The n-Category Café

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.

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.

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.

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.

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.