How long have the respective fundamental theorems, for example the fundamental theorem of arithmetic and the fundamental theorem of calculus, been referred to as such? Who coined these names, was it the mathematician who originally discovered them, or was it by consensus over a period of time?
It’s most likely a consensus process, but I don’t know any of the specifics.
AFAIK, calculus is the only branch of mathematics that is considered to have been developed by a single person (actually two people, Newton and Liebniz) so the Fundamental Theorem of Calculus (and the Second Fundamental Theorem of Calculus, whose name still sound like an oxymoron to me) was probably discovered by them, although I couldn’t tell you whether they called it that or not.
That’s a very good question.
The only reference I’ve been able to come up with is the following from Ivor Grattan-Guinness’s The Norton History of the Mathematical Sciences: “The modern name, ‘the fundamental theorem of algebra’, seems to have come in late in the 19th century; Gauss himself [who proved the theorem] never used it.”
The fundamental theorem of projective geometry
I think this is by consensus, but as with many terms like this, ‘fundamental theorem of x’ eventually came to mean something on its own. I’m not a very good lexicographer, but it means something like “theorem without which you could not even start talking about the subject as we know it”
Does the fundamental theorem of algebra follow that pattern? It seems like Lagrange’s theorem, or the theorem that every group is isomorphic to a subgroup of S[sub]n[/sub], would be much better candidates.
By that logic, the FT of Calculus might be more appropriately called the FT of Integral Calculus, since you can do a fair number of problems (though admittedly not that many) with derivatives only. The FTC still unites the two major halves of calculus, I guess, so it does make some sense.
Can I make up a new theorem? TJ’s Theorem: The simpler the integrand, the harder the integral.
Ah, thanks all.
Relevent quote from my topology prof: “You probably think that the Fundamental Theorem of Algebra is a theorem of algebra. Wrong.”
Study a bit of classical algebraic geometry, which is really the study of algebras over C. You’ll quickly come to appreciate that the complex numbers are algebraically closed.
More generally, though, the notion of an algebraically closed field predates group theory. In fact, groups were first described as transformations of extension fields in Galois theory. That such a well-established concept as the complex numbers is already algebraically closed turns out to be very nice for a lot of things.
First of all, the Hilbert nullstellensatz needs algebraic closure. That is, not only does the FToA tell you about solutions of polynomials in one variable, but it tells you a lot about solutions in many variables (remember, btw, that “algebra” used to be about solving equations, not studying algebraic structures).
It turns up in representation theory to say that all complex representations of finite groups are completely reducible. Your suggestions of group-theoretical theorems tell us that a finite group (not every group) has a representation on its own complex group algebra and a bit about the combinatorics of the irreducible subrepresentations, but you wouldn’t even know you had the subreps without algebraic closure.
Lie groups… Don’t even get me started. Basing them on a field of zero characteristic (C) takes care of a little less than half the big headaches. A little more than half are taken care of by? Did you say algebraic closure? Great.
As you keep looking, complex numbers are just more and more unimaginably convenient, and most of the time they’re nice directly because of algebraic closure.
The unification is exactly the point. Move to multivariate calculus and the FToC gets bootstrapped into Stokes’ Theorem (parts of which are called Green’s and Gauss’ theorems in most calc 3 texts). Move up to more general manifolds than R[sup]n[/sup] and you’re not going anywhere without Stokes’ Theorem, which doesn’t itself get off the ground without FToC.
Differential calculus is nice for finding certain geometric answers, but all the real power of the calculus comes from the interplay of differential and integral methods, and that all works because of the FToC.
A good source for answers from researchers on history of mathematics questions, especially these vexing terminology points, is the Historia-Matematica (no “h” in Matematica, it’s Spanish) Archives at the Math Forum. Because it’s a discussion list, sometimes you only find a question with no substantive answers, but a lot of times there’s very detailed information, if not always conclusive.
On the Fundamental Theorem of Calculus: