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Psy-Kosh comments on 0 And 1 Are Not Probabilities - Less Wrong

34 Post author: Eliezer_Yudkowsky 10 January 2008 06:58AM

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Comment author: Psy-Kosh 10 January 2008 07:53:48PM 0 points [-]

"1, 2, and 3 are all integers, and so is -4. If you keep counting up, or keep counting down, you're bound to encounter a whole lot more integers. You will not, however, encounter anything called "positive infinity" or "negative infinity", so these are not integers."

This bothered me, more to the point, it hit on some stuff I've been thinking about. I realize I don't have a very good way to precisely state what I mean by "finite" or "eventually"

The above, for instance, basically says "if infinity is not an integer, then if I start at an integer and move an integer number of steps away from it, I will still be at an integer that's not infinity, therefore infinity isn't an integer"

But if we allowed infinity to be considered an integer, then we allow an infinite number of steps...

How about this: if N is a non infinite integer, SN is N's successor, PN is N's predecessor, neither SN nor PN will be infinite. Great, no matter where we start from, we can't reach an infinity in one step, so that seems to make this notion more solid.

but... if N is an infinity, then neither SN nor PN (thinking about ordinals now, btw, instead of cardinals) will be finite. Doh.

So the situation seems a bit symmetric here. This is really annoying to me.

I have as of late been getting the notion that the notions of "finite" and "eventually" are so tied to the idea of mathematical induction that it's probably best do define the former in terms of the latter... ie, the number of steps from A to A is finite if and only if induction arguments starting from A and going in the direction toward B actually validly prove the relevant proposition for B.

This is a vague notion, but near as I can tell, it comes closes to what I actually think I mean when I say something like "finite" or "eventually reach in a finite number of steps" or something like that.

ie, finite values are exactly those critters for which mathematical induction arguments can be used on. (maybe this is a bad definition. I'm more stating it as a "here's my suspicion of what may be the best basis to really represent the concept")

Anyways, as far as 0,1 not being probabilities... While I agree that one should't believe a proposition with probability 0 or 1, I'm not sure I'd consider them nonprobabilities. Perhaps "unreachable" probabilities instead. Disallowing stuff like sum to 1 normalizations and so on would seem to require "unnatural" hoops to jump through to get around that.

Unless, of course, someone has come up with a clean model without that. (If so, well, I'm curious too.)