Registered: Sep 2005
Posts: 2

The Copycat Press:

"Take seashells. One of the most esteemed documents of modern paleontology is Stephen Jay Gould's doctoral thesis on shells. According to Gould, the fact that there are thousands of potential shell shapes in the world, but only a half dozen actual shell forms, is evidence of natural selection. Not so, says Wolfram. He's discovered a mathematical error in Gould's argument, and that, in fact, there are only six possible shell shapes, and all of them exist in the world.
Michael S. Malone, Forbes ASAP, 11.27.00"

I couldn't find a copy of Gould's doctoral thesis anywhere online.
It would probably be a lot of trouble to find it, maybe have to contact Harvard directly. It's not clear if Wolfram read his original thesis (as implied) or merely some other more recent work of his (more likely given how hard it is to find the dissertation that is from 1967 and out of print). Nothing in wolfram's book seems to mention this shell thing with Gould. So why is this quote found in nearly every single article about Wolfram?

Also, It is not true as stated in the quote that there are a half-dozen actual shell forms, actually there are a huge number. What Wolfram's book talks about is that you can model most of the basic shape features with only 6 variables. That is a much weaker statement. Also, Wolfram is not like the lone investigator who discovered shell modeling. There are lots of other people modeling them with only a few variables and writing papers on it too. His online bibliography does not mention Gould at all. -

http://www.wolframscience.com/refer...bliography.html

Why is this story so ubiquitous in the press, and has no contact with reality. What is the "mathematical error" that Gould is reported to have made?

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Registered: Aug 2003
Posts: 712

First, the section of the NKS book that discusses this subject starts on page 414 and runs for 4 pages or so. There are also relevant notes, including a history note explaining the relation of the model in the book to previous work, on pages 1008 and 1009. As the history note explains, Wolfram's parameterization of shell shapes is an extension of a model proposed by Henry Moseley in the 19th century. Page 1008 includes the actual model in an earlier note, the one entitled "shell model". That note also leaves a role for selection effects, though minor ones, with most of the parameter space occupied.

Second, the "online bibliography" is not a list of works consulted while working on NKS. The source for such cites is the index, and the history notes in the back of the book. The forward also includes a list of people consulted in the course of the work. The online bibliography is a different creature entirely and comes after the book in time. It contains works that cite NKS, not the other way around. The relevant sourcing for the shell model is the index entry Moseley, Henry and the note on page 1008 cited above. Gould's work was not the basis of Wolfram's shell shape parameterization model.

Third, Wolfram certainly knew of Gould's work including his dissertation. He also knew Gould personally. The idea that natural selection had pruned a much larger possibility space to arrive at the shells shapes we see, was well known to him. He believes that impression is created by overly complicated parameterizations that allow too many possibilities (discussed later in the history note), and that when a simpler one is used, most of the parameter space is occupied, leaving only a minimal role to selection (weeding out shapes that leave too little space for the organism, e.g.).

As he says on page 417, top, "therefore natural selection cannot reasonably be considered the source of the elaborate forms we see", in this case. He traces said elaborate form to a simple growth process instead, their variety traced to different values of a small number of parameters. (Of course he still thinks natural selection is operating. But not that it is the primary source of the variety of form seen. Instead different shell types have different fixed values in the growth-parameter space and their various shapes automatically result).

The quote you start from makes a serious error, however, in putting a claim that there are only 6 shapes in Wolfram's mouth. He said no such thing, shows populated parameter spaces, and notes in a caption on page 416 that this is the space for just 2 of his 5 parameters and that one could make similar plots for other parameter pairs. The whole space is five dimensional and certainly has more than 6 entries.

A fine question by the way. I hope this helps.

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Registered: Sep 2005
Posts: 2

Thanks for the thoughtful informative reply, Jason. Journalists are successful because the are good and enterntaining writers, not because they are careful with the facts.

It's true that Natural Selection isn't single-handedly responsible for everything. Obviously it doesn't create the raw material of changes and mutations. There is some drift and historical accident component. Certainly NS doesn't cause atoms to exist or have the chemistry that they do. NS doesn't make the circumference be pi * diam. OK. Who said that it did? I would grant that some people have not appreciated enough how many of those things are physics, chemistry, geometry, etc. rather than being due to complex optimizations by NS. Of course Mr. Wolfram is not the first to point this out.

On the other hand, just observing that you can find some species somewhere in each "fillable" niche of param space doesn't explain why one species occupies the one niche that it does. If one species occupied all the niches equally then one might argue that any shell shape will do. But if not, then the most likely explanations are NS and historical accident/drift. I grant you that if there are only a few important params involved then NS has a much easier job of it.

Wolfram's contribution is reminding us that maybe the rules aren't that complex after all, and helping us model and think about the possibilities.

Does anybody have cellular automata or the like that act like growing shells?

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Registered: Aug 2003
Posts: 712

Yes, the basic question is about how much can be done by the "raw material of changes and mutation" step alone. I've always appreciated about Gould the fact that he was quite open minded about causes other than natural selection operating within evolution and having important effects. Serendipity being his favorite.

What Wolfram and no doubt others in complexity work have noticed is how many similarities there are between complex forms seen in non-biological systems and in biological ones. There is no serious question of natural selection operating in the non-biological cases - though one can occasionally export some of its concepts, as analogies. If a complex form can arise in weathering or fracture there is no apriori reason to reach for a NS explanation for a similar form seen in a biological system. Common formal factors can be operating in both and give rise to similar outcomes, and since NS is not essential to them in the former there is no reason to assume it is essential in the latter.

What is involved is explaining not life or every phenomenon within it, but specifically complexity, and to a lesser extent, variety. Many biologists have tried to explain both through NS. It is useful to analyse the form of that argument. One sees that it formally imports diversity from outside the biological subsystem, in the form of varied environments acting as separated "attractors". NS explains "homing" on something. That the something homed in on is varied is imported from outside. Coevolution ecosystem studies might get a little more "endogenous" about variety growth. But it is a lot of hacky work to explain variation from NS. It always needs variation in input e.g. from mutation, and also in attractors in "fitness space". If random in and varied out, then NS can sit between and laboriously try to connect one to the other.

But this is not necessary to evolutionary theory. Darwin's original formula was descent with variation as the input to NS. How much variety can be explained by the prior step? Now, from our intuition of complicated systems, prior to our mastering the idea of computation, we might have thought just about any change would "crash", and a lot of special aiming would be needed to "tune" any random input to any noticable variation in output. But when we look at simple enough formal systems this is not what we see.

Instead, point changes in simple programs produce a characteristic diversity of outputs, some simple, many repetitive, many mixed and seemingly random, a few involving semi-periodic persistent structures. So, if some simple program already exists that produces form X, and we imagine untuned random changes thrown at it, we expect a "spray" of forms to come out. If a simple growth process with a few parameters makes a given form, we expect to find "neighboring" forms in the resulting parameter space, before any tuning or any niche-attracting.

Niche-attracting might well then "lock on" to some of the resulting variation and favor this one rather than that, sure. But it is not the underlying cause of the variety, as such. Nor do we formally require some tower of tuned optimizations to arrive at some complex output form, as best fitted for some narrow set of constraints. It is in fact rather difficult to satisfy complicated constraints by successive improvement. But it is quite easy to get complexity from simple programs.

Overall, then, Wolfram thinks he has found some of the principles of an underlying diversity and complexity "motor" that can operate in biology, and feed its results up to NS. NS becomes a winnower and optimizer, expected to prune an already diverse tree of forms. It no longer has to do the job of creating all complexity or all diversity from scratch, so to speak. If you look at the fossil record, this fits rather better in my opinion. Invent a body type "switch" at some point, and you expect a wide variety of body types all at once, followed by slow optimizing pruning. The diversity motor can be algorithmic, the expected result of "poking" an existing body of form-making code.

There is some implicit stuff about the scale of typical effects here. When a progam is very complicated, we tend to expect the impact of a small change to be small, or to "crash" it completely. But when a program is simple, a small change typically "runs" just fine, while producing an output that can be dramatically different from the previous version, or similar to it but different in the details. Essentially, when there aren't that many bits in the formal set up, change to any one of them can have a large effect. While, if that set up is simple enough, the basic surrounding facts about it (what it operates on, the range it can vary things within) may be set by factors external to that bit of the "code". So you expect bounded variation - large changes in a definite aspect, other characteristics (set by other programs, presumably) remaining unchanged.

For those with Mathematica, you can see the typical behavior of all the ECAs that differ from a given ECA at one location in their rule table, using this function -

nearby[rl_] :=  With[{sample = 
  Flatten[{#, # + Reverse[2^Range[0, 7]] 
  (IntegerDigits[#, 2, 8] /. {1 -> -1, 0 -> 1})}] &@rl,
    initial = Table[Random[Integer],{50}]},
  ArrayPlot[CellularAutomaton[#, initial, 30], 
PlotLabel -> #] & /@ sample] 

Just ask for e.g. nearby[30] to see plots of rule 30 and those of all the rules that differ by one bit in their rule table, from the same initial condition. nearby[45] for the "neighbors" of rule 45 in "mutation space", etc. For those without, the upshot is qualitatively distinct behaviors lie quite close to one another in rule-mutation space, when the form of the rule is simple enough.

The place where this idea has to prove itself is in the micro-study of growth, finding the genetic "switches" involved and seeing how they operate to create this body plan or that pigmentation pattern. Some problems there are easier than others. Shell pigmentation is darn close to a CA in the seen outputs and the known process, as discussed in the book. Leaf shape from L systems and the like is another bit of low hanging fruit. The book has some results there, they can be improved (as I've seen from others working from it, since) and mated more thoroughly to mutation studies and the like.

Another place where people have used CA models in modeling growth is things like models of cell differentiation forming a limb, typically CA style models mixed with growth promoter gradients or diffusion points. Just getting a good formal module that produces the observed effect is the first step. People work on that at e.g. the Biocomplexity Institute at Indiana University, among other places -

http://biocomplexity.indiana.edu/

I hope this helps.

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