Modern Science, Metaphysics, and Mathematics

τὴν ἐπιστήτην (Meno 85d), "bringing up and taking up—above and beyond the other—taking the knowledge itself from out of himself."

There is a prior grasping together in this mente concipere of what should be uniformly determinative of each body as such, i.e., for being bodily. All bodies are alike. No motion is special. Every place is like every other, each moment like any other. Every force becomes determinable only by the change of motion which it causes—this change in motion being understood as a change of place. All determinations of bodies have one basic blueprint, according to which the natural process is nothing but the space-time determination of the motion of points of mass. This fundamental design of nature at the same time circumscribes its realm as everywhere uniform.

Now, if we summarize at a glance all that has been said, we can grasp the essence of the mathematical more sharply. Up to now we have stated only its general characteristic, that it is a taking cognizance of something, what it takes being something it gives to itself from itself, thereby giving to itself what it already has. We now summarize the fuller essential determination of the mathematical in a few separate points:

1.The mathematical is, as mente concipere, a project of thingness which, as it were, skips over the things. The project first opens a domain where things—i.e., facts—show themselves.

2. In this projection is posited that which things are taken as, what and how they are to be evaluated beforehand. Such evaluation and taking-for is called in Greek ἀξιόω. The anticipating determinations and assertions in the project are ἀξιώματα. Newton therefore entitles the section in which he presents the fundamental determinations about things as moved Axiοmata, sive leges motus [The Axioms or Laws of Motion]. The project is axiomatic. Insofar as every science and cognition is expressed in propositions, the cognition that is taken and posited in the mathematical project is of such a kind as to set things upon their foundation in advance. The axioms are fundamental propositions.

Martin Heidegger (GA 41) Basic Writings (1993)