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 »

I've recently finished a draft of the first chapter for my masters thesis, on the topic of mathematical structuralism. I argue that the philosophical discussion of mathematical structuralism has, by and large, misrepresented mathematical attitudes on the subject. This dialogue, beginning with Benacerraf’s classic What Numbers Could Not Be, has focused on the ontology of structures; such an attitude fails to do justice to a circle of ideas designed to explain and exploit the application of certain mathematical ideas across a wide range of domains. This oversight is compounded by an inattention to mathematical practice, which we find already in Benecerraf’s treatment of rival ordinal systems in ZFC.

At the same time mathematical ideas on structuralism, especially as de veloped in category theory, may offer philosophical dividends by helping us to understand the mathematical value judgements neglected by Benacerraf and his successors. This is because the mathematician’s structuralism addresses the “How?” rather than the “What?” of mathematics. Rather than invalidating alternative approaches, structuralists argues that theirs is a better way of doing mathematics. In the case of the ordinal systems in ZFC, Von Neumann’s ordinals provide a better reduction of the natural numbers than Zermelo’s because they require shorter and simpler definitions and proofs. This is because we are able to use the existing structural machinery of set theory (i.e., products, disjoint unions, etc.) both in building the internal theory of N and in the extension
to transfinite ordinals.

The Red Herring of Ontology (right click & save)

In a recent joint paper with Jeremy Avigad and John Mumma (forthcoming in the Review of Symbolic Logic, preprint available at the arXiv), we devise a formal system that is intended to faithfully capture the notion of Euclidean geometric proofs. Specifically, is meant to be a formal counterpart to Books I through IV of Euclid's Elements, including a formal codification of Euclid's diagrammatic reasoning; we want proofs in to mirror Euclid's actual proofs. With this post I just want to briefly "set the scene" of the paper. In a later post, I will discuss some proof theory that comes up naturally in our analysis.

Continue reading »

For my first few blog posts here at unwanted capture, I'm going to talk about a subject that I will return to often: rethinking our mathematical foundations aesthetically. This time, I want to look at our definitions for the continuum; I think that a non-standard approach can give us a cleaner presentation which is closer to our basic intuitions. This has been said before, but most approaches tend to pirate off of existing set theoretic definitions, instead of taking advantage of the added flexibility a non-standard predicate gives us.

We're all probably familiar with the usual constructions of the real numbers via either cauchy sequences or dedekind cuts. Add to this the theory of limits and we have a foundation sufficient for all the calculus and analysis that we need to do. However, these definitions don't exactly correspond to our basic intuitions; they're actually the product of a long struggle to eliminate the infinitesimals of Newton and Leibniz.

Continue reading »

To get the blog rolling, here's something I came across in my recent reading.

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.

Continue reading »

Thanks for stopping by. If you're curious what you've stumbled upon, please see the About page. Posts with real content are on the horizon, so do come back soon.