Ante rem structuralism
The view in philosophy of mathematics that numbers simply are positions in a structure — there is nothing more to the number 2 than the place it occupies. It is the purest available version of “the slots are the individuals”, and it has been in trouble for twenty-five years over exactly that.
Latin ante rem, “before the thing”.
Ask what the number 2 is, and the structuralist answers: nothing but the second position in the natural-number structure. There is no further object hiding behind the position. Mathematical objects are places, and places are all there is.
It is a serious and widely held view, and it is the purest form of the assumption a slot-counting Trinitarian argument needs — that occupying distinct places is enough to be distinct. Its own literature has spent twenty-five years on the discovery that this does not work, because structures routinely contain places nothing inside them tells apart. The field's repair is the same one that shows up everywhere else in this material: appeal to a relation nothing can bear to itself.
Where this comes up
Sources
- Jukka Keränen, “The Identity Problem for Realist Structuralism”, Philosophia Mathematica 9.3 (2001), 308–330