[ moved from a conversation on ElizabethWiethoff ]
If you liked the concept of groups, you should definitely read about its inventor, a french boy named Galois (http://en.wikipedia.org/wiki/Galois). Goedel aside, he's IMHO the most interesting person in recent mathematics. -- PhilipBusch
Stillwell in *Mathematics and its History* says that, to have published what he did, at the age he did, given that we do know when he started studying math (late...age 17 or something), he must have learned more math faster than anyone else in history, including up through the present. That's not **necessarily** literally true, since an argument can also be made for Gauss, Ramanujan, etc, but still, Galois certainly is a prime candidate. -- Doug
I'm a bit cautious regarding such statements as cited above. In Galois' times, the science called "mathematics" was completely different from what we have today. There were no sets (!), no infinity, obviously no groups, hence no fields, natural numbers couldn't be proved or derived, real numbers were mysterious, irrational numbers plain magic, complex numbers a joke, there was no such thing as "function theory" and so on. Besides, every math student today graduates within about three years and knows about ten times more (think about TaniyamaShimuraWeilConjecture). Nevertheless, Galois is an interesting person; I actually wonder how he could define a "group" without a proper definition of "set", as a group merely is a set with an associated operation and some trivial properties (don't know the proper english terms for them). But my point actually is Galois' life, not his discoveries. Most people seem to think that mathematicians are boring nerds and couldn't imagine that they get killed in duels (Galois), kill themselves (Taniyama), kill others (Newton), go completely mad (Cantor, Goedel), have strong political opinions (Einstein) or actually have some other life besides mathematics. I particularly like Galois as I primarily see the revolutionary rather than the mathematician. His early death fascinates me, as it is not clear what actually pushed him to participate in a duel. One might think that such a bright person would know better... Maybe it got stuck because I'm in vaguely the same age (23) and have vaguely equal political perceptions and share his deep interest in mathematics (though I haven't finished university yet, because I first studied computer science). -- PhilipBusch
The particular examples you give are methods for adding rigor to math (e.g. Dedekind's formalization of the reals). Not a single one of those topics was outright missing from math in Galois' day, they merely had not been rigorously formalized yet. I also don't see why you include irrationals in that list; in 1832, no one had been afraid of irrationals for hundreds of years. Complex numbers had been manipulated routinely for a long time by then, and Gauss and already created algebraic numbers. Infinity sure as hell had been around for thousands of years -- perhaps you meant to say higher orders of infinity.
As to how he could possibly define groups without sets, that's an odd question unless you've been exposed to a strangely too-abstract definition of groups. He considered them as blobs being permuted, and permutations had been around for a long time (1321 C.E., Levi ben Gershon). Permutation was applied to the roots of polynomials by Vandermonde (1771) and Lagrange (1771). Galois built on such former work, and the motivation was polynomials. Group theory as he created it applied only to polynomials. The current abstract view of applying to anything and everything, like the classic crystallography example, came after Galois' death.
As for what actually fascinates you about Galois, that romantic stuff about his tragic early death is the whole reason he's famous with anyone but mathematicians. It's pretty much a cliche (a fun one), due to the romanticized story about him in the extremely famous "Men of Mathematics" by Bell. -- Doug

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