[drdeca]

Why is it that when I have a stack of business cards, each with a picture of a different finger on my left hand, then when I arrange them in a grid, there’s only one way to do it, but when I instead have each have a picture of either a different finger from either of my left or right hand, there is now two different arrangements of the cards in a grid?

I claim the reason is that 5 is prime, while 10 is composite (10 = 5 times 2).

Therefore, 5 and 10, and 2, exist.

[zmgsabst]

You’re abstracting to connect the math: 2, 5, 10, multiplication, and primality are all abstract concepts that don’t exist.

What you’ve pointed out is that the interactions of your cards, when confined to a particular set of manipulations and placements, is equivalent to a certain abstract model.

[griffzhowl]

You've already assumed 5 exists in order to assert that it's prime.

In any case existence of mathematical objects is a different meaning of existence to physical objects. We can say a mathematical object exists just by defining it, as long as it doesn't lead to contradiction.

[naasking]

I think your closing paragraph holds the key. 5 doesn't really exist, it's a constructor that parameterizes over something that does exist, eg. you never have "5", you have "5(something)". Saying 5 is prime is then saying that "for all x, 5(x) has the same structural properties as all other primes".

[drdeca]

Yes, the answer to the question does assume that 5 exists.

You try answering the question without speaking of 5 or 10.

That is my argument.

[griffzhowl]

Numbers are definitely essential concepts for some kinds of reasoning. If that's what you're saying then fine.

The thing is assuming that 5 exists to conclude that 5 exists is obviously circular.

[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”.