Wittgenstein's Philosophy of Mathematics

I'm back from snowy Kent with a much clearer idea of Wittgenstein's views on mathematics. Some of the speakers there were the kind of hard-core Wittgensteinians who can recognise a quotation of the master from 100 yards and tell you its section number. An important point that was insisted upon is not to treat all the published writings equally. In the days before word-processors, Wittgenstein would jot aphoristic comments in his notebooks, type up the best of them, and then cut out and paste them into what he took to be the best order. It is very dangerous, then, to take writings from any of the stages of this process as having the same status. Would you be able to stand by everything you've written in notebooks, e-mails, or anything that a student has taken down from your lectures?

With this priviso in place, we come to the assessment of Wittgenstein's views. And they seemed to fall into two camps: a stricter reading, and a more generous reading. The stricter reading finds Wittgenstein realising he is in "a bit of a pickle" by the late 1930s. He has wanted to have us see mathematics through different lenses, so that the charm of certain parts of mathematics, e.g., transfinite set theory, dissipates. He wants to point to concrete demonstrations of arithmetic facts, such as showing 3 X 4 = 12 with pebbles, as paradigmatic examples of doing mathematics. The meaning of such propositions is precisely the proof act of manipulating and counting the pebbles. What then of universal arithmetic statements? Well, their meaning is constitued by their proof, typically a proof by mathematical induction. Don't get lured into believing in the completed infinity of natural numbers. Instead, make the conventional decision, if you wish to join this language game, of being able to assert f(n) for a natural number n when you have seen an inductive proof of for all n f (n).

But this idea of a statement taking its meaning from its proof meets with problems. It suggests that a proposition has no meaning before it has been proved, and that we can't say we have two different proofs of the same proposition. Can it really be that we don't understand Goldbach's conjecture, that any even number greater than 2 is the sum of two primes, since we don't yet have a proof? We might want to say that we don't fully understand this area of number theory, and that a proof, and further proofs, would augment this understanding. But it seems to me to be too generous a reading to say that this is what Wittgenstein was driving at.

Other generous moves are to suggest that the later Wittgenstein was pointing to new things to do in the philosophy of mathematics in an era dominated by logicism, formalism and intuitionism: to look at picture proofs, to look at applied mathematics, to wonder why pieces of mathematics are 'interesting' or 'surprising'. All good suggestions, I agree, but I'm not sure Wittgenstein has carried out much in the way of useful groundwork. The problem, it seems to me, for him and many others, is the tendency to restrict oneself to assessing mathematics purely at the level of propositions and proofs. I think it is necessary to think of individual statements as part of a greater system. Mathematics comes in larger sizes - projects, programmes, traditions.

My favourite programme - higher dimensional algebra (aka n-category theory) - is bubbling along nicely. Aside from being drawn to it aesthetically, my interest has been sustained over the years by the wonderful web-publishing of John Baez. He has been joined in this activity by another mathematical physicist Urs Schreiber who has largely made The String Coffee Table his own. What adds to the interest is that where Urs comes from a background in string theory, John favours its rival, loop quantum gravity, and yet they have worked together on developing a categorified gauge field theory. The last few entries and comments on the blog point you to the latest moves. If you're a beginner who wants to join the higher-dimensional algebra party, try papers 39, 49, 52, 53, 68 & 73 from here, or chapter 10 of my book.

One small contribution philosophers could make to mathematics would be to pay attention to the expository efforts of the exponents of research programmes to encourage this activity. A further obvious candidate for philosophical treat is Alain Connes' noncommutative geometry, another programme with a strong articulated sense of direction. This case has the additional intriguing feature that there are other rival takes on what noncommutative geometry should be. I'm a great believer that the study of rivalry and disputation can be very revealing.

Wittgenstein conference

I'm attending a conference on Wittgenstein's Philosophy of Mathematics at the end of the week, and will report back on it next week. I've always been a little wary of reading much Wittgenstein for fear of being drawn too far into his world, without much assurance of intellectual profit. Mathematicians have generally not looked too kindly on his work. Saunders Mac Lane certainly didn't think too highly of it, as mentioned here on page 2 of this obituary, (see also his 'Mathematics: Form and Function'), but then Mac Lane was dismissive of much post-war philosophy of mathematics.

Translations and Duality

Imagine playing a game in which cards numbered from 1 to 9 are placed face up on the table, and players take it in turns to pick up a card. Your aim is to be the first to collect a hand in which there are three cards which sum to 15. Now, if you know an optimal strategy for noughts and crosses, or tic-tac-toe, you'll be able to play this game optimally too. The solution is at the end of the post.

This translation between problems I found in a paper by Offer Shai to introduce more useful kinds of transformations between engineering problems. For example, to check whether a complicated configuration of trusses is stable, you form the dual of its graph, thereby yielding a new graph which represents a certain mechanism. Your question has now become whether this mechanism is mobile, something you already know how to solve. (See p. 16 for a more thorough table of the analogy.) Shai goes on to use this translation to find unthought of analogues to known concepts.

In recent years string theorists have pointed to extraordinarily rich mathematical dualities thrown up by their understanding that certain physical models should be dual. Cumrun Vafa's Geometric Physics is a gentle introduction (at least to begin with) to some of these. Again, analogues can end up looking very different: "...the question of quantum corrections for one manifold gets transformed to the question involving the variation of complex structure on the other, which is classical."

I've been chatting with John Baez about another opportunity for analogizing, this time between using different rigs. Rigs are essentially rings but without necessarily having inverses for addition. like the natural numbers. If you follow the link, you'll be able to see how sums of amplitudes of paths from quantum mechanics, the path of least action from the calculus of variations, and path connectedness from homotopy theory are all related.

Answer to puzzle: Each time you or your opponent takes a card imagine a O or an X being placed in the corresponding square in this grid:

2 7 6
----------
9 5 1
----------
4 3 8

Why Do People Get Ill?

The finishing touches are completed for the second draft of 'Why Do People Get Ill?', due to be published by Penguin in October 2006. Intriguing stuff. The psychoanalyst/physician Michael Balint realised over 50 years ago that so many kinds of chronic disease arise from the immune system's dysregulated inflammatory response, and it seems that he was right. Then over the past few decades a huge amount has been discovered about interactions between the neural, endocrine and immune systems. The central question then is whether life's difficulties can sufficiently perturb these systems.

I was talking last summer to Niels Birbaumer, who worked in the early days on conditioning the autonomic nervous systems of rats, with psychsomatic theories in mind. He didn't need any convincing that our immune system may be significantly affected by our mental states. Niels is in charge of a brain imaging unit in Tübingen, and has done work on imaging musicians while they imagined they were playing. We discussed the possibility of doing similarly for mathematicians. We sketched a plan to conduct a pilot study around April/May. Are there any volunteers from not too far away? We would need professional mathematicians/ mathematical physicists and controls with a reasonable level of mathematics.

Mathematical seed bank

The Norwegians are building a seed bank to safeguard the world's crops in case of various catastrophes - nuclear war, climatic chaos, pandemic, etc. Rather than wasting all that effort of farmers over the millennia, two million carefully stored varieties of crop seed will allow future survivors the opportunity to reconstruct agriculture by recovering the hoard from Spitzbergen. Reading about this put me in mind of Michael Polanyi's warning:

The transmission of mathematics has today been rendered more precarious than ever by the fact that no single mathematician can fully understand any longer more than a tiny fraction of mathematics. Modern mathematics can be kept alive only by a large number of mathematicians cultivating different parts of the same system of values: a community which can be kept coherent only by the passionate vigilance of universities, journals and meetings, fostering these values and imposing the same respect for them on all mathematicians. Such a far-flung structure is highly vulnerable and, once broken, impossible to restore. Its ruins would bury modern mathematics in an oblivion more complete and lasting than that which enveloped Greek mathematics twenty-two centuries ago. (Personal Knowledge, 1958: 192-3)

As we now know, 'oblivion' is rather too strong a word to use for the fate of Greek mathematics. Although there were some shaky moments in the passing on of the baton, a continuous line can be traced. "Medieval Europe learned a lot of Greek science by reading Latin translations of Arab translations of Syriac translations of second-hand copies of the original Greek texts!" was John Baez's summary.

But what if we had a cultural meltdown? Polanyi was worrying here about a less catastrophic onset of an intellectual Dark Ages, but whatever the cause of the breakdown, perhaps we should start planning for the storage of our intellectual products. Storage on hitech devices would clearly be risky, but presumably acid-free paper would be safe enough, although the Babylonian clay tablet has its advantages. But more to the point, what should we store of our mathematics? Would you deposit works in the form of Bourbaki's Elements? Surely what would be more useful would be crates full of commentary, informal exposition, and the history of conceptual development. This is precisely what would be suggested by taking mathematics as a tradition of enquiry, and as such a socially-embodied argument seeking to further human understanding. Perhaps different seed banks would have to be set up to reflect different views of the best ways of organising mathematical priorities.

Someone I imagine who would have a distinctive idea about how to do this is Doron Zeilberger. I met Doron at the 'Mathematics and Narrative' conference in Mykonos last summer. He's a thoroughly likable mathematician, based at Rutgers, and famous for his opinions. I agree with the spirit of many of them, but feel most distance between us when he takes what humans have achieved to date as being of a trivial level of complexity compared to what computers will be doing in the future (e.g., opinion 69). Certainly, humans with computers are capable of more than humans alone. But I can't agree that, catastrophe permitting, "In fifty years (at most) human mathematicians will be like lamp-lighters and ice-delivery men. All serious math will be done by computers." No, unless their activity is embedded within the ongoing histories written by the mathematical community, it is not mathematics.

Doron will no doubt think this a glorification of the weakness of our minds, but I end with a quotation, copied from Alissa Cran's webpage:

Many people who have never had occasion to learn what mathematics is confuse it with arithmetic and consider it a dry and arid science. In actual fact it is the science which demands the utmost imagination. One of the foremost mathematicians of our century says very justly that it is impossible to be a mathematician without also being a poet in spirit... It seems to me that the poet must see what others do not see, must see more deeply than other people. And the mathematician must do the same.-Sofya Kovalevskaya, 1890

On Beyond Zebra

Mathematicians and scientists looking for a new symbol may have been tempted by Dr. Seuss' letters beyond Z, like Yuzz. No need. It appears that they can use a real letter - the Old English and Scottish, Yogh.

Yogh explains why we have to pronounce the first name of the contender for the LibDem party leadership, Menzies Campell, as MING-IS. The trouble is that the army slang word ming has swept into the vocabulary of the nation's youth, "It's really minging" and "You minger!". Still, at least my kids find the news interesting at the moment.

By the way, all you MacKenzies out there should be pronouncing your name MacKenyie.

Some speculative floating of loosely related ideas

(1) A Bayesian interpretation of quantum mechanics

Choice quotations from 'Quantum Mechanics as Quantum Information (and only a little more)', Christopher A. Fuchs :

The theory prescribes that no matter how much we know about a quantum system—even when we have maximal information about it—there will always be a statistical residue. There will always be questions that we can ask of a system for which we cannot predict the outcomes. In quantum theory, maximal information is simply not complete information. But neither can it be completed. (11)

...it turns out to be rather easy to think of quantum collapse as a noncommutative variant of Bayes’ rule. (35)

In this connection, it is interesting to note that the quantum de Finetti theorem and the conclusions just drawn from it work only within the framework of complex vector-space quantum mechanics. For quantum mechanics based on real Hilbert spaces, the connection between exchangeable density operators and unknown quantum states does not hold.(47)

One is left with a feeling—an almost salty feeling—that perhaps this is the whole point of the structure of quantum mechanics. Perhaps the missing ingredient for narrowing the structure ofBayesian probability down to quantum mechanics has been in front of us all along. It finds no better expression than in taking account of the challenges the physical world poses to our coming to agreement.(48)

The tensor-product rule for combining quantum systems can be thought of as secondary to the structure of local observables.(52) (See section 5)

(2) Monoidal categories as arenas for mathematics

Fuchs wants to deflate the mystery of quantum mechanics, including entanglement. Entanglement seems to have something to do with the (noncartesian) tensor product of Hilbert spaces, which for Fuchs is just a consequence of having local observables. Something I once posed to John Baez, who admittedly didn't look totally convinced, is whether one could see through some of the supposed strangeness of entanglement by looking away from the category of Hilbert spaces to another noncartesian monoidal category, Sets and Relations, whose objects are sets, and whose arrows from A to B are relations, i.e., subsets of the product A x B.

So, your sister goes to live in Australia. It is not surprising if someone observing her there knows about you by noting whether she marries (you're an in-law) or has a child (you're an aunt/uncle). What is different is that in the classical world a state of maximal information involves you knowing every property of an entity, in the quantum world this is no longer possible. If observables don't commute, then you can't know their values simultaneously.

(3) How much of the mathematics used in physics is describing our knowledge and ways of observing and intervening, and how much the physical world itself?

Fuchs sees the majority of the apparatus of quantum mechanics as representing how we can gamble wisely on how quantum measurements will turn out. Omnes was careful in 'Converging Realities' to identify mathematics and the laws of physics, rather than the physical world. Wigner in his 'On unitary representations of the inhomogeneous Lorentz group', Ann. Math. (2) 40, 149-204 (1939), gets far by thinking of invariances relative to different observers.

If Baez is right about QM and GR having something in common in that they both use symmetric monoidal categories with duals, do they share a common separation between world and knower?

(4) If for the Bayesian, probability theory is a generalised logic, what happens when you deform it?

Free probability theory forms noncommutative analogues of constructions from classical probability theory. E.g., Wigner's semicircle distribution is the analogue of the normal distribution.

Just as the tropical analogue of the fourier transform is the legendre transform, so the analogue of probability theory is optimisation theory.

Update: Abstracts from a conference studying Fuchs' ideas.

Deformations of mathematics

The review mentioned yesterday is finished. As I had forgotten the word limit, it had to be hacked back, so I've made the original available here.

I noticed this paper on Tropical Geometry today on the ArXiv. The idea of tropical mathematics is that many of the constructions of ordinary mathematics usually carried out over the reals or complex numbers can be profitably transferred to various tropical semirings. The term 'tropical' was chosen supposedly to commemorate a Brazilian mathematician, Imre Simon. An example of one of these semirings is R È {-¥}, where "x + y" = max{x, y} and "x.y" = x + y. A good introduction to tropical mathematics is this paper by Litvinov, where he tells us:

Idempotent mathematics can be treated as a result of a dequantization of the traditional mathematics over numerical fields as the Planck constant h tends to zero taking imaginary values... In other words, idempotent mathematics is an asymptotic version of the traditional mathematics over the fields of real and complex numbers.

The basic paradigm is expressed in terms of an idempotent correspondence principle. This principle is closely related to the well-known correspondence principle of N. Bohr in quantum theory. Actually, there exists a heuristic correspondence between important, interesting, and useful constructions and results of the traditional mathematics over fields and analogous constructions and results over idempotent semirings and semifields (i.e., semirings and semifields with idempotent addition).

A systematic and consistent application of the idempotent correspondence principle leads to a variety of results, often quite unexpected. As a result, in parallel with the traditional mathematics over fields, its “shadow,” the idempotent mathematics, appears. This “shadow” stands approximately in the same relation to the traditional mathematics as does classical physics to quantum theory, see Fig. 1. In many respects idempotent mathematics is simpler than the traditional one. However the transition from traditional concepts and results to their idempotent analogs is often nontrivial.

Another example of the deformation of large tracts of mathematics also invokes the quantum language. This historical paper gives a detailed account of this 'q-disease'. This Week's Finds aficionados will recall the series on q-mathematics (weeks 183-188).

Vladimir Arnold has made some fascinating suggestions concerning systematic transformations of blocks of mathematics in his 'Polymathematics'. (See also another account of these ideas in Lecture 2 of the Toronto Lectures on this page.) Arnold suggests that there is a way of thinking systematically about the mysterious relations between apparently diverse fields so frequently noted by mathematicians via various informal processes:

The informal complexification, quaternionization, symplectization, contactization etc., described below, are acting not on such small things, as points, functions, varieties, categories or functors, but on the whole of mathematics. I have successfully used these ideas many times as a method to guess new results. I hope therefore that in the future this method of the multiplication of mathematics will be as standard, as is now the transition from finite-dimensional linear algebra to the theory of integral equations and to functional analysis.

And

The main dream (or conjecture) is that all these trinities are united by some rectangular "commutative diagrams". I mean the existence of some "functorial" constructions connecting different trinities. (Arnold lecture 2: 10)

Converging Realities

Happy New Year to everyone! Hampered by the overindulgence of the past few days, but envigorated by blasts of Atlantic air along the coastal path of Cornwall (please don't switch off the Gulf stream - ten degrees Centigrade is very pleasant at Christmas when you're fifty degrees North of the Equator), I managed to finish Roland Omnes' 'Converging Realities' (Princeton). I have to review this book very soon. In fact, I should already have reviewed it, two deadlines having passed. I feel a certain sympathy with Omnes, not least because his choice of subtitle for this English translation - 'Toward a Common Philosophy of Physics and Mathematics' - conveys the same kind of desire for philosophy of mathematics to be something else as that expressed by the title of my book. Not that Omnes seems to be aware of the agenda that grips Anglophone philosophers of mathematics. And here lies a problem.

Something akin to a complementarity principle applies in the way philosophy of mathematics operates in the English speaking world. Either you buy into an established agenda which traces a lineage back to Quine and Putnam and ponders questions such as whether it is right to say that we are committed to the existence of mathematical objects because we use them in our best science. This gives you the advantage that work can be conducted in scholarly fashion, there are a set of recognised contributions to the debate, a sense of progress is generated. The drawback, if you care, is that just about everyone else, and most especially mathematicians, think what you're doing is beside the point. The alternative, then, is to write an ambitious, unworked out thesis. This is usually done by 'outsiders'. Insiders may enjoy reading this work, but are unlikely to do the kind of detailed filling in necessary for it to become part of the discipline.

Omnes' book suffers from a host of errors concerning the theories of various philosophers, while others are treated far too briefly. But what is good about the book is that its positive thesis - physism: mathematics and the laws of physics are one and the same thing - forces us to attend to the changing relationship between mathematics and physics over the centuries. Some when asked suggest that it is not at all surprising that our mathematics fits the world, no more surprising than that our lungs are adapted to the world's atmosphere. But Omnes has a different story to tell. Where mathematics and physics grow together to generate the classical science of Newtonian mechanics, they will later part through the early twentieth century, only to be reunited later in that century. The separation happens as each partner independently goes through a series of crises: physics through the relativistic and quantum revolutions, mathematics through the discovery of pathological functions, the incompleteness of Euclid's axioms, the foundational paradoxes, and so on. Given this divergence and convergence of paths, Omnes believes his 'physism' to be the most reasonable explanation. You might put it thus: string theorists don't learn to study noncommutative tori from dunking doughnuts in their coffee.

The best part of the book, perhaps through my ignorance, is an explanation of how physicists recover our everyday classical world from the quantum substrate. Omnes worked himself on this program. 'Decoherence' is the buzzword. I'll leave an explanation to those better informed, but you can see how much philosophy would be changed if Omnes' thesis were to triumph and anyone who wanted to contribute to philosophy of mathematics had to know quantum mechanics.

I was reminded reading this book of John Baez's Quantum Quandaries. Here, sets as the basis of mathematics are taken to have arisen from our encounter with a classical world of discrete, identifiable, located things. As we depart from this world to that of quantum mechanics and general relativity, we develop different categories of objects, Manifolds and cobordisms, Hilbert spaces and operators, which are more like each other than either is to Sets and functions, both being symmetric monoidal categories with duals, where the monoid operation is noncartesian. A question to finish on then. Can one characterise decoherence in terms of the way a symmetric monoidal category with duals 'looks' cartesian when one forms the tensor product of sufficiently many objects?