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.

I'm also reasonably comfortable with basic ring theory and the manipulation of ideals and modules. I'd like to go up to Artin's reciprocity theorem and at least be able to discuss it at the conceptual level. That means I should understand at least that much, and that will be the subject of my next post or two here.

I'm going to assume some level of comfort with rings and modules. In particular, the set of ideals has sums, products and the obvious -action

The story of class field theory begins when we ask what division could mean in the context of ideals. Everything takes place in a integral domain, where we can build a field of fractions for our ring, . A fractional ideal ought to be a submodule . We need more, though, because we also want to relate relate to via the multiplication, so we require that there is some for which . Note that this holds trivially in case is finitely generated.

It is easy to see that the multiplication of integral ideals extends to the fractions, and this gives us an associative operation with the unit . It is not the case that every fractional ideal has an inverse, but we do have a sort of "best approximation"

Then the product is trivially contained in . Moreover, if anything more were included that condition would fail, so if has an inverse in the semigroup, is it.

We also have the notion of a principle ideal , generated by a single element. This generalizes to fractional ideals as well, now allowing to range over the field . Clearly, every principle ideal is invertible, with inverse . We immediately have the notion of "principle fractional domain", where all fractional ideals are principle. This would certainly imply that itself was a principle ideal domain, but the converse turns out to be true as well, since two principle ideals and are equal iff for some unit .

We finally get to the notion of class fields by taking the quotient of all fractional ideals by the subgroup principles. In general we only get back another monoid; if we want to get a class field group, we have to pass to the more restricted notion of a Dedekind domain. That will be the topic of my next post.

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.

My research has centered around implicature theory, a topic in the field of pragmatics.  This series will walk through the research I have done and the theory proposed in my thesis.  In doing so, it will start from the absolute basics; the series will be self-contained, presupposing only basic knowledge of logic and naive set theory.

This first post will serve to introduce pragmatics, the study of non-literal meaning.

What is pragmatics?

Pragmatics is the study of language in use. It can also be considered the study of non-literal meaning. Taken in the second sense, pragmatics carves the study of meaning in half with its sister science, semantics. Between characterizing literal and non-literal meaning, the science of linguistic interpretation is (ideally) fully spanned.

Pragmatics may best be understood as the complement of semantics. Semantics is the study of literal, cross-contextual meaning.

Characterizing semantics as the science of “meaning”, however, means nothing if we don’t properly understand the term “meaning”. Any theory of meaning must first pin the meaning of “meaning” if the enterprise is to get off of the ground.

Properly and fully analyzing the word “meaning” in a way that matches our pre-theoretic intuitions about the concept is a philosophical problem that still has no satisfactory solution. Any attempt to flesh out the concept either misses some important component of what we consider meaning or falsely attributes extra, incorrect properties to it. However, a working definition that has worked quite well for systematic scientific study is one promoted by philosopher Donald Davidson; if meaning is viewed as “truth-conditional meaning”, then familiar, rigorous methods in formal semantics (typically reserved for logical, constructed languages) can be used to study natural-language.

Viewing the meaning of a sentence as its truth-conditions (the conditions under which the sentence is true) fails to capture many aspects of meaning. It doesn’t capture shades of meaning, for instance, that separate one poetic statement from another. While other aspects of meaning are still important and worth explaining, there’s something to be said about the truth-conditional viewpoint. If a speaker knows the meaning of a particular statement, then it’s reasonable to say that the speaker knows when the statement is true or false. In other words, knowing the meaning of a sentence means knowing what makes the sentence true; if someone didn’t know what made a sentence true, then debatably the person doesn’t know what it means.

Thus, truth-conditional meaning is a proper subset of meaning at large. To explain and study meaning as a whole, one must explain and study the truth-conditions of sentences and how they are acquired. Though this is but a sub-part of the total problem of meaning, this sub-part is far from being adequately solved; even so, it is the most fruitful and quantified area of investigation into linguistic interpretation.

When we say things such as “Sally had a baby and got married”, semantics’ job is to tell us the truth-conditions of this sentence that are contained solely in the words used (and not in their connotations). Any semantic theory worth its salt will tell us that “Sally had a baby and got married” is true just in case Sally, in fact, had a baby and got married. Nothing shocking here; semantics, viewed in this way, seems like a trivial topic. Needless to say, as we consider general theories of interpretation that must account for the behavior of complicated logical and intensional operators, accurate semantic theorizing gets a lot harder.

Semantics cannot distinguish between the following two sentences:

A: “Sally had a baby and got married.”

B: “Sally got married and had a baby.”

Literally speaking, A and B both say the same thing. A is true just in case Sally, in fact, had a baby and got married. B is true in the exact same conditions, and so A and B have the same semantic content (written ||A|| = ||B||, where “||A||” is read “the interpretation of A”).

A good way to characterize pragmatics is to point out that, from the pragmatic point of view, A and B say quite different things. Sentence A suggests that Sally had a baby before she got married, while sentence B suggests things the other way around. The order of appearance for the conjuncts in these sentences matters; we tend to understand the order of appearance in a list of conjuncts as a temporal ordering, though nothing about the word “and” itself mandates this interpretation.

A pragmatic theory of the behavior of “and” should account for this non-literal difference between the meaning of A and B, whereas a semantic theory isn’t on the hook for such a thing.

Why care?

The difference between A and B above is so natural that it hardly seems to call for an explanation. However, there are plenty of reasons to care about characterizing such linguistic behavior mathematically.

The ease at which human beings incorporate non-literal speech in discourse is a fact worth explaining. Hardly anything humans utter contains purely literal meaning; what we utter doesn’t merely borrow from the words we utter, but also from general facts about human reasoning. In other words, context matters. Where, when, and how we say things play systematically into constructing the meaning of what we say.

Crafting an explanation of how language in use spans nearly every unique cognitive aspect of humankind. It means crafting a model of human reasoning as it applies to language and communication; this bridges what we find salient and what we expect others to find salient in the immediate context, shared knowledge, and other domains of belief. Explaining how we engage in linguistic reasoning involves explaining facts about reasoning at large.

For those with a more practical streak, pragmatics is an essential sub-problem in constructing artificial intelligence. Sophisticated artificial intelligence will require sophisticated communicative ability. Without being able to understand non-literal speech, our artificial agents will have serious trouble communicating reliably with humans. Such limitations necessarily limit their ability to perform; if artificial agents are to serve wider and more useful roles, their ability to converse naturally with laypeople will be absolutely essential.

Pragmatics then, broadly construed, is the attempt to predict natural, systematic linguistic inferences. Along with a good theory of semantics, a complete theory of pragmatics would afford us a truth-conditionally complete theory of meaning. Such a theory would be able to take any utterance and fully decode its meaning-in-context.

The problem in pragmatics that this series will be investigating is that of conversational implicature. The next post will be concerned with defining implicature and some of its sub-phenomena, one in particular to which my theory applies.

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.

This describes Moebius transformations, which are maps of the complex plane which have the form

for fixed . This video really emphasizes how anything having to do with the complex numbers is really about rotation.

www.youtube.com/watch?v=JX3VmDgiFnY

I don't think we have a plug-in for embedded video yet. Maybe I'll work on that later. Until then, enjoy the link. [UPDATE: We now have embedded youtube functionality, as can be seen. -- Ed]