### meta

There is no abiding thing in what we know. We change from weaker to stronger lights, and each more powerful light pierces our hitherto opaque foundations and reveals fresh and different opacities below. We can never foretell which of our seemingly assured fundamentals the next change will not affect.

H.G. Wells, A Modern Utopia

So there’s a recent paper by some physicists, two of whom work just across the campus from me at IST, which purports to explain the Pioneer Anomaly, ultimately using a computer graphics technique, Phong shading. The point being that they use this to model more accurately than has been done before how much infrared radiation is radiating and reflecting off various parts of the Pioneer spacecraft. They claim that with the new, more accurate model, the net force from this radiation is just enough to explain the anomalous acceleration.

Well, plainly, any one paper needs to be rechecked before you can treat it as definitive, but this sort of result looks good for conventional General Relativity, when some people had suggested the anomaly was evidence some other theory was needed.  Other anomalies in the predictions of GR – the rotational profiles of galaxies, or redshift data, have also suggested alternative theories.  In order to preserve GR exactly on large scales, you have to introduce things like Dark Matter and Dark Energy, and suppose that something like 97% of the mass-energy of the universe is otherwise invisible.  Such Dark entities might exist, of course, but I worry it’s kind of circular to postulate them on the grounds that you need them to make GR explain observations, while also claiming this makes sense because GR is so well tested.

In any case, this refined calculation about Pioneer is a reminder that usually the more conservative extension of your model is better. It’s not so obvious to me whether a modified theory of gravity, or an unknown and invisible majority of the universe is more conservative.

And that’s the best segue I can think of into this next post, which is very different from recent ones.

Fundamentals

I was thinking recently about “fundamental” theories.  At the HGTQGR workshop we had several talks about the most popular physical ideas into which higher gauge theory and TQFT have been infiltrating themselves recently, namely string theory and (loop) quantum gravity.  These aren’t the only schools of thought about what a “quantum gravity” theory should look like – but they are two that have received a lot of attention and work.  Each has been described (occasionally) as a “fundamental” theory of physics, in the sense of one which explains everything else.  There has been a debate about this, since they are based on different principles.  The arguments against string theory are various, but a crucial one is that no existing form of string theory is “background independent” in the same way that General Relativity is. This might be because string theory came out of a community grounded in particle physics – it makes sense to perturb around some fixed background spacetime in that context, because no experiment with elementary particles is going to have a measurable effect on the universe at infinity. “M-theory” is supposed to correct this defect, but so far nobody can say just what it is.  String theorists criticize LQG on various grounds, but one of the more conceptually simple ones would be that it can’t be a unified theory of physics, since it doesn’t incorporate forces other than gravity.

There is, of course, some philosophical debate about whether either of these properties – background independence, or unification – is really crucial to a fundamental theory.   I don’t propose to answer that here (though for the record my hunch at he moment is that both of them are important and will hold up over time).  In fact, it’s “fundamental theory” itself that I’m thinking about here.

As I suggested in one of my first posts explaining the title of this blog, I expect that we’ll need lots of theories to get a grip on the world: a whole “atlas”, where each “map” is a theory, each dealing with a part of the whole picture, and overlapping somewhat with others. But theories are formal entities that involve symbols and our brain’s ability to manipulate symbols. Maybe such a construct could account for all the observable phenomena of the world – but a-priori it seems odd to assume that. The fact that they can provide various limits and approximations has made them useful, from an evolutionary point of view, and the tendency to confuse symbols and reality in some ways is a testament to that (it hasn’t hurt so much as to be selected out).

One little heuristic argument – not at all conclusive – against this idea involves Kolmogorov complexity: wanting to explain all the observed data about the universe is in some sense to “compress” the data.  If we can account for the observations – say, with a short description of some physical laws and a bunch of initial conditions, which is what a “fundamental theory” suggests – then we’ve found an upper bound on its Kolmogorov complexity.  If the universe actually contains such a description, then that must also be a lower bound on its complexity.  Thus, any complete description of the universe would have to be as big as the whole universe.

Well, as I said, this argument fails to be very convincing.  Partly because it assumes a certain form of the fundamental theory (in particular, a deterministic one), but mainly because it doesn’t rule out that there is indeed a very simple set of physical laws, but there are limits to the precision with which we could use them to simulate the whole world because we can’t encode the state of the universe perfectly.  We already knew that.  At most, that lack of precision puts some practical limits on our ability to confirm that a given set of physical laws we’ve written down is  empirically correct.  It doesn’t preclude there being one, or even our finding it (without necessarily being perfectly certain).  The way Einstein put it (in this address, by the way) was “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”  But a lack of certainty doesn’t mean they aren’t there.

However, this got me thinking about fundamental theories from the point of view of epistemology, and how we handle knowledge.

Reduction

First, there’s a practical matter. The idea of a fundamental theory is the logical limit of one version of reductionism. This is the idea that the behaviour of things should be explained in terms of smaller, simpler things. I have no problem with this notion, unless you then conclude that once you’ve found a “more fundamental” theory, the old one should be discarded.

For example: we have a “theory of chemistry”, which says that the constituents of matter are those found on the periodic table of elements.  This theory comes in various degrees of sophistication: for instance, you can start to learn the periodic table without knowing that there are often different isotopes of a given element, and only knowing the 91 naturally occurring elements (everything up to Uranium, except Technicium). This gives something like Mendeleev’s early version of the table. You could come across these later refinements by finding a gap in the theory (Technicium, say), or a disagreement with experiment (discovering isotopes by measuring atomic weights). But even a fairly naive version of the periodic table, along with some concepts about atomic bonds, gives a good explanation of a huge range of chemical reactions under normal conditions. It can’t explain, for example, how the Sun shines – but it explains a lot within its proper scope.

Where this theory fits in a fuller picture of the world has at least two directions: more fundamental, and less fundamental, theories.  What I mean by less “fundamental” is that some things are supposed to be explained by this theory of chemistry: the great abundance of proteins and other organic chemicals, say. The behaviour of the huge variety of carbon compounds predicted by basic chemistry is supposed to explain all these substances and account for how they behave.  The millions of organic compounds that show up in nature, and their complicated behaviour, is supposed to be explained in terms of just a few elements that they’re made of – mostly carbon, hydrogen, oxygen, nitrogen, sulfur, phosphorus, plus the odd trace element.

By “more fundamental”, I mean that the periodic table itself can start to seem fairly complicated, especially once you start to get more sophisticated, including transuranic elements, isotopes, radioactive decay rates, and the like. So it was explained in terms of a theory of the atom. Again, there are refinements, but the Bohr model of the atom ought to do the job: a nucleus made of protons and neutrons, and surrounded by shells of electrons.  We can add that these are governed by the Dirac equation, and then the possible states for electrons bound to a nucleus ought to explain the rows and columns of the periodic table. Better yet, they’re supposed to explain exactly the spectral lines of each element – the frequencies of light atoms absorb and emit – by the differences of energy levels between the shells.

Well, this is great, but in practice it has limits. Hardly anyone disputes that the Bohr model is approximately right, and should explain the periodic table etc. The problem is that it’s largely an intractable problem to actually solve the Schroedinger equation for the atom and use the results to predict the emission spectrum, chemical properties, melting point, etc. of, say, Vanadium…  On the other hand, it’s equally hard to use a theory of chemistry to adequately predict how proteins will fold. Protein conformation prediction is a hard problem, and while it’s chugging along and making progress, the point is a theory of chemistry alone isn’t enough: any successful method must rely on a whole extra body of knowledge.  This suggests our best bet at understanding all these phenomena is to have a whole toolbox of different theories, each one of which has its own body of relevant mathematics, its own domain-specific ontology, and some sense of how its concepts relate to those in other theories in the tookbox. (This suggests a view of how mathematics relates to the sciences which seems to me to reflect actual practice: it pervades all of them, in a different way than the way a “more fundamental” theory underlies a less fundamental one.  Which tends to spoil the otherwise funny XKCD comic on the subject…)

If one “explains” one theory in terms of another (or several others), then we may be able to put them into at least a partial order.  The mental image I have in mind is the “theoretical atlas” – a bunch of “charts” (the theories) which cover different parts of a globe (our experience, or the data we want to account for), and which overlap in places.  Some are subsets of others (are completely explained by them, in principle). Then we’d like to find a minimal (or is it maximal) element of this order: something which accounts for all the others, at least in principle.  In that mental image, it would be a map of the whole globe (or a dense subset of the surface, anyway).  Because, of course, the Bohr model, though in principle sufficient to account for chemistry, needs an explanation of its own: why are atoms made this way, instead of some other way? This ends up ramifying out into something like the Standard Model of particle physics.  Once we have that, we would still like to know why elementary particles work this way, instead of some other way…

An Explanatory Trilemma

There’s a problem here, which I think is unavoidable, and which rather ruins that nice mental image.  It has to do with a sort of explanatory version of Agrippa’s Trilemma, which is an observation in epistemology that goes back to Agrippa the Skeptic. It’s also sometimes called “Munchausen’s Trilemma”, and it was originally made about justifying beliefs.  I think a slightly different form of it can be applied to explanations, where instead of “how do I know X is true?”, the question you repeatedly ask is “why does it happen like X?”

So, the Agrippa Trilemma as classically expressed might lead to a sequence of questions about observation.  Q: How do we know chemical substances are made of elements? A: Because of some huge body of evidence. Q: How do we know this evidence is valid? A: Because it was confirmed by a bunch of experimental data. Q: How do we know that our experiments were done correctly? And so on. In mathematics, it might ask a series of questions about why a certain theorem is true, which we chase back through a series of lemmas, down to a bunch of basic axioms and rules of inference. We could be asked to justify these, but typically we just posit them. The Trilemma says that there are three ways this sequence of justifications can end up:

1. we arrive at an endpoint of premises that don’t require any justification
2. we continue indefinitely in a chain of justifications that never ends
3. we continue in a chain of justifications that eventually becomes circular

None of these seems to be satisfactory for an experimental science, which is partly why we say that there’s no certainty about empirical knowledge. In mathematics, the first option is regarded as OK: all statements in mathematics are “really” of the form if axioms A, B, C etc. are assumed, then conclusions X, Y, Z etc. eventually follow. We might eventually find that some axioms don’t apply to the things we’re interested in, and cease to care about those statements, but they’ll remain true. They won’t be explanations of anything very much, though.  If we’re looking at reality, it’s not enough to assume axioms A, B, C… We also want to check them, test them, see if they’re true – and we can’t be completely sure with only a finite amount of evidence.

The explanatory variation on Agrippa’s Trilemma, which I have in mind, deals with a slightly different problem.  Supposing the axioms seem to be true, and accepting provisionally that they are, we also have another question, which if anything is even more basic to science: we want to know WHY they’re true – we look for an explanation.

This is about looking for coherence, rather than confidence, in our knowledge (or at any rate, theories). But a similar problem appears. Suppose that elementary chemistry has explained organic chemistry; that atomic physics has explained why chemistry is how it is; and that the Standard model explains why atomic physics is how it is.  We still want to know why the Standard Model is the way it is, and so on. Each new explanation gives an account for one phenomenon in terms of different, more basic phenomenon. The Trilemma suggests the following options:

1. we arrive at an endpoint of premises that don’t require any explanation
2. we continue indefinitely in a chain of explanations that never ends
3. we continue in a chain of explanations that eventually becomes circular

Unless we accept option 1, we don’t have room for a “fundamental theory”.

Here’s the key point: this isn’t even a position about physics – it’s about epistemology, and what explanations are like, or maybe rather what our behaviour is like with regard to explanations. The standard version of Agrippa’s Trilemma is usually taken as an argument for something like fallibilism: that our knowledge is always uncertain. This variation isn’t talking about the justification of beliefs, but the sufficiency of explanation. It says that the way our mind works is such that there can’t be one final summation of the universe, one principle, which accounts for everything – because it would either be unaccounted for itself, or because it would have to account for itself by circular reasoning.

This might be a dangerous statement to make, or at least a theological one (theology isn’t as dangerous as it used to be): reasoning that things are the way they are “because God made it that way” is a traditional answer of the first type. True or not, I don’t think you can really call an “explanation”, since it would work equally well if things were some other way. In fact, it’s an anti-explanation: if you accept an uncaused-cause anywhere along the line, the whole motivation for asking after explanations unravels.  Maybe this sort of answer is a confession of humility and acceptance of limited understanding, where we draw the line and stop demanding further explanations. I don’t see that we all need to draw that line in the same place, though, so the problem hasn’t gone away.

What seems likely to me is that this problem can’t be made to go away.  That the situation we’ll actually be in is (2) on the list above.  That while there might not be any specific thing that scientific theories can’t explain, neither could there be a “fundamental theory” that will be satisfying to the curious forever.  Instead, we have an asymptotic approach to explanation, as each thing we want to explain gets picked up somewhere along the line: “We change from weaker to stronger lights, and each more powerful light pierces our hitherto opaque foundations and reveals fresh and different opacities below.”

So for my inaugural blog post of 2009, I thought I would step back and comment about the big picture of the motivation behind what I’ve been talking about here, and other things which I haven’t. I recently gave a talk at the University of Ottawa, which tries to give some of the mathematical/physical context. It describes both “degroupoidification” and “2-linearization” as maps from spans of groupoids into (a) vector spaces, and (b) 2-vector spaces. I will soon write a post setting out the new thing in case (b) that I was hung up on for a while until I learned some more representation theory. However, in this venue I can step even further back than that.

Over the Xmas/New Year break, I was travelling about “The Corridor” (the densely populated part of Canada – London, where I live, is toward one end, and I visited Montreal, Ottawa, Toronto, Kitchener, and some of the areas in between, to see family and friends). Between catching up with friends – who, naturally, like to know what I’m up to – and the New Year impulse to summarize, and the fact that I’m applying for jobs these days, I’ve had occasion to think through the answer to the question “What do you work on?” on a few different levels. So what I thought i’d do here is give the “Cocktail Party Version” of what it is I’m working on (a less technical version of my research statement, with some philosophical asides, I guess).

In The Middle

The first thing I usually have to tell people is that what I work on lives in the middle – somewhere between mathematics and physics. Having said that, I have to clear up the fact that I’m a mathematician, rather than a physicist. I approach questions with a mathematician’s point of view – I’m interested in making concepts precise, proving facts about them rigorously, and so on. But I do find it helps to motivate this activity to suppose that the concepts in question apply to the real world – by which I mean, the physical world.

(That’s a contentious position in itself, obviously. Platonists, Cartesian dualists, and people who believe in the supernatural generally don’t accept it, for example. For most purposes it doesn’t matter, but my choice about what to work on is definitely influenced by the view that mathematical concepts don’t exist independently of human thought, but the physical world does, and the concepts we use today have been selected – unconsciously sometimes, but for the most part, I think, on purpose – for their use in describing it. This is how I account for the supposedly unreasonable effectiveness of mathematics – not really any more surprising than the remarkable effectiveness of car engines at turning gasoline into motion, or that steel girders and concrete can miraculously hold up a building. You can be surprised that anything at all might work, but it’s less amazing that the thing selected for the job does it well.)

Physics

The physical world, however, is just full of interesting things one could study, even as a mathematician. Biology is a popular subject these days, which is being brought into mathematics departments in various ways. This involves theoretical study of non-equilibrium thermodynamics, the dynamics of networks (of chemical reactions, for example), and no doubt a lot of other things I know nothing about. It also involves a lot of detailed modelling and computer simulation. There’s a lot of profound mathematical engagement with the physical world here, and I think this stuff is great, but it’s not what I work on. My taste in research questions is a lot more foundational. These days, the physical side of the questions I’m thinking about has more to do with foundations of quantum mechanics (in the guise of 2-Hilbert spaces), and questions related to quantum gravity.

Now, recently, I’ve more or less come around to the opinion that these are related: that part of the difficulty of finding a good theory accomodating quantum mechanics and general relativity comes from not having a proper understanding of the foundations of quantum mechanics itself. It’s constantly surprising that there are still controversies, even, over whether QM should be understood as an ontological theory describing what the world is like, or an epistemological theory describing the dynamics of the information about the world known to some observer. (Incidentally – I’m assuming here that the cocktail party in question is one where you can use the word “ontological” in polite company. I’m told there are other kinds.)

Furthermore, some of the most intractable problems surrounding quantum gravity involve foundational questions. Since the language of quantum mechanics deals with the interactions between a system and an observer, so applying it to the entire universe (quantum cosmology) is problematic. Then there’s the problem of time: quantum mechanics (and field theory), both old-fashioned and relativistic, assume a pre-existing notion of time (either a coordinate, or at least a fixed background geometry), when calculating how systems (including fields) evolve. But if the field in question is the gravitational field, then the right notion of time will depend on which solution you’re looking at.

Category Theory

So having said the above, I then have to account for why it is that I think category theory has anything to say to these fundamental issues. This being the cocktail party version, this has to begin with an explanation of what category theory is, which is probably the hardest part. Not so much because the concept of a category is hard, but because as a concept, it’s fairly abstract. The odd thing is, individual categories themselves are in some ways more concrete than the “decategorified” nubbins we often deal with. For example, finite sets and set maps are quite concrete: here are four sheep, and here four rocks, and here is a way of matching sheep with rocks. Contrast that with the abstract concept of the pure number “four” – an element in the set of cardinalities of finite sets, which gets addition and multiplication (abstractly defined operations) from the very concrete concepts of union and product (set of pairs) of sets. Part of the point of categorification is to restore our attention to things which are “more real” in this way, by giving them names.

One philosophical point about categories is that they treat objects and morphisms (which, for cocktail party purposes, I would describe as “relations between objects”) as equally real. Since I’ve already used the word, I’ll say this is an ontological commitment (at least in some domain – here’s an issue where computer science offers some nicely structured terminology) to the existence of relations as real. It might be surprising to hear someone say that relations between things are just as “real” as things themselves – or worse, more real, albeit less tangible.  Most of us are used to thinking of relations as some kind of derivative statement about real things. On the other hand, relations (between subject and object, system and observer) are what we have actual empirical evidence for. So maybe this shouldn’t be such a surprising stance.

Now, there are different ways category theory can enter into this discussion. Just to name one: the causal structure of a spacetime (a history) is a category – in particular, a poset (though we might want to refine that into a timelike-path category – or a double category where the morphisms are timelike and spacelike paths). Another way category theory may come in is as the setting for representation theory, which comes up in what I’ve been looking at. Here, there is some category representing a specific physical system – for example, a groupoid which represents the pure states of a system and their symmetries. Then we want to describe that system in a more universal way – for example, studying it by looking at maps (functors) from that category into one like Hilb, which isn’t tied to the specific system. The underlying point here is to represent something physical in terms of the sort of symbolic/abstract structures which we can deal with mathematically. Then there’s a category of such representations, whose morphisms (intertwiners in some suitably general sense) are ways of “changing coordinates” which get along with what’s important about the system.

The Point

So by “The Point”, I mean: how this all addresses questions in quantum mechanics and gravity, which I previously implied it did (or could). Let me summarize it by describing what happens in the 3D quantum gravity toy model developed in my thesis. There, the two levels (object and morphism) give us two concepts of “state”: a state in a 2-Hilbert space is an object in a category. Then there’s a “2-state” (which is actually more like the usual QM concept of a state): this is a vector in a Hilbert space, which happens to be a component in a 2-linear map between 2-vector spaces. In particular, a “state” specifies the geometry of space (albeit, in 3D, it does this by specifying boundary conditions only). A “2-state” describes a state of a quantum field theory which lives on that background.

Here is a Big Picture conjecture (which I can in no way back up at the moment, and reserve the right to second-guess): the division between “state and 2-state” as I just outlined it should turn out to resolve the above questions about the “problem of time”, and other philosophical puzzles of quantum gravity. This distinction is most naturally understood via categorification.

(Maybe. It appears to work that way in 3D. In the real world, gravity isn’t topological – though it has a limit that is.)

Meta

I like writing things for this blog, and so far I’ve tended to go in for long-ish comments or explanations of things. I’d definitely like to keep doing that, since it’s helpful to put some thoughts in order and get them out there. I also would like to try writing some shorter update-style entries about little things. One reason is that it might be useful to sit down more often with the plan of writing something here.

I’m also thinking it might be interesting to broaden out the range of things I write about, since my original theme is pretty open-ended. We’ll see how that goes.

Update

I started sitting in on Masoud Khalkhali’s Noncommutative Geometry seminar for this term, which is mostly attended by his graduate students. This term, at least to begin with, he’s planning to fill them in on some basics of operator algebras, (C*-algebras, von Neumann algebras, classification theorems, etc.) I’d like to learn some of this in any case – not least because it’s used in a lot of the standard work on groupoids. The first seminar was a refresher on the basics of Hilbert spaces – also well worth going through from the beginning again since I’ve been thinking about what’s involved in categorification of infinite-dimensional Hilbert spaces.

So we looked at the basic definitions and a little of the history of how von Neumann came up with the idea of a Hilbert space (hence the name) in the late 1920’s under the influence of Hilbert’s work on classical analysis, and the
then-recent work on quantum mechanics. (The reason Hilbert spaces seem to do just what quantum mechanics needs them to do (at the beginning) is that they were specifically designed for it). Some basic tools, like the (complex) polarization identity, the Cauchy-Schwartz inequality and its corollaries like the triangle inequality, and the important theorem that closed convex sets in Hilbert space have unique minimum-norm elements.

There was one question that came up which I’m not sure of the answer to. Remember that there’s a difference between a “Hilbert basis” and a vector-space basis for a Hilbert space $\mathcal{H}$. A Hilbert basis for $\mathcal{H}$ is just a maximal orthonormal set of vectors (which always exists, by an argument involving Zorn’s Lemma). A vector-space basis has to span $\mathcal{H}$: in other words, any vector must be expressible as a finite linear combination of basis vectors. So, in particular, in the infinite-dimensional case, the linear span of a Hilbert basis $B$ will only be a subspace of $\mathcal{H}$, since a vector might only be in the completion of the set of finite combinations of vectors in $B$.

So, for example, the space of square-summable sequences, $l^2(\mathbb{N})$, has a Hilbert basis consisting of sequences like $(0, \dots, 0, 1, 0 \dots)$ with just a single nonzero entry. This is a countably infinite set. However, not all square-summable sequences can be written as a finite linear combination of these. Using the axiom of choice (!) it’s possible to prove that there is some vector-space basis, but it will be a uncountably infinite. In fact, no Hilbert space has countably-infinite vector-space dimension.

Now, Hilbert spaces are determined up to isomorphism (which in this context means isometry) by their Hilbert dimension – the cardinality of a Hilbert basis. For finite dimensional spaces, this is the same as the vector space dimension, so there’s no trouble here. The question is: are Hilbert spaces determined up to isometry by their vector space dimension?

In other words: is the relationship between the cardinalities of a vector-space basis and a Hilbert basis for $\mathcal{H}$ a 1-1 relationship? Or can there be Hilbert spaces with different Hilbert dimension (therefore non-isometric) which have the same vector-space dimension? If there is such a counterexample, it will have at least one of the Hilbert spaces non-separable (i.e. no countable Hilbert basis).

In fact, since this question involves large infinite cardinals, I wouldn’t be terribly surprised to learn that the answer is indeterminate in ZFC, but neither do I see a good argument that it should be. A quick web-search doesn’t reveal the answer at once. Does anyone know?

I recently got back to London, Ontario from a trip to Ottawa, the first purpose of which was to attend the Ottawa Mathematics Conference. The other purpose was to visit family and friends, many of whom happen to be located there, which is one reason it’s taken me a week or so to get around to writing about the trip. Now, the OMC was a general-purpose conference, mainly for grad students, and some postdocs, to give short talks (plus a couple of invited faculty from Ottawa’s two universities – the University of Ottawa, and Carleton University – who gave lengthier talks in the mornings). This is not a type of conference I’ve been to before, so I wasn’t sure what to expect.

From one, fairly goal-oriented, point of view, the style of the conference seemed a little scattered. There was no particular topic of focus, for instance. On the other hand, for someone just starting out in mathematical research, this type of thing has some up sides. It gives a chance to talk about new work, see what’s being done across a range of subjects, and meet people in the region (in this case, mainly Ottawa, but also elsewhere across Eastern and Southern Ontario). The only other general-purpose mathematics conference I’ve been to so far was the joint meeting of the AMS in New Orleans in 2007, which had 5000 people and anyone attending talks would pick special sessions suiting their interests. I do think it’s worthwhile to find ways of circumventing the various pressures toward specialization in research – it may be useful in some ways, but balance is also good. Particularly for Ph.D. students, for whom specialization is the name of the game.

One useful thing – again, particularly for students – is the reminder that the world of mathematics is broader than just one’s own department, which almost certainly has its own specialties and peculiarities. For example, whereas here at UWO “Applied” mathematics (mostly involving computer modelling) is done in a separate department, this isn’t so everywhere. Or, again, while my interactions in the UWO department focus a lot on geometry and topology (there are active groups in homotopy theory and noncommutative geometry, for example), it’s been a while since I saw anyone talk about combinatorics, or differential equations. Since I actually did a major in combinatorics at U of Waterloo, it was kind of refreshing to see some of that material again.

There were a couple of invited talks by faculty. Monica Nevins from U of Ottawa gave a broad and enthusiastic survey of representation theory for graduate students. Brett Stevens from Carleton talked about “software testing”, which surprised me by actually being about combinatorial designs. Basically, it’s about the problem of how, if you have many variables with many possible values each, to design a minimal collection of “settings” for those variables which tests all possible combinations of, say, two variables (or three, etc.). One imagines the variables representing circumstances software might have to cope with – combinations of inputs, peripherals, and so on – so the combinatorial problem is if there are 10 variables with 10 possible values each, you can’t possibly test all 10 billion combinations – but you might be able to test all possible settings of any given PAIR of variables, and much more efficiently than just an exhaustive search, by combining some tests together.

Among the other talks were several combinatorial ones – error correcting codes using groups, path ideals in simplicial trees (which I understand to be a sort of generalization to simplicial sets of what trees are for graphs), heuristic algorithms for finding minimal cost collections of edges in weighted graphs that leave the graph with at least a given connectivity, and so on. Charles Starling from U of O gave an interesting talk about how to associate a topological space to an aperiodic tiling (roughly, any finite-size region in an aperiodic tiling is repeated infinitely many times – so the points of the space are translations, and two translations are within $\epsilon$ of one another if they produce matching regions about the origin of size $\frac{1}{\epsilon}$ – then the thing is to study cohomology of such spaces, and so forth).

The talk immediately following mine was by Mehmetcik Pamuk about homotopy self-equivalences of 4-manifolds, which used a certain braid of exact sequences of groups of automorphisms (among other things). I expected this to be very interesting, and it was certainly intriguing, but I can’t adequately summarize it – whatever he was saying, it proved to be hard to pick up from just a 25 minute talk. I did like something he said in his introduction, though: nowadays, if a topologist says they’re doing “low-dimensional” topology, they mean dimension 3, and “high-dimensional” means dimension 4. This is a glib but indicative way to point out that topology of manifolds in dimensions 1 and 2 is well understood (the connected components are, respectively, circles and n-holed tori), and in dimension 5 and above have been straightened out more recently thanks to Smale.

There were some quite applied talks which I missed, though I did catch one on “gravity waves”, which turn out not to be gravitational waves, but the kind of waves produced in fluids of varying density acted on by gravity. (In particular, due to layers of temperature and pressure in the atmosphere, sometimes denser air sits above less dense air, and gravity is trying to reverse this, producing waves. This produces those long rippling patterns you sometimes see in high-altitude clouds. Lidia Nikitina told us about some work modelling these in situations where the ground topography matters, such as near mountains – and had some really nice pictures to illustrate both the theory and the practice.)

On the second day there were quite a few talks of an algebraic or algebra-geometric flavour – about rings of algebraic invariants, about enumerating lines in special “blow-up” varieties, function fields associated to hyperelliptic curves, and so on – but although this is interesting, I had a harder time extracting informative things to say about these, so I’ll gloss over them glibly. However, I did appreciate the chance to gradually absorb a little more of this area of math by osmosis.

The flip side of seeing what many other people are doing was getting a chance to see what other people had to say about my own talk – about groupoids, spans, and 2-vector spaces. One of the things I find is that, while here at UWO the language of category theory is widely used (at least by the homotopy theorists and noncommutative geometry people I’ve been talking to), it’s not as familiar in other places. This seems to have been going on for some time – since the 1970’s if I understand the stories correctly. After MacLane and Eilenberg introduced categories in the 1940’s, the concept had significant effects in algebraic geometry/topology, homological algebra, and spread out from there. There was some deep enthusiasm – possibly well-founded, though I won’t claim so – that category theory was a viable replacement for set theory as a “foundation” for mathematics. True or not, that idea seemed to be one of those which was picked up by mathematicans who didn’t otherwise know much about category theory, and it seems to be one that’s still remembered. So maybe it had something to do with the apparent fall from fashion of category theory. I’ve heard that theory suggested before: roughly, that many mathematicians thought category theory was supposed to be a new foundation for mathematics, couldn’t see the point, and lost interest.

Now, my view of foundations is roughly suggested in my explanation of the title of this blog. I tend to think that our understanding of the world comes in bits and pieces, which we refine, then try to stick together into larger and more inclusive bits and pieces – the “Atlas” of charts of the title. This isn’t really just about the physical world, but the mathematical world as well (in fact I’m not really a Platonist who believes in a separate “world” of mathematical objects – though that’s a different conversation). This is really just a view of epistemology – namely, empirical methods work best because we don’t know things for sure, not being infinitely smart. So the “idealist”-style program of coming up with some foundational axioms (say, for set theory), and deriving all of mathematics from them without further reference to the outside doesn’t seem like the end of the story. It’s useful as a way of generating predictions in physics, but not of testing them. In mathematics, it generates many correct theorems, but doesn’t help identify interesting, or useful, ones.

So could category theory be used in foundations of mathematics? Maybe – but you could also say that mathematics consists of manipulating strings in a formal language, and strings are just words in a free monoid, so actually all of mathematics is the theory of monoids with some extra structure (giving rules of inference in the formal language). Yet monoid theory – indeed, algebra generally – is not mainly interesting as foundations, and probably neither is category theory.

On the whole, it was an interesting step out of the usual routine.

In “The Fabric of Reality”, David Deutch gives a refutation of solipsism. I’m not entirely sure it works – all he really tries to do is to show that the difference between solipsism and realism is more nearly a mere semantic distinction than is generally assumed. But in any case, along the way, there’s an anecdote about a solipsist professor lecturing his (imaginary?) class merely to help him clarify his ideas. The idea being that, even if the imaginary students don’t really exist, it helps to clarify the professor’s own ideas by lecturing to them, answering questions, and so forth. In this view, you don’t really understand your own opinions – let alone justifiably believe in them – unless you’ve argued for them against a variety of possible criticisms. (J.S. Mill gave a defense of full-fledged freedom of speech, even for grossly offensive and even “dangerous” opinion, on this ground.)

I mention this because, when I told Dan about the blog, he seemed dubious about blogging as a way of communicating math. It’s certainly more solipsistic than a usenet newsgroup, or a mailing list. Those are channels devoted to a particular subject, with many participants. A blog, comments notwithstanding, is mainly a channel devoted to one voice, on many particular subjects. It’s true that half the point of communicating ideas is to get feedback on them from other people. You make your thinking part of one of those great processes like cathedral-building – ad-hoc, gradual, and (significantly) collective. Even so, relatively solipsistic channels are not entirely pointless.

To wit: by working through my theorems about transporting 2-vectors through spans – both for this blog, and for my talk at Groupoidfest, I discovered some problems. Nobody pointed them out, but discovering them was a consequence of approaching the material again from a new angle, with an audience in mind.

The problem is a conceptually important one – mistaking an n-dimensional space for a 1-dimensional space. I’m fairly sure, for various reasons, that the theorem that there is a 2-functor $V : Span(\mathbf{Gpd}) \rightarrow \mathbf{Vect}$ is still true, but the proof I have in my thesis (in the special case where the groupoids are flat connection groupoids on spaces) has a problem. Since that affects the Part 4 of “Spans and Vector Spaces” which I was going to post, I’ll put that off for a while as I get the proof straightened out.

Here is the issue in a nutshell, however:

The proof I have involves a construction of a functor by a particular method, which I’ve been describing in the last three posts. The final step I was going to describe involved what the contstruction does for 2-morphisms – spans between spans. (There is more to the proof, but the remainder is technical enough to be fairly unenlightening – basically, to be a 2-functor, there need to be specified natural isomorphisms replacing the equations for preserving identities and composition in the definition of a functor, and these have to obey some equations which need to be checked.)

The construction given in my thesis is supposed to give a way to take a span of spans of groupoids, and give a natural transformation between a pair of 2-linear maps. But a 2-linear map can be written as a matrix of vector spaces, and a natural transformation is then written as a matrix of linear operators which act componentwise. So one way to look at the problem is to construct a linear map between vector spaces from a span of groupoids.

That is, we have spans $A \leftarrow X_1 \rightarrow B$ and $A \leftarrow X_2 \rightarrow B$. Picking basis objects for $V(A)$ and $V(B)$ (namely, objects $a \in A$ and $b \in B$, plus representations $U, W$ of their automorphism groups) gives a subgroupoid of of $X_1$, consisting of those objects $x \in X_1$ which are sent to $a$ and $b$ under the maps in the span. It also gives a vector space which is built as a colimit of some vector spaces associated to these objects. Assuming $X_1$ is skeletal, this works out (as I described before) to $W^{\ast} \otimes_{\mathbb{C}[Aut(x)]} U$ for each of the $x \in X_1$ in question. The same holds for $X_2$.

Now suppose we have a span-of-spans $X_1 \leftarrow Y \rightarrow X_2$ making the obvious diagram commute. Then because of that commutation, we also have a span of groupoids over each of the choices $(a,b)$ of objects, and so then the question becomes, partly, how to get a linear map between the vector spaces we just constructed. If you have bases for all the vector spaces here, it’s not too bad: vectors can be seen as complex-valued functions on the basis. We can push these through the span just as we’ve been talking about in the last few posts here: first pull back a function along one leg by composition, then push forward along the other leg. The push-forward will involve a sum over some objects, and some normalizing factors having to do with the groupoid cardinalities of the groupoids in the span.

However, I won’t go too far into detail about this, because the construction I actually outlined doesn’t adequately specify the basis to use. In fact, it will really only work if all the vector spaces $W^{\ast} \otimes_{\mathbb{C}[Aut(x)]} U$ is one-dimensional. Then there is a basis for the combined space which just consists of all the objects $x$. I’d hoped that Schur’s lemma (that intertwiners from $W$ to itself, or from $U$ to itself, have to be multiples of the identity) would get out of this problem, but I’m not sure it does. So there is a problem with the construction I was trying to use.

As I say, I’m fairly sure the theorem remains true – it’s just the proof needs fixing, which I don’t expect to be too hard. However, I’ll refrain from getting sidetracked until I know I have it worked out.

Instead, next time I’ll describe some of the things I learned at Groupoidfest 07 when I presented a talk on this stuff. (At first I was nervous, having discovered this flaw while preparing the talk – but then, a lot of people were talking about work-in-progress, so I don’t feel too bad now. Plus, the meeting was a lot of fun.)

You may be wondering about the title: “Theoretical Atlas”. Both words have a double meaning here.

First, Atlas: originally, this was the name of a Titan in Greek mythology, who was condemned by Zeus to stand at the Western edge of the world and hold up the sky on his shoulders forever. The Western edge of the Greek world – the Mediterranean – is indeed where the Atlas mountains are found, in the Maghreb. Also named for him is the Atlantic Ocean (and, therefore, Atlantis, a continent once speculated to be located somewhere in it). You can see a picture of the Atlas mountains in the banner at the top of this blog’s main page.

So one meaning comes from a notion that tends to crop up fairly often when one talks about the project of finding a quantum theory of gravity. This is the prospect of a complete unified theory of physics, a Theory of Everything (TOE), or some such name. People peering into the mist of our limited knowledge sometimes seem to see prospects of a single theory that unifies every aspect of the physical world in one single model – all forms of matter, energy, forces, gravity, etc. The name “M-theory” is popular in some circles for this idea – an as-yet undiscovered theory which might go beyond what string theory can do today. Other prospects have been proposed, but the image I have is of a single, immensely powerful theory, holding up the entire world on the strength of its explanatory power – a theoretical Atlas holding up this enormous burden.

But this great Atlas of a theory has never been written down – alas. For myself, I’m quite skeptical if it even could be: why should there be a short, pithy idea that encodes the whole huge, complex, endlessly surprising universe? Even if we had a theory which accounted for all particles and forces in nature, would that be a theory of everything? The point of a theory, after all, is to help us understand things: we’d still need, at the very least, a theory to explain how chemistry emerges from physics, what life is and how it can come into being – all just to account for even our most basic experience. Then there are whole areas of the world that open up from there. So this great single Atlas of an idea that accounts for the entire world of experience is, as they say, just a theory. It’s a (merely) theoretical Atlas.

(Of course, this use of the phrase “just a theory”, often used to dismiss the insights of Darwin, and much less prominently used any other way, is simply wrong. The meaning of “theory” depends on context, but it always means something more than a mere guess. Still, as I said before, I’m not going to worry TOO much about being wrong now and then – and the more accurate hypothetical Atlas just didn’t sound as good.)

The other meaning of the word “atlas” has to do with maps. The other element of the banner above mentions the Bellman’s map from The Hunting of the Snark. It had no markings on it at all – “purely conventional signs”. But mathematics is all about using purely conventional signs as a reference point in describing the features of the world. The Bellman’s map showed no land – only sea – and so it left out not only the conventional reference points, but also anything definite to refer to.

A “theory” can be seen as a way of taking some standard, pre-existing structure, and trying to “map” it onto the features we see in the real world. In a way, a literal map is an example of a theory: it imposes a regular grid of coordinates on some convoluted shape, which is itself a model of some territory off elsewhere in the world. It’s an artificial imposition – but it allows us to find our way around. Assuming it’s accurate enough, and we know how to read it.

In the case of a literal atlas, we have a collection of – usually flat, generally rectangular – drawings of the surface of a sphere (more or less). Each one is a little bit distorted, because the Earth isn’t flat (no, no, I know – that’s just a theory – but I think it’s accurate enough). In the study of manifolds, these are called “charts” – each one is a map from some open subset of Rn to a subset of the manifold. Generally – and, for instance, on the surface of the Earth – one chart won’t be enough. You need several charts, and an understanding of how they fit together. The collection of charts is an atlas, and one imagines a big book filled with these charts, each one imposing a rectilinear grid of coordinates onto some underlying terrain. “Transition maps” tell you how they fit together to cover the whole surface.

So the other meaning of theoretical atlas is the notion that we may need many theories to properly account for the world. Each one may describe some part of it fairly well – maybe with a bit of distortion, but certainly not so much that it doesn’t help to find our way around. None by itself explains everything – but given enough, and some knowledge of how to manage the transition between the domain of one theory and the domain of another, they can tell us a lot. This is my image of what our researches into physics, and the world in general, are aiming at: an atlas of theories that covers everything.

Mind you, I realize that such an atlas, like the other kind of Atlas, is purely theoretical.

Oh, all right: hypothetical.

Here is an apology – with apologies to the Unapologetic Mathematician

One inspiration for starting this blog is the fact that Dr. Baez has a great abundance of stuff on the Web. Some of the better-known include the ever-popular This Week’s Finds in Mathematical Physics, and the newer n-Category Café, which is a group venture together with Urs Schreiber and David Corfield. Between the three of them, they write on “math, physics, and philosophy”. That’s more or less what I propose to do here.

Why the redundancy?

The n-Category Café has turned out to be a very productive way of sharing ideas informally over long distances, and without being too confined by a narrow topic or the strictures of publishability. The participants have also adopted the ethic that it’s better to share ideas than keep them secret until they’re perfected. One essential reason is that science, math, and philosophy are cultural products – discussion is like oxygen for culture. This is a lesson that has been learned many times in the past, and, I suspect, will have to be learned many times again in the future. Publication, peer review, giving public talks – the whole essence of research is communicating ideas. Of course, you need to develop good ideas to communicate, but the point is to share and discuss them. One more voice in a conversation like that may be a drop in a bucket, but it’s not redundant.

So I aim for this to be my particular drop in our great collective bucket. I’ll relate things that I’ve been thinking about; explain things I’ve figured out; express confusion over things I haven’t; describe the experience of starting a research career; muse; investigate; and, if possible, not bloviate. And I won’t worry too much about being incomplete, tentative, or even (a little bit) wrong. That’s all part of investigating things.

This is as much “apology” (in the sense of a justification of one’s actions – quite the opposite of what we moderns usually mean by “apologize”) as I suppose the minor nuisance of starting yet another blog really requires.

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