Once upon a time, mathematical axioms were considered to be statements of self-evident truth. At least since development of the so-called axiomatic method, however, they have mostly been treated more like stipulative definitions chosen for convenience.

Between the respective times of Aristotle and Kant, allegedly self-evident truths found their way into philosophy. Descartes and others pulled principles like rabbits out of a hat, much as many of the scholastics had done earlier. There was a good deal of sound reasoning based on the dubious establishment of principles, but needless to say, the selection of principles drove the results.

Kant tried very hard to do better, and largely succeeded. In the Transcendental Dialectic section of the Critique of Pure Reason, arguments are examined that do seem to appeal to self-evident truths, but invariably, something is found by Kant himself to be wrong with the arguments.

I’ve previously referred to his antinomies as artificially staged. There, contradictory conclusions are derived from conflicting assumptions about self-evident truth, and Reason itself is apparently blamed for the result. Some commentators have thought that the arguments on the non-creationist side actually seem stronger than those on the creationist side, which raises the possibility that the apparently symmetrical presentation was an exercise in diplomacy on these sensitive matters. On the other hand, a counter-argument could be made that Kant had sympathy for both sides. Regardless, both sides of the antinomies appeal to allegedly self-evident truths, and it is the combination of these premises that leads to contradictory conclusions.

In the introduction to the antinomies, Kant himself says that “every transcendental illusion of pure reason rests on dialectical inferences” (Cambridge edition, p. 459). This seems to invoke either the common early modern pejorative sense of “dialectic” used in denunciations of scholasticism, or an actual Aristotelian distinction between dialectic and demonstration, where they are said to use the same inferential structures, but whereas demonstration aims at showing the reasons behind conclusions that are considered to be true, dialectic particularly examines the inferential consequences of using various premises, without regard for their actual truth. In any case, it seems Reason is found guilty of leading to contradiction when it works from such “dialectical” premises. There is nothing particularly surprising about that. (See also Kantian Discipline; Dialectical Illusion?)