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I am just curious about some theorems that I had been used for some time.

We have Fundamental Theorem on Calculus the Fundamental Theorem of Calculus. In number theory we have Fundamental Theorem of Arithmetic. In other fields we have other fundamental theorems as well.

My question then is why they are called fundamental theorems? And compared to other theorems are they the most important?

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  • $\begingroup$ Usually they form the backbone of the subject. However, I wouldn't be surprised if some of the names are mostly just historical now. $\endgroup$ – Hayden Feb 16 '15 at 22:41
  • $\begingroup$ Thanks @Hayden. I have an idea now. $\endgroup$ – Jr Antalan Feb 16 '15 at 22:43
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    $\begingroup$ Sometimes indeed it is just historical. The Fundamental Theorem of Algebra (that $\mathbf C$ is algebraically closed) is certainly not fundamental for algebra in the modern sense of that term. Back in the 1800s when "all numbers" were inside the complex numbers, that theorem could perhaps have been considered fundamental for explaining why $\mathbf C$ sufficed for roots of polynomials. But nowadays, while it's still important, it is by no means fundamental. $\endgroup$ – KCd Feb 16 '15 at 23:02
  • $\begingroup$ Got it @KCd thanks $\endgroup$ – Jr Antalan Feb 16 '15 at 23:29
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The importance of so-called fundamental theorems is related to who labelled them "fundamental" and to an extent, to how many people care about their subject matter. Let's look at 9 examples, spanning a good deal of variety, and one that is not called "fundamental" but perhaps should be.

  • The fundamental theorem of arithmetic is truly important and a building block of number theory. It also contains the seeds of the demise of prospects for proving arithmetic is complete and self-consistent because any system rich enough to allow for unique prime factorization is subject to the classical proof by Godel of incompleteness.

  • The fundamental theorem of calculus was the "aha" moment discovered by Leibniz and by Newton (and perhaps vaguely realized by others before them). It is the different ingredient between what newton could do with Calculus and what Archimedes (himself a really sharp guy) could do without it: any area or volume calculations before calculus were a tour de force; after the FTC many became almost trivial.

  • The fundamental theorem of algebra (that every degree-$n$ equation has roots of total multiplicity $n$) is also quite important. It was kind-of known many years earlier, but Gauss made his mathematical bones by being the first to prove it, in his doctoral dissertation. It is perhaps less deserving of the title fundamental in the sense of "it all depends on this one"," since it is a pinnacle of achievement, moving forward from notions hinted at by de Moivre's theorem about roots of unity.

  • The fundamental theorem of equivalence relations, which says loosely speaking that the set of equivalence classes partitions a set, is also worthy of the word fundamental. Much of the subsequent theory in that field relies on this concept.

  • The fundamental theorem of matrix games is absolutely essential because in some sense it allows you to put the notions of the "solution" and the "value" of a 2-player game on a firm footing. To appreciate it, you have to see how disquieting it is to study a 2-player (non-matrix) game that violates this theorem. I have studied a lot of non-matrix 2-player zero-sum games, and even in those cases I start with the existence of a solution in the sense of the fundamental theorem, although in the end I have to go back to prove that the purported solution does work.

  • The "fundamental theorem of game theory", put forth by John Nash, says that any $n$-player game based on a finite set of pure strategies for each player has an equilibrium point. While beautiful and surprising, this theorem might not be considered fundamental; in fact, von Neumann considered it "a triviality". The point is that (unlike in the 2-player case) the equilibrium point is often not so relevant to the optimal solution of the game, since it is usually unstable against the notion of more than one player colluding.

  • The "fundamental theorem of card counting" unifies some observations of gaining advantage in Blackjack, Baccarat, and other such games, based on the notion that the fluctuations in a process based on selection without replacement grow as the remaining pool size decreases. The name was pinned on by just a few authors. The theorem is true, and a nice observation, but I'd hesitated to call it fundamental.

  • The "fundamental theorem of gambling" appears to me to be a one-author naming of a pretty trivial fact in probability, that this one person considers important in the construction of gambling strategy systems. I'd give this one thumbs down.

  • The fundamental theorem of Galois group theory is indeed a fundamental starting block for rich further development (and yet non-trivial to demonstrate). As opposed to all the other examples I have given, this one is not easy to explain to somebody without some study of Galois group theory first. So does that make it less fundamental? I would say not...

  • Finally, consider the prime number theorem. If anything was ever a fundamental theorem, this one is. But it is hard to title it as a "fundamental theorem of whatever because what would be a good name for the whatever? If you say "the fundamental theorem of prime numbers," well, that equally describes the fundamental theorem of arithmetic. If you say "the fundamental theorem of prime number distribution" then that sounds too narrow. So this theorem, just as important as many of the others, is never called by the name fundamental.

The one rule I think I could count on is that you should never have more than one fundamental theorems of some specific field.


Added later:

The Fundamental Theorem of Symmetric Polynomials says that any symmetric polynomial in $n$ variables can be expressed as a polynomial in the elementary symmetric polynomials in those $n$ variables. [An elementary symmetric polynomial is a symmetric polynomial of degree $k$ containing no terms of degree more than one in any single variable, e.g. $(x_1x_2 + x_1x_3 + x_2 x_3)$ but not $(x_1^2 + x_2^2 + x_3^2)].$

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  • $\begingroup$ Oops: there are in fact two "fundamental theorems of economics". But they are related being converses of one another. Also, I am embarrassed not to have included the fundamental theorem of beam loading, because I am supposed to have been an expert in particle accelerator dynamics. $\endgroup$ – Mark Fischler Feb 16 '15 at 23:37
  • $\begingroup$ The name "Galois group theory" is not used by anyone. It's called Galois theory. $\endgroup$ – KCd Feb 16 '15 at 23:53
  • $\begingroup$ Thanks for the comprehensive answer @MarkFischler. I learned a lot. $\endgroup$ – Jr Antalan Feb 16 '15 at 23:54
  • $\begingroup$ KCd is of course right about Galois theory. I used the wording I did because the reference I saw to that theorem used the word "group" in the title of the fundamental theorem. $\endgroup$ – Mark Fischler Feb 17 '15 at 0:03

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