“Let no one ignorant of geometry enter here!” (inscription above the entrance to Plato’s Academy)
In his essay “Modern Science, Metaphysics and Mathematics” (1962; from Basic Writings, ed. David Farrell Krell [San Francisco: Harper Collins, 1977] 247–82, an excerpt from “What is a Thing?” [1967; Chicago: Regnery, 1969] 66-108), Heidegger wrote:
In its formation the word mathematical stems from the Greek expression tà mathémata, which means what can be learned and thus, at the same time, what can be taught; manthanein means to learn, mathésis the teaching, and this is a twofold sense. First, it means studing and learning; then it means the doctrine taught. (249-50)
- mathésis: teaching and learning
- tà mathémata: what is teachable or learnable
Learning is a kind of grasping and appropriating. But not every taking is a learning. [. . .] To take means in some way to take possession of a thing and have disposal over it. Now, what kind of taking is learning? Mathémata—things, insofar as we learn them. . . .
The mathémata are the things insofar as we take cognizance of them as what we already know them to be in advance, the body as the bodily, the plant-like of the plant, the animal-like of the animal, the thingness of the thing, and so on [verbatim from “The Age of the World-Picture“]. This genuine learning is therefore an extremely peculiar taking, a taking where he who takes only takes what he basically already has. Teaching corresponds to this learning. Teaching is a giving, an offering; but what is offered in teaching is not the learnable, for the student is merely instructed to take for himself what he already has. If the student only takes over something that is offered he does not learn. He comes to learn only when he experiences what he takes as something he himself really already has. True learning occurs only where the taking of what one already has is a self-giving and is experienced as such. Teaching therefore does not mean anything else than to let the others learn, that is, to bring one another to learning. (251)
Heidegger continues:
Teaching is more difficult than learning; for only he who can truly learn . . . can truly teach. The genuine teacher differs from the pupil only in that he can learn better and that he more genuinely wants to learn. In all teaching, the teacher learns the most. (251-52)
I’m reminded of Plato’s discussion of amamnesis, of learning as remembering in the Meno and Phaedo—though Heidegger most often employs this characterisation of ta mathémata in his critique of the pseudo-circular nature of modern scientific research, which is almost tautological in its foreclosure of knowledge by its use of deductive or hypothetico-deductive method and its pursuit of objectivity.
In “The Age of the World-Picture,” he argues that scientific research is a type of rigorous knowledge (Erkennen, a.k.a. “judgement”) that relies on a procedure (Vorgehen, a.k.a., “priority, lead”) that establishes its field of operation by the projection (Entwurf, “design, project, plan, outline”) in advance of a ground-plan (Grundriss, a.k.a, “framework”): projection → procedure → knowledge.
But for him, to speak in the most general terms,
[t]he mathémata, the mathematical, is that “about” things which we really already know. Therefore we do not first get it out of things, but, in a certain way, we bring it already with us. (252)
Te Ipu Pakore: The Broken Vessel 