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Anatoly_Vorobey comments on The Emergence of Math - Less Wrong
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There aren't any apples in reality. There are, arguably, atoms, and let's stop there and not go further to quantum states, elementary particles, virtual particles, etc. We can allow for the sake of the argument that there are atoms. But there certainly aren't any apples in reality. There're lots of atoms everywhere, and some of them are bunched together in a somewhat different way than some other ones, and the boundary between them is inherently fuzzy. Apples exist in your mind.
Discreteness on a micro level is arguable, but irrelevant to your everyday perception; discreteness on a macro level only exists in your mind. Out there in reality, there are just atoms.
For the sake of argument you are willing to keep atoms irreducibly, even though in the same reality in which there are no apples there are also no atoms, just quarks and neutrinos and a few 100 other "fundamental" particles.
What do we gain by deciding "for the sake of argument" to deny apples but accept atoms?
We gain simplicity of argument. The "messiness" of atoms is already enough for my argument that apples as discrete entities exist in the mind but not in reality; the reality of quarks, leptons and bosons (perhaps 16-18, rather than a few hundred particles) is even more "messy", which makes my argument stronger, not weaker, but also more complicated to explain.
I know. But it's easier to talk about apples than atoms. And the apples are just another level of abstractions. From atoms emerge apples, and from apples emerge [natural] numbers.