Die Schwester, die Freundin, die Mutter.

But it was hot too. Last week I visited Oftersheim, Germany in order to see meine Mutter. It's a small town just a bit south of Mannheim and Heidelberg. My lady came with me, and as luck would have it, my sister was able to travel over from Boston the same time we did. As you can see in the picture of the three of them, there's quite a bit of construction going on at my mother's abode; but we had beds and air conditioning, so we were all set.

I was hoping to be present for a German victory in the World Cup finals. I was eager to compare the scene with that of being in Boston in 2004, when the curse was broken most emphatically. But alas, they lost their semifinal to Spain as we were crossing the Atlantic; too bad they had to play that one without Thomas Müller. At least we still got to enjoy the victory over Uruguay for third place, and it was just a nice all-around week with family.

Oftersheim Bahnhof

Speaking of sports, I want to just take a moment to say how impressed I am with my Braves. I was pretty confident coming into this season that they could get back to the playoffs, hoping for the NL wild-card berth. And here they sit, leading their division, with the fourth best record in the majors, and an upcoming series against the Padres that could put them back atop the NL. Not too shabby.

Anyway, now that I'm back in Pittsburgh, I've got plenty to keep me busy. (Which is why I just wrote this post instead of course.) So it's back to working on my dissertation, interspersed with myriad thoughts about 19th century mathematics. When did I become such a thrill-seeker?

Gino Severini, Suburban Train Arriving in Paris (1915)

Perimeter Institute in Waterloo, Ontario

The Perimeter Institute for Theoretical Physics hosted a conference last month on a topic which is very much at the intersection of physics and philosophy: the nature of laws of nature. Carnegie Mellon Philosophy's own Kevin Kelly was there to give a talk related to one of his favorite pet subjects, Ockham's razor.

Scientific American's Observations blog has a very nice write-up of the conference as a whole. Interesting stuff, even if - like me - you don't often concern yourself with the philosophy of science.

www.youtube.com/watch?v=-_tSuKAOGSY

I don't know if these videos "should" be on YouTube, but they're there for now, and there's plenty of interesting material.

First of all, let's consider an example of a classical measure-theoretic result. Suppose we have an integrable function . We call a point a Lebesgue point of provided that

Lebesgue Differentiation Theorem. Let be an integrable function. Then almost every point is a Lebesgue point of .

To say that "almost every" point is a Lebesgue point is to say that the set of Lebesgue points has measure . Recently, Pathak proved a version of the Lebesgue differentiation theorem in the spirit of algorithmic randomness. Her result follows a pattern that has been seen before, e.g. in V'yugin: (1) take some probabilistic or measure-theoretic result that holds almost everywhere, (2) add some computability-related hypothesis, (3) conclude that the result in fact holds for every Martin-Löf random point in the space.

OK, fine. So what is a Martin-Löf random point? To answer that, let's consider a fuzzy moral question: what should count as a "random" element in our measure space? We might say that a random point shouldn't be too special; so we might make this try:

Attempted Definition. A random point in a measure space is one that doesn't satisfy any properties of measure , i.e. it is not contained in any null set.

This runs into the problem that, well, for any . Martin-Löf's 1966 definition is based on the same general idea, but it gives an account of randomness that can be satisfied and turns out to have all sorts of interesting interactions with computability theory:

Definition. A random point is one that isn't contained in any effectively null set.

Definition. An effectively null set is one of the form

where is a sequence of uniformly effectively open sets, for which for all .

Definition. A set is effectively open if it is a union of balls

with a computably enumerable subset of .

Basically, the sets narrow in on the null set in an effective manner. Any point in the space that cannot be pinned down in such an effectively null is what we call a Martin-Löf random point.

Alright, so returning to Pathak's version of the Lebesgue differentiation theorem, what is her additional hypothesis? She restricts attention to functions that are not just integrable, but also:

Definition. A function is -computable if there is a computable sequence of polynomials such that

for all .

So, we only consider that can be effectively approximated by polynomials in the -norm. Pathak's result is then:

Theorem. If is -computable, then every Martin-Löf random point is a Lebesgue point for .

I have put up some rough working notes in the Miscellania section of my web page that situate Pathak's result in a conceptual framework developed by Mathieu Hoyrup and Cristóbal Rojas for working with algorithmic randomness in spaces other than Cantor space (where classical computability theory lives). Their work brings a unifying, systematic approach to results like Pathak's and V'yugin's (linked to above). I wrote the notes for my own benefit, as a way to get clear on some of the structure and details of the papers I've been talking about; perhaps someone else might find them helpful too.

Cover from Logicomix

Papadimitriou might be best known to logicians from his textbook with Harry Lewis on the theory of computation. But he already has an earlier foray into fiction. In 2003, his Turing: A Novel about Computation was published. Yes, it really is a novel, complete with a love triangle and digressions about, well, computation. (Note: the title refers to a character that is not the Alan you know.) I actually own a copy of that book; my mother the computer scientist gave it to me one birthday. At the risk of her seeing this, I only read the first little bit of that book. But that reflects more on my general relationship with novels than it does on the readability of that particular tome.

In any case, while I have never been one for comics or graphic novels, I'm sufficiently curious about Logicomix: An Epic Search for Truth to give it a shot. For one thing, all of the art by Alecos Papadatos and Annie Di Donna looks very nice. I won't say anything about the contents of the book, having not had my hands on it. A preview is available at the Logicomix site, and Richard Zach gives his largely favorable take in the post linked to above.

I've just put a draft of one such translation up on my web page. It is an excerpt from the 1894 edition of the Dirichlet-Dedekind Vorlesungen über Zahlentheorie. The selected portion focuses on the structure of the lattice of subfields of , and it spells out how Galois theory can be construed in the field-theoretic framework laid out by Dedekind (i.e. the modern take on Galois theory).

Any comments on the draft are welcome.

Sequential compactness theorem: Let be a countable language, and let be a sequence of -structures. Then there is a subsequence which elementarily converges to a limit -structure (with a countable universe).

The sequence elementarily converges to if, for any sentence of , implies that for sufficiently large . This notion and the stated result struck me as unfamiliar, but I figured maybe it was just me. However, model theorist John Goodrick also indicated that he'd never come across this notion of elementary convergence either.

Tao gives a quick proof of the theorem as a corollary to compactness, which goes roughly as follows:

Proof. The set of all theories in can of course be identified with where is the (countable) set of sentences from . Moreover, the space with the product topology is sequentially compact because it is a countable product of the sequentially compact . (See Proposition 9 here for instance.) So the sequence of theories in has a subsequence converging to some theory . Using the compactness theorem, it follows that is in fact a consistent theory, and we get a countable model . Without too much ado, one can verify that is indeed an elementary limit of the subsequence.

Note that the restriction on 's countability is essential for this proof to go through, as an uncountable product of sequentially compact spaces need not be sequentially compact. Consider, say, . For let be the -th digit of 's binary expansion, and let be the function . One can check that the sequence has no convergent subsequence.

One reason this formulation might not have gotten much play in model theory itself is that ultraproducts already give us elementary limits of sequences of structures, via Łoś' theorem. Tao's interest in elementary limits arises from his combinatorial pursuits. From his post:

The sequential compactness theorem also lets us construct infinitary limits of various sequences of finitary objects; for instance, one can construct infinite pseudo-finite fields as the elementary limits of sequences of finite fields. I recently discovered that other several [sic] correspondence principles between finitary and infinitary objects, such as the Furstenberg correspondence principle between sets of integers and dynamical systems, or the more recent correspondence principles concerning graph limits, can be viewed as special cases of the sequential compactness theorem;

Let's consider an example of the kind of correspondence principle that Tao has in mind, one which was discussed in detail in an earlier post from Tao's blog. First we have what is essentially a quantitative version of the statement that bounded, monotone sequences of real numbers converge:

Infinite Convergence Principle: For any and any , there exists an such that for all .

Tao goes through some machinations in order to finally arrive at a finitary version of the same principle, which he proves to be quickly interderivable with the infinitary version:

Finite Convergence Principle: If and , and with sufficiently large, then there is an such that for all .

As pointed out by Henry Towsner and Ulrich Kohlenbach in the comments on the post, the finite convergence principle is nothing more than the no-counterexample interpretation of the infinite convergence principle. (See this paper by Kohlenbach and Gaspar for more details.) There has been much recent work applying proof theory to the fields of analysis, combinatorics and ergodic theory. Kreisel's no-counterexample interpretation comes up, for instance, in the formulation of a constructive mean ergodic theorem in the paper "Local stability of ergodic averages" by Avigad, Gerhardy and Towsner. Kohlenbach has a multitude of papers in this area, as well as the monograph Applied Proof Theory: Proof Interpretations and their Use in Mathematics.

We saw above that Tao indicated that the sequential compactness theorem subsumes things like this correspondence between the infinite and finite convergence principles (which is just an instance of the no-counterexample interpretation, or at its heart, Herbrand's theorem) and correspondence principles concerning graph limits à la Elek and Szegedy (who use an ultraproduct construction). So it distills a common logical thread that runs through these results. I might write another post about the logical status of this sequential compactness theorem later. (Then again, I might very well not.)

By modern logical lights, geometry did not receive a properly rigorous foundation until works such as Hilbert's informal Grundlagen der Geometrie, and later formal work by the likes of Tarski. To be sure, Euclid certainly leaves certain necessary assumptions unstated (a minor problem which is easily remedied); but Euclid's methods of proof have also been derided for relying on spatial intuition rather than precisely formulated logical rules. If one is not careful when proving with diagrams, one can easily misstep and use features of a particular diagram that are not general in order to make unwarranted deductions (as in classic "proofs" such as that all triangles are isosceles).

But of course, Euclid does not prove absurdities such as these! As Ken Manders has stressed, Euclidean geometry was a remarkably stable practice for thousands of years, and its practitioners exhibited a certain diagram discipline. One central idea is this:

What makes Euclid's proofs (as opposed to those of Hilbert, say) diagrammatic has nothing to do with an improper reliance on spatial intuition; rather, it is merely that Euclid's one-step inferences are of a very different nature than those of Hilbert.

While Euclid's Elements might appear very far from rigorous when looking at it with systems like Hilbert's or Tarski's in mind, the point is that Euclid's methods of proof can be given a solid formal foundation in and of themselves. (And in this light Euclid's text is much closer to formalized mathematics than typical informal-yet-rigorous mathematical works, which goes well with the fact that Euclid was considered to be a paragon of mathematical rigor for many centuries.)

I don't want to get into the nitty gritty details of spelling out the construction of , and I can get to the topic I want to discuss without doing so. As indicated above, we want single inference steps in to match the sort of "diagrammatic" inferences that Euclid makes. Consider:

Proposition I.12: Given a point off of line , construct line through perpendicular to .

Diagram for Proposition I.12

Proof. Let be a point on the other side of from . Let be the circle through with center . Let and be the intersection points of and . By Proposition I.10, let bisect the segment from to . Let be the line through and . By Proposition I.8, , hence and are perpendicular.

Inferences like the highlighted one above are the sort of step where we might worry that we could misuse a diagram to reach unwarranted conclusions (though this case is rather obviously benign). Our particular instantiation of our diagram has such intersection points, but we have given no reasoning to indicate why it must, i.e. why this isn't merely an artifact of the particular way we drew it. A Hilbertian or Tarskian proof would provide a chain of logical reasoning in order to arrive at the existence of these intersection points. That is all well and good, but we end up with very different proofs than Euclid's of course. In fact, in too we will have such chains of reasoning that look nothing like moves Euclid would explicitly carry out, but we have these only on a "background" or "internal" level. The proof given above (N.B. this has nothing to do with the picture) actually is the proof of Proposition I.12. (I led you to believe I was giving Euclid's proof; one can check that ours has come out essentially identical to Euclid's.)

In order for to properly license these kinds of diagrammatic inferences, but not go too far, there is a formalized notion of "direct diagrammatic consequence" in our paper. Next time, I will talk about some of the interesting proof-theoretic points surrounding its design.

My old advisor Peter Koellner, in a paper on pluralism in mathematics, raises the kind of interesting point (based on a simple observation) that I'm often disappointed didn't occur to me already. The main issue Koellner is considering is the problem of selection of mathematical theories, analogous to the problem of selection in physics. For physics, the problem is to select from among classes of empirically equivalent theories; for math, it is to select from among various interpretability degrees, i.e. equivalence classes under the relation of mutual interpretability. (Don't worry about the precise definition of that for now.) The point I have in mind is an aside Koellner makes.

He writes:

The structure of the hierarchy of interpretability is more disorderly than one might expect---it forms a distributive lattice that is neither linearly ordered nor well-founded. This is shown via the construction of non-standard theories via coding techniques. Remarkably, however, when one restricts to the natural theories that occur in mathematical practice the theories are well-behaved---they are well-ordered under interpretability. (28)

(NB: The single best source I know for such material is Lindström's Aspects of Incompleteness, where he investigates the structure of interpretability degrees for theories containing arithmetic.) Koellner has in mind the hierarchy of large cardinal axioms---e.g. inaccessibles, Mahlos, measurables, Woodins, and so on---for extending . Similarly, in the context of second-order arithmetic instead of set theory, there are the extensive results in reverse mathematics showing that many theorems of ordinary mathematics fall directly into the well-ordered progression of subsystems , , , , - and so on. (Simpson's Subsystems of Second Order Arithmetic is the standard reference.)

As Koellner points out, there is some imprecision in the notion of what counts as a "large cardinal" axiom. But setting that aside, we can just talk of the finitely many instances of axioms that currently go by the name. Or in the context of we can talk of the finitely many subsystems in the list above. The fact is that "natural" theories tend to fall into the corresponding well-orderings of interpretability. But now there is the notion which would be interesting to examine and perhaps try to make somewhat precise, that of a natural theory.

Let's focus for the moment on subsystems of , taking as a base theory. We know that theorems from countable algebra and separable analysis tend to fall into the hierarchy of subsystems listed above, that is, along one particular chain from to in the lattice of interpretability degrees. Here are a couple of questions I would ask; they are rather vague, and I do not know what form satisfactory answers would take.

Question 1: What is so special about this particular chain? Why does ordinary mathematics lie on it? (How could we characterize this in a precise manner?)

Question 2: Would ordinary mathematics be different (and in what ways) if it, say, lived on a different chain? Could one argue that it necessarily resides where it does, in some interesting sense of the word?

Again, these are pretty vague, but I could imagine rigorous logico-mathematical work that would inform a consideration of these philosophical questions and others like them.

In sum, while I've long been aware of the fact that "natural" theories tend to fall along the well-ordering of large cardinal axioms (or, within , the well-studied subsystems), and also of the rather wild structure of the lattice of interpretability degrees, for some reason I never thought about examining the inherent tension there. I agree with Koellner that this is "a mystery that calls for clarification" (28).