which is your favourite mathematical prof/teorem?

[perceval]

I always liked the integral of a gaussian over the reals.

Integral(e^(-x^2)) from -Inf to Inf

1) Square the integral,
2) Then change from cartesian to polar,
3) Solve integral
4) Take the square root
5) Square again
6) Multiply by 2
7) Have dessert

love that one too

[GoRo]

1. Assume natural numbers p and q such that (p/q)^3 = 2.
2. Then p^3 / q^3 = 2,
3. i.e. p^3 = 2 * q^3 = q^3 + q^3.
4. Let a = q, b = q, c = p.
5. Then a^3 + b^3 = c^3 for some natural numbers a, b and c.
6. That contradicts Fermat.
7. Thus the assumption led to a contradiction.
8. q.e.d.

The first error is in line ....?

6. (Or are you being pedantic about the fact that no step in the proof assumes Fermat, so 6 -> 7 is non sequitur?)

Being pedantic is not so bad in mathematical context, no?
"Fermat" is short for the (meanwhile proven) fact that there
are no natural numbers a, b, c such that a^n + b^n = c^n for
natural numbers n > 2.
Thus step 5. proposes something which is impossible.
That is, what my lines #6 and #7 are telling.

I really cannot detect a circular reasoning. That would be the case
if I wrote propositions p_1, p_2, p_3 etc. such that not all p_i
could be proven from the p_j with j < i, but at least one of the
p_i needed some p_j to be true, where j > i.

I completely agree that the unsolvability of (p/q)^3 = 2 is such
an easy thing to prove that it is part of "Fermat". But I doubt
you can call that a circulus vitiosus. At least I see your
point: you are trying to see "Fermat" not as a monolithic "fact",
but you glance into the how-and-why of this proven truth.

Greetings,
Rainer

[Dusk Eagle]

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

[jts]

But I doubt you can call that a circulus vitiosus.

If you do not assume that Fermat's theorem is true, then our proof is false by non sequitur. If, on the other hand, you treat step 6 as an enthymeme which implies "... and we have a proof that Fermat's theorem is true," then our proof is false by petitio principii, since the proof of the theorem itself assumes our demonstrandum.

[GoRo]

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

[jts]

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.

[GoRo]

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

[HermanHiddema]

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.

[GoRo]

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

[TheBigH]

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.

[cyndane]

The Gauss–Bonnet theorem.

[Discipline]

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.

[Samura]

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.

[Redundant]

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.

[hyperpape]

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.