M Theory Lesson 270

In M Theory, we often talk about classical structures in the topos Set and quantum (and classical) structures in a category of vector spaces, Vect.

Thus it shouldn't be surprising that quantum mechanics might have something to do with arithmetic. Consider finite dimensional vector spaces over the field with two elements. If we were to sum a plane with another plane, the four dimensional resultant space would have $2 + 2 = 4$ basis vectors. On the other hand, there would be $4 \times 4 = 16$ elements in the full space. The basis cardinality was added while the vector space cardinality was multiplied. This is not so obvious when working with spaces over infinite fields.

Now recall that the adjunction from Set to Vect was well behaved for finite fields. And we can talk about Set as the category of vector spaces over the field with one element. The forgetful functor from Vect takes the abovementioned product into a set with that many elements. So if we started with two two element sets, and chose two two dimensional spaces over the field with two elements, then we would end up with a $16$ element set of vectors in Set, along with a natural map to the four element set $2 + 2$ with which we started.