In this paper, `What Are the New Implications of Chaos for
Unpredictability?’, Werdnl proposes a new definition for chaos, defends it, and
uses it to show that chaotic systems (now precisely defined) have a particular unpredictability
property---the new implication of chaos for unpredictability.
Werndl spends the majority or the paper defending her
proposed definition, that a system is chaotic just when it exhibits mixing (aka strong mixing) which is precisely defined in section 3.2, equation
(4). It’s easiest to see what mixing means once you’ve made the move of
equating a physical measure defined
on a phase-space to the probability
that an arbitrary system will be in the region of phase space (given an
appropriate normalisation of the measure). Werndl makes this move without
arguing for it except for saying that `it is quite natural under certain
assumptions’. Nevertheless, with this in hand we can gloss what it means for a
system to exhibit mixing: given two regions of the phase space A, and B; let T∞A be
the region of points resulting from allowing the points in A evolve, under the dynamics of the system for a very long time;
then the probability that an arbitrary system will be in both T∞A and B is equal to the product of the probabilities that a system is in A and B respectively. Another picture of this is that after large amounts
of time-evolution, any bundle of initial conditions becomes spread out evenly
over the whole phase space (although becoming highly filamentous in order to
preserve its initial measure).
Remembering, from elementary probability theory, that the
probability of two independent events occurring is the product of the
probabilities of each of them occurring in isolation, one is quickly led to
Werndl’s new implication for predictability. Suppose you knew that a system
started out in a particular region of the phase space, and that it had evolved
for a large amount of time. Suppose,
also, that you wanted to know what the likelihood is of it being in some other
region after the time-evolution. When the system is mixing, knowing the initial
conditions is no help whatsoever for finding where the system will end up. Since
the initial phase-space bundle has been spread all around the phase-space, the
probability of it ending up in some region is just the same as the probability
of an arbitrary system being in that region. This (along with her defence of
mixing as a definition of chaos) justifies her claim that: `a general new
implication of chaos to unpredictability is that for predicting any event … all
sufficiently past events are approximately probabilistically irrelevant.’
Finally, a note on Werndl’s project of defining chaos:
In this week’s seminar we argued about what worth there was
in defining chaos or given a vague
definition of chaos what worth there is in the concept at all. Werndl defends the
adequacy of her definition using the criteria due to Brin, Stuck, and Devaney
that `(i) [a proposed definition] captures the main pretheoretic intuitions about chaos, and (ii) it is extensionally correct’ [her emphasis]. I
think she successfully shows that her definition meets both of these criteria,
but this doesn’t provide support for redefining
chaos in the first place. I say redefining as, I submit, chaos is already
defined (albeit vaguely) by its use.
Werndl’s motivation for this redefinition seems to be to
answer the question: `What are the new implications of chaos for
unpredictability?’ which, in the abstract, she suggests ought already to have
been answered based on the commonplace views of chaos-theorists (of all
flavours). In my mind this isn’t sufficient, as it would be perfectly
permissible for the new implication to be exemplified by only some chaotic
systems (under a more permissive definition) or some chaotic systems and some
systems that aren’t (under a more restrictive definition).
The new implication could equally well (if not better) be
thought of as a `new implication of strongly mixing, measure preserving,
dynamical systems for predictability’ as an implication of chaos. It seems to
me that the real motivation for the redefinition is political, in that chaos is
a more sexy name with which to describe ones field---after all Jurassic Park didn’t
have a theorist of strongly mixing measure preserving dynamical systems, but
rather a chaos theorist.
http://xkcd.com/1399/
ReplyDeleteHi all. The URL in my preceding comment points to a relevant comic; don't be afraid to click.
ReplyDeleteJon, very nice exposition. I would just pick at two points.
The account of strong mixing is intuitively helpful but potentially a *little* misleading. You write, "let T∞A be the region of points resulting from allowing the points in A evolve, under the dynamics of the system for a very long time; then the probability that an arbitrary system will be in both T∞A and B is equal to the product of the probabilities that a system is in A and B respectively." If by T∞A you literally mean the the region of points resulting from allowing the points in A evolve, under the dynamics of the system for a very long, *finite* time, then this is *approximately* correct: the probability that an arbitrary system will be in both T∞A and B is *approximately* equal to the product of the probabilities that a system is in A and B, but not exactly. On the other hand, if by T∞A you mean the result of letting the points in A evolve under the dynamics *forever*, as '∞' would suggest, this makes no sense. As I'm sure you realise, those points won't settle down anywhere, and where mixing holds, the sequence of sets will not have a limit in any natural sense. Only the measures involved have limits.
Second, I think it might be a little unfair to chalk up Werndl's project to politics or marketing instincts. I see your point, but it's also true that many people before Werndl have thought that there was something distinctly special and enlightening about *chaotic* dynamical systems. That's part of what made it sexy in the first place. Werndl can be seen as earnestly trying to see how well such views could be sharpened and justified. And it's rather surprising that she is able to argue so successfully that in fact they can be made very sharp and very well justified. That is not to say the argument is flawless, of course, but I find it surprisingly strong.
Thanks for the comments.
DeleteRE my gloss of strong mixing:
I take your point that what I wrote is potentially misleading. My intention in saying `a very long time' was to avoid discussion of limits, but of course my gloss can't be exactly accurate without referring explicitly or implicitly to limits.
RE redefinition:
I may have phrased my point overstrongly. I think my objections comes from a personal preference against having arguments about definitions of words. It seems to me that everything of philosophical interest in the New Implication can be summed up as: ``there exist systems (either mathematically or physically) that exhibit this particular unpredictability phenomena''. This can be just as easily (or more easily) coached in terms of strong mixing as in terms of chaos. I agree that Werndl is very successful in terms of showing the adequacy of her new definition for chaos, I just think that that is a distraction from the philosophically interesting point.
I said click; I mean copy and paste into your browser address bar.
ReplyDeleteThank you Jon for your nice and clear summary. I was thinking about your objection to redefinition and was wondering when it is the case that a term in a language needs definition (or redefinition). I guess this issue is more related to philosophy of language of which I don’t have much knowledge but I will try to convey my thoughts about the issue in the following:
ReplyDeleteI think the need for definition arises when the users of a language cannot decide whether to apply a term to a set of phenomena (objects). In this case they try to find a description of the term in terms of other terms about which such ambiguity doesn’t exist. This description has to have two important features: first it has to include all those phenomena (objects) to which language users apply the term without hesitation before the attempt for defining, and second, it needs to exclude all those phenomena (objects) to which language users don’t apply the term without hesitation before the attempt for defining.
I don’t think that finding such a description is straightforward but it seems that when you find it you achieve something more than labeling a set of undecided phenomena (objects). In this way, you may discover a certain new feature common to all those phenomena to which the term can be applied and I suppose it is this outcome that makes the act of defining worth trying.
I believe what I said is the case for the term “chaos” i.e. this term exists in scientific language and scientists use it for certain systems without hesitation but there are other systems to which they cannot decide whether to apply this term. So finding a definition seems necessary. Werndl managed to propose a definition which meets the two important criteria I mentioned above and also to discover a new common feature between the systems classified under this term by her definition. If it hadn’t been for those ambiguities and hesitations in the usage of the term “chaos” and its relation to the notion of “unpredictability”, this new relation between the mathematical notion of mixing and the notion of “unpredictability” wouldn’t have been discovered. That’s why I think Werndl’s attempt is a precious one.