The Universe of Discourse
Mon, 24 Nov 2008

Variations on the Goldbach conjecture

  1. Every prime number is the sum of two even numbers.
  2. Every odd number is the sum of two primes.
  3. Every even number is the product of two primes.

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Wed, 12 Nov 2008

Flag variables in Bourne shell programs
Who the heck still programs in Bourne shell? Old farts like me, occasionally. Of course, almost every time I do I ask myself why I didn't write it in Perl. Well, maybe this will be of some value to some fart even older than me..

Suppose you want to set a flag variable, and then later you want to test it. You probably do something like this:

        if some condition; then


        if [ "$IS_NAKED" == "1" ]; then
          flag is set
          flag is not set
Or maybe you use ${IS_NAKED:-0} or some such instead of "$IN_NAKED". Whatever.

Today I invented a different technique. Try this on instead:

        if some condition; then


        if $IS_NAKED; then
          flag is set
          flag is not set
The arguments both for and against it seem to be obvious, so I won't make them.

I have never seen this done before, but, as I concluded and R.J.B. Signes independently agreed, it is obvious once you see it.

[ Addendum 20090107: some followup notes ]

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Tue, 11 Nov 2008

Another note about Gabriel's Horn
I forgot to mention in the original article that I think referring to Gabriel's Horn as "paradoxical" is straining at a gnat and swallowing a camel.

Presumably people think it's paradoxical that the thing should have a finite volume but an infinite surface area. But since the horn is infinite in extent, the infinite surface area should be no surprise.

The surprise, if there is one, should be that an infinite object might contain a merely finite volume. But we swallowed that gnat a long time ago, when we noticed that the infinitely wide series of bars below covers only a finite area when they are stacked up as on the right.

The pedigree for that paradox goes at least back to Zeno, so perhaps Gabriel's Horn merely shows that there is still some life in it, even after 2,400 years.

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Mon, 10 Nov 2008

Gabriel's Horn is not so puzzling
Take the curve y = 1/x for x ≥ 1. Revolve it around the x-axis, generating a trumpet-shaped surface, "Gabriel's Horn".

Elementary calculations, with calculus, allow one to show that although the Horn has finite volume, it has an infinite surface area. This is considered paradoxical, because it says that although an infinite amount of paint is required to cover the interior surface of the horn, the entire interior can be filled up with a finite amount of paint.

The calculations themselves do not lend much insight into what is going on here. But I recently read a crystal-clear explanation that I think should be more widely known.

Take out some Play-Doh and roll out a snake. The surface area of the snake (neglecting the two ends, which are small) is the product of the length and the circumference; the circumference is proportional to the diameter. The volume is the product of the length and the cross-sectional area, which is proportional to the square of the diameter.

Elementary Calculus: An Infinitesimal Approach
Elementary Calculus: An Infinitesimal Approach
with kickback
no kickback
Now roll the snake with your hands so that it becomes half as thick as it was before. Its diameter decreases by half, so its cross-sectional area decreases to one-fourth. Since the volume must remain the same, the snake is now four times as long as it was before. And the surface area, which is the product of the length and the diameter, has doubled.

As you continue to roll the snake thinner and thinner, the volume stays the same, but the surface area goes to infinity.

Gabriel's Horn does exactly the same thing, except without the rolling, because the parts of the Horn that are far from the origin look exactly the same as very long snakes.

There's nothing going on in the Gabriel's Horn example that isn't also happening in the snake example, except that in the explanation of Gabriel's Horn, the situation is obfuscated by calculus.

I read this explanation in H. Jerome Keisler's caclulus textbook. Keisler's book is an ordinary undergraduate calculus text, except that instead of basing everything on limits and on limiting processes, it is based on nonstandard analysis and explicit infinitesimal quantities. Check it out; it is available online for free. (The discussion of Gabriel's Horn is in chapter 6, page 356.)

[ Addendum 20081110: A bit more about this. ]

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Fri, 07 Nov 2008

Addenda to recent articles 200810

  • I discussed representing ordinal numbers in the computer and expressed doubt that the following representation truly captured the awesome complexity of the ordinals:

            data Nat = Z | S Nat
            data Ordinal = Zero
                         | Succ Ordinal
                         | Lim (Nat → Ordinal)
    In particular, I asked "What about Ω, the first uncountable ordinal?" Several readers pointed out that the answer to this is quite obvious: Suppose S is some countable sequence of (countable) ordinals. Then the limit of the sequence is a countable union of countable sets, and so is countable, and so is not Ω. Whoops! At least my intuition was in the right direction.

    Several people helpfully pointed out that the notion I was looking for here is the "cofinality" of the ordinal, which I had not heard of before. Cofinality is fairly simple. Consider some ordered set S. Say that an element b is an "upper bound" for an element a if ab. A subset of S is cofinal if it contains an upper bound for every element of S. The cofinality of S is the minimum cardinality of its cofinal subsets, or, what is pretty much the same thing, the minimum order type of its cofinal subsets.

    So, for example, the cofinality of ω is ℵ0, or, in the language of order types, ω. But the cofinality of ω + 1 is only 1 (because the subset {ω} is cofinal), as is the cofinality of any successor ordinal. My question, phrased in terms of cofinality, is simply whether any ordinal has uncountable cofinality. As we saw, Ω certainly does.

    But some uncountable ordinals have countable cofinality. For example, let ωn be the smallest ordinal with cardinality ℵn for each n. In particular, ω0 = ω, and ω1 = Ω. Then ωω is uncountable, but has cofinality ω, since it contains a countable cofinal subset {ω0, ω1, ω2, ...}. This is the kind of bullshit that set theorists use to occupy their time.

    A couple of readers brought up George Boolos, who is disturbed by extremely large sets in something of the same way I am. Robin Houston asked me to consider the ordinal number which is the least fixed point of the ℵ operation, that is, the smallest ordinal number κ such that |κ| = ℵκ. Another way to define this is as the limit of the sequence 0, ℵ00, ... . M. Houston describes κ as "large enough to be utterly mind-boggling, but not so huge as to defy comprehension altogether". I agree with the "utterly mind-boggling" part, anyway. And yet it has countable cofinality, as witnessed by the limiting sequence I just gave.

    M. Houston says that Boolos uses κ as an example of a set that is so big that he cannot agree that it really exists. Set theory says that it does exist, but somewhere at or before that point, Boolos and set theory part ways. M. Houston says that a relevant essay, "Must we believe in set theory?" appears in Logic, Logic, and Logic. I'll have to check it out.

    My own discomfort with uncountable sets is probably less nuanced, and certainly less well thought through. This is why I presented it as a fantasy, rather than as a claim or an argument. Just the sort of thing for a future blog post, although I suspect that I don't have anything to say about it that hasn't been said before, more than once.

    Finally, a pseudonymous Reddit user brought up a paper of Coquand, Hancock, and Setzer that discusses just which ordinals are representable by the type defined above. The answer turns out to be all the ordinals less than ωω. But in Martin-Löf's type theory (about which more this month, I hope) you can actually represent up to ε0. The paper is Ordinals in Type Theory and is linked from here.

    Thanks to Charles Stewart, Robin Houston, Luke Palmer, Simon Tatham, Tim McKenzie, János Krámar, Vedran Čačić, and Reddit user "apfelmus" for discussing this with me.

    [ Meta-addendum 20081130: My summary of Coquand, Hancock, and Setzer's results was utterly wrong. Thanks to Charles Stewart and Peter Hancock (one of the authors) for pointing this out to me. ]

  • Regarding homophones of numeral words, several readers pointed out that in non-rhotic dialects, "four" already has four homophones, including "faw" and "faugh". To which I, as a smug rhotician, reply "feh".

    One reader wondered what should be done about homophones of "infinity", while another observed that a start has already been made on "googol". These are just the sort of issues my proposed Institute is needed to investigate.

    One clever reader pointed out that "half" has the homophone "have". Except that it's not really a homophone. Which is just right!

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Thu, 06 Nov 2008

Election results
Regardless of how you felt about the individual candidates in the recent American presidential election, and regardless of whether you live in the United States of America, I hope you can appreciate the deeply-felt sentiment that pervades this program:


        my $remain = 1232470800 - time();
        $remain > 0 or print("It's finally over.\n"), exit;

        my @dur;
        for (60, 60, 24, 100000) {
          unshift @dur, $remain % $_;
          $remain -= $dur[0];
          $remain /= $_;  

        my @time = qw(day days hour hours minute minutes second seconds);
        my @s;
        for (0 .. $#dur) {
          my $n = $dur[$_] or next;
          my $unit = $time[$_*2 + ($n != 1)];
          $s[$_] = "$n $unit"; 
        @s = grep defined, @s;

        $s[-1] = "and $s[-1]" if @s > 2;
        print join ", ", @s;
        print "\n";

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Mon, 03 Nov 2008

Atypical Typing
I just got back from Nashville, Tennessee, where I delivered a talk at OOPSLA 2008, my first talk as an "invited speaker". This post is a bunch of highly miscellaneous notes about the talk.

If you want to skip the notes and just read the talk, here it is.

Talk abstract

Many of the shortcomings of Java's type system were addressed by the addition of generics to Java 5.0. Java's generic types are a direct outgrowth of research into strong type systems in languages like SML and Haskell. But the powerful, expressive type systems of research languages like Haskell are capable of feats that exceed the dreams of programmers familiar only with mainstream languages.

In this talk I'll give a brief retrospective on the history of type systems and an introduction to the type system of the Haskell language, including a remarkable example where the Haskell type checker diagnoses an infinite loop bug at compile time.

I did not say in the abstract that the talk was a retread of a talk I gave for the Perl mongers in 1999 titled "Strong Typing Doesn't Have to Suck. Nobody wants to hear that. Still, the talk underwent a major rewrite, for all the obvious reasons.

In 1999, the claim that strong typing does not have to suck was surprising news, and particularly so to Perl Mongers. In 2008, however, this argument has been settled by Java 5, whose type system demonstrates pretty conclusively that strong typing doesn't have to suck. I am not saying that you must like it, and I am not saying that there is no room for improvement. Indeed, it's obvious that the Java 5 type system has room for improvement: if you take the SML type system of 15 years ago, and whack on it with a hammer until it's chipped and dinged all over, you get the Java 5 type system; the SML type system of the early 1990s is ipso facto an improvement. But that type system didn't suck, and neither does Java's.

So I took out the arguments about how static typing didn't have to suck, figuring that most of the OOPSLA audience was already sold on this point, and took a rather different theme: "Look, this ivory-tower geekery turned out to be important and useful, and its current incarnation may turn out to be important and useful in the same way and for the same reasons."

In 1999, I talked about Hindley-Milner type systems, and although it was far from clear at the time that mainstream languages would follow the path blazed by the HM languages, that was exactly what happened. So the HM languages, and Haskell in particular, contained some features of interest, and, had you known then how things would turn out, would have been worth looking at. But Haskell has continued to evolve, and perhaps it still is worth looking at.

Or maybe another way to put it: If the adoption of functional programming ideas into the mainstream took you by surprise, fair enough, because sometimes these things work out and sometimes they don't, and sometimes they get adopted and sometimes they don't. But if it happens again and takes you by surprise again, you're going to look like a dumbass. So start paying attention!

Haskell types are hard to explain

I spent most of the talk time running through some simple examples of Haskell's type inference algorithm, and finished with a really spectacular example that I first saw in a talk by Andrew R. Koenig at San Antonio USENIX where the type checker detects an infinite-loop bug in a sorting function at compile time. The goal of the 1999 talk was to explain enough of the ML type system that the audience would appreciate this spectacular example. The goal of the 2008 talk was the same, except I wanted to do the examples in Haskell, because Haskell is up-and-coming but ML is down-and-going.

It is a lot easier to explain ML's type system than it is to explain Haskell's. Partly it's because ML is simpler to begin with, but also it's because Haskell is so general and powerful that there are very few simple examples! For example, in SML one can demonstrate:

        (* SML *)
        val 3 : int;
        val 3.5 : real;
which everyone can understand.

But in Haskell, 3 has the type (Num t) ⇒ t, and 3.5 has the type (Fractional t) ⇒ t. So you can't explain the types of literal numeric constants without first getting into type classes.

The benefit of this, of course, is that you can write 3 + 3.5 in Haskell, and it does the right thing, whereas in ML you get a type error. But it sure does make it a devil to explain.

Similarly, in SML you can demonstrate some simple monomorphic functions:

	  not : bool → bool
	 real : int → real
	 sqrt : real → real
	floor : real → int
Of these, only not is simple in Haskell:

            not :: Bool → Bool
    fromInteger :: (Num a) ⇒ Integer → a    -- analogous to 'real'
           sqrt :: (Floating a) ⇒ a → a
          floor :: (RealFrac a, Integral b) ⇒ a → b
There are very few monomorphic functions in the Haskell standard prelude.


I'm still using the same slide-generation software I used in 1999, which makes me happy. It's a giant pile of horrible hacks, possibly the worst piece of software I've ever written. I'd like to hold it up as an example of "worse is better", but actually I think it only qualifies for "bad is good enough". I should write a blog article about this pile of hacks, just to document it for future generations.

Conference plenary sessions

This was the first "keynote session" I had been to at a conference in several years. One of the keynote speakers at a conference I attended was such a tedious, bloviating windbag that I walked out and swore I would never attend another conference plenary session. And I kept that promise until last week, when I had to attend, because now I was not only the bloviating windbag behind the lectern, but an oath-breaker to boot. This is the "shameful confession" alluded to on slide 3.

On the other hand...

One of the highest compliments I've ever received. It says "John McCarthy will be there. Mark Jason Dominus, too." Wow, I'm almost in the same paragraph with John McCarthy.

McCarthy didn't actually make it, unfortunately. But I did get to meet Richard Gabriel and Gregor Kiczales. And Daniel Weinreb, although I didn't know who he was before I met him. But now I'm glad I met Daniel Weinreb. During my talk I digressed to say that anyone who is appalled by Perl's regular expression syntax should take a look at Common Lisp's format feature, which is even more appalling, in much the same way. And Weinreb, who had been sitting in the front row and taking copious notes, announced "I wrote format!".

More explaining of jokes

As I get better at giving conference talks, the online slides communicate less and less of the amusing part of the content. You might find it interesting to compare the 1999 version of this talk with the 2008 version.

One joke, however, is too amusing to leave out. At the start of the talk, I pretended to have forgotten my slides. "No problem," I said. "All my talks these days are generated automatically by the computer anyway. I'll just rebuild it from scratch." I then displayed this form, which initialliy looked like this:

I then filled out the form appropriately for OOPSLA:

I pushed the button, and poof! Instant slides.

Wadler's anecdote

I had the chance to talk to Philip Wadler, one of the designers of Haskell and of the Java generics system, before the talk. I asked him about the history of the generics feature, and he told me the following story: At this point in the talk I repeated an anecdote that Wadler told me. After he and Odersky had done the work on generics in their gj and "Pizza" projects, Odersky was hired by Sun to write the new Java compiler. Odersky thought the generics were a good idea, so he put them into the compiler. At first the Sun folks always ran the compiler with the generics turned off. But they couldn't rip out the generics support completely, because they needed it in the compiler in order to get it to compile its own source code, which Odersky had written with generics. So Sun had to leave the feature in, and eventually they started using it, and eventually they decided they liked it. I related this story in the talk, but it didn't make it onto the slides, so I'm repeating it here.

I had never been to OOPSLA, so I also asked Wadler what the OOPSLA people would want to hear about. He mentioned STM, but since I don't know anything about STM I didn't say anything about it.

View it online

The slides are online.

[ Addendum 20081031: Thanks to a Max Rabkin for pointing out that Haskell's analogue of real is fromInteger. I don't know why this didn't occur to me, since I mentioned it in the talk. Oh well. ]

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