Sorry, I was going on someone else's claim that Wallace was a minister. It may have come from his interest in spiritualism, which caused him to believe that natural selection alone could not account for life's existence or our intelligence. According to Wikipedia he's also the one we can blame for the word Darwinism...

There are things that we clearly discover --mountains, continents, etc. There are things we clearly build/create --houses, saws, works of art.

But math doesn't unambiguously go in either category. Do we create math, or just discover it? Are mathematical entities just mental constructs, or do they have some sort of free-standing reality? To me it seems that the expansion of mathematical knowledge is much more like exploring a continent than it is like building a house. We didn't decide that pi would be irrational, we discovered that fact. Any given graph of the Mandelbrot is a purely human construction (aided by computer) but the shape itself is anything but made to order.



I think a lot of the persuasive force of Dawkins' argument comes from the fact that evolution is a single, simple, easy to understand process that, in Dawkins' view, wholly explains the most important question of existence. The fact that this is important to Dawkins is shown by the fact that he has attempted to create Darwinian evolutionary theories of things as disparate as human psychology and the initial creation of the universe. Evolution, to Dawkins, is the single principle that explains everything (important). Once you weaken that claim, you may still have an explanation for biological life, but you no longer necessarily have a persuasive argument against God.

Mathematics was developed as tool. There are some restraints in what it can look like, given for the uses we had in mind for it, but that is true for saws too.

And now that we have it, we can explore, what else it can do, beyond what we had in mind, when we built it. In the case of the saw, other uses are pretty straightforward and how the properties neccessary as wood cutting instrument relate to properties neccessary for other uses, though the use as musical instrument is indeed suprising.

Mathematics is a far more complex and versantile tool and thus we have a lot less an idea, of what we really have created here. But i see a difference in complexity here, not in principle.

This seems utterly wrong to me. I believe it's generally assumed that if some utterly different alien race on the opposite side of the universe possessed intelligence, they would discover largely the same math that we did, even though they might develop different branches of it. To me, that suggests two observers looking at the same "object," not simultaneous invention.

I'm curious though --if I could poll the other readers of this thread, do your intuitions match mine, arcosh's or some other point of view entirely?

I would also assume, that if thoose aliens have similar enough bodies as we do, they would develop similiar saws, shovels, picks ect. (Especially for saws, there is also the assumption, that their enviroment gives them the same tasks, if there would be no wood or similiar substance, a lot of saws will get useless)

Because they will have similiar problems to solve, and the condition "solves this problem" gives restrictions on how the tool looks like.

I also assume they will have largely the same math, because i assume they will want to solve problems similar to thoose, we have solved with inventing out math.

Hmm, I'm having trouble finding good words to crystallize my own feelings on mathematics. The concept, the ideas, the rules, the laws, the thing itself is outside human creation, timelessly existing, unalterable. But the disciple of mathematics that we use to uncover and describe and navigate that thing is a tool of human invention.

Frequencies of sound exist unto themselves, but we can invent instruments to play them and names (e.g. Middle C) to describe them.

Elements exist unto themselves, but we can create a discipline (chemistry) to study them and a periodic table to classify them.

What happens when one moving billiard ball slams into another is always going to happen no matter what names humans give it, but we can name the laws of motion and use variables and theorems to study that collision and predict other ones.

Etc.

To me, the word mathematics refers to one or the other or both. It can be the trowel and brush and pick that dig in the dirt, and it can be the name and the placard and the museum that display the find, but it can also be the bones and fossils themselves under the earth that we didn't put there and couldn't create if we wanted to.

So to speak.

My intuition on math is that we discover the consequences of the axioms we choose. In many cases, like basic algebra giving rise to the Mandelbrot set, those consequences can turn out to be much more intricate than our assumptions had seemed. On the other hand, it really seems to me that saying this is something in need of an explanation is really just a way of refusing to accept explanations.

I mean, most people would agree that when we find genuinely complex artifacts in our universe they're something to account for. Those accounts can be two things: they're the result of other complex things, like a Rembrandt painting, or they're really the result of simple processes, like a tree-like growth of minerals. It seems to you, kitoba, would be taking both as evidence something has been designed, but in that case you're not actually using either as evidence at all, because if we're honest there just aren't alternatives to it you would accept.

For myself, I think "being that creates potential in mathematics" is far less of an answer than "complex result of simple processes", and don't understand why anyone would suppose the former is less in need of explanation than the latter. That math is something out there to discover only reinforces that.

Now that I think about it, the Mandelbrot set seems very strange from an information-theoretic point of view.

Consider. Numbers start out with the natural numbers; a measure of quantity. One, and another, and another, that makes three. Addition is an easy step from there - if I have two hundred men in my army, and you have two hundred men in yours, then if we work together we can deal with that guy over that way who's been lording it over us with his three-hundred-man army.

Multiplication is an easy step from addition; it's just addition repeated. And, as a bonus, it allows us to count up the size of our armies more quickly; we divide the men into squads of ten, then count the squads. And squaring - that's multiplying a number by itself. Or, alternatively, arranging a square of men on the parade ground (ten men per row, ten rows, that's a hundred men in a square, and my army has two squares...)

So that's all fairly quick and easy mathematics. Square roots might take a bit longer to get to, and imaginary numbers are another hill to climb; but these concepts can be derived from squaring pretty straightforwardly, once someone has a chance to stop going to war and think about numbers for long enough.

And then we take a simple equation; x_(n+1)=x_n^2+C, and we see for which value of C it is finite. The equation is simple and straightforward. You can write it on a postage stamp, and have a lot of space to spare. That looks like very little information. The equation is recursive, but aside from that it's no more complicated than the equation for a circle; x^2+y^2=r^2. And a circle's about as simple as shapes go.

...and then you throw some computing power at that equation, and out pops the Mandelbrot set. Bumps, squiggles, corners, fiddly bits all over the place. It looks like something that takes a ton of information to draw up.

But it's not.

The Mandelbrot set, I'd say, was discovered and not constructed. Taking derivatives and integrating, on the other hand was very much a constructed part of mathematics.

Hmm. I would say the Mandelbrot has more to do with that sort of calculus than with squaring numbers. It's not something that just falls out of the basic equation, after all, it's something you get by looking at convergence or divergence of infinite sequences based on that equation.

I'm going to regret getting into this again, but kitoba, I think, is right. I've heard a convincing variant of this argument from Roger Penrose in The Road to Reality, among others. Consider something as esoteric as the rules for cross multiplying vectors. If one multiplies vector A by vector B, the result, vector C, is orthogonal to both with a length equal to the area described by the parallelogram created by displacing vector A to the edge of vector B and displacing vector B to the edge of vector A. When one first comes across the rules, they seem somewhat arbitrary and counterintuitive. How can multiplication result in a vector orthogonal to the original plane?

But one can cross multiply physically with a spun up bicycle wheel and a barstool. Take the spinning bicycle wheel and twist it through 90 degrees and one will start rotating on the stool. Twist the wheel back and one stops.

I can go through an entire list of mathematical oddities and point out real world physical analogs, no matter how oddball the math. Even complex numbers pop up with disturbing frequency. The eigenvector representing the solution to a quantum mechanical problem is an imaginary quantity. One must multiply the wavefunction by its complex conjugate to compute the real probability distribution of the resulting waveform. Thus, there is a physical reality to both imaginary numbers and complex conjugates.

I want to be clear: I'm not a personal fan of the "argument from design" because for me, a simple thing is equally evidence of God as is a complex thing. Either everything that exists is evidence for God, or nothing is. I've focused in on the argument from design solely because it is so central to Dawkins.

I would like to address the "complex result of simple processes" comment, however. The graph of the Mandelbrot Set is the complex result of simple processes. And self-similar fractals such as the fern are themselves are the complex result of simple processes. But the Mandelbrot Set itself is quite different. It may seem as though the Set itself must somehow be generated through the millions of calculations we use to view it, but the more we examine that assumption the less it seems to hold. Unlike the ruleset for a --for instance --branching fractal, where the final structure is clearly related to the generating code, there's no discernible relationship between the rule that defines the Mandelbrot and the shape that its graph takes.

Also, there is the fact that the Mandelbrot is only there if we look in the right place. If we look too close to the origin of the graph, we see nothing of interest. And if we look too far from the origin of the graph we see nothing of interest. The only interesting area is somewhere around the circle of radius two around the origin. If it was the calculations themselves generating the complexity, why wouldn't we see complexity no matter where we looked?

My point is this: Dawkins didn't simply assume life had no designer, but instead considers one as a possible source for complex design, with its emergence from simple rules as another. That puts him in a position where it is meaningful to ask about the two alternatives. Treating emergent complexity as something to have a creator as well, on the other hand, is simply presupposing one and so doesn't touch on what he is looking at at all. We all know you can assume a creator; the question is more if there is anything you can't explain without one.


I'm surprised anyone could say such a thing after studying it in any depth.

The Mandelbrot Set is made out of pieces corresponding to different period attractors. Taken together it's complex, and higher periods certainly become hard to follow. But for instance it's not too hard to see how the main heart-shaped piece comes from the equation, and so why there is no detail close to the origin. It's even more obvious that z(n+1) = z(n)^2 + c won't converge when |c| is large, and so there will be no detail far away.

I've agreed I think the set is a discovered consequence of axioms, and things about it like its fractional dimension are surprising. But it's not a mystery that the set is simple in regions where the defining iteration is really obvious, unless your expectations come without any regard to what it actually represents. And even though the little bubbles with littler bubbles wouldn't be something I could anticipate, I can still appreciate that they relate to different iterative periods.

Edit: in fact, I feel like it would be helpful to spell this out, because as I said the Mandelbrot set does not just fall out of z^2+c. It is based on iterating that function an infinite number of times and seeing where points settle. Some, like the ones near the origin, settle on giving 1 value again and again. Some, like the ones near -1, settle on giving 2 values in alternation. And you can have patterns of 3 repeating values, or 4, or 5, and so on forever.

So when we look at all these together, we have basic shape from points that settle on 1 repeating value, and then a second level of detail where they settle on 2 repeating values, and then a third where they settle on 3 repeating values. And so on forever - since we are allowing an infinite number of iterations, we end up with infinite levels of details. It is shaped by our simple equation, but is still getting built in a way very similar to the fractal tree.

Dawkins frames the situation as "God or Darwin." He even puts it that bluntly in one of his essays:



Now I happen to think that's a false choice. But accepting it on Dawkins' terms, it seems to me that he overreaches science in his claims for Darwin just far enough that it's possible to say to him "Darwinism doesn't meet all the claims you made for it. You're the one who set up the binary opposition. If Darwinism had met all your claims, you would have claimed that as sufficient evidence for disbelieving in God. Since it doesn't, how are you able to avoid the conclusion that you no longer have sufficient evidence for disbelieving in God?"

I don't seriously think this argument could convert Dawkins, but it might force him to dial back his rhetoric.



I'll admit you do have a strong point contra my argument in terms of the fact that viewed in this light, the Mandelbrot is not irreducible complexity, but reducible complexity. You can build it up, period by period.

From another point of view, however, you're missing the forest for the trees. The Mandelbrot has a shape that illuminates the period of various values related to the initial equation, true. It doesn't come out of nowhere. But --to adapt Dawkins' comment about people who don't appreciate the complexity of biological life --if you aren't struck by awe and wonder when contemplating the Mandelbrot, then there's not much else I can say to you. There's just no comprehensible reason it should be that beautiful.

But from what I can tell your arguments are essentially based on the supposition that all roads lead to God - inherent simplicity would be as good evidence as inherent complexity. Of course any dilemma is going to run into contradictions if you combine it with a presupposition that there can only be one answer. But if you only consider it in those terms, you can't show anything about how valid it might be from a perspective where there could be different answers, which is obviously where it is meant to apply.


I think you simply might not care to find reasons, because there are certainly lots of things about it that make it wonderful, and it is no surprise to me that such objects in math should be so. But you know that's not the matter I was considering, because you weren't talking about how beautiful it was.

On the contrary, you were saying how it's quite different from the fractal tree, because the structure has no discernable relationship to its generation, and how if the complexity were from the calculations you would expect it everywhere, maybe to imply it comes from somewhere else. I'm pointing out that this is all complete BS if you actually care to understand what the Mandelbrot set is and how it's built. If your reply to that is simply to change what the point was and then accuse me of missing it, well, I certainly can think harder before discussing such things in the future.

He's not saying here that Darwinism is a sufficient reason for believing in God. I know you say you're not that interested in design, but if not you're attacking the wrong person, because that's what Dawkins is talking about, in all that you've quoted of him. He is saying he can see three explanations why a male elephant seal is big enough to defend it's beaches - either because the biggest elephant seal males are the ones who successfully defended their beaches in the past and passed on their genes; because fighting elephant seals build up their muscles and brawn and pass this on to their offspring; or because a creator made them as big as they need to be. The Lamarckist answer is demonstrably false and insufficient, so if it turned out that natural selection didn't work either - only God is left. Maybe there is some other potentially satisfactory answer we've missed, but to fair to Dawkins, I don't think anyone's thought of it yet.

This is not an argument meant to apply outside anything other than the apparent design in living things. Arguments for God's existence which do not rest on the apparent design in living things are irrelevant to and unaffected by this argument. He's simply making the point that a naturalistic process which can account for the apparent design of organisms does away with what he considers the most convincing argument for a creator.

If you do want to talk about things other than apparent design, then your argument has little to do with Dawkins.