Categories in physics and the Trivium

I mentioned last time that our understanding of even simple mathematical notions such as the natural numbers or integers is far from complete ("The sphere spectrum is the true integers"). More evidence for this comes in this paper, 'The Dirichlet Hopf algebra of arithmetics', ArXiv: math-ph/0511079, which studies the combinatorial intricacy of co-addition and co-multiplication operations, dual to addition and multiplication, in the natural numbers. Dual operations correspond to ways of decomposing entities. Amongst other goals, Fauser and Jarvis are trying to understand why number theoretic terms, such as values of the Riemann zeta function, appear in work on renormalization in quantum field theory, such as those found by Kreimer and Connes. There appears to be a shared combinatorial structure between these two fields, which was treated in this workshop.

As with the work of Baez and Coecke (25 October), the idea is afoot to strip quantum theory down to its combinatorial bones. Fauser and Jarvis argue

The appearance of complex numbers, an algebraically closed field, is often argued to be a key feature, e.g. for producing interference effects, but we doubt this. A ‘phase’ may be modelled by a finite cyclic group also. The expectation values can be obtained in a topos theoretic setting using more general truth objects and therewith related subobject classifiers. This will be explored elsewhere. We think that the present work shows at least, that for the identification of the algebraic structures involved in quantum mechanics and quantum field theory, a characteristic free approach to quantum mechanics would be of great help. Especially the interpretation of combinatorial factors, normalizations etc. would benefit from such a view, even if the complex number field is finally adopted. The appearance of number theoretic functions in renormalization supports this point of view. (p. 41)

The authors also mention that: "The present structure is more compatible with a 2-category picture, but we refrained here from exploring this in a first exposition." (p. 3)

The Hopf structures the authors are studying are closely related to tensor categories, which seem to be cropping up all over mathematical physics. Perhaps Levin and Wen know why. In their paper, 'A unification of light and electrons through string-net condensation in spin models', they claim:

In a crystal, atoms organize themselves into a very regular pattern - a lattice. Since different lattice structures are distinguished by their symmetries, we can use group theory to classify all the 230 crystals in three dimensions. In much the same way, string-net condensed states are highly structured. The different possible structures are described by solutions to (3). Tensor category theory provides a classification of the solutions of (3), which leads to a classification of string-net condensates. Thus tensor category theory is the underlying mathematical framework for understanding string-net condensed phases, just as group theory is for symmetry breaking phases.

Someone who has done interesting things with tensor categories is Michael Müger. You can read about why one of his papers was a fast breaking paper. He conjectures that "all existing (and future) applications of subfactor theory to low-dimensional topology ‘factor through category theory’—as is by now well known for the knot invariants of Jones and HOMFLY" (p. 154). (Many of the interviews for fast breaking papers across a wide range of disciplines are interesting.)

On a different subject, I came across this interesting advocacy of a return to a mediaeval syllabus by Dorothy Sayers. The word 'trivial' derives from the term 'Trivium', the first part of the syllabus, covering Grammar, Dialectic, and Rhetoric, but the Trivium was anything but trivial. Is this grounding what is needed to lay the foundations for the life of learning Alasdair MacIntyre wishes us to enjoy:

...we have to stop thinking of teaching and learning as activities restricted to specialized, compartmentalized area of life within schools, colleges and universities. Of course, schools, colleges and universities have their own highly specific tasks, but these tasks need to be defined in terms of their contribution to lifelong learning and teaching, most of it carried out in nonscholastic and nonacademic contexts. We need, that is, to think of formal academic education not primarily as a preparation for something else, a life of work, which terminates when that life of work begins, but rather as itself the beginning of, and the providing of skills, virtues and resources for, a lifelong education directed toward and informed by the achievement of the good. We need, for example, to teach our students to read, so that they go on reading throughout their lives. We need to make such reading a way of illuminating their social relationships, so that their familial and communal lives continue to be enriched by a stock of common reading. We need to rethink the time-scale of education so that we make one of the tests of the adequacy of what we teach now the answer to the question: "What will our students be reading when they are forty, sixty, seventy-five?" and to accept that if they are not then returning to the Republic and the Confessions, to Don Quixote and Dostoievski and Borges, we will have failed as teachers. (The Privatization of Good: An Inaugural Lecture 359, The Review of Politic 52(3), 344-377, 1990, available on JSTOR)

Finally, a good 'Bibliography for Philosophical Materials Pertaining to Mathematics and Proof'.

R. G. Collingwood and the Historical Stance

I'm spending a lot of my time at the moment wondering what Statistical Learning Theory (see John Langford's informative blog Machine Learning) can teach philosophy. Here's a small sample of possibilities: (1) There's more to learning than finding the generative mechanism producing observed data; (2) Choosing a hypothesis and then testing it is not the only way to gain confidence in its accuracy; (3) Complicated averages of hypotheses are often the more accurate; (4) It is possible to factor quantitively into the expected accuracy of a hypothesis how well other hypotheses in the class under consideration perform during training.

Parts of philosophy concerned with empirical inference can expect to be transformed by integrating these ideas. Here, as elsewhere, this should give us pause for thought. What to do when science starts to deal with traditionally philosophical topics? Judea Pearl (Causality, CUP, 2000) integrates some of the research of philosophers (e.g., David Lewis) on causality with ideas from statistics and graphical modelling into a powerful calculus for treating causal predictions and counterfactuals. What next? Must the philosopher wishing to continue thinking about causality take on board Pearl's work? Consider other traditional philosophical topics, such as time, which have been treated by other disciplines for longer. Must a philosopher who wants to discuss time keep abreast of the latest physical theories, the latest neuropsychology of the perception of time, or the latest historical research on the transformation of the time of everyday life by, say, the time-keeping of the monasteries or the advent of national railways?

It seems to me that we need a conception of philosophy as inseparable from the historical study of forms of thinking and acting, which can be expected to continue to develop. A philosopher who worked out such a position was the Waynflete Professor of Metaphysical Philosophy at Oxford Univerity, R. G. Collingwood (1889-1943). Here he is on logic:

Logic an Historical Science
The aim of logic is to expound the principles of valid thought. It is idly fancied that validity in thought is at all times one and the same, no matter how people are at various times actually in the habit of thinking; and that in consequence the truths which it is logic's business to discover are eternal truths. But all that any logician has ever done, or tried to do, is to expound the principles of what in his day passed for valid thought among those whom he regarded as reputable thinkers. This enterprise is strictly historical. It is a study in what is called contemporary history = history of the recent past in a society which the historian regards as his own society...Logic as 'theory of scientific method' is in effect, at any given time, a fragment of a history of scientific method. (The Principles of History, pp. 242-3)

We might take the recent results of statistical learning theory to be such a fragment.

Collingwood extended this historical interpretation to the whole of philosophy, including metaphysics. Metaphysics flourishes today, not least through the attention of the late aforementioned David Lewis, in a way that Collingwood would have seen as wrong-headed. Modern metaphysicians aim to give timelessly true accounts of identity, time, cause, necessity, property, etc. They thus become targets for empirically-inclined philosophers. Let's consider a contemporary critic of metaphysics - Bas van Fraassen. Van Fraassen in his recent book outlines what he calls 'The Empirical Stance'. Now, a stance is something like a philosophical orientation. It isn't composed of a set of factual beliefs, but rather a kind of epistemic 'policy'. E.g., the 'Metaphysical Stance', the target of his first chapter, advocates:

M1: Accept demands for explanation in terms of things underlying the phenomena.
M2: Attempt to answer such demands for explanation by postulation.

The Empirical Stance has been summarised by Anjan Chakravartty as follows:

E1: Reject demands for explanation in terms of things underlying the phenomena.
E2: A fortiori, reject attempts to answer such demands for explanation by postulation.
E3: Follow, as a model of inquiry, the methods of the sciences.

Notice how E’s position is defined in terms of M's, as a reaction. In the battle between stances, the only resource van Fraassen seems to have is E3 which he uses to deny that metaphysicians are behaving like scientists. This is just what he does. But it's hardly a killer punch.

Van Fraassen describes the metaphysician as engaging in at least one of the following two tasks: (1) Postulating entities behind phenomena; (2) Reading off the ontological commitments of theories. But to set the term of the debate in this way is surely to reason ahistorically if it is the case that metaphysics was once a different activity. Consider again Collingwood:

I have tried to dispel certain misconceptions about it [metaphysics] which have led (and, had they been true, would have led with perfect justice) to the conclusion that metaphysics is a blind alley of thought into which knaves and fools have combined these many centuries past to lure the human intellect to its destruction. (Collingwood, Author’s preface, p. civ, An Essay on Metaphysics, OUP, revised ed. 1998)

His way of dispelling such misconceptions takes him back to Aristotle:

In writing about metaphysics it is only decent, and it is certainly wise, to begin with Aristotle.” (ibid., p. 1)

For Collingwood, Aristotle has two conceptions of metaphysics, the first he considers as misguided:

(i) Metaphysics is the science of pure being.
(ii) Metaphysics is the science which deals with the presuppositions underlying ordinary science.
Adopting the second of these conceptions, he can then claim

... there are no ‘eternal’ or ‘crucial’ or ‘central’ problems in metaphysics. (ibid., p. 72)

How can metaphysics become a science? By becoming more completely and more consciously what in fact it has always been, an historical science. (ibid., p. 77)

It is important to note that Collingwood uses the term 'science' with a wider scope than is the norm today. For example, history for him is counted a science. This is in contrast to van Fraassen who takes it in the contmporary sense of natural science.

The metaphysics of an Aristotle is altogether different from that of a David Lewis. For Aristotle,

...no science examines the principles which are a presupposition of its having a subject matter to study; e.g. geometry does not consider whether there are points nor arithmetic whether numbers exist. These are questions for another study, which Aristotle calls first philosophy (metaphysics). But he thinks of this higher study as delivering conclusions which the sciences subordinate to it can use as first principles. Whereas twentieth-century philosophy has usually thought of science and metaphysics as quite distinct kinds of inquiry (because in our world they usually are), for Aristotle natural philosophy is simply 'second philosophy' (e.g. Metaphysics 1037a 14-15). It is a less abstract and less general enterprise than first philosophy, because it deals with one part of the subject matter of first philosophy, and secondary to it, because first philosophy has access to the ultimate principles of explanation (Metaphysics E. 1). That is all. (Miles Burnyeat, 'The sceptic in his time and place', in R. Rorty et al. (eds.) Philosophy in History, CUP, 1984: 246-7)

The rise of disciplinary boundaries between philosophy and the natural sciences, and the increasing implausibility that philosophers could dictate first principles to scientists, has led to metaphysics becoming an inquiry which aims to tell a story of how the world must be for science to work as it does, but with no pretensions to affect science. Van Fraassen's complaint is that it models itself on science - employing inference to the best explanation, etc. - but without exposing itself to any of the same kinds of risk that a scientist does. As we have seen, an alternative view of metaphysics, e.g., that of R. G. Collingwood, is closer to Aristotle's in taking it to be the study of the presuppositions of the sciences (taken in the broad original sense). A philosopher's contribution to the metaphysics of a discipline is not restricted to their arguing against, or finding tensions amongst, current presuppositions, although this does typically take place in philosophy of physics. (Some contemporary work in philosophy of physics is very much closer in outlook to Collingwood's 'The Idea of Nature' than it is to analytic metaphysics.) One can also help by providing a scaffold to support discussions of fundamental issues, frame debates, provide historical insight, make comparisons to other fields, etc.

Collingwood extends his historical stance to ethics, currently a boom industry in philosophy

Ethics as an Historical Science
(a) Ethics as an account of the principles of action depends for its content on the structure of the moral world of which it tries to give an account. Thus ancient Greek ideas of conduct are different from Christian ideas and consequently Aristotle's ethics (say) differ[s] widely from any seventeenth- to twentieth-century ethics, without this implying error on either side. Any ethical theory is an attempt to state what kind of a life is considered worth aiming at, and the question always arises-by whom? (b) There are departmental ethical sciences like politics, economics. These, at any given time and place, describe the political and economic principles accepted at that time and place. For economics, this has been seen by the Marxists, and it has been admitted by J. M. Keynes, with the odd result that he has tried to construct a 'general' economic theory, stating the supposedly permanent general principles of which any 'special' economic theory, like Adam Smith's, is a special case. This of course is illusory. (c) Even the distinction between logic and ethics is an historical one and no more. As we inherit it from the Greeks, it certainly has no permanent validity: the Indians or the Chinese do not make the distinction between thought and conduct in any such way as that which we presuppose when we make it. (The Principles of History: 249)

And this brings us to Alasdair MacIntyre who agrees:

From a methodological point of view, it is today clear to me that while I was writing A Short History of Ethics I should have taken as a central standpoint what I learned from R. G. Collingwood: that morality is an essentially historical subject matter and that philosophical inquiry, in ethics as elsewhere, is defective insofar as it is not historical. (A. MacIntyre, 'An Interview with Giovanna Borradori' in The MacIntyre Reader, Kelvin Knight (ed.), Polity Press, 1998: 261)

This point of view he adopted in 'After Virtue', 'Whose Ratioanlity? Whose Justice?' and 'Three Rival Versions of Moral Enquiry'.

Is this Historical Stance plausibly the one to adopt as regards mathematics? Surely we don't need to think of '2 + 3 = 5' as something in need of historical treatment, but rather should consider it in timeless fashion in a way orthogonal to the interests of the mathematician by answering the questions: To what kind of thing do the terms of the proposition refer? What kind of truth does it express? Well, I would argue we should. Insofar as we are considering this statement in the context of research mathematics, there's an awful lot behind even something as simple as this, and yet more behind 2 + 3 = 3 + 2, as you can see from the first of John Baez's talks here. Our conceptual understanding of the integers is still changing. And if you really want your intellect to be adequate to the integers, you'd better gain enough background to understand this claim 'The sphere spectrum is the true integers', even if it is only to criticise it from another perspective within mathematics. For an inkling of what André Joyal means by this claim try week 102 of Baez's This Week's Finds.

I'll leave the last word to Collingwood. Writing in 1935 about the conception of nature worked out by philosophers such as Bergson, Alexander and Whitehead, he remarks:

As in the time of Descartes, so again today physics and metaphysics are working hand in hand, and one of the most remarkable features of this new cosmology is just this fact: the fact that the separation of philosophy from science, of which we have been so long been conscious that we have come to regard it as a necessary evil, has disappeared. The results of this new situation ought to be extremely fertile both for science and for philosophy, strengthening and enriching both of them: and perhaps it is not too much to hope that the alliance and cross-fertilization of these two streams of thought will have a beneficial effect on the future of civilization, which has suffered greatly in the last few generations from the fact that those who ought to be its intellectual leaders have spent much of their time in a mutual warfare, damaging to the prestige of both sides and of no advantage to either. (The Principles of History: 253)

Thales and Friends

Some of the papers from the 'Mathematics and Narrative' meeting held in Mykonos this July are now available at the Thales and Friends website. I have tweaked mine a little since I submitted it, and this November 21 version should appear there soon, otherwise find it here. I won't adjust it for a while now.

If you have reading access to Nature on-line, you can find a 2 page report (August 4) on the meeting by Sarah Tomlin.

John Baez has agreed to join the advisory board of Thales and Friends. His latest edition - week 223 - of This Week's Finds in Mathematical Physics deals with the idea of entities fibred over other entities:

And here, as usual, the n-category theorists meet up with the topologists - and find that the topologists have already done everything there is to do with ω-groupoids ... but usually by thinking of them of them as spaces, rather than ω-groupoids!

It's sort of like climbing a mountain, surmounting steep cliffs with the help of ropes and other equipment, and then finding a Holiday Inn on top and realizing there was a 4-lane highway going up the other side.

The extra difficulties the n-category theorists have to deal with arise from the lack of 'reversibility' in their more general context. In spaces, if there is a path from A to B, there will be one that runs in the opposite direction. If a path can be deformed onto another path, the deformation can take place in reverse, and so on. Computer scientists studying the computation paths of parallel computers care about this lack of reversibility. Two processors can find themselves following a path to a dead-end, along which they cannot backtrack. See the talks by Philippe Gaucher, Lisbeth Fajstrup and Eric Goubault at the IMA n-categories workshop.

Mathematics and Ethics

Several philosophers have recently been seeking analogies between mathematics and ethics. For example, Brendan Larvor (University of Hertfordshire) has given talks on 'Particularism and the exact sciences', stressing how both the mathematician and the moral agent cannot rely on general principles to conduct their reasoning, and Jim Franklin in his On the parallel between mathematics and morals argues that relativist objections to an objectivist ethics would work equally well against an objectivist mathematics. For both of these philosophers, the important parallels between mathematics and ethics do not support the mathematicisation of ethics. Indeed, Franklin makes the interesting observation that: "It is a strange fact that whereas objectivist ethics has tended to avoid mathematics, reductive attempts to replace ethics by something else have been highly mathematical." (p. 109)

Time and again, I'm struck reading Alasdair MacIntyre's writings on ethics, by the parallels between his account of moral enquiry, and my account of mathematical enquiry. For example, the final sentence of 'The Magic in the Pronoun "My"', a review of Bernard Williams, Moral Luck (CUP, 1991) in Ethics 94: 113-125 (available on JSTOR),

although premature systematization is always the enemy of truth in philosophy, delaying systematization for too long can be equally injurious. (p. 125),

perfectly encapsulates the conclusion of chapter 7 of my book, in which I criticise Lakatos for overemphasising the equivalent of the first half of MacIntyre's sentence at the expense of the second half.

As for the quest to understand Lie algebroids, Kirill Mackenzie has dedicated a page to them, and has made available the very interesting Introduction to his forthcoming book 'General Theory of Lie Groupoids and Lie Algebroids', Cambridge University Press.

Time to enter the 21st century

If "...physics blogs, in particular, are democratizing the process of scientific research, providing equal access to everyone from amateur enthusiasts to grad students and Nobel Prize winners, helping to sharpen debates" Blogs: a new force in physics, let's see what a philosophy of mathematics blog can do.

November 1-12

November 2005

Saturday 12

When I mentioned before (Nov 5) Alexander Borovik's notion of 'vertical integration', I had it slightly wrong. His term is 'vertical unity', which he contrasts with the usual form of unity so beloved by mathematicians:

Many eloquent speeches were made, and many beautiful books written in explanation and praise of the incomprehensible unity of mathematics. In most cases, the unity was described as a cross disciplinary interaction, with the same ideas being fruitful in seemingly different mathematical disciplines, and the technique of one discipline being applied to another. The vertical unity of mathematics, with many simple ideas and tricks working both at the most elementary and at rather sophisticated levels, is not so frequently discussed— although it appears to be highly relevant to the very essence of mathematics education.

Were this form of unity commonplace, it might give us hope that David Hilbert was correct when he said:

A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.

Perhaps, though, people in the streets of 1920s Gottingen were particularly bright. So often the ingredients of sophisticated ideas are individually graspable while their composition seems opaque.

I recently leafed through the paper 'A Survey of Lagrangian Mechanics and Control on Lie algebroids and groupoids', Jorge Cortes et al., ArXiv: math-ph/0511009, interested to see what new was happening with groupoids, the subject of chapter 9 of my book. One of the lines of advocacy for groupoids over groups is that they interact with other structures in novel ways. This is the case for Lie groupoids. So I was happy to see that the authors of the paper wanted to:

...show how the flexibility provided by Lie algebroids and groupoids allows us to analyze, within a single framework, different classes of situations such as systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and Field Theory

But it's no easy matter to take these Lie algebroids onboard. I'm not sure the authors of this paper did the best job of it. They did not follow Israel Gelfand's advice of giving the simplest nontrivial example straight after the definition. Eventually a case is presented of a ball rolling on a rotating plate, and one can start to see the motion of the ball fibred over the space of points of contact between ball and plate, and the 'anchor map' to the motion of the centre of the ball.

A little more insight on Lie algebroids came from Urs Schreiber's blog entry. Schreiber is a string theorist working now in Hamburg, who has linked up with John Baez recently to write a couple of papers on categorified gauge theory. Lie algebroids appear on this blog entry in the vector bundle version of Baez's n-categories table. Landsman's 'Lie Groupoids and Lie algebroids in physics and noncommutative geometry' ArXiv: math-ph/0506024 is also helpful.

A philosopher who can throw some light on the difficulty of grasping mathematical concepts, even though their components are simple, is Michael Polanyi. In his article Tacit Knowing: Its Bearing on Some Problems of Philosophy (Reviews of Modern Physics, 34 (4)Oct. 1962, 601-616), Polanyi explains his idea that much of our grasping of things requires tacit knowledge of their constituents. For example, to understand a sentence one has to have tacit knowledge of its constituent words. If we choose to focus instead on the constituents themselves, we will not be able to comprehend the whole. Just concentrate on the individual words of this sentence to see what he means. Mathematical constructions involve towers of blended concepts, and one must have the constituents sufficiently well-understood that one can flick between different focus points, allowing 'encapsulation' and 'de-encapsulation' to use terms from Borovik.

I also like Polanyi for his account of mathematical reality. You may be able to glimpse something of his notion from these two quotations taken from his 1958 book Personal Knowedge:

A new mathematical conception may be said to have reality if its assumption leads to a wide range of new interesting ideas. (Personal Knowledge: 116),

...while in the natural sciences the feeling of making contact with reality is an augury of as yet undreamed of future empirical confirmations of an immanent discovery, in mathematics it betokens an indeterminate range of future germinations within mathematics itself. (Personal Knowledge: 189)

This is an exemplification of his idea of reality as "that which may yet inexhaustibly manifest itself".

Tuesday 8

It's going to be a lot of fun and a huge amount of hard work for future historians and philosophers of science to make sense of the development of a theory of quantum gravity. Perhaps the most informative debate I've read about the current status of string theory is here. Some genuine mutual understanding seems to be achievable if participants debate reasonably charitably.

I've been keeping an eye on contributions to the notion of a field with one element. On the face of it the idea is absurd. Fields by definition must have at least two elements. But there's plenty of evidence that there has to be something like it. Here's Anton Deitmar in his paper 'Cohomology of F1-schemes' ArXiv: NT/0508642:

The analogy between number fields and function fields is one of the most striking phenomena in number theory. Unfortunately, it does not go all the way. In order to use methods of algebraic geometry for the integers, number theorists would like to view Spec Z as a geometrical object (a curve) over a 'field of one element' F_1. A field of one element does not exist. So one has to look for a replacement that would grant the desired geometrical methods for number theoretical problems.

This analogy was the principal one I studied in the chapter on analogy in my book. The idea of a field with one element goes back at least to Tits in 1957. Lots of formulas concerning fields of order pⁿ make sense when p = 1, so long as you give them the right interpretation. For example a vector space over F_1 should be seen as a finite pointed set. Some more motivation from Deitmar:

The F_1-viewpoint as it stands won't solve any problems in number theory, because, for instance, all prime numbers look the same from F_1. It is clear that something has to be added to make this theory useful to arithmetic. Based on the philosophy that all problems in arithmetic stem from the entanglement of addition and multiplication, this is an attempt to disentangle them, respectively, to investigate multiplication alone. Later on they will have to be joined again.

See weeks 184 and 187 of John Baez's This Week's Finds for some of his typical user-friendly exposition on this issue.

As a confirmation of the rule that in mathematics you are never more than a couple of steps away from any other field, Angus MacIntyre in Model Theory: Geometry and Set-theoretic Aspects and Prospects, The Bulletin of Symbolic Logic 9(2), June 2003, is advocating to model
theorists that they look to Grothendieck for inspiration:

Van den Dries' insights are certainly close to those of Grothendieck in [Esquisse d'un programme - see link below in Sept 30], though my feeling is that the potential of[Esquisse] is far from exhausted...For my taste, he [Grothendieck] is unrivalled in terms of ability to select notions, axioms and theorems of maximum potential... This is a kind of "atomic" model theory, where set theory is again largely irrelevant. (203)

[Tameness, one of Grothendieck's issues in Esquisse, is also discussed by MacIntyre.] Then:

I sense that we should be a bit bolder by now. There are many issues of uniformity associated with the Weil Cohomology Theories, and major definability issues relating to Grothendieck's Standard Conjectures. Model theory (of Henselian fields) has made useful contact with motivic considerations, including Kontsevich's motivic integration. Maybe it has something useful to say about "algebraic geometry over the one element field", ultimately a question in definability theory. (211)

Then, Lo and Behold, in the same article MacIntyre discusses the VC-dimension, just the thing I'm working on here in Tuebingen. See this Technical Report if you want to know what it has to do with Karl Popper. Please note that this is still just a draft.

Saturday 5

I'm out of the country so can't see if special efforts are being made to 'celebrate' the 400th anniversary of the foiling of the plot to kill James I and the aristocracy. More than 200 years later Catholics were still not allowed to vote in elections to Parliament.

Why are so many of the world's top mathematicians Russian? (Manin, Kontsevich, Drinfeld, Gelfand, Beilinson, Voevodsky, ...) Presumably, much must be attributed to desirability for very intelligent people to work in an area with little state interference, when other disciplines such as economics are controlled. Lack of opportunity for money-making outside the university must be another factor. But presumably the largest contributor was a policy of carefully selecting and hot-housing promising youngsters. They must have got something right as regards their teaching techniques.

It is not surprising then that one of the most important contributions to the conference 'Where will the next generation of UK mathematicians come from?' held at the university of Manchester in March 2005 came from the pen of the Russian emigre, Alexander Borovik. His piece is entitled What is It That Makes a Mathematician? I like this description of the life of the mathematician:

Mathematicians are sometimes described as living in an ideal world of beauty and harmony. Instead, our world is torn apart by inconsistencies, plagued by non sequitur, and worst of all, made desolate and empty by missing links between words, and between symbols and their referents; we spend our lives patching and repairing it. Only when the last crack disappears, are we rewarded by brief moments of
harmony and joy. And what do we do then? We start to work on a new problem, descending again into chaos and mental pain. We do that to earn the next fix of elation. (p. 3)

His discussion of 'vertical integration' is very important. It gives you hope that even the most advanced concepts are explicable to lesser mortals. Borovik's diagnosis of the crisis in British mathematics education is given here.

For more about scientific bets (Nov 3) see this New Scientist article.

Thursday 3

When I included a chapter on Bayesianism in Mathematics in my book, I did so with the hope that it would draw a few more philosophers to look at mathematical practice. There are many considerations affecting the plausibility of mathematical statements, from the verification of cases to the establishment of subtle analogies. Here's an example to reconstruct in Bayesian terms:

...it is my view that before Thurston's work on hyperbolic 3-manifolds and his formulation of the general Geometrization Conjecture there was no consensus amongst experts as to whether the Poincare Conjecture was true or false. After Thurston's work, notwithstanding the fact that it has no direct bearing on the Poincare Conjecture, a consensus developed that the Poincare Conjecture (and the Geometrization Conjecture) were true. Paradoxically, subsuming the Poincare Conjecture into a broader conjecture and then giving evidence, independent from the Poincare Conjecture, for the broader conjecture led to a firmer belief in the Poincare Conjecture. (John W. Morgan, 'Recent Progress on the Poincare Conjecture and the Classification of 3-Manifolds', Bulletin of the American Mathematical Society 2004,42(1): 57-78)

It doesn't sound at all paradoxical to me, if you take Polya's "hope for a common ground" into account, see chapter 5 of my book.

There seems to be a resistance though to these ideas. George Polya had already worked out most things by the 1940s, but was largely ignored by Bayesian philosophers. When I was talking about this idea in 1999, nobody remembered a 1987 article written by James Franklin, a mathematician at the University of New South Wales, entitled "Non-deductive Logic in Mathematics", British Journal for the Philosophy of Science 38: 1-18 (available here).

Some Bayesian philosophers object to mathematics being treated in this 'quasi-empirical' way. They take it as a tenet that it is irrational to accord logically equivalent statements different degrees of belief. If A follows logically from B, and Pr(A) is less than Pr(B) then you are incoherent, even if you do not know this relation.

A much more interesting response is the Lakatosian one. Putting it in my own terms, it would run like this: the whole point of Proofs and Refutations was to show that mathematical concepts change their meanings. Imagine if in the early 1800s you bet someone that the relation V - E + F = 2 holds for all polyhedra. They accept, then point out that the cyclinder has V = 0, E = 2, F = 2, so is a counter-example to the relation. But you don't accept the cylinder as a polyhedron, and fighting breaks out. What can you do to formulate a precise bet? Appeal to the long term: I bet that 100 years from now the majority of mathematicians will understand the term 'polyhedron' in such a way that V - E + F = 2 holds for it? It would be a shame that, had you lasted that long, you would have lost. You sensed that there was an important relation in the air and that it was worth refining a definition of polyhedron within a theoretical framework with the resources to understand the relation. You just overlooked that 'polyhedron' might come to embrace torus-shaped entities, and so on.

Replying to Lakatos, one can agree that concept-stretching is in many ways more important than the plausibility of results, but that there are many situations with enough of a solid framework to allow for precise bets. If someone asks you to bet on whether the 10^30th zero of the zeta function satisfies the Riemann Hypothesis, you have Saunders MacLane's immortal poetry to guide you:

Norm Levinson managed to show, better yet,
At two-to-one odds it would be a good bet,
If over a zero you happen to trip
It would lie on the line and not just in the strip.

I seem to recall von Neumann taking part in a mathematical bet. I'm sure that must have been others. In a sense, any research career involves a series of gambles as to what is likely to work, what is likely to prove important, etc. For some scientific ones, see Wikipedia's Scientific wager article.

Update
It wasn't von Neumann, it was Hermann Weyl. In his article, Predicativity, Solomon Feferman explains:

A story here, recounted in my book [In the Light of Logic, Oxford Univ. Press], is apropos:

...a famous wager was made in Zurich in 1918 between Weyl and George Polya, concerning the future status of the following two propositions: (1) Each bounded set of real numbers has a precise upper bound. (2) Each infinite subset of real numbers has a countable subset. [The latter requires the Axiom of Choice.] Weyl predicted that within twenty years either Polya himself or a majority of leading mathematicians would admit that the concepts of number, set and countability involved in (1) and (2) are completely vague, and that it is no use asking whether these propositions are true or false, though any reasonably clear interpretation would make them false... . the loser was to publish the conditions of the bet and the fact that he lost in the Jahresberichten der Deutschen Mathematiker Vereinigung... (Feferman 1998, p. 57)

The wager was never settled as such, for obvious political reasons. According to Polya (1972) ['Eine Erinnerung an Hermann Weyl', Mathematische Zeitschrift 126, 296-29], “The outcome of the bet became a subject of discussion between Weyl and me a few years after the final date, around the end of 1940. Weyl thought he was 49% right and I, 51%; but he also asked me to waive the consequences specified in the bet, and I gladly agreed.” Polya showed the wager to many friends and colleagues, and, with one exception, all thought he had won.

I'd still like to know if there has been a straightforward case of odds being offered on a conjecture.

Tuesday 1

Peter Woit's blog Not Even Wrong is the most prominent space on the Web for criticisms of string theory. Take the October 26 entry and its 90+ replies. The contributors there are expressing their philosophy of science as they wrestle with the problem of the right way to go about a theory of quantum gravity. Speaking about the ways researchers ought to conduct their studies, terms referring to the virtues or their lack, such as 'honest' or 'arrogant', naturally appear. Is there anything philosophers can contribute to the debate?

There's a battle-line weaving through the disciplines that study science - philosophy, history, sociology - between those who by and large believe science to be a rational process in some absolute sense and those who do not. Reacting to a simplistic tale of the confirmation of theories, sociologists submitted scientific episodes to close scrutiny and found 'rationality' nowhere to be seen, except as a word bandied about by participants who give it their own gloss. Anthropological studies in the laboratory would find each of two groups calling the other 'unscientific', but when questions were raised about the scientificity of their own work would reply with "it doesn't matter, truth will out in the long run." Clearly, a more sophisticated rationalist position is needed.

Such a position must recognise, as do participants of the Woit discussion, that rival programmes need not have precisely the same aims. Results that group A produces may be taken to be important by them, while group B takes them to be insignificant. A clear case of this where I'm working now involves getting machines to classify hand-written digits. At one point, the frequentist camp had the most accurate classifier. But the Bayesian camp had a classifier which although its error rate was higher could tell you which digits it was least certain of. If its least certain 2% were excluded, it achieved extremely good error rates. And this performance could not be emulated by the frequentists. So, even in what appears to be a very narrow field, where it would appear that there could be little disagreement as to the goals, we find conflicting appraisals of achievements.

So what can we hope for? Surely great care in characterising the goals and achievements of a programme, with the understanding that this characterisation will need to be rethought as the programme unfolds. But along with this we need other intellectual virtues, such as honesty, and a lack of pride preventing the acknowledgement of current weaknesses of one's own programme, along with recognition that the other programme might have resources to comprehend these weaknesses. This wedding of the language of the virtues to the rationality of enquiry is characteristic of the philosopher I have mentioned here before, Alasdair MacIntyre. The rivalry that has most influenced his work is that between the Aristotelians and Augustinians in 13th century Paris. He has attempted to characterise what was necessary for Aquinas to be in a position to reconcile the two doctrines, and use each to resolve the other's weaknesses. For one thing, it required someone to learn both languages as 'first' languages, something often frowned upon by native speakers of each language.

Perhaps MacIntyre's most difficult advice to put into effect is that a research programme lay out what it considers to be its current weaknesses. For this to be possible, it would require a further virtue from any rival groups, that they are sufficiently just not to exploit unfairly such a confession for their own self-promotion.