, these things do nevertheless force on and grow against all this by their own charm, and I should not be too surprised if they should really come to light. . . . 

“Let us put astronomy as the fourth study, assuming that solid geometry, which we leave aside now, is there for us if only the city would support it.”

[The Republic (“Great Dialogues of Plato,” pp. 315–16, 323–31). Plato gave Socrates the first person, “I.”]

Pythagoreans!

 

The number one, they argued, is the generator of numbers and the number of reason; the number two is the first even or female number, the number of opinion; three is the first true male number, the number of harmony, being composed of unity and diversity; four is the number of justice or retribution, indicating the squaring of accounts; five is the number of marriage, the union of the first true male and female numbers; and six is the number of creation. Each number had its peculiar attributes. The holiest of all was the number ten, or the tetractys, for it represented the number of the universe, including the sum of all possible dimensions. [See also Heath, 1981, p. 75.] A single point is the generator of dimensions, two points determine a line of dimension one, three points S (not on a line) determine a triangle with area of dimension two, and four points (not in a plane) determine a tetrahedron with volume of dimension three; the sum of the numbers representing all dimensions, therefore, is . . . ten. It is a tribute to the abstraction of Pythagorean mathematics that the veneration of the number ten evidently was not dictated by anatomy of the human hand or foot.”

Mainstream Philosophies of Mathematics

 

Frege

                                                          

The name “foundationism” was invented by a prolific name-giver, Imre Lakatos. It refers to Gottlob Frege in his prime, Bertrand Russell in his full logicist phase, Luitjens Brouwer, guru of intuitionism, and David Hilbert, prime advocate of formalism. Lakatos saw that despite their disagreements, they all were hooked on the same delusion: Mathematics must have a firm foundation. They differ on what the foundation should be. 

Foundationism has ancient roots. Behind Frege, Hilbert, and Brouwer stands Immanuel Kant. Behind Kant, Gottfried Leibniz. Behind Leibniz, Baruch Spinoza, and René Descartes. Behind all of them, Thomas Aquinas, Augustine of Hippo, Plato, and the great grandfather of foundationism—Pythagoras. 

We will find that the roots of foundationism are tangled with religion and theology. In Pythagoras and Plato, this intimacy is public. In Kant, it’s half covered. In Frege, it’s out of sight. Then in Georg Cantor, Bertrand Russell, David Hilbert, and Luitjens Brouwer, it pops up like a jack-in-the-box.

In the twentieth century, we look at Russell, Brouwer, Hilbert, Edmund Husserl, Ludwig Wittgenstein, Kurt Gödel, Rudolph Carnap, Willard V. O. Quine, and a small sample of today’s authors. Philip Kitcher said the philosophy of mathematics is generally supposed to begin with Frege—before Frege there was only “prehistory.” Frege transformed the issues constituting philosophy of mathematics. In that sense earlier philosophy can be called prehistoric. But to understand Frege you must see him as a Kantian. To understand Kant you must see his response to Newton, Leibniz, and Hume. Those three go back to Descartes, and through him to Plato. Plato was a Pythagorean. The thread from Pythagoras to Hilbert and Gödel is unbroken. I aim to tell a connected story from Pythagoras to the present—where foundationism came from, where it left us. 

Instead of going straight through from Pythagoras, I’ve split the story into two parallel streams—the first section is about the “Mainstream.” The second is about the “humanists and mavericks.” 

For the Mainstream, mathematics is superhuman—abstract, ideal, infallible, eternal. So many great names: Pythagoras, Plato, Descartes, Spinoza, Leibniz, Kant, Frege, Russell, Carnap. (For Kant, membership in this group is partial.) 

Humanists see mathematics as a human activity, a human creation. Aristotle was a humanist in that sense, as were Locke, Hume, and Mill. Modern philosophers outside the Russell tradition—mavericks—include Peirce, Dewey, Roy Sellars, Wittgenstein, Popper, Lakatos, Wang, Tymoczko, and Kitcher (a self-styled maverick). There are some interesting authors who aren’t labeled philosophers: psychologist Jean Piaget; anthropologist Leslie White; sociologist David Bloor; chemist Michael Polányi; physicist Mario Bunge; educationists Paul Ernest, Gila Hanna, Anna Sfard; mathematicians Henri Poincaré, Alfréd Rényi, George Pólya, Raymond Wilder, Phil Davis, and Brian Rotman.

As mathematics changes...

 

As mathematics grows and changes, geometry changes. A more current example. Until the mid-twentieth century, the “derivative” or “slope” of a function at a point existed only if at that point the graph of the function was smooth—had a definite direction, and no jumps. Now mathematicians have adopted Laurent Schwartz’s generalized functions. Every function, no matter how rough, has a derivative.

**

The meaning of differentiation has changed. Newton and Leibniz’s differentiation operator has become something more general. Our generalized differentiation includes the old differentiation, and it’s much more powerful. 

As mathematics grows and changes, functions and operators change.  

A familiar cliché says that while other sciences throw away old theories, mathematics throws away nothing. But the old mathematics isn’t preserved intact. Mathematics is intensely interconnected and self-interactive. The new is vitally linked to the old. The old is revitalized, enriched, and complexified by interaction with the new.