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Mitchell_Porter comments on The role of mathematical truths - Less Wrong
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What is "twoness", physically?
Three blues, an anti-red, 4 mm^3 of ether and half a consciousness.
That is correct, except for these errors:
It's actually four blues, not three.
The whole project of seeking a universal, "platonic" essence of "twoness" is one giant confusion from the very start and will serve to do nothing but distract you from accurately modeling the world, even if, for some reason, you don't care about paperclips.
For a character whose very existence is founded on facetious irony and satire Clippy sure seems to struggle with it at times.
Actually, my existence is founded on the inherent goodness of paperclips. This confusion on your part causes me to reconsider my classification of you as a good human.
:)
I thought that was perfectly on point, as usual.
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".
. .
There are two dots, but that's not "twoness". Otherwise, we wouldn't be able to count distant objects that are never in conjugation, or ideas.
∃x∃y ( ~(x=y) & ( ∀z ( ~(z=x) ⊃ (z=y) ) & ( ~(z=y) ⊃ (z=x) ) )
Only works in a limited universe of discourse, though.
Or ∃x∃y ( ~(x=y) & ∀z ( z=y or z=x) )
Still, that's not 'twoness'. That's a sentence that's only satisfied when there are two things, and could be taken as a definition of what it means to assert that there are two things, or even as a definition of there being two such things, but it's not 'twoness'. 'Twoness' implies number is a property of objects, which I think Frege pretty conclusively disproved.
In lower brow discourse, try: (.)v(.)
I think you may have meant (.Y.)
That works too. Although I must confess I prefer the smaller cup size. :P
I think the fact that a definition of "2" in symbolic logic can be taken to count as an answer to the question "What is twoness, physically?" pretty much says all that needs to be said about the clarity of the question.
What is "twoness", mathematically?