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Monday, July 31, 2006

Feeling the master's superiority

There can be but two reasons for a philosopher to spend time, as I have, rethinking fundamental concepts of geometry. The first of these derives from the thesis that philosophy may legitimately contribute to the formulation of certain fundamental concepts, which include those of geometry. The second derives from the Collingwoodian thesis that to do the philosophy of a discipline well one must be 'thoroughly at home' with it. As for the first, the list of such legitimate concepts, as conceived by contemporary analytic philosophy, includes causation, probability, necessity, time, consciousness, identity, number, set, and yet would appear to exclude mathematical concepts such as (mathematical) space, point, dimension. I have no objection to the drawing of a distinction - it is hardly within the philosopher's brief to explore the nature of being igneous or of being crystalline - , but I have yet to see a principled way of separating the legitimate from the illegitimate. Let me, here, pursue the second reason.

As I observe in my book,
For Collingwood, ... a capacity to experience the force of the absolute presuppositions of the contemporary form of the discipline about which one is philosophising is vital. While describing which qualities someone should possess to be able to answer the questions of philosophy of history, he remarks acidly that:
No one, for example, is likely to answer them worse than an Oxford philosopher, who, having read Greats in his youth, was once a student of history and thinks that this youthful experience of historical thinking entitles him to say what history is, what it is about, how it proceeds, and what it is for. (Collingwood 1946: 8)
A similar conclusion could be formulated for philosophy of mathematics, and indeed Kant is praised for dealing with the presuppositions of mathematics 'rather briefly' for 'he was not very much of a mathematician; and no philosopher can acquit himself with credit in philosophizing at length about a region of experience in which he is not very thoroughly at home' (Collingwood 1940: 240). Returning to history, he continues:
An historian who has never worked much at philosophy will probably answer our four questions in a more intelligent and valuable way than a philosopher who has never worked much at history. (Collingwood 1946: 9)
Evidence for the equivalent statement about mathematics is provided by the very many important contributions made by mathematicians thinking about their discipline, several of which I shall lean on in the course of this book. (pp. 17-18)

This Collingwood, Robin George, Waynflete Professor of Metaphysics at Oxford, was the son of William Gershom Collingwood, who worked with John Ruskin at Brantwood, the latter's home on the shores of Coniston Water. There could have been little direct personal influence on the young Collingwood, Ruskin dying in 1900 shortly before he was 11, and suffering greatly from mental illness in his final years, but most likely his father, who completed a biography of Ruskin in 1893, provided the necessary immersion in Ruskinian principles.

In earlier posts (2 May & 5 May), I mentioned Ruskin's Unto This Last. I am currently reading Ruskin's autobiography Praeterita, where we read:

"Mostly a quiet stream there, through the bogs, with only a bit of step or tumble a foot or two high on occasion; above which I was able practically to ascertain for myself the exact power of level water in a current at the top of a fall. I need not say that on the Cumberland and Swiss lakes, and within and without the Lido, I had learned by this time how to manage a boat - an extremely different thing, be it observed, from steering one in a race; and the little two-foot steps of Tummel were, for scientific purposes, as good as falls twenty or two hundred feet high. I found that I could put the stern of my boat full six inches into the air over the top of one of these little falls, and hold it there, with very short sculls, against the level [Distinguish carefully between this and a sloping rapid.] stream, with perfect ease for any time I liked; and any child of ten years old may do the same. The nonsense written about the terror of feeling streams quicken as they approach a mill weir is in a high degree dangerous, in making giddy water-parties lose their presence of mind if any such chance take them unawares. And (to get this needful bit of brag, and others connected with it, out of the way at once), I have to say that half my power of ascertaining facts of any kind connected with the arts, is in my stern habit of doing the thing with my own hands till I know its difficulty; and though I have no time nor wish to acquire showy skill in anything, I make myself clear as to what the skill means, and is. Thus, when I had to direct road-making at Oxford, I sate, myself,with an iron-masked stone-breaker, on his heap, to break stones beside the London road, just under Iffley Hill, till I knew how to advise my too impetuous pupils to effect their purposes in that matter, instead of breaking the heads of their hammers off, (a serious item in our daily expenses). I learned from an Irish street crossing-sweeper what he could teach me of sweeping; but found myself in that matter nearly his match, from my boy-gardening; and again and again I swept bits of St Giles' foot-pavements, showing my corps of subordinates how to finish into depths of gutter. I worked with a carpenter until I could take an even shaving six feet long off a board; and painted enough with properly and delightfully soppy green paint to feel the master's superiority in the use of a blunt brush."
How much more important for Ruskin, then, to devote years to drawing, so as to be able to write intelligently about art.


Anonymous said...

David --

Is contemporary geometry about concepts such as "space, point, dimension"? Since the work of Mario Pieri in 1895, taken up by David Hilbert in 1899, geometry has been "about" the consequences of a collection of abstract axioms. Although possibly motivated by intuitions of spaces, points, dimensions, etc, these axioms and the resulting theory are most emphatically not ABOUT such concepts. You will recall the famous statement of Hilbert in a bar, that his geometric axioms could apply to the furniture they were sitting on, as well as to points and lines. (Something Gottlob Frege, for all his other genius, never understood.)

-- Peter McB.

July 31, 2006 3:10 PM  
david said...

Mathematicians are still reasoning about space, point and dimension. E.g., in Cartier's paper he entitles section 3 'On the nature of space and its points', and on p. 404 says "We begin to suspect that not all points are alike - there are several species of monads."

For dimension, see Manin's The Notion of Dimension in Geometry and Algebra.

July 31, 2006 6:29 PM  
david said...

Perhaps I should mention another motivation for the geometric exercise, which is to put to the test the capacity of the higher-dimensional algebra program to reformulate mathematics, see e.g. here, here, and here.

August 09, 2006 10:42 AM  

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