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Grothendieck, who is famous inter alia for his capacity/tendency to look for the most general formulation of a problem, introduced a number of new concepts (with topos maybe the most famous ?) that would generalize existing ones and provide a unified, more elegant and more efficient way to think of a class of objects.

What are your favorite examples of such generalizations and their authors ?

(NB: this is a soft question and largely an excuse to commemorate once more the passing of Alexander Grothendieck this week.)

Grothendieck, who is said to have been both very humble and at times very difficult to cope with, reportedly had (source, the quote comes from L. Schwartz's biography) an argument with Jean Dieudonné who blamed him in his young years for "generalizing for the sole sake of generalizing":

Dieudonné, avec l'agressivité (toujours passagère) dont il était capable, lui passa un savon mémorable, arguant qu'on ne devait pas travailler de cette manière, en généralisant pour le plaisir de généraliser.

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    $\begingroup$ The anonymous article you refer to is of doubtful reliability. I wonder what source there is behind the scandalous assertion that Grothendieck slightly despised Dieudonné ( "Il méprisait légèrement Dieudonné"). The assertion that he left Bourbaki because of his rows with Weil ("ses prises de becs avec Weil causèrent son départ de Bourbaki") is demonstrably false: Grothendieck, in a letter to Serre dated 3-5 October 1964, explains that Bourbaki bores him and that it is difficult for him not to doze off in their meetings! Consequently he writes that he will attend the meetings less often. $\endgroup$ – Georges Elencwajg Nov 17 '14 at 9:22
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    $\begingroup$ The compliant by Dieudonné is of course a very ... generalized statement $\endgroup$ – Hagen von Eitzen Nov 17 '14 at 16:39
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    $\begingroup$ Huh. I did not hear about his death. Thanks. Could you translate this part into English? I'm curious about it (true or not). $\endgroup$ – tomasz Nov 17 '14 at 23:35
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    $\begingroup$ Dear @Alex: thanks for this information. Anyway Grothendieck's judgments have zero reliability. His absurd accusations in Récoltes et semailles of plagiarism against most of the mathematicians he had worked with are particularly saddening when addressed to Deligne, his most loyal admirer and the most modest and honest mathematician conceivable. Grothendieck himself apologizes (in the introduction) for his slandering of Kashiwara, admitting that he had just relayed without any checking Mebkhout accusations. Reading Récoltes et Semailles is an interesting but also very depressing experience. $\endgroup$ – Georges Elencwajg Nov 18 '14 at 13:04
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    $\begingroup$ I am not so sure topoi are more famous than schemes... but there are people more knowledgeable than I better prepared to comment on this. $\endgroup$ – user98602 Nov 19 '14 at 17:04

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To modern eyes the definition of metric space by Frechet in 1906 may not seem much, but it paved the way to modern analysis and was a huge leap forward. The ubiquity of metric spaces in virtually all realms of mathematics also shows how profound Frechet's contribution was.

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    $\begingroup$ And then, of course, the generalization to topology... $\endgroup$ – Thomas Andrews Nov 25 '14 at 21:34
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Surely the step from numbers to groups and fields (which is due mostly to 19th-century mathematicians such as Abel, Galois and Dedekind) must count as one one of the greatest leaps forward in history.

Much of what was already known about numbers was quickly reproven for abstract algebraic structures – and thus for an infinite number of concrete structures with sheer limitless applications.

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  • $\begingroup$ Agreed. Btw, who exactly came with the modern definition/axiomatization and naming of "a group"? (maybe that's two different persons) $\endgroup$ – Alexandre Halm Nov 17 '14 at 8:40
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    $\begingroup$ The abstract definition of group is attributed to von Dyck, en.wikipedia.org/wiki/Walther_von_Dyck $\endgroup$ – arsmath Nov 17 '14 at 8:49
  • $\begingroup$ Thanks, didn't know that. I was pretty sure it was Abel or Galois (probably a bias from French education ...). $\endgroup$ – Alexandre Halm Nov 17 '14 at 9:35
  • $\begingroup$ @AlexH.: The history, development and conception of the field of algebra, from Galois to Grothendieck, is throughout reported on in the PhD-thesis-turned-400-book Modern Algebra and the Rise of Mathematical Structures by Corry. I really enjoyed it! (The book happens to be on the web somewhere, btw.) $\endgroup$ – Nikolaj-K Nov 20 '14 at 9:24
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My 5 cents in favor of category theory by Samuel Eilenberg and Saunders Mac Lane.

Not only did it provide a unifying language for very diverse groups of objects/relations, it was the first (to my very limited knowledge) abstract theory to put the emphasis on morphisms preserving structure (rather than structures themselves), and it paved the way to Grothendieck's Topos theory.

Also, at first sight it can seem quite over-abstract and pointless (see for instance http://en.wikipedia.org/wiki/Abstract_nonsense), which makes its power even more surprising.

Edit: I also need to include Laurent Schwartz for his theory of distributions, which (at least in common terms) are generalizations of $L^1$ functions and Radon measures. The theory finally gave a proper context to write things like $H' = \delta$ (with $H$ the Heavyside function and $\delta$ the Dirac "function"). Anecdotally, it helped me understand the point of taking the smallest possible set (here test functions) to build the largest possible dual space.

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The jump form length/area/volume to the abstract measure theory, that encompasses this basic measures and much stranger measures and is the essential tool of probability theory.

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I think the introduction of complex numbers (that can be seen as a generalization of real numbers) by Gerolamo Cardano in 1545 is noteworthy.

Note: Introduction of "nonnatural" quantities like $0$ and negative numbers are of the same kind of huge leap in the maths concepts.

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Ramsey theory, beginning with Ramsey's theorem itself, or before that with the pigeonhole principle, and generalized by Erdős and Rado et al. to transfinite cardinal and ordinal numbers, order types, graphs, etc.

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Riemann's epoch making paper on Number Theory which connected two seemingly unrelated areas of Mathematics, Number Theory and Complex Analysis.

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    $\begingroup$ But did he do that by generalizing? $\endgroup$ – Thomas Andrews Nov 25 '14 at 21:36
  • $\begingroup$ Yes. He generalized the definition of $\zeta$ function. $\endgroup$ – user 170039 Nov 26 '14 at 4:17
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I would have to say the invention of set theory by Cantor. At first, there was a huge amount of resistance to it, which, if anything, is a very good measure of how big of a leap it was at the time. Grothendieck, in the introduction to EGA, says:

«Il sera sans doute difficile au mathématicien, dans l’avenir, de se dérober à ce nouvel effort d’abstraction, peut-être assez minime, somme toute, en comparaison de celui, fourni par nos pères, se familiarisant avec la théorie des ensembles. »

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    $\begingroup$ Nice quote and perfectly in tune with the question :) $\endgroup$ – Alexandre Halm Nov 25 '14 at 13:42
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I am always impressed by mathematicians who were able to make statements beyond our everyday experience, which is shaped by the mostly finite physical world, especially when going from the finite to the infinite, like

  • the folks who went from ten fingers to larger numbers
  • Archimedes approximating the area of a circle with $n$-polygons
  • Euclid proofing the existence of infinite many prime numbers
  • the mathematicians who came up with infinite sequences, sums, products and continued fractions
  • Cantor finding different levels of infinity
  • the mathematicians who extended finite dimensional vector spaces to infinite function spaces

I am not sure if that is rather extension than generalisation, but in these cases the domain of existing concepts were extended.

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Surely something in the history of probability would qualify.

Be it:

  • Blaise Pascal and Pierre Fermat - "credited with founding mathematical probability because they solved the problem of points, the problem of equitably dividing the stakes when a fair game is halted before either player has enough points to win." i.e., they took to mathematizing games of chance.

  • James Bernoulli, Montmort, and De Moivre, who "investigated many of the problems still studied under the heading of discrete probability, including gambler's ruin, duration of play, handicaps, coincidences, and runs..."

  • "Daniel Bernoulli, Joseph Louis Lagrange, and Laplace, [who] derived methods for combining observations from various assumptionsabout error distributions. The most important fruit of this probabilistic work was the invention by Laplace of the method of inverse probability—what we now call the Bayesian method of statistical inference."

Or many other examples.

(Source: http://www.glennshafer.com/assets/downloads/articles/article50.pdf)

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    $\begingroup$ Actually you're right. This made me think of Kolmogorov, who I believe was among the first to propose the modern axiomatisation of probabilities, which in turn made me think of Lebegue, who surprisingly hasn't been mentioned. $\endgroup$ – Alexandre Halm Nov 20 '14 at 6:07
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I'd like to introduce my favorite example by citing a question I found in an email in the net:

  • When I asked many people what is the most difficult areas of math to understand, and also what is the most important unsolved mathematical problem, they always responded with two words: Langlands Conjecture (or Langlands philosophy or Langlands Program).

In this email we can also read from kubo@brauer.harvard.edu (Tal Kubo):

  • The Langlands program is a system of conjectures connecting number theory and the representation theory of Lie groups.

  • It predicts that a large class of the $\zeta$- and $L$-functions coming from number theory and algebraic geometry coincide with $L$-functions coming from representation theory.

  • $L$-functions have been used for over a century in number theory; Langlands isolated the correct analogue from representation theory (so-called "automorphic" $L$-functions) and was the first to understand the general picture.

As an example of why one might expect some sort of connection between Lie groups and number theory, consider the Galois group $$G = \text{Gal}(K/L)$$ where $K$ is an algebraic closure of a number field $L$.

Number theorists are very interested in representations of $K$. $G$ is a profinite group (projective limit of finite groups). Representation theory of profinite groups is not so well-developed but there is at least one situation where there is some hope: algebraic groups over p-adic rings. Lie groups over p-adic fields turn out to be prominent actors in Langlands' conjectures.

All these theories are (of course) far beyond my knowledge. But I'm deeply fascinated about it.

Here's a (so-called) elementary introduction and an interesting page with refs. You might also find some interesting posts about the Langlands conjecture in math overflow.

To get a quick impression of this extremely difficult and extraordinary exciting field you may also visit Geometric Langlands and QFT where you can find some milestones from the beginning of the theory in the early $1970$s up to $2006$.

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  • $\begingroup$ Interesting. Thanks, I must say I had never heard of Langlands, makes an interesting read. $\endgroup$ – Alexandre Halm Nov 25 '14 at 5:03
  • $\begingroup$ @AlexH.: You're welcome! Langlands program is exciting, tremendously far-reaching and extremely difficult. I suppose it will keep top mathematicians busy for this and the next century! Somewhere I've read that Andrew Wiles' proof of the Taniyama-Shimura-Weil conjecture from which the FLT follows was just a small exercise within the framework of Langlands program! :-) $\endgroup$ – Markus Scheuer Nov 25 '14 at 8:02
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Perhaps Claude Shannon deserves a mention for his work, such as defining Entropy in the 1948 paper "A Mathematical Theory of Communication" and generally spurring the field of information theory. He created very general models for communication and gave some profound results about them. I don't know much about Shannon's other contributions (besides that they are numerous) so hopefully someone can inform.

Alan Turing's general computing device, the "Turing Machine", could be mentioned. In a world without electronic computers, "this machine can compute anything that you can" seems to me to be a monumental generalization, even to the foundations of mathematics.

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  • $\begingroup$ True. At least he created the formal context which permitted to handle real-life problems, that's probably an example of generalisation (maybe more axiomatisation or conceptualisation, but that's only terminology). Makes me think Gödel would probably deserve a quote also... $\endgroup$ – Alexandre Halm Nov 25 '14 at 12:25
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I want to highlight the Aristotelian logic.

In human language there was discovered the most general mathematical system which even today forms the basis of science and especially mathematics. From tautologies, combinations of sentences which independently of the context seems to be true, truly non-trivial systems of logical deductions was evolved.

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May be it is not the most generalized theory, but I think Lagrangian approach had the most general impact on science.

Modern physics, modern chemistry, are all based on such approach.

But that is not math, or was it ?

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