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March 2, 2007

Dynamics of Mathematical Reason

Posted by David Corfield

I’m having a spot of bother getting this paper published. It’s about the philosopher Michael Friedman’s treatment of mathematics in his Dynamics of Reason. I’d be grateful for any comments from the Café clientele.

As I’m writing for a philosophical audience, I have been rather brief with my description of Friedman’s views. But I hope enough is conveyed about them for you to gain a sense of his position. I’m arguing that despite Friedman’s greater sympathy for the role of mathematics in physics, greater than say an old style logical empiricist who might have seen mathematics merely as a bunch of tautologies, something important about the internal life of mathematics is still being overlooked.

Posted at March 2, 2007 11:00 AM UTC

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11 Comments & 1 Trackback

Re: Dynamics of Mathematical Reason

Take these comments with a large bucket of salt, because I know from embarrassing experience that I am sometimes a very peculiar reader, but:

What is the central line of argument of this paper? It seems very digressive and discursive, and I’m having trouble keeping track of what you’re arguing for at various points.

Given that your review is kind of a negative reaction to the book, I’d feel more comfortable if I felt you engaging more directly with Friedman’s arguments. I realise you don’t want to be incorporating a review of his book inside your paper, but at the moment it gives me the feeling of “Friedman didn’t write about all these cool things that I’m interested in!” This is a legitimate reaction, but it doesn’t necessarily give a paper a strong narrative drive, so it’s possibly not the main thing that you want to come across to the reader.

There’s a lot of interesting material in there, but at the moment I’m finding it difficult to see what the strategy is behind the way you’re deploying it.

But this may just be me being dense. Keep the buckets of salt on hand.

Posted by: Tim Silverman on March 2, 2007 10:32 PM | Permalink | Reply to this

Re: Dynamics of Mathematical Reason

Good, thanks for this. Maybe one of the problems is that I’ve set myself a near impossible task.

If you’ve read what I’ve written in philosophy of mathematics, you’ll know that a constant refrain of mine has been that philosophy, especially the English-language tradition of the past century, has wrongly not wanted to know anything about the inner life of mathematics. This attitude has largely grown out of the founding act of analytic philosophy, the apparent reduction of mathematics to logic, often presented as the completion by Frege and Russell of the nineteenth century project to rigorise mathematics by their reduction of arithmetic to logic.

Where the majority wish to work over the fall-out from this founding act, perhaps working on incompleteness results, or reviving Frege with Hume’s principle, the consequence that this inner life is not to be treated has struck me as a reductio, a proof by contradiction that something must be missing in what frames the agenda.

We’ve headed down this track for so long now that it’s hard to envisage internally-driven change. What I rather suspect will happen is that philosophers of science will prompt some movement, and to be specific I think it will come from philosophers of physics taking onboard the kind of mathematics we talk about here.

But there’s still a problem, since even if philosophers of science become au fait with what we do here, English-language philosophy of science largely derives from a world view founded on the reduction of mathematics to logic. From this perspective there’s mathematics/logic as a meaningless calculating apparatus, which only gains philosophically interesting meaning when interpreted physically.

Now Michael Friedman is someone I consider to be one of the very best of contemporary philosophers of science, someone who understands physicists as historically situated thinkers, trying to create the right story for the next chapter in the narrative of physics. And while in this book Dynamics of Reason mathematics is allowed to play a more substantial role than mere scaffolding for physical understanding, it never quite extends to being the fully fledged partner that it is in reality. There’s still no sense of the dynamic interplay between mathematics and physics, each with their rich inner lives. So I focused especially on one claimed point of divergence between the disciplines from the lower half of page 10.

I think that probably the space of a typical journal article is not sufficient to do what is necessary, in that too much meta-philosophical work must first be done to explain its purpose.

Posted by: David Corfield on March 2, 2007 11:29 PM | Permalink | Reply to this

Re: Dynamics of Mathematical Reason

David said:

Maybe one of the problems is that I’ve set myself a near impossible task.

I think that probably the space of a typical journal article is not sufficient to do what is necessary, in that too much meta-philosophical work must first be done to explain its purpose.

Gee, David, you do give up easily.

If ‘what is necessary’ is to launch a whole new research programme, and have people flock to your standard, then yeah, probably this is not a good format to do it in. (Maybe something more on the lines of “Once more unto the breach, dear friends, once more!” or, better still, “gentlemen in England now-a-bed shall think themselves accurs’d they were not here” would do to stir up enuthusiasm.)

However, I’ve read the paper once again, and I still feel that, rather than attempting the impossible, it would be better to cut down on the rather large number of ideas currently in it and focus more tightly on one of them. Or maybe it needs to be more explicit about the ideas and how they are related. Or something else to make it clearer what you are for, as well as what you are against and what you are worried about. When I skim a paper, deciding whether to read it, I usually try to construct my own sort of skeleton or abstract of its basic line of argument, but I haven’t been able to do that with yours. I could go through it with a pencil and try to extract what I think you’re saying at each point, I guess. And then try to piece it together in my own mind.

Or maybe you think this wouldn’t be very productive? I seem to have been rather negative. I don’t think I’ve ever really properly understood your outlook. It seems kind of alien to me. So maybe I’m not the best person to be trying to offer help. :-(

Posted by: Tim Silverman on March 3, 2007 10:16 PM | Permalink | Reply to this

Re: Dynamics of Mathematical Reason

And I remember seeing somewhere a mathematician comment that Goedel’s incompleteness theorem pried logic out of the domain of philosophy and put it into mathematics. There’s a fundamental difference in these worldviews.

Posted by: Peter Shor on March 3, 2007 12:02 AM | Permalink | Reply to this

Re: Dynamics of Mathematical Reason

David, I agree with you when you say that the world view based on mathematics being reduced to logic is where the rot set in. It seems what people are missing is the emergent structure that appears on top of basic systems.

The emergent structures are what we call research projects, or schools of thought. Some schools are very closely tied in with the mathematics used to describe the (real?) world; others follow their own evolutionary path, perhaps under the aegis of a luminary figure, or aiming at some goal. Various schools have come and gone in fashion, some never to be revived.

Perhaps string theory is going to be one of those schools; it was given cohesion by some interesting results that tantalised those working with the idea that it described quantum gravity. A social network arose with guiding luminaries and the research project is still going.

Perhaps these emergent properties, which I know I haven’t really discussed at adequate length, are what you call the “inner life” of mathematics and the “internally-driven change”. But something like this perspective has to be recognised, otherwise all mathematical activity is just some sort of proof machine, starting with logic and gradually proving more and more “tautologies” with no particular direction. We know mathematics is not this sort of machine - there is an emergent structure, and in France (maybe it’s the wine) the emergent structure can even spring to life, as with M. Bourbaki.

Posted by: Frank Ashe on March 3, 2007 11:13 AM | Permalink | Reply to this

Re: Dynamics of Mathematical Reason

But something like this perspective has to be recognised, otherwise all mathematical activity is just some sort of proof machine, starting with logic and gradually proving more and more “tautologies” with no particular direction.

That’s exactly it. But how to convince the mainstream that what goes beyond the proof machine is worth philosophical attention?

As with any human enterprise, value judgements are an integral part of mathematics. I find it hard to conceive that one could treat this subject philosophically without needing to treat its values.

Can there really be an argument against this necessity? Philosophers of science study scientific values.

I should add that some work has been done on the idea of a proof being ‘explanatory’. But I want those very high level, research programme-level, values to be treated too.

The emergent structures are what we call research projects, or schools of thought.

We had a discussion about schools of thought in maths here.

Posted by: David Corfield on March 4, 2007 10:04 AM | Permalink | Reply to this

Re: Dynamics of Mathematical Reason

p. 5: “Freedman Dyson” should be “Freeman Dyson”

p.6: Feynman admitted that Feynman diagrams were somewhat ad hoc, but he had no great interest in the attempts of orthodox mathemasticians to “clean up” the mathematics, which attempts have proceeded in several different directions. Similarly, once could say that Newton’s fluxions were somewhat ad hoc, but that Calculus has been cleaned up since then.

Feynman did not necessarily believe or disbelieve in von Neumann’s quantum logic, but had his own “internal” reasons for suggesting that quantum computers might be the proper substrate in which to ds quantum calculations.

Worth mentioning the von Neumann universe? “…In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets is the class of all sets, divided into a transfinite hierarchy of individual sets. It is also sometimes called the cumulative hierarchy….”

As wikipedia comments on this:

“There are two distinct approaches to understanding the relationship of the von Neumann universe V to ZFC (and many variations of each approach, and shadings between them). Roughly, formalists will tend to view V as something that flows from the ZFC axioms (for example, ZFC proves that every set is in V). On the other hand, realists are more likely to see the von Neumann hierarchy as something directly accessible to the intuition, and the axioms of ZFC as propositions for whose truth in V we can give direct intuitive arguments in natural language. A possible middle position is that the mental picture of the von Neumann hierarchy provides the ZFC axioms with a motivation (so that they are not arbitrary), but does not necessarily describe objects with real existence.”

On Grothedeick, is it worth mentioning his “inner life” moved from Math to leftist French politics?

Lakatos on polyhedra might need comment on star polyhedra, infinite polyhedra, sections of polytopes, and other things that Plato or Pythagoras might not have taken as Math.

Today (3 March) is the birthday of Georg Cantor, by the way.

Worth going from Ising models to Lattice models in general, and of higher dimensionality?

p.15: I’d be interested to see more on Habermas, not just as filtered through Friedman. Habermas appeals to people I know who are trying to develop meta-theories of communicative networks in online collaboration, random graphs in the in social network theory, and phase changes in random networks somewhat different from the Renormalization Group.

Posted by: Jonathan Vos Post on March 4, 2007 2:24 AM | Permalink | Reply to this

Re: Dynamics of Mathematical Reason

David,

Here is a very simple take on the subject, perhaps so simple that it will expose the admitted fact that I know very little of mathematical philosophy.

The view that mathematics is just a machine for demonstrating that complicated theorems and surprising truths are reducible to logical tautology misses the real creative work of guessing just what axioms should be included in just which definitions. If I read you right, you are saying that philosophers should take up the challenge of playing a part in this process. The argument for this is as follows:

1) The fact that philosophers played an important role in both fomenting and interpreting revolutions in physics.

2) The fact that the work of theoretical physics has intimate connections with mathematics in two ways:

a) Historically this work of choosing axioms is where physics and mathematics really intersect. Observations of physical systems, either real or ideal, are often the fertile ground in which axiomatic systems are born. Why do we want to be able to do arithmetic anyhow, if not to be able to add the number of rocks you hold to the number of those in my hand? This extends directly to the fancy mathematics of string theory, where we want to define the right sort of structures to model particle interactions.

b) Of course much of the creative work of mathematicians (here we ignore the often even greater creativity required in order to find proofs) is analogous to theoretical physics, in the sense that we look for axioms for definitions which allow us to model yet other mathematical systems. This is exemplified by the homotopy hypothesis, in which the correct definitions of n-groupoids are sought in order to model the existing definitions of homotopy types and loop spaces. The example you give is that of the axioms of category theory being developed in order to organize homology theories. Hence we speak of theoretical mathematics–a fancy term for the work of creating definitions which allow useful connections to be made between existing mathematical definitions (and the theorems which they imply) as well as useful connnections to physics.

Perhaps one way of interpreting your thesis is to say that current mathematical philosophy pays not enough attention to theoretical mathematics, thus missing the chance to play an important role in helping to chart its course. Is this the right idea? I imagine that you want to say even more than this, since it is largely commmon sense so far. Perhaps though, modern mathematical philosophy is in bad shape, fallen far from the days of Hans Reichenbach–although his (and Friedman’s) strength is probably in physics.

Posted by: Stefan on March 6, 2007 5:32 PM | Permalink | Reply to this

Re: Dynamics of Mathematical Reason

This is very useful, thanks.

As I discussed in the talk I mentioned here, I set for myself a two-fold task with regard to mathematics.

1) To see what mathematics is like as an example of a tradition of rational enquiry. To make comparisons with other traditions - e.g., the natural sciences - with respect to the elaboration of disciplinary values. To determine whether intellectual virtues are necessary, etc.

2) To participate in a particular research program of foundational interest, ‘foundational’ taken in the broad sense of dealing with fundamental concepts.

Neither of these is much treated at present in philosophy. As regards (1), the sole value to be given much attention is ‘truth’. This bring about ‘worries’ such as what are the objects of mathematics that I can know truths about them.

As regards (2), most activity has centred around different logical calculi, with no regard for the conceptual ‘grain’ of mathematics.

What excites me about n-category theory is that it does appear to run along the grain (or one of the grains, at least). But what role for a philosopher in connection to it? Back at this post, I was wondering what happens when a topic which has been a subject of philosophical attention takes on a life of its own. While philosophers of physics keep quite well up-to-date with developments in physics, and so can carry on sensibly talking about the arrow of time, etc., will they have to keep up with, say, Judea Pearl’s ‘Causality’ to continue contributing to that topic?

If we accept that philosophers can keep up-to-date with such topics, why not mathematical ones? If fundamental changes are taking place to notions of sameness/identity, I don’t see why philosophers shouldn’t be prodding about trying to find a coherent story. E.g., where is the mathematical concept of ‘space’ heading?

As to whether we should expect philosophers to be of much service, we might note that the candidates Friedman puts forward for the meta-scientific work leading to GTR were philosophically-interested scientists rather than philosophers (Poincaré, Helmholtz, Hertz). Perhaps when we look back at the current time the corrsponding names will be Rovelli, Sorkin, Witten, Isham or Smolin, (not to leave out Schreiber and Baez). But it wouldn’t surprise me if whoever it is was happy hanging out with philosophers.

Maybe this might also be the case in mathematics. Mac Lane was certainly interested in philosophy while in Germany.

Posted by: David Corfield on March 7, 2007 4:57 PM | Permalink | Reply to this

Re: Dynamics of Mathematical Reason

These comments may not help in getting your paper published, David, but they may help me understand the points your driving at. I ramble a bit because I don’t really know what I’m going to say until I type it.

One of the problems you allude to is that in your point (1) “truth” is just about the only value to which philosophical attention is given. For point (2), activity is concentrated on different logical calculi. It seems to me that n-category theory is producing results that philosophers need to come to speed on to tackle their own foundational ideas; in much the same way that philosophers need to understand biology, especially evolution, to start to make headway with epistemology and ethics.

N-categories are giving us insights into how we can define similarities in a much richer framework than the usual schools of logic. As you say, this then starts to morph the ideas of “truth” into a much richer structure. Philosophers need to work on this concept a bit and feed it back into category theory.

Baez’s paper on Quantum Quandaries should be a good introduction for a philosopher to see why they’d better get on-board, by showing that the world is not like the category Set, which I’m sure is the unquestioned assumption of 99.99% of philosophers. But if the world is like nCob and Hilb then what does this mean? What parts of metaphysics do we need to reinterpret? Any of it?

So, as well as running along a natural grain in mathematics, n-category theory is extending philosophy along the ideas of “truth”, “resemblance”, as well as positing a fundamentally different metaphysical foundation for the world than the naive ideas of sets that are the current underlay. This should be philosopher’s heaven.

Perhaps the reason we’re not seeing much interest from the philosophers is akin to Smolin’s argument as to how string theory took over physics in US schools - it’s difficult for a maverick to get a tenured position, and to understand the mathematics and philosophy to sufficient depth you need to be a bit of a maverick (and have difficulty getting papers published!).

Posted by: Frank Ashe on March 9, 2007 1:18 PM | Permalink | Reply to this

Re: Dynamics of Mathematical Reason

An excellent summary.

This should be philosopher’s heaven.

Exactly! That’s what’s so frustrating.

Thinking about the parallels you mention between physics and philosophy with regard to being a maverick, it’s not that I feel out on a limb with regard to all philosophers. Strangely I find greater affinity with those apparently quite distant and this seems to be reciprocated, see e.g., this post by Brandon, a philosopher who works on theology.

Also worth considering is whether Bertrand Russell would succeed today. Or whether he would have succeeded if he hadn’t been an Earl. I can imagine the dons at Cambridge, “So you’ve been abroad in Germany, and you’ve come back to tell us that this new ‘logic’ devised by an obscure mathematician from Jena, rewritten in the notation of an obscure Italian, is to be the answer to all our problems.”

Posted by: David Corfield on March 9, 2007 5:56 PM | Permalink | Reply to this
Read the post Where is the Philosophy of Physics?
Weblog: The n-Category Café
Excerpt: Should we devote more time to the philosophy of physics?
Tracked: May 19, 2009 2:18 PM

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