That's an awesome essay! Thanks. Of course, I never say anything unless I have something to disagree with... So, I have 2 points to argue about:
1) Your finiteness is illusory because you assume crisp sets, and 2) The problem of induction is about the origins of a conjecture, not merely about the evaluation of candidate set members. I've beaten (1) quite to death. So, I'll just try to lay out (2). Alternatives for the origins of conjecture abound. Not all conjecture is based on previous experience. Or, at least, we can conjecture that not all conjecture is based on previous experience, if we believe in illusion, hallucination, and that ideas are more plastic than things. I don't intend to dicker about the meanings of words like "based". But there is a difference between, say, a griffin and a unicorn. The griffin is more fantastic than the unicorn. It's pretty easy for me to imagine all horses are unicorns who have lost their horns. But it would be difficult to think that lions are griffins with mutated heads and no wings. Conjecture can come from _anywhere_ ... from dreams, fevers, and sci-fi novels. They need not be based on previous experiences in the (rigorous?) way we tend to think of them being based on previous experiences. In that sense, the finite nature of our past experiences is fuzzified ... not so finite after all. I can read Heinlein or Vern and form a conjecture based on a fuzzy mix of those "experiences". From this "alphabet" of ideas, I can generate an infinite (or at least indefinite) number of words and sentences. This seems to provide plenty of origins for alternative conjecture beyond induction. Eric Smith wrote at 03/28/2012 12:02 PM: > I agree, Wow, to Eric Charles's summary. > > Can I ask, is there any role for finiteness in this discussion? > There seem to me to be two places the constraints of being finite > enter, and the specific point at which they seem forced by one of the > questions that has been asked (Why would you accept the inductive > hypothesis, except on faith?) is to offer the alternative answer > (Because I am incapable of doing otherwise). The latter answer seems > more operational than invoking the word "faith", which for all I know > may not mean anything more than "I am incapable of doing otherwise > because I am finite", or the circularity that the inductive > hypothesis is the only premise from which to defend its own use. > > Here is one argument for finiteness, which I have pestered Nick with > in conversations long ago. > > Suppose I am an input-output machine with a finite number of outputs > (and for that matter, of internal states). (I intend here to be in > the general domain of Ross Ashby's notions of "requisite variation", > but let me not get off on that.) If a philosopher objects that I > have no _right_ to be finite, and that, like Paris Hilton, I must > "from now on, promise to pay complete attention to everything", then > don't consider me, consider a single type of cell-surface receptor on > any of my cells. Surely, for practical purposes, we want to model it > as having a finite repertoire of behaviors, whether it has a right to > or not. It's only a protein. > > But a finite thing, in an indefinitely variable world, must then > produce at least one of its outputs as the response to an indefinite > set of distinct circumstances. (Here, I would be happy to say > "infinite", but I mean "indefinite" as that which "might grow toward > infinity if we could keep accumulating cases".) In practice, of > course, for most cases I could think of, _each_ of the output states > occurs as a response to indefinitely many distinct circumstances. > Thus, my input-output box partitions an indefinite environment into > equivalence classes, at least one, and probably many of them, of > indefinitely large size. It cannot do otherwise, being finite. > > But my cell-surface receptor has been (naturally-)selected from a > finite history of events. Again, a practical man, being honest about > noise and the erasure of memory, about costs, etc., would say that it > has probably been selected by a rather short recent history from a > rather narrow set of cases. Thus, the response it will generate, for > an indefinite range of circumstances, has been chosen from a finite > set of selection criteria. It appears that survival-under-selection > of finite objects, then, is perforce a commitment to one form of > induction. The generation, by the device for however long it > survives, of a potentially-indefinitely-large set of input-output > pairs is being conditioned on a finite set of input-output pairs. > > Of course, natural selection is just Bayes's theorem for model > selection, with the fitness being the log-likelihood, and selection > on spaces with a finite number of types is then Bayesian induction on > probability distributions over a finite number of tokens. I think > (?) we believe in statistics that Bayesian updating is the best > general-purpose method we can formalize (Cosma Shalizi, sometimes > solo and sometimes with Andrew Gelman, has written on this). So > either in thinking or in science, can we claim to be capable of any > better actual, implementable mechanism than an algorithm that (if I > am not wrong in the last par) instantiates the inductive hypothesis? > If we can't be capable of anything better, than what is the nature of > an objection against such algorithms, besides reminders that they are > finite in an infinite world (which, being unable to fathom an > infinite world, we may tend to forget)? > > The second argument for caring about finiteness is related. I am > still a finite input-output machine. > > I have a choice. I can either suppose that a future event will be > the same as -- or structurally close to, in some way I can > operationalize -- one of my past-experienced events under > suitably-identified conditions, or I can suppose it will "not be". > My past is finite. The number of its histories that I can actually > distinguish may be (for me, _is_) even smaller. Let's suppose that > the set of all non-comparable futures to anything I have experienced > and can distinguish and remember is, again, indefinitely large. > > If I suppose the future will repeat the past, and I am in a > circumstance where that is true, since the past has finitely-many > tokens, then I have non-zero probability to guess the right future. > If I am not in that circumstance, then I am sure to be wrong. If I > suppose the future will _not_ resemble the past -- and, CRUCIAL POINT > HERE , if I am at all honest -- then I must make my bets over an > indefinite set of futures with non-singular measure, meaning I have > measure zero on everything. Whether or not my environment will > repeat its past, my chances of being right are zero either way. > Again, among my options, where do I find an argument against choosing > induction from the past? That is, fine, we recognize the logical > limits of induction, but do we have any argument that any specific > alternative could be supported in any better way, or by any better > premises? > > Of course, this sounds like Pascal's wager, and the structure of the > argument is similar, but its topic is not similar, because the > premises for which one is arguing in the two cases are not > comparable. A punishing God, whom one doesn't want to cross -- > whether one believes He is a an emotional imprint of punishing > parents, punishing spouses, punishing colleagues, any of whom may > hold grudges forever in response to any act of disobedience, or one > believes that the universe and existence are a subset of human > social-psychology (rather than the other way around) -- is still a > particular and arbitrary premise. The previous paragraph was about > search and methodology. Search in finite spaces, which may be > feasible, versus search infinite spaces, which is almost-surely > infeasible, addresses a specific problem with which we are already > specifically stuck, and which it is not ours to arbitrarily choose to > escape from. > > So the rhetorical argument about the religious person who claims that > the grass-is-green induction is an expression of the same flavor of > faith as his support of a cultural dogma, seems to me > methodologically unclear in a way that usually gets used > disingenuously. If we admit that we are all finite, then the only > remaining question is which templates we are inductively generalizing > _from_, in a particular situation. If you want to generalize from > human-social emotion in discussions about plate tectonics, dark > matter, dark energy, or whatever (I have relatives who do), then that > is your option out of finitely many. If instead you want to narrow > the arbitrariness and try to bundle the constraints that give the > polynomial expansion for gravitational action functionals (and then > just try to identify parameters), or to do the more complicated > version of that for planet formation, then at least we can say by > Occam's razor that you have made a smaller search space that overlaps > with a larger set of more-apparently similar evidence. > > > I realize that my email here seems (and presumably is) stupid, since > every child knows there can be no discussion of induction that is not > predicated on the availability of infinities. But, since I am > finite, I have tended to think that the point of these abstractions > is to extract salient features of a class of (actual, realizable) > problems which may not fulfill the abstraction, but for which the > abstraction can serve as the appropriate index of a class of problems > that can be extended in a particular direction. So, in looking at > these "grass-is-grue" arguments, or at almost anything Rorty writes > when he wants to be a gadfly against someone who is getting something > useful done (notably, I do not extend this criticism to his > constructive comments about desirable societies), and in trying to > figure out where there is a productive discussion about scientific > methods or claims, it is hard for me to feel other than confused > unless I look at the specific role that finiteness has played in > creating the structure of any of the specific propositions being > discussed. > > > There are, of course, other classes of propositions where one could > well argue against induction, but again, it would be for reasons that > can be made operational and thus comprehensible. Economists would > dislike David Wolpert's no-free-lunch theorems if they felt them > worth reading, since David samples over complete sets of > combinatorial search and optimization problems, and shows that any > expectation that can be satisfied by some cases in such spaces, must > be violated on an equal measure of cases. The economists (the > particular ones I know) don't want to pursue this, because the > discipline wants preferred solution concepts, preferred equilibria, > etc. But of course, if we were talking about the design of > regulatory systems, in which any set of published rules becomes a > target for gaming, by those who wish to set up an expectation and > violate it, then the need to guard against anti-induction solutions > becomes large. All this, though, I can do on finite spaces, and > compare the severity of simplified inferences to the size of the > violation set, and then look at limits under scaling of the number of > cases. > > > Also, if I had read Pierce, I expect I would have found that he shows > in the first few pages that arguments of this type are trivial and > don't address anything. In another life... -- glen e. p. ropella, 971-222-9095, http://tempusdictum.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
