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#1 2008-08-18 07:20:36

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

The Book

Paul Erdõs liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics.  Erdõs also said that you need not believe in God but, as a mathematician, you should believe in The Book.

So begins the book titled, "Proofs from THE BOOK" by Martin Aigener and Gunter M. Ziegler.  It was written for Erdõs and the proofs in it are largely influenced by him.  One of my favorites is a topological proof of the infinity of primes.

But as I thought about that opening, something seemed off.  As mathematicians, we don't believe in the existence of things which aren't axioms.  We prove that they exist.  Can we prove the existence of a perfect proof?

If the number of possible "good" proofs is finite, then the assertion should follow immediately after.  After all, we only have a finite number to choose from.  Now for every statement s_i, I conjecture the existence for a positive integer k_i such that any proof with more than k_i pages is not "good".  This is an axiomatic conjecture, as there is clearly no way to prove it.  It fits the idea of an axiom as it should be immediately obvious to anyone, even those who are unreasonable.  A proof spanning 1000 pages that an even integer times an even integer is in fact even is absurd, let alone "not good".  Furthermore, any proof for which a single human can not read in a lifetime can not be considered "good" either.  A proof that spans 10^100 pages can not be considered good if no one single person can read the entire thing.  We shall pick this as our absurd upper bound for all such proofs.

However, there are a rather infinite amount of symbols one can use in a proof.  We may once again limit the complexity of symbols used.  For example, one can not consider a proof "good" if the two symbols are not decipherable to a human without the use of tools such as a microscope.   That is to say the "pixels" of the paper are limited to what a human can see.  Also, our dealing in pages was rather arbitrary.  Why couldn't one use a 3x5" note card or a 10x10' poster?  Certainly it is only the area of the entire piece that matters.  As such, the area on which the proof is written is our only concern, not the size of the medium.  This limitation on size limits how large our symbols may be.  As yet another absurd upper bound, we choose that symbol "pixels" may only be as small as atoms, and that the symbol may be (obviously) no larger than the area for which we have to write them proof.  The end result of this is that the dictionary of symbols as well as the number of them used must be finite.

The number of "good" proofs is therefore, by the above, finite.  As a result, there is a perfect proof for every statement.  However, there is something that has been ignored all this time.  How exactly does one rate a proof?  The majority of this is aesthetic, and as such, is incompatible with current mathematics.  However, what we have shown is that once there is a method for determining how "good" a proof is, there must exist the best proof.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#2 2008-08-18 09:32:17

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,651

Re: The Book

Interesting idea. What we consider "good" is influenced by the structure of our brain, and akin to our sense of beauty. A computer (or other) intelligence may define it differently.

So it makes no sense for us to insist that a proof be "nice", "good" or "perfect", because it somehow pleases our sense of beauty.

However I WANT to believe that proofs should be elegant!

And the search for more elegance does seem to help ... !


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#3 2008-08-18 09:42:10

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: The Book

So it makes no sense for us to insist that a proof be "nice", "good" or "perfect", because it somehow pleases our sense of beauty.

I'm not certain how you came to this conclusion.  Mathematics is not meant to be done by computers.  To paraphrase Hardy, mathematics is an art and its material is ideas.  Creativity and aesthetic arguments are a must.  I'll find some good Hardy quotes in a little bit.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2008-08-18 13:55:50

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,651

Re: The Book

OK, instead of computers, substitute aliens. The aliens *may* think a proof to be ugly that we think elegant and vice versa.

Or it may be that elegance is universal.

But how are we to know when we are the ones thinking about it?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#5 2008-08-19 02:09:45

LQ
Real Member
Registered: 2006-12-04
Posts: 1,285

Re: The Book

Your question was: Does a thing1 necessarily have to think the same thing1 about a thing2 as another thing3?
The integral of thing 1, which is all its properties, thinks "all the properties in thing2" about thing2. So does Thing3. how the gods reacts on turnips, however, is somewhat different since how they think about a thing is not always how they react on it. Whether or not they think the thing they see is positive depends on if their product with the turnips is positive or negative. For all i know, it might be neither, in which case, they don't exist!


I see clearly now, the universe have the black dots, Thus I am on my way of inventing this remedy...

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