A brief note: I did some research before posting to see if I could find any direct evidence in the fossil record to support this mechanism. I was not able to obtain that --and the types of articles I did find made me far more sympathetic to your fears about the way arguments of this sort can be misused!

With that said, here's what I think Dawkins is not accounting for: As I read his interpretation of Darwin, a shape like the fern would have to emerge gradually, over the course of a long period of time, as a series of small incremental phenotypal changes that would build out each individual frond and subfrond.

But once we know about the fractal mechanism, we might instead expect that one small change to the DNA could produce the entire fern shape, since the subfronds and sub-subfronds are all defined by the same basic generator. Another small change could produce a radically different overall shape with the same branching fractal structure.

So we would thus expect that even if the genotype displays tiny incremental, gradual change, the phenotype might still change in large jumps. It's that kind of phenotypal jump that I think poses a problem for Dawkins because it suggests that evolution didn't "climb Mount Improbable" alone, but with a substantial assist.

Again, this is only a problem for Dawkins solely because of the philosophical mileage he gets out of the idea that steady gradual evolution is the universe's only important creative process. If he abandons that contention, then this ceases to be a problem for him.

I expect you will find that's what happened. A fern isn't actually a fractal like your image; the leaflets aren't actually smaller versions of the leaves, they are solid objects with a great deal of structure to them. On the other hand, nobody would expect evolution to build a tree by adding one branch after another. It's obvious they are made of repeating parts, and since the genetic code is identical in every cell, the harder question is how mutations could apply to some parts differently than others.

I know Dawkins talks about that a little in The Ancestor's Tale - how animals primitively have a series of identical segments, and hox genes provide the tools necessary to turn them into ones with differentiated functions. As such, I can't imagine why he would be surprised to find mutations could have effects repeated on every segment, and the same applies to leaves, leaflets, and branches. Those effects shouldn't be too big, though, or the chances of it still being viable are going to be really low.


That's all right so far as it goes, but I'm not sure we have because it doesn't feel like you've appreciated what I've been saying. Part of that is even subjective calls like significance should be informed by what we know, and it seems like yours are from lots of love for chaos theory but almost no familiarity with the rest of biology. Even more, though, is that as caffeine says you genuinely keep confusing concepts like "complex design" as Dawkins and other use it, and so your arguments mostly don't concern his. That's not just seeing things a different way to me.

I'm sorry you didn't feel I appreciated what you said. I thought you had some of the strongest points against my argument, and I tried to acknowledge that. I explicitly conceded a number of your points. I'm not sure what else it would take.

Sorry, kitoba, I should have said something like "all that I've been saying". I didn't mean to imply you haven't been responsive, and possibly you've been even more so than it seems at first glance, since it's been hard for me to notice acknowledgments like in this post. But again, I don't think we're at a full understanding because there are some things you're just not touching, most importantly caffeine's point that "complex design" is a term used by people like Dawkins to refer to things that show structure with teleology or teleonomy rather than any type of detail.

As a creative force? Evolution is the thing doing the creating there.

Imagine a space, where every point in the space corresponds to a viable organism, and they're arranged in a high-dimensional fashion so that evolution's explorations are comparatively local. Now consider a bunch of blobs containing all living things.

Evolution is the engine and the driver that spreads these blobs through this space. The mathematics of complexity just makes the terrain they're traversing more interesting.

OK, here's the full argument on the concept of design:

1. How do you know a structure shows teleology --purpose or design? A generic theist --or me, if you prefer --would say that all things have teleology. In fact, that's the chief reason the "Argument From Design" as rendered by Dawkins or Paley isn't a good argument from a theological perspective. The Bible doesn't just say God created life, it says God created all things. From a theological perspective there's no reason to think only living things serve a purpose, all things serve a purpose, we just don't always know what that purpose is. So if Dawkins wants to open a conversation with a theist, he needs to first delineate what he means by design in such a way that a theist can't just reply "yes, all things have that."

This is how we get from "design" to "appearance of design". Appearance is empirical, there is something to measure, so it offers a putatively neutral way to resolve the "yes it does" "no it doesn't" argument over possessing teleology.

2. However Dawkins cannot simply say at the start that "things that appear to display design are things created either by evolution or by human beings," because that assumes his conclusion. He needs to instead give a definition of the appearance of design that
a) presents as having natural boundaries, ie. a definition not custom gerrymandered to suit certain ends
b) doesn't immediate reduce to his conclusion

3. In "Blind Watchmaker" Dawkins tackles the problem with the following hallmarks of appearance of design:

a) heterogeneity - the complex object cannot just be an undifferentiated mass
b) statistical improbability --the complex object must be extremely unlikely to be created by chance
c) proficient, that is to say possessing an ability "highly unlikely to have been acquired by random chance alone".

As it stands, the Mandelbrot Set clearly meets a and b. Unlike simpler fractals, it is entirely heterogeneous. It is statistically improbable that a randomly generated shape would be the Mandelbrot (perhaps even impossible, since the Mandelbrot is infinitely detailed). It also arguably meets c, since it is proficient at delineating the period of points in successive iterations of the generating equation, a quality highly unlikely to arise by chance. (I would personally also include the Set's aesthetic beauty as an "unlikely proficiency," but that's a harder sell due to the difficulty of pinning down the concept of beauty).

Whether or not this represents a true counterexample to Dawkins' attempt to confine "appearance of design" to the realm of living creatures (and the things they create) depends entirely on whether you view the Mandelbrot as something we built or as something that we discovered.

(By the way, thank you for pushing me on this point. I think this is a much stronger, more clear and more succinct representation of my core argument than the one I started with)



If you drive on the moon, it's a dead environment. If you drive on the earth, the terrain is much more interesting. The theological question we're exploring is whether evolution is exploring an enriched terrain, and if so, why?

Well, you already know how I feel about the Mandelbrot set, and how far short it falls in such respects, discovered or not. But this does address the concern about definition, if we really can drop apparent purpose so fast. It seems odd to do so, though.

I mean, I understand entirely that most theists presume everything has a purpose in the end. But again, that's a supposition of the sort Dawkin's argument is not meant to work with, because it's about where simple rules are enough explanation on their own. And I would hope you can see that there is a substantial difference between purpose in, say, a phosphofructokinase and a weather pattern. Let me spell out where I think it is.

Theoretically the hexagon on Saturn could be for something, but it's definitely not apparent. Winds on earth might be easier to invent purposes for. Even here, though, a meteorologist can get by just fine without them. If you want to understand the structure of weather patterns, you have to understand the physics that generate them, and some math about how such rules work, and then you're done. Teleology is a separate philosophical question.

The case is the opposite for something like a calculator or an enzyme. The purpose is the most apparent thing about them, which is to say, both structures are actually much easier to describe in terms of it than how they are actually put together. In fact they're both arbitrary combinations of components put together in an arbitrary way, and the only real way to explain them is what they do. This is so different from the above I would expect anyone to recognize them as having at least a different kind of relationship with their function.

It's odd that there isn't anything like this in the hallmarks you quote. I am guessing it's because in practice, it ends up equivalent to having the exceptionally high levels of information that also characterize these objects, what we've already discussed. But I think for instance it would definitely confirm the Mandelbrot set isn't an example. Does he really not offer the usual definition of design as concerning apparent purpose elsewhere?

If the universe is both {regular enough and has enough moving parts} that the notion of it containing life isn't farcical, then mathematical complexity is available. So no, it's not especially enriched above any other viable world.

Above non-viable worlds, well, we'd need a distribution over hypothetical worlds to see how it stacks up. Got one?

He definitely does offer that definition, but he correctly realizes that it isn't neutral enough to stand on its own.

Let's look at your example of the calculator. "Calculator" as a category is only meaningful in light of already knowing the purpose of a calculator --that's what lets us see that an abacus and a TI-82 are variations on a theme. It's possible that the hexagon on Saturn and the Great Spot on Jupiter could both be examples of some analogous category most easily defined with reference to its function, but that we don't know what that function is.

So Dawkins is forced to offer the hallmarks as empirical ways of discerning design assuming no comprehensive foreknowledge of what the purpose of something is. In essence, what the hallmarks say is if an object is really good at doing something, and it defies probability that it would be randomly good at doing that thing, then we can categorize it as likely designed to do that thing. It's a very sensible definition, but it can lead to some unexpected results.



It begs the original question to limit the consideration to viable worlds. Yet if this is an appeal to the anthropic principle, it would only ease the situation if we actually knew there was a range of universes and furthermore that the availability of mathematical complexity varied over that range. Otherwise we've only sharpened the mystery rather than reduced it.

That's really speculative but I guess a fair argument. I mean, I do think the two are genuinely different in the sense I gave, whether a function is much more or much less explanatory than the generation. But I suppose that's ultimately because the one can readily be explained by simple rules and the other can't, which means it isn't actually a separate question from the one we discussed like I thought.

It's not question-begging since I'm explicitly addressing both prongs.

The point is, basically, the conditions on mathematical complexity being possible are much much weaker than life being possible. It's not like you can be almost there on life being viable but mathematical complexity is the one thing you're missing.

True, but you could arguably be almost there on an explanation of life being viable, but mathematical complexity is the one thing you're missing.

To use your metaphor of the grid, Dawkins' claim is that there's nothing special about the grid, you're just autofilling it with a range of possibilities, and that the evolutionary process of expanding throughout the grid can get you everywhere you want to go (i.e. to every actual living or extinct species) without any special interventions, and in the specified amount of time.

If the grid isn't uniform, however, if it has special properties that play a shaping role in what terrain is actually reached and how fast, then that represents a consequential change to the explanation of the origins of the characteristics of life, even if evolution remains the immediate driver.

If you want to bring this back to an "Intelligent Designer" argument, then you argue that God designed the grid. But you don't have to take that approach in order to yield consequential differences with Dawkins' account. For instance, you could describe the "punctuated equilibrium" theory --that waffle likes, but Dawkins loathes --as a conceptualization under which the grid itself evolves over time, with different areas of terrain opening and closing in response to events such as mass extinctions.

What happens to mathematical oddities, that some mathematican comes up with? If they are usefull in a mathematical model somewhere, they get widely (at least among scientists) known. If they make good examples or counterexamples, they get used in teaching mathematics, but are seldom mentioned elsewhere. They are well preserved for the time, when a mathematical model gets needed, that has thoose properties. If they have have no use for the time being, they are in danger to fall into obscurity even among mathematicans. Or it has some aestethic or similiar property, like the mandelbrot, then it get's remembered for that.

So (admitedly without having checked the history of the cross product in depth) i assume, that the cross product is widely taught, because it has applications.
BTW does your example rely on Newton mechanics, which would mean that the physical multiplication is not actually the same as the mathematical one, only close enough, that the mathematical model is good enough, that you are hard pressed to find an example, where it is not sufficient?

Complex numbers are a logical if counterintuitive derivate of real numbers. You use the +,* and ^ operators and their inverses you natrually get to complex numbers. So while i still can't grasp complex numbers intuitivly, i am not really suprised at them turning up in mathematical models. As in my experience, nature cares nothing for my intuition, but seems to have a tendendcy for simple and logical rules.

Kitoba, I'm pretty sure Dawkins is aware of the way life takes advantage of mathematical complexity. He just sees it as a part of particular solutions, not as a part of the algorithm - and the questions he's trying to answer are about the algorithm.

If you have a paint-by-numbers painting, who is the artist, the person who applies the paint according to the algorithm, or the person who designed the picture and set the numbers? Is "follow the numbers and the picture emerges" a valid explanation of where the picture came from, or merely an effective description of the process that reveals the picture?