tag:blogger.com,1999:blog-4987609114415205593.post2636949417690792881..comments2020-07-13T20:26:53.037+01:00Comments on M-Phi: It's ComplicatedJeffrey Ketlandhttp://www.blogger.com/profile/01753975411670884721noreply@blogger.comBlogger19125tag:blogger.com,1999:blog-4987609114415205593.post-47947380143873683342013-04-25T04:49:44.576+01:002013-04-25T04:49:44.576+01:00Glad we sorted that out! Yes, I'll be in Oxfo...Glad we sorted that out! Yes, I'll be in Oxford two weeks from Friday. See you then.David Chalmershttp://consc.net/chalmers/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-57846649159700528532013-04-25T02:34:49.474+01:002013-04-25T02:34:49.474+01:00David, right - glad it dawned on me! The options y...David, right - glad it dawned on me! The options you give for "the world's *-structure is m-complicated" are then reasonable ways of interpreting "the world is complicated". (I was only worried about the case where * is the empty set.)<br /><br />I think you're in Oxford in a few weeks' time for the Jowett/Phil Society? See you then, hopefully.<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-69252399350380265772013-04-25T01:16:38.054+01:002013-04-25T01:16:38.054+01:00Jeff: Yes, that's one way to translate what I&...Jeff: Yes, that's one way to translate what I'm saying into your terminology (that is, a terminology that assumes that 'complicated' is to be defined in purely logical/mathematical terms). <br /><br />Put neutrally, we could use the term 'm-complicated' for a purely mathematical notion of complexity, and 'p-complicated' for a physical notion of complexity such as the notion at play when we say that that world is complicated. We can't simply assume that m-complexity and p-complexity are the same notion. Even if one defines p-complexity in terms of m-complexity, there will be many ways of bridging the gap here. E.g. one can say that the world is p-complicated if the world's *-structure is m-complicated. Then the candidates for *-structure include (i) property/relation structure (the definition you start with), (ii) natural property/relation structure (your alternative definition), (iii) causal relation structure (one idea I suggest), and many others. Clearly (i) and (ii) aren't the only options here, and the issue very much depends on how you define physical complexity.David Chalmershttp://consc.net/chalmers/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-20575953789322097692013-04-24T23:15:05.321+01:002013-04-24T23:15:05.321+01:00David, it's clicked with me: the idea is that ...David, it's clicked with me: the idea is that you're suggesting is that we interpret,<br /><br />(1) The world is complicated,<br /><br />to mean:<br /><br />(6) The causal relation on the world is complicated.<br /><br />(I've removed "mind-independent" as well, as that's an orthogonal issue.) <br /><br />If that's it, I understand your points above better. For (6) looks like a reasonable way to interpret (1) (modulo Russell-style scepticism about causation). And unlike (4) in the original post, (6) isn't ramsified, so doesn't succumb to there being too many gerrymandered relations around (a problem solved by moving to physical or natural relations). And (6) doesn't contain any reference to causation being "physical" or "natural", as you say. <br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-7626021045161941302013-04-24T18:43:46.074+01:002013-04-24T18:43:46.074+01:00David, for the isomorphism-invariance issue, for a...David, for the isomorphism-invariance issue, for any measure that I'm familiar with in algorithmic complexity, Shannon entropy, graph complexity, etc., the measure is a structural invariant, i.e., isomorphism-invariant. For example, for graphs $G = (V,E)$, we can assign many different measures $c(G)$ of its complexity. But, for any such measure $c$, if $G_1 \cong G_2$ then $c(G_1) = c(G_2)$.<br /><br />If I understand right, your objection is that even if the causal relation $R$ (on physical events) is isomorphic to some causal* relation $R^{\ast}$ (on physical events), then we can have $c(R) \neq c(R^{\ast})$?Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-24485934362083252322013-04-24T14:46:50.332+01:002013-04-24T14:46:50.332+01:00David, I still don't quite get your objection!...David, I still don't quite get your objection! <br />The complexity measure, call it $Comp(.)$, that I am considering *is* defined in terms of some specific relation, or probability function, maybe in terms of information, or computational complexity, or Kolmogorov complexity, etc. For example, given a relation $R$, then $Comp(R) = c$, say, where $c$ is some real. Or, given a probability space $(\Omega, E, Pr)$, then $Comp(Pr) = c$.<br /><br />The reductio argument, that shows that the world can be as complicated as we like modulo cardinality, is:<br /><br />(P1) The world has cardinality at least $\kappa$.<br />(P2) Suppose $\mathcal{A} = (A,R)$ is a structure with $|A| = \kappa$ and $Comp(R) = c$.<br />Therefore:<br />(C) There is a relation $S$ on the world such that $Comp(S) = c$.<br /><br />Is the idea that (C) doesn't follow from the premises?Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-57192357770557040852013-04-24T04:08:09.024+01:002013-04-24T04:08:09.024+01:00Really, I'm not. The term 'physical' ...Really, I'm not. The term 'physical' plays no essential role in my response; nor does any other predicate attached to 'property' or 'relation' or 'probability function'. To avoid your conclusion, all that's needed is that complexity is defined in terms of some specific (nonlogical, nonmathematical) property or relation or probability function -- which can be as natural or as unnatural as you like.David Chalmershttp://consc.net/chalmers/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-24821997337338847642013-04-23T15:58:43.916+01:002013-04-23T15:58:43.916+01:00David, but I think you're using "physical...David, but I think you're using "physical" as a near synonym for "natural".<br />So, e.g., if I define causation* by a peculiar permutation of events, you might put this by saying causation* is not physical (I agree), whereas I might put it by saying causation* is not natural. <br />But I don't mind using either, to be honest! The main point is that certain relations are non-physical, and it seems to me that this is Lewis, Sider et al mean.<br /><br />Here's a toy example (LaTeX didn't work first time ...). <br />Suppose I consider the simple probability space $(\Omega,E,Pr)$, where $\Omega$ is, say, $\{0,1\}$, and $E=\mathcal{P}(\Omega)$ and <br /><br />$Pr(\varnothing)=0$.<br />$Pr(\Omega)=1$.<br />$Pr(\{0\})=0.5$.<br />$Pr(\{1\})=0.5$.<br /><br />(We have four events, and their probabilities.) Consider a bijection $f : \{0,1\}$ to the set {John, Yoko}. Now use this to define<br /><br />$Pr^{\ast}(\varnothing)=0$.<br />$Pr^{\ast}({John,Yoko})=1$.<br />$Pr^{\ast}({John})=0.5$.<br />$Pr^{\ast}({Yoko})=0.5$.<br /><br />Everyone usually agree that $Pr^{\ast}$ is not a physical probability function. And I think this amounts to saying that $Pr^{\ast}$ is "unnatural".Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-36117125847663110112013-04-23T14:53:44.766+01:002013-04-23T14:53:44.766+01:00Jeff: Your last two sentences involve a non sequit...Jeff: Your last two sentences involve a non sequitur, for reasons outlined above. Nothing here requires natural properties and relations. It just requires that complexity is defined in terms of a specific relation (e.g. causation) or a specific probability function (e.g. chance).David Chalmershttp://consc.net/chalmers/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-11509712370359093742013-04-23T10:17:27.822+01:002013-04-23T10:17:27.822+01:00David, here goes again - hopefully LaTeX will work...David, here goes again - hopefully LaTeX will work!! <br /><br />The Newman argument here isn't one that permutes properties and relations (that's a Quine-Putnam style argument), and I agree causation is not invariant under permutations! The Newman argument is that, if there is no distinguished subclass of special relations (e.g., the physical/natural one), then the world has any structure/complexity one likes (modulo cardinality). Suppose: <br /><br />(1) The world W has no distinguished relations but has cardinality $\kappa$.<br /><br />Let $\mathcal{A} = (A, R_1, \dots)$ be some mathematical structure. Then,<br /><br />(2) If $\kappa = |A|$, then there are relations on the world such that the world has structure $\mathcal{A}$.<br /><br />Let $\kappa = |A|$. This gives a bijection $g : A \to W$. Then define relations $S_i = g[R_i]$. Then $(W, S_1, \dots) \cong \mathcal{A}$. QED.<br /><br />It's this conclusion (2) which states that the world has any complexity (or structure) one likes, on the "glubby" assumption that there are no distinguished physical/natural relations.<br /><br />If, on the other hand, the relations $S_i$ must be physical, then one cannot get the conclusion. I.e., one can't prove,<br /><br />(3) If $\kappa = |A|$, then there are *physical* relations on the world such that the world has structure $\mathcal{A}$.<br /><br />So, e.g., your chances are physical, but "schmances" are not physical. I agree that this is the right response. Amongst *all* the relations there are, some are physical and some are not. This is then the Lewisian view that some relations are natural and some are not.<br /><br />In the end, I think by "physical" you mean more or less what David Lewis meant, or what Ted Sider means, or what I mean, by "natural".Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-82085981683987097312013-04-23T09:52:22.769+01:002013-04-23T09:52:22.769+01:00David, thanks - I wrote a reply, but the LaTeX was...David, thanks - I wrote a reply, but the LaTeX was wrong, and there's no preview here, so one can't fix it!Jeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-86723459844383147482013-04-22T23:37:44.121+01:002013-04-22T23:37:44.121+01:00P.S. Last sentence should read "The latter cl...P.S. Last sentence should read "The latter claim leads to the former claim...".David Chalmershttp://consc.net/chalmers/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-76297278134891916402013-04-22T23:34:25.138+01:002013-04-22T23:34:25.138+01:00Jeff: When complexity measures of this sort are de...Jeff: When complexity measures of this sort are defined with a probabilistic framework, the probabilities have to be construed as physical probabilities or chances. Then the issues are the same as above but with "chance" playing the role of "causation". Arbitrary permutations of properties and relations don't preserve causal structure and likewise they don't preserve chances. For an isomorphism here, one would have to permute chances to "schmances" (some other sort of probability measure). But on this approach it is chances that matter to complexity, not schmances.<br /><br />Regarding your second paragraph: this isn't insisting that only physical probabilities are real probabilities. Rather, it's insisting that the sort of probabilities that complexity is defined in terms of are physical probabilities (chances). As before, this certainly doesn't requires the claim that physical probability (or chance) is natural. All that's required is that there's a definitional link between complexity and physical probability. The former claim leads to the latter claim only if one accepts the additional premise that complexity is natural.David Chalmershttp://consc.net/chalmers/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-18018425313819441362013-04-22T16:13:45.556+01:002013-04-22T16:13:45.556+01:00Thanks, David.
I was not aware of such approaches...Thanks, David. <br />I was not aware of such approaches. Certainly have to familiarize myself with those. Panu Raatikainennoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-90242256342476763052013-04-22T11:39:46.550+01:002013-04-22T11:39:46.550+01:00David, thanks, very interesting - I don't know...David, thanks, very interesting - I don't know this particular work, but it's hard to see how a measure like this, based on the notion of a probability space, could fail to be isomorphism-invariant (maybe it would involve haecceities?). For example, in computing probabilities with a probability space $(\Omega, E, Pr)$, it doesn't matter if the "outcomes" in the sample space $\Omega$ are experimental outcomes or are, say, real numbers. One will get the same probabilities either way. Any bunch $X$ of objects in the world can be turned into a probability space, even though the resulting probability function $Pr$ will usually be non-physical. <br /><br />So, if the world has sufficiently large cardinality $\kappa$, say, then one can define relations on things in the world, which have any complexity one likes, unless one insists that only special properties/relations may be considered (e.g., causal relations or physical ones). So, e.g., if one objects to calling $Pr$ a genuine probability because it is non-physical, it seems now that "physical" has just become a synonym for "natural". <br /><br />But this is fine, though, with me!<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-8385245068442832012013-04-22T09:41:15.378+01:002013-04-22T09:41:15.378+01:00Jeff: All the work here is done by your parentheti...Jeff: All the work here is done by your parenthetical remark "(at least if complexity is invariant under isomorphism)". You're assuming that physical complexity is definable in terms of mathematical complexity in a certain highly constrained way, so that arbitrary permutations of physical properties and relations yield exactly the same complexity. That will certainly be false for the notions of complexity I'm discussing (permuting causation for causation* will change complexity). I hazard that it will be false for any reasonably intuitive notion of physical complexity. Certainly it will be false for those notions that generate the initial intuition that the mind-independent world is complex.<br /><br />Panu: An example of defining complexity in terms of causation is Tononi's information-integration measure of complexity "phi", which is in effect defined by causal/counterfactual relations among the components of a system. (More generally, if one applies notions of informational or algorithmic complexity to physical (rather than mathematical) systems one typically invokes causal/counterfactual notions in order to determine the informational/algorithmic structure of a physical system.) Note that this definition certainly doesn't require that one take causation to be a natural relation or even that one believe in natural relations.David Chalmershttp://consc.net/chalmers/noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-8752122968869030692013-04-22T07:35:07.544+01:002013-04-22T07:35:07.544+01:00David, thanks.
I'm thinking primarily in term...David, thanks.<br /><br />I'm thinking primarily in terms of computational complexity of sets of numbers and Kolmogorov complexity for strings. But suppose there's a function F such that F(R1, ..., Rn) = k is some way of assigning complexity to some relations R1, ..., Rn (i.e., sets of objects, sets of pairs, etc.).<br /><br />And fix some complexity k, say a very big one, such that for some purely mathematical relations R1, ..., Rn we have F(R1, ..., Rn) = k. So, we have a mathematical structure A with distinguished relations R1, ..., Rn, and with complexity k. <br /><br />Then a ramsified claim like (4)<br /><br />(4) There are mind-independent relations R1, ..., Rn such that F(R1, ..., Rn) = k<br /><br />will reduce to a cardinality claim<br /><br />(3) The mind-independent world W has at least $\kappa$ things.<br /><br />holds, where $\kappa$ is the cardinality of A. The important direction is (3)->(4). Suppose (3), and take the bijection g: A -> W, and use g to project onto W the relations R1, ..., Rn in A. I.e., R1* = g[R1], etc. This gives us relations R1*, R2*, ... on the worldly things, with overall complexity k, and therefore (4) is true (at least if complexity is invariant under isomorphism). <br /><br />But these worldly relations R1*, etc., will usually be highly gerrymandered. So, we can block the reasoning from (3) to (4) by insisting that we replace (4) with (4)*,<br /><br />(4)* There are mind-independent *natural* relations R1, ..., Rn such that F(R1, ..., Rn) = k<br /><br />Then we can't infer (4)* from the cardinality of W.<br /><br />I'm not sure about the other suggestion, by measuring complexity via causality. (There is a programme in quantum gravity of causal sets associated with Sorkin.) But I guess I'd say what you guessed I'd say! I do think this would require that causality be a natural relation. And if there is a basic, natural, mind-independent causal relation, say C, then that's certainly enough to sustain the conclusion. <br /><br />Then the revised interpretation of (1) is<br /><br />(6) The complexity of C is k.<br /><br />(I'd think though that a complexity function F should be isomorphism invariant: if R is a mathematical relation (say a relation on the reals), and C is this natural causal relation, and R and C are isomorphic, then F(C) = F(R).)<br /><br />JeffJeffrey Ketlandhttps://www.blogger.com/profile/01753975411670884721noreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-21756921093669506322013-04-22T06:19:58.233+01:002013-04-22T06:19:58.233+01:00David, can you tell a bit more, or give a referenc...David, can you tell a bit more, or give a reference, about complexity in terms of causation? Panu Raatikainennoreply@blogger.comtag:blogger.com,1999:blog-4987609114415205593.post-80130880936433994742013-04-22T05:18:43.165+01:002013-04-22T05:18:43.165+01:00Actually the argument you've given isn't i...Actually the argument you've given isn't independent of how one understands 'complexity'. The Newman-inspired argument you've given for the claim that (4) reduces to (3) assumes in effect that 'complexity' is definable using logical expressions (including property quantifiers) alone. But there are natural ways of understanding 'complexity' that are incompatible with this assumption: e.g. understand 'complexity' to be defined in terms of causation, as with various notions of causal or computational complexity. Then the argument that follows doesn't go through. There's in effect a gap in the argument, concerning the relation between logical structure and complexity, that needs to be filled by an explicit premise.<br /><br />Of course many other nonlogical notions could also play the role of causation here, e.g. defining complexity in terms of spatiotemporal properties and relations or in terms of fundamentality. Indeed your own way of avoiding an entailment from (4) to (3) takes this form, defining complexity in terms of the presumably nonlogical notion (pace Carnap) of naturalness.<br /><br />I suppose you might reply that to understand 'complexity' this way is in effect to assume that causation is a natural relation, but that doesn't seem obvious. All that you're really entitled to is the claim that if complexity so defined is to be natural then causation must be natural. The corresponding argument then yields the conclusion that if (i) the world is complex and (ii) complexity is natural then (iii) there are natural properties and relations. That's a less surprising conclusion.David Chalmershttp://consc.net/chalmersnoreply@blogger.com