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        general >> truth >> Math: road to perfection??
        
(Message started by: LeoYard on Jun 10th, 2008, 1:04pm)
Title: Math: road to perfection??
Post by LeoYard on Jun 10th, 2008, 1:04pm
I thought of a catchy phrase today and would like to hear views on it.

"There are many roads to Rome but there is only one road to perfection - mathematics. "

Maybe just "There is only one road to perfection - mathematics. " would suffice.

If this is true, then why do we not call 1 a prime number?

Also, Wiles' proof of Fermat's Last Theorem completely makes me an unhappy camper.

How can we say this is perfect when maybe 100 people in the world understand it?

I wished Fermat had said there are no integers a,b,c,n>2 such that a^n+b^n=c^n can be proved to be true or false for all n using elementary methods.

I would have thought that Physics was the road to perfection - and the scenery along the way.

So many of the "weird" mathematical theorms and derivations seem to be developed to cover up shortcomings in the mathematical ideas we have defined to model the physical world - then the wild imaginings that have followed from those qualifications/ cover-ups.

So it seems to me.

Another thing that cracks me up is when mathematicians say "It is so simple and elegant that it has to be true!" or"The demonstration or proof is so elegant that it cannot be untrue."

What do you think? Is mathematics really the road to perfection?

Surely, more ideas are out there.
Title: Re: Math: road to perfection??
Post by towr on Jun 10th, 2008, 1:32pm

on 06/10/08 at 13:04:25, LeoYard wrote:
If this is true, then why do we not call 1 a prime number?
Because number theory is prettier when it isn't. For example, now every number has a unique prime number factorization, which wouldn't be the case if 1 were a prime number. And that'd have ugly consequences for numerous other theorems and results.


Quote:
I would have thought that Physics was the road to perfection - and the scenery along the way.
Without mathematics physics wouldn't really get anywhere.


Quote:
Another thing that cracks me up is when mathematicians say "It is so simple and elegant that it has to be true!" or "The demonstration or proof is so elegant that it cannot be untrue."
It's wishful thinking, but it pans out often enough; and it's as much the case in mathematics as in physics or biology.
Title: Re: Math: road to perfection??
Post by LeoYard on Jun 10th, 2008, 2:54pm

on 06/10/08 at 13:32:23, towr wrote:
Because number theory is prettier when it isn't. For example, now every number has a unique prime number factorization, which wouldn't be the case if 1 were a prime number. And that'd have ugly consequences for numerous other theorems and results.


Yes, Your argument is the classical one. I just think we should be able define 1 in contextof the problem being worked on. Sometimes it could be prime other times composite and othertimes what it has been decreed to today. There are only a handful theorems 1 prime violates.

A prime number is a number that is divisible by itself and one only. This is true for 1. To take 1 out of the set of primes, we have to stipulate number > 1. So we have to say 1or "one" every time we express the definition of a prime number. Just think of it. The oddity of the first prime number being even would be relaxed. I guess we could take 2 out of the set of primes for certain theorems out there.

It is a moot point but it still ruffles my feathers.
Title: Re: Math: road to perfection??
Post by towr on Jun 10th, 2008, 3:10pm

on 06/10/08 at 14:54:30, LeoYard wrote:
Yes, Your argument is the classical one. I just think we should be able define 1 in context of the problem being worked on.
But isn't that a bit confusing? Admittedly it's sometimes unavoidable to have some confusion, but I really see no benefit in changing what 'prime' means from one occasion to the next.


Quote:
Sometimes it could be prime other times composite
Composite? Which "at least two distinct prime factors" would it have?


Quote:
A prime number is a number that is divisible by itself and one only.
Or, "a prime is a number that is divisible by exactly two positive integers". Which excludes one and includes all primes perfectly.
Title: Re: Math: road to perfection??
Post by JiNbOtAk on Jun 10th, 2008, 6:26pm

on 06/10/08 at 13:32:23, towr wrote:
Without mathematics physics wouldn't really get anywhere.


What's the point of mathematics without physics ?
Title: Re: Math: road to perfection??
Post by towr on Jun 11th, 2008, 12:22am

on 06/10/08 at 18:26:18, JiNbOtAk wrote:
What's the point of mathematics without physics ?
Economics, biology, encryption, puzzles/recreation, etc.
There's a lot more mathematics you can do without physics than vice versa. The only physics you could do without mathematics is taking measurements and making qualitative statements. And even in the latter case you risk treading on the area of topology.
Title: Re: Math: road to perfection??
Post by towr on Jun 11th, 2008, 4:33am
Sometimes I wonder how he manages to always be so topical.

http://imgs.xkcd.com/comics/purity.png (http://xkcd.com/435/)
mouseover text: [hide]On the other hand, physicists like to say physics is to math as sex is to masturbation.[/hide]
Title: Re: Math: road to perfection??
Post by mikedagr8 on Jun 11th, 2008, 4:40am
Hahaha. SO true. Especially the physics on top.

Nice find towr, and good comment.
Title: Re: Math: road to perfection??
Post by Jigsaw on Jun 11th, 2008, 5:35am

on 06/11/08 at 04:33:41, towr wrote:
Sometimes I wonder how he manages to always be so topical.
. . . [hide]On the other hand, physicists like to say physics is to math as sex is to masturbation.[/hide]
Haha. Good comment.  ;D
Title: Re: Math: road to perfection??
Post by towr on Jun 11th, 2008, 5:46am

on 06/11/08 at 05:35:51, Jigsaw wrote:
Haha. Good comment.  ;D
Just to make sure there's no confusion, that's the original text you get when you move your mouse over the image at xkcd.

You can do sociology without worrying much about psychology; you can do psychology without worrying much about biology; you can do biology without worrying much about chemistry; and you can do chemistry without worrying much about physics.
Suffice it to say, trying to explain someone's dreams in terms of elementary particles is a hopelessly futile undertaking. I'm not quite sure what is meant by "purity" in science; it sounds like a tainted concept to me.
Title: Re: Math: road to perfection??
Post by LeoYard on Jun 11th, 2008, 1:06pm
Thank you very much for the feedback. I'm enjoying this a lot.

What's the point of physics without mathematics?

Can we really do physics without maths?
Title: Re: Math: road to perfection??
Post by LeoYard on Jun 12th, 2008, 12:54pm

on 06/11/08 at 13:06:07, LeoYard wrote:
Thank you very much for the feedback. I'm enjoying this a lot.

What's the point of physics without mathematics?

Can we really do physics without maths?



I see that no one cares to continue. Did I ask a dumb question?
Title: Re: Math: road to perfection??
Post by ThudanBlunder on Jun 12th, 2008, 1:19pm

on 06/11/08 at 13:06:07, LeoYard wrote:
What's the point of physics without mathematics?

Can we really do physics without maths?



on 06/12/08 at 12:54:20, LeoYard wrote:
Did I ask a dumb question?

And which question is the other one.  :D
Title: Re: Math: road to perfection??
Post by LeoYard on Jun 12th, 2008, 1:34pm

on 06/12/08 at 13:19:00, ThudanBlunder wrote:
And which question is the other one?   :D


Grammatical error on my part. There are 2 questions, so i should have written

Did i ask dumb questions? or were the last 2 questions dumb?
Title: Re: Math: road to perfection??
Post by ThudanBlunder on Jun 12th, 2008, 1:59pm

on 06/12/08 at 13:34:41, LeoYard wrote:
Did i ask dumb questions? or were the last 2 questions dumb?

Dumb? No, but towr had already opined that...


on 06/11/08 at 00:22:47, towr wrote:
The only physics you could do without mathematics is taking measurements and making qualitative statements.

....so maybe he just lost interest.
Title: Re: Math: road to perfection??
Post by towr on Jun 12th, 2008, 3:11pm
Well, I wouldn't really know what more to say on the subject. So I thought I'd let someone have a go first.
Of course, in measuring you also use numbers, typically. So you are already nominally using math again. And even qualitative statements in physics have a good chance of falling under geometry or topology.
Title: Re: Math: road to perfection??
Post by LeoYard on Jun 12th, 2008, 3:29pm
Ok. There are lots of debates between experimental physicists, theoretical physicists and mathematicians going on regarding the importance of mathematics. Experimental physicists see math just as a tool to use. They  take great care to not get too hung up in the whole "proof" thing. Math is a tool. They view it that way.

I would like to discuss the unreasonable? effectiveness of mathematics in describing the universe, the reality around us.

I need to take a pause and think about what I would like to post here.

Thank you.
Title: Re: Math: road to perfection??
Post by towr on Jun 13th, 2008, 12:21am
Anything that's governed by law-like rules can be (unreasonably) effectively described by maths, I think.

And is math a tool? Sure; and so is physics. Or any science. You can like them well enough in their own right, but also for what they can do for you. Ultimately science is about modeling the world around us so that we can better predict and control it.
Title: Re: Math: road to perfection??
Post by denis on Jun 13th, 2008, 4:52pm
http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_whathappened;action=display;num=1170530331;start=40#40
Title: Re: Math: road to perfection??
Post by BenVitale on Jul 24th, 2008, 11:18am
Many philosophers, such as Plato, Descartes, Leibniz have attached importance to logical deduction as the path to certain knowledge. Russell hoped modern logic would make it possible.

Has anyone read Principia mathematica?

Why would any attempt to reduce mathematics to logic fail?
Title: Re: Math: road to perfection??
Post by BenVitale on Jul 24th, 2008, 11:54am
This is the document I'm looking at

http://plato.stanford.edu/entries/principia-mathematica/

But I still don't feel satisfied
Title: Re: Math: road to perfection??
Post by Grimbal on Jul 25th, 2008, 9:51am

on 06/10/08 at 13:04:25, LeoYard wrote:
Also, Wiles' proof of Fermat's Last Theorem completely makes me an unhappy camper.

How can we say this is perfect when maybe 100 people in the world understand it?


Mathematics might be perfect, our understanding of it is not.

No, in fact, I don't even think Mathematics is prefect.  Goedel proved it to be incomplete.*  To me mathematics is like a scaffolding that will help our limited brain to explore a bit further what is true.  The further we go, the more complicated the scaffolding.  Mathematics will never describe all the subtlety of truth.

[edit]In fact, it depends what you call mathematics, whether it goes beyond what you can prove formally.[/edit]
Title: Re: Math: road to perfection??
Post by BenVitale on Jul 25th, 2008, 10:38am

Quote:
Why would any attempt to reduce mathematics to logic fail?


Well, at least Russell's attempt failed.

Whether or not mathematics can be reduced to logic is an open question. So it is an undecidable question.
Title: Re: Math: road to perfection??
Post by rmsgrey on Jul 26th, 2008, 4:24am

on 07/25/08 at 10:38:18, BenVitale wrote:
Whether or not mathematics can be reduced to logic is an open question. So it is an undecidable question.

Not all undecided questions are undecidable.

FLT was undecided for centuries, but, as Wiles' proof shows, it was decidable all along.


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