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]

In this post I gave a brief sketch of what I claim is a better approach to modelling the continuum. In this post I am going to develop the ideas a little bit more thoroughly and also explain why I this is a better approach than the usual set-theoretic definitions of the real numbers.

Here are a few of the points which I think distinguish this approach:

Beneath the fold I will hint at a few of these points, and give some small detail for the first. Although the details are by no means worked out, I am certain that most of our methods for the real numbers (e.g., calculus, topological groups, measure theory) will carry over in a natural way. A significant exception may be  descriptive set theory (and other theories of the fine logical detail of ); in fact it seems to me that my notion of the continuum doesn’t even have a well-defined cardinality. Rather than a failing of this approach, I see this as a sign that set theory has injected unnecessary baggage into our theory of the number system.

These ideas are mostly idle musings, not in active development, and I’m releasing them to the wind. If anyone who happens by wants to develop these ideas, go ahead. You can take them for yourself (with an attribution) or I would be happy to work on a joint project. Otherwise, I’ll continue a haphazard development in these pages from time to time.

Continue reading »

Over at Terence Tao’s blog, the Fields medalist produces prodigious volumes of posts; some detail his major research work, some are course lecture notes, and then some are expository bits that he writes up for his own edification. I just noticed a post from April about Gödel’s completeness and compactness theorems. From a logician’s point of view, the post’s contents are by and large pretty vanilla. But early in the post, Tao mentions a notion of elementary convergence and proves the following:

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.

Continue reading »

This post, and a few to follow, aren’t going to be about mathematics at all, but rather another subject close to my heart: education. I am going to share some thoughts on education in general and, in particular, the sorry state of public education (in this country and elsewhere, and from kindergarten through college). In this first post I will discuss elementary education, while the second and third will address high school and university, respectively.

A significant theme at all levels is the promise of technology to flatten barriers and cut costs in education. At the same time, using these new methods to their fullest will require modifying fundamental attitudes about the role of educators and the purpose of institutions. There is no doubt that many of my prescriptions will sound counter-intuitive in a modern world where education seems so important. I urge you to resist the fallacy of considering proposed changes against what the current system promises, rather than what it delivers.
Continue reading »