Monday, April 23, 2012

Callahan Again

Gene Callahan has once again visited the EPJ comments section in an attempt to "prove" his point on Hypothetical Frequentism. His "proof" seems to be that "everyone" buys into his "infinite dart"  (An appeal to mass authority "proof", I guess). Callahan writes:
This is not a contrarian point: it is absolutely standard probability theory.
Standard, or not, Gene baby, not everyone buys into it. Aren't we supposed to test theory based on the logic of a theory, rather than by vote?

But, since Gene apparently likes to argue based on authority, lets throw Alan Hájek at him.

Hájek studied statistics and mathematics at the University of Melbourne (B.Sc. (Hons). 1982), where he won the Dwight Prize in Statistics. He took an M.A. in philosophy at the University of Western Ontario (1986) and a Ph.D. in philosophy at Princeton University (1993), winning the Porter Ogden Jacobus fellowship. He has taught at the University of Melbourne (1990) and at Caltech (1992-2004), where he received the Associated Students of California Institute of Technology Teaching Award (2004). He has also spent time as a visiting professor at MIT (1995), Auckland University (2000), and Singapore Management University (2005). Hájek joined the Philosophy Program at Research School of Social Sciences Austrailian National University College of Arts & Social Sciences, as Professor of Philosophy in February 2005.

What does he think of Hypothetical Frequentism  and stuff  like Callahan's infinite dart? He thinks its nuts, but aside from credentials that make Callahan look like a ditch digger in comparison, he also has the logical arguments to back up his view . In fact, he wrote a paper titled, Fifteen Arguments Against Hypothetical Frequentism. Of the fifteen Hájek arguments against HF, here is one I really liked:

The Counterfactuals Appealed to are Utterly Bizarre

For HF isn’t just some innocent, innocuous counterfactual. It is infinitely more farfetched than the solubility counterfactual. To focus our discussion, let us think of counterfactuals as being analysed in terms of a Stalnaker/Lewis-style possible worlds semantics (Stalnaker 1968, Lewis 1973). Taking hypothetical frequentism’s statement literally—and I don’t know how else to take it—we are supposed to imagine a world in which an infinite sequence of the relevant attribute occurs. But for almost any attribute you can think of, any world in which that is the case would have to be very different from the actual world. Consider the radium atom’s decay. We are supposed to imagine infinitely many radium atoms: that is, a world in which there is an infinite amount of matter (and not just the 1080 or so atoms that populate the actual universe, according to a recent census). Consider the coin toss. We are supposed to imagine infinitely many results of tossing the coin: that is, a world in which coins are ‘immortal’, lasting forever, coin-tossers are immortal and never tire of tossing (or something similar, anyway), or else in which coin tosses can be completed in ever shorter intervals of time… In short, we are supposed to imagine utterly bizarre worlds—perhaps worlds in which entropy does not increase over time, for instance, or in which special relativity is violated in spectacular ways. In any case, they sound like worlds in which the laws of physics (and the laws of biology and psychology?) are quite different to what they actually are. But if the chances are closely connected to the laws, as seems reasonable, and the laws are so different, then surely the chances could then be quite different, too.

Note also a further consequence for the world that we are supposed to consider here. If there are infinitely many events of a given sort in a single world, then either time is continuous (as opposed to quantized), or infinite in at least one direction, or infinitely many events are simultaneous, or we have a Zeno-like compression of the events into smaller and smaller intervals. In any case, I find it odd that there should be such extravagant consequences for the truth-makers of probability statements in the actual world.

So what goes on in these worlds seems to be entirely irrelevant to facts about this world. Moreover, we are supposed to have intuitions about what happens in these worlds—for example, that the limiting relative frequency of Heads would be 1/2 in a nearest world in which the coin is tossed forever. But intuitions that we have developed in the actual world will be a poor guide to what goes on in these remote worlds. Who knows what would happen in such a world? And our confidence that the limiting relative frequency really would be 1/2 surely derives from actualworldly facts—the actual symmetry of the coin, the behavior of other similar coins in actual tosses, or what have you—so it is really those facts that underpin our intuition.
There are 14 more arguments here, Gene baby, against your infinitesimal dart probability model. All based on logic. It's not an infinite amount of arguments against your theory, but if that is what you want, I can conjure that up for you.


  1. He's just an attention whore. Ignore him. No one takes him seriously.

  2. "it is absolutely standard probability theory"
    ... note the last word :)
    Actually, randomness and probability take their roots from ignorance. If you have to resort to probability, then you really don't know what is happening. Same with randomness, it is a measure of ignorance. Now, maybe there is something called "infinitely small", but I would say Nature defines that as "non existent" :)

  3. This Callahan fellow seems very confused.

  4. Gene Callahan = Obvious Butt Boy

  5. Callahan's brain has been poisoned by pretensions of God-like concepts. Of course he'll be hoodwinked by counter-factual arguments where infinity is "valid". In his brain, God exists in some other dimension and is infinite in its attributes.

    He's just confused.

  6. FINALLY!!!!

    Thanks so much, Wenzel, for directing attention to Hájek. I've never read him before, but after reading his paper, I see that his intuition is the exact same as my own, and I just couldn't articulate well enough to counter the common use of "infinity." Now I know a good way to approach it.

    Thank you!!

  7. I use to think that Gene was an interesting and intelligent provocateur but have since realized that he is just an annoying douchey antagonist.

  8. I thought about the infinitely small dart problem some more, and refuting the conclusion of zero probability is actually just common sense. For the probability to actually reach zero, the dart itself would also have to be zero width, and a dart of zero width can't "hit" anything. So the premise of the problem is nonsense right from the start.

    1. Agreed. Another way to approach it is to grant the possibility of an infinitely fine pointed dart, and then realize that this doesn't mean the dart's point is infinitely fine the entire length of the dart itself, and so eventually, as the dart travels forward, the cross-sectional width of the dart will become non-zero, and that is when the dart will lodge into the number line.

      Then there is also the fact that humans cannot discern lengths, sizes, or time increments that are smaller than Planck sizes, so it's not only logically absurd, it's empirically unscientific as well.

  9. I thought that Gene Callahan was an Austrian/Libertarian. I read an excellent primer on austrian economics by him called Economics for Real People, a book that also has writings from Bob Murphy, Israel Kizner, Rothbard, Mises, Hayek. So what the heck happenned with Callahan? Has he gone mad? What's the beef with Wenzel? Aren't they both Austrians?

    1. You're definitely not the first person to ask that question "what happened to Callahan?"

      I've never been able to procure a definitive answer, but there's plenty of speculation.

      You'll probably get as much info here and here as anywhere else.

    2. That second link was supposed to go here.