Sunday, December 12, 2010

On Scope and Difficulty

Literature and philosophy are fortunate in object, deficient in methodology. You can create whatever you like, and it can be quite unexpected and wonderful; but there’s no generally accepted method to check that you aren’t going off the deep end – or the shallow. So we wind up with diffuse or frivolous productions (like much of fiction and metaphysics) or even noxious ones (nihilism, postmodernism, nouveau roman). Attempts to add some formal rigor, some “constraints”, may produce abortions (like sestinas, or leipogrammatic novels), or shrunken offspring of once-great ideals (the nigglings of the Linguistic Turn, vice the Big Bow-Wow issues of Aquinas or Plato).

Physics is fortunate in methodology, restricted in its object – the actual physical world. In principle this might have led to a rather thin, grey science, had the structure of the cosmos been much simpler than that of the mind. (If you study universes other than the actual one, you’re doing mathematical cosmogony, not physics.) And indeed there were times when physicists thought they were already close to a Theory of Everything and could soon shut up shop. In practice, the universe turns out to have even a minimum intricacy that beggars our comprehension and stretches our mathematics.
Psychology, like physics, aspired to be a science, and here its object was no real limitation to the ambitions of the human mind, since it was itself coextensive with the human mind. But methodological bad luck seems to have produced some nonsense – cf. Frederick Crews on psychoanalysis, passim.

Mathematics is supremely fortunate in methodology, and in one sense unrestricted in object: you create the space you explore. Initially much of the space was suggested by aspects of Nature; but already with the hypothetico-deductive method, math had in principle cut its ties. As MacLane puts it, “The axiomatic method is a declaration of independence for mathematics. … A scientific theory can be falsified by factual data, while a mathematical theory cannot be so falsified.” So the actual angles of your triangle of photonic geodesics do not total 180 degrees? No skin off Euclid; go talk with Riemann or Bolyai down the hall.

The result has been an exfoliation of astonishing depth and invention. What has been dreamt up and perfected is far wilder than anything a poet or novelist has ever thought of attempting. Yet as the field becomes fragmented, there may develop a lack of check. It was rumored already at the birth of Relativity that only a dozen men around the world could understand it; how much more so as we clamber ever higher on the shoulders of those who stand on the shoulders of those who stand (or totter) on the shoulders of giants. In principle you could have some subspecialty in which the going theory – the generally acknowledged state of the art – was understood by only one person, its originator – and perhaps not even by him, once the inspiration had passed.  (No mere hypothetical -- cf. the sad confession of Mordell: 
"I have a poor memory, and cannot remember many of my results or proofs, let alone prove them again."

The case is different in literature. There is the class of “poet’s poet”, but it is not thus homotopic to zero. Once the circle of admirers shrinks to just the poet and his mom, we conclude, “He’s a bad poet.” In the case of some theoretical development which, say, only Noam Elkies understands and can vouch for, the rest of us may be uncertain what to think.
There have been dead ends in mathematics, like the “unit fractions” which are the theme of the Rhind Papyrus. Perhaps there are likewise now what may prove to be some overactive meristems.


Difficulty is not itself a measure of depth. Some things are hard in a bad way: we might call it “complication without genuine complexity”. Example: tax law; etiquette at the court of the Pooh-Bah.

Learning fifty languages fluently is hard – doubtless no one’s ever done it – and also broadening (“quot linguae, tot homines”, though the relationship is really sublinear), but after the second or third foreign language, you’re not going appreciably deeper. By contrast, an extra layer of genuine depth is attained when a specifically restricted set is studied by the methods of Indo-Germanistik. Here the languages don’t just sit side by side, but form a parallax, are projected backwards and forwards again according to rules – diachronic rules qualitatively different from the rules of any given languages – of reconstruction.

A major thrust of Chomsky’s movement was to go deeper, much deeper – “deep structure” wasn’t the half of it. A number of American students, ahistorical by inclination in any case, fancied that his focus on ordinary sentences of modern American English freed them from the drudgery of philology, and from considering any work done more recently than last year; and were startled when he published Cartesian Linguistics, with its serious consideration of Port-Royal. And bemused to discover that he is the same N. Chomsky of computer-science Chomsky-hierarchy fame. But for this deep thinker, it did all hang together: the rationalist philosophy, the intense algebraic abstraction (arriving finally at rules for which we no longer have conscious linguistic intuition), and the rich deductive structure of the working theory itself. This latter indeed posed problems for the would-be Chomskyan. No doubt inevitably, given their penchant for depth rather than breadth, many of its luminaries were somewhat deficient in workaday polyglot knowledge; and in accord with their abstract and theoretical bent, were not above manhandling the odd awkward fact, and massaging ill-fitting data along the edges:  Galileo as against Tycho Brahe. Now, if the theory had been superficial, errors might easily have been blacked out, like typos. But tout se tenait. Erroneous data would contribute to a particular formulation of, say, c-command, which itself then participated in myriad new structures and deductions. The student would have to carry forward through all this a mental footnote of doubt, adding little fuzzy-logic indices of partial truth to each succeeding syllogism. It was all hard enough to follow as a true believer; as a skeptic, I found it exhausting.

Contemporary mathematics seems weirdly, wildly deeper than anything else we do. By the time you get (as I have not) to, say, cohomology of sheaves (or its further offspring, which I cannot even name), you have passed through several layers of qualitatively different, structurally rich mentation. The further this development continues, and the more recherch√© the mental capacities required, the fewer people will have even a potential neurological capacity to follow where it leads. Our ancestors having survived the sabretooth, there is no qualitative difference in athletic capacities over mankind; despite the gushing of scribblers over this or that “sports genius”, there is really no analog of a Chomsky or an Elkies – such a person would have to actually be able to fly. But these exotic exaptations of more ethereal cognitive capacities may give rise to truly incommunicable insights, like different species which, with the best will in the world – man and faithful dog, or chimp, or dolphin – fail to communicate to one another anything at all complex. Something of the sort apparently happened some time ago in the field of music.

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Postscript on difficulty: quotations from an author of unusual scope.

Ian Hacking, "Historical Ontology" (2002), p. 32. He engagingly admits, "I do find it very hard to make sense of Descartes, even after reading commentaries, predecessors, and more arcane texts of the same period. The more I make consistent sense of him, the more he seems to me to inhabit an alien universe." Accordingly, those who fancy they understand him, are living in a fools' paradise:
Ian Hacking, "Historical Ontology" (2002), p. 56: "It is paradoxical that Descartes can speak directly to sophomores whose conception of the world seems to be distressingly achronic -- yet the better you know the text, the more you realize that only the most arduous hermeneutical scholarship can make much sense out of it at all."
Ian Hacking, "Historical Ontology" (2002), p. 220. More modesty: "Chunks of Wittgenstein's writings have appeared every year or so since his death in 1951. They have the effect of time-release capsules. This is salutary if you think, as I do, that many of us skim his words and forget."

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