In 1973, the philosopher Paul Benacerraf published a paper called "Mathematical Truth" that is, by general philosophical consensus, the most important single contribution to philosophy of mathematics of the 20th century. Its impact comes not from proposing a solution but from demonstrating precisely why all existing solutions fail.
Benacerraf identified two constraints that any satisfactory philosophy of mathematics must meet:
First, the semantic constraint: the semantics of mathematical statements should be the same as the semantics of ordinary statements about the world. "7 is prime" should work semantically like "the cat is black", it should refer to an object (7) and predicate a property of it (primeness). Mathematical statements appear to be about objects, and a good philosophy of mathematics should respect that appearance rather than treat it as systematically misleading.
Second, the epistemological constraint: our theory of mathematical knowledge should mesh with a credible account of how human beings come to know things. We are physical beings in a physical world, and we come to know things through causal interaction with our environment, seeing, hearing, touching, measuring, doing experiments.
Here is the problem: if mathematical objects are abstract, non-physical, non-causal entities (satisfying the semantic constraint, they are real objects that mathematical statements refer to), then it is completely mysterious how we could know anything about them. We can only know about things we can causally interact with, and abstract objects are causally inert.
But if we try to satisfy the epistemological constraint, by saying mathematical objects are actually physical, or that mathematical statements don't really refer to objects, we end up with semantics that treats mathematical language as systematically misleading. Seems wrong.
Think of it this way: it's like being told that to know your neighbor's name, you have to have met her, but also that she lives in a dimension you cannot enter, has no physical form, and cannot causally interact with you in any way. Either you can know the name (but then how?) or she's not really there (but then what are you doing when you do mathematics?).
None of the standard positions escapes both horns cleanly. Platonists satisfy the semantic constraint but struggle with the epistemological one, Gödel's "mathematical intuition" is evocative but philosophically underdeveloped. Nominalists satisfy the epistemological constraint but struggle with the semantic one, explaining away apparent reference to abstract objects requires increasingly heroic paraphrase. Structuralism (the view that mathematics is about structures rather than objects) and fictionalism (the view that mathematical objects are like characters in a well-developed fiction) are modern attempts to thread the needle, with ongoing debate about whether they succeed.
Structuralism, associated with philosophers like Stewart Shapiro and Michael Resnik, holds that mathematics is not about particular objects (the number 2 is not a particular thing) but about structures, abstract patterns of relations. What matters about 2 is not what it is intrinsically but its position in the structure of the natural numbers: it comes after 1, before 3, is the only even prime. Mathematics describes structural possibilities, and the question of what the objects of mathematics "really are" is a bad question.
The analogy: a chess king is not a particular piece of wood or plastic, what makes something a chess king is its role in the game, the moves it can make, how it relates to other pieces. You could play chess with bottle caps. The game-role is what matters, not the intrinsic nature of the token filling that role. Similarly, the natural numbers are positions in a structure, not intrinsically characterized objects.