Abram, II use constructivist's and intuitionist's (and for that matter finitist's) methods when they seem useful to me. I often make mistakes when I am not wary of constructivist issues. Constructist criticisms are interesting because they can be turned against any presumptive method even though they might seem to contradict a constructivist criticism taken from a different presumption.
I misused the term "computable" a few times because I have seen it used in different ways. But it turns out that it can be used in different ways. For example pi is not computable because it is infinite but a limiting approximation to pi is computable. So I would say that pi is computable - given infinite resources. One of my claims here is that I believe there are programs that will run Solomonoff Induction so the method would therefore be computable given infinite resources. However, my other claim is that the much desired function or result where one may compute the probability that a string will be produced given a particular prefix is incomputable. If I lived 500 years ago I might have said that a function that wasn't computable wasn't well-defined. (I might well have been somewhat pompous about such things in 1510). However, because of the efficacy of the theory of limits and other methods of finding bounds on functions, I would not say that now. Pi is well defined, but I don't think that Solmonoff Induction is completely well-defined. But we can still talk about certain aspects of it (using mathematics that are well grounded relative to those aspects of the method that are computable) even though the entire function is not completely well-defined. One way to do this is by using conditional statements. So if it turns out that one or some of my assumptions are wrong, I can see how to revise my theory about the aspect of the function that is computable (or seems computable). Jim Bromer On Thu, Jul 22, 2010 at 10:50 PM, Abram Demski <abramdem...@gmail.com>wrote: > Jim, > > Aha! So you *are* a constructivist or intuitionist or finitist of some > variety? This would explain the miscommunication... you appear to hold the > belief that a structure needs to be computable in order to be well-defined. > Is that right? > > If that's the case, then you're not really just arguing against Solomonoff > induction in particular, you're arguing against the entrenched framework of > thinking which allows it to be defined-- the so-called "classical > mathematics". > > If this is the case, then you aren't alone. > > --Abram > > > On Thu, Jul 22, 2010 at 5:06 PM, Jim Bromer <jimbro...@gmail.com> wrote: > >> On Wed, Jul 21, 2010 at 8:47 PM, Matt Mahoney <matmaho...@yahoo.com>wrote: >> The fundamental method is that the probability of a string x is >> proportional to the sum of all programs M that output x weighted by 2^-|M|. >> That probability is dominated by the shortest program, but it is equally >> uncomputable either way. >> Also, please point me to this mathematical community that you claim >> rejects Solomonoff induction. Can you find even one paper that refutes it? >> >> You give a precise statement of the probability in general terms, but then >> say that it is uncomputable. Then you ask if there is a paper that refutes >> it. Well, why would any serious mathematician bother to refute it since you >> yourself acknowledge that it is uncomputable and therefore unverifiable and >> therefore not a mathematical theorem that can be proven true or false? It >> isn't like you claimed that the mathematical statement is verifiable. It is >> as if you are making a statement and then ducking any responsibility for it >> by denying that it is even an evaluation. You honestly don't see the >> irregularity? >> >> My point is that the general mathematical community doesn't accept >> Solomonoff Induction, not that I have a paper that *"refutes it,"*whatever >> that would mean. >> >> Please give me a little more explanation why you say the fundamental >> method is that the probability of a string x is proportional to the sum of >> all programs M that output x weighted by 2^-|M|. Why is the M in a bracket? >> >> >> On Wed, Jul 21, 2010 at 8:47 PM, Matt Mahoney <matmaho...@yahoo.com>wrote: >> >>> Jim Bromer wrote: >>> > The fundamental method of Solmonoff Induction is trans-infinite. >>> >>> The fundamental method is that the probability of a string x is >>> proportional to the sum of all programs M that output x weighted by 2^-|M|. >>> That probability is dominated by the shortest program, but it is equally >>> uncomputable either way. How does this approximation invalidate Solomonoff >>> induction? >>> >>> Also, please point me to this mathematical community that you claim >>> rejects Solomonoff induction. Can you find even one paper that refutes it? >>> >>> >>> -- Matt Mahoney, matmaho...@yahoo.com >>> >>> >>> ------------------------------ >>> *From:* Jim Bromer <jimbro...@gmail.com> >>> *To:* agi <agi@v2.listbox.com> >>> *Sent:* Wed, July 21, 2010 3:08:13 PM >>> >>> *Subject:* Re: [agi] Comments On My Skepticism of Solomonoff Induction >>> >>> I should have said, It would be unwise to claim that this method could >>> stand as an "ideal" for some valid and feasible application of probability. >>> Jim Bromer >>> >>> On Wed, Jul 21, 2010 at 2:47 PM, Jim Bromer <jimbro...@gmail.com> wrote: >>> >>>> The fundamental method of Solmonoff Induction is trans-infinite. >>>> Suppose you iterate through all possible programs, combining different >>>> programs as you go. Then you have an infinite number of possible programs >>>> which have a trans-infinite number of combinations, because each tier of >>>> combinations can then be recombined to produce a second, third, fourth,... >>>> tier of recombinations. >>>> >>>> Anyone who claims that this method is the "ideal" for a method of >>>> applied probability is unwise. >>>> >>>> Jim Bromer >>>> >>> >>> *agi* | Archives <https://www.listbox.com/member/archive/303/=now> >>> <https://www.listbox.com/member/archive/rss/303/> | >>> Modify<https://www.listbox.com/member/?&;>Your Subscription >>> <http://www.listbox.com/> >>> *agi* | Archives <https://www.listbox.com/member/archive/303/=now> >>> <https://www.listbox.com/member/archive/rss/303/> | >>> Modify<https://www.listbox.com/member/?&;>Your Subscription >>> <http://www.listbox.com/> >>> >> >> *agi* | Archives <https://www.listbox.com/member/archive/303/=now> >> <https://www.listbox.com/member/archive/rss/303/> | >> Modify<https://www.listbox.com/member/?&;>Your Subscription >> <http://www.listbox.com/> >> > > > > -- > Abram Demski > http://lo-tho.blogspot.com/ > http://groups.google.com/group/one-logic > *agi* | Archives <https://www.listbox.com/member/archive/303/=now> > <https://www.listbox.com/member/archive/rss/303/> | > Modify<https://www.listbox.com/member/?&;>Your Subscription > <http://www.listbox.com/> > ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
