Discourse, Life and Criticism on Heidegger on the grounds of mathematics
This is a disagreement about the mathematical. According to Heidegger, the mathematical is what can be learned from what one already has. Wittgenstein and Cantor disagree about what diagonalization tells us because they do not share the same presuppositions. Cantor is eager to have the absolute show itself. Wittgenstein determines that the finitude of mathematics derives from its being a human endeavor. These are radically different origins or foundations from which mathematics unfolds. From these varying axioms or guiding principles, Wittgenstein and Cantor (with is robust set theory) diverge on what is the same tactile situation. The evidencing of diagonalization occurs in drastically different ways, and the evidence, informed by the mathematical project of both men, becomes something other than itself. The evidence is something other than a method for constructing a sequence. It becomes operational in the correctness of both mathematical projects, and its truth is covered over by the initial projection of the essential determinations of each projector. Which is just to say: the truth subsists and is carried along in the praxis of projecting. Truth, though, always subsists. Within the project, correctness reigns, and truth lies underneath, covered over the by the work of the mathematical thinker.
All of this is simply to repeat a thought from Heidegger, but not for the sake of repetition. The repetition has its place for us vitally, for what is more important at our present stage then the necessity of understanding genuine, foundational disagreement? What have we shown ourselves, what have we seen, other than the damnation of not being honest about the origins of our reasoning and of not striking out to, at bottom, reconcile the pace between origins? Reason-giving is itself a practice whose force and validity comes from itself. Its power, beyond this, is instituted in history, but this should not be confused with some kind of necessity. For, where reasons do not come forth there is not, then, “merely nothing”. For in reasons’ stead may come a more primordial force that we take care not to name, an immutable groundswell… Yet this we must attend, and we fail to do so at great peril. We must attend the groundswell, in our way, which is, at least in one way, by giving reasons, by setting forth to find that which was already there, but whose disclosure and existence must be worked into place.