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Daniel Greco

I think the defender of the BSA should welcome the CTL as a precisification of her view. Along those lines, though, I think you overstate some of the differences between the views. You seem to suggest that the BSA defender thinks the nomological accessibility relation is symmetric and transitive, while according to CTL, it's not. But the BSA defender (or at least, the most prominent recent one--David Lewis) doesn't think this, and his reasons for not thinking it are essentially the same ones that are formalized in the CTL. Here's a quote from on the plurality of worlds:

"Then, indeed, puzzling questions about the logic of iterated nomological necessity turn into more tractable questions about the relation of nomological accessibility. Is it symmetric? Transitive? Euclidean? Reflexive?...A theory of lawhood can be expected to answer these questions, and we can see how different theories would answer them differently. (For instance, my own views on lawhood answer all but the last in the negative.)" (p. 20)

I'm inclined to say the same thing about your response to the central anti-humean argument. They strike me more as presifications of things that Humeans standardly say in response to such examples (see, e.g., Helen Beebee's "The Non-Governing Conception of Laws of Nature") rather than fundamentally different responses to such examples that the Humean can give.

Lastly, on simplicity/relativity, I'm just inclined to agree. I think talking about computational complexity lets us relocate and precisify (but not bypass) older debates about the relativity of simplicity. Think, e.g., Goodman on grue--lots of people want to say that the theory that all emeralds are grue is somehow less simple than the theory that all emeralds are green, though familiar arguments from Goodman suggest that whether we get to say this will depend on what language we use. Theories of inductive inference that depend on computational copmlexity (e.g., Solomonoff induction) don't avoid this problem, though they do relocate and precisify it.

tomkow

Daniel,

Thanks for your thoughtful comment! 

I would be very happy if proponents of BSA welcomed CTL. 

I am absolutely content with saying  that CTL  is only a refinement of the guiding ideas behind the Best System Theory.   As I say in the post, you are welcome to think of the CTL as only an updated version of the Best System analysis; one which replaces Euclid's conception of  "System" with Turing's.

And I entirely agree that a BSA theorist can-- and some already have--  endorse the conclusions about the logic of laws that I argue for here.   I hope though that we also agree that CTL gives us useful new ways to think about laws and to argue for these conclusions. . 

I like to think Lewis would have welcomed CTL.  Had he had it to hand, I think he might not (as he did increasingly in his later work) have taken the course of saying  that Humeanism might only be a contingent truth about "worlds like ours".  He was, you will recall, driven to this odd position by cases like Armstrong's spinning sphere. That thought experiment was supposed to show there could be worlds which were indistinguishable in Humean terms but in which had different counterfactual/dispositional properties.  Lewis thought he had to concede the possibility, but he didn't: I think it clear that Armstrong's story fails in the same way as  Carroll's.

And, with all due respect, I do think that CTL does a bit more than just "precisify" the issues confronting BSA. A central problem for BSA is to say not just what, precisely, simplicity is but also to say why it matters.  CTL gives an answer to that question not obviously available to BSA.

I am very pleased to that you are not shocked by the prospect of Nomological Relativity (I can report that some philosophers of my acquaintance find it very shocking indeed).  But here I again I think that CTL does a bit more than just restate the old problems more precisely. 

The problem of "grue" and "green" presents itself as a problem of underdetermination: why inductively project one predicate not the other given the available evidence.  It is, in turn, only stepchild  to the larger problem of Quinean underdetermination: how to choose between contradictory theories equally compatible with all possible evidence. 

As I shall argue in a future post, I think CTL gives us a way of solving these problems by understanding that what is at work here is not underdetermination but relativity.  Relative to some reference machine, a Gruesome description of the world might be simpler than the one we find natural. What properties we find natural may be an evolutionary accident. As Steve Petersen puts it, it is a matter of what Turing Machine evolution has put between our ears. But, as I argue in the post, we need not take this relativity in our judgments as undermining their objectivity

Except relative to some reference machine or other, there is no theorizing about the order of the world. There is no computational, or cognitive, "view from nowhere". And while different points of cognitive/computational view point of can lead to apparently contradictory claims about laws, counterfactuals and dispositions, the appearance of contradiction goes away when we understand that the claims are consistent and commensurable when understood as relative to our cognitive/computational perspective.

Paul Torek

Terrance,

I only just now discovered this excellent post. I have two questions. First, why is Nomological Relativity not relevant to the issue at hand, or in other words what *is* the issue at hand? Second, is this supposed to be an ontological reduction of laws, or simply a set of logical implications to and from law-statements from/to other statements?

tomkow

Thank you, Paul!

I do think that last section is too telegraphic. I was trying to short-circuit an objection to the effect that since algorithmic complexity is language/computer relative that "it's all relative, so we can say what we like."

The very narrow issue I had in mind was this: the Humean says that the Humean facts about a world determine a unique set of laws for it. The anti-Humean says contrary sets of laws might have equal claims to governing the same (under a Humean description) world .

The point about relativity, by analogy, goes like this: imagine two objects in motion with respect to one another. The question, "which is moving faster" can only be answered relative to a frame of reference and by picking different frames you can get different answers; but there is no frame of reference with respect to which they are not in motion with respect to one another. Likewise, relative to different measures you might get different orderings of complexity, but systems aren't going to be tied in some orderings but not others.

Certainly, the whole topic of relativity and its upshots deserves a more extended discussion. I hope to return to it in future posts.

As to what "this is supposed to be". I'm not sure exactly what "ontological reduction" might mean in this context. I would simply say that, like The Best System theory, The Computational Theory of laws offers an explanation of what makes some true propositions laws of nature.

Best
tomkow

Paul Torek

Ah, thank you, "what this is supposed to be" is clear now. I should have read the previous post, although now that I did, I think I'm having a Far Side moment (Mr. Osbourne, may I be excused? My brain is full.)

I think you can live with some relativity. Suppose there are two Turing Machines one of which optimizes complexity with a few more "initial conditions", and the other of which has a few more laws. Still, they make the same predictions, and it just seems intuitively likely that most laws in one system would have rough or exact counterparts in the other.

I'm not convinced by the argument in the other post for dropping the O(1) constant, though. You made an enumeration of Turing machine specifications, written as data on tape, to be computed by a given universal Turing machine (UTM) U. You then pointed out that another UTM U' could simulate machine T(i) when just given the number i. But had you started from the different UTM V, the complexity order of Turing machines (as data on tape) would typically differ (by at most a constant, I guess), and what we used to call T(i) would now have a different number j.

tomkow

Paul,
I think you have the gist of it but I encourage you to read others on the topic of algorithmic information theory. The field is very new and full of interesting philosophical upshots.

Dropping the "O" does not imply that the relativity goes away, it's just a matter of taking it as given to make the equations a bit tidier.

Your remarks don't make it clear if you buy my arguments for "the objectivity of complexity" above, but that is how I "live with the relativity".
Best
T

Paul Torek

Oh, I misunderstood about the dropped constant (but I feel like it should be kept, to smooth the segue into relativity). Sure, there are objective facts about the complexity of a laws+conditions system relative to any given UTM, and moreover the complexity measures are strongly related, those points seem straightforward.

tomkow

Actually, I am making a stronger claim. Not just that there objective facts about complexity relative to any given UTM but that complexity is an objective feature of the world... as objective as velocity. What is relative to UTM is only its *measure* (again, cf. velocity).

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