The article considers structuralism as a philosophy of mathematics, as based on the commonly accepted explicit mathematical concept of a structure. Such a structure consists of a set with specified functions and relations satisfying specified axioms, which describe the type of the structure. Examples of such structures such as groups and spaces, are described. The viewpoint is now dominant in organizing much of mathematics, but does not cover all mathematics, in particular most applications. It does not explain why certain structures are dominant, not why the same mathematical structure can have so many different and protean realizations. ‘structure’ is just one part of the full situation, which must somehow connect the ideal structures with their varied examples.

Very nice philosophy paper by one of the progenitors of category theory on structure. The idea to show a correspondence between Bourbaki and category theory seems like a nice grad school project.

This is how we do math in the 21st century.

So when you see a no-go theorem that’s being given a very broad interpretation, you may do well to ask whether there is, after all, a way to get around the theorem, by achieving what the theorem is informally understood to preclude without doing what the theorem formally precludes.