I don’t. I believe that there are things that can only be described in terms of stupendously huge numbers, but I believe that everything that exists can be described without reference to infinities.
Really, when I think about how incomprehensibly enormous a number like BusyBeaver(3^^^3) is, I have trouble believing that there is some physical aspect of the universe that could need anything bigger. And if there is, well, there’s always BusyBeaver(3^^^^3) waiting in the wings.
Eliezer calls this infinite-set atheism, which is as good a name as any, I suppose.
Infinities exist as concepts, yes. They’re even useful in math. But I have never encountered anything that exists (for any reasonable definition of “exists”) that can’t be described without an infinity. MWI describes a preposterously large but still finite multiverse, as far as I understand it. And if our physical universe is infinite, as some have supposed, I haven’t seen proof of it.
Really, like any other form of atheism, infinite-set atheism should be easy to dispel. All anyone has to do to change my mind is show me an infinite set.
That’s not the only proof I’d accept, but given that I do accept conceptual infinities, I don’t think my brain is necessarily the limiting factor here.
Another form of acceptable evidence would be some mathematical proof that begins with the laws of physics and demonstrates that reality contains an infinity. I’m not sure if a similar proof that demonstrates that reality could contain an infinity would be as convincing, but it would certainly sway me quite a bit.
Unfortunately, observations don’t have epistemic power, so we’d have to live with all possible concepts. Besides, it’s quite likely that reality doesn’t in fact contain any infinities, in which case it’s not possible to show you an infinity, and you are just demanding particular proof. :-)
I distinguish between “believing in X” and “believing reality contains X”. I grew to dislike the non-mathematical concept of reality lately. Decision theory shouldn’t depend on that.
I’m not sure I understand. Part of it is the use of BusyBeaver—I’m familiar with Busy Beaver as an AI state machine, not as a number. Second: So you say you do not believe in infinity … but only inasmuch as physical infinity? So you believe in conceptual infinity?
The BusyBeaver value I’m referring to is the maximum number of steps that the Busy Beaver Turing Machine with n states (and, for convenience, 2 symbols) will take before halting. So (via wikipedia), BB(1) = 1, BB(2) = 6, BB(3) = 21, BB(4) = 107, BB(5) >= 47,176,870, BB(6) >= 3.8 × 10^21132, and so on. It grows the fastest of all possible complexity classes.
OK, have to make technical corrections here. Busy Beaver is not a complexity class, complexity classes do not grow. Busy Beaver function grows faster than any computable function, but I doubt it’s the “fastest” at anything, seeing as you can always just take e^BB(n), e.g.
Ugh, thank you. I seem to have gotten complexity classes and algorithmic complexity mixed up. Busy Beaver’s algorithmic complexity grows asymptotically faster than any computable function, so far as considerations like Big-O notation are concerned. In those sorts of cases, I think that even for functions like e^BB(n), the BB(n) part dominates. Or so Wikipedia tells me.
ETA: cousin_it has pointed out that there uncomputable functions which dominate Busy Beaver.
As Eliezer pointed out on HN, there is a way to define numbers that dominate BB values as decisively as BB dominates the Ackermann function, but you actually need some math knowledge to make the next step, not just stack BB(BB(...)) or something. (To be more precise, once you make the step, you can beat any person who’s “creatively” using BB’s and oracles but doesn’t know how to make the same step.) And after that quantum leap, you can make another quantum leap that requires you to understand another non-trivial bit of math, but after that leap he doesn’t know what to do next, and I, being a poor shmuck, don’t know either. If you want to work out for yourself what the steps are, don’t click the link.
I don’t. I believe that there are things that can only be described in terms of stupendously huge numbers, but I believe that everything that exists can be described without reference to infinities.
Really, when I think about how incomprehensibly enormous a number like BusyBeaver(3^^^3) is, I have trouble believing that there is some physical aspect of the universe that could need anything bigger. And if there is, well, there’s always BusyBeaver(3^^^^3) waiting in the wings.
Eliezer calls this infinite-set atheism, which is as good a name as any, I suppose.
See also: Finitism
Concepts don’t have to be about “reality”, whatever that is (not a mathematically defined concept for sure).
Infinities exist as concepts, yes. They’re even useful in math. But I have never encountered anything that exists (for any reasonable definition of “exists”) that can’t be described without an infinity. MWI describes a preposterously large but still finite multiverse, as far as I understand it. And if our physical universe is infinite, as some have supposed, I haven’t seen proof of it.
Really, like any other form of atheism, infinite-set atheism should be easy to dispel. All anyone has to do to change my mind is show me an infinite set.
Considering your brain is finite, I don’t think you’re entitled to that particular proof.
(Perhaps you’re just saying it would be a sufficient but not a necessary proof, in which case...okay, I guess.)
That’s not the only proof I’d accept, but given that I do accept conceptual infinities, I don’t think my brain is necessarily the limiting factor here.
Another form of acceptable evidence would be some mathematical proof that begins with the laws of physics and demonstrates that reality contains an infinity. I’m not sure if a similar proof that demonstrates that reality could contain an infinity would be as convincing, but it would certainly sway me quite a bit.
Unfortunately, observations don’t have epistemic power, so we’d have to live with all possible concepts. Besides, it’s quite likely that reality doesn’t in fact contain any infinities, in which case it’s not possible to show you an infinity, and you are just demanding particular proof. :-)
Wait… he’s already saying he believes reality doesn’t contain any infinities…
And you say that you can’t show proof to the contrary because it’s likely reality doesn’t contain any infinities…
I don’t think I followed you there.
I distinguish between “believing in X” and “believing reality contains X”. I grew to dislike the non-mathematical concept of reality lately. Decision theory shouldn’t depend on that.
My disbelief in infinities extends only to reality; I make no claims about the question of their existence elsewhere.
I’m not sure I understand. Part of it is the use of BusyBeaver—I’m familiar with Busy Beaver as an AI state machine, not as a number. Second: So you say you do not believe in infinity … but only inasmuch as physical infinity? So you believe in conceptual infinity?
The BusyBeaver value I’m referring to is the maximum number of steps that the Busy Beaver Turing Machine with n states (and, for convenience, 2 symbols) will take before halting. So (via wikipedia), BB(1) = 1, BB(2) = 6, BB(3) = 21, BB(4) = 107, BB(5) >= 47,176,870, BB(6) >= 3.8 × 10^21132, and so on. It grows the fastest of all possible complexity classes.
OK, have to make technical corrections here. Busy Beaver is not a complexity class, complexity classes do not grow. Busy Beaver function grows faster than any computable function, but I doubt it’s the “fastest” at anything, seeing as you can always just take e^BB(n), e.g.
Ugh, thank you. I seem to have gotten complexity classes and algorithmic complexity mixed up. Busy Beaver’s algorithmic complexity grows asymptotically faster than any computable function, so far as considerations like Big-O notation are concerned. In those sorts of cases, I think that even for functions like e^BB(n), the BB(n) part dominates. Or so Wikipedia tells me.
ETA: cousin_it has pointed out that there uncomputable functions which dominate Busy Beaver.
Sure, but my point is it’s not the “fastest” of anything unless you want to start defining some very broad equivalences...
As Eliezer pointed out on HN, there is a way to define numbers that dominate BB values as decisively as BB dominates the Ackermann function, but you actually need some math knowledge to make the next step, not just stack BB(BB(...)) or something. (To be more precise, once you make the step, you can beat any person who’s “creatively” using BB’s and oracles but doesn’t know how to make the same step.) And after that quantum leap, you can make another quantum leap that requires you to understand another non-trivial bit of math, but after that leap he doesn’t know what to do next, and I, being a poor shmuck, don’t know either. If you want to work out for yourself what the steps are, don’t click the link.
Ah, excellent, so I’m not so far off. Then what’s 3^^^3, then?
3^^^^3 on Less Wrong wiki.
Oh, good golly gosh, that gets big fast. Thank you!