[Undated] [12]. Cogitations

It has just come to me that the mathematician is one who is capable of α and that it has this consequence: he can see the essential characters in, say, the paranoid-schizoid and depressive positions, or the Oedipal situation, or primal scene, or any other. These situations, or complexes, are seen concretely and, at some point, in visual images. For example, Euclid's geometrical patterns can easily be visual images that are made abstract. But the mathematician is able to produce abstractions, and signs for these abstractions, and can then replace the abstraction or its sign (reversing the direction of α) with a particular concretization.

If my idea is right, then Euclid could say good-bye to the ‘elements’ (1.5, Elefuga). That is to say, the ‘elements’ could be worked – transformed, in his case – into an abstract visual image (the theorem) and then still further abstracted by Descartes (or rather the use of the system of co-ordinates he worked out) so as to form the whole system of projective geometry, and then abstracted still further to form the system of algebraic geometry. Now the calculus thus elaborated lies ready to use.

Leaving aside the question, for the time being, of how this process of abstraction is to take place and how α affects it, we can see that the calculus that has been produced is available for the solution of all kinds of problems which, on the face of it, are far removed from the basic facts (‘elements’, in this case) from which the original abstraction was effected. But it is probable that it will only serve effectually for the solution of problems in which the concrete facts, which are to be substituted for the variables of the calculus, have a value relative to each other which is similar to the values the original basic facts had relative to each of the other basic original facts. Put in other terms, a Newton explores astronomical space, elaborating his laws of motion in doing so by virtue of an already existing calculus, or calculi, which de