Mathematics is the strangest thing. On one hand, it is the most reliable knowledge we have, more certain than empirical science, more precise than any other discipline. Mathematicians do not run experiments and update their theories when results come in. They prove theorems, and proved theorems stay proved. The Pythagorean theorem was true in ancient Greece, and it will be true after the heat death of the universe.
On the other hand, it is completely mysterious. Where are mathematical objects? You cannot see a number. You cannot trip over a set. You cannot weigh a function or measure the temperature of a triangle. Mathematical objects, if they exist at all, are not in space, not in time, not made of matter or energy, not observable by any scientific instrument. And yet mathematicians speak of discovering them, not inventing them, with the same confidence a geographer speaks of discovering a mountain range.
This is the scandal at the heart of mathematics, and it has occupied philosophers for as long as philosophy has existed. Here are the main positions.
Platonism (mathematical realism): mathematical objects really exist, independently of human minds, human cultures, and the physical universe. The number 7 is an abstract object that exists necessarily, eternally, and independently of anyone thinking about it. Mathematicians discover truths about these objects in the same way scientists discover truths about physical objects. The epistemic question, how can we know about objects that are non-physical and non-causal?, is harrowing, but Platonists bite the bullet. Gödel himself held this view: he spoke of mathematical intuition as a kind of perception of abstract objects, analogous to sensory perception of physical ones.
The analogy that makes Platonism feel compelling: the number π is not a human invention. It is the ratio of any circle's circumference to its diameter, in any possible universe. Every sufficiently advanced civilization in the galaxy, if such there be, will discover the same π. When different mathematicians work on the same problem independently and arrive at the same proof, it feels like they converged on a pre-existing fact, not that they happened to make the same invention. Mathematics feels discovered, not made up.
Nominalism: there are no abstract objects. Only concrete, physical things exist. Numbers, sets, and functions are not real entities, they are convenient fictions, useful shorthands for talking about patterns in the physical world. When we say "2 + 2 = 4" we are not asserting the existence of abstract entities called 2 and 4; we are saying something about what happens when you aggregate concrete collections of things. The advantage: no mysterious non-physical realm. The problem: mathematics seems to be about much more than just physical aggregations. The empty set, infinite cardinals, imaginary numbers, none of these obviously refer to physical things, and yet they are essential to mathematics that describes physical reality.
Formalism: mathematics is a formal game played with symbols according to rules. Mathematical statements are neither true nor false in any deep metaphysical sense, they are moves in a formal system. The advantage: no ontological commitments at all. The devastating problem: Gödel's incompleteness theorems showed that no consistent formal system sufficient for arithmetic can prove all true arithmetic statements. Truth in mathematics outruns provability in any formal system. This is fatal to any view that equates mathematical truth with formal provability.
Mathematical truth and formal provability are not the same thing. This is not a minor technical result, it is a deep philosophical fact about the nature of mathematical knowledge.
Quick reflection
When you think about the number 7 — not the numeral '7', not seven apples, but the abstract mathematical object — does it feel like you're thinking about something that exists independently of your mind? Or does it feel like a useful mental tool? Your intuition here places you on one side or the other of a 2,500-year debate.