Monday, 22 April 2013

It's Complicated

[This is a post using Newman-style reasoning to argue for the existence of natural properties and relations.]

Consider a claim like:
(1) The mind-independent world is complicated
One might deny that there is a mind-independent world (Idealism) or one might accept that there is, while insisting that it is "unknowable", while adding that what is known is mentally constituted (Kantian Transcendental Idealism). Here, in asserting the latter, one does merely mean representations are mentally constituted, for this is a truism that no one denies. One means that what knowledge is about is also mentally constituted (e.g, that physical objects are representations; that space and time are representations). Idealism is not the truism that our thoughts and representations are somehow in, or connected with, our minds; it is the much stronger metaphysical claim that everything (Idealism) or almost everything (Kant) is mind-dependent.

Assuming that we're not Idealists, what might this statement (1) mean? It might mean:
(2) The cardinality of the mind-independent things is quite large (e.g., $>10^{50}$).
If this is what (1) means, then the complexity of the world is solely its cardinality. Therefore, a sound and complete description of the mind-independent world consists in a statement of the form:
(3) The cardinality of the mind-independent world $= \kappa$,
where $\kappa$ is some cardinal number. It should strike anyone as surprising that the ultimate goal of physics, chemistry, biology, etc., is simply to identify this number $\kappa$. (Cf., the punchline of Douglas Adams's joke "42".) So, I take it that this is not what the statement (1) means.

So, perhaps (1) means,
(4) There are mind-independent properties and relations amongst the mind-independent things and their relations (e.g., scientific laws) are complicated.
Here "complexity" may mean something like the structural complexity of the truth set for a language containing predicates for these properties and relations. For example, the truth set for full arithmetic is more complicated than the truth set for arithmetic with just addition. For the latter is a recursive set, while the former is not recursive -- and in fact not even arithmetically definable. There are other ways of measuring complexity, notably Kolmogorov complexity, for finite strings, and various notions of computational complexity. Perhaps, if the world is finite, "complexity" might involve the Kolmogorov complexity of the simplest program that answers soundly all questions about the world.

However, independently of how one understands the concept of "complexity", one has to be careful. Suppose that by "property" or "relation" one means just any set of things, or any set of ordered pairs of things. These are properties in a very broad sense. It then follows, by Newman-style reasoning, that (4) is reducible to (3). For any structure (or classification, if you like) $\mathcal{A}$ can be imposed on some collection $C$ of things so long as there are enough of them.

To illustrate: consider a finite set $X = \{1, \dots, n\}$ of numbers, and partition it any way you like. Let the partition be $(Y_i \mid i \in I)$, where $I$ is the index set. I.e., the sets $Y_i$ are non-empty and disjoint, and $X = \bigcup_i Y_i$. Now, suppose that we have a collection $C$ of $n$ things, or physical objects, or what have you. Then it is easy to define a partition $(C_i \mid i \in I)$ of these things which is isomorphic to $(Y_i \mid i \in I)$. For since $C$ and $X$ have the same cardinality, let $f : C \to X$ be a bijection (this function enumerates the elements of $C$). Then, for each $i \in I$, define $C_i$ by:
$c \in C_i$ iff $f(c) \in Y_i$. 
By construction, this gives us an isomorphism. So, if we have a partition of $n$ natural numbers (the "mathematical model") and collection $C$ of physical things of size $n$, we can partition $C$ isomorphically to the original partition. If there are no independent constraints built into $C$ itself beyond cardinality, we can impose any structure $\mathcal{A}$ we like onto $C$, modulo $C$ having cardinality at least as large as that of $\mathcal{A}$.

Consequently, if the reasonable sounding (4) is not to trivialize down to (3), the quantifier in "there are ... properties" must range over a special subset of the set of all properties in the broader sense. In principle, this might be any special subset. But, usually, what is intended is what metaphysicians call "natural properties". This is because what "selects" that subset as special is not the mind, but Nature. If one intends it to mean "there is a mind-dependent subset of properties ...", then one is back to Idealism, this is almost certainly not what (1) is taken to mean by anyone.

So, if this reasoning is right, the most reasonable interpretation of "the mind-independent world is complicated" is:
(5) There are mind-independent natural properties and relations amongst the mind-independent things and their relations (e.g., scientific laws) are complicated.
And this is much more in keeping with scientific inquiry. However, note that (5) implies the existence of mind-independent natural properties and relations.

So, if there is a mind-independent world (Idealism is incorrect) and the mind-independent world is complicated, then either this mind-independent complexity consists merely in its cardinality, or it consists in the complexity of the laws and relations amongst natural properties and relations. In particular, if Idealism is incorrect but there are no natural properties or relations, then the complexity of the mind-independent world consists solely in its cardinality.

(I'm inclined to think that this latter position is, more or less, Kant's metaphysical view.)

19 comments:

  1. 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.

    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.

    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.

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  2. Panu Raatikainen22 April 2013 at 06:19

    David, can you tell a bit more, or give a reference, about complexity in terms of causation?

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  3. David, thanks.

    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.).

    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.

    Then a ramsified claim like (4)

    (4) There are mind-independent relations R1, ..., Rn such that F(R1, ..., Rn) = k

    will reduce to a cardinality claim

    (3) The mind-independent world W has at least $\kappa$ things.

    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).

    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)*,

    (4)* There are mind-independent *natural* relations R1, ..., Rn such that F(R1, ..., Rn) = k

    Then we can't infer (4)* from the cardinality of W.

    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.

    Then the revised interpretation of (1) is

    (6) The complexity of C is k.

    (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).)

    Jeff

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  4. 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.

    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.

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    1. Panu Raatikainen22 April 2013 at 16:13

      Thanks, David.
      I was not aware of such approaches. Certainly have to familiarize myself with those.

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  5. 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.

    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".

    But this is fine, though, with me!

    Jeff

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  6. 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.

    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.

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    1. P.S. Last sentence should read "The latter claim leads to the former claim...".

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  7. David, thanks - I wrote a reply, but the LaTeX was wrong, and there's no preview here, so one can't fix it!

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  8. David, here goes again - hopefully LaTeX will work!!

    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:

    (1) The world W has no distinguished relations but has cardinality $\kappa$.

    Let $\mathcal{A} = (A, R_1, \dots)$ be some mathematical structure. Then,

    (2) If $\kappa = |A|$, then there are relations on the world such that the world has structure $\mathcal{A}$.

    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.

    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.

    If, on the other hand, the relations $S_i$ must be physical, then one cannot get the conclusion. I.e., one can't prove,

    (3) If $\kappa = |A|$, then there are *physical* relations on the world such that the world has structure $\mathcal{A}$.

    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.

    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".

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  9. 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).

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  10. David, but I think you're using "physical" as a near synonym for "natural".
    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.
    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.

    Here's a toy example (LaTeX didn't work first time ...).
    Suppose I consider the simple probability space $(\Omega,E,Pr)$, where $\Omega$ is, say, $\{0,1\}$, and $E=\mathcal{P}(\Omega)$ and

    $Pr(\varnothing)=0$.
    $Pr(\Omega)=1$.
    $Pr(\{0\})=0.5$.
    $Pr(\{1\})=0.5$.

    (We have four events, and their probabilities.) Consider a bijection $f : \{0,1\}$ to the set {John, Yoko}. Now use this to define

    $Pr^{\ast}(\varnothing)=0$.
    $Pr^{\ast}({John,Yoko})=1$.
    $Pr^{\ast}({John})=0.5$.
    $Pr^{\ast}({Yoko})=0.5$.

    Everyone usually agree that $Pr^{\ast}$ is not a physical probability function. And I think this amounts to saying that $Pr^{\ast}$ is "unnatural".

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  11. 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.

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  12. David, I still don't quite get your objection!
    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 $Comp(Pr) = c$.

    The reductio argument, that shows that the world can be as complicated as we like modulo cardinality, is:

    (P1) The world has cardinality at least $\kappa$.
    (P2) Suppose $\mathcal{A} = (A,R)$ is a structure with $|A| = \kappa$ and $Comp(R) = c$.
    Therefore:
    (C) There is a relation $S$ on the world such that $Comp(S) = c$.

    Is the idea that (C) doesn't follow from the premises?

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  13. 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 $c(G_1) = c(G_2)$.

    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 $c(R) \neq c(R^{\ast})$?

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  14. David, it's clicked with me: the idea is that you're suggesting is that we interpret,

    (1) The world is complicated,

    to mean:

    (6) The causal relation on the world is complicated.

    (I've removed "mind-independent" as well, as that's an orthogonal issue.)

    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.

    Jeff

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  15. 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).

    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.

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  16. 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.)

    I think you're in Oxford in a few weeks' time for the Jowett/Phil Society? See you then, hopefully.

    Jeff

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  17. Glad we sorted that out! Yes, I'll be in Oxford two weeks from Friday. See you then.

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