Concluding Summary

Abstract

In the beginning mechanism seemed as untenable for our reproducing bodies as for our calculating minds. It took centuries to find mechanisms for the two simplest computations of our minds, but it seemed unlikely that they could be found for the higher mental calculations, and inconceivable that a single mechanism could ever encompass all of its calculations. At the center of these calculations lay the finest flower of the constructive mind, the intuitive number sequence (N). Dedekind formulated the mathematical foundation (DT) for this flower — as well as a theory of recursions on it undreamt of in earlier theories of calculation — by means of the finest flower of the non-constructive mind, set theory. But for Dedekind this latter flower, especially its infinite petals, grew only in the mind itself, leading to a temporary engagement of mathematics and idealism. But paradoxes were also discovered to grow here, causing Dedekind to remark that “human reason totters”. Constructivists like Kronecker reclaimed their flower, claiming now that it — together with its effective functions — should provide the foundation for the rest of mathematics. Cantor’s diagonal argument seemed to refute this, but Richard showed that this also led to paradox. Yet it also led to recursions far transcending any formal principle like Dedekind’s.