Simplicity

Can someone tell me why it is said that “other things being equal, the simplest theory is the best”?
I have some thoughts about it, but I’d like to hear what other people think.

Extra credit question: Is simplicity in philosophical theories a virtue in the same way (or in any way) as simplicity in scientific theories?


Comments

Simplicity — 43 Comments

  1. My collegue Kevin Kelly has various non-trivial things to say about simplicity and its role in science. The two more recent manuscripts are:

    “Simplicity, Truth, and the Unending Game of Science”, 2005, manuscript.
    “Learning, Simplicity, Truth, and Misinformation”, 2005, manuscript.

    which can be downloaded in his CMU web page. But of course this is a topic with a long pedigree, some of which is Bayesian (a view that Kevin does not adopt).

    H.

  2. Here’s a flat-footed response:

    If we put truth to the side, then it might be just for the same reason that we’d prefer to take the shortest route to the grocery store: why make things more complicated for ourselves?

    Of course, one might say that we precisely can’t put truth to the side, and so degrees of complication should have nothing to do with it. But we shouldn’t forget where in the process of knowledge acquisition these questions come up; if we don’t know what the truth is, then there is a sense in which the truth is put to the side for us (we can’t refer to it). And then we might as well theorize in ways which are less complicated if we can.

    Then again, some people like to take the scenic route, and I’m not sure we can say anything which should persurade them otherwise.

  3. Could “Simple theories are better” be something like “The future will resemble the past”? (Or might “The future will resemble the past” be a special case of “Simple theories are better,” if better-ness is understood to mean (at least) likely to be true?)

    I’m inclined to take a Humean/Reidian/Wittgensteinian position about “The future will resemble the past” – this is just something that we all believe, as a matter of fact (whether it’s hardwired or not), and we don’t have any evidence or an argument for it, but we do (and are permitted to) believe it anyway. And I don’t see any reason not to treat “Simple theories are better” in the same way.

    But putting that particular position aside, here’s my suggestion: that we treat “Simple theories are better” the way we treat the principle of uniformity and other such basic commitments (“hinge propositions,” “first principles,” whatever). SO whatever your theory of those is, just apply it mutatis mutandis to “Simple theories are better.”

    (Why use the same theory? Because simplicity is better, of course. Seriously.)

  4. Some minimal thoughts about: Is simplicity in philosophical theories a virtue in the same way (or in any way) as simplicity in scientific theories?

    Some recent theories of simplicity like Kevin’s tend to show that there is an a priori structural invariance linking the notions of EFFICIENCY and simplicity. For example, Kevin shows that among the methods that converge to the truth in an empirical problem, the ones that do so with a minimum number of reversals of opinion prior to convergence (i.e. the methods that are in a sense more efficient) are exactly the ones that prefer simple theories.

    Neo-Bayesians (especially those of pragmatist conviction like current incarnations of van Fraassen or Putnam) have in addition considered efficiency as one of the foremost epistemic virtues (although they are less worried about convergence in the long run and therefore they tend to focus on efficiency in one-shot changes rather than in long sequences of changes). And this, in turn, might just be the consequence of thinking about knowledge from an economical point of view. This view was already present in `Two Dogmas’ where theory change is seen (in the last section of the essay) as a particular kind of choice where loses of valuable information have to be minimized (changes have to obey the principle of Conservatism). Quine concludes that: `Conservatism figures in such choices, and so does the quest for simplicity’.

    In a way the idea here is that epistemologists should ask what an agent should do next, given his current background knowledge, in order to render her knowledge more efficient in performing its functions. But it seems less clear that efficiency (either in its static or dyanmic forms) has to be considered as a salient epistemic virtue when the goal is UNDERSTANDING. To be sure the `enterprise of understanding’ (to borrow an expression coined by Howard Stein) is deeply intertwined with the goal of increasing the amount of error free information in the next step of inquiry (or with the Peircean goal of learning the truth in the long run). But these epistemic goals are not identical either. For example, historical pedigree and redundant information might play an important role in understanding which might be less central in pure scientific learning. Understanding might require to take u-turns and long detours that in pure empirical research might seem otiose. And understanding might play a central role in areas like, say, philosophy of mathematics (about which a means-end epistemology has little to say).

    H.

  5. Horacio, thank you for the tip about Kevin Kelly’s papers — I’ve just started the “Unending Game” paper, and not only is it very helpful, it also has the cutest figures I’ve ever seen in a philosophy paper!

    My own take: simplicity is one criterion among several that we have learned are such, as a matter of contingent fact, that if we prefer theories that maximize on those criteria to ones that don’t, we do a better job of getting at the truth. One upshot of this is that it gives a reason to think that simplicity might not be such a valuable criterion in much of philosophy: if simplicity’s value is only a contingent (or at best nomologically necessary) fact, then one cannot rely on that value when one’s theorizing is meant to be modally stronger than that.

  6. Part of it may be the asymmetry of complexity. Theories can get infinitely more complex while still maintaining the same results, but there is a ground floor to their simplicity. So, if one is generating a principle about how much complexity to posit in absence of particular evidence about what is going on, a complexity minimizing rule works in a way that other complexity oriented rules would not.

    Since sometimes, arguably, the only difference between competing theories is complexity, it makes sense to have a way to decide between them.

    Combining this with support in the form of theories we’ve had confirmed not usually tending towards the unnecessary complications, might explain it to some extent.

  7. Jonathan:

    Glad you found the references useful.

    >>it also has the cutest figures I’ve ever seen in a philosophy paper!

    Oh, Kevin loves diagrams and he seems to have some irrational faith in the didactic virtues of pictures and even cartoons. His first book features two little cartoon characters (which he draws himself): a Cartesian demon and a scientist who fights him. Now the scientist is a Monk (Ockham) who fends off the evil one by appealing to … simplicity! He has an animation for talks which is hilarious.

    Aside from these cute things I think that in a way the mathematical part of his account is now more elegant and direct and therefore more accessible to readers who have less enthusiasm than him for learning and applying the intricacies of recursion theory.

  8. Last week I was at a Miami University’s Grad Students’ Epistemology conference, which was a great event, and I remember asking a fellow student what connection truth has to simplicity. I’ve wondered for a long time about why philosophers invoke Ockham’s Razor (OR) in theory choice. I think what is being discussed here has something to do with this mysterious Razor, esp., when it comes to adjucating why we think simplicity is connected to truth.

    I think Steve Hales’ paper is a good antidote to some of us who think that OR should be taken as a given. I do not think that is the case. I think it’d be a good idea to read Steve’s paper here to further this discussion beyond a given such as OR:
    http://www.bloomu.edu/departments/philosophy/pages/content/hales/articlepdf/ockham.pdf

    Tedla

  9. Besides Kevin Kelly’s interesting work on simplicity in science, Elliott Sober has also done a lot of work on simplicity (in fact, he’s had about 3 different theories over the years, beginning with his dissertation book _Simplicity_, which he now considers unsalvageable.)

    A few representative works: “Let’s Razor Ockham’s Razor”, which is reprinted in _From a Biological Point of View_

    and chapter 2 of his book _Reconstructing the Past_ (entitiled “Simplicity”) which discusses simplicity, Hume’s problem of induction, the uniformity of nature, etc.

    Also, in his more recent work on model selection (e.g., his paper with Malclom Forster “How to tell when simpler, more unified, less ad hoc theories are predictively accurate” BJPS 1994 (available on Sober’s website).

    Sober’s take on simplicity in the first two works is that when simplicitiy is legitimately used in science, it is not a “superempricial” virtue – it is, instead, a stand in for some empirical claim (so it can’t legitimately help decide between predictively equivalent theories_).

    In his more recent work, he uses model selectdion criteria to talk about how simplicity (understood as number of adjustable paradmeters) should be traded off with goodenss of fit in order to maximize predictive accuracy.

    Sober argues for his first claim through a detailed examination of how simplicity is used in real biological cases – especially the problem of phylogenetic inference and the units of selection problem.

    Of course, that leaves what is perhaps the more difficult question. Even if you think he has made a strong case that leigitimate use of simplicity in science always depends on a stand in empirical claim or theory (that is, simplicity is not a legitimate tie breaker for predictively equivalent theories), what should we think about simplicity. Here I’d be very interested in to know what you think, Michael. If philosophy should look to science to see how simplicity should be used, we get one answer (perhaps). But if simplicity can be defended as a tie breaker on some purely philosophical or a priori grounds that would be really interesting.

  10. Thanks Chris for these references to Sober’s work. I read with some attention the chapter in Recontructing the Past a while ago.

    >In his more recent work, he uses model selectdion criteria to talk about how >simplicity (understood as number of adjustable paradmeters) should be traded off >with goodenss of fit in order to maximize predictive accuracy.

    What would be a good reference for this more recent work? Thanks,

    H.

  11. Thanks, everyone, for the helpful suggestions — just what I was looking for. I’ll see what I have to say about Sober, Hales, and Kelly after I’ve read some of their work.

    Somewhat along the lines of Lewis Powell’s suggestion, maybe the asymmetry of complexity & simplicity (the fact that there is a lower bound but no upper bound on possible degrees of complexity) has something to do with the virtue of simplicity. I take it that we want to show that simpler theories are (ceteris paribus) more likely to be true. So here’s a tricky (possibly annoying) suggestion:

    The mere fact that complexity has a lower but no upper bound forces us to assign decreasing probabilities to higher degrees of complexity, because that is the only way that our probabilities can add up to 1 (or: the only way our pdf can be normalizable).

    For ease of exposition, suppose that degrees of complexity are discrete, and they can be measured as 1, 2, 3, etc. If you try to assign the same probability to each possible degree of complexity, you get a total probability of either zero (if each Pr. = 0) or infinity (if each Pr. is nonzero).

    Or, if we suppose that degree of complexity is a continuous variable, the probability density, integrated from 0 to infinity, has to sum to 1. That can happen only if the probability density approaches 0 as degree of complexity increases.

  12. I suppose, though, that someone who dislikes this sort of view could argue that complexity of a theory bears no correlation to its likelihood of being true. So, maybe, if we want to assign the function, the function will have to favor simplicity, but perhaps there is no function (or at least, no useful/characterizable function) to assign.

  13. Ok, all the non-Bayesian accounts (+ some Bayesian accounts) of simplicity fail. Responses to some that were mentioned here:

    (1) The Pragmatic Account (Aaron): The problem with the pragmatic account is that, even purely on pragmatic grounds, having the true or false theory frequently has consequences that swamp the pragmatic value of having a theory that’s easy to grasp, easy to work with, or aesthetically pleasing. You have theories T1 and T2 that each accomodate the data so far and are approximately alike on the other criteria of theory-evaluation, but they make different predictions for the future. T1 is much simpler than T2. (If you think of “grue” vs. “green” hypotheses, or curve-fitting problems, as examples, this situation is extremely common.) Typically, having the right prediction is, pragmatically, far more important than having an easier time working with a theory. If you pick T1 because it’s easier to draw predictions out of, but T1 is in fact false, it’s not going to be worth it.

    You might say that if T1 and T2 are equally likely, then the value of an easy-to-work-with theory — small as that value is — could still be the tie-breaker. But that’s wrong because (again, pragmatically speaking) it would be best to withhold judgement. So, for example, you refrain from sending the astronauts into space if can’t decide on which physical theory that bears on what’s going to happen is true.

    (2) Efficient Convergence (Horacio, Kevin Kelly): Kelly argues that Occam’s Razor is good, roughly, because it represents a methodology that minimizes the number of times the scientist can be forced to change his opinion.

    I’m unpersuaded of this account for much the same reason as in the case of (1). We need an account of why we are justified in believing some theory, now, and that requires an account of why the theory is probably true. The fact that we plan actions (like sending people into space, building nuclear power plants, etc.) on the basis of our theories is one reason why we need that. Kelly’s account is only concerned with minimizing changes of opinion in the long run; it doesn’t give any reason for thinking that the simplest theory to accomodate the data is, now, most likely to be true. I wouldn’t want to plan the design of a nuclear power plant on theories that merely follow a methodology that, in an infinitely long game, mimimizes changes of opinion; I need theories that are likely to be true.

    Another objection to Kelly’s account is that even if one were concerned only with minimizing changes of opinion in the long run, what one should really be asking is what method gives you the smallest expected number of changes in opinion (so you have to talk probabilities to assess that).

    However, in Kelly’s defense, he does have extremely cute diagrams, and that has considerable persuasive power.

    (3) Anti-Occamism (Tedla, Steven Hales): I didn’t find Hales’ article persuasive, because he doesn’t address the sort of approach that I favor to explaining why simplicity is probability-conducive.

    (4) The Cheap Bayesian Account: This is the view that you should just assign a higher a priori probability to simpler theories. Swinburne says this. “Theft over honest toil” comes to mind — we need an account of why one should do this. It’s not self-evident.

    (5) The Empiricist Account (Weinberg): This is the view that we learn that simplicity is truth-conducive by induction from the history of science.

    My problem with this is essentially the circularity/skepticism worry. I think one can’t actually know that science has gotten to the truth a lot, unless one presupposes the virtue of simplicity (we believe “Science has gotten to the truth a lot” because that is the simplest theory to explain certain data that we now have).

    I also wonder whether “Simple theories tend to be true” would be a projectible generalization, if we assume that there is no general Bayesian-type account of why it is true. Essentially, I think that induction is only good when an inference to the best explanation stands behind it (or at least when we justifiably believe that there is an explanation underlying the regularity).

    (6) Good Bayesian Accounts: These include my comment #11.

    Also, the point that it is harder for a simple theory to fit a given amount of data (unless it is true) than for a complex theory to do so. For example, say you are trying to determine the relationship between two variables (assuming there is a lawlike relationship). If you have three data points, it is guaranteed that there exists a 2nd-degree polynomial fitting those points. But the prior probability of there being a 1st-degree polynomial fitting three randomly-chosen points is zero. (There are infinitely many more triples of points that are non-colinear than triples that are colinear.) I think one can therefore do a kind of Bayesian inference:

    • If it were false that the true relationship was linear, it would be highly unlikely that the data could be accomodated by a hypothesis that says the true relationship is linear.
    • Therefore, if the data in fact can be accomodated by such a hypothesis, this confirms that the true relationship is linear.

    Some complication would need to be introduced to accommodate the case of noisy data, but essentially this argument will still apply.

  14. Mike, you say, ” I think one can’t actually know that science has gotten to the truth a lot, unless one presupposes the virtue of simplicity”. But I took the initial question to be an _explanatory_ one, not a _justificatory_ one. The explanatory challenge here would be something like, given that things are pretty much as we take them to be, what explains the epistemic virtue of simplicity? In the process of answering this challenge, we may decide to refine or give up some bits of what we take ourselves to know, but that’s something that has to fall out of our investigations.

    If the challenge, in fact, is to justify simplicity without any appeal to anything that we might have learned by using it — i.e., almost all of our empirical knowledge — then my official position would be: the challenge is probably unmeetable, but also (fortunately) probably not well-founded.

    I’m also not at all sure I’m committed to it being an _induction_ from the structure & history of our scientific practices, but maybe that’s not an important part of your argument against my view.

    (It occurs to me that, since I believe in the real possibility of apriori justification for strongly contingent propositions, that I might could try to meet the justificatory challenge that way — simplicity’s virtue is a contingent fact, but one we’re apriori justified in accepting. But since part of my defense of that view relies on some particular scientific claims, I’d probably still run afoul of your circularity worries. Oh, well.)

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  16. Jonathan Weinberg says:

    “If the challenge, in fact, is to justify simplicity without any appeal to anything that we might have learned by using it — i.e., almost all of our empirical knowledge — then my official position would be: the challenge is probably unmeetable, but also (fortunately) probably not well-founded.”

    If the challenge is well-founded, then I think it is unmeetable. Or, at least, the evidence suggests that is so. Formal justifications of Ockham’s Razor have been attempted over and over, and generally have either ended up begging the question or changing the rules of the game. E.g., Sober’s work since 1988 has changed the rules of the game by reconceptualizing what parsimony is supposed to be. Much of the work in statistics has begged the question by relegating parsimony to a bias reflected in prior probabilities.

    Anyway, I’m not convinced that Michael Huemer’s circularity-skepticism objection to subjecting scientific method itself to the tribunal of experience hits its mark. I worry that the level of description, using the apparent success of Total Science, is the wrong one.

  17. Robert —

    I have a short argument that the challenge is meetable:

    1. If the challenge is unmeetable, then skepticism is true;
    2. Skepticism is obviously false; so
    3. The challenge can be met.

    You, Jon W, and I probably share premise (2). I accept (1) because

    a) Unless reliance on simplicity is somehow justified, beliefs that are largely based on appeal to simplicity are unjustified.
    b) It’s not foundational that simplicity is truth-conducive.
    c) Circular reasoning is illegitimate, however big the circle is.
    It follows that we need a non-circular argument for relying on simplicity, to avoid skepticism.

    I also have a longer argument that the challenge can be met: It would consist in defending my proposals in comments 11 and 13 (point #6). Do you think those proposals are no good?

  18. Jon W —

    I didn’t understand the distinction between the explanatory and the justificatory question — explaining why simplicity is a virtue sounds to me a lot like justifying the use of simplicity. Anyway, what I’ve been wondering is why we’re justified in relying on simplicity (I assume that we are in fact justified).

    I think the explanation has to be noncircular (I’m afraid I’m not sympathetic to coherentism and such). That doesn’t necessarily mean it has to be a priori (though in fact I think the correct account is a priori). How much of our empirical knowledge you get to rely on depends on just how much really depends essentially on appeals to simplicity, which I’m unsure of.

    Anyway, I’m not sure why you said the challenge was probably not well-founded. ?

  19. I think the best Bayesian stuff on simplicity is Roger Rozenkrantz’s stuff. Specifically, “Simplicity” in his (1977) book, “Inference, Method, and Decision”. Forster and Sober’s (1994) BJPS paper is a non-Bayesian approach along similar lines (initiated by Akaike in the late 60’s).

  20. Michael–

    I don’t accept premise (1) in your short argument. I presume your (1) is attached to your assumption that our argument for the truth of Total Science involves an appeal to simplicity. But must it?

    I think we need to be clear about what we mean by “simplicity.” I take simplicity to be a methodological desideratum of theory construction and assessment. (Crucially, it’s not a principle about the simplicity of nature.) Now, there’s not much to choose from for a simplicity metric. But the best stuff going seems come from information theory (Rissanen, Li and Vitanyi).

    The most recent stuff (as far as I know) runs a Bayesian justification for the Li and Vitanyi simplicity metric. But it appears to be question-begging because it just fixes a preference in the priors.

    If you’ve got a Bayesian solution that’s not question-begging, I’d love to see it. From looking at your 11 and 13 (#6), you seem to share some insights with Li and Vitanyi. So I’m suspicious. (I’m also a diehard anti-Bayesian, so I may not be able to be convinced!)

    Note: I believe that Sober thinks a simplicity metric isn’t required since he thinks “parsimony is a virtue that does not speak its name.” Sober’s line is interesting, especially for understanding cladistic parsimony in phylogenetic inference. But cladistic parsimony is quite different from the familiar situation in which one appeals to simplicity to choose between two apparently observationally equivalent explanations. I think Sober’s analysis of parsimony distorts how that kind of reasoning goes.

  21. Mike, suppose someone asked me, “why is visual perception of good source of evidence?” In most contexts, that isn’t a plea for a noncircular defense of visual perception — we pretty much never, in our real practices, challenge such basic sources of evidence — but rather a request for an explanation of perception’s evidential goodness. My answer would surely make significant reference to scientific work on vision, but I only know about most of that stuff because of my own uses of vision (and surely the same will go for the scientists themselves). But that characteristic, while potentially damning in a justification that p, is just not a problem when one is offering an explanation of p. Explanations of p take p as given, so there’s nothing at all fishy in implicitly or explicitly using p in the explanation itself. (You surely can’t let p be the entirety of its own explanation, but that’s not a circularity worry.)

    As for the challenge not being in good order: well, goodness, can the skeptic give me some real reason to suspect that pretty much everything we take ourselves to know from science is in fact mistaken? If not, then I don’t see why I need, or should, take the challenge seriously. The skeptic here is asking for something that she has no right to ask for.

  22. Perhaps Jonathan grants too much here: we also know quite a bit about how our visual system fails us, too. This point won’t satisfy a radical skeptic, but it should satisfy the rest of us. For it gives us a grip on what ‘most contexts’ means by telling us some specific things about when the system is not a good source of evidence. This in turn gives us a basis for judging how well the system performs, not just an explanation for why it (typically) performs well.

    The point generalizes. We manipulate things under experimental conditions not only to find (or refute) causal links between events and to explain the effect under study; we also (often) learn about the reliability of the system we are studying, how to manipulate it: when it breaks; where it breaks; how it breaks; all this in addition to why it breaks. And this gives us an understanding of the specific conditions under which we may err. This seems epistemic, to me.

  23. Like Robert, I tend to be anti-Bayesian. Gilbert Harman has a paper “Simplicity as a Pragmatic Criterion for Deciding What Hypotheses to Take Seriously”, that is perhaps worth a look – he is dissatisfied with the Sober 1988 view that scientists ever (legitimately) appeal to a domain independent account of simplicity. So, in a way, he thinks that Sober has “changed the subject” too. Harman wants a way of ruling out evil demon and grue like hypotheses. He discusses what he calls a “computational ” theory of simplicity.

    There is an exchange in Philosophical studies 1999, I think, that may be of interest. Sober applies the Akaike inpsired account of simplicity to the problem of physicalism (“Physicalism from a Probabilistic Point of View”) and Peter Godfrey-Smith responds. In particular, Peter asks about the apparent tension betwen the Sober 1988 view and his newer view, and Sober responds.

    Oh Yeah, the Harman piece is in Grue_: the new riddlie of induction, edited by Stalker.
    best,
    chris

  24. I’ve always thought the problem of parsimony is at least as important as the problem of induction, probably more so because I think it might be possible to ground parsimony on induction ( a world in which all X’s happen If Y is more simple than a world in which sometimes X happens if Y, or X happens if Y at certain times).

    I am just an undergraduate, that said the only remotely useful thought I’ve ever had on the subject is the weird idea that maybe simplicity could be founded on the concept of possible worlds, there are more possible worlds which fit all our experience in which X ( a simple hypothesis) is true then there are possible worlds in which Y ( a complex hypothesis ) is true. The only problem with this line of reasoning is that it implies our scientfic beliefs, and more disturbingly our beliefs about other minds, are almost certainly false because there are still far, far more worlds in which X doesn’t hold then there are worlds in which X holds.

  25. Jonathan,
    I’m not sure how you’re understanding “evidence”. We can distinguish at least three things:

    1. A request for an argument that x is a reliable belief-forming method.
    2. A request for an explanation of why x is a reliable belief-forming method.
    3. A request for an explanation of why x is a source of justified (rather than unjustified) beliefs.

    Since I’m an internalist, I take (2) and (3) to be importantly different. It sounds like you may be using “evidence” in an externalist sense.

    Which of these requests might the induction from the success of science address? I don’t see that it addresses (2)–the success of science might support the claim that simplicity is truth-conducive, but I still don’t see that an explanation of why it is truth-conducive has been given. I suggest that it addresses (1). But what I wanted was (3).

    I doubt that it addresses (3) because of the circularity concern. Now, since (3) is a request for an explanation, does circularity matter? Yes. The account of how we’re justified must impute to us a non-circular justification. That is, if we are justified in doing something because of some argument we have, it must be a non-circular argument.

    You don’t have to be a skeptic to want (3) addressed. As far as I can tell, all I’m presupposing is (1) that reliance on the simplicity criterion requires justification, (2) that it’s not justified by being self-evident, and (3) that circular reasoning is not a source of justification. I’m definitely not presupposing that simplicity is not truth-conducive, nor that science is generally mistaken (since in fact I think there is a good response to request #3).

    Btw, having just looked at the Rosenkrantz thing suggested by Fitelson, I think it’s great.

  26. I’m not convinced Fitelson has led you down a productive path.

    With Rosenkrantz, you’ll have to buy into objective Bayesianism and the view that simplicity is “sample coverage.” Schaffner (in his 1993 Discovery and Explanation in Biology and Medicine) has an interesting critique of the latter idea (but develops a Bayesian line himself). Critiques of objectivism are pervasive. Better I think to go back to Harold Jeffreys’ (1957) Scientific Inference, which inspired Rosenkrantz.

    William Jeffreys and Jim Berger have a quite useful paper on Bayesian simplicity and model selection called, “Ockham’s Razor and Bayesian Analysis”, from American Scientist 80 (1992): 64-72.

    But is there an insistence for a Bayesian solution?

  27. Thanks for your suggestions, Robert. I’m pretty sympathetic to Bayesianism, though I don’t think I’m fully Bayesian. Anyway, here’s what I want from a solution:

    a. It should account for the epistemic, not merely pragmatic justification of simple theories.
    b. It should be non-circular.
    c. It should be more than just an arbitrary preference for simpler theories (as in just saying, “We assign higher priors to simple theories” without explaining why we should do so).
    d. Simplicity should come out as a virtue ceteris paribus and in most cases, but not necessarily in all cases.

    With regard to Rosenkrantz: I’m already sympathetic to objective Bayesianism; indeed, I think the subjectivism of subjective Bayesianism is its most serious flaw (I think if you can’t constrain the priors, there’s no point in the whole Bayesian programme; it won’t explain our intuitions about what is [in fact] rational to believe, about induction, etc.)

    Also, though Rosenkrantz does so, I don’t think one has to define simplicity in terms of sample coverage (nor should one do so), to use the general approach. Rather, I think one just has to say that in the clear cases where simplicity is a virtue, simplicity correlates with sample coverage. (If it only does so in some of the cases, then we have a partial account of the virtue of simplicity.)

  28. Some interesting reasons for being skeptical about objective bayesianism were offered early on in:

    T. Seidenfeld. Why I am not an objective Bayesian; Some reflections prompted by
    Rosenkrantz. Theory and Decision, 11:413–440, 1979.

    These issues are controversial and subtle and to a large extent depend on the coherence of update rules used by OB. I am sure that an unadulterated version of Bayesian theory might be hard to use for epistemological purposes (especially versions of it relying on some garden variety of radical probabilism ). But the objective version of Bayesianism is hardly free from these kind of problems. Some of the old problems remain and many new problems appear.

    H.

  29. Michael, I wonder if something along these lines might be amenable to you (after necessary cleaning up).

    The principle

    (S1) When multiple theories cover the data, the simplest is the most reasonable to believe.

    seems to me to be of the same basic (synthetic a priori?) status (or relevantly similar status) as

    (PC) When it seems as if P and there is no evidence to the contrary, it is reasonable to believe P.

    In fact, it seems plausible to me that (S1) partly constitutes (or is entailed by) our concept of reasonable belief or inference or explanation (I think the same for PC). In fact, I think it is defensible on similar grounds as (PC) in some cases. For example, it seems to me that any (non-skeptical) argument that some other factor F is the *real* deal in inference, explanation, or reasonable belief will *inevitably* invoke (at some level) the notion of simplicity in its defense.

    Like you, I’m quite sympathetic to bayesianism and hope especially for a successful objective account. I find that many writers tend to forget that Williamson 2000 addresses this in Chapter 10. When I’m at a dead end sometimes I read that and it spurs my thoughts even though he doesn’t say very much.

    I think that if a broadly Chisholmian approach like the one I suggest above is accepted, another layer of reflection might uncover the particular nature of the simplicity which is part of the concept. I don’t see any reason for grave doubts that a domain-independent notion of simplicity can be had, and if it can, then I have hope that it can be given a bayesian analysis. I even have hopes for a Kornblith-style derivation of ought from is. All this, though, is icing on the cake and most will doubt the existence of the cake (though I host moist and delicious qualia when I savor (S1)).

  30. Mike,

    I think your statement of what a solution to the simplicity problem should do is right on target. Although, we disagree about how to satisfy those desiderata. I don’t think confirmation theory is even the right kind of approach. (In this sense I agree with Kevin Kelly & Clark Glymour.)

    The justification problem is the crucial one. But the problem of a metric and of the trade-off between simplicity and evidence are also priorities.

    I’m working on simplicity as well. Would love to exchange solutions when we think we’ve got them. (Appears like your a bit further along than I am.)

  31. Trent,

    I’m reluctant to accept S1 as an ultimate a priori principle (as Swinburne does), for the simple reason that it does not strike me as self-evident.

    When one puts forward S1 as a priori, I think one should have to answer further questions, e.g., does S1 presuppose that in general, the world is simple? If so, that doesn’t seem very self-evident.

    Swinburne also says that other approaches to defending S1 rely on simplicity in some way, but I’m not convinced of that. At any rate, if the Bayesian approach I’m leaning toward relies on simplicity, I think it does so in a more understandable and less arbitrary way than the original appeals to simplicity that it is designed to explain. If, for example, it turns out that all other appeals to simplicity come down to some form of simplicity exhibited by uninformative priors, I think that’s progress and potentially highly explanatory.

    I look forward to seeing what Williamson says. He’s a genius.

    Robert,

    I’ll post something when I have more of what I’ve been working on done.

    My ultimate interest is actually this: philosophers are constantly appealing to ‘simplicity’, especially in metaphysics. I regularly find these appeals totally unpersuasive. Q: Is this because simplicity considerations do not apply to philosophical theories in the way they apply to scientific theories, or is it just because my reasons for rejecting the theories on behalf of which simplicity is appealed to would, if correct, swamp the consideration of simplicity? To answer that, I need to know why simplicity is a virtue of scientific theories.

    On behalf of the Bayesian likelihoods solution:

    I’ve read a little more on it; it seems like other people have already figured out the sort of thing I was thinking, and done so more thoroughly and rigorously.

    I note that one doesn’t have be a Bayesian to accept a “Bayesian account” of simplicity. Some of the more dubious assumptions of strict Bayesianism can be rejected. E.g., I don’t think that people always have degrees of belief, I think sometimes our degrees of belief may be indeterminate, I doubt that ‘belief’, ‘disbelief’ and ‘suspended judgement’ supervene on degrees of belief, and I think there are rational ways to update one’s beliefs other than by conditionalization (as when one discovers that one’s previous degrees of belief were based on an error or oversight).

    As far as I can tell, all you really have to accept is that sometimes ‘likelihood reasoning’ is cogent, where this is the kind of reasoning in which you argue that E favors H1 over H2 because E would be more likely given H1 than given H2. If that kind of reasoning is sometimes cogent, it seems like a good example of it is the Bayesian reasoning for a simple model that fits the data reasonably well being better supported than a complex model (with more adjustable parameters).

  32. Mike,

    I think the following are 6 pervasive considerations scientists take into account in their preference for simple theories. As the complexity of a theory increases,

    1. predictive accuracy decreases;
    2. falsifiability decreases;
    3. amenability to error localization decreases;
    4. array of observationally equivalent alternatives grows;
    5. systematicity decreases (ad hoc assumptions increase);
    6. manipulability and comprehendability decreases.

    Do philosophical simplicity considerations map onto these? Not clear. My interest is specific to science, particularly to biology. (Lots of epistemic problems with this list.)

    I think likelihood reasoning doesn’t pertain to simplicity appeals. Instead, it appeals to evidential considerations (that may bear on simple theories). Sober’s “Let’s Razor Ockham’s Razor” (in his 1994 collection) runs this kind of line for cladistic parsimony. But I’m not certain cladistic parsimony isn’t a special case of parsimony reasoning.

    Tough nut to crack.

  33. Mike, I don’t think the issue here is one of internalism/externalism, since I’m not a thorough-going externalist and what externalism I do endorse does not seem to me relevant here. Rather, it seems to me that we’re disagreeing about the scope of circularity worries. I see no reason to think that such worries extend past specific chains of reasoning: if I need to defend my belief that p, and I cannot do so without appealing to p itself, then I’m not justified in my belief that p. But, if there are no legitimate challenges to p, then worries about circularity simply fail to arise.

    So, in the case at hand, I have granted for the sake of argument that the appeal to scientific successes is such that, if one could legitimately challenge those successes, then there may be no way to defend them without an appeal to simplicity (and to the norms of good explanation more generally). But there is no such legitimate challenge here! And so there is nothing wrong with me (or anyone else) appealing to those successes, in explaining simplicity’s virtue.

    Now, your original objection was that we couldn’t know that science has had these successes without appeal to simplicity, and my response has been that that premise of scientific success is just not at stake. However, if I had to appeal to simplicity in the process of getting from the premise of science’s successes to the conclusion of simplicity’s virtue, then there might indeed be a circularity worry. But it’s not obvious to me that one would have to so appeal to simplicity, in making that step.

  34. Mike,

    I think some accounts of “reasonable” will require that the simplicity-a priorist answer questions about the way the world is and some will not. For example if one takes reasonable belief to entail belief objectively likely to be true, then one is committed to the world being simpler (or at least most relevant worlds being simpler). However, I think all such theories land in skepticism (actually there my only hope is coherentism of all things).

    However, on a deontological reading of “reasonable” I doubt this is the case. It seems like a basic fact of rationality to me that we don’t have a right to posit any more (kinds or tokens of) entities than are required to cover the data. Much of _Skepticism and the Veil of Perception_ seems to work best in a deontological setting, so I wonder what you think of this latter version of the proposal. I know it’s much more boring than probabilistic accounts (Jon can tell you how enthralled I am with “the beautiful game”), but it does strike me as a plausible account of the epistemic role of simplicity in inference.

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  36. Jon,

    Actually, it seems to me like the questions you and I are asking are the reverse of what you said earlier: I’m asking for an explanation, while you’re providing a justification. I don’t want someone to convince me that simplicity is a virtue; I am already convinced of that. I want someone to explain to me how I and others are justified in believing that.

    Now, if the request were for a justification, then I think it would be to the point to talk about whether there is a legitimate challenge to scientific knowledge, whether the premise of scientific success is at stake, and so on. But I don’t think those things are relevant to my explanation request. I’m not in any doubt that science has been successful. (I’m also not hypothetically positing someone who is in doubt about that, or anything like that.) I’m just looking for the best epistemological theory of how we know that simplicity is a virtue.

    I’m not sure I’ve successfully articulated my concern. Consider a simplistic analogy. Say I want to know how we know of the existence of external objects. You offer the following account: “We know of the existence of external objects by means of the following argument:
    1. External objects exist.
    2. Therefore, external objects exist.”

    Now, I think I would be right to object to the circularity of the argument imputed to us. Suppose you said in response:

    “But Mike, what reason is there for thinking external objects don’t exist? Sure, if there was a legitimate challenge to the existence of external objects, then maybe you’d be right that I’d have to give a non-circular argument for it. But there is no legitimate challenge.”

    But now I’d just say you must be misunderstanding my objection. My objection didn’t hinge on the plausibility of the denial of premise (1).

    Now, I could imagine a view you might hold such that your reply would be on target. Suppose you held the following pair of views:

    a) If S has no grounds for doubting P, then S may be justified in believing Q on the basis of a circular argument with premise P.
    b) But if S has grounds for doubting P, then S may not be justified in believing Q on the basis of a circular argument with premise P.

    That would explain the exchange we just had: you might be thinking that by raising the circularity concern, I must be assuming that S has grounds for doubting P, and thus that I’m trying to apply principle (b). You then point out that we have no grounds for doubting the success of science, so (a) applies. But I can’t see why one would believe those views (particularly (a)) to begin with. I hold:

    c) S may not be justified in believing Q on the basis of a circular argument, no matter what.

  37. Mike, I think the way my view diverges from yours is more radical but, I hope, less silly-sounding than that. Basically, if a proposition is inside the sphere of propositions that are not challengeable, then they can be known without argument. Turned into something like a sufficient condition for knowledge, we get: if (i) S justifiably believes herself to share in a community-wide belief that p, and (ii) S justifiably believes that her community is an epistemically responsible one, and (iii) S justifiably believes that she is not epistemically impoverished by comparison to their community, and finally (iv) S has no grounds for doubting p, then S is justified in believing p.

    Looking at the principle a) that you considered last post:
    a) If S has no grounds for doubting P, then S may be justified in believing Q on the basis of a circular argument with premise P.
    it would be better to say that I endorse
    d) If S has no grounds for doubting P, then S may be justified in believing Q without any argumentative basis whatsoever, if other appropriate conditions obtain. (Note that we are not to understand the clauses in the antecedent above as giving S the means for, say, an argument from the reliability of her epistemic community. If that were the case, then S would just be giving a different kind of argument; I take her to be recognizing, rather, that she owes no argument.)

    This shows, btw, why the question of the challengeability of things like 1. in your last post really matter. If 1. (or, in the main case under consideration, the general success of science) is beyond challenge, then one might simply need not track down some justificatory basis for it. And so circularity concerns won’t arise.

    By comparison with a view that one needs arguments all the way down to some firmer sort of bottom, I would note that the picture I am presenting fits our practices much more closely than the full-argument view, since those practices never endorse requirements of argument past the point of reasonable challenge. Moreover, I don’t see any reason why it would make sense for us to want to adopt the set of practices that would be involved, if we wanted to switch over to the full-argument view of justification.

  38. JW,
    You’re, right, that does sound considerably less silly. The four conditions that you mention strike me as sufficient for S to have available a non-demonstrative argument that p. It just doesn’t seem to me as though they suffice for P to be foundationally justified. Think of all the things that could be foundational. My belief that humans evolved from ape-like primates about 2 million years ago would be foundational.

    By the way, do you know of any references on the argument from the success of science that simplicity is truth-conducive? Have you or someone you know written about it? (This would help with the paper I’m writing.)

  39. Few people have opined on whether simplicity appeals are evidentially valuable in philosophy. So here I’ll state my view: in the typical cases in which philosophers appeal to parsimony, the appeal has no evidential weight.

    My argument for this is that of all the accounts of the value of simplicity in empirical reasoning that I have heard, none of them applies naturally to typical philosophical cases, such as dualism vs. physicalism or realism vs. nominalism.

    To take just one example, it is unclear how one could argue that dualism has more adjustable parameters than physicalism, or that realism has more adjustable parameters than nominalism.

  40. Mike,

    Chris Stephens knows more about this than I. He mentions a Sober piece in Philosophical Studies from 1999 on a probabilistic approach to physicalism. In 1996, Sober published a similar paper in Erkenntnis, with more philosophical theories as examples. Both use Akaike’s Theorem as a justification of simplicity.

    For my part, I’m on your side.

  41. Just looked at Sober’s Phil Studies article on his web page. I think he’s mistaken because he defines physicalism as the view that:

    For any mental property M, if a system has M at t, then there exists a set P of physical properties that the system also has at t, such that Pr (M|P) = 1.

    But that view is consistent with (even strongly suggested by) interactionist dualism. Sober’s formulation does not distinguish between causal connections and some metaphysically stronger relation.

  42. My own sense (recently) is that “intrinsic probability” may be analyzed in terms of complexity. For example,

    ‘x is less likely than y’ =def ‘x has more parts than y, and x and y are propositions’.

    According to this account the intrinsic probability of impossible propositions need not be zero. I distinguish “intrinsic probability” from “prior probability”, where prior probability may be analyzed in terms of ratios of possible worlds.

    I believe that given a structured view of propositions (i.e. propositions have parts) and a correspondence theory of truth (according to which a prop is true if and only if it’s parts are exemplified by the parts of an arrangement of things in the right order–see a draft of my dissertation ), contingent propositions that have a relatively low intrinsic probability will thereby have a relatively low prior probability. And if it’s intrinsic probability is relatively high, then so is its prior probability. That’s a desirable result.

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