Abstract

What on Earth does metamathematics have to do with dialectics? One may justifiably find the association of these concepts dubious. Metamathematics, the foundational science of mathematics, seeks to grasp the conditions under which a contradiction-free science of quantity is possible. The following essay would like to show, in taking up the foundational reflections proper to metamathematics, that the principal anchoring of consistent mathematical objectivations presupposes a ground which not only founds these, but at the same time surpasses them in principle. Such an immanent movement of the principal presuppositions of a spiritual construct is what Hegel understood by a dialectical development. Yet it is not my concern here to provide a Hegel-interpretation, but rather to demonstrate a development in the Hegelian sense with respect to a science. Only what is astonishing here is that it should be possible to show such a development precisely with respect to mathematics, a form of the human spirit extremely remote from dialectics. And yet this development also has to be understood dialectically: knowledge can neither avoid the quantitatively unambiguous nor be content with it.