A Book Acquired: Synthesis and an Optimistic Viewpoint.

At last.

I've finally snagged a copy of Be11aiche's $ub-Riεmannian Ge0metry¹, a Birkhauser text and an all-around recommended and cited text for those who study CC (Carn0t-Carathe0dory) spaces. After months of harassing the library staff at UM and steadfastly refusing to pay the $100+ for my own copy of the text, it eventually occurred to me that I could just as easily take advantage of some other library's archive through Inter-Library Loan .. and here I have it: the earthy green hardcover and all.

So wonderful .. I might even be getting a little misty-eyed.

Half of the book is composed of an article by M. Gr0mov, and browsing through the first section (it seems only mathematicians and computer scientists are willing to name the first section: 'Section Zero'), I'll repeat the description which I had written to a friend of mine, just a few hours before:

It's like sitting down for a lecture without knowing how short or long it will last. The speaker begins by turning on several projectors to multiple, polygonal-shaped transparency screens, one devoted to a branch of mathematics.

During the introduction he narrates, pointing to this screen, then that one, often to two or three at the same time. Provided you recognise the content of most of the screens, it becomes a kaleidoscope where all the screens form prism-faces, and you find yourself looking through the ultimate looking-glass. The object in sight is the same, and it is only your perception which changes.

Well, that was more of a paraphrase than a repetition .. but you get the idea. (;

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At moments like these it is easy to revert from thinking in mathematics to thinking about mathematics, or more precisely, thinking about mathematicians and mathematical thought.

What is it that we are doing exactly, when we study and make mathematics, and good mathematics, at that? What is 'good' mathematics, anyway?

I'd be foolish to give an answer, but I'll do it anyway - that is, I'll give a partial and personal answer to what sort of mathematics I like. One of the great beauties in pure maths is the sheer variety and at the same time, the surprising condensation of ideas. There seem countless ways to describe and discuss all sorts of concept and phenomenon, real or theoretic .. yet on many occasions, there is a deeper fact or idea which governs it all, which manifests itself in that multitude of ways.

I find it sufficiently difficult to understand even the fundamental ideas in a single field of mathematics. There seems no end to the possible intricacy and subtlety in a single definition or concept.

That is why it's so fascinating to me, when I learn a relation or correspondence between seemingly separate and disparate ideas. These persons who explain these things to me, they really know something of substance: they understand this and they undeerstand that, and now they tell me that the two ideas are really the same! No, no, you must simply look at it from this perspective, or perhaps through this funny pair of glasses, and when I do, the resolution is amazing. I see what they see, so very clearly in that single instant ..

It's a wonderful reduction when it occurs, an efficiency of memory and brainpower, a convenient structure one may fall back upon and it will not collapse. On a more philosophical level, it assures us, we bearers of human intellect, that perhaps the world is knowable, if only this world of ideas. We may know much more after having learned old lessons and how they fit into new context. Perhaps we need only understand a few things, and the rest will follow.

Now I'm rambling. Better to spend time reading and learning, rather than ranting and supposing. I've waited for this volume of Be11aiche long enough, and I might as well enjoy it before the late fees arrive.

¹More accurately, A. Be11aiche is one of the editors to $ub-Riem@nnian Ge0metry, which is a collection of several articles on the topic. He is, however, the author of one of the articles therein.