Great Theorems of Mathematics

[arthwollipot]

Well, I can hardly be called a mathemetician. The most I know about higher maths (as opposed to just arithmetic) is what I've read in Gödel, Escher, Bach. But once I'd finally managed to absorb what was in it, the Incompleteness Theorem really made me wonder.

I'm also a fan of what I know about Euclid - specifically his geometry, and I thought that his proof of the infinitude of the primes was very elegant. I've brought that out as a party trick on a couple of occasions. It didn't go down very well.

 

[PixyMisa]

That's because you're going to the wrong kind of party.

 

[arthwollipot]

I've always suspected something like that...

 

[ddt]

Both the proof that sqrt(2) is irrational, and the proof that there are infinitely many primes are fine, and early, examples of proof by contradiction. As a simple yet very useful theorem I further offer the pigeonhole principle.

 

[Almo]

e^i(pi) + 1 = 0

Five of the most important numbers in mathematics in one simple equation.

Explanation:
http://www.math.toronto.edu/mathnet/questionCorner/epii.html

 

[pnerd]

e^i(pi) + 1 = 0

.
Is that a theorem or an identity?

 

[AvalonXQ]

.
Is that a theorem or an identity?

From the link above, it seems to be a theorem -- meaning that based on the definition of e^n for real n, it can be demonstrated that e^(i*n) for real n holds to the equation e^(a*i) = i*sin(a) + cos(a), for which e^(i*pi) is a case that simplifies nicely to i*sin(pi) + cos(pi) = i*0 + -1 = -1.

 

[ctamblyn]

The Fundamental Theorem of Calculus. Beautiful and practical.

 

[jadey]

Don't mean to whine, but it would be helpful if links were included with the theorems. I realize that I can google them all, but the posters might tend to provide more succinct links.

 

[Doctor Evil]

Gauss's Theorema Egregium deserves a mention. It connects the integral over the Guassian curvature of a closed surface to thenumber of wholes in said surface. (This is the lay person's way of describing it.)

 

[ddt]

Don't mean to whine, but it would be helpful if links were included with the theorems. I realize that I can google them all, but the posters might tend to provide more succinct links.

Pigeonhole principle. When you have m boxes, and n > m objects, at least one box contains 2 objects. Generalized statement: if you have m boxes and n > k*m objects, at least one box contains k+1 objects.

Another nice theorem: Brouwer's fixed point theorem. Simple form: if you have a function f : [0,1] -> [0,1], i.e., f maps numbers between 0 and 1 (both inclusive) to numbers between 0 and 1, there is at least one x such that f(x) = x. Generalized, it applies to circles, spheres and any compact set.

 

[Philosaur]

The four-color theorem. I like it because its proof-by-computer (Appel and Haken 1977) raised an important question about the nature of mathematical proof.

The question was: is a proof which is practically (though not in principle) unverifiable a proof at all? That is, if we have to take the computer's word on it--so to speak--is it really a proof?

I also like the ham sandwich theorem and the hairy ball theorem, mostly because of the names.

Oh yeah, I recognize the obvious bias toward geometry in my list--I tend to think visually and spatially.

 

[Doctor Evil]

Gauss's Theorema Egregium deserves a mention. It connects the integral over the Guassian curvature of a closed surface to the number of wholes holes in said surface. (This is the lay person's way of describing it.)

Doh.

 

[boooeee]

Fermat's Little Theorem

If a is coprime to a prime number p, then a^(p-1) - 1 is divisible by p. Unlike Fermat's other theorem, its various proofs are layman-accessible. The Bracelet Proof is especially neat.

It's also an early example of key group theory concepts.

 

[dfranke]

For those that most shook up the field, I would nominate, in this order:

1. Lobachevsky and Bolyai discovering non-Euclidean geometry, and particularly their demonstration that Euclid's 5th postulate can be neither proved nor disproved as a theorem derived from the first four.
2. Russell's demonstration of the inconsistency of naive set theory (Russell's paradox).
3. Gödel's second incompleteness theorem.

 

[Roboramma]

I'm also a fan of what I know about Euclid - specifically his geometry, and I thought that his proof of the infinitude of the primes was very elegant. I've brought that out as a party trick on a couple of occasions. It didn't go down very well.

I remember bringing up the same proof to my dad one time. I thought it was cool and started talking to him about it. I think it was a part of a discussion and I was making about point about how you can know something about a whole category of things without having access to every particular example of those things.

After I finished explaining the proof he said something like, "Well, that's pretty cool Rob, but until we find that next bigger prime number, how can we actually know that there is one?"

I couldn't believe he said that! But I calmly started over again, and this time he listened.

 

[PixyMisa]

Pigeonhole principle. When you have m boxes, and n > m objects, at least one box contains 2 objects. Generalized statement: if you have m boxes and n > k*m objects, at least one box contains k+1 objects.

Yep. My experience with boxes an objects bears this out.

Another nice theorem: Brouwer's fixed point theorem. Simple form: if you have a function f : [0,1] -> [0,1], i.e., f maps numbers between 0 and 1 (both inclusive) to numbers between 0 and 1, there is at least one x such that f(x) = x. Generalized, it applies to circles, spheres and any compact set.

Showing that I haven't completely lost it, I said to myself, that's only true for continuous functions. Followed the link to Mathworld, and yep, the formal definition specifies a continuous function. It's an interesting one. Thanks!

 

[PixyMisa]

For those that most shook up the field, I would nominate, in this order:

Lobachevsky and Bolyai discovering non-Euclidean geometry, and particularly their demonstration that Euclid's 5th postulate can be neither proved nor disproved as a theorem derived from the first four.

Yes, an excellent one. Two.

Russell's demonstration of the inconsistency of naive set theory (Russell's paradox).

The set of sets that are not members of themselves.

Gödel's second incompleteness theorem.

Ah, yes. I was going to ask why the second, because I couldn't remember the difference, but yes.

For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Upsets our Seekers After Truth no end, that does.

A fine selection!

 

[ddt]

Showing that I haven't completely lost it, I said to myself, that's only true for continuous functions. Followed the link to Mathworld, and yep, the formal definition specifies a continuous function.

Ahem, yes, of course.

It's an interesting one. Thanks!

You're welcome!

Another nice theorem is the "Marriage Theorem". Suppose you have a set of girls and a set of boys, and every girl likes a number of boys. Now, if for every finite set of girls, the set of boys one of them likes is as least as big (e.g., for every 2 girls, there are at least 2 boys that one of them likes; for every 3 girls, there are at least 3 boys that one of them likes, etc.) - then you can arrange marriages so that every girl has a boy to her liking.

Neat, uh? It also holds when you have infinitely many girls and boys.

 

[ddt]

After I finished explaining the proof he said something like, "Well, that's pretty cool Rob, but until we find that next bigger prime number, how can we actually know that there is one?"

Your father is an intuitionist? Cause, you know, proofs by contradiction are EVIL!