Philosophy of Real Mathematics page
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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".
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.
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.
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.
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.
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.
It seems that I was somewhat unfair to number theorists (October 19) - they are making use of the Web. Also take a look at the incomparable number theory and physics archive, a website you may find hard to exit once you have entered.
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”. Some mathematicians I have spoken to seem sheepish when they use the word 'philosophy' in this sense, as though they have no right to use it thus. I disagree. Tao also claims that “a subfield of mathematics has a better chance of staying dynamic, fruitful, and exciting if people in the area do make an effort to make good surveys and expository articles that try to reach out to other people in neighboring disciplines and invite them to lend their own insights and expertise to attack the problems in the area.” This supports my argument that by not making this effort mathematicians are failing to be fully rational.
One talk I haven't mention from the 'Impact of Categories' conference was Bob Coecke's very enjoyable 'Kindergarten Quantum Mechanics'. Bob works in the Computing Laboratory in Oxford and is trying to strip down quantum information theory to its bare category theoretic bones. What results is an extraordinarily simple diagrammatic calculus. Proofs in this calculus can be seen as flips from one slide to the next. This is work very much in line with the kind of diagrammatic reasoning I discuss in chapter 10 of my book. In the October 4 entry I mentioned this paper. Lauda is showing here with glorious diagrams that the 2-category of open strings, planar open string worldsheets, and isotopy classes of worldsheets embedded in the cube, known as the 2-category of ‘3-dimensional thick tangles’, is the (semistrict monoidal) 2-category freely generated by a ‘categorified Frobenius algebra’.
Lauda, now in Cambridge working with Martin Hyland, is a student of John Baez, who has devoted much of his research effort to n-category theory, along with its role in reconciling quantum mechanics and general relativity. This is very evident through many weeks of his 'Quantum Gravity Seminar', but you can get a taste of his ideas in 'Quantum quandaries: a category-theoretic perspective', in S. French et al. (Eds.) Structural Foundations of Quantum Gravity, Oxford University Press, also here. The terminology may be different, symmetric monoidal with duals rather than strong compact closedness, but Baez and Coecke are talking about very similar things.
As a philosopher you can either study scientific movements from a distance, perhaps to see how they clash with each other, or else you commit yourself to a particular position. My interest in n-category theory began in the first vein, but it's very easy to slip over to the second. Aside from higher-dimensional category theory, I am also rooting for Bayesianism to succeed. I haven't studied these closely but there are responses, amongst others by Fuchs and Cave, to the charge against Bayesianism that quantum mechanics shows that at least some probabilities are out there in the world.
There have been some category theoretic attempts to capture probability theory, e.g., by Lawvere and Giry, but I don't think this has taken off yet. Fields medallist Vladimir Voevodsky appears to be trying to do something similar. Perhaps then we could put together all the ideas in this post. For what it's worth, I have never seen it observed elsewhere that a Bayesian network can be seen as an arrow in a symmetric monoidal category with duals. The dual of an arrow between A and B is given by Bayes' rule.
Update: Voevodsky's lectures 'Categories, Population Genetics and a Little of Quantum Physics' have the following abstract:
In these lectures I will tell about my work on two related but separate subjects. The first one is mathematical population genetics. I will describe a simple model which is useful for the study of the relationship between the history of a population and its genetic properties. While the positive results obtained in the framework of this model may have little use because of the model's simplicity the negative results are likely to remain valid for more complex real world populations. The second subject can be described as a categorical study of probability theory where "categorical" is understood in the sense of category theory. Originally, I developed this approach to probability to get a better understanding of the constructions which I had to deal with in population genetics. Later it evolved into something which seems to be also interesting from a purely mathematical point of view. On the elementary level it gives a category which is useful for the work with probabilistic constructions involving complicated combinations of stochastic processes of different types. On a more advanced level, applying in this context the old idea of a functor as a generalized object one gets a better view of the relationship between probability and the theory of (pre-)ordered topological vector spaces. This leads to the third topic mentioned in the title. But I am only beginning to understand this connection.Unfortunately, there appears to be no other trace of the contents of these lectures on the Web.
Naming mathematical entities is an important business. Charles Peirce argued that a scientist had the right to name their discoveries, but that this right would be overturned if the naming turned out to be unwise. Mathematicians have tended to be rather conservative. Only occasionally does a term convey associated imagery well, such as 'sheaf' with its paper and wheat connotations matching the covering and fibre imagery. It is very common to use a mathematician's name as a token of gratitude, although Peirce might have seen this as a failure on the part of the discoverer/ inventor to find a suitable term. Alan Weinstein and colleagues in a paper today, looking to name a generalisation of Hopf algebras, explain their choice:
We call our new objects hopfish algebras, the suffix “oid” and prefixes like “quasi” and “pseudo” having already been appropriated for other uses. Also, our term retains a hint of the Poisson geometry which inspired some of our work.We await the new children's book 'One fish, two fish, red fish, hopfish'. It's good to avoid "quasi" and "pseudo". These versions often become the ones you care about, leaving you with the lengthier names as the norm. "Weak" n-categories may be a case in point. Some just want to drop the "weak", and specify "strict" in the other case. As to whether names have effects on the careers of concepts, Alain Connes in 'Noncommutative Geometry' suggested that the "-oid" ending of "groupoid" had a detrimental effect, and led to the concept being "despised".
Returning to the Paris conference, it is noticeable how Anglophone philosophers interested in category theory are very well-informed about the history of mathematics. They seem to be united in their love of mathematics as a quest, so cannot rest happy with straightening out a timeless conception of what mathematics is about and what kind of entities it deals with. In Colin McLarty's excellent 'Mathematical Platonism’ Versus Gathering the Dead: What Socrates teaches Glaucon', Philosophia Mathematica 13, 115–134, he shows that this quest conception is to be found in Plato's thinking, especially in The Republic. According to McLarty, it is Glaucon whose views are closest to what we mean today by 'platonism'. His paper requires a subtle reading of the dialogue to avoid misattributing to Plato views expressed by some of the characters. Less excusably this kind of misattribution has occurred in commentary on Lakatos's dialogue 'Proofs and Refutations'.
To refute a simplistic conception of an opposition between Aristotle and Plato on the Forms, Alasdair MacIntyre's "The Form of the Good, Tradition and Enquiry," (Value and Understanding: Essays for Peter Winch, Raimond Gaita, ed. (London: Routledge, 1990), pp. 242-62) presents Aristotle's response to the tensions in Plato's work as part of a continuous tradition of enquiry. This sense of philosophy as quest partially explains my interest in MacIntyre.
Not all branches of mathematics use the Web to the same extent. For example, number theorists are known for their reluctance to use the ArXiv. Perhaps they have most closely identified with Gauss' sentiment pauca sed matura, on balance not a very helpful policy. Philosophers as might be expected tend to behave like the number theorists, one exception being philosophers of science (and mostly of physics) and their PhilSci Archive. Even here the deposition rate since it started in 2001 has wavered 149 (2001), 223 (2002), 136 (2003), 215 (2004), and this year so far 115.
I mentioned in the last entry my sense that MacIntyre's tradition-constituted enquiry needs some of form of supplementation in terms of the personal or individual. Jack Russell Weinstein has some thoughts along these lines here, where he looks to Adam Smith. As you might imagine, such a reconciliation could only occur if Smith had been seriously misunderstood. Weinstein believes he has.
For MacIntyre's account of a way through Mill and Kant's differences on lying, read his 1994 Tanner Lectures.
What might a merging of Alasdair MacIntyre and Michael Polanyi resemble? Seeking to locate MacIntyre's traditions within mathematics, they appear a little monolithic, like Lakatos's research programmes. Theoretical commitments are surely more flexible and varied. Perhaps we need something of Polanyi's focus on the personal. My latest version of How Mathematicians May Fail to be Fully Rational points to this need.
Once upon a time synthetic differential geometry threatened to show how category theoretic thinking could aim right for the heart of as central an area of mathematics as differential geometry. Then the feeling grew that it was never quite going to make it. Evidence, at last, of its potential is here.
Concerning yesterday's comment on 2-vector spaces, John Baez writes:
I designed my 2-vector spaces for the purposes of higher linear algebra - Lie 2-algebras, but also associative 2-algebras and so on. I couldn't resist telling Nils about them and suggesting that he try those as a replacement of the K-V 2-vector spaces in studying 2-K-theory. Aaron Lauda went to visit him and explained a bunch of stuff. But, apparently they don't do the job.
It's a pity, and a bit mysterious since Lie 2-algebras ARE related to the string group, which plays a key role in Stolz and Teichner's work on elliptic cohomology, which Baas was trying to simplify in his work on 2-K-theory.
So, I don't think we've gotten to the bottom of things. In particular, Baas' work on the K-theory of the ku spectrum does not capture all features of elliptic cohomology. So, he hasn't gotten to the bottom of it either.
Add Jacob Lurie's version of elliptic cohomology by doing algebraic geometry in a monoidal omega-category into the mix, and the 'bottom' should be quite impressive when located.
I'm just back from Paris and 'The Impact of Categories' conference held at the ENS. While mathematics may approximate an international activity, philosophy is far from being so, but we still managed to learn a little from each other. Many of the Anglophone talks were aimed implicitly or explicitly at the issue of how to think about foundations. There are some subtle issues here. I agree with Jean-Pierre Marquis (Montreal) that merely saying set theorists/logicians' sense of the term 'foundations' is different from category theorists' and so they need not be seen as rivals is not the right way to look at things. Rather the different senses of the term must be viewed as interconnected. Where people differ is over the best way to organise this interconnection. It would be worth comparing Marquis' 1995 paper "Category Theory and the Foundations of Mathematics: Philosophical Excavations", Synthese, 103, 421-447, with Alasdair MacIntyre's 'First principles, final ends and contemporary philosophical issues' to appear in The Tasks of Philosophy (it already has appeared in Kevin Knight's 'MacIntyre Reader').
Steve Awodey (Carnegie-Mellon) told us that Saunders Mac Lane was 12 years earlier than Quine in finding that Carnap had failed to find a principled way to distinguish the logical from the non-logical in a formal theory. This is very important when you recall the role this observation plays in 'Two Dogmas of Empiricism'. We were left to draw our own conclusions from the fact that Mac Lane and Quine were well enough acquainted to go sailing together.
Nils Baas (mathematician from Oslo) stood in at the last moment for Alain Badiou. He reported interesting results in trying to form a 2-K theory using 2-vector bundles, as a way of approaching elliptic cohomology and homotopy data at chromatic filtration level 2 (see Baez on this). Baas claims his 2-vector spaces work here where Baez's don't. The next step is to sort out the 2-functor from the 2-category of surface elements of a space to 2-Vect, before pushing on to 3-vector spaces. Once we have this, and a 3-category of cobordisms with corners, edges and boundaries, more will be convinced that mathematics beyond 2-categories exists "in nature".
On his Algebraic Topology Problem List, Mark Hovey remarks “…even if the problems we work on are internal to algebraic topology, we must strive to express ourselves better. If we expect our papers to be accepted in mathematical journals with a wide audience, such as the Annals, JAMS, or the Inventiones, then we must make sure our introductions are readable by generic good mathematicians. I always think of the French, myself--I want Serre to be able to understand what my paper is about. Another idea is to think of your advisor's advisor, who was probably trained 40 or 50 years ago. Make sure your advisor's advisor can understand your introduction. Another point of view comes from Mike Hopkins, who told me that we must tell a story in the introduction. Don't jump right into the middle of it with "Let E be an E-infinity ring spectrum". That does not help our field.” And naturally Mike Hopkins abides by his own advice. He gave an interesting introduction to homotopy theory at the ICM 2002. The following passage is very relevant to chapter 3 of my book, where I discuss Ronnie Brown and Tim Porter's schema:
geometry-->underlying process-->algebra-->algorithms-->computer implementation
from page 8 of The intuitions of higher dimensional algebra for the study of structured space.
For some questions the homotopy theoretic methods have proved more powerful, and for others the geometric methods have. The resolutions that lend themselves to computation tend to use spaces having convenient homotopy theoretic properties, but with no particularly accessible geometric content. On the other hand, the geometric methods have produced important homotopy theoretic moduli spaces and relationships between them that are difficult, if not impossible, to see from the point of view of homotopy theory. This metaphor is fundamental to topology, and there is a lot of power in spaces, like the classifying spaces for cobordism, that directly relate to both geometry and homotopy theory. It has consistently proved important to understand the computational aspects of the geometric devices, and the geometric aspects of the computational tools.
My fascination with 'Higher dimensional algebra', the study of n- and omega-categories continues. Something it's especially good at is providing a framework to understand theories which use algebra to extract topological information. Not just the 'old-fashioned' algebraic topology, but also 'quantum topology', where the algebra has to tailored to the dimension of the topological object. This framework is being applied in physics and computer science. As John Baez puts it, "we're starting to see a unification of logic, singularity theory/topology, and physics."
An interesting recent paper in this programme is Aaron Lauda's 'Frobenius algebras and planar open string topological field theories'. The diagrammatic proofs such as the one on p. 61 will become increasingly common over the coming years. Those 2-arrows could be represented by two-dimensional surfaces. Can we expect papers of the future to contain java applets showing surfaces and higher-dimensional entities transforming as parts of calculations and proofs?
Grothendieck's 'Sketch of a Programme' is available here. Besides the mathematics, it is full of fascinating methodological points:
There are people who, faced with this, are content to shrug their shoulders with a disillusioned air and to bet that all this will give rise to nothing, except dreams. They forget, or ignore, that our science, and every science, would amount to little if since its very origins it were not nourished with the dreams and visions of those who devoted themselves to it. (251)
…the moment seems ripe to rewrite a new version, in modern style, of Klein’s classic book on the icosahedron and the other Pythagorean polyhedra. Writing such an expose on regular 2-polyhedra would be a magnificent opportunity for a young researcher to familiarise himself with the geometry of polyhedra as well as their connections with spherical, Euclidean and hyperbolic geometry and with algebraic curves, and with the language and the basic techniques of modern algebraic geometry. Will there be found one, some day, who will seize this opportunity? (255-256)
This situation, like so often already in the history of our science, simply reveals the almost insurmountable inertia of the mind, burdened by a heavy weight of conditioning, which makes it difficult to take a real look at a foundational question, thus at the context in which we live, breathe, work – accepting it, rather, as immutable data. It is certainly this inertia which explains why it took millennia before such childish ideas as that of zero, of a group, of a topological shape found their place in mathematics. It is this again which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom “wildness” is a fatal necessity, rooted in the nature of things. (259)
Let us note in relation to this that the isomorphism classes of compact tame spaces are the same as in the “piecewise linear” theory (which is not, I recall, a tame theory). This is in some sense a rehabilitation of the “Hauptvermutung”, which is “false” only because for historical reasons which it would undoubtedly be interesting to determine more precisely, the foundations of topology used to formulate it did not exclude wild phenomena. It need (I hope) not be said that the necessity of developing new foundations for “geometric” topology does not at all exclude the fact that the phenomena in question, like everything else under the sun, have their own reason for being and their own beauty. More adequate foundations would not suppress these phenomena, but would allow us to situate them in a suitable place, like “limiting cases” of phenomena of “true” topology. (276)
It will perhaps be said, not without reason, that all this may be only dreams, which will vanish in smoke as soon as one sets to work on specific examples, or even before, taking into account some known or obvious facts which have escaped me. Indeed, only working out specific examples will make it possible to sift the right from the wrong and to reach the true substance. The only thing in all this which I have no doubt about, is the very necessity of such a foundational work, in other words, the artificiality of the present foundations of topology, and the difficulties which they cause at each step. (262)
…around the age of twelve, I was interned in the concentration camp of Rieucros (near Mende). It is there that I learnt, from another prisoner, Maria, who gave me free private lessons, the definition of the circle. It impressed me by its simplicity and its evidence, whereas the property of “perfect rotundity” of the circle previously had appeared to me as a reality mysterious beyond words. It is at that moment, I believe, that I glimpsed for the first time (without of course formulating it to myself in these terms) the creative power of a “good” mathematical definition, of a formulation which describes the essence. (274)
The sempiternal question “and why all this?” seems to me to have neither more nor less meaning in the case of the anabelian geometry now in the process of birth, than in the case of Galois theory in the time of Galois (or even today, when the question is asked by an overwhelmed student...); the same goes for the commentary which usually accompanies it, namely “all this is very general indeed!”. (275)
In any case, this application will have been the occasion for me to write this sketch of a programme, which otherwise would probably never have seen the light of day. I have tried to be brief without being sybilline and also, afterwards, to make it easier reading by the addition of a summary. If in spite of this it still appears rather long for the circumstances, I beg to be excused. It seems short to me for its content, knowing that ten years of work would not be too much to explore even the least of the themes sketched here through to the end (assuming that there is an “end”...), and one hundred years would be little for the richest among them!(273)
I came across Predrag Cvitanovic's work on birdtracks while researching diagrammatic techniques for mathematical calculation, but he has a much wider range of interests in mathematical physics summarised here. The article 'Search without a plan' is worth reading.
Two pages of interesting links from Yemon Choi here and here
The website for a conference I'm attending The Impact of Categories is up and running. It will be interested to compare notes with others on the way philosophy should engage with category theory. Later in the same week I'll be talking to a mathematics group in Paris VII.
In July I attended a conference entitled Mathematics and Narrative, involving some prominent mathematicians and 'paramathematicians'. The meeting was featured in the 4 August edition of Nature. A comment of mine is reported, which should become clearer after reading the notes for my contribution. Here are the abstracts of the other talks.
Check out the new discussion group Workshop on Mathematical Practice. It's only just started and will need enthusiastic support to succeed.
I've started a page to discuss reviews of my book.
For a very modest 12.99 sterling you can purchase Mathematical Reasoning and Heuristics edited by Carlo Cellucci and Donald Gillies, a collection of articles in the spirit of the philosophy of real mathematics. Anyone wishing to avoid the overcharging of a publisher like Kluwer might consider King's College Publications.
I have deposited some 'Reflections on Michael Friedman's Dynamics of Reason at the Pittsburgh archive. Abstract: Friedman's rich account of the way the mathematical sciences ideally are transformed affords mathematics a more influential role than is common in the philosophy of science. In this paper I assess Friedman's position and argue that we can improve on it by pursuing further the parallels between mathematics and science. We find a richness to the organisation of mathematics similar to that Friedman finds in physics.
On 6 May I spoke to some mathematicians in Warwick (slides).
Before Easter 2005 I went on a mini-tour of the Western Lake Erie region, giving talks in Wayne State, Toledo, and Case Western. The latter talk was a response to 'The Way We Think: Conceptual Blending and the Mind's Hidden Complexities' by Gilles Fauconnier and Mark Turner (Basic Books, 2003). Mathematicians say things like:
Just as the concept of Lie 2-algebra blends the notions of Lie algebra and category, the concept of 'L∞-algebra' blends the notions of Lie algebra and chain complex.
Might Fauconnier and Turner's notion of blending throw some light on this way of speaking (talk slides)?
At Wayne and Toledo I discussed a MacIntyrean future for the philosophy of mathematics. notes
From February 2005 I have been based in the Max Planck Institute for Biological Cybernetics in Tübingen attempting to generate a MacIntyrean analysis of the rivalry between Bayesian and non-Bayesian statistical learning theory. Very early results are here.
In 2004 I deposited a couple of articles (Word documents) with the Pittsburgh Philosophy of Science Archive. The first was a paper I gave in Rome in September 2004, discussing how higher-dimensional algebra guides you from important constructions to important constructions. It turns out that starting with propositional logic, you jump to predicate logic and then modal logic. Might this be a clue to why Charles Peirce termed those parts of his theory of existential graphs, his diagrammatic logical theory, alpha (propositional), beta (predicate), gamma (modal)? Talking of rival traditions, the rivalry between higher-dimensional algebra (or higher-dimensional category theory) and set theory to be the basic language of mathematics is probably the one to watch out for over the next decades. The second paper (postscript) starts out from Poincaré's claim that:
In my book I claimed that the philosophy of mathematics should concern itself with "what leading mathematicians of their day have achieved, how their styles of reasoning evolve, how they justify the course along which they steer their programmes, what constitute obstacles to these programmes, how they come to view a domain as worthy of study and how their ideas shape and are shaped by the concerns of physicists and other scientists". Andrew Arana has observed to me that the description above sounds somewhat elitist. Perhaps 'leading' isn't the best choice of words. I certainly wasn't intending to restrict attention to the top few dozen mathematicians of an era.