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SilasBarta comments on The role of mathematical truths - Less Wrong
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Where is the chess in Deep Blue?
What I should have said:
The nature of mathematics is one instance of the problem of universals. (The opposite of a "universal" is a "particular".) Platonism says that universals exist independently of particulars - and extreme platonism says that universals are all that exists (example). Aristotelian realism says that universals always occur in association with a particular. Nominalism says there are no universals, just words.
I cannot tell if you are a nominalist or an Aristotelian realist. For a physical process involving rocks to have an isomorphism onto the equation 2+2=4, it seems like there has to be some actual twoness in the physical reality, which maps onto the abstraction '2'. So I want to know your views on the nature of this physical twoness.
Okay. What I have advocated here is a species of nominalism.
Well, that's where I disagree. For the isomorphism, it's only necessary that I have a working model; I needn't endorse any more abstract or universal concept of "twoness".
I continually ask myself: for whatever truth I posit, how justifiably surprised can I be if nature refused to play along? (Following the heuristic here.) All the "twoness" that I need to accept the existence of, is contained in physical agents' physical models of reality. To give it any greater role is to attach myself to a premise for which I have no contradiction to complain about if nature were to refuse to yield any other instance of "twoness".