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cousin_it comments on Consistent extrapolated beliefs about math? - Less Wrong Discussion
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I don't know. All occurrences of "exist" in my post are inside quotes, referring to intuitions that I don't know how to explain. For what it's worth, these intuitions are similar to what's described in Landsburg's post.
if you don't mean any ontological import you might want to phare it as "integers is system that works". The negation would be things that dont' work. For example a triangle with a right corner with angles summing over 180 degrees. But even then you have to specify the background assumptions as those kinds of triangles actually work out. Usually a mind defaults to a euclidean mindset while the applied concepts could apply to non-euclidean context too.
That integers exist could mean a number of things. Like that x + y = y + x for every x and y that is a value. However the logic of non-commuting values has been figured out. Therefore that sentence would be false. There are things that don't fall under this rule meaning this is not an universal rule. Having some assumtion that you need not have violated in your life doesn't mean such a violation would be impossible. The only way back would be to explicitly declare the delineation of that context ie the various properties needed. But then "integers exists" becomes just "assuming integers integers is all there is" which isn't very surprising or would need any explanation.
As long as you aren't clear what you mean with "exist" the rest of the argument is build on quicksand.
How so? The argument is about disbelieving intuitions about "existence" when they seem to contradict well-known math results.
If you start to reject naive assumptions about 'existence' the straightforward way is to do ontology and get a concept of 'existence' that's not naive.
In logical positivism there the attitude that you don't need to do any ontology, but when it comes to issues like this that just isn't helpful.
Landsburg writes:
But the set of all possible minds is so vast that the fact that numbers feel real to us feels to my mind as extremely weak evidence that numbers are real.
I think (it's not my post) that it's supposed to be evidence that numbers are real because the set of all possible minds is vast. Because there are so many possible minds, it's unlikely that a mind chosen randomly from that set would have similar intuitions about numbers to mine. It's even more unlikely that a third mind would also have those intuitions. Yet, for some reason, this vastly unlikely thing happens anyway. This implies that there is some reason which is responsible for all the minds feeling the same way. One such reason would be "the intuition is correct, and the minds have correctly figured it out".
(Other possible reasons could be "all the minds are biased in the same way" or some other reason unrelated to truth of the idea, but nobody's saying it's proof, it's just evidence.)
Though the number of possible minds is vast, I think the likelihood of two minds sharing an intuitive concept of number is high, because minds (or perhaps I should say consciousnesses) process information sequentially. Perhaps it is akin to the shared perception of rhythm, which is not limited to human minds. I suppose you have already seen it, but this video is amazing: the dancing cockattoo.