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/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
