Affine Connections, and Midpoint Formation

It is a striking fact that differential calculus exists not only in analysis (based on the real numbers R), but also in algebraic geometry, where no limit processes are available. In algebraic geometry, one rather uses the idea of nilpotent elements in the “affine line” R; they act as infinitesimals. (Recall that an element x in a ring R is called nilpotent if xk = 0 for suitable non-negative integer k.) Synthetic differential geometry (SDG) is an axiomatic theory, based on such nilpotent infinitesimals. It can be proved, via topos theory, that the axiomatics covers both the differential-geometric notions of algebraic geometry and those of calculus. I shall provide a glimpse of this synthetic method, by discussing its application to two particular types of differential-geometric structure, namely that of affine connection and of midpoint formation. I shall not go much into the foundations of SDG, whose core is the so-called KL1 axiom scheme. This is a very strong kind of axiomatics; in fact, a salient feature of it is: it is inconsistent – if you allow yourself the luxury of reasoning with so-called classical logic, i.e. use the “law of excluded middle”, “proof by contradiction”, etc. Rather, in SDG, one uses a weaker kind of logic, often called “constructive” or “intuitionist”. Note the evident logical fact that there is a tradeoff: with a weaker logic, stronger axiom systems become consistent. For the SDG axiomatics, it follows for instance that any function from the number line to itself is infinitely often differentiable (smooth); a very useful simplifying feature in differential geometry – but incompatible with the law of excluded middle, which