Because being exposed to ordered sensory data will rapidly promote the hypothesis that induction works

Not if the alternative hypothesis assigns about the same probability to the data up to the present. For example, an alternative hypothesis to the standard "the sun rises every day" is "the sun rises every day, until March 22, 2015", and the alternative hypothesis assigns the same probability to the data observed until the present as the standard one does.

You also have to trust your memory and your ability to compute Solomonoff induction, both of which are demonstrably imperfect.

There's an infinite number of alternative hypotheses like that and you need a new one every time the previous one gets disproven; so assigning so much probability to all of them, that they went on dominating Solomonoff induction on every round even after being exposed to large quantities of sensory information, would require that the remaining probability mass assigned to the prior for Solomonoff induction be less than exp(amount of sensory information), that is, super-exponentially tiny.

1DaFranker7yBut... no. "The sun rises every day" is much simpler information and computation than "the sun rises every day until Day X". To put it in caricature, if hypothesis "the sun rises every day"is: XXX1XXXXXXXXXXXXXXXXXXXXXXXXXX (reading from the left) then the hypothesis "the sun rises every day until Day X" is: XXX0XXXXXXXXXXXXXXXXXXXXXX1XXX And I have no idea if that's even remotely the right order of magnitude, simply because I have no idea how many possible-days or counterfactual days we need to count, nor of how exactly the math should work out. The important part is that for every possible Day X, it is equally balanced by the "the sun rises every day" hypothesis, and AFAICT this is one of those things implied by the axioms. So because of complexity giving you base rates, most of the evidence given by sunrise accrues to "the sun rises every day", and the rest gets evenly divided over all non-falsified "Day X" (also, induction by this point should let you induce that Day X hypotheses will continue to be falsified).

You only need faith in two things

by Eliezer Yudkowsky 1 min read10th Mar 201386 comments


You only need faith in two things:  That "induction works" has a non-super-exponentially-tiny prior probability, and that some single large ordinal is well-ordered.  Anything else worth believing in is a deductive consequence of one or both.

(Because being exposed to ordered sensory data will rapidly promote the hypothesis that induction works, even if you started by assigning it very tiny prior probability, so long as that prior probability is not super-exponentially tiny.  Then induction on sensory data gives you all empirical facts worth believing in.  Believing that a mathematical system has a model usually corresponds to believing that a certain computable ordinal is well-ordered (the proof-theoretic ordinal of that system), and large ordinals imply the well-orderedness of all smaller ordinals.  So if you assign non-tiny prior probability to the idea that induction might work, and you believe in the well-orderedness of a single sufficiently large computable ordinal, all of empirical science, and all of the math you will actually believe in, will follow without any further need for faith.)

(The reason why you need faith for the first case is that although the fact that induction works can be readily observed, there is also some anti-inductive prior which says, 'Well, but since induction has worked all those previous times, it'll probably fail next time!' and 'Anti-induction is bound to work next time, since it's never worked before!'  Since anti-induction objectively gets a far lower Bayes-score on any ordered sequence and is then demoted by the logical operation of Bayesian updating, to favor induction over anti-induction it is not necessary to start out believing that induction works better than anti-induction, it is only necessary *not* to start out by being *perfectly* confident that induction won't work.)

(The reason why you need faith for the second case is that although more powerful proof systems - those with larger proof-theoretic ordinals - can prove the consistency of weaker proof systems, or equivalently prove the well-ordering of smaller ordinals, there's no known perfect system for telling which mathematical systems are consistent just as (equivalently!) there's no way of solving the halting problem.  So when you reach the strongest math system you can be convinced of and further assumptions seem dangerously fragile, there's some large ordinal that represents all the math you believe in.  If this doesn't seem to you like faith, try looking up a Buchholz hydra and then believing that it can always be killed.)

(Work is ongoing on eliminating the requirement for faith in these two remaining propositions.  For example, we might be able to describe our increasing confidence in ZFC in terms of logical uncertainty and an inductive prior which is updated as ZFC passes various tests that it would have a substantial subjective probability of failing, even given all other tests it has passed so far, if ZFC were inconsistent.)

(No, this is *not* the "tu quoque!" moral equivalent of starting out by assigning probability 1 that Christ died for your sins.)