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?

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.

I just stumbled across a cornucopia of Quine videos on YouTube. Specifically, they are videos coming from this series. From the looks of things, nothing from the Boolos, Dreben or Dennett panels is up on YouTube. But the Fara interview, as well as the Block, Fogelin and Goldfarb panels, are all there. Here's the first excerpt from the Goldfarb panel:

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

In the very near future I'm going to be writing a paper on reciprocity theorems in the history of algebra. I really ought to describe a bit of that history here now, because it's a really great story, but I'm more interested in trying to figure out how the modern approach to the subject looks. I'm fairly clear on the basic form of a reciprocity law (quadratic, most famously, but also quartic and more generally).

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.

I have (very) recently gotten into the study of algorithmic randomness, and figure that airing some things out here on the blog might do me some good. I will not be providing an in-depth introduction to the fundamentals of the area here. What I will do in this post is give some basic definitions, and spell out the statement of one recent example from the field, for the sake of illustrating the kind of work that I am interested in.

Continue reading »

Here's something I just read about over at LogBlog. It's a graphic novel with a logical focus, written by Apostolos Doxiadis (author of the mathematically-tinged 1992 novel Uncle Petros and Goldbach's Conjecture) and computer scientist Christos Papadimitriou.

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.

This past summer, I finished and defended my Master's thesis.  Though my work has moved from linguistics and philosophy of language to causal and statistical reasoning (the Tetrad project), I'm still researching natural-language in my spare time. Therefore, I will be writing a series of posts here on Unwanted Capture concerning my linguistic work, both to get my ideas out there and to encourage myself to continue thinking about linguistic issues.

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.

Continue reading »

My advisor and I are currently crafting plans for a book focusing on Dedekind's style of mathematics and the manner in which things like Galois theory and algebraic number theory evolved in his hands. Part of the book would consist of some translations (with commentary) of pieces by Dedekind.

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.

I just found this linked from Wikipedia, and wanted to share/tag it for myself.

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.

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]