Bruno Marchal skrev:
> I have to think. I think that to retrieve a Leibniz rule in discrete 
> mathematics, you have to introduce an operator and some non 
> commutativity rule. This can be already found in the book by Knuth on 
> numerical mathematics. This has been exploited by Kauffman and one of 
> its collaborator, and they have published a book which I have ordered 
> already two times ... without success. It is a very interesting matter. 
> Dirac quantum relativistic wave equation can almost be retrieved form 
> discrete analysis on complex or quaternion. It is worth investigating 
> more. Look at Kauffman page (accessible from my url), and download his 
> paper on discrete mathematics.

I will look closer at the Kauffman paper on Non-commutative Calculus and 
Discrete Physics.  It seems interesting, but not quite what I am looking 
for.  Kauffman only gets the ordinary Leibniz rule, not the extended 
rule I have found.

What I want to know is what result you will get if you start from the 
axiom that *everything in universe is finite*.

For this you will need a function calculus.  A function is then a 
mapping from a (finite) set of values to this set of values.  Because 
this value set is finite, you can then map the values on the numbers 
0,1,2,3, ... , N-1.

So a function calculus can be made starting from a set of values 
consisting of the numbers 0,1,2,3, ... , N-1, where N is a very large 
number, but not too large.  N should be a number of the order of a 
googol, ie 10^100.  Because the size of our universe is 10^60 Planck 
units, and our universe has existed for 10^60 Planck times.  As the 
arithmetic, we can count modulo N, ie (N-1) + 1 = 0.  This makes it 
possible for the calculus to describe our reality.

A function can then be represented as an ordered set of N numbers, namely:

f = [f(0), f(1), f(2), f(3), ... , f(N-1)].

This means that S(f) becomes:

S(f) = [f(1), f(2), f(3), ... , f(N-1), f(0)].

The sum or the product of two functions is obtained by adding or 
multiplying each element, namely:

f*g = [f(0)*g(0), f(1)*g(1), f(2)*g(2), ... , f(N-1)*g(N-1)].

and to apply a function f on a function g then becomes:

f(g) = [f(g(0)), f(g(1)), f(g(2)), ... , f(g(N-1))].

Exercise: Show that the extended Leibniz rule in the discrete 
mathematics: D(f*g) = f*D(g) + D(f)*g + D(f)*D(g), is correct!

Torgny Tholerus

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