Keränen's identity problem
Jukka Keränen's 2001 result that structures usually contain positions nothing inside the structure can distinguish, and that every account of identity available to the structuralist then makes those positions the same thing — so the theory ends up saying that 1 = −1 in the integers under addition.
If mathematical objects are just positions in structures, something has to say when two positions are the same position. Keränen's argument is that every answer available to the structuralist ends up identifying positions that the structure cannot distinguish — and structures are full of those. The integers under addition cannot internally distinguish 1 from −1, so the theory says they are the same number.
Two features make this useful far outside philosophy of mathematics. First, it was reached with no theological stake whatever, by people arguing about groups and complex numbers. Second, its repair is the same one that shows up in Augustine, in Aquinas and in the physics of indistinguishable particles: find a relation nothing can bear to itself.
Note also which way the failure runs. The default outcome is *collapse* — distinct-looking positions turn out to be one. Applied to the Trinity, that is the direction of modalism.
Where this comes up
Sources
- Jukka Keränen, “The Identity Problem for Realist Structuralism”, Philosophia Mathematica 9.3 (2001), 308–330
- Tim Button, Analysis 66.3 (2006), 216–222