Writing sizeable chunks of math blog takes longer than I expected. Here are a few non-intensive things that occurred to me.
While I was walking home from the UWO campus, I was reminded of the nature of Canada in late November: everything, from sky to plantlife to earth, is in shades of grey, brown, ochre, and the occasional desaturated greenish-whatever. Autumn leaves have pretty much stopped falling, and are on the ground turning greyish versions of whatever colours they were before. There are whole vistas of bare branches, dead underbrush, and so on.
Which seems dreary for a while, until you’re immersed in it, as I am on the particular route I walk home, along London, Ontario’s Thames River (not to be confused with the River Thames in London, England), which is lined with parks. Then, after a while, all the subtle differences in shading and texture start to jump out at you more and more, until brownish moss on a tree under overcast late-afternoon light is vibrant green, a patch of snow is glowing bluish white, the occasional flicker of sunset through the cloud cover is warm pumpkin-orange, that one particular bush’s leaves look startlingly red… and then you see something artificial, like someone’s nylon jacket, or a kid’s plastic play-structure, and their colours look implausibly oversaturated, like a badly photoshopped picture.
Which got me to thinking about fine distinctions that seem drab outside their context – the way these colours look at first. Or nitpicky, like having 30 different words for “cold” and the different qualities it can have, or recognizing 15 different types of snowflake from a distance. Coming back to Canada after several years in California, I noticed all this specialized knowledge I’d forgotten about, and seems terribly arcane outside its native habitat. It occurred to me that this is how mathematics probably seems to outsiders – like physicists, or statisticians… (I jest)
For instance: I often have the experience of using the term “categorification” in describing something I’m doing – often in scare-quotes, followed by some kind of explanation – only to have it echoed back as “categorization”, and wonder whether to risk pedantry and explain that they’re not the same thing at all. “Categorification, not to be confused with…”
On another note, I went looking for this paper by Carter, Kauffman and Saito, on a kind of invariant of 4-manifolds which generalizes 3D Dijkgraaf-Witten invariants, on the supposition that it would be closely related to some things I’ve been thinking about, from a diagrammatic point of view I’ve not paid much attention to in the last year or so. As I was looking through seach results, I noticed a paper from about 10 years ago by Kauffman and Smolin with an interesting sounding title, A Possible Solution to the Problem of Time in Quantum Cosmology. Since Lee Smolin has written on linking topological field theory and quantum gravity, I guessed it would also be interesting to look at. Only after reading the first few pages did I notice that the first listed author was not Louis Kauffman, who studies knot theory (and things tangent thereto), but Stuart Kauffman, who studies biocomplexity and complex systems.
I happen to be interested in the work of both Kauffmans – more immediately and professionally that of Louis, but I also read a couple of Stuart’s more accessible books, “At Home in the Universe”, and “Investigations” – and since the paper was short, I finished reading it. The basic premise is that the configuration space for 4D quantum gravity may not be constructible by any finite procedure (classifying spin networks, they say, might present a problem; doing path integrals over all 4-manifold topologies certainly does). So the “problem of time”, that there’s no role for time in describing dynamics in terms of paths through a configuration space, wouldn’t make sense – at least for a constructivist. (Or indeed a constructivist, though of course they shouldn’t be confused.) One thing that threw me off in noticing which Kauffman was involved was that part of this portion of the argument was about classifying knots.
That cleared itself up when they got to the part proposing a solution – that the total space of possible states isn’t a-priori given, but time re-enters the situation as the universe evolves, at each time step having some amplitude to move into each configuration in a (newly defined!) space called the adjacent possible. Having read Stuart K.’s books, this was when I realized my mistake – he describes this concept in “Investigations” in the context of a biosphere, or an economy, where a theorist also doesn’t have an explicit description of all possible future states given in advance.
It seems like this idea has a lot in common with type theory as a solution to Russell’s paradox: the collection of all sets isn’t a set, and so to get at it, sets are generated starting with nothing in successive stages. Whether this also doubles as a solution to the problem of time, I don’t know. In any case, it’s an interesting idea. It definitely would be a problem to have to do path integrals over a space of all topologies for 4-manifolds, when these can’t be classified, so some sort of suggestions are definitely a good thing here.