"A is A, Existence exists..."

My instinct is that numbers exist, and that they are at least similar to objects. They require consciousness to perceive them, but they do seem to exist in the world whether we notice them or not.

You might see a group of apples on the table and notice that they have the property of five. You might just ignore the number 18,156,965,975,191. Everyone else seems to, it was the amount of the national debt in dollars early this afternoon. But I have the notion that it exists anyway.

Personally, I am not sure how far I can push that argument.

Does Infinity exist? Science says we live in a finite universe, with a finite age, and a finite measure of mass in it. It is very hard to point to a concrete example of infinity.

Go down in the other direction, at certain small intervals of space of time, nothing is there and nothing happens. Can I say that the infinitely small exists?

But the concept of infinity is central to all technological progress mankind has made in the last 500 years. How can it not exist?

I know, I know, this is all rather pedantic. But the fact is philosophy and reason lie at the heart of all mathematics.

Students of philosophy often are not exposed to the fact that mathematicians are forced to look at very philosophical questions concerning the nature of the concepts they are dealing with.

Philosophy leads to logic and reason, logic and reason lead to math and science, math and science describe the natural laws governing our existence. I find it interesting that mathematics now turns back to philosophy for inspiration and understanding.

If you have never come across these ideas, I think the video is a very accessible introduction to the subject. Are numbers objects? Is math just a formal game? If non-physical A can exist, does A therefore exist?