Ronald,

On 21 Dec 2008, at 15:40, Bruno Marchal wrote:

>>
>> How is there any mathematics with nothing to
>> conceive of it?



Let me try a straightest answer from math, with an example. Take the  
digital or discrete line. You can map it on the integers. It is the  
symmetrical extension of the natural numbers. Is there a diophantine  
equation capable of breaking that symmetry?
Could *that* be true by convention?

Actually there is one: an integers is a positive integers if and only  
if it can be written as the sum of four squares. This has been  
discovered by Diophantus and rediscovered and proved by Legendre  
centuries later. The sum of four squares property break the digital  
line symmetry.

Then you can ask yourself the natural question: is there a formula or  
algorithm giving the numbers of ways a natural numbers can be written  
as a sum of four squares.

This is a difficult exercise. Jacobi has found the solution. Then odd  
numbers have 8 * sum of its divisors ways to be written as a sum of  
four squares (the order counts). The even numbers, the feminine  
numbers for some Greeks!, have 24 * sum of its male, well odd, divisors.

What do you mean by nothing to conceive of it? Just the numbers gives  
plenty questions at different levels. For example, at another level,  
the four squares sum property shows that the notion of natural numbers  
can be represented in the theory of integers. It can be shown in  
similar ways that first order theory of positive integers, integers  
and rational are mainly equivalent. First order theory of the reals  
leads to a decidable theory. The reals realm looses turing  
universality. The reals simplifies the numbers too much. But the real  
+ trigonometry gives the waves and their stationnary quanta. The waves  
reintroduce the integers, digitality, and Turing universality in the  
realm of the reals.

There is a mathematical reality, and it contains many processes and  
relations, transformation and fixed points, symmetries and breaking   
of symmetries, even galaxy collisions, brane collisions, why not taxes  
and other  local relative financial crisis ...

The mathematical reality kicks back would say David Deustch.

Jacobi theorem is not easy to prove, in a book by Kak, I found a proof  
of Jacobi theorem which uses ... the bosonic string theory.

In the same manner the study of the distribution of the prime numbers  
leads to a surprising sequence of surprises, I tend to believe that  
such distribution could encoded a key in our inquiries as I have  
described in some of my Riemann Zeta TOE old posts.

With addition and multiplication, shit happens, already. Assuming  
comp, all the possible shit happen, but the beauty is that "we" are  
not eliminated, and "we" can keep a *partial* control. Of course the  
price here is that "we" are also responsible of some amount of that  
shit. If I can say.

Bruno
http://iridia.ulb.ac.be/~marchal/




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