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
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.
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