better and that he more genuinely wants to learn. In all teaching, the teacher learns the most.
The most difficult learning is coming to know actually and to the very foundations what we already know. Such learning, with which we are here solely concerned, demands dwelling continually on what appears to be nearest to us, for instance, on the question of what a thing is. We steadfastly ask the same question—which in terms of utility is obviously useless—of what a thing is, what tools are, what man is, what a work of art is, what the state and the world are.
In ancient times there was a famous Greek scholar who traveled everywhere lecturing. Such people were called Sophists. This famous Sophist, returning to Athens once from a lecture tour in Asia Minor, met Socrates on the street. It was Socrates' habit to hang around on the street and talk with people, for example, with a cobbler about what a shoe is. Socrates had no other topic than what the things are. "Are you still standing there," condescendingly asked the much-traveled Sophist of Socrates, "and still saying the same thing about the same thing?" "Yes," answered Socrates, "that I am. But you who are so extremely smart, you never say the same thing about the same thing."
The μαθήματα, 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. From this we can now understand why, for instance, number is something mathematical. We see three chairs and say that there are three. What "three" is the three chairs do not tell us, nor three apples, three cats, nor any other three things. Rather, we can count three things only if we already know "three." In thus grasping the number three as such, we only expressly recognize something which, in some way, we already have. This recognition is genuine learning. The number is something in the proper sense learnable, a μάθημα, i.e., something mathematical. Things do not help us to grasp "three" as such, i.e., threeness. "Three"—what exactly is it? It is
Modern Science, Metaphysics, and Mathematics
GA 41 p. 74