FOOD FOR THOUGHT

“I still remember a lecture by the famous American probabilist Joe Doob, in Paris, in which he discussed probabilities on the non-locally compact space of continuous functions; he was talking about Wiener’s probability and Brownian motion. The members of Bourbaki present at the conference interrupted him constantly, loudly and unpleasantly, on the pretext that his space was not locally compact so what he was saying ‘didn’t make sense’. I was really bothered by their attitude and indignant because of their rudeness. I already mentioned how Paul Lévy was ostracized. Bourbaki should have recognized the legitimate equality of the two categories of measures and thus neutralized his ostracism, especially as its members individually respected and esteemed Paul Lévy and considered him one of the best mathematicians of his generation. The biased influence of Bourbaki and the ostracism of Paul Lévy, together, caused a historical slowdown in French probability theory which, fortunately, has recovered today.

Laurent Schwartz, “A mathematician grappling with his century”, p. 164


“My life closed twice before its close;

It yet remains to see

If Immortality unveil

A third event to me,

So huge, so hopeless to conceive

As these that twice befell.

Parting is all we know of heaven,

And all we need of hell”.

Emily Dickinson. From the book “Dickinson: Selected Poems and Commentaries”, by Helen Vendler


“But here are three of the most remarkable theorems of number theory:

I (Euclid, around 300 B.C.) There exists an infinity of prime numbers (or, in the original formulation, each natural number is smaller than some prime number).

II (Lagrange, 1770) Each natural number can be written as the sum of one, two, three, or four squares of natural numbers (for example: 7 = 2×2 + 1×1 + 1×1 + 1×1).

III (Vinogradov, 1937) There exists a (very large) natural number N such that each odd number greater than N is a prime number or the sum of three odd prime numbers [footnote]”.

[…]

“As for the origins of the problems a mathematician sets out to solve, they must almost always be sought in his contacts with other mathematicians, whether through reading classic works or explanatory texts, or through conversation or exchange of letters, or again through listening to talks. But this is not enough for the historians of science, who are critical of the concept that mathematicians form of their own science, and undertake to bring to bear their universal ‘explanation’, the influence of social environment. This is known to be a dogma proclaimed by many intellectuals in the desire to minimize the contribution and the originality of individuals and to combat what they call ‘élitism’. I am not competent to judge of the validity of this dogma in other sciences, although I cannot see how the societies in which they lived can have influenced the discoveries of Newton or of Einstein. But in mathematics, not counting those parts which serve as models for the other sciences, the dogma seems to me absolutely absurd, in view of what is known of the way in which mathematicians of the past have worked and of the behaviour of our [their] contemporaries. How can it be thought that social environments as diverse as the Alexandria of the Ptolemys for Euclid, the Paris of Louis XV or the Berlin of Frederick II for Lagrange, or the Soviet union under Stalin for Vinogradov, could have anything whatever in common with their theorems in number theory cited above?”.

Jean Dieudonné, “Mathematics—The music of reason”, p. 26-27 (original title: “Pour l’ honneur de l’ esprit humain”)


By way of contrast, let us take one of the most fruitful theories of modern mathematics, the one known as ‘sheaf cohomology’. Thought of in 1946, it is more or less contemporary with the ‘double helix’ of molecular biology, and has led to progress of comparable magnitude. I should, however, be quite unable to explain in what this theory consists to someone who had not followed at least the first two years of a course in mathematics at a university. Even for an able student at this level, the explanation would take several hours; while to explain the way in which the theory is used would take a good deal more time still. This is because here we can no longer make use of explanatory diagrams; before we can get to the theory in question, we must have absorbed a dozen notions equally abstract: topologies, rings, modules, homomorphisms, e.t.c, none of which can be rendered in a ‘visual’ way”.

Jean Dieudonné, “Mathematics—The music of reason”, p. 8 (original title: “Pour l’ honneur de l’ esprit humain”)


 

“But a witty chess master once said that the difference between a master and a beginning chess player is that the beginner has everything clearly fixed in his mind, while to the master everything is a mystery”.

N. Ya. Vilenkin, “In search of infinity”, p. 89


At the end of the nineteenth century, not much more could be credited to exterior algebra, and Engel concluded his biography on a melancholy note, relegating Grassmann to the rank of unheeded prophet of what was then the fashion among mathematicians (and has remained as popular with present day physicists), the horrible ‘Vector analysis’, which we now see as a complete perversion of Grassmann’s best ideas. [Footnote:] It [“Vector analysis”] is limited to three dimensions, replaces bivectors by the awful ‘vector product’ and trivectors by the no less awful ‘mixed product’, notions linked to the Euclidean structure and which have no decent algebraic properties”.

Jean Dieudonné (1979): “The tragedy of Grassmann”, Linear and Multilinear Algebra, 8:1, 1-14


“Only the stubborn conservatism of academic tradition could freeze it [the Riemann integral] into a regular part of the curriculum, long after it had outlived its historical importance”.

Jean Dieudonné, “Foundations of Modern Analysis”, Ch. VIII.


“Except for boolean algebra, there is no theory more universally employed in mathematics than linear algebra; and there is hardly any theory which is more elementary, in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices”.

Jean Dieudonné, “Foundations of Modern Analysis”, Appendix. 

[…] Cartan objected. ‘Listen’, he told me, ‘you mustn’t try to work with these smooth functions with compact support, they’re monstrous’ “.

Laurent Schwartz, “A mathematician grappling with his century”, p. 229. 


“[…] les mathématiciens qui font de l’ abstraction pour l’ amour de l’ abstraction sont le plus souvent des médiocres”.

Jean Dieudonné, “Calcul infinitésimal”, p. 13. 

 

Advertisements