Thursday, 15 August 2013

Landsburg vs Dawkins- a self inflicted debacle.

Steve Landsburg, a man who has made a profession of being cleverer at stupidity than anybody else on call for Fox News, is seeking to square his own extreme Mathematical Platonism with the Creationism of the great unwashed.

With typical fatuity he attempts to take down Richard Dawkins with a wholly specious, seemingly mathematical, argument.
'Here is a quote from Dawkins’s book “The God Delusion”:
Time and again, my theologian friends returned to the point that there had to be a reason why there is something rather than nothing. There must have been a first cause of everything, and we might as well give it the name God. Yes, I  said, but it must have been simple and therefore, whatever else we call it, God is not an appropriate name….The first cause that we seek must have been the simple basis for a self-bootstrapping crane which eventually raised the world as we know it into its present existence.
I could provide additional quotes, but this one should suffice. Dawkins believes, unless I have misunderstood him completely, that he has a quite general argument, not tied in any way to biology (because the above quote, for example, has nothing to do with biology) to establish that complex structures must have simple causes. That argument, whatever it might be, cannot be correct because the natural numbers stand as a counterexample.
Obviously, Landsburg has misunderstood Dawkins completely because that's his metier, it's what he does best, and moreover he has worked hard at it- devoting a whole chapter to doing so in one of his worthless books.

Just to be clear, Dawkins views are summarized thus-
  1. One of the greatest challenges to the human intellect, over the centuries, has been to explain how the complex, improbable appearance of design in the universe arises.
  2. The natural temptation is to attribute the appearance of design to actual design itself. In the case of a man-made artefact such as a watch, the designer really was an intelligent engineer. It is tempting to apply the same logic to an eye or a wing, a spider or a person.
  3. The temptation is a false one, because the designer hypothesis immediately raises the larger problem of who designed the designer. The whole problem we started out with was the problem of explaining statistical improbability. It is obviously no solution to postulate something even more improbable. We need a "crane," not a "skyhook;" for only a crane can do the business of working up gradually and plausibly from simplicity to otherwise improbable complexity.
  4. The most ingenious and powerful crane so far discovered is Darwinian evolution by natural selection. Darwin and his successors have shown how living creatures, with their spectacular statistical improbability and appearance of design, have evolved by slow, gradual degrees from simple beginnings. We can now safely say that the illusion of design in living creatures is just that—an illusion.
  5. We don't yet have an equivalent crane for physics. Some kind of multiverse theory could in principle do for physics the same explanatory work as Darwinism does for biology. This kind of explanation is superficially less satisfying than the biological version of Darwinism, because it makes heavier demands on luck. But the anthropic principle entitles us to postulate far more luck than our limited human intuition is comfortable with.
  6. We should not give up hope of a better crane arising in physics, something as powerful as Darwinism is for biology. But even in the absence of a strongly satisfying crane to match the biological one, the relatively weak cranes we have at present are, when abetted by the anthropic principle, self-evidently better than the self-defeating skyhook hypothesis of an intelligent designer.  

Dawkins doesn't say what philosophy of mathematics he subscribes to. Suppose he is a constructivist. Then the complexity displayed by the natural numbers has evolved along with the minds that do mathematics but otherwise has no independent existence.
Suppose, on the contrary, Dawkins is a Platonist of some sort. Then, since the Platonic idea of Complexity pre-exists, he could scarcely say that 'complex structures must have simple causes'. Thus we know Dawkins isn't a Platonist. It is simply not credible to attribute to him the  belief that  individual horses participate in some Platonic idea of the horse.

Something else we know is that Dawkins is an Evolutionary Biologist. This doesn't force him to be nothing but a Constructivist in the Philosophy of Mathematics. He could be a Piercian Instrumentalist or a Heytig type Intuitionist or even a Formalist of a certain sort. What he can't be is a Mathematical Platonist of an extreme Tegmark type- believing 'All structures that exist mathematically also exist physically'- because that implies 
a) either accepting Godel’s proof of God, because mathematical structures would have what Godel calls ‘positive properties’- i.e. match to real world objects and so a principal ultrafilter of the St. Anselm type exists at least in some possible world- and if we are in the wrong world that’s just tough titty- we’re foredoomed to damnation
b) or denying mathematical structures can have positive properties- but this means minds have no way to ‘match’ with mathematical structures and so the human claim to be able to apodictically differentiate orders of complexity- P, NP, exponential etc- fails as these may be phenomenological artifacts.
What is true of Dawkins is true of each one of us.
In other words- though all the facts about the world don’t change, admitting Tegmark’s proposition alters what we can good faith affirm as being our own rational beliefs arrived at in accordance with best Public Justification practice.
Suppose we lower the bar and limit ourselves to what Quine & Putnam call ‘indispensable arguments’- i.e. stuff it makes sense to believe exists because our technology is based on it- is it unreasonable for Dawkins to a priori reject extreme Tegmarkism in favour of Constructivism (in which case the complexity of mathematical objects co-evolves with the minds that construct them)? Not at all, because currently there’s no way around ‘Beenakker’s boundary’. (http://arxiv.org/abs/physics/0702072v1)and there’s no Instrumental reason to counterbalance this objection.
Indeed, the reverse is the case, and what’s more Tegmark knows it.
What about Landsburg?
In this post, he writes ‘I claim to have explained in The Big Questions exactly how the first cause of our Universe could be a mathematical structure that is far more complex than the Universe itself; of course others, like Max Tegmark, have demonstrated this possibility in far more detail than I have. Whether or not Tegmark and I are correct in our beliefs, I claim we’ve at least demonstrated that (as far as we can tell) those beliefs could be true, which, once again, refutes Dawkins’s position.’

The problem here is that Computability puts a limit on the Complexity of what can be believed, other than by a leap of faith, about a mathematical structure . Either we have to believe things we can never prove or else there is a limit to the complexity only of mathematical but not of physical objects in Tegmark’s world (in which only computable mathematical objects are considered to be as real as physical objects ). Since we know the Red Queen effect drives co-evolved complexity for living things but not for mathematical objects considered as existing outside living minds and since we know some minds can grasp some complex mathematical objects but don’t know if the reverse is the case, it follows that it is impossible to demonstrate that a mathematical structure can be more complex than some currently living thing. Put another way, we know that some complex mathematical objects are ‘reducible’ for living organisms (otherwise we couldn’t say a living brain knows that a given object is complex) but we don’t know that the reverse is also the case- otherwise we’d have a proof that P equals NP.
For Economists, as opposed to philosophers, there is a big penalty involved in failing to grasp the central insight of Complex Adaptive Systems theory- viz. free agents, even with zero intelligence, create the Math that describes their behavior-that’s why we see real time polynomial complexity in price determination but not Walrasian fixed point algorithms running in exponential time (Axtell).
Mathematical Platonism is a bad habit for Econ, because it lets us off the hook for showing real time tractability for fixed point computation (necessary for things like the assumption of Ricardian Equivalence). True, one can adopt an an Austrian praxeology but the risk is of being ignored or sidelined by the mainstream.
The problem with Platonism is that it can cash out as faith in propositions known to be Mathematically unprovable. This is something worse than what Philip Mirowski decries- viz. neo-classical Econ’s tendency to take dis-confirmation as deep confirmation- it is a refusal to update the neo-classical Research Program in line with developments in Maths and computing and thus running the risk of becoming irrelevant.

Returning to Landsburg's post- first he makes a bizarre claim about what Dawkins is saying- viz. Dawkins isn't actually rebutting the Creationist 'argument from design' instead he's a mathematician working in Descriptive Complexity Theory who claims he has a proof that everything mathematicians call complex evolved from something simple. 
 Then Landsburg, by bringing up Tegmark, gives us the ammunition to prove, not Dawkins case (he wasn't talking about Tegmark universes) , but Landsburg's  Strawman-Dawkins case. At the same time, the atheist Landsburg, all unknowingly shows a way to save (or at least not destroy) the Creationist, Michael Behe's 'Irreducible Complexity' thesis. This is like Robin Hood aiming an arrow at the Sheriff of Nottingham and ending up killing Maid Marion and Little John and Good King Richard and Little Bo Peep.  But this is typical of Landsburg. His blog is pure intellectual slapstick.
Time and again we see Landsburg destroy the cause he claims to be defending with his weird logic and general air of illiteracy.  
Landsburg's wish to run with the hares and hunt with the hounds makes him a dangerous friend- in this case, he hasn't helped level the playing field for the Theists- only Narada's veena plucked by many fingered Time can do that- but he is a suicidal enemy. Aiming a blow at Dawkins he cuts off his own Tegmarkian head.

What is the moral of this story?
Survival of the fittest?
No, it is-  'more sex is safer sex when your name is Steven Landsburg and you are pleasuring the one great love of your life while typing up your blog with your other hand.'

Update
Landsburg latest comment is as follows- (my remarks are in bold)
Dawkins makes the statement that everything which is complicated arises from something that is simple. Dawkins does no such thing.  Dawkins doesn't say that Evolution always goes from the simple to the complex. The reverse happens if selection pressure eases up- for e.g. with Spiegelman monsters.  However, he does say that we human beings have evolved from relatively simple life forms. We weren't created by a more perfect, unimaginably more intelligent and complex being in the manner suggested by Genesis. Dawkins fails to define “complicated”, “simple” and “arises from”, so we have to guess what he means. Dawkins is a Public Intellectual writing in English. To find out what he means, you don't HAVE to guess, you can just ask himIn order to construct a potential counterexample to his statement, I have to come up with something that is complicated, but did not arise from a simple precursor, and the thing I have to come up has to satisfy this condition for a broad class of possible interpretations of Dawkins’s language. Dawkin's statement was domain specific. To refute him you have to come up with, not a potential, but an actual counterexample from that very same domain- viz. evolutionary biology. A
I claim that if the set of all true first-order statements about a structure X (in some appropriate formal language) is non-recursive, that’s a pretty good definition of complex — in the sense that no matter what everyday interpretation Dawkins intends of the word “complex”, anything that satisfies this definition will probably also satisfy his. See below
So: Dawkins says that the current state of the universe is complex, hence  must have a simple precursor. I observe that he hasn’t told us what he means by complex, but we can almost surely capture it by the difficulty of axiomatizing the set of true statements about the state of the universe. Therefore the right analogy is: No matter what you mean by complex, the natural numbers are almost surely complex by the same definition, and therefore the natural numbers — not any individual natural number, but the natural numbers, must have a simple precursor.
Landsburg is making two different mistakes
1) A mistake about natural language- He assumes that the conventions determining the truth value of statements in English about a formal language arise from something intrinsic to the formal language and not to customary English usage. This is not true. By English convention, this statement is true-  'Number theory is simple.  Aesthetics is complex.' Arguments in Number Theory get resolved quickly with relatively short papers deciding the issue. Arguments in Aesthetics just run and run and involve vast and expanding universes of discourse. This is an example of a meaning of 'complex' which is in accordance with English convention. It suffices to refute Landsburg. 
2)  A mistake about formal language-  he says ' if the set of all true first-order statements about a structure X (in some appropriate formal language) is non-recursive, that’s a pretty good definition of complex.  A first order formal language includes the Existential Predicate, thus, it can feature impredicativity and hence non-recursivity.  But by the Godel's 'compactness' & the upward Löwenheim–Skolem theorem nothing relevant to complexity can be said in the first order language as opposed to the second order language.
Fagin's theorem is a result in descriptive complexity theory that states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP. Thus Landsburg's claim that he has a 'pretty good' counter-example is refuted because we don't know if it is computable. We don't know if unicorns exist somewhere in the Universe. But referring to something which may or may not exist is not a 'pretty good' counterexample to a theory about what exists.

In any case, something whose descriptive complexity can be proved to be nameable in English by just two capital letters is pretty damn Kolmogorov simple if English is our universal descriptive language. 
In fact, as time goes by we all expect to see increased compressibility in model theory- i.e. proofs will get shorter and have more generality. However,  we don't currently expect to see increased compressibility in real world objects- e.g. I don't expect to see vat grown clones of myself which have the same biological age and memories and tastes and phobias. It may happen but there is no proof that it will happen. The same may be said about Landsburg one day speaking other than through his arse.

3 comments:

  1. The Robson Red Queen article you linked to is behind a pay-wall. This is the pdf- http://privatewww.essex.ac.uk/~scher/My%20Home%20Page_files/Dec2004%20Special%20Issue/RobsonEJFEature2005.pdf

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    1. Thanks! Fixed.
      We need a modern day Kapila to give content to Red Queen Prakriti's Purusha.

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    2. Such a Purusha would have exponential negative complexity. OMG I'm Buddhist!

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