Responses to comments are finally up. (Go get your attention fix- after all, it’d be a shame if all that attention went to waste.) I’m doing a lot worse lately at this organization business. For instance, this is not the posts I wanted to write. This is merely the one I thought I could get together in the twenty minutes I’ve (shamefully) allowed for creative activity today.
I think there is something wrong with the following proof, but I can’t quite figure it out.
Let A be any entity for which there is no evidence (like Russell’s teapot). Imagine an entity A’, which is intrinsically mutually exclusive to A. If A’ is no more or less likely to exist than A, the probability that A exists is reduced by half. Imagine an entity A”, which is mutually exclusive to both A’ and A…and so on. An infinite number of such entities can be postulated, which means that, so long as there is no evidence for the existence of A, the probability of its existence is infintesimally small.
Proof of (a variant of) Occam’s razor
I responded thus, which I’ll quote largely to give the appearance of productive blogging although I’ll only be expanding on the latter point.
Imagine I propose the question of aliens to Joe Average, who is not interested in science or fun estimations and has never heard of the Drake equation. If I ask him the odds of the existence of extraterrestrial life somewhere in the universe, he might say 50% (with some justifiable hesitation).
In this case, it is nonsense to say that he could imagine an entity that somehow contradicts the existence of extraterrestrial life. Perhaps a nonliving superpredator; at any rate he can imagine a large number of things that could snuff out life. He could his our imagination to create universes with different rules in which life is more or less likely. He could do various other thought experiments.
But these imaginary entities are all unrelated to the actual probability of the statement “Aliens exist.” It is not the existence or nonexistence of these entities that can change the probability of the statement, but rather the application of our reason. Probability is the science of measuring our ignorance.
This proof strikes me as remarkably similar to Zeno’s paradox. I suspect my reply might be more understandable if I explained this with mathematical terminology, e.g. Statement A has probability p, B has probability 1-p, and so on.
Which raises another objection in my oft-clouded mind: even if we can, by mere induction, subject such infinitesimal probabilities to limits a la Calculus, how could we say that the final limit must be zero and not 1/4 or 1/10?
Anyway, thanks. This was fun.
He seemed to think I’d made a good point, which is swell because he’s a lot smarter than I am.
I gave some thought to the proposition that probabilities can be subjected to limits. My book on calculus-based probability certainly seems to think so, but I’ve noticed that my meatputer make fewer errors in this domain than most meatputers.
Let’s use an example probability density function where the random variable is just a horizontal axis. Maybe I’m throwing a football over a football field and the function describes the distance thrown. Certainly seems continuous on some interval of that horizontal axis.
And, unless I’m mistaken, any continuous interval must contain a differentiable subdomain. (Which is to say, I can’t imagine a continuous interval made up entirely of nondifferentiable points. Could be my imagination fails me, so a proof would be better here.)
But are physical distances continuous? Let’s be unnecessarily sceptical and say that the universe is pixelated, but the resolution is very high…
Ah dammit, I’ve already run out of time. More later.