False ID

Bob Murphy, always my favorite theist, posts a defense of Intelligent Design theory, or at least an attack on its attackers, who, he claims, have largely failed to grasp what the ID theorists, such as Michael Behe, are claiming:

Behe is fine with the proposition that if we had a camera and a time machine, we could go observe the first cell on earth as it reproduced and yielded offspring. There would be nothing magical in these operations; they would obey the laws of physics, chemistry, and biology. The cells would further divide and so on, and then over billions of years there would be mutations and the environment would favor some of the mutants over their kin, such that natural selection over time would yield the bacterial flagellum and the human nervous system.

Yet Behe’s point is that when you look at what this process spits out at the end, you can’t deny that a guiding intelligence must be involved somehow.

(Emphasis added.)

Perhaps Bob has forgotten that I disposed of this argument in Chapter 4 of The Big Questions , with a single counterexample that refutes both Behe and his polar opposite Richard Dawkins in one fell swoop. Let’s recall their positions, stated as simply as possible:

Behe: Irreducible complexity requires an intelligent designer.

Dawkins: Irreducible complexity requires evolution. (This is Dawkins’s stated position in his book The God Delusion.)

Landsburg: The natural numbers are irreducibly complex, moreso (by any reasonable definition) than anything in biology. But the natural numbers were not designed and did not evolve, so Behe and Dawkins are both wrong.

This is usually the point when someone with all the sophistication of a sophomore jumps in to object that the natural numbers are not “real”; they are merely figments of our imagination or some such thing. To which I reply: Hey, maybe you’re just a figment of my imagination. Maybe I’m the only sentient being in the Universe, and I’ve just imagined up all these flagella, cells, and organisms, and that’s the sum total of everything you need to know about biology. Also, there’s no need ever to debate the causes of the Civil War because that might have been imaginary too. The point here is that you can always shut down any discussion about anything by retreating to a position of extreme skepticism, but that’s never been a good way to make intellectual progress.

(And no, that’s not an argument for admitting the reality of unicorns and flying spaghetti monsters. If you need convincing that the natural numbers — unlike unicorns — are at least as likely as your furniture to exist outside your mind, then do buy yourself a copy of The Big Questions today!).

And if you doubt the complexity of the natural numbers, then you are probably unaware of the vast mathematical literature on how unfathomably complex the natural numbers (provably) are. For starters, the set of true statements about the natural numbers is too complex (in a very precise sense) to be derived from any system of axioms. The set of true statements about your brain, by contrast, is probably easy in principle to axiomatize (though to make this precise, you’d have to be precise about what sort of statements you’re allowing).

So what responses might we expect from Behe or Dawkins? Behe, I think, would have the easier time of it, since he can always contend that the natural numbers must have been intelligently designed. But that paints him into a pretty isolated corner. In my experience, even the most committed theists are reluctant to believe that an equation like “2 plus 2 equals 4” is the result of a design choice.

For Dawkins, I think, the game is up. Surely he won’t want to argue that the natural numbers have evolved, so that their properties today are different from their properties at some time in the distant past. (Of course, our knowledge of the natural numbers has evolved, but that’s a different matter entirely.)

Not everything that’s complex — not even everything that’s irreducibly complex — was designed. Not everything that’s complex evolved from something simpler. Whether any given complex structure was designed, or evolved, or was just plain complex from the get-go, is a question that requires more than an appeal to glib, and demonstrably false, generalities.

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64 Responses to “False ID”


  1. 1 1 Mike H

    “Hey, maybe you’re just a figment of my imagination. Maybe I’m the only sentient being in the Universe….

    Solipsism!

    The point here is that you can always shut down any discussion about anything by retreating to a position of extreme skepticism, but that’s never been a good way to make intellectual progress…

    “Oh, there used to be a solipsist, but he solipsided away!
    Solipsiding away, solipsiding a-a-waaaay….!
    There used to be a solipsist, but he solipsided away!”

  2. 2 2 Mike H

    In my experience, even the most committed theists are reluctant to believe that an equation like “2 plus 2 equals 4″ is the result of a design choice

    Ah, but what of the fact that the first English word in the base 27 digits of pi is “PI”? http://www.dr-mikes-math-games-for-kids.com/your-name-in-pi.html

    Nobody’s observed this yet: http://xkcd.com/10/

  3. 3 3 AB

    Is it Solipsistic in here? Or is it just me?

  4. 4 4 EricK

    There’s no ‘U’ in ‘solipsism’.

  5. 5 5 suckmydictum

    The integers, bob Murphy, opine
    Watchmakers are not very divine.
    Since natural selection
    stands up to dissection,
    Your superstitions are by design.

  6. 6 6 Harold

    A few things, which will all probably be recaps of previous posts and commmnents. By irreducible complexity do we mean something that cannot be described in fewer “points” than itself? Initial thought is that natural numbers can be described by a simple algorithm. Start at 1 and add one. The only problem is that there is no stopping, but in a sense we have described the natural numbers. I am sure you have dealt with this before, but the answer escapes me for now. The set of true statements about the natural numbers is a different thing altogether.

    I am not sure that your depiction of Dawkins position is accurate, but I would have to go back to the God Delusion to check. As with all extreme summaries, they may miss things. A string of random numbers is complex, but does not require evolution.

  7. 7 7 Scott H.

    So if I said to you “there are no perfect circles in nature”, your reply would be “nope! there’s one right in my head.”

  8. 8 8 Dave B

    I would suggest that Dawkins argument would be better phrased as: The only way (that we know of) to get something complex starting from ony something simple is by evolution. Presumably you don’t start from something simple to get the Natural Numbers – they just are.

  9. 9 9 RPLong

    The natural numbers are a concept. I don’t think it can be taken as a foregone conclusion that concepts aren’t a product of social evolution.

    We can apply this same reasoning to any abstract concept. Here’s an easy example: Complexity itself is irreducibly complex. But complexity was not designed and did not evolve.

    To be clear, I am sympathetic to Landsburg’s view on this, but I think it is yet to be proven. It’s clear enough that abstract concepts are something. In one sense they have always existed, independent of the mind’s ability to think about them. In another sense, though, they are ideas that would cease to occur to sentient beings the moment our species is eradicated by an asteroid.

    I’m not sure what that says about their “existence.” But in any case, I think to suggest that they do not really “exist” is more than just skepticism.

  10. 10 10 Dave

    “Yet Behe’s point is that when you look at what this process spits out at the end, you can’t deny that a guiding intelligence must be involved somehow.”

    woosh

  11. 11 11 Ken B

    I think Steve’s argument depends on misinterpreting Dawkins by ignoring the context. The context is functioning mechanisms and life forms. I’m pretty sure Steve wrote once “people respond to incentives” but of course that is false: people in comas, people unaware of the incentives, dead people. Have I refuted Steve? No, I have lost the context. I doubt Dawkins would deny the naturall numbers are as complex as you like, but he’d point out that the Sylow Theorems don’t walk around eating and reproducing. It’s a cute literary conceit on Steve’s part, but it’s not much of an argument.

  12. 12 12 TjD

    Came here to write about context, or rather the lack of it. But Ken B beat me to it.

  13. 13 13 EricK

    I think Dawkins means that the only way to go from a situation within this universe where there isn’t irreducible complexity to a situation where there is is via evolution (i.e. either it evolved, or it was designed by a designer who evolved, or that designer was designed by an evolved designer etc).

    There was never a time when the natural numbers didn’t exist, but there was time that life didn’t exist.

  14. 14 14 suckmydictum

    @13

    Do you think the natural numbers, taking the Landsburg line, predate the physical universe?

  15. 15 15 Bob Murphy

    Steve, your whole case rests on the mere assertion on your part that complex properties in the natural numbers could not possibly be there by design. Let’s see how far you are willing to push that intuition.

    Suppose I announced on my blog that I had found a new bit of mind-blowing evidence of the existence of God. If you use a base-26 system, so that we can convert numbers into letters of the English alphabet, then (I have somehow discovered) if you take your date of birth (expressed in our normal base-10 format as yyyymmddhhmm) and multiply it by your location of birth, expressed as latitude and longitude, and then go find that particular entry in a sequence of prime numbers, then in base-26 that particular prime number spells out a message for the person.

    You think this is absurd and go look up your personal info, then run the algorithm. You come up with something like (and I have no idea how long the message would be, just tinker with the algorithm to make it ballpark):

    “Steven, I was there for you as a child at the wagon, I am here for you now.”

    Now you’re telling me Steve that if *that* were to happen, you would still be 100% certain that the structure of mathematics wasn’t designed?

    OK, I hope you are willing to concede that an outcome like *that* would at least give you pause. You’d probably start thinking that the only reason you perceive the natural numbers the way you do, is the structure of your mind which (you currently believe) is due to the accidents of mutations and natural selection.

    Now if you agree that a seemingly hand-tailored message to Steve Landsburg, embedded in the structure of natural numbers themselves, would be evidence of intelligent design, why wouldn’t this be? *That* evidence is far more beautiful than a Post-It note to Steve Landsburg, isn’t it?

  16. 16 16 Steve Landsburg

    Ken B:

    I think Steve’s argument depends on misinterpreting Dawkins by ignoring the context.

    But since Dawkins (as best I can recall) never uses that context in making his argument, a counterexample that ignores the context is still a counterexample.

    If someone asserts that all numbers are less than 10, and offers a proof that involves fiddling with square roots and Taylor approximations, and if I say “but 11 is not less than 10”, it is no defense to say “But *in context* I was referring only to single-digit numbers”. If your proof never mentions single-digit numbers, then the existence of 11 is still a counterexample to your argument.

    Now to my best recollection (and I really should doublecheck this), Dawkins does not attempt to give a proof, but he does give some sort of vague heuristics that don’t invoke eating and walking around and reproducing. Therefore a counterexample that doesn’t involve eating or walking around or reproducing is still a relevant counterexample.

  17. 17 17 Ken B

    @16:
    I believe Dawkins is referring to the probabilty arguments deployed against natural selection. A cellular mechanism that requires the co-ordination of 20 chemicals is wildly unlikely to assemble spontanteously, it is too complex. But it can be assembled step by step (by evolution) if each step is “stable”. If a pre-cursor with 19 can be contrived from a precursor with 18, can be contrived by combining a precursor with 10 and another with 8, etc, then a complex mechanism can evolve.

    Let me forestall a foolish argument I see from posters over at Murphy’s (where I will spank them later): YES the 20 molecule machine could evolve from a 21 molecule more complex precursor. That is not a counter to dawkins, as the 21 had to come from something, and the chain cannot go on forever.

  18. 18 18 Steve Landsburg

    Bob Murphy (#15): As usual, you’ve gone right to the heart of the matter. Yes, my entire argument against Behe (though, of course, not my entire argument against Dawkins) rests on the mere assertion that the natural numbers could not have been designed. If you reject that assertion, I have no argument. (Or, more precisely, I no longer have *this* argument.)

    But at least in my own experience, the ID folks, and theists generally (and yes, I realize there might be, in principle, ID folks who are not theists) are very reluctant to allow that the natural numbers could have been designed, once you start pointing out what that means (e.g. that the designer could have arranged for 2+2 to equal 7, or for there to be no number 3).

    So I should have said that my argument rests not merely on my own assertion that the natural numbers can’t have been designed, but on (many of) my opponents’ own squeamishness about allowing that possibility.

  19. 19 19 Bob Murphy

    Steve,

    Actually that last comment by me came off too strident; I’m really not trying to “win the argument,” I want to show you why I think you are missing something really important (to say the least).

    I shouldn’t have said that you are merely “asserting” the natural numbers are undesigned; you actually said:

    “In my experience, even the most committed theists are reluctant to believe that an equation like “2 plus 2 equals 4″ is the result of a design choice.”

    So you weren’t merely asserting it, you thought it was a premise everybody endorsed so that you didn’t need to dwell on it or justify it.

    OK, and I’m now saying that that relies on a little bit of a bait-and-switch. You’re right, nobody would say “2 plus 2 equals 4” is evidence of design, because it’s straightforwardly the result of definitions and (I guess) our basic concept of numbers.

    One way of rephrasing what we mean by this is to say, “I can’t *imagine* a reality in which 2 + 2 does *not* equal 4. That just doesn’t even make sense; it’s inconceivable. So, I don’t think the equation is the result of a design choice; it couldn’t be otherwise; it just has to be like this.”

    Now that’s great for 2+2=4, but what about the implications of the Axiom of Choice? I think most people would say, “Of course I could imagine these results being false; in fact that’s exactly what I imagined yesterday, before you proved them to me! I am still in shock. This makes no sense to me.”

    Of this type of result, most mathematicians would still presumably say, “It couldn’t be otherwise; it just has to be like this” but they would not feel it in their bones. It would be like a minister telling a grieving family that their infant “had to” die from cancer as part of God’s plan. Yeah, intellectually he is committed to that, but he doesn’t really feel it the way he can joyously talk about how great God is when someone is “miraculously” cured of cancer.

    Anyway, my point in this is that you are loading the deck a bit when you jump from 2+2=4 to saying that there in principle, we could never see a result in mathematics that would make us think its structure had been intelligently designed.

  20. 20 20 Bob Murphy

    (Our comments were being posted simultaneously; I backed down, Steve, before seeing your #18 comment.)

  21. 21 21 Neil

    Dawkins was pointing out that the intelligent designer of ID “theory” would need to be more complex than the biological mechanism it supposedly designed, therefore in need of an explanation itself. Thus, where a process creating complexity is to be explanatory, it must be from simpler to more complex, not from more complex to less complex. And evolution is such a process.

    As for the natural numbers, in my sophomoric view they evolved in the minds of human beings through the process of induction, but this is irrelevant to Dawkins’ argument.

  22. 22 22 Patrick R. Sullivan

    ‘ But the natural numbers were not designed and did not evolve’

    How can you KNOW this?

  23. 23 23 RPLong

    Another way to look at concepts is to acknowledge that they are nothing more than words.

    If the Earth exploded tomorrow, the word “two” would be eradicated forever. If so, how many moons would Mars have? Still two, right? But is that because the second natural number is a real thing that exists, or is it because humans have developed a word that signifies duality?

    I think Landsburg is highlighting something really profound here. What is the existence of an abstract concept? Is its existence tied to the sentient beings who have figured it out, or does it exist outside of all such sentient beings.

    Think of it this way: If I say “Odysseus,” everyone immediately identifies a concept – an old Greek guy who fought in an ancient war and spent 20 years trying to get back home. Does Odysseus exist in the same sense that the number two exists? And if not, why not?

  24. 24 24 RPLong

    One more thing I just realized after submitting that comment: Instead of saying “Odysseus,” I could have said, “God.”

  25. 25 25 Tony N

    I could be wrong, but I think Steve is using the phrase ‘irreducible complexity’ to mean “not able to be axiomatized” while ID advocates, at least in my experience, use it to refer to something different, i.e. “not able to be simplified, as the composition of certain systems demands each and every component in order to function. Simpler version of these systems could not exist and still serve a purpose.”

    Hence their (ID’ers) inclination to invoke the phrase as something contra evolution, as, in their view, some biological systems are simply all or nothing.

  26. 26 26 Steve Landsburg

    Bob:

    Now that’s great for 2+2=4, but what about the implications of the Axiom of Choice?

    1) For the record, 2+2=4 *is* an implication of the axiom of choice.

    2) Anything I can prove about the integers using the axiom of choice must be true, because the axiom of choice is true in the constructible universe and the integers (unlike, say, the real numbers), don’t change when I go from one set-theoretic universe to another.

    3) But in any event, what I’m concerned with is the true properties of the natural numbers, not the logical tools we humans use to discover those properties. We can make all kinds of mistakes, from assuming axioms that are actually false to simple calculational errors, and convince ourselves of wrong things. I do it all the time. But I don’t see why our limited knowledge of the natural numbers has anything to do with this.

    I could try to refute Behe by saying “Hey, some of those things you believe about flagella might be false. Maybe your microscope lens was dirty.” I don’t think that gets to the heart of the argument.

  27. 27 27 Bob Murphy

    OK Steve maybe the Axiom of Choice was a cul de sac. Let me try it like this:

    (1) Everybody from Landsburg to Murphy to Aquinas agrees that “2+2=4” does not seem to be designed, because we can’t even imagine what it would otherwise be. It hardly makes sense to even ask, “Could God have designed it differently?” in the same way we could imagine a giraffe having a different structure.

    (2) Even if we restrict our attention to branches of mathematics (natural numbers? something else?) that Steve Landsburg agrees have an independent existence from human foibles, there are results that strike as us counterintuitive or unexpected. We definitely *could* imagine them being false, in a qualitatively different way from us trying to imagine “2+2=4” being false. Yes, we agree they are true, but it does not jump out as being *necessarily* true the way “2+2=4” does.

    (3) Therefore, focusing on “2+2=4” and concluding that something as complex as the natural numbers is “obviously” not designed, is a bait-and-switch. The concepts needed to evaluate “2+2=4” are actually not complex at all, and that’s why it doesn’t seem designed. In contrast, some of the truly beautiful and complex things that exist in the world of natural numbers might indeed strike some people as evidence of intelligence. They might not strike Steve Landsburg, but he can at least concede that focusing on “2+2=4” as the representative of “the natural numbers” and whether they exhibit signs of design, is a bit of stacking the deck. It would be like looking at a piece of dirt on a car’s hood instead of the engine inside, and saying the whole thing must not be designed.

    So Steve, with which part(s) of my above argument do you disagree?

  28. 28 28 Jonathan Kariv

    Does Dawkins actually say that complexity REQUIRES evolution? As opposed to simply the complexity does not require a designer?

    If his claim is that complexity doesn’t require a designer then pointing at evolution finishes the job. I was under the impression that this was pretty much all he was claiming.

    OK he does also claim that Darwinian evolution almost certianly produced the biological complexity we see today but this is very different to claiming that it’s the only thing that can produce complexity.(Disclaimer: I never finished the god delusion but I have watched a few of his debates on youtube, so if I’m misrepresenting Dawkins’ view someone tell me so).

    Of course if Dawkins is making the stronger claim that we need evolution for complexity then of course he needs to rule out every alternative. Given how long it took for us to come up with the idea of Darwinian evolution I’m somewhat skeptical of anyone having a list of all conceivable (and inconceivable)ways to produce complexity. Even without Steve’s example of the naturals I don’t see any particularly good reason to believe that design and evolution are the only possible ways of producing complexity. Or even more so I don’t see any good reason to believe that the ways we know of producing complexity (whatever list of ways you believe we have) are the only possible ways.

  29. 29 29 Keshav Srinivasan

    Steve, a few points about #26:

    1. You keep insisting that the natural numbers stay the same, while the real numbers can vary. Yet as I’ve told you in other threads, the natural numbers and the real numbers are on equal footing as far as technical results go: the second-order theory of real numbers is categorical, just as the second-order theory of natural numbers is categorical. You can define natural numbers by quantiying over sets of real numbers, and you can define real numbers by quantifying over natural numbers. There are models in which N is standard and R is nonstandard, and there are models in which R is standard and N is nonstandard. Now you may say that natural numbers are more fundamental because axiom systems presuppose the idea of ordered lists, but that’s a separate argument.

    2. You say that the arithmetical consequences of the Axiom of Choice must be true. But couldn’t your argument apply equally well to the arithmetical consequences of the negation of Choice? What gives a model of ZFC more weight than a model of ZF + not C? What gives Godel’s construction of a model of ZFC within ZF more weight than Paul Cohen’s construction of a model of ZF + not C within ZF?

  30. 30 30 Ken B

    @ JK 28:
    I don’t know what Dawkins said in the book Steve cites. I read it quickly and didn’t much like it. But I do know, because it’s a consistent part of his writings, that Dawkins generally means *biological* things. He’s a biologist. I have never seen him discuss Godel. Here is a quote from The Blind Watchmaker:

    “Biology is the study of complex things that appear to have been designed for a purpose. Physics books may be complicated, but …The objects and phenomena that a physics book describes are simpler than a single cell in the body of its author. And the author consists of trillions of those cells, many of them different from each other, organized with intricate architecture and precision-engineering into a working machine capable of writing a book.”

    Dawkins was responding to religious arguments about Darwinism. His Climbing Mount Improbable is all about this idea. Evolution does need all the parts to come together at once as it can build on previous forms.

  31. 31 31 Steve Landsburg

    Keshav Srinivasan:

    Thanks for this, and I apologize if I’ve failed fully to internalize some of your past comments.

    Re your point 2, I think the answer is that, in light of the existence of Cohen’s model, that the negation of AC can’t imply an arithmetical falsehood any more than AC can. In other words, the answer to your “But couldn’t your argument apply equally well….?” is: Yes, it could. I’m not sure why you see a problem here.

    Re your point 1, I think I want to digest this a bit before responding. I hope I find time for this.

  32. 32 32 maznak

    I am fairly sure that Dawkins does not mean that irreducible complexity requires evolution. In fact, irreducible complexity is a favorite ID argument against evolution. Where Dawkins rightly gives examples and parallels of ramps and scaffoldings to explain that seemingly irreducibly complex systems MIGHT evolve. I believe that if you accept that Dawkins is talking about biology most of the time, you will have very hard time trying to catch him making logical (or other) mistake.

  33. 33 33 Keshav Srinivasan

    Steve, there are arithmetical consequences of the axiom of choice that contradict arithmetical consequences of the negation of choice. To take another example, there are models of ZFC + the continuuum hypothesis and there are models of ZFC + not CH, even though CH can be written as a statement of third-order arithmetic.

    As far as my point 1 goes, you can see the proofs of some of the results I mentioned in one of my old comments:

    http://www.thebigquestions.com/2012/09/13/simple-as-abc/#comment-63796

  34. 34 34 Xorph

    To continue with Maznak’s excellent point : Dawkins’ point is that evolution is a way to go from a very simple to a very complex state, and the very fact that one way has been found (evolution) is enough to discard all the irreducible complexity arguments of creationists.

    Obviously only living things (or their “extended phenotypes”) can evolve : things which can replicate, with enough copying errors, and compete for ressources. Numbers don’t compete, don’t have copying errors, so they can’t evolve.

  35. 35 35 Tony N

    Emergent complexity (sans agency) and irreducible complexity are not mutually exclusive ideas.

    Now, I’m certainly no apologist for the ID crowd, but I do think their notion of irreducible complexity is being contorted a bit here.

    As I indicated earlier, in my experience, the argument isn’t that ‘significantly high levels’ of complexity cannot be achieved without agency, but that some ‘types’ of complexity, certain complex structures,cannot be the product of evolution since there is no way such structures can evolve step by step, piece by piece. I think their argument is along the lines of “among other things, engines have pistons, push rods, valves, and spark plugs. Remove any one of those components and the thing simply doesn’t work. It must exist in its final form or you’ve got nothing. How then could something like an engine, or an eye, ever evolve?” Personally, I don’t find it persuasive, but it’s a better argument than the one being attributed to them here.

  36. 36 36 Steve Landsburg

    Keshav: Yipes! Obviously I need to think about this more carefully. Thanks.

    Edited to add: Now that I’ve thought about it a little more, I stand by my argument.

  37. 37 37 Mike H

    (e.g. that the designer could have arranged for 2+2 to equal 7, or for there to be no number 3).

    Everybody from Landsburg to Murphy to Aquinas agrees that “2+2=4″ does not seem to be designed, because we can’t even imagine what it would otherwise be

    I remember seeing an argument on a usenet newsgroup. One the one side was a nutcase who claimed he’d found solutions to Fermat’s equation a^n+b^n=c^n for n>2 and a,b,c positive. On the other was a bunch of people who knew some mathematics, and were pointing out why he was wrong.

    The nutcase, however, was not mathematically wrong, his mistake was insisting on his own semantics. He had unilaterally defined whole numbers as having, perhaps, infinite sequences of digits, instead of only finite. Effectively, he’d found solutions for a^3+b^3=c^3 for 10-adics a,b,c, but insisted on calling them integers.

    I pointed this out, but neither side paid any attention.

    (PS, if you’re wondering why I still call him a nutcase, it’s because he was. He insisted, for example, that the solar system was a giant Plutonium atom)

    Now, I can easily imagine a universe – or at least, a system of numbers where 2+2 was 7, or where 3 didn’t exist. I remember multiple abstract algebra assignment in university where we had to muck about with wierd forms of addition and multiplication. Some of those wierd forms stuck, and we even gave the resulting number systems names – modular arithmetic, for example, where 2+2 might be 0. Or the quaternions. Finite fields. There’s a lot more that we didn’t give names to. They aren’t any less real than the natural numbers or the octonions, they’re just less useful to Homo Sapiens.

    This focus on the natural numbers seems irrational. There are multiple number systems that could be invented. The rational approach is to not be negative, but acknowledge that arithmetic is modular – we plug in the bits we need into the problems we have at hand. An algebraic approach to the field implies that, when discussing transcendental questions about whether the natural numbers are real or imaginary, we need to conclude also that all these systems enjoy the same Platonic magnitude. To decide whether that magnitude is zero or nonzero would require a complex argument.

  38. 38 38 Keshav Srinivasan

    Steve, to be clear, when I was talking about arithmetical consequences of AC and its negation, I meant consequences for statements in second-order arithmetic. You can’t use the axiom of choice or the continuum hypothesis to prove any statements in first-order arithmetic that you couldn’t already prove using ZF, due to Schoenfield’s theorem:

    http://en.wikipedia.org/wiki/Absoluteness#Shoenfield.27s_absoluteness_theorem

  39. 39 39 Steve Landsburg

    Keshav Srinivasan: I still need to think about this more carefully, but it seems to me that what you’re calling “arithmetical consequences” are in fact statements in third order arithmetic, i.e. not statements about numbers but statements about sets of sets of numbers — and so, in essence, talking about the real numbers, or even the power set of the real numbers. I’m disinclined to call statements about the real numbers “arithmetic”.

    So I want to say that the (standard) natural numbers do not change from one universe to another, but their power set does. Is that a fair statement?

  40. 40 40 Keshav Srinivasan

    Steve, responding to your comment:

    1. There are consequences of ZFC in second-order arithmetic which contradict consequences of ZF + not C in second-order arithmetic. The same can be said about ZFC + CH and ZFC + not CH. But as I said, the same cannot be said if you change second-order arithmetic to first-order arithmetic, so you’re right that this result is in some sense only about the power set of N and not N itself.

    2. However, I would disagree with your statement that N stays fixed from model to model while R varies, but the reason isn’t because of the stuff involving choice, but because of the results I mentioned in point 1 of my comment #29.

  41. 41 41 Scott H.

    It seems with Behe the new relationship between the “Intelligent Designer” and evolution is very analogous to the relationship between “Intelligent Causer” and the Big Bang (for those that believe in an Intelligent Causer).

    However, the Big Bang isn’t denied by proponents of the Intelligent Causer theory. There aren’t huge groups advocating teaching Intelligent Causer theory side by side Big Bang theory. Big Bang theory isn’t seen as a threat to religion. There isn’t an attempt to create an Intelligent Causer faux science discipline.

    This begs the question as to why there is such a different approach with ID and evolution. I would say that it means the majority of ID theory devotees don’t agree with Behe (and Murphy’s?) interpretation of ID theory.

  42. 42 42 iceman

    41 – part of it may involve Tony N #35 — I presumed this would be ID101 to many of you, but what of the argument that it’s hard to see how some biological features evolved incrementally if en route they would seem to represent biological disadvantages?

  43. 43 43 Ken B

    @iceman 42:
    1) appearances may be deceiving.
    2) as ever the issue is what advantage accrues to the genotype. A phenotypic feature might be a drawback if it comes with other advantages. A peacock’s tail has disadvantages.

  44. 44 44 Harold

    The peacocks tail is a good example. It has huge cost, and contributes nothing to the direct survival of the animal. The only advantage is in attracting a mate. This huge and complex structure may evolve into something unexpected – perhaps an ability for peacocks to build bridges. Future generations of biologists may be mystified how peacocks evolved their incredibly long bridge building tail feathers. After all, there would have been no advantage when the feathers weren’t quite long or strong enough for the bridge building.

  45. 45 45 Ken B

    @Harold 44:
    Sexual selection in general is a great illustration of natural selection and a problem for ID. From the point of view of the species, or the individual, traits like the peacock’s tail make no sense. But from a selfish gene/extended phenotype point of view they make perfect sense.

    What you describe is (as I am sure you know) “pre-adaptation”. An unfortunate term seized upon by IDers to suggest planning and forethought. It means no such thing.

  46. 46 46 Tony N

    @Harold

    Agreed, sort of. The problem with your example though, as I see it at least, is that you assume there was no advantage gained through the tail feathers prior to their reaching the point where they could be used to build bridges. But we know that isn’t the case. A penis doesn’t really contribute to the direct survival of animals either–many don’t even have one– but it does provide obvious benefits with respect to reproducing. We also know a peacock’s tail feathers confer a mating advantage that seems to overcome the disadvantages correctly alluded to by Ken B. We can assume future biologists would know the same.

    The idea behind irreducible complexity, at least as I tend to encounter it, is that the future tail-feather structure that allows peacocks to build bridges cannot arrive via evolution if when a component is removed you are left with something that confers no advantage at all (hence we expect the appendix to eventually vanish, as it seems to be doing, rather than evolve into something more complex). But we know that’s not the case re the peacock. Before they built bridges, thier tails, in a simpler form, attracted mates.

    That said, I have reservations about the entire concept. But even if I didn’t, I’m not sure ID has produced a valid example of an I.C. biological structure anyway.

  47. 47 47 Ken B

    “We can assume future biologists would know the same. ”
    The depends on whether we let Bob Murphy teach them biology.

  48. 48 48 iceman

    43 – I presume you meant “a phenotypic feature might NOT be a drawback…”

    Actually the peacock’s tail seems not the best example of something that couldn’t have evolved gradually (mine’s longer than yours again). That’s why we can say “they’d be better off if they all had shorter tails.” I believe more common examples for IDers are flagella tails or eyeballs, things that may seem to represent pure disadvantages ‘in-process’.

    PS Lots of bird species have one gender brightly colored for the good of the genotypic village (to decoy predators from the young or young-bearers), no? I’ve learned something potentially useless from this blog today — not everyone agrees with our characterization of the poor peacock.

  49. 49 49 Tony N

    @Ken B

    Heh. And all viruses would be referred to as “Krugmans”

  50. 50 50 iceman

    46 – Tony N beats me to the punch again (this time I didn’t even notice b4 posting)

  51. 51 51 Ken B

    @49: Ha! Then I could trot over to FA and write “God is a Krugman”!!

  52. 52 52 Scott H.

    35 & 43 — The human eye is also not a good candidate to demonstrate I.C.. I’ve seen films about the evolution of the eye. The analysis struck me as totally plausible.

  53. 53 53 Ken B

    @iceman 48: Not quite, I was sloppy though. I meant something might seem to have obvious disavantages but haver other less obious but crucial advantages. Sexual selection features are good candidates. I recall once seeing the argument that the allele for sickle cell anemia is a disadvantage, since early death is a downside, but that ignores that it also protects against malaria when only one copy of the allele is present.

  54. 54 54 Ken B

    @Scott H:
    The eye is about the worst imaginable example for the ID crowd, but they love it. http://www.youtube.com/watch?v=KJ-6TCPdbl0

  55. 55 55 Tony N

    Ken B: Then I could trot over to FA and write “God is a Krugman”!!

    Funny, Ken B. Although Krugman might insist the part in quotes is backwards.

  56. 56 56 Harold

    #46 “We can assume future biologists would know the same.” Well, probably in this case, but the point is that the advantages may not be obvious, or even discernable to the future (or present) biologist. Thus ID’ers would have a case if they could catagorically point to a feature that could not have evolved. However, it is pretty much impossible for them to do so, because we can claim there may be an advantage in the intermediate form that we are unaware of.

    This may be presenting evolutionists as relying on a tautology, except that the this is not the basis of th eevidence on which evolution is based.

    “A penis doesn’t really contribute to the direct survival of animals either–many don’t even have one” Indeed – I have discovered my wife is one of these!

  57. 57 57 Ken B

    @Harold 56: See, this is the difference between the net and conversation. In conversation I’d say “So have I.”
    Ba-dum-ba!
    :)

  58. 58 58 Tony N

    @ Harold and Ken B

    I’d say that qualifies as good news!

    I tell mine she needs to let me investigate regularly. No such thing as too thorough.

  59. 59 59 Harold

    Hey Ken B, keep your hands off my wife – even if it is in the interests of science!

  60. 60 60 Ken B

    @TonyN:”No such thing as too thorough.”

    Indeed. A second opinion never hurts you know. If you need help, I’m available.

  61. 61 61 Tony N

    Booya! Hah!

    You should see her, then you’d really mean it :)

  62. 62 62 Ken B

    A random act of music. Moby http://www.youtube.com/watch?v=1vHrXBNwW60

  63. 63 63 Harold

    I may have to re-evaluate. It seems lots of people say my wife is often seen with a prick.

  64. 64 64 iceman

    Now that’s a pretty good line Harold. You’re like an onion.

  1. 1 A Story Illustrating Intelligent Design (ID) Theory

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