The nature of number

[Categories+Sheaves]

Is there an accepted standard definition of what a number is?
Do you think it could be defined within maths or does it have to be externally evaluated?

There are more systems of numbers than you can shake a stick at--you've got natural numbers, integers, ordinals, cardinals, surreals, rationals, reals, hyperreals, p-adics (one such system for every prime p!), hyper(rationals/integers/naturals/p-adics), every integral extension of the integers, every algebraic extension of the rationals... all of these systems are given formal logical/axiomatic foundations within mathematics, (esp. modern set theory) but each of these systems is motivated by some concern outwith mathematics.

There is a tremendous plurality of ways we can interpret (and use) numbers. Which ones do you want to explore?

Yes you see this is my problem. I am not a mathematician, I am an artist and thus am trained to look.

As long as you don't feel obligated to argue 1 != 0.999... or some supposed inconsistency in modern set theory, we'll get along just fine.

The idea that the number is a fixed point does not seem consistent with the results I am getting from my geometry.

You'll have to explain that further. But for now you can enjoy this quote:

The introduction of numbers as coordinates is an act of violence.

Moving along...

Also Russell's Paradox, it seems to me would naturally arise when both the observer and the observed are within the same system, in this case maths.

Russell's paradox arises when a model of set theory (call it M) can take itself (or the totality of the sets it contains) as an set for mathematicians to fiddle with (the axiom of restricted comprehension can only protect set theory from Russell's when you don't have M contained in M). This whole observer/observed thing sounds like some watered-down pop-QM... it looks ill-founded and I'm a little leery of it.

As an artist I know to observe the nature of a thing I must be to some extent external. As such to understand the nature of number I would have to have an external tool which is not affected by the number.

You're able to observe and reflect on the way you act in everyday life, so this issue has to be at least a little more nuanced and complicated than you're making it out to be.

In short I feel mathematical proofs of number cannot provide a definition of what number is.

Well, there are two ways to take this: we have some number-intuition from handling enumerable things in our lives, and sometimes we write out a list of axioms that dictate how a certain system (in this case, numbers) should function. Do you want the axiomatization or the phenomenology of math?

Am I barking, or do you think there could be a scintilla of logic in my position?

Well, I'm game to kick this stuff around.