WHAT IS A THING?
we can grasp the essence of the mathematical more sharply. Up to now we said 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 (Entwurf) of thingness (Dingheit) which, as it were, skips over the things. The project first opens a domain (Spielraum) where things--i.e., facts--show themselves.
2. In this projection there is posited that which things are taken as, what and how they are to be evaluated (würdigt) beforehand. Such evaluation (Würdigen) and taking-for (Dafürhalten) is called in Greek άξιάω. The anticipating determinations and assertions in the project are αξιώματα. Newton therefore entitles the section in which he represents the fundamental determinations about things as moved: axiomata, sive leges motus. The project is axiomatic. Insofar as every science and cognition is expressed in propositions, the cognition which 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.
3. As axiomatic, the mathematical project is the anticipation (Vorausgriff) of the essence of things, of bodies; thus the basic blueprint (Grundriss) of the structure of every thing and its relation to every other thing is sketched in advance.
4. This basic plan at the same time provides the measure for laying out of the realm, which, in the future, will encompass all things of that sort. Now nature is no longer an inner capacity of a body, determining its form of motion and place. Nature is now the realm of the uniform space-time context of motion, which is outlined in the axiomatic project and in which alone bodies can be bodies as a part of it and anchored in it.