Jim, An example reference on the theory of computability is "Computability and Logic" by Boolos, Burgess and Jeffrey. For those who accept the church-turing thesis, this mathematical theory provides a sufficient account of the notion of computability, including the space of possible programs (which is formalized as the set of Turing machines).
--Abram On Mon, Jul 19, 2010 at 6:44 AM, Jim Bromer <jimbro...@gmail.com> wrote: > Abram, > I feel a responsibility to make an effort to explain myself when someone > doesn't understand what I am saying, but once I have gone over the material > sufficiently, if the person is still arguing with me about it I will just > say that I have already explained myself in the previous messages. For > example if you can point to some authoritative source outside the > Solomonoff-Kolmogrov crowd that agrees that "full program space," as it > pertains to definitions like, "all possible programs," or my example > of, "all possible mathematical functions," represents an comprehensible > concept that is open to mathematical analysis then tell me about it. We use > concepts like "the set containing sets that are not members of themselves" > as a philosophical tool that can lead to the discovery of errors in our > assumptions, and in this way such contradictions are of tremendous value. > The ability to use critical skills to find flaws in one's own presumptions > are critical in comprehension, and if that kind of critical thinking has > been turned off for some reason, then the consequences will be predictable. > I think compression is a useful field but the idea of "universal induction" > aka Solomonoff Induction is garbage science. It was a good effort on > Solomonoff's part, but it didn't work and it is time to move on, as the > majority of theorists have. > Jim Bromer > > On Sun, Jul 18, 2010 at 10:59 PM, Abram Demski <abramdem...@gmail.com>wrote: > >> Jim, >> >> I'm still not sure what your point even is, which is probably why my >> responses seem so strange to you. It still seems to me as if you are jumping >> back and forth between different positions, like I said at the start of this >> discussion. >> >> You didn't answer why you think program space does not represent a >> comprehensible concept. (I will drop the "full" if it helps...) >> >> My only conclusion can be that you are (at least implicitly) rejecting >> some classical mathematical principles and using your own very different >> notion of which proofs are valid, which concepts are well-defined, et >> cetera. >> >> (Or perhaps you just don't have a background in the formal theory of >> computation?) >> >> Also, not sure what difference you mean to say I'm papering over. >> >> Perhaps it *is* best that we drop it, since neither one of us is getting >> through to the other; but, I am genuinely trying to figure out what you are >> saying... >> >> --Abram >> >> On Sun, Jul 18, 2010 at 9:09 PM, Jim Bromer <jimbro...@gmail.com>wrote: >> >>> Abram, >>> I was going to drop the discussion, but then I thought I figured out why >>> you kept trying to paper over the difference. Of course, our personal >>> disagreement is trivial; it isn't that important. But the problem with >>> Solomonoff Induction is that not only is the output hopelessly tangled and >>> seriously infinite, but the input is as well. The definition of "all >>> possible programs," like the definition of "all possible mathematical >>> functions," is not a proper mathematical problem that can be comprehended in >>> an analytical way. I think that is the part you haven't totally figured out >>> yet (if you will excuse the pun). "Total program space," does not represent >>> a comprehensible computational concept. When you try find a way to work out >>> feasible computable examples it is not enough to limit the output string >>> space, you HAVE to limit the program space in the same way. That second >>> limitation makes the entire concept of "total program space," much too >>> weak for our purposes. You seem to know this at an intuitive operational >>> level, but it seems to me that you haven't truly grasped the implications. >>> >>> I say that Solomonoff Induction is computational but I have to use a >>> trick to justify that remark. I think the trick may be acceptable, but I am >>> not sure. But the possibility that the concept of "all possible programs," >>> might be computational doesn't mean that that it is a sound mathematical >>> concept. This underlies the reason that I intuitively came to the >>> conclusion that Solomonoff Induction was transfinite. However, I wasn't >>> able to prove it because the hypothetical concept of "all possible program >>> space," is so pretentious that it does not lend itself to mathematical >>> analysis. >>> >>> I just wanted to point this detail out because your implied view that you >>> agreed with me but "total program space" was "mathematically well-defined" >>> did not make any sense. >>> 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/> >>> >> >> >> >> -- >> 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> > <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 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
