But I certainly don't accept your argument that they pop into existence when first summoned! That sounds silly.
I do not think they pop into existence.
I think you misunderstand what I mean by existence. A number can be written down, thought about, expressed on a computer and so on. It needs somewhere to appear, even if this is just somewhere in the human brain.
The same goes for any abstract idea. Abstractions are not physical objects that have an existence of their own in the real world. They are ideas we create and document.
I do not believe there is an infinite number of numbers existing in some invisible eternal realm somewhere. To me, this is a philosophical idea that is no use in practical terms because we have no way of accessing this realm, or do we (if so, how)?
I do not believe that this abstract collection of infinite numbers exists at all. Thus I do not believe that numbers just pop into this type of existence when first summoned.
OK, but v8archie's underlying point remains. If you take some big enough random number B, then with very high probability no person or computer has processed, looked at, thought of, etc. B, B+1, or B+2. So if you have a computer pick B and print B and B+2 (thus bringing them into existence), you have two consecutive integers which are both even or both odd, at least until someone computes B+1. Doesn't that strike you as strange?
Again, our understandings of 'existence' are very different indeed.
When a computer or human does a subtraction, they simply follow the rules explaining how to perform subtraction for the relevant number type.
There is no need for infinite sets.
There is no need to fetch numbers that already exist from an intangible infinite realm somewhere.
And it all works fine.
I could define my own formal system of objects and operations. I could define apples and pairs and work with these, not numbers. I could define what addition means and other operations. Or I could just create my own functions and operations such as â€˜blend_texturesâ€™. Would I have discovered this from the infinite realm of mathematical truths or does it only contain numbers?