Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect.

Physicists and philosophers stick stubbornly to the principles of a mechanistic interpretation of the world after physics has, in its factual structure, already outgrown the latter. They have the same excuse as the inhabitant of the mainland who for the first time travels on the open sea: he will desperately try to stay in sight of the vanishing coast line, as long as there is no other coast in sight, towards which he steers.

We now come to a decisive step of mathematical abstraction: we forget about what the symbols stand for... [The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for.

In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.

The introduction of numbers as coordinates by reference to the particular division scheme of the open one dimensional continuum is an act of violence whose only practical vindication is the special calculatory manageability of the ordinary number continuum with its four basic operations. The topological skeleton determines the connectivity of the manifold in the large.

On a certain level of generality A which I call the ground level, you have certain theorems that have been proved and certain unsolved problems P of recognised interest. Suppose you discover a generalisation of one of these theorems and thereby rise to a higher level of generality A'. Write it up, but lock it away in a drawer - unless or until it serves to solve one of the problems P on the ground level... But the deeper one drives the spade the harder the digging gets; maybe it has become too hard for us unless we are not given some outside help, be it even by such devilish devices as high-speed computing machines.

Cartan developed a general scheme of infinitesimal geometry in which Klein's notions were applied to the tangent plane and not to the n-dimensional manifold M itself. On the foundations of general infinitesimal geometry.

Important though the general concepts and propositions may be with which the modern industrious passion for axiomatizing and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity constitute the stock and core of mathematics; and to master their difficulties requires on the whole the harder labor.

We cannot hope to give here a final clarification of the essence of fact, judgement, object, property; this task leads into metaphysical abysses; about these one has to seek advice from men whose name cannot be stated without earning a compassionate smile-e.g.