[drdeca]

With numbers, I can give an explanation for the phenomenon I described above. If such reasoning cannot be done without reference to numbers, then, if such reasoning is correct, numbers must exist. If there is no other reasoning can be given that provides a good explanation, and as the explanation I gave for the phenomenon is compelling, then I think that a good reason to conclude that the reasoning is correct, and that therefore those particular numbers exist.

In particular, I would expect that if numbers don’t exist, the explanation I gave of the phenomenon I described, couldn’t be correct.

[griffzhowl]

You could say they exist as concepts, that are necessary to use for some reasoning processes, without having any kind of independent existence.

It's similar for the case of programs or algorithms. We can say that a sorting algorithm exists, or a chess-playing program or whatever, which means we know how to implement the logical process in some physical system, but it doesn't mean that they have some kind of existence which is independent of the physical systems. It's just a way of talking about patterns that can be common to many physical systems

[drdeca]

My view is that something exists iff there is any statement that is true of it.

I of course don’t mean that mathematical objects (such as the number 2, or some sorting algorithm) have the same kind of existence as my bed. To make the distinction, I would say that my bed “physically exists”.

[griffzhowl]

That sounds like the same circularity, since you'll have to assume numbers exist before proving any statements about them.

Physical objects aren't like that because you can discover that they exist by empirical investigation.

In mathematics the discoveries are about the logical implications of sets of axioms. Some of those axioms contain assertions of existence, like a number 0 in Peano arithmetic or the empty set in set theory, and then you can prove statements about these objects based on the axioms. It's circular to infer from these conclusions that the axioms are true.

What's interesting is why certain axiom systems are so useful and fruitful. Personally I think it's because they evolved that way from our investigations of the physical world, but that's another matter