Friday, August 18, 2017

New Paper: The Frontloading Argument

Forthcoming in Philosophical Studies.
Maybe the most important argument in David Chalmers's monumental book Constructing the World is the one he calls the 'Frontloading Argument', which is used in Chapter 4 to argue for the book's central thesis, A Priori Scrutability. And, at first blush, the Frontloading Argument looks very strong. I argue here, however, that it is incapable of securing the conclusion it is meant to establish. My interest is not in the conclusion for which Chalmers is arguing. As it happens, I am skeptical about A Priori Scrutability. Indeed, my views about the a priori are closer to Quine's than to Chalmers's. But my goal here is not to argue for any substantive conclusion but just for a dialectical one: Despite its initial appeal, the Frontloading Argument fails as an argument for A Priori Scrutability.
You can find the paper here.

New Paper: Speaker's Reference, Semantic Reference, and Intuition

Forthcoming in The Review of Philosophy and Psychology.
Some years ago, Machery, Mallon, Nichols, and Stich reported the results of experiments that reveal, they claim, cross-cultural differences in speakers' `intuitions' about Kripke's famous Gödel-Schmidt case. Several authors have suggested, however, that the question they asked they subjects is ambiguous between speaker's reference and semantic reference. Machery and colleagues have since made a number of replies. It is argued here that these are ineffective. The larger lesson, however, concerns the role that first-order philosophy should, and more importantly should not, play in the design of such experiments and in the evaluation of their results.
You can find the paper here.

Tuesday, July 4, 2017

DLNA Output for Linux

Many of my favorite bands stream their concerts these days, sometimes live, sometimes afterwards. It's fine to listen on the computer sometimes, but other times I'd like to listen to the show over something a bit better-sounding, like my stereo. I figured there had to be a way to do this, and it turns out that, indeed, there is. The Logitech Transporter I use as a digital source will function as a DLNA renderer (i.e., you can send it a DLNA audio signal). And I know that Linux plays nice with DLNA, since I often stream video to my TV that way (using minidlna, aka, ReadyMedia). So the only question is: How can I convince Linux to send audio from the computer to the Transporter?

Saturday, March 18, 2017

LaTeX Notation for Numerals

It is common in meta-mathematics to use the notation n to mean the numeral for the number n, that is: S...S(0), where S is a symbol for successor and there are n S's in the numeral for n. In LaTeX, one can typeset this notation using \overline{n} in math mode. Unfortunately, this does not always look very good: The height of the bar will vary with the height of the contained character(s), so the heights of the bars in \overline{n} and \overline{k} will not match.

The solution is to use a 'strut': an invisible (because 0 width) rule that functions only to set the height of the bar:

It would perhaps be better to use the current font in \numheight, but I've never had a problem with this in practice.

Sunday, March 5, 2017

A Bound in Gödel 1931

When teaching Gödel's famous 1931 paper on the incompleteness theorems this semester, I got hung up on one of the bounds he gives in the course of the 45 definitions of primitive recursive notions. This is the case of concatenation. Recall that Gödel here codes finite sequences via prime factorization, so the sequence <a1, ..., an> is coded as: 2a1 × ... ×pnan, where pn is the nth prime. The 'star function' is then defined as follows:
x * y = μz≤Pr[l(x) + l(y)]x+y {∀n≤l(x)(n Gl z = n Gl x) &

     ∀n≤l(y)(0<n → (n + l(x)) Gl z = n Gl y)}
Here, l(x) is the length of the sequence x; n Gl x is the nth element of that sequence. So the definition says that x * y is the least number coding a sequence that agrees with x on its first l(x) elements and agrees with y on the next l(y) elements. Of course, there is such a number (and, actually, given how "Gl" works, there are infinitely many). The bound is needed to guarantee that * is primitive recursive.

The question, though, is how the bound is supposed to work. Gödel does not often discuss his bounds, which tend to be pretty loose, but he does explain one of them in footnote 35. And if one follows the sort of reasoning Gödel uses there, then it is difficult to see how to get the bound in the above.

I asked a question about this on the Foundations of Mathematics mailing list, and Alasdair Urquhart took the bait and replied with an elegant proof showing why Gödel's bound works. I thought I'd record a version of it here, in case anyone else has a similar question.

First, we show, by a straightforward induction on n, that 2a1 × ... ×pnan ≤ pna1 + ... + an.

Now let <a1, ..., an> and <b1, ..., bm> be two sequences. The code of their concatenation is:
2a1 × ... ×pnan × pn+1b1 × ... × pn+mbm ≤ pn+ma1 + ... + an + b1 + ... + bm
a1 + ... + an ≤ 2a1 + ... + pnan ≤ 2a1 × ... × pnan = x
and similarly for the other sequence. (Note that the last inequality depends upon the fact that none of the ai = 0, but Gödel's coding of sequences only works for positive integers.) So
a1 + ... + an + an+1 + ... + an+m ≤ x + y
which gives Gödel's bound.