Great Theorems of Mathematics

[ctamblyn]

Most of the "great theorems" don't have much of an application outside abstract math that doesn't connect with any useful application. Interesting is that at the time of the conception, Pythagorians guarded the idea as a secret, and I believe that they killed someone who was talking too much.

They sound like an irrational bunch.

 

[Cuddles]

Oh come on. A² + B² = C² is about the dullest of the things Pythagoras achieved.

Especially since he probably didn't achieve it at all. It was never attributed to him until several centuries after he died, and there's evidence that at least the Egyptians and Babylonians, and possibly others, knew it well before Pythagoras was born. Of course, even if the theorem could be genuinely attributed to the Pythagorean school, there is a lot of question over how much was actually done by Pythagoras himself and how much by his followers.

 

[Perpetual Student]

Especially since he probably didn't achieve it at all. It was never attributed to him until several centuries after he died, and there's evidence that at least the Egyptians and Babylonians, and possibly others, knew it well before Pythagoras was born. Of course, even if the theorem could be genuinely attributed to the Pythagorean school, there is a lot of question over how much was actually done by Pythagoras himself and how much by his followers.

I believe the relationship was known for many centuries by other cultures before Pythagoras, but the first proof (that we can confirm) was discovered by the Pythagoreans.

 

[Cuddles]

I believe the relationship was known for many centuries by other cultures before Pythagoras, but the first proof (that we can confirm) was discovered by the Pythagoreans.

But that's the point - there's no evidence that that was actually the case. No-one attributed the proof to them until centuries later, but there are at least a few proofs known to exist in other regions dating from the same time or earlier, as well as evidence that proofs could well have been known a lot earlier again.

According to Wiki, the earliest known proof is actually an Indian one from around 600BC. A Chinese proof may also pre-date Pythagoras, although the dating seems to be rather more uncertain for that one. And the earliest European proof is actually from Euclid a couple of centuries after Pythagoras' death. The only evidence we actually have that Pythagoras had a proof at all is later historians claiming that he did. There doesn't appear to be any direct evidence, or even any contemporary accounts, to support those claims.

 

[epix]

Especially since he probably didn't achieve it at all.

Especially? So the theorem has been deemed "dull" by a poster, but since there is an uncertainty regarding Pythagoras' authorship, the theorem is farther up on the dull scale. Well, different millennia; different reasoning . . .

 

[arthwollipot]

Especially? So the theorem has been deemed "dull" by a poster, but since there is an uncertainty regarding Pythagoras' authorship, the theorem is farther up on the dull scale. Well, different millennia; different reasoning . . .

An admittedly uninformed poster... who is finding the discussion fascinating.

Has anyone read The Crest of the Peacock: Non-European Roots of Mathematics? Fascinating book, but I can't follow a lot of it.

 

[CriticalThanking]

When I saw all the brackets above, I thought we were going all Doronshadmi for a minute.

I have a thing for (nearly) useless proofs. Fermat's Last Theorem was my favorite when I was young. Then I stumbled onto the one that that proves there does exist a function that generates only primes. We may never know what it is, but it must exist. I will try to dig that up.

Godel's incompleteness has spawned more useless philosophical musings in my experience, often leading to "this proves god".

CT

 

[Soapy Sam]

I'm fairly sure I once read a theorem proving that any person who understands mathematics has either a very small penis, or none at all.
It would be boastful to remark that I have Pi tattoed upon my manly member to 1300 decimal places in 12 point Times New Roman. Of course I have to be excited to read beyond the first couple of hundred....

Not that I'm bitter or anything.

Somewhat more seriously- Which of you smart Alecs is familiar with George Spencer Brown's "Laws of Form" and what do you make of it?

 

[epix]

I'm fairly sure I once read a theorem proving that any person who understands mathematics has either a very small penis, or none at all.
It would be boastful to remark that I have Pi tattoed upon my manly member to 1300 decimal places in 12 point Times New Roman. Of course I have to be excited to read beyond the first couple of hundred....

That makes it real handy to compute the girth.

Now I see why mathematicians decided to use Greek letter Pi instead of Omega: "Honey, what does that Om mean?"

 

[!Kaggen]

The Fundamental Theorem of Projective Geometry

http://www.youtube.com/watch?v=3BRtaHlUItQ

 

[Perpetual Student]

Somewhat more seriously- Which of you smart Alecs is familiar with George Spencer Brown's "Laws of Form" and what do you make of it?

I read it almost 40 years ago so my recollections are a bit fuzzy. I recall being quite fascinated by the fact that one could construct the laws of logic using little more than the concept of marking a "difference." I should take a look at it again sometime.

 

[readme.txt]

e^i(pi) + 1 = 0

e^(i*theta) = cos(theta) + i*sin(theta)

Ask Euler and De Moivre

Another theorem I like is La Grange's on group theory :

Let G a finite group and H a subgroup of G, then |H| divides |G|.

 

[Floyt]

An admittedly uninformed poster... who is finding the discussion fascinating.

Has anyone read The Crest of the Peacock: Non-European Roots of Mathematics? Fascinating book, but I can't follow a lot of it.

Read that a couple years ago, and that's exactly how it went for me Highly fascinating, and left a crumble trail of beautiful trivia in my mind, but I did conk out on many of the more involved ideas. Probably did me good to realize that a Mandarin priest with time on his hands and lots of dedication can leave my not-particularly-mathematics-focussed science education in the dust...

 

[Carefulplease]

When I attended my first undergraduate course in analysis we proved the theorem that 2 + 2 = 4 from the Peano axioms. I thought it was neat because prior to attending to the proof I hadn't realized that such an obvious proposition could admit of a formal proof at all.

 

[Carefulplease]

Pixy already named my two favorites in the OP.

Third place is the already-mentioned Euclid proof of infinite prime numbers.

The Goldbach Conjecture will always have a special place for me as well, because as a youth I spent a good deal of time searching for a proof I even came up with a reversed form of mathematical induction to try to cope with it.

I also tried once; tough nut to crack. I thought I had maybe one chance in a billion to succeed in proving the conjecture. After reading a bit about contemporary attempts and the amount of mathematical knowledge that's required to properly tackle the problem, I realized I was optimistic.

 

[readme.txt]

The mathematical proof that uses Graham's number might be a good one.

http://en.wikipedia.org/wiki/Graham's_number

 

[Nonpareil]

To the tune of "Pop Goes the Weasel"

x is equal to negative b
plus or minus the square root
of b squared minus four a c
all over two a

 

[I Ratant]

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.

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Nope!
I don't always put my stuff away!
.

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.

.
Man, I'd hate to see the complex form of that one!

 

[AvalonXQ]

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.

How does it apply to a circle?
f(x) = x + pi [rotate the whole circle 180 degrees]
... seems to map one circle to the other with all x != f(x). What am I missing?

 

[ben m]

How does it apply to a circle?
f(x) = x + pi [rotate the whole circle 180 degrees]
... seems to map one circle to the other with all x != f(x). What am I missing?

My guess: when you think of the circle's perimeter as a line from [0-1], that rotation is not a continuous function. The point at 0.499... maps to 0.999... but the point at 0.50000 maps to 0.0 all the way on the other side of the interval.