Mathematicians From The United States

"Bertrand Russell found Frege's famous error: Frege had overlooked what is now known as the Russell paradox. Namely, Frege's rules allowed one to define the class of x such that P(x) is true for any "concept" P. Frege's idea was that such a class was an object itself, the class of objects "falling under the concept P." Russel used this principle to define the class R of concepts that do not fall under themselves. This concept leads to a contradiction... argument: (1) if R falls under itself then it does not fall under itself; (2) this contradiction shows that it does not fall under itself; (3) therefore by definition it does fall under itself after all."

"Alan Turing was the first to make a careful analysis of the potential capabilities of machines, inventing his famous "Turing machines" for the purpose. He argued that if any machine could perform a computation, then some Turing machine could perform it. The argument focuses on the assertion that any machine's operations could be simulated, one step at a time, by certain simple operations, and that Turing machines were capable of those simple operations. Turing's first fame resulted from applying this analysis to a problem posed earlier by Hilbert, which concerned the possibility of mechanizing mathematics. Turing showed that in a certain sense, it is impossible to mechanize mathematics: We shall never be able to build an "oracle" machine that can correctly answer all mathematical questions presented to it with a "yes" or "no" answer. In another famous paper Turing went on to consider the somewhat different question, "Can machines think?." It is a different question, because perhaps machines can think, but they might not be any better at mathematics than humans are; or perhaps they might be better at mathematics than humans are, but not by thinking, just by brute-force calculation power. These two papers of Turing lie near the roots of the subjects today known as automated deduction and artificial intelligence."

"Gottfried Leibniz is famous... for his slogan Calculemus, which means "Let us calculate." He envisioned a formal language to reduce reasoning to calculation, and he said that reasonable men, faced with a difficult question of philosophy or policy, would express the question in a precise language and use rules of calculation to carry out precise reasoning. This is the first reduction of reasoning to calculation ever envisioned. ...he actually designed and built a working calculating machine, the Stepped Reckoner ...inspired by the somewhat earlier work of Pascal, who built a machine that could add and subtract. Leibniz's machine could add, subtract, divide, and multiply, and was apparently the first machine with all four arithmetic capabilities."

"George Boole took up Leibniz's idea, and wrote a book he called '. The laws he formulated are now called ... Boole seems to have had a grandiose vision about the applicability of his algebraic methods to practical problems—his book makes it clear that he hoped these laws would be used to settle practical questions. William Stanley Jevons heard of Boole's work, and undertook to build a machine to make calculations in Boolean algebra. He successfully designed and built... the Logical Piano... the first machine to do mechanical inference."

"Gottlob Frege created modern logic including "for all," "there exists," and rules of proof. Leibniz and Boole had dealt only with what we now call "propositional logic" (that is, no "for all" or "there exists"). They also did not concern themselves with rules of proof, since their aim was to reach truth by pure calculation with symbols for the propositions. Frege took the opposite track: instead of trying to reduce logic to calculation, he tried to reduce mathematics to logic, including the concept of number."

"R. A. Fisher, J. Neyman, R. von Mises, W. Feller, and L. J. Savage denied vehemently that probability theory is an extension of logic, and accused Laplace and Jeffreys of committing metaphysical nonsense for thinking that it is."

"It was only when L. J. Savage arrived on the scene, and championed the work of Ramsey and de Finetti that the work of these two pioneers in subjective probability first received serious philosophical attention."

"[Leonard Jimmie Savage is] one of the few people I have met whom I would unhesitatingly call a genius."

"My dissertation for the Ph.D. degree at the University of Michigan was on applications of vectorial methods to metric geometry (in the sense of the Menger school), especially with a view to the merging of metric geometry in that sense with differential geometry. Professor S B Myers at the University of Michigan sponsored my dissertation, but I was particularly close to R L Wilder there."

"For a person who wants to do original, realistic, and critical work in statistics there is no atmosphere anywhere in the world today to compare with this Department."

"[The George E. P. Box paper Fitting empirical data (1960) is] a mature exposition of an important branch of statistics, to which the author has made great contributions. One feature of particular interest is practical discussion of genuinely nonlinear fitting problems and their solution with the help of tact and a special, publicly available, IBM-704 program. Another is insightful comments on the role of prior distributions in statistics."

"John Tate and I were asked by Nature magazine to write an obituary for Alexander Grothendieck. Now he is a hero of mine, the person that I met most deserving of the adjective "genius". I got to know him when he visited Harvard and John, Shurik (as he was known) and I ran a seminar on "s". His devotion to math, his disdain for formality and convention, his openness and what John and others call his naiveté struck a chord with me."

"I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate."

"There is only one other survey, Datta and Singh’s 1938 History of Hindu Mathematics, recently reprinted but very hard to obtain in the West (I found a copy in a small specialized bookstore in Chennai). They describe in some detail the Indian work in arithmetic and algebra and, supplemented by the equally hard to find Geometry in Ancient and Medieval India by Sarasvati Amma (1979), one can get an overview of most topics."

"In the winter of 2008, Jenifer and I visited Chennai Mathematical Institute. This remarkable Institute is the creation of Seshadri. It is a unique blend of an American style liberal arts college with traditional Indian guru one-on-one teaching, adding physics, computer science, history and music to its maths curriculum. Only in India could an intellectual with no business or management experience, who spends all his spare time singing classical south Indian music, have been the catalyst for such a unique educational experiment."

"On a more personal note, I see many similarities between India's Dalit problems and the African-American problems that have rocked the US since its beginnings. For this reason, I personally take Dr. Ambedkar as one of my heroes."

"The preface to the French edition of that work contains the following passage: "To me the French edition of my work is not a mere translation, but a transcription of ideas into a language in which it should have been written in the first place... I proudly acknowledge... my master. His words are among the most brilliant recollections of my youth; his piercing wisdom and potent prose have inspired my efforts of a riper age. To the memory of Henri Poincaré, the intellectual giant who was the first to recognize the role which the idiosyncrasies of the race play in the evolution of scientific ideas, I dedicate this book."

"The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. ...The conic sections, invented in an attempt to solve the problem of doubling the alter of an oracle, ended by becoming the orbits followed by the planets... The imaginary magnitudes invented by Cardan and Bombelli describe... the characteristic features of alternating currents. The absolute differential calculus, which originated as a fantasy of Riemann, became the mathematical model for the theory of Relativity. And the matrices which were a complete abstraction in the days of Cayley and Sylvester appear admirably adapted to the... quantum of the atom."

"The arithmetization of mathematics... which began with Weierstrass... had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics. These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought."

"Between the philosopher's attitude towards the issue of reality and that of the mathematician there is this essential difference: for the philosopher the issue is paramount; the mathematician's love for reality is purely platonic."

"But how can we avoid the use of human language? The... symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason—only thus may we hope to build mathematics on the solid foundation of logic."

"The great Cartesian invention had its roots in those famous problems of antiquity which originated in the days of Plato. In endeavoring to solve the problems of the trisection of an angle, of the duplication of the cube and of the squaring of the circle, the ruler and compass having failed them, the Greek geometers sought new curves. They stumbled on the conic sections...There we find the nucleus of the method which Descartes later erected into a principle. Thus Apollonius referred the parabola to its axis and principal tangent, and showed that the semichord was the mean propotional between the latus rectum and the height of the segment. Today we express this relation by x2 = Ly, calling the height the ordinate (y) and the semichord the abscissa (x); the latus rectum being... L. ...the Greeks named these curves and many others... loci... Thus the ellipse was the locus of a point the sum of the distances of which from two fixed points was constant. Such a description was a rhetorical equation of the curve..."

"Greek thought was essentially non-algebraic, because it was so concrete. The abstract operations of algebra, which deal with objects that have been purposely stripped of their physical content, could not occur to minds which were so intently interested in the objects themselves. The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and reflects all the phases under which the human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations."

"In the history of mathematics, the "how" always preceded the "why," the technique of the subject preceded its philosophy."

"The progress of mathematics has been most erratic, and... intuition has played a predominant rôle in it. ...It was the function of intuition to create new forms; it was the acknowledged right of logic to accept or reject these new forms, in whose birth in had no part. ...the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied."

"‘…the transition [to the Hindu number system], far from being immediate, extended over long centuries. The struggle between the Abacists, who defended the old traditions, and the Algorists, who advocated the reform, lasted from the eleventh to the fifteenth century and went through all the usual stages of obscurantism and reaction. In some places, Arabic numerals [more precisely, Hindu numerals] were banned from official documents; in others, the art was prohibited altogether. And, as usual, prohibition did not succeed in abolishing, but merely served to spread bootlegging, ample evidence of which is found in the thirteenth century archives of Italy, where, it appears, merchants were using the Arabic numerals as a sort of secret code.’"

"There exists among the most primitive tribes of Australia and Africa a system of numeration which has neither 5, 10, nor 20 for base. It is a binary system, i.e., of base two. These savages have not yet reached finger counting. They have independent numbers for one and two, and composite numbers up to six. Beyond six everything is denoted by “heap.”"

"It has the very commendable aim of contributing towards stressing the cultural side of mathematics. ...there appears the widespread interchange of the definitions of excessive and defective numbers. ...it is stated that Euclid contended that every perfect number is of the form 2n-1(2n -1). It is true that Euclid proved that such numbers are perfect whenever 2n - 1 is a prime number but there seems to be no evidence to support the statement that he contended that no other such numbers exist. ...it is stated that the arithmetization of mathematics began with Weierstrass in the sixties of the last century. The fact that this movement is much older was recently emphasized by H. Wieleitner... it is stated that the arithmos of Diophantus and the res of Fibonacci meant whole numbers, and... we find the statement that in the pre-Vieta period they were committed to natural numbers as the exclusive field for all arithmetic operations. On the contrary, operations with common fractions appear on some of the most ancient mathematical records."

"Despite the vociferous claims of the Platonists and Neoplatonists, Plato was not a mathematician. To Plato and his followers mathematics was largely a means to an end... they viewed the technical aspects of mathematics as a mere device for sharpening one's wits, or at most a course of training peparatory to handling the larger issues of philosophy. This is reflected in the very name "mathematics,"... a course of studies or... a curriculum. ...in the Dialogues... such topics as harmony, triangular numbers, figurate numbers... which we view today as more or less irrelevant, if not trivial, were taken up at length. ...the guiding motive behind the... Pythagoreans and Platonists was... metaphysical ...which for the nonprofessional have all the earmarks of the occult. ...We also discover in the Pythagorean speculations more than a mere germ of... the scientific attitude."

"d'Alembert, who wrote the introduction to the Encyclopédie, resigned his editorship with the scathing remark that the work was like a harlequin's coat: some good stuff, but mostly rags."

"Science... may be viewed as man's supreme effort to find himself in that perplexing pattern which he calls Nature. ...Has he succeeded in achieving some measure of harmony with Nature? Or has he merely managed to transfer to Nature the irreconcilable duality within himself?"

"Barely a hundred and fifty years had passed since Galileo's experiment at Pisa had ushered in the new order of things; a mere instant as compared with the previous life of the race. Yet, this brief span had witnessed a complete shift in the outlook of the intellectual leaders of humanity: from blind adherence to authority and dogma towards a healthy habit of facing facts and an enlightened faith in the efficacy of reason. Few doubted that this buoyancy and self-reliance of the leaders would eventually reach the masses, thus causing a profound metamorphosis in the attitude of the common man towards his own life and the destinies of his race. ...Led by thinkers, and under the banners of liberty, happiness, and truth, humanity was to emerge into a Golden Age, free from oppression and strife. Alas! The French Revolution... resembled more a convention of inquisitors and hangmen than it did an assembly of enlightened emancipators. ...After twenty years of adventure, the humanitarian aspirations bequeathed by the Encyclopedists, tattered and trampled first by a bloody republic, then by a still bloodier empire, were finally declared dead by the Holy Alliance."

"One part of the dreams of the eighteenth century intellectuals was realized: the resources of nature did yield a magnificent harvest. But the thinkers who helped to tap these resources... failed to attend, detained in their studies and laboratories, lost in their dreams and calculations, seeking new fields, co-ordinating old and new, spinning abstract theories... the thinkers were unequal to the task of developing these vast resources, most of which they had themselves uncovered. The shrewd declassés, who had... the world to gain, pioneered this development and took possession of the earth."

"The Industrial Revolution, too, failed to introduce a reign of freedom and happiness: it converted the medieval serf into an industrial slave; replaced the feudal baron by the industrial mogul, created in its wake an ever-growing, ever-shifting class of declassés, who had neither pride of ancestry nor love of tradition... The age of machine and competition, of capital, class-struggle, and demagogy was upon man."

"To describe means to classify, and the man Poincaré defies classification, as does indeed his philosophy."

"His essays on the foundations of science are cases in point. They strike one as extemporaneous speeches rather than edited articles. ...those who knew him best insisted that he rarely, if ever, would revise a manuscript, even if he was fully aware of its stylistic shortcomings."

"Poincaré was an artist par excellence. Estheticism with him was not a mere creed: it was a way of life."

"Poincaré's mind was not subject to hysteresis or hibernation. He had the unique faculty of dismissing an idea from his mind, the instant the stimulus was gone, and to supplant it immediately with another creative idea."

"He was an iconoclast. But even in this category he defies classification. For, he fits no pattern, and is beyond all norm. He sought no followers, he shunned confederates, he hewed no tablets to replace those which he had shattered."

"The evolution of scientific thought is inseparable from the history of man's efforts to resolve the perplexities of his own existence."

"The mathematical activity of Ancient Greece reached its peak during the glorious era of Euclid, Eratosthenes, Archimedes and Apollonius, a time when Greek letters, art and philosophy were already on the decline. ...it was not Greece proper but its outposts in Asia Minor, in Lower Italy, in Africa that had contributed most to the development of mathematics."

"Mathematics fluourished as long as freedom of thought prevailed; it decayed when creative joy gave way to blind faith and fanatical frenzy."

"Among students and faculty alike, any mention of his name is usually accompanied by a quick glance around, a devious smile, and hunched-over, half-whispered anecdotes about the man."

"Noam is unlike anyone we've ever had."

"One of the things about math and chess is that we’re better at doing it than philosophizing about it."

"Indeed, Elkies admits he can improvise fugues, "at least on reasonable themes in a slow enough tempo.""

"One does not have to have experience raising children through school, dealing with family tragedies, and so forth, to be able to find three numbers whose fourth powers add up to another one."

"I think I’m starting to resolve a puzzle that’s been bugging me for awhile. Pop economists (or, at least, pop micro-economists) are often making one of two arguments:1. People are rational and respond to incentives. Behavior that looks irrational is actually completely rational once you think like an economist. 2. People are irrational and they need economists, with their open minds, to show them how to be rational and efficient. Argument 1 is associated with “why do they do that?” sorts of puzzles. Why do they charge so much for candy at the movie theater, why are airline ticket prices such a mess, why are people drug addicts, etc. The usual answer is that there’s some rational reason for what seems like silly or self-destructive behavior. Argument 2 is associated with “we can do better” claims such as why we should fire 80% of public-schools teachers or Moneyball-style stories about how some clever entrepreneur has made a zillion dollars by exploiting some inefficiency in the market. The trick is knowing whether you’re gonna get 1 or 2 above. They’re complete opposites!"

"I’m not saying that arguments based on rationality are necessarily wrong in particular cases. (I can’t very well say that, given that I wrote an article on why it can be rational to vote.) I’m just trying to understand how pop-economics can so rapidly swing back and forth between opposing positions. And I think it’s coming from the comforting presence of rationality and efficiency in both formulations. It’s ok to distinguish economists from ordinary people (economists are rational and think the unthinkable, ordinary people don’t) and it’s also ok to distinguish economists from other social scientists (economists think ordinary people are rational, other social scientists believe in “culture”). You just have to be careful not to make both arguments in the same paragraph."

"[O]ne reason why we care about the Riemann hypothesis is that if... we knew it was false, ...it would mean there was a secret pattern to the primes that we were not aware of, and I think we would very rapidly abandon any cryptography based on the primes, because if there was one pattern... there are probably more, and these... can lead to exploits in crypto."