Geometry, Topology, Global Analysis and General Structures

Fortunately analysis is not the only way to resolve inner conflicts. Life itself still remains a very effective therapist... The therapy effected by life itself is not, however, within one's control. Neither hardships nor friendships nor religious experience can be arranged to meet the needs of the particular individual. Life as a therapist is ruthless; circumstances that are helpful to one neurotic may entirely crush another.

My work in the future must be devoted entirely to pure mathematics in its abstract meaning. I shall apply all my strength to bring more light into the tremendous obscurity which one unquestionably finds in analysis. It lacks so completely all plan and system that it is peculiar that so many can have studied it. The worst of it is, it has never been treated stringently. There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has led to so few of the so-called paradoxes. It is really interesting to seek the cause.

By this way of Analysis we may proceed from Compounds to Ingredients, and from Motions to the Forces producing them; and in general, from Effects to their Causes, and from particular Causes to more general ones, till the Argument end in the most general. This is the Method of Analysis: and the Synthesis consists in assuming the Causes discover'd, and establish'd as Principles, and by them explaining the Phænomena proceeding from them, and proving the Explanations.