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.