M Theory Lesson 100

On page 127 of their book [1], Lambek and Scott discuss the unifying properties of topos theory, as seen from the perspective of type theory. The three traditional mathematical philosophies in question are intuitionism, Platonism and formalism (in the sense of concrete symbolism). They note that a fourth philosophy is somewhat neglected, namely neologicism: the idea that all mathematics can be formulated in terms of logic. Strict logicism would require deriving the properties of the natural numbers from more foundational principles.

But isn't that just what the physics has been telling us we must do? Lambek and Scott take the viewpoint that categorical logic treats logic pragmatically, as a part of ordinary mathematics, thus refuting logicism. Should logos theory refuse to accept this point of view? Recall that in M theory any useful logical statement has a physical meaning pertaining to a given experiment and its constraints. The numbers associated with logical statements (or rather diagrams) also take on a physical meaning (loosely speaking, a collection of measurements). Thus both logic and number theory should be derived from the physical principles of measurement, expressed as a higher dimensional topos theory.

There is a chicken-and-egg objection: that the formalism of higher toposes requires the mathematics of logic before acquiring a physical interpretation. However, since physics is undoubtedly guiding the axioms in question, this argument appears weak. Ironically, logicism in M Theory would be the ultimate in reductionism, despite the intuition of emergent schemes on different scales.

[1] J. Lambek and P.J. Scott, Introduction to higher order categorical logic, Cambridge U.P. 1986