Largest "leap-to-generality" in math history?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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)

Answer 7

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.

Answer 8

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

Answer 9

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. »

Answer 10

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.

Answer 11

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).

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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 $2006$.

Answer 12

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.

Answer 13

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.

Answer 14

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 ?