mathematicians-from-france

"This present art, in which we use those twice five Indian figures, is called algorismus."

"It was not exactly as you say. In practice, the 12 of us never lived at home at the same time, because the family grew up with 22 years between the eldest to the youngest (the last one was born in 1968). My elder sisters left early, and I am the tenth. In my first memories, there were only five or six siblings at home. The worst was not to have a room of my own. The house was big, but the rooms were not many, however large."

"In this respect I recall with much sympathy the writer Virginia Woolf who wrote about this need for private space (A Room of One’s Own) as a condition for liberty and creativity. I had a room of my own only by the end of high school, of which I have excellent memories."

"Hermite was born in 1822, four years before Riemann. Now in his sixties, he had become one of the standard-bearers for Cauchy and Riemann's work on functions of imaginary numbers. Cauchy's influence on Hermite went beyond mathematics. As a young man Hermite had been an agnostic, but Cauchy, a devout Roman Catholic, had caught Hermite in a weak moment during a severe illness and had converted him to Catholicism. The result was a strange mix of mathematical mysticism, akin to the cult of the Pythagoreans. Hermite believed that mathematical existence was some supernatural state which mortal mathematicians were only occasionally allowed to glimpse."

"It’s the same kind of pleasure as with hiking: You hike uphill and it’s tough and you sweat, and at the end of the day the reward is the beautiful view. Solving a math problem is a bit like that, but you don’t always know where the path is and how far you are from the top. You have to be able to accept frustration, failure, your own limitations."

"The Indian people were the only civilization to take the decisive step towards the perfection of numerical notation. We owe the discovery of modern numeration and the elaboration of the very foundations of written calculations to India alone.""

"The Indian mind has always had for calculations and the handling of numbers an extraordinary inclination, ease and power, such as no other civilization in history ever possessed to the same degree. So much so that Indian culture regarded the science of numbers as the noblest of its arts ... A thousand years ahead of Europeans, Indian savants knew that the zero and infinity were mutually inverse notions."

"In India, an aptitude for the study of numbers and arithmetical research was often combined with a surprising tendency towards metaphysical abstractions; in fact, the latter is so deeply ingrained in Indian thought and tradition that one meets it in all fields of study, from the most advanced mathematical ideas to disciplines completely unrelated to 'exact sciences.""

"It is clear how much we owe to this brilliant civilization, and not only in the field of arithmetic; by opening the way to the generalization of the concept of the number, the Indian scholars enabled the rapid development of mathematics and exact sciences. The discoveries of these men doubtless required much time and imagination, and above all a great ability for abstract thinking. These major discoveries took place within an environment which was at once mystical, philosophical, religious, cosmological, mythological and metaphysical.""

"One of the first pieces of advice I got as I was starting my Ph.D. was from Tristan Rivière (a previous student of my adviser, Fabrice Béthuel), who told me: People think that research in math is about these big ideas, but no, you really have to start from simple, stupid computations — start again like a student and redo everything yourself. I found that this is so true. A lot of good research actually starts from very simple things, elementary facts, basic bricks, from which you can build a big cathedral. Progress in math comes from understanding the model case, the simplest instance in which you encounter the problem. And often it is an easy computation; it’s just that no one had thought of looking at it this way."

"In short, Indian science was born out of a mystical and religious culture and the etymology of the Sanskrit words used to describe numbers and the science of numbers bears witness to this fact.""

"I tell myself that there are always very bright people who have thought about these problems and made very beautiful and elaborate theories, and certainly I cannot always compete on that end. But let me try to rethink the problem almost from scratch with my own little basic understanding and knowledge and see where I go. Of course, I have built enough experience and intuition that I sort of pretend to be naive. In the end, I think a lot of mathematicians proceed this way, but maybe they don’t want to admit it, because they don’t want to appear simple-minded. There is a lot of ego in this profession, let’s be honest."

"Ancient Indian culture has regarded the science of numbers as the noblest of its arts … A thousand years ahead of Europeans, Indian savants knew that zero and infinity were mutually inverse notions. In short, Indian science was born out of a mystical and religious culture and the etymology of the Sanskrit word used to describe numbers and the science of numbers bears witn The early passion which Indian civilization had for high numbers was a significant factor contributing to the discovery of the place-value system, and not only offered the Indians the incentive to go beyond the calculable physical world, but also led to an understanding much earlier than in our civilization of the notion of mathematical infinity itself."

"Sanskrit means "complete", "perfect" and "definitive". In fact, this language is extremely elaborate, almost artificial, and is capable of describing multiple levels of meditation, states of consciousness and psychic, spiritual and even intellectual processes. As for vocabulary, its richness is considerable and highly diversified. Sanskrit has for centuries lent itself admirably to the diverse rules of prosody and versification. Thus we can see why poetry has played such a preponderant role in all of Indian culture and Sanskrit literature." 604"

"...If you have enough ability, then you cultivate it and build on it, just as a musician plays scales and practices to get to a top level."

"From the axiomatic point of view, mathematics appears thus as a storehouse of abstract forms -- the mathematical structures; and it so happens -- without our knowing why -- that certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation. Of course, it can not be denied that most of these forms had originally a very definite intuitive content; but, it is exactly by deliberately throwing out this content, that it has been possible to give these forms all the power which they were capable of displaying and to prepare them for new interpretations and for the development of their full power. The unity which it gives to mathematics is not the armor of formal logic, the unity of a lifeless skeleton; it is the nutritive fluid of an organism at the height of its development, the supple and fertile research instrument to which all the great mathematical thinkers since Gauss have contributed, all those who, in the words of Lejeune-Dirichlet, have always labored to "substitute ideas for calculations.""

"At the center of our universe are found the great types of structures... they might be called the mother-structures... Beyond this first nucleus, appear the structures which might be called multiple structures. They involve two or more of the great mother-structures simultaneously not in simple juxtaposition (which would not produce anything new), but combined organically by one or more axioms which set up a connection between them... Farther along we come finally to the theories properly called particular. In these the elements of the sets under consideration, which, in the general structures have remained entirely indeterminate, obtain a more definitely characterized individuality. At this point we merge with the theories of classical mathematics, the analysis of functions of a real or complex variable, differential geometry, algebraic geometry, theory of numbers. But they have no longer their former autonomy; they have become crossroads, where several more general mathematical structures meet and react upon one another."

"... in a single view, it sweeps over immense domains, now unified by the axiomatic method, but which were formerly in a completely chaotic state... In place of the sharply bounded compartments of algebra, of analysis, of the theory of numbers, and of geometry, we shall see, for example, that the theory of prime numbers is a close neighbor of the theory of algebraic curves, or, that Euclidean geometry borders on the theory of integral equations."

"But in the eyes of contemporary structuralist mathematicians, like the Bourbaki, the Erlanger Program amounts to only a partial victory for structuralism, since they want to subordinate all mathematics, not just geometry, to the idea of structure. Classical mathematics is a quite heterogeneous collection of algebra, theory of numbers, analysis, geometry, probability calculus, and so on. Each of these has its own delimited subject matter; that is, each is thought to deal with a certain “species” of objects... Transformations may be disengaged from the objects subject to such transformation and the group defined solely in terms of the set of formations. The Bourbaki program consists essentially in extending this procedure by subjecting mathematical elements of every variety, regardless of the standard mathematical domain to which they belong, to this sort of “reflective abstraction” so as to arrive at structures of maximum generality., since they want to subordinate all mathematics, not just geometry, to the idea of structure."

"It can now be made clear what is to be understood, in general, by a mathematical structure. The common character of the different concepts designated by this generic name, is that they can be applied to sets of elements whose nature has not been specified; to define a structure, one takes as given one or several relations, into which these elements enter (in the case of groups, this was the relation z = xτy between three arbitrary elements); then one postulates that the given relation, or relations, satisfy certain conditions (which are explicitly stated and which are the axioms of the structure under consideration.) To set up the axiomatic theory of a given structure, amounts to the deduction of the logical consequences of the axioms of the structure, excluding every other hypothesis on the elements under consideration (in particular, every hypotheses as to their own nature)."

"The "structures" are tools for the mathematician; as soon as he has recognized among the elements, which he is studying, relations which satisfy the axioms of a known type, he has at his disposal immediately the entire arsenal of general theorems which belong to the structures of that type. Previously, on the other hand, he was obliged to forge for himself the means of attack on his problems; their power depended on his personal talents and they were often loaded down with restrictive hypotheses, resulting from the peculiarities of the problem that was being studied... each structure carries with it its own language, freighted with special intuitive references derived from the theories from which the axiomatic analysis described above has derived the structure. And, for the research worker who suddenly discovers this structure in the phenomena which he is studying, it is like a sudden modulation which orients at one stroke in an unexpected direction the intuitive course of his thought and which illumines with a new light the mathematical landscape in which he is moving about."

"Nicolas Bourbaki is the nom-de-plume adopted in the 1930s by a group of out standing young French mathematicians who undertook the monumental task of reorganizing mathematics in terms of basic structural components. This enterprise is an ongoing effort, whose members must resign at age fifty according to Bourbaki's bylaws."

"Now one final remark on the term spectrum. In physics each type of atom or molecule possesses a characteristic spectrum formed by its emission or absorption lines. Quantum mechanics interprets these as the characteristic values of an operator, the Hamiltonian, acting on a certain Hilbert space. It is thus natural to speak of the discrete spectrum of the Hamiltonian. The emission or absorption bands correspond to a continuous spectrum. In the early 1930s von Neumann succeeded brilliantly in defining the concept of a self-adjoint (unbounded) operator H on a Hilbert space 𝔥 and its spectrums. The contribution of Gelfand in 1940 was in associating a commutative Banach algebra A with the operator H and an isomorphism of A onto C0(S;\mathbb{C}). From that point on the evolution of the meaning of the word spectrum can be understood. For Grothendieck the spectrum of a commutative ring consists of its prime ideals (as in the case of Dedekind)."

"... Formal groups. This topic is by far the deepest and most imaginative creation of Dieudonné, realized when Dieudonné was nearing 50, supposedly the term for an active mathematical life. It can be seen as the creation of a differential calculus for groups over a field of characteristic p > 0 (possibly finite). The methods of calculus do not work, and one has to resort to pure algebra. There were a number of forerunners: a version of Taylor’s formula in characteristic p > 0 due to Dieudonn ́e himself, the ideas of Delsarte about convolution operators (as explained in Book IV, chapter 6 of Bourbaki’s Éléments), a definition of the Lie algebra of a Lie group and its enveloping algebra in terms of distributions on the group (by L. Schwartz). But the impetus came from the book by Chevalley, in 1951, about algebraic groups. Chevalley had developed a purely algebraic version of Lie theory, but restricted to fields of characteristic 0. The case of characteristic p > 0 was “terra incognito”. In a long series of papers, published between 1954 and 1958, later on collected into a book ... Dieudonné explored in depth this new world."

"Let us now consider toposes. ... Unlike schemes, toposes generate geometry without points. In fact, nothing prevents us from proposing an axiomatic framework for geometry in which points, lines, and planes would all be on the same footing. Thus we know axiomatic systems for projective geometry (George Birkhoff) in which the primitive notion is that of a plate (a generalization of lines and planes), and in which the fundamental relationship is that of incidence. In mathematics, we consider a class of partially ordered sets called lattices; each of these corresponds to a distinct geometry. ... In the geometry of a topological space, the lattice of open sets plays a starring role, while points are relatively minor. But Grothendieck’s originality was to reprise Riemann’s idea that multivalued functions actually live not on open sets of the complex plane, but on spread-out Riemann surfaces. The spread-out Riemann surfaces project down to each other and thus form the objects of a category. However, a lattice is a special case of a category, since it includes at most one transformation between two given objects. Grothendieck thus proposed replacing the lattice of open sets with the category of spread-out open sets. When adapted to algebraic geometry, this idea solves a fundamental difficulty, since there is no implicit function theorem for algebraic functions. Sheaves can now be considered as special functors on the lattice of open sets (viewed as a category), and can thus be generalized to étale sheaves, which are special functors of the étale topology. Grothendieck would successfully play many variations on this theme in the context of various problems of geometric construction (for example, the problem of modules for algebraic curves). His greatest success in this regard would be the étale “ℓ-adic” cohomology of schemes, the cohomological theory needed to attack the Weil conjectures."

"Has not printing freed the education of the people from all political and religious shackles? It would be vain for any despotism to invade all the schools ... The instruction that every man is free to receive from books in silence and solitude can never be completely corrupted. It is enough for there to exist one corner of free earth from which the press can scatter its leaves. How with the multitude of different books, with the innumerable copies of each book, of reprints that can be made available at a moment’s notice, how could it be possible to bolt every door, to seal every crevice through which truth aspires to enter?"

"Enjoy your life without comparing to that of others. It is enough for you to know that you are good, without examining whether others are as good as you."

"All those who were acquainted with Hadamard know that until the end of his very long life, he retained an extraordinary freshness of mind and character: in many respects, his reactions remained those of a fourteen-year-old boy. His kindness knew no bounds. The warmth with which Hadamard received me in 1921 eliminated all distance between us. He seemed to me more like a peer, infinitely more knowledgeable but hardly any older; he needed no effort at all to make himself accessible to me."

"... Professor Hadamard concludes that the general pattern of invention, or, as it might also be put, of original work, is three-fold : conscious study, followed by unconscious maturing, which leads in turn to the moment of insight or illumination. Thereupon another period of conscious work ensues, the purpose of which is to achieve a synthesis of several elements: the novel idea, its logically deduced consequences including proof, and the traditional knowledge to which the new item is added."

"We are going to speak of the role of analysis situs in our modern mathematics. This theory is also called the geometry of situation. It is the study of connection between different parts of geometrical configurations which are not altered by any continuouse deformation. For instance, a sphere and a cube are considered as one and the same thing from the point of view of the geometry of situation, because one can be transformed into the other without separating parts, or uniting parts which formerly were separated."

"... In the case of ordinary differential equations, the arbitrary elements being numerical parameters, we have to determine them by an equal number of numerical equations, so that, at least theoretically, the question may be considered as solved, being reduced to ordinary algebra; but for partial differential equations, the arbitrary elements consist of functions, and the problem of their determination may be the chief difficulty in the question. ..."

"Just after the discovery of infinitesmal calculus, physicists began by needing only very simple methods of integration, the problems in general reducing to elementary differential equations. But when higher partial differential equations were introduced, the corresponding problems almost immediatelly proved to be far above the level of those which contemporary mathematics could treat."

"The systematic study of the singularities of analytic functions was begun by Hadamard. In 1901, a very valuable account of his own investigations together with those of other early workers, as Fabry, Leau, LeRoy, Borel and others, was presented by Hadamard in his now classic little book La Série de Taylor et son Prolongement Analytique published in the Collection Scientia (No. 12)."

"In the case of partial differential equations employed in connection with physical problems, their use must be given up in most circumstances, for two reasons: first, it is in general impossible to get the general solution or general integral, and second, it is in general of no use even when it is obtained."

"... Let a perturbation be produced anywhere, like sound; it is not immediately perceived at every other point. There are then points in space which the action has not reached in any given time. Therefore the wave, in that sense a surface, separates the medium into two portions (regions): the part which is at rest, and the other which is in motion due to the initial vibration. These two portions of space are contiguous. It was only in 1887 that Hugoniot, a French mathematician, who died prematurely, showed what the surface of the wave can be; and even his work was not well known until Duhem pointed out its importance in his work on mathematical physics."

"‘Mons. Bailly, the celebrated author of the History of Astronomy, may be regarded as beginning the concert of praises, upon this branch of the science of the Hindus. The grounds of his conclusions were certain astronomical tables; from which he inferred, not only advanced progress in the science, but a date so ancient as to be entirely inconsistent with the chronology of the Hebrew Scriptures. [...] Another cause of great distrust attaches to Mons. Bailly, Voltaire, and other excellent writers in France, abhorring the evils which they saw attached to catholicism, laboured to subvert the authority of the books on which it was founded. Under this impulse, they embraced [...] the tales respecting the great antiquity of the Chinese and Hindus as disproving, entirely, the Mosaic accounts of the duration of the present race of men. [...] The argument [...] by Mons. Bailly, was [...] for a time regarded as a demonstration in form of the falsehood of Christianity.’"

"‘It follows, therefore, that the astronomers of Alexandria take from the Indians the primitive and fundamental knowledge of the theory of the moon.’"

"Even before Jones's announcement, Bailly stated that "the Brahmans are the teachers of Pythagoras, the instructors of Greece and through her of the whole of Europe" (51)."

"The motion of the stars calculated by the Hindus some 4500 years before vary not even a single minute from the modern tables of Cassini and Meyer."

"The astronomer and onetime mayor of Paris, Jean-Sylvain Bailly, in his Histoire de I'astronomie ancienne et moderne (1805), felt that "these tables of the Brahmana are perhaps five or six thousand years old" (53;). Bailly approved of the traditional date of the Kali Yuga, and seemed to have convinced at least some of his colleagues such as Laplace and Playfair of the accuracy of the Indian astronomical claims (Kay, [1924] 1981, 2). This was bitterly opposed by another astronomer, John Bentley ([1825] 1981), with a concern that we have seen was typical for the times: "If we are to believe in the antiquity of Hindu books, as he would wish us, then the Mosaic account is all a fable, or a fiction" (xxvii)."

"That Hindu astronomical lore about ancient times cannot be based on later back-calculation, was also argued by Playfair’s contemporary, the French astronomer jean-Sylvain Bailly: “The motions of the stars calculated by the Hindus before some 4500 years vary not even a single minute from the [modem] tables of Cassini and Meyer. The Indian tables give the same annual variation of the moon as that discovered by Tycho Brahe - a variation unknown to the school of Alexandria and also the Arabs.”"

"The Hindu systems of astronomy are by far the oldest, and that from which the Egyptians, Greeks, Romans, and even the Jews derived Hindus their knowledge."

"These observations must therefore have been made elsewhere, and one can hardly refuse to believe that they were made in India where the Chaldeans seem to have borrowed the first elements of their Astronomy."

"The theory of functions of several complex variables has gone from its infancy with the work of Hartogs, Levi and Poincaré shortly after the turn of the century to its current role as a central field of modern mathematics, much as its predecessor, function theory in one complex variable, did in the 19th century. A central figure in this development has been Henri Cartan, whose series of papers in this field starting in the 1920's dealt with fundamental questions relating to Nevanlinna theory, generalizations of the Mittag-Leffler and Weierstrass theorems to functions of several variables, problems concerned with biholomorphic mappings and the biholomorphic equivalence problem, domains of holomorphy and holomorphic convexity, etc. The major developments in the theory from 1930 to 1950 came from Cartan and his school in France, Behnke's school in Münster, and Oka in Japan. The central ideas up to that time were synthesized in Cartan's Séminaires in the early 1950's, and these were very influential to the next several generations of mathematicians. Cartan's accomplishments were broad and he influenced mathematics through his writing, his teaching, his seminars, and his students in a remarkable manner."

"Speak little of what you know, and not at all of what you do not know."

"No useless discourse. All conversation which does not serve to enlighten ourselves or others, to interest the heart or amuse the mind, is hurtful."

"Speak to every one of that which he knows best. This will put him at his ease, and be profitable to you."

"Iron and heat are... the supporters, the bases, of the mechanic arts."

"Nature, in providing us with combustibles on all sides, has given us the power to produce, at all times and in all places, heat and the impelling power which is the result of it. To develop this power, to appropriate it to our uses, is the object of heat-engines."

"Question thyself to learn what will please others."