What would you say if I were to ask you, "do numbers exist"? You might say yes: we can count things so there is such a thing as number. On the other hand, you might say no. There are objects but there are no numbers; we make up numbers to make life easier.
This is a big ongoing debate in the philosophy of mathematics. What is maths? Nominalists say that its like a game of checkers. There are pieces (numbers) and there are rules (methods of calculation) but there is no meaning. This view is not widely accepted. If maths was just a made up game, then why does it have such useful applications in the real world? And why does each culture end up following the same rules?
A more plausible position is Aristotelianism. This view says that numbers exist as concepts. So we can't see them in the real world but they still exist because they exist in our minds. But what would happen if there were no people in the world, and no other intelligent life which could count? Would numbers still exist?
Aristotelianism is actually part of a bigger debate over the Universals. It tries to answer the following question: if all red things in the world were destroyed, would the colour red still exist? Aristotelianism says yes! The colour exists in our minds as a concept. But this faces the same problem as before. What happens if there are no minds to hold the concept? Does the concept die?
A popular position in the debate over the existence of numbers is platonism. This view holds that numbers and colours and all other properties (any adjectives you can think of) exist in a separate world - plato's heaven. So mathematics has an abstract existence. Unfortunately, this view is also problematic. If numbers do exist in this separate realm, then how do we know about it? Usually, if we know something, we had some sort of causal connection with it. For example, if I said that I knew there was a chair in the room, I would know because I saw it with my own eyes (or someone else who saw it told me about it). But in the case of abstract objects, we have no causal connection. Therefore, if numbers are abstract objects, we cannot know about them.
But we do know about them. We have an established discipline called mathematics. There is one more thing that is worth mentioning about platonism. It is called the Indispensibility Argument. It was formulated by Quine-Putnam and basically says that mathematics must exist because it is indispensible (it cannot be eliminated from) science. Field tried to show that Newton's gravitational theory could be proven without mathematics but it is doubtful whether he has succeeded in his endeavour. Even if he has, it is even more doubtful that he could achieve the same for quantum theory (our best theory of very small things). So it seems that mathematics must exist. But where?
Maddy has recently attempted to solve the problem. She suggests that mathematical entities have an abstract existence, but they do not reside in a separate realm. Rather, they are down here, with us. This solves the problem of our knowledge of mathematics. Maddy says that we know about mathematics because we have causal connections with the abstract objects. When we see three eggs for example, we can see the "threeness" as a set (group of objects). She attempted to prove this by showing that we have set-perceptive mechanisms. Also, once we understand the concept of number, we are able to deduce the rest through our use of logic.
So we have finally arrived at a view which allows for the existence of numbers as abstract objects and explains our knowledge of them. Please feel free to challenge this view...