The first page of text in Morris Kline’s Mathematics and Western Culture quotes Descartes:
…..I was not surprised that many people, even of talent and scholarship, after glancing at these sciences, have either given them up as being empty and childish or, taking them to be very difficult and intricate, been deterred at the very outset from learning them…..But when I afterwards bethought myself how it could be that the earliest pioneers of Philosophy in bygone ages refused to admit to the study of wisdom any one who was not versed in Mathematics….I was confirmed in my suspicion that they had knowledge of a species of Mathematics very different from that which passes current in our time.
And ours! Part of what motivated me to create this blog is my conviction that we haven’t given mathematics the opportunity to reflect, as it does, something about ourselves, our biology and even rudimentary aspects of life itself.
Although the question of whether mathematics is something we’ve constructed or discovered can seem stale, like an overused cliche, it is still debated. And in this context I’ve wondered about generalities. Generalities are the way math ideas grow into specialized complex systems of thought – symbols for quantity grew into the real number system, an analysis of three-dimensional space grew into, among other things, infinite-dimensional vector spaces. But what does it mean to find a generality? It’s not exactly constructed. And it doesn’t only happen in mathematics. Every time we name an aggregate of objects, like trees, or dogs, insects, molecules, cells, subatomic particles, we have found some sameness among a multitude of particulars.
Today the work of Semir Zeki again got my attention again. Zeki is a neurobiologist who has worked extensively on the visual brain. He is currently a professor of Neuroesthetics at University College London and founder of the Institute of Neuroesthetics. (For some reason, word press won’t link the page for the Institute at http://neuroesthetics.org) The Institute means to establish and investigate the way art is an extension of the brain’s inherent function, which Zeki would say is to know something when faced, in every sense, with permanent change. The brain (or the body) accomplishes this with its capacity to abstract or “to emphasize the general at the expense of the particular.”
Within the Institute’s statement is the following claim:
Abstraction, which arguably is a characteristic of every one of the many different visual areas of the brain, frees the brain from enslavement to the particular and from the imperfections of the memory system.
To describe what he means by abstraction, Zeki often uses what is called orientation selectivity, the talent of many cells in the primary visual cortex to respond to lines of a certain orientation, less well to other lines, and not at all to lines orthogonal (or perpendicular) to their preferred orientation. The cells’ responses are unaffected by any particulars about the line (like color or width or context). This is a generality at the cellular level! – just a hint of what the body does.
Mathematics investigates what appear to be more conscious or more willful abstractions. Yet when scrutinized, we find more hidden within the abstraction than we started with. And these discoveries are what have made mathematics into the powerful and bewildering body of knowledge it has become.
On their web page the Institute quotes Paul Klee:
Art does not represent the visual world, it makes things visible.
Is this not what mathematics does? My interest in the biology of mathematics does not come from a reductionist tendency but more from, as Hermann Weyl says in a 1932 lecture, a desire to find the world looking “more and more as an open one, as a world not closed but pointing beyond itself.” I hope to see us break our ‘subjective’ vs. ‘objective’ habits of thought, and erase some of the boundaries we have drawn between ourselves everything else.
Zeki doesn’t address the abstractions in mathematics, but he found himself in the unexpected company of some string theorists and he wrote about the questions that came to mind in one of his blogs.
I’ll end this with another quote from Kline:
Mathematics has brought life to the dry bones of disconnected facts and, acting as connective tissue, has bound series of detached observations into bodies of science.
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