It doesn't exactly come from counting, but mathematics like that is simply a matter of definition. We have ourselves defined what all the included symbols mean and how they relate to eachother.
The tricky bit comes when you try to make a system which you can argue relates to the real world. When it comes to linear things like simple counting, that's rather easy but it's still based mostly on the fact that it works; there is nothing in Reality that seems to *require* linearity but we know from experience that it works.
I think this is what he means by intuition. Why do we know what linear math applies to, say, apples? Because our experiences so far leads us to consider this something that works and seems right.
We still can't say why the world is exactly like it is, and such a question is rather odd in general.
Then you come to the nice bits in math where you can describe worlds and such that don't directly correspond to our real world. 54-dimentional geometry comes to mind.
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