Clippings

It used to be considered important to examine, for each consequence of the axiom of choice, the extent to which the axiom is need in the proof of the consequence. An alternative proof without the axiom of choice spelled victory; a converse proof, showing that the consequence is equivalent to the axiom of choice … meant honorable defeat. Anything in between was considered exasperating.

Until lately the best thing that I was able to think of in favor of civilization, apart from blind acceptance of the order of the universe, was that it made possible the artist, the poet, the philosopher, and the man of science. But I think that is not the greatest thing. Now I believe that the greatest thing is a matter that comes directly home to us all. When it is said that we are too much occupied with the means of living to live, I answer that the chief worth of civilization is just that it makes the means of living more complex; that it calls for great and combined intellectual efforts, instead of simple, uncoordinated ones, in order that the crowd may be fed and clothed and housed and moved from place to place. Because more complex and intense intellectual efforts mean a fuller and richer life. They mean more life. Life is an end in itself, and the only question as to whether it is worth living is whether you have enough of it.

[Quoting a speech by Oliver Wendell Holmes, Jr.]

This is not a Zen koan. There are actually fields and fields of fish beside the road that leads to Korea’s oldest ski slope, YongPyong.

… The fish, he says, are carted here from the docks to cure in the brutal winds that right now are blowing down from Siberia. “Specialty of the region,” he says. The fish? I wonder. The wind? Outside, a million pairs of shriveling eyes stare up at the quilted sky. The air smells of snowflakes and low tide on a hot day. This combination strikes me as unpropitious.

It was in July, I don’t remember the year. I was participating in a summer meeting on category theory at the Isles of Thorns, in Sussex. Somebody was actually giving a talk on the history of Eilenberg and Mac Lane’s collaboration in the forties, making clear what the exact contribution of the two authors was. At some point, somebody in the audience started to complain about the speaker giving credit to Eilenberg and Mac Lane for some basic aspect of their work which – he claimed – they borrowed from somebody else. A very sophisticated and animated discussion followed, which I was too ignorant to follow properly. The only things I can remember are the names of the two opponents: the speaker was Saunders Mac Lane and his opponent was Samuel Eilenberg. I was not born when they invented category theory. With my little story in mind, maybe you will forgive me for not having tried to give credit to anybody for the notions and results presented in this Handbook.

What, then, impels us to devise theory after theory? Why do we devise theories at all? The answer to the latter question is simply: because we enjoy “comprehending,” i.e., reducing phenomena by the process of logic to something already known or (apparently) evident. New theories are first of all necessary when we encounter new facts which cannot be “explained” by existing theories. But this moivation for setting up new theories is, so to speak, trivial, imposed from without. There is another, more subtle motive of no less importance. This is the striving toward unification and simplification of the premises of the theory as a whole…

…the world of mathematical truth has infinite complexity, even though any given [formal axiomatic system] only has finite complexity.

That is amazing said Rachel He doesn’t even care what cupboard he puts it in. Reprehensible said Fisher. I’m the only tidy person here said Rachel. And you have beautiful lustrous black hair thought Fisher But you have devised a system for using sponges which ranks with the most gruesome inventions of man.