Have Fun With René, Jerry

I did not know Jerry Fodor at all, personally. But when you spend as much time with someone's work as I did with his, you feel like you did know him.

When I got to MIT in 1987, Jerry had already left, a year before. But his influence was still ubiquitous. There were a lot of older graduate students who were still working with him, and many of those a year before me clearly seemed like they wished they were.

Interview in 3AM Magazine

Thanks to Richard Marshall for the invitation. The interview is here.

Statement Concerning Allegations By Heidi Howkins Lockwood

Last week, Heidi Howkins Lockwood publicly accused me of groping her after a colloquium at Yale in October 2007.

I categorically and unequivocally deny having groped Lockwood on that occasion or on any other.

Lockwood asserts that I was so drunk that night that I would have been unable to find my way to my hotel, apparently implying that this is why I had "no recollection" of groping her. Both claims are false. I have vivid memories of the entire evening. When I deny groping Lockwood, it is not because I do not remember doing so; it is because I positively remember not doing so.

When Lockwood confronted me with these allegations (almost six and a half years later), I did apologize to her, but not because I thought I might have done what she alleged. She was clearly distraught, and it is possible to apologize for the role you played in causing someone to be upset, even if you know that you did not do anything wrong. This is something that decent people do. My apology was intended in that spirit, as an expression of sympathy, not an admission of guilt.

Let me emphasize that I am not accusing Lockwood of lying. Nonetheless, she is mistaken. I did not grope her on that occasion or any other.

Questions and Answers About the Lockwood Allegations

A few people have asked me not unreasonable questions about my response to the Lockwood allegations. Here they are, with answers.

• Why did you wait so long to issue a denial?
  Please try to imagine being publicly accused of such a thing yourself. I was shocked, upset, angry, and confused. It took me three days to calm down enough to think straight.
• Lockwood claims you took no interest in her work after this incident.
  The following spring, I read and commented on material that Lockwood intended to include in her dissertation: a formal argument in provability logic. I had no obligation to do this. She was not (and never has been) my student, and I was not on her committee. Lockwood has not shared any of her work with me since that time. Over the next few years, though, I did occasionally correspond with her about sexual harassment and related issues in the profession. But I stopped having any contact with Lockwood in March 2014, when she accused George Boolos of molesting her.
• Why would Lockwood fabricate such a story about you?
  If "fabricate" means "intentionally invent something in order to deceive", which it does, then I repeat that I am not accusing Lockwood of lying. Why would she have such a false belief about me? That is an interesting question, but she has many views about things in this vicinity that I personally find it difficult to believe are true. (Again, I'm not saying she's lying about those things, either.) And many of them concern a man about whom Lockwood clearly has very strong feelings and with whom I am closely identified. Someone once joked that I'm George Jr. (I wish.) Why Lockwood decided to share all this stuff with someone she barely knew, that's what I find puzzling.
• Do you have anything else to say?
  People who know me know that I not only did not grope Lockwood, but that I would not and could not do such a thing. If you don't believe me, ask them.

I shall have no more to say about this matter publicly (though I may add a question or two, and already have). If you are a friend of mine, and would like to talk about it, however, feel free to contact me. As always!

Further Remarks on the Lockwood Allegations

Elsewhere, I have categorically and unequivocally denied Heidi Howkins Lockwood's allegation that I groped her in October 2007. Of course, I do not expect people to take my denial at face value. Anyone can deny anything. Well, there is a great deal more I could say here. For now, however, I offer just the following. My intention is to offer evidence that Lockwood is not a credible accuser. She has made many other "interesting" claims about her own experience of sexual misconduct.

A Truly Incredible Story

From the Huffington Post UK. The title is "The Week My Husband Left And My House Was Burgled I Secured A Grant To Begin The Project That Became BRCA1". Just read it. And make sure you read the whole thing. You really have no idea.

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? (Note that something like this will work with any DLNA renderer.)

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.

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:

\newlength{\numheight}
\setlength{\numheight}{\fontcharht\font`0}
\newcommand\numeral[1]{\overline{\rule{0pt}{\numheight}#1}

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

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, ..., a_n> is coded as: 2^a₁ × ... ×p_n^a_n, where p_n is the n^th 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 n^th 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 2^a₁ × ... ×p_n^a_n ≤ p_n^{a₁ + ... + a_n}.

Now let <a₁, ..., a_n> and <b₁, ..., b_m> be two sequences. The code of their concatenation is:

2^a₁ × ... ×p_n^a_n × p_n+1^b₁ × ... × p_n+m^b_m ≤ p_n+m^{a₁ + ... + a_n + b₁ + ... + b_m}

Moreover,

a₁ + ... + a_n ≤ 2^a₁ + ... + p_n^a_n ≤ 2^a₁ × ... × p_n^a_n = x

and similarly for the other sequence. (Note that the last inequality depends upon the fact that none of the a_i = 0, but Gödel's coding of sequences only works for positive integers.) So

a₁ + ... + a_n + a_n+1 + ... + a_n+m ≤ x + y

which gives Gödel's bound.