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Friday, September 30, 2005

Before October

September 2005

Friday 30

Grothendieck's 'Sketch of a Programme' is available here. Besides the mathematics, it is full of fascinating methodological points:

There are people who, faced with this, are content to shrug their shoulders with a disillusioned air and to bet that all this will give rise to nothing, except dreams. They forget, or ignore, that our science, and every science, would amount to little if since its very origins it were not nourished with the dreams and visions of those who devoted themselves to it. (251)

…the moment seems ripe to rewrite a new version, in modern style, of Klein’s classic book on the icosahedron and the other Pythagorean polyhedra. Writing such an expose on regular 2-polyhedra would be a magnificent opportunity for a young researcher to familiarise himself with the geometry of polyhedra as well as their connections with spherical, Euclidean and hyperbolic geometry and with algebraic curves, and with the language and the basic techniques of modern algebraic geometry. Will there be found one, some day, who will seize this opportunity? (255-256)

This situation, like so often already in the history of our science, simply reveals the almost insurmountable inertia of the mind, burdened by a heavy weight of conditioning, which makes it difficult to take a real look at a foundational question, thus at the context in which we live, breathe, work – accepting it, rather, as immutable data. It is certainly this inertia which explains why it took millennia before such childish ideas as that of zero, of a group, of a topological shape found their place in mathematics. It is this again which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom "wildness" is a fatal necessity, rooted in the nature of things. (259)

Let us note in relation to this that the isomorphism classes of compact tame spaces are the same as in the "piecewise linear" theory (which is not, I recall, a tame theory). This is in some sense a rehabilitation of the "Hauptvermutung", which is "false" only because for historical reasons which it would undoubtedly be interesting to determine more precisely, the foundations of topology used to formulate it did not exclude wild phenomena. It need (I hope) not be said that the necessity of developing new foundations for "geometric" topology does not at all exclude the fact that the phenomena in question, like everything else under the sun, have their own reason for being and their own beauty. More adequate foundations would not suppress these phenomena, but would allow us to situate them in a suitable place, like "limiting cases" of phenomena of "true" topology. (276)

It will perhaps be said, not without reason, that all this may be only dreams, which will vanish in smoke as soon as one sets to work on specific examples, or even before, taking into account some known or obvious facts which have escaped me. Indeed, only working out specific examples will make it possible to sift the right from the wrong and to reach the true substance. The only thing in all this which I have no doubt about, is the very necessity of such a foundational work, in other words, the artificiality of the present foundations of topology, and the difficulties which they cause at each step. (262)

…around the age of twelve, I was interned in the concentration camp of Rieucros (near Mende). It is there that I learnt, from another prisoner, Maria, who gave me free private lessons, the definition of the circle. It impressed me by its simplicity and its evidence, whereas the property of "perfect rotundity" of the circle previously had appeared to me as a reality mysterious beyond words. It is at that moment, I believe, that I glimpsed for the first time (without of course formulating it to myself in these terms) the creative power of a "good" mathematical definition, of a formulation which describes the essence. (274)

The sempiternal question "and why all this?" seems to me to have neither more nor less meaning in the case of the anabelian geometry now in the process of birth, than in the case of Galois theory in the time of Galois (or even today, when the question is asked by an overwhelmed student...); the same goes for the commentary which usually accompanies it, namely "all this is very general indeed!". (275)

In any case, this application will have been the occasion for me to write this sketch of a programme, which otherwise would probably never have seen the light of day. I have tried to be brief without being sybilline and also, afterwards, to make it easier reading by the addition of a summary. If in spite of this it still appears rather long for the circumstances, I beg to be excused. It seems short to me for its content, knowing that ten years of work would not be too much to explore even the least of the themes sketched here through to the end (assuming that there is an "end"...), and one hundred years would be little for the richest among them!(273)

Saturday 24

I came across Predrag Cvitanovic's work on birdtracks while researching diagrammatic techniques for mathematical calculation, but he has a much wider range of interests in mathematical physics summarised here. The article 'Search without a plan' is worth reading.

Two pages of interesting links from Yemon Choi here and here

Friday 23

The website for a conference I'm attending The Impact of Categories is up and running. It will be interested to compare notes with others on the way philosophy should engage with category theory. Later in the same week I'll be talking to a mathematics group in Paris VII.

Older material

In July I attended a conference entitled Mathematics and Narrative, involving some prominent mathematicians and 'paramathematicians'. The meeting was featured in the 4 August edition of Nature. A comment of mine is reported, which should become clearer after reading the notes for my contribution. Here are the abstracts of the other talks.

Check out the new discussion group Workshop on Mathematical Practice. It's only just started and will need enthusiastic support to succeed.

I've started a page to discuss reviews of my book.

For a very modest 12.99 sterling you can purchase Mathematical Reasoning and Heuristics edited by Carlo Cellucci and Donald Gillies, a collection of articles in the spirit of the philosophy of real mathematics. Anyone wishing to avoid the overcharging of a publisher like Kluwer might consider King's College Publications.

I have deposited some 'Reflections on Michael Friedman's Dynamics of Reason at the Pittsburgh archive. Abstract: Friedman's rich account of the way the mathematical sciences ideally are transformed affords mathematics a more influential role than is common in the philosophy of science. In this paper I assess Friedman's position and argue that we can improve on it by pursuing further the parallels between mathematics and science. We find a richness to the organisation of mathematics similar to that Friedman finds in physics.

On 6 May I spoke to some mathematicians in Warwick (slides).

Before Easter 2005 I went on a mini-tour of the Western Lake Erie region, giving talks in Wayne State, Toledo, and Case Western. The latter talk was a response to 'The Way We Think: Conceptual Blending and the Mind's Hidden Complexities' by Gilles Fauconnier and Mark Turner (Basic Books, 2003). Mathematicians say things like:

Just as the concept of Lie 2-algebra blends the notions of Lie algebra and category, the concept of 'L-algebra' blends the notions of Lie algebra and chain complex.

Might Fauconnier and Turner's notion of blending throw some light on this way of speaking (talk slides)?

At Wayne and Toledo I discussed a MacIntyrean future for the philosophy of mathematics. notes

From February 2005 I have been based in the Max Planck Institute for Biological Cybernetics in Tübingen attempting to generate a MacIntyrean analysis of the rivalry between Bayesian and non-Bayesian statistical learning theory. Very early results are here.

In 2004 I deposited a couple of articles (Word documents) with the Pittsburgh Philosophy of Science Archive. The first was a paper I gave in Rome in September 2004, discussing how higher-dimensional algebra guides you from important constructions to important constructions. It turns out that starting with propositional logic, you jump to predicate logic and then modal logic. Might this be a clue to why Charles Peirce termed those parts of his theory of existential graphs, his diagrammatic logical theory, alpha (propositional), beta (predicate), gamma (modal)? Talking of rival traditions, the rivalry between higher-dimensional algebra (or higher-dimensional category theory) and set theory to be the basic language of mathematics is probably the one to watch out for over the next decades. The second paper (postscript) starts out from Poincaré's claim that:

...the mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law, just as experimental facts lead us to the knowledge of a physical law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another.

In my book I claimed that the philosophy of mathematics should concern itself with "what leading mathematicians of their day have achieved, how their styles of reasoning evolve, how they justify the course along which they steer their programmes, what constitute obstacles to these programmes, how they come to view a domain as worthy of study and how their ideas shape and are shaped by the concerns of physicists and other scientists". Andrew Arana has observed to me that the description above sounds somewhat elitist. Perhaps 'leading' isn't the best choice of words. I certainly wasn't intending to restrict attention to the top few dozen mathematicians of an era.


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