Moving along, Heidegger lists his six points on the mathematical essence of modern science.

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 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 sumarize the fuller essential determination of the mathematical in a few seperate 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 axiow. The anticipating determinations and assertions in the project are axiwmata. Newton therefore entitles the section in which he repesents 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) fo 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 emcompass 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.

5. This realm of nature, axiomatically determined in outline by this project, now also requires for the bodies and corpuscles within it a mode of access (Zugangsart) appropriate to the axiomatically predetermined objects. The mode of questioning and the coginitive determination of nature are now no longer ruled by traditional opinions and concepts. Bodies have no concealed qualities, powers and capacities. Natural bodies are now only what they show themselves as, within this projected realm. Things now show themselves only in the relations of places and time points and in the measures of mass and working forces. How they show themselves is prefigured in the project. Therefore, the project also determines the mode of taking in and studying of what shows itself, experience, the experiri. However, because inquiry is now predetermined by the outline of the project, a line of questioning can be instituted in such a way that it poses conditions in advance to which nature must answer in one way or another. Upon the basis of the mathematical, the experientia becomes the modern experiment. Modern science is experimental because of the mathematical project. The experimenting urge to the facts is a necessary consequence of the preceding mathematical skipping (Überspringen) of all facts. But where this skipping ceases or becomes weak, mere facts as such are collected, and positivism arises.

6. Because the project establishes a uniformity of all bodies according to relations of space, time, and motion, it also makes possible and requires a universal uniform measure as an essential determinant of things, i.e., numerical measurement. The mathematical project of Newtionian bodies leads to the development of a certian "mathematics" in the narrow sense. The new form of modern science did not arise because mathematics became an essential determinant. Rather, that mathematics, and a particular kind of mathematics, could come into play and had come into play is a consequence of the mathematical project. The founding of analytical geometry by Descartes, the founding of the infinitesimal calculus by Newton, the simultaneous founding of the differential calculus by Leibniz--all these novelties, this mathematical in a narrower sense, first became possible and, above all, necessary, on the grounds of the basically mathematical character of the thinking.

P. 91-94

So, to do modern science, one must think mathematically, about abstractions, rather than about the things themselves.

Continued.  ¶ 2:37 PM