Alternatively, we can have synthetic differential geometry or smooth Infinitesimal analysis with its roots in category theory. This approach departs dramatically from the classical logic used in conventional mathematics by denying the law of the excluded middle--i.e., NOT ("a" ≠ b) does not have to mean a = b. A nilsquare or nilpotent Infinitesimal can then be defined. This is a number x where x ² = 0 is true, but x ≠ 0 can also be true at the same time. With an Infinitesimal such as this, algebraic proofs using infinitesimals are quite rigorous, including the one given above.

number x where x ² = 0 is true, but x ≠ 0 can also be true at the same time (no excluded middle law).