Let's discuss Mathematics

[ParadigmShifter]

I guess, I submitted to peer review and you found a flaw that's all

I won't bother posting my proof of the Riemann Conjecture then

 

[Mise]

Have we found the difference between mathematicians and physicists?

Physicists take 5 minutes to get the right answer, Mathematicians take 20 minutes to get the wrong answer?

 

[dutchfire]

Wait, you can also express it as 7+sum_n=1->5 (12n-n^2)(6-n)/216

 

[ParadigmShifter]

Hey I'm out of practice it's 15 years since I finished my degree

 

[Atticus]

Have we found the difference between mathematicians and physicists?

That's the topic of numerous jokes told in maths and physics departments all around the world. Typical for these jokes is that only mathematicians and physicists find them amusing (and usually not even them, and that's what makes them funny, little like dirty limericks).

I'll reproduce one of them which I heard when I was a student:

Engineer's, physicist's and mathematician's houses caught fire in the middle of the night and all three woke up when the alarm went off. The engineer took the hose and extinguished the fire generously causing bad water damages to his apartment. The physicists in his apartment calculated the exact amount of water needed for the job, and extuinguished the fire. The mathematician woke up, said "solution exists for the problem", and went back to sleep.

EDIT:

I guess, I submitted to peer review and you found a flaw that's all

Yes! Why do have to be so negative Dutchfire? Why do you seek flaws in other people's proofs and not what's good in them?

EDIT2: Just in case somebody missed it, with no smileys and so on, the EDIT#1 was a joke...

 

[Ammar]

Hey I'm out of practice it's 15 years since I finished my degree

What sort of degree do you have by the way?

 

[ParadigmShifter]

Maths, first year pure/applied/stats, 2nd year pure/stats, 3rd pure only.

 

[Gooblah]

We just finished DeMoivre's Theorem in school a few weeks ago. The idea that e ^ (ipi) + 1 = 0 is pretty sweet. However, a proof was not included, sadly.

 

[ParadigmShifter]

Well e^it is

cos t + i sin t

(by the Taylor expansion of e^x, and by looking at the expansions for sin, cos)

so with t = pi we have

e^i*pi = cos pi + i sin pi
= -1 + 0

and the result follows.

 

[sanabas]

Engineer's, physicist's and mathematician's houses caught fire in the middle of the night and all three woke up when the alarm went off. The engineer took the hose and extinguished the fire generously causing bad water damages to his apartment. The physicists in his apartment calculated the exact amount of water needed for the job, and extuinguished the fire. The mathematician woke up, said "solution exists for the problem", and went back to sleep.

I sat down across the street to watch it burn, what does that make me?

 

[ParadigmShifter]

A reporter?

 

[Truronian]

A chemist?

 

[dutchfire]

A chemist?

That would be if you set it on fire yourself.

 

[sanabas]

Hmm. I think I'll decide I woke up, said 'solution doesn't exist for the problem' and so sat down across the street. Which puts me safely in the mathematician class.

 

[ParadigmShifter]

The Banach-Tarski paradox was mentioned in the quiz thread. Is it evidence against the axiom of choice or do the 'impossible' fractal dissections (i.e. can't be done with real matter) make it meaningless for real-world discussion?

 

[dutchfire]

By work of Kurt Gödel and Paul Cohen, the axiom of choice is logically independent of the other axioms of Zermelo–Fraenkel set theory (ZF). This means that neither it nor its negation can be proven to be true in ZF, if ZF is consistent. Consequently, if ZF is consistent, then ZFC is consistent and ZF¬C is also consistent. So the decision whether or not it is appropriate to make use of the axiom of choice in a proof cannot be made by appeal to other axioms of set theory. The decision must be made on other grounds.

One argument given in favor of using the axiom of choice is that it is convenient to use it: using it cannot hurt (cannot result in contradiction) and makes it possible to prove some propositions that otherwise could not be proved. Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact. Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.

Well, I don't really know enough about this to really argue either way, but it seems to me that using it can't really hurt in most cases.

 

[Atticus]

Some do consider it as a counter-evidence. Most of the people don't really care about it, usually they say it's impossible in real word to cut things so finely, and the parts are nonmeasurable anyways. Latter explanation isn't very good, since it doesn't really explain the paradox.

As far as I know, BT-paradox is the best intuitive argument against the axiom of choice, which on the other hand is pretty much universally accepted. Much of the best known and most powerful theorems rely on it, Hahn-Banach and Radon-Nikodym to name few (they might be provable otherwise, but not as easily), so life without it would be nasty, brutish and short.

Often people use it even without noticing it. Imagine how hard it woud be to teach evertybody new maths without axiom of choice.

 

[LulThyme]

A related well-known joke:
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

 

[LulThyme]

Often people use it even without noticing it. Imagine how hard it woud be to teach evertybody new maths without axiom of choice.

You don't need the axiom of choice that often at all in "basic" mathematics. For example, you don't need it when "choosing" from a finite set.

People in non-math majors probably never see a result that depends on the axiom of choice and math majors probably might only see a few in say 3rd or 4th year of undergrad (usually related to Zorn's Lemma)

 

[ParadigmShifter]

Every vector space has a basis (specifically, infinite-dimensional ones) is probably the most important result. I think we assumed fi-di when proving it in 1st year undergrad algebra however.