NOT YET POSTED Newsgroups: comp.ai.philosophy,comp.ai References: <4bs41p$fo4@mp.cs.niu.edu> <820108318snz@longley.demon.co.uk> <4bt8gp$n9s@mp.cs.niu.edu> Subject: Improbable theories and scientific progress. (Was: Cognitive Science ??) Keywords: science, Popper, probability, learning, AI Summary: Notes on philosophy of science and learning. LONG GENES, MEMES, PROBABILITY, CONTENT, FITNESS OF THEORIES AND GENE POOLS An interchange between Neil Rickert and David Longley has prompted some thoughts about science, Popper, probability, evolution, learning and AI. In response to David Longley rickert@cs.niu.edu (Neil Rickert) wrote: > Date: 27 Dec 1995 23:04:25 -0600 > Organization: Northern Illinois University [NR] > >> David quotes from Popper, "Truth, Rationality, and the Growth of > >> Knowledge" and includes this paragraph: > > >> > This trivial fact has the following inescapable > >> > consequences: if growth of knowledge means that we > >> > operate with theories of increasing content, it must > >> > also mean that we operate with theories of decreasing > >> > probability (in the sense of the calculus of > >> > probability). Neil objected > >> That is a remarkable claim by Popper. And it is remarkable because > >> it is so obviously mistaken. The history of the growth of scientific > >> completely contradicts this thesis. [DL] > >The point here *is* irrefutable.... [NR] > 'Irrefutable' or not, it is absurd. > .... > Two hundred years ago weather predictions were not much better than > "if the sunset is red it might rain tomorrow." Today predictions are > more refined, more precise, and have a higher probability of being > correct. The improvement is due to the growth of scientific > knowledge. Thus Popper's absurd conclusion is contradicted by the > evidence. Like so many apparent disagreements in these newsgroups, I think this one arises because different people unwittingly attach different meanings to the same words and phrases. Analysis of the concept of "probability" is an old and difficult philosophical problem. There are very different ways in which the term can be used, including at least the following: (a) expressing a degree of strength of belief (b) describing a feature of a mechanism or situation (adding a weight to a die can increase the probability of a five being thrown - Popper called this a "propensity") (c) describing the long run limiting frequency (if one exists) with which events of type P will occur in circumstances of type C, in the world (This concept is different from (b) though the two are related: propensities may explain long run frequencies) (d) describing the long run limiting frequency with which a certain property is present in a randomly generated mathematically defined sequence (e.g. the limiting frequency of occurrence of two fives in pairs of random selections of numbers between one and six with equal average frequency) NB: (d) refers to a mathematical property of a mathematically defined system, and can often be calculated exactly (e.g. proving that the probability of two fives is 1/36, given assumptions about the equiprobability of all individual numbers) whereas (b) and (c) both refer to a property of the actual world and can never be calculated by purely mathematical analysis. Instead (b) and (c) require estimations based on measurement and observation, and they are subject to error even when no mathematical mistake has been made, unlike case (d): biased dice don't disprove a mathematical result. If you are talking about the probability of producing two fives with a real pair of dice you are not referring to case (d). There are deep philosophical problems about whether ANY method can *reliably* detect probabilities of type (b) and (c) even though many methods are actually used and seem to work much of the time. Incidentally, unlike some philosophers, Popper claimed that there is a useful concept of type (b). But that's not the concept he was using in making the claim quoted by David, which I'll analyse shortly. Another source of muddle is ambiguity regarding what counts as a measure of the CONTENT of a proposition. Popper was using the idea that for Pa to say more about something than Pb does, Pa has to be more precise, i.e. it has to specify a narrower range of possible states of affairs in which it would be true, than Pb does. I.e. a proposition with more content rules out more possibilities. Increasing content in this sense implies increasing precision. As I'll explain below that's not the only possible measure of content, and so we have another ambiguity producing spurious disagreements. When making the claim that increasing content (in the sense just decribed) reduced the probability of a theory, Popper was using a mathematical concept of probability something like type (d). If Pa has more content than Pb (in that sense) this mathematically implies that there are potentially fewer instances satisfying Pa than satisfy Pb, and so that long run frequency of situations in which Pa could be true is smaller, and that's why Pa's having lower probability than Pb follows "trivially" from its having more content. Think of it this way: if you add content to Pb to get Pa, in such a way that Pa -> Pb remains true, then whenever Pa is true Pb will be, thus the probability of Pb is at least as high as that of Pa. [NR] > ....Popper is assuming that all knowledge growth > amounts to taking conjunctions of components of existing knowledge. It's true that this is ONE way of increasing the content (in Popper's sense) of propositions, but not the ONLY way. Another way is to reduce the number of disjuncts in a disjunctive proposition. There are other ways which don't depend on logical relations, but geometric or topological relations. E.g. if I tell you Fred is in the kitchen I say something with more content (in Popper's sense) than if I tell you Fred is in the house, and I also thereby reduce the probability of what I say being true (if nothing else is known). Note that these are NOT two conjunctive expressions. (You might think of them as disjunctive, where each says something of the form "Fred is in location a, or in location b or in ... " with "Fred is in the kitchen" expressing fewer disjuncts. But that's not a good general interpretation of spatial concepts. For example, I don't have to be able to form a disjunction specifying all the locations in a house, in order to believe that Fred is in the house.) I don't believe Popper was presupposing the rather silly notion that all scientific theories have to be expressed in some particular logical form using conjunctions (or even conjunctions and disjunctions, or even first order pedicate calculus). Rather, he was relying on the intuitive notion that the more information I give you about some state of affairs the more possibilities I am ruling out. I.e. he was linking increasing content with increasing precision. Notice that there are perfectly good senses of `increased content' for which it would NOT be true that having more content means ruling out more possibilities. E.g. you can say that the proposition P1. All physical objects are solids, liquids, gases or plasma has more content than the proposition P2. All physical objects are solids, liquids or gases. In some sense P1 has more content (assuming a plasma is not a gas): it implies a richer ontology. But it does not have more content in Popper's sense because of the disjunctive form: P1 does not allow you to be as precise about the next object you'll come across as P2 does, and in that sense it says less. This is a case where the theory with LESS content in Popper's sense is better, because of how the world is. More generally, every time you add an existential statement to your theory, of the form "It's possible for entities of type X to exist" the more content you add in one sense, for you say more about the types of things that the universe can contain, but the less you say in Popper's sense, for less can now be inferred about the next object you'll come across. It no longer has to be one of the previously specified types. So now, in some sense the probability of the theory has increased: it's true in more cases. However, if you consider the set of possible universes, and for each of them describe the types of things they contain, then universes satisfying P1 will be a subset of universes satisfying P2, so in that context the probability of P1 is higher. Whether the probability is higher or lower in the mathematical sense, depends on the choice of space within which to compare frequencies. I don't think Popper thought about that. (I suspect Popper's link, in a subset of cases, between increased semantic content and ruling possibilities out, i.e. reduction of uncertainty, is what lies behind the irresponsible misuse of the word "information" in connection with mathematical so-called "information theory", which has nothing to do with semantics or meaning, but only with the syntactic properties of sets of signals. There is an indirect link with real (semantic) information in that this syntactic measure is related to how much semantic information a signal could carry in a given context. But that's a measure of potential information, not actual information, in the ordinary sense. This mathematical misuse of the ordinary word "information" has caused *enormous* amounts of confusion -- even leading some silly composers to link randomness in music with musical significance, using arguments like this: "The less predictable music is, the more possibilities each new note rules out, and therefore the more meaningful it is. Therefore random sequences are most meaningful!!" I can remember a time when some music students were taught that kind of stuff.) Anyhow, unlike Popper in the quotation, who refers to mathematical probability of type (d), Neil, I suspect, is talking about empirical probability of type (b) or possibly (c), when he compares two situations, 18th century meteorology and 10th century meteorology, where meteorologists are armed with powerful theories that have withstood a lot of testing. He is claiming that in the latter case the probability of a prediction being correct is higher. In Popper's terms I guess this could also be described by saying that modern meteorologists have an increased propensity to get their predictions right. And of course Neil is correct in arguing that modern meteorologists with their improved higher-content theories not only have more precise predictions (i.e. theories and predictions with more Popperian content) but also a higher probability of being correct. And of course Popper is right in claiming that IN GENERAL adding content (in his sense) to a theory reduces the probability of its predictions being correct (in all possible situations), simply because adding more content (in Popper's sense) reduces the set of possible states of affairs in which the theory is true. I.e. Popper's claim is not mistaken or absurd. It is correct, but that's because it is not a claim about the world but about mathematically defined proportions, using the concept of probability of type (d). (Whether that supports any of David Longley's arguments is a different matter: it's perfectly possible to quote a correct theory in support of an incorrect one.) All this raises a problem for philosophers of science, namely to explain what sort of scientific process can bring about a state of affairs in which scientists produce richer and more precise theories with a HIGHER probability of correctness in their predictions, even though increasing the precision and content (in Popper's sense) of a theory mathematically implies a REDUCED probability of it, or predictions based on it, being true. I think Popper was aware of this problem and as far as I recall his answer was based on an analogy between the development of science (or ideas and knowledge in general) and biological evolution. He argued (or could have argued, if he didn't actually do so!) that the processes of scientific testing (which would include implicit testing through engineering applications), like biological selection processes, weeds out theories that are relatively "unfit" in various ways. One way in which a theory can be unfit is in not matching how the world is when tested through predictions. There are others, e.g. not being compatible with the current world view, or locally dominant religion. These different fitness criteria are frequently in opposition, and can get in the way of assessing scientific theories. Because he had so much faith in the scientific selection process, Popper (unlike David Longley) imposed NO restrictions on where theories came from or what formalisms could be used to express them, so long as they were capable of being tested so that relatively unfit ones could be rejected (though he also knew that such rejection might prove mistaken and relative fitness illusory, e.g. because of poor measurements, or too narrow a range of experiments, or whatever. More on this below.) There is still an implicit assumption behind the analogy between theory selection and biological selection, namely the assumption that a process of selection, whether biological or scientific, will in general go on producing organisms or theories that are better and better fitted to their niche. Whether that is so depends on how the world is. If the world is capable of changing in arbitrary ways (so that niches change) then what is fit can change in unpredictable ways. And that is the case with biological fitness. E.g. a volcanic eruption can radically alter fitness requirements in the neighbourhood of the volcano, and rapidly replace one set of successful species with another set, including allowing what was previously less successful to become dominant. Is the process of scientific theory selection liable to such disruption through changing fitness conditions? It is often assumed that at least some sciences (physics, chemistry, cosmology, though NOT geology, biology, psychology, sociology) are concerned with matters that are universal (the same everywhere) and immutable (the same at all times), and do not depend on the contingent circumstances of a particular planet, or galaxy, or historical epoch. If there are any such universal sciences, then, the scientific fitness criteria, properly applied, could in general (apart from temporary aberrations) lead continually to better and better theories, at least in these sciences, without any possibility of dead-end dinosaur theories that are soon to be doomed because the ultimate nature of physical reality is about to change. Of course, because of limitations and aberrations in the testing processes mentioned above, dead-end theories can occur, even on this view. That's partly because scientific testing is inherently error prone because the quality of the tests are themselves limited by the current state of scientific knowledge. Testing can give erroneous or misleading results because of several factors, e.g. - tests may fail to take account of `hidden' features of the testing situation - tests may be limited to too narrow a range of circumstances (like many psychological experiments) - tests may use inadequate measuring devices - e.g. devices that have been carelessly made, or worn out, or damaged in some way, - tests may be based on wrong theories that are assumed not to be under test. (E.g. tests that assume that only the topology of electronic circuits in measuring instruments and not the physical layout of the components can make a difference to their performance may give misleading measurements because of the poor layout of circuits in those instruments, leading to the wrong rejection of a theory, or wrong non-rejection. Or a test of theory A assuming a theory B linking size of pulsars to their pulse frequency might wrongly support or reject theory A because of errors in theory B.) By contrast, evolutionary selection is not done by an intelligent agent that might be misinformed. If a bunch of collaborating genes fail to get themselves reproduced in a particular context then they have just failed. Excuses are irrelevant. [Sexual selection pressure is a counter-example, leading to things like peacocks. Here part of the gene pool driving the peahen information processing is part of the niche for other parts of the gene pool shared with peacocks.] Scientific selection is not the only form of cultural selection, as shown by the spread of religions, waves of fashion, tastes in food, etc. How scientific selection differs from the rest, and why it is more concerned with truth, is a long and complex story, and still a matter of controversy. Now let's return to Neil's point about weather forecasting: theories now have more content than they used to have, and predictions based on them have higher probability of being correct than they used to have. Is this a paradox? Is meteorology like physics? Is physics in fact investigating universal and immutable aspects of reality? Or is it more like a species selected in balmy days, and liable to be dislodged following a volcanic eruption. Neil, I think, assumed (at least when he wrote the above) that meteorology is concerned with an unchanging type of reality. Even if physics is the study of an unchanging reality (which some cosmologists may dispute) I think it is not true of meteorology, e.g. because derivation of principles of meteorology from those of physics is computationally intractable, or at least beyond the wit of current scientists. In that case, the meteorological principles used at present will have been selected not by derivation from physics but only by direct testing of rival theories in the earth's weather system. [Actually, I am making a general point about how some sciences might work which could be correct even if I am wrong about current meteorology: e.g. if all its forecasts are based directly on principles of quantum mechanics.] If theories are selected not by derivation from physics but by testing in the earth's atmosphere, this will make them much more fragile than current physical theory, which has survived a far wider variety of tests. If changes in physical circumstances, e.g. changes in the chemical composition of the atmosphere (perhaps caused by a massive volcano, or industrial by-products), or changes in the radiation emitted by the sun, were to change the dynamics of the earth's atmosphere, it might turn out that 18th Century meteorological theories again gave a higher proportion of correct predictions than those used now. In that case it would be best to revert to depending on the older, less precise, theories. They would have a higher probability of giving correct results, partly because they have less content in Popper's sense. Of course, after a while the meteorological theories might again be revised, as a result of much selective testing, until newer, more precise, less intrinsically probable, theories were developed which again had both a higher content and a higher probability of successful predictions in the new context than the earlier theories. There's no guarantee of this, however. The weather system involves much nonlinear feedback and is inherently chaotic. The post-traumatic weather system might or might not have some high level patterns discernable by human minds. Such patterns could provide a basis for a new system of precise predictions based on the accuracy with which aspects of the system could be measured. But there might be none (too many "butterfly effects"), in which case we'd be stuck with the older, less precise, systems of prediction. Thus insofar as modern meteorology is both more precise and more often correct than two hundred years ago, that's a matter of luck. In physics it can also be necessary to abandon current theories, but not because the world has changed, rather because our creative abilities and selection processes have led us down a bad trail. Again, when that happens it's not provable apriori that that we'll find another better set of patterns in nature. For the sciences, as for biological gene pools, it may be essential not to reject completely all the old `refuted' ideas, but to keep them around as a basis for generating future theories that may supersede the short term winners. One reason for that is that some theories, like biological species, are inherently fit only for local conditions that might change, like theories of the weather. Another reason is that imperfections in the scientific selection process can lead us temporarily into blind alleys from which we need to escape. E.g. particle theories of light (propounded by Newton, for example) were refuted by Young's diffraction experiments supporting only the fitness of wave theories. But later on the idea of a photon, or light `particle', again became important, as a wider range of experiments produced selection pressure favouring deeper, more general, conceptually richer, theories. (NB: the theories were never DERIVED from the observations, contrary to what some people think.) I hope I've clarified the extent to which both Popper and Neil are correct, and the difference in status of their claims. Popper's claim about the inverse relation between content (in his sense) and probability (of type (d)) is mathematical and can never be refuted by observation. Neil's claim is an empirical claim about the relative fitness of meteorological theories, and might turn out wrong if the world changed in certain ways. However, it seems to be true, at present. There's a moral here for AI learning theories. I have no idea how many of them fully address the issue of recovering from mistaken commitments. It may be that individual human learning mechanisms cannot deal fully with the problem because commitments have to be made on account of how the brain works, or because keeping options open (e.g. using assumption based truth maintenance systems) is computationally intractable. Of course there are specific cases of things learnt by humans apparently being retracted: e.g. children learn how to express the past tense of words like "run" ("ran"), "hit" ("hit"), "catch" ("caught"), and then they learn a rule which contradicts what they have previously learnt, so they let the rule over-ride their previous information, and then they re-learn examples which over-ride the rule in selective cases, perhaps because they have by then also developed a new mechanism for allowing generalisations to have exceptions. I suspect human learning does not use a general back-tracking mechanism but something that simply allows new specific information to selectively supersede old, including cases where the `new' information is information that has been encountered previously, and was the basis of some earlier learning, which was then superseded. Even if individual human ability to backtrack to previously stored choice points during learning is limited, it may nevertheless be possible for social learning (e.g. scientific development) to address the problem of escaping from local maxima in scientific hill climbing because communities can include a mixture of coexisting individuals with different commitments. (Like a mixed gene pool). Some of the individuals may still hold views that were previously rejected by the community at large, and this could support backtracking of a sort, though not necessarily the systematic and rigid sort of backtracking found in a Prolog system. If individual learning can use `genetic' mechanisms, such as classifier systems, then perhaps individuals can in principle have as much flexibility as social systems or species. But I don't know whether such highly flexible learning mechanisms can operate on fast enough time scales for use by individuals. (NB such genetic mechanisms differ from truth-maintenance mechanisms in that when current beliefs are refuted they don't immediately attempt to compute the best alternative on the basis of stored records of previous decisions. Instead the genetic mechanisms allow new replacements to evolve on the basis of continued interactions with the environment. This may or may not lead back to an earlier theory. Similar comments would apply to a neural net that does not separate a training phase and an application phase, but constantly adjusts itself during use on the basis of feedback.) I've not kept up with all the work on learning in AI, and I wonder to what extent the issue of "run-time" mechanisms for escaping from entrenched errors has been addressed. I doubt that it has been addressed by experimental psychologists, for their conception of learning tends to be very restricted (e.g. something they can measure in repeatable experiments in a laboratory, unlike most actual human learning, which is highly individualistic and dependent on prior knowledge, like scientific progress, and often not expressible in numeric terms, like the growth of a set of concepts or a grammar). [NR] > If an apparently correct logical argument leads to an absurd > conclusion, you should treat the argument as a reductio ad absurdum, > and then look for what is wrong with the premises. Alternatively, you can look, as I have done above, for a semantic ambiguity that causes people to argue endlessly at cross-purposes. It's a very frequent occurrence. [NR] > ....Popper is assuming that all knowledge growth > amounts to taking conjunctions of components of existing knowledge. I don't think so. That may be David's interpretation, because of his commitment to a restricted language for science. I don't think Popper has any such commitment. In fact he suggested that there was a kind of objective social knowledge that was expressed in many aspects of the humanly constructed environment, including libraries, museums, etc. This was his `third world'. The `first world' was the world of physical objects events and processes. The `second world' was the world of subjective private mental events and processes. He did not say that only those bits of the third world using predicate calculus were knowledge. [NR] > That is an extreme nativist position, for it implies that we already > know everything and only need to logically derive any missing > details. Only someone in the grip of an ideological committment to > formalism could make such an absurd assumption. You should not judge Popper by the contexts in which he is cited. As far as I know he never did make restrictive claims about the forms in which knowledge could be expressed. By the way, just for the record, though I learnt much from Popper I don't agree with everything he wrote. In particular, in Chapter 2 of `The Computer Revolution in Philosophy' I strongly criticised him e.g. for rejecting discussion of meanings or concepts as irrelevant to the study of science, and for his falsificationist criterion of scientific content. What Popper should have said was that a criterion of merit in scientific theories is having objectively decidable consequences. Being falsifiable is a special case of this, and too restrictive. I also criticised Popper, and most other philosophers of science, for emphasising discovery and testing of laws in science, thereby ignoring deeper forms of scientific creativity. The latter includes extending our grasp of what is possible or conceivable, i.e. extending our grasp of the `form' of reality, which defines the space of possible laws worth investigating. By comparison with that, finding a particular law is much easier and far more commonplace. This is related to the over-simple link between increased content and increased precision, criticised above. Sometimes vision-expanding theories have less precision (in the technical sense) than the older theories they replace. But they are still semantically richer for they refer to an enlarged ontology. The really great theories combine this with increased precision in particular applications. This Popperian requirement is what constrains the potential explosion of ontology expanding theories. But that's another long story. Aaron PS [Margaret Boden's book on creativity also stresses the growth of knowledge of what is possible, as opposed to laws.] --- e