Thinking loudly

Blogs are great places to think aloud, or to think loudly, as Imre Lakatos used to say. I heard one of his former students John Worrall once lament that there was so much more of great substance that Lakatos had said than appears in his surviving writings. It will be interesting to see whether the keeping of a blog effects one's 'proper' writing, whether more of those small but important insights, which the brevity of a blog post can support, sharpened by readers' comments, will be preserved when one writes more formal pieces.

After our discussion following this post, Denis Lomas has written to me to say:

Set theory universalism is a great overarching story of the past century of mathematics. (It's likely part myth.) Of course, it shouldn't override or erase other stories.

This is a useful corrective if I seemed a little dismissive of set theory. I couldn't agree more with this comment, but time stops still for no-one, and now that the set theoretic impetus has led it far from the centre of mathematical concerns, we should keep our eyes open for other grand stories. These we can help articulate by pushing them to extremes, or by forcing them to confront rival stories. The previous blog post was an exercise in the former. If you take the categorification program seriously, basic theory like Euclidean geometry ought to be categorifiable (not the prettiest word). I wish the noncommutative geometry program had a resident philosopher to probe it, and help us understand the tensions between rival accounts (besides Connes, try this blog). Then we might link up to see how Grothendieck's and Connes's conceptions of space face up to each other.

Other fields allow philosophers to do this. In research for the illness book (publication date now pushed back to March 2007), I came across an interesting paper by Kenneth F. Schaffner, 'Assisting immunologists to examine the philosophical foundations and implications of the new theories of tolerance', in Singular selves, Historical issues and contemporary debates in immunology, AM Moulin, A Cambrosio (eds.), Elsevier. pp. 86-93, 2001. As the title suggests, exponents of rival conceptions of immunity, such as self/nonself and danger theorists, were brought together in a debate run by Schaffner. One of the advantages of having a philosopher well-informed in the history of immunology to host the discussions is that they can pick up on a loose use of a term of, say, Popper or Kuhn. Any philosopher who has studied earlier scientific debates will understand what can be disappointing about them, and so try to steer contemporary ones in productive directions. What might have been achieved if someone had been there to mediate relations between Hilbert and Frege or between Hilbert and Brouwer?

n-categories: logic and geometry

The 2006 Reith Lectures are being aired at the moment. This year they're on the importance of music, and are written and delivered by the musician Daniel Barenboim. I've heard only the first so far, in which there's a wonderful passage where he explains how he came to understand in terms of a badly timed piece of music why the Oslo peace process would never work. The current domination of the visual over the aural, and the anaestheticisation of the later through exposure to muzak are other themes he touches on, and are to be developed in later lectures. In sum, we are letting down our children by failing to help them to understand and make serious music, and so learn to integrate discipline and passion.

This seems reminiscent of the teachings of Plato and Aristotle, who had plenty to say about the place of music in education. Later in a person's education they must meet mathematics, something Plato in particular stresses is vital for the formation of political leaders. I hope that a mathematician will be chosen as a future Reith lecturer to talk to us about the transformational role of mathematics. Perhaps John Baez would be a good choice. Recently he has been delivering some Lectures on n-categories and cohomology in Chicago. These notes include an appendix by Michael Shulman which moves the program on a little in an interesting direction. It's fascinating to see how logic and topology are so thoroughly interwoven in the n-categories setting. For something else on this see the end of Categorification as a Heuristic Device, where a link to modal logic is touched upon. (I know there's a mistake in that equation explaining permutations as sets of cycles.)

Here's a brief exchange I had with Baez in recent days:

Hi,

Here's a thought: gaining new insights by categorifying the very simplest entities seems a good way to bring in the punters. You've treated various kinds of number, natural, rational, etc. [E.g., From finite sets to Feynman diagrams] Now, if you ask anyone what else they first learned at school, they'll say 2-dimensional Euclidean geometry (if they're old enough to have been taught properly). Where, they may ask, are categorified lines and circles? If all interesting equations are lies, what of Pythagoras' theorem? Or, when we say the intersection of any pair of altitudes of a triangle is the *same* point as that of any other, is there room for weakening?

I suppose you might give two responses:

1) Euclidean geometry although it came first is actually a complicated affair. First you need to categorify a stripped down 'geometry' such as differential topology.

2) OK. 2d-Euclidean space is a homogeneous space, the points corresponding to cosets of the quotient of the Lie group of Euclidean transformations by the stabilizer of a point. All we need is a Lie 2-group version.

Either way what prevents a Erlangen program for 2-groups?

Best, David


Hi -

> Where, they may ask, are categorified lines and
> circles?

Interesting idea; one could take it in various directions, I suppose.

> 1) Euclidean geometry although it came first is actually a
> complicated affair. First you need to categorify a stripped
> down 'geometry' such as differential topology.

Mainly you need to see where the categories are, so you can see if there are interesting n-categories lurking beneath them.

> 2) OK. 2d-Euclidean space is a homogeneous space, the
> points corresponding to cosets of the quotient of the Lie
> group of Euclidean transformations by the stabilizer of a
> point. All we need is a Lie 2-group version.

> Either way what prevents a Erlangen program for 2-groups?

Nothing! Especially since the Erlangen program is just the flip side of Galois theory:

http://math.ucr.edu/home/baez/namboodiri/nam1.pdf

(see especially the slide about the icosahedron), and Galois theory has already been n-categorified to powerful and still growing effect.

But you're right - nobody seems to have thought hard about Klein geometry with Lie 2-groups (or higher) replacing Lie groups. Somehow people have skipped straight to categorifying principal bundles, even though principal bundles are a stripped-down way of thinking about Cartan geometries, which generalize Klein geometries! Sometimes ontogeny fails to recapitulate phylogeny. So, maybe the "punters" should be handed a nice specific Lie 2-group, some 2-spaces on which it acts, and be asked to study the "incidence relations" between these figures. Incidence geometry could be given a whole new lease on life!

Best, jb

In a pleasanter world I'd be funded to think longer about such things. For those with the leisure time, you can read about incidence geometry at TWF 178, which treats incidence relations in projective geometry in terms of the Dynkin diagrams A_n. Perhaps we should start with projective rather than Euclidean geometry. Is there an obvious candiate for a Lie 2-group one step up from SL(n, C)? What then is projective 2-geometry? What are Dynkin 2-diagrams?

Or doing things axiomatically, perhaps we can categorify the axioms of projective geometry, such as those for the projective plane, taken from week 145:

A) Given two distinct points, there exists a unique line that both points lie on.
B) Given two distinct lines, there exists a unique point that lies on both lines.
C) There exist four points, no three of which lie on the same line.
D) There exist four lines, no three of which have the same point lying on them.

Three announcements

The conference organisers of Perspectives on Mathematical Practices 2007 are inviting abstracts.

What is Category Theory? has now been published by Polimetrica.

The way this blog works, if anyone comments on a post not appearing in the list of previous posts, I will be alerted but it will not appear in recent comments. If I reply to that comment, no-one but me will be alerted.

More about MacIntyre

In the previous post I mentioned how MacIntyre’s philosophy was considerably more integrated than is the norm in contemporary English-language philosophy, and that it is integrated around ethics. ‘Epistemological Crises, Dramatic Narrative and the Philosophy of Science’ makes an analogy between individuals facing a moral crisis and those engaged in communal forms of rational enquiry facing an intellectual crisis. It opens by considering the crisis Hamlet confronts on his return to the Danish court, and what it would mean to resolve this crisis:

When an epistemological crisis is resolved, it is by the construction of a new narrative which enables the agent to understand both how he or she could intelligibly have held his or her original beliefs and how he or she could have been so drastically misled by them. The narrative in terms of which he or she at first understood and ordered experiences is itself now made into the subject of an enlarged narrative. The agent has come to understand how the criteria of truth and understanding must be reformulated. He has had to become epistemologically self-conscious and at a certain point he may have come to acknowledge two conclusions: the first is that his new forms of understanding may themselves in turn come to be put in question at any time; the second is that, because in such crises the criteria of truth, intelligibility, and rationality may always themselves be put in question – as they are in Hamlet – we are never in a position to claim that now we possess the truth or now we are fully rational. The most that we can claim is that this is the best account which anyone has been able to give so far, and that our beliefs about what the marks of "a best account so far" are will themselves change in what are at present unpredictable ways. (p. 455)

MacIntyre continues by contrasting Hamlet with Jane Austen’s Emma, in which the eponymous heroine comes to realise the error of her interpretation of the social position of her friend Harriet, but does so only to arrive at what she conceives to be the right interpretation, that of Mr. Knightly. No suggestion is given in the book that this new view may later find itself challenged.

Philosophers have customarily been Emmas and not Hamlets, except that in one respect they have often been even less perceptive than Emma. For Emma it becomes clear that her movement towards the truth necessarily had a moral dimension. Neither Plato nor Kant would have demurred. But the history of epistemology, like the history of ethics itself, is usually written as though it were not a moral narrative, that is, in fact as though it were not a narrative. For narrative requires an evaluative framework in which good or bad character help to produce unfortunate or happy outcomes. (p. 456).

This was written in 1977 and seems to be much influenced by debates which took place in the philosophy of science in the late 60s and early 70s between Popper, Polanyi, Kuhn, Lakatos, and Feyerabend. This is the kind of connectivity that interests me. Not that science through its theories – genetics, cosmology, etc. – has a bearing on philosophical theses, but that a moral philosopher may learn from philosophers of science, as may a philosopher of mathematics from moral philosophers. Let’s see what we can glean from ‘Relativism, Power and Philosophy’ (details in previous post).

MacIntyre poses the problem of someone in Ireland in 1700 who is able to inhabit both the community of indigenous Irish, and also the English community of plantation owners. To one group it must speak of Doire Colmcille - St. Columba’s oak grove – which "names – embodies a communal intention of naming – a place with a continuous identity ever since it became in fact St. Columba’s oak grove in 546". To the other it refers to Londonderry which "names a settlement made only in the seventeenth century and is a name whose use presupposes the legitimacy of that settlement and of the English language to name it" (p. 7). Londonderry was a plantation enforced on the Irish population by the English with the foreign concept of individual property rights – "what is from one point of view an original act of acquisition, of what had so far belonged to nobody and therefore of what remained available to become only now someone’s private property, will be from the other point of view the illegitimate seizure of what had so far belonged to nobody because it is what cannot ever be made into private property – for example, common land." (8)

MacIntyre’s point is that these two communities hold deeply incommensurable views about justice and the rationality of actions, and that someone coming to see this, and failing to find a neutral position from which to judge them, might well lapse into a form of relativism. One solution he rejects is that with modern languages we are in a happier position. To us in the early twenty-first century, used to reading accounts of people from different times and places, with all manner of practices and belief systems, we believe we can readily understand each side, and understand their differences. The Irish of 1700 didn’t have the concept of ‘right’. The English didn’t understand the sacred importance of the spot for the Irish. But we can congratulate ourselves today for having a language into which we can render the positions of the two peoples. However, MacIntyre then undermines our triumph by pointing out the obverse of possessing this flexible language: We can’t rationally settle major disagreements, since we don't share sufficient by way of background.

For MacIntyre, what is notable about modern languages is the lack of reference to canonical texts, and the ease of inter-translation suggestive of a presuppositionlessness. By contrast,

The Attic Greek of the fifth and fourth centuries, the Latin of the twelfth to fourteenth centuries, the English, French, German and Latin of the seventeenth and eighteenth centuries were each of them neither as relatively presuppositionless in respect of key beliefs as the languages of modernity were to become, nor as closely tied in their use to the presuppositions of one single closely knit set of beliefs as some premodern languages are and have been… Such languages-in-use, we may note, have a wide enough range of canonical texts to provide to some degree alternative and rival modes of justification, but a narrow enough range so that the debates between these modes is focused and determinate. (p. 18)

It is in languages of this kind that we can expect debates to be eventually decisive.

Now let us try to relate what we have seen so far to mathematics. Instead of someone torn between the two communities of early eighteenth century Ireland, imagine a Chinese mathematician, thoroughly versed in the Nine Chapters and other classic texts, who in 1600 comes to understand Euclidean geometry from Jesuit missionaries. They would surely experience incommensurability as they attempted to decide for themselves the superiority of one of these very different conceptions of mathematics. (See Joseph W. Dauben, Ancient Chinese mathematics: The Jiu Zhang Suan Shu vs. Euclid’s Elements. Aspects of proof and the linguistic limits of knowledge, International Journal of Engineering Science 36 (1998) 1339-1359. See also this paper.) As a mathematical parallel to what MacIntyre conceives of as a modern language the obvious choice is set theory. Today we could render each variety of mathematics as set theory. But such flexibility comes at a cost. It can’t help us choose in which direction to proceed. Rewriting mathematics within set theory we can say that something has a correct proof or that something is calculated correctly, but we can’t say that it was worth proving or calculating in the first place.

For this we need stories, and, fortunately, the condition of modern mathematics is not one of presuppositionless set theoretic universalism, but rather is scored across by countless stories, which in printed form are easiest to locate in book reviews, such as those of the Bulletin of the American Mathematical Society, or in articles in the AMS Notices. Count the number of times Atiyah uses the words ‘story’ or ‘stories’ in his Mathematics in the 20th Century, Bulletin of the London Mathematical Society 34(1), 1-15, 2002. Turning your web browser to the Arxiv every morning most closely resembles tuning into a daily soap opera. It's not easy to discern the story lines from this tapestry of articles, but if you do succeed, you find that individual stories range through: daring quests for single gems using whatever resources can be found; Hausmannian reconstructions of higgledy-piggledy slums; beautiful irrigation projects using water from distant streams; attempted take-overs, and resistance to subsumption within a broader theory.

But with all these subplots is there any danger that mathematics itself may take on the attributes of the modern condition of endless, unresolvable debates about which direction to take? Not if the traditions represented by these stories are carefully maintained and passed on, and their advocates remain open to what other traditions can offer them. I mentioned several months ago that in a Clay Mathematics Institute interview, Terence Tao speaks of the importance of "being exposed to other philosophies of research, of exposition, and so forth". The danger to ward against is dissipation in a kind of heat death.

"Epistemological Crises, Dramatic Narrative and the Philosophy of Science," The Monist, 60 (1977), 453-72. Reprinted in Paradigms and Revolutions: Appraisals and Applications of Thomas Kuhn's Philosophy of Science, Gary Cutting, ed. (Notre Dame: University of Notre Dame Press, 1980) pp. 54-74. For excerpt see previous post.

Philosophy and politics

The difficulties faced by philosophers of mathematics of my persuasion stem, I think, from our reluctance to fit in with the dominant analytic tradition, while not being readily classifiable in that catch-all category ‘continental philosophy’. If mathematics assisted at the birth of analytic philosophy, the latter now grown up doesn’t look favourably upon anyone thinking about mathematics in terms other than its own. Consider the questions analytic philosophy asks of mathematics today: If we use mathematics in our science, are we committed to the existence of mathematical entities? What are mathematical entities and how do we come to know about them? Are the natural numbers anything more than a structure? What is a structure? Like-minded colleagues and I are not happy to place ourselves in a framework that would take these to be the primary questions to ask of the discipline Harvard mathematician Barry Mazur calls "mankind’s longest conversation". Where perhaps we have most grievously failed is in properly articulating a shared philosophical stance which would require the study of what we take to be more important matters, rather relying too heavily on what we imagine to be obvious, that mathematical thought through the ages should be our central concern.

Recently I have turned to the writings of Alasdair MacIntyre to help with this articulation. Here I have selected three quotations:

It thus turns out that, just as the achievements of the natural sciences are in the end to be judged in terms of achievements of the history of those sciences, so the achievements of philosophy are in the end to be judged in terms of the achievements of the history of philosophy. The history of philosophy is on this view that part of philosophy which is sovereign over the rest of the discipline. This is a conclusion which will seem paradoxical to some and unwelcome to many. But it has at least one merit: it is not original. Vico, Hegel and Collingwood all at various points come very close to theses remarkably, and indeed not at all by coincidence, similar. (The relationship of philosophy to its past, 47)

(MacIntyre acknowledge the possibility that such a history will have to tell of centuries long reversals in the fortunes of a tradition of enquiry.)

Greek thought, like Greek practice, understands morals-and-politics as a unified object of enquiry; modern moral theory distinguishes itself both from political philosophy and even more sharply from political science. So the academic division of labour allows us to pretend that our pupils can understand Aristotle’s Ethics without reading the Politics and vice versa. Greek moral thought make central to its concerns issue of the nature of human psychology which are as alien to characteristically modern moral philosophy as are some of its central issues, the fact/value distinction and the relationship of morality to utility, for example, to Plato and Aristotle. (The relationship of philosophy to its past, 38-39)

…philosophy, like all other institutionalized human activities, is a milieu of conflict. And the conflicts of philosophy stand in a number of often complex and often indirect relationships to a variety of other conflicts. The complexity, the indirectness and the variety all help to conceal from us that even the more abstract and technical issues of our discipline – issues concerning naming, reference, truth and translatability – may on occasion be as crucial in their political or social implications as are theories of the social contract or of natural right. The former no less than the latter have implications for the nature and limitations of rationality in the arenas of political society. All philosophy, one way or another, is political philosophy. (Relativism, Power and Philosophy, 12-13)

MacIntyre is certainly no marginal figure in Anglo-American philosophy. Indeed, the last of these quotations comes from a lecture delivered during his presidency of the American Philosophical Association. But while these quotations hint at an integrated philosophy which is political to its core, it is very unlikely that you will hear his name outside ethics classes. For instance, I very much doubt students electing for lectures in the philosophy of science will hear of ‘Epistemological Crises, Dramatic Narratives and the Philosophy of Science’ (excerpt from a soon to be published collection of essays). Such transgressions of boundaries are not encouraged. Despite the ructions mentioned here, the dominant post-Quinean analytic tradition is generally happy to cede some departmental space to allow ‘Continent Philosophy’ to be taught. Although many ‘continental’ philosophers have, like MacIntyre, had very different conceptions of what philosophy should be like, generally a more closely integrated and political discipline, such courses tend to be self-contained, and don’t spill over to challenge analytic philosophy’s tidy way of treating ethics, aesthetics, metaphysics, mind, language, knowledge, etc. as largely disjoint subjects. ‘Continental’ approaches to science are generally limited to certain kind of history of science programme, which enjoy exposing the ebb and flow of power in scientific communities.

Commentators (this book and this book) have noted how peculiarly apolitical analytic philosophy in the second half of the 20th century has been. If we think of the activism of the Vienna Circle, this clearly stands in need of explanation. Whether McCarthyism is responsible, or an increasing desire to make philosophy a professional academic discipline with a standard set of problem areas, what is beyond dispute is that contemporary philosophy of mathematics has become an insular affair without a whiff of politics associated to it. So, were we to adopt a MacIntyrean stance, clearly much would have to change. My understanding of his ideas suggests that epistemology and metaphysics would have to be treated very differently. Instead of the epistemology of the individual - ‘How do I know that 2 + 2 = 4 ?’, it would develop an account of rational traditions of mathematical enquiry – ‘Why was the mathematical community justified in adopting, and making precise, Riemann’s conception of a manifold?’. Instead of a timeless metaphysics - ‘What is a number?’, it would question the underlying presuppositions of the discipline - ‘Are there tensions between the conceptions of space of a Connes and a Grothendieck?’. Its gift to other branches of philosophy would be a clearer understanding of what participation in a form of rational enquiry involves, and of what it means for theoretical and practical advances to be rationally justified.

And this gives us a clue as to how the philosophy of mathematics may be related to politics and ethics. While the political role of Plato’s theory of Forms was apparent to Popper, and while Plato managed to discuss mathematics in a political/ethical context in the Republic, can we, who surely do not recommend that our future rulers learn mathematics for a decade, do likewise? Well, what if we consider our lives to be formed of a series of interlocking practices, including the very important ones of maintaining a thriving family and community? Then we might learn from a practice with the pedigree of mathematics – mankind’s longest conversation – about the necessity of certain intellectual and moral virtues. Saying so, we must not remain blind to its faults. While I think it is fair to say that mathematics is flourishing at present, this is not because of a complete absence of institutional irrationality. Its most obvious failing is that exposition is not justly rewarded. Lakatos realised this in the 1960s, and so have mathematicians such as Rota and Thurston more recently. This is troubling. If the mathematical community cannot conduct its affairs both rationally and justly, it seems unlikely that other communities will fare better. In the bluntest terms, unless irrationality and injustice are remediable, can we imagine that humankind will be able to learn to govern itself well enough to avert ecological catastrophe?

To end on an encouraging note, read the Wednesday, February 15 entry of this blog to see that the spirit of Plato lives on: "Using the word job in a pure math course is nothing but mortifying."

References:
"The Relationship of Philosophy to its Past," in Philosophy in History, Richard Rorty, J.B. Schneewind and Quentin Skinner, eds. (Cambridge: Cambridge University Press, 1984) pp. 31-48.

"Relativism, Power and Philosophy," in Proceedings and Addresses of the American Philosophical Association (Newark, Delaware: APA, 1985) pp. 5-22. Reprinted in After Philosophy: End or Transformation, Kenneth Baynes, James Bohman and Thomas McCarthy, eds. (Cambridge: MIT Press, 1987) pp. 385-411. Available on JSTOR.

Natural distributions

I've been thinking of late about what we can say about the probability distributions we might expect the world to throw at us. This is very important in machine learning. An algorithm which, knowingly or not, 'expected' to meet with certain specific kinds of distribution which in fact did occur is clearly at an advantage. But what kinds of consideration are important here? The universality of the Gaussian normal distribution as made explicit in the Central Limit Theorem might provide a pointer. Are there other distributions which arise robustly in different situations?

In Universality for mathematical and physical systems, Percy Deift describes distributions which govern data as far removed as the spacings between parked cars and the (scaled) spacings between the zeros of the Riemann zeta function. These distributions are studied in a field known as random matrix theory, which considers, for example, the eigenvalues of a random orthogonal matrix. Answers are beginning to emerge as to why these distributions are encountered, which parallel the three components of the central limit theorem: a statistical component (take independent, identically distributed random variables, centered and scaled), an algebraic component (add the variables), and an analytic component (take the limit in distribution as n → infinity). He signs off with the intriguing comment:

Our final comment/speculation is on the space D, say, of probability distributions. A priori, D is just a set without any "topography". But we know at least one interesting point on D, the Gaussian distribution F_G. By the central limit theorem, it lies in a "valley", and nearby distributions are drawn towards it. What we seem to be learning is that there are other interesting distributions, like F₁ or F₂, etc., which also lie in "valleys" and draw nearby distributions in towards them. This suggests that we equip D with some natural topological and Riemannian structure, and study the properties of D as a manifold per se.

My soul’s an amphicheiral knot

This is the first line of a poem written by Maxwell, in the style of Shelley, to tease gently his fellow physicist Peter Guthrie Tait's efforts to build a theology on the basis of a knot theoretic account of atoms. I wrote a chapter (figures omitted) on this theologically-motivated program of research, in which Tait was joined by Thomson (Kelvin) and an amateur mathematician, the Reverend Thomas Penyngton Kirkman. I intended this as part of a popular book on mathematicians. However, since this project is on hold, I thought I'd make it available.

Update: I've scanned in the figures in two 1.8 MB tif files here and here.

Update: Here's the first stanza of Maxwell's poem in full:

My soul's an amphicheiral knot
Upon a liquid vortex wrought
By Intellect in the Unseen residing,
While thou dost like a convict sit
With marlinspike untwisting it
Only to find my knottiness abiding,
Since all the tools for my untying
In four-dimensioned space are lying,
Where playful fancy intersperses
Whole avenues of universes,
Where Klein and Clifford fill the void
With one unbounded, finite homoloid,
Whereby the infinite is hopelessly destroyed.

And a Homage to James Clerk Maxwell, which mentions the verse, by someone who taught me electromagnetism.