Life In 19x19

GoRo wrote:

jts wrote:If you do not assume that Fermat's theorem is true, then our proof is false by non sequitur.

Fermat's theorem *is* true.

... an enthymeme ... false by petitio principii ...

I see. I thought we were talking about mathematics, not rhetoric.
In rhetoric you may be right, in mathematics you are not.

Cheers,
Rainer

Have you ever done a math proof before? I'm not trying to be mean, I'm just not sure what level of specificity to go into.

"Enthymeme" is just a formal name for a situation where you do not explicitly assert "X", but your listeners will know that you are tacitly asserting X. That's true whether the speaker is a mathematician, an orator, or a troll on the internet.

jts wrote:Have you ever done a math proof before? I'm not trying to be mean, I'm just not sure what level of specificity to go into.

I admit I did not do the proof of Fermat. But 24 years before that exciting
proof I made my maths diploma in Berlin, Germany.
So I did not only learn what a proof is, but had to provide proofs alot.

If you have a certain set of axioms and deduction rules, then
any regular deduction sequence leads to valid propositions (theorems).
Moreover, any regular deduction sequence starting with and using
axioms and valid theorems, leads to valid theorems. Any such sequence
of regular deductions is called a proof.

I showed you such a sequence of regular deductions where one of the
theorems used was Fermat.
In short and without the assumption/contradiction technique:
Fermat ==> never a^3+b^3=c^3 ==> never a^3+a^3=c^3
==> never 2*a^3 = c^3 ==> never 2 = c^3/a^3 ==> never 2^(1/3) = c/a.

Please feel free to add the "for all natural numbers ..." bla-bla.
This is the heart of the proof, and it remains a proof, even if you
believe otherwise.

(Sorry for my English, as I am not a native speaker of English.)

Cheers,
Rainer

Seems like a lot of miscommunication happening? As far as I understand it, Wiles' proof requires that 2^1/n is irrational, therefore using it in a proof that 2^1/n is irrational is circular.

HermanHiddema wrote:Seems like a lot of miscommunication happening? As far as I understand it, Wiles' proof requires that 2^1/n is irrational, therefore using it in a proof that 2^1/n is irrational is circular.

You are right. It seemed to me as if circular was meant in the sense of "false".
So it's a "circulus", but not a "circulus vitiosus". OK?

Thanks for intervening.

Cheers,
Rainer

I like this "proof" that 1=2, because it's such an elegant demonstration of how you can get absurd answers if you apply mathematical operations without thinking.

x^2-x^2 = x^2-x^2
x(x-x) = (x+x)(x-x) by common factor on the LHS, and difference of squares on the RHS
x = x + x cancel common factor
1 = 2 cancel common factor


On similar lines, 16/64 = 1/4 because you can cancel a "6" from numerator and denominator! Completely stupid reasoning, but the answer is correct.

The Gauss–Bonnet theorem.

Among my all time favorite theorems in Mathematics/Physics is the `Noether's theorem'.

http://en.wikipedia.org/wiki/Noether's_theorem


In crude language it says- corresponding to every continuous symmetry of a system there is a conserved quantity (eg: Energy conservation in a system is a consequence of time translational symmetry). I still remember how intrigued I was when I first heard about this during my undergrad.

Apart from the mathematical and physical profoundness of the theorem itself, Emmy Noether's life and the prejudices she had to overcome at that time also make a very inspiring story.

Dusk Eagle wrote:Mine would either have to be Cantor's diagonalization proof for uncountable sets, or Gödel's first incompleteness theorem.

Seconded!

I deem the distinction between countable and uncountable infinity as high point on the history of humankind's intelligence. And Turing's and Gödel's theorems on undecidibility and incompleteness (which can be viewed from the point of view of uncountable sets) are so beautiful that I feel I will cry right now.

My favorite theorem in set theory is probably the reflection theorem. It says that for every "hierarchy" of sets and every property that's true about the whole hierarchy, there's a point in the hierarchy such that that property is true at that point.

My favorite theorem in model theory is the lowenheim skolem theorem. It says that for every structure and every subset of the structure, there's a substructure containing that subset that is relatively small compared to the subset, and this substructure and the structure itself agree on statements about the substructure.

After some thought, I like that you can well-order the reals (of course, a consequence of the fact that any set can be well ordered in ZFC), but Cantor's diagonalization argument is probably the first thing that struck me.

From set theory, my favourite result is probably the Banach-Tarski paradox.

Why?

It gives rise to this joke:
q: What's an anagram on the the Banach-Tarski paradox?
a: Click to show

Did someone watch University Challenge last night?

They asked about the Banach Tarski paradox on the show, or just told the joke? (I didn't watch it.)

Banach Tarski was the answer to a question about doubling the sphere, with the clue that it was named after two people; the team of classicists guessed "Smith Wilson".

My favorite theorem ... math, though nestled wholly within physics ... is Noether's theorem. Informally stated, it's that for any symmetry of a physical system, there is a corresponding conservation law. Translational invariance means conservation of momentum, rotational means conservation of angular momentum, invariance in translation through time means conservation of energy, which is a bit mind-blowing. It can be extended to QFT in some way my lack of understanding of which is even beyond me, but which gives a series of "guage symmetries" implying conservation of electric charge and other such quantities.

When I properly read about it, I became very angry at my old college tutors for not doing mechanics with generalized coordinates and the principle of least action in year 1 and QFT in year 3. It's like they told me I'd learned mountain climbing, then drove me up Pike's Peak, without suggesting I give Everest a try. I mean, I could have pursued it all myself, but I was a callow and foolish young man then.

I also have a little affection for the ABC conjecture, but only because Shinichi Mochizuki has apparently proved it by the method of reductio ad nauseam.