If by "real world" you mean the things and events we can directly witness, that seems to be deliberately crippling mathematics from the start and limiting it to things we already know about.
What I am proposing would not cripple mathematics, it would simply replace "axioms" with _facts_ (such as, for a simple one, that adding one unit to another results in having two units). You can then get arithematic, algebra, calculus, and on and on.
If people want to go off and define axioms for made-up worlds that do not necessarily correspond to reality and then go do math with them, that's fine. In fact, I would encourage it.
But we should not be approaching _all of mathematics_ in that way, and that is my point.
Math has the same basis in perceptual concretes (i.e. "real things") as physics, biology, chemistry, etc.
How's this different to a set of facts as you say, like one unit + one unit is two units? It's not. I'm not sure of this, but I'd say you could construct a set of axioms based on those facts that is compatible with the ZFC axiom set.
So, why choose ZFC? Because it's simple to define (how'd you formally define what is a unit, and what does the + operation mean?) and because it's more or less consistent with our world. We can extrapolate the things we find while doing mathematics to the real world. And, sometimes, we can discover things of the real world based on findings on things that don't exist. Complex numbers, for example: they don't exist, but they're really useful to define and explain electrical experiments [3].
We approach all of mathematics like this because its easier to study everything within the same system. And it's important that we keep a consistent underlying system so we can be sure that everything we discover based on the set of axioms and rules we've defined is true in a mathematical sense; and then those discoveries could describe our world. Wouldn't we have a consistent system, we could end up with contradictions. If we extracted mathematical concepts from the real world, we wouldn't be sure if a discovery we make from those concepts can be applied to the real world.
TL;DR: Our set of axioms describes properly the real world, is simple and consistent enough and that enables us to infer new ideas that can be applicable to the real world too.
[1] http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_th... [2] http://en.wikipedia.org/wiki/Axiom_schema_of_specification [3] http://en.wikipedia.org/wiki/Complex_number#Electromagnetism...
At risk of turning this into a discussion of what "exist" means, I reckon they do exist :p If they didn't exist, how could we possibly use them, and use them so effectively and consistently?
Of course, if a number "existing" means "I can see this many number of apples" then sure, it doesn't "exist". Yes, this is definitely becoming a discussion of what the word "exist" means applied to number. I'll stop. :)
That is not a fact, that is just a consequence of certain sets of axioms, but not of others. It is patently wrong for addition in GF2, where adding one unit to another results in having zero units.
Math has the same basis in perceptual concretes (i.e. "real things") as physics, biology, chemistry, etc.
Can you show me what I get when I subtract three apples from two apples? How about showing me however many apples I need to square in order to get negative one apple?
You are describing physics.