Let's begin with a simple question: what is a number? Anyone who has ventured into mathematics and philosophy would agree that this question does not seem to have a universally satisfactory answer. In fact, it’s precisely here that the boundary between mathematics and philosophy starts to blur. If we try to trace the origins of mathematics without investigating too deeply, we quickly find it branching out from philosophy. Some people even say that mathematics is just philosophy made precise, or rather, abstracted.
If we want to answer the question we started with, we face two broad possibilities: are numbers discovered as part of reality, or are they invented as useful structures of thought? To approach this, it helps to ask which problems the concept of number was originally introduced to solve. The simplest story starts from the top down: numbers arose to address the problem of counting. So numbers might have emerged at the moment when “how many” became separable from the things being counted. When we look at three books or three tables, we recognize that each collection has a count of three, and we mentally separate that quantity from the objects themselves. This raises a further question: how did “three objects” become the abstract idea behind the number 3, and this strange symbol, which would never have made sense if we had never felt the need to talk about numbers at all?
A natural digression here (and one I’m happy to take) is to Descartes. He asked a fundamental question: what does it mean to know something? Any account of numbers must eventually explain how we can claim to “know” them, especially if they are abstract objects rather than physical things.
Perhaps the question, essentially, is: how much of our own knowledge do we actually understand?