I am now 8 days into teaching a six-week, every-morning-at-8:30 summer intro calculus class. In order to (1) make all the material fit into such a short time, (2) leverage the student’s (theoretically) good grasp of algebraic manipulation and (3) because I visualize infinitesimals more often than limits, I decided to teach as much of the course as possible using a somewhat ill-defined version of the real numbers that includes infinitesimals. In particular, the students have been working over the ring where . We additionally extend the ordering on to by defining for any positive real (this is vital if we want to extend piecewise-defined real functions to since we need the order predicates to extend).
These nilsquare infinitesimals lead to some really nice calculations once you get used to them. For example, here is one way to find the derivative of directly: for some finite . Squaring both sides gives , so by equating infinitesimals we find that .
But what I want to talk about is the obvious but puzzling connection to Greg’s notion of complexes as -modules. Now, I thought I understood calculus. I thought I understood complexes. But for the life of me, I can’t figure out how to think about complexes as “vector spaces with infinitesimals”, which is to say -modules. What the heck is going on here, morally speaking?
A few more points relating all this to Greg’s posts on complexes as -modules: suppose we adjoin two nilsquare infinitesimals and to and specify no relations between them. Then the sum is not nilsquare: its square is . So if we want to adjoin two or more nilsquare infinitesimals and keep out any higher order (nilcube, etc.) infinitesimals, we also need the anticommutation relations . Note the relationship between this and Greg’s definition of bicomplexes. I reiterate: what the heck is going on here?