Postscript in 1985 Springer Verlag book:

Analysis of a Finite Element Method: PDE/PROTRAN

Granville Sewell

Mathematics Dept.

University of Texas El Paso

As I begin my 12th year of work on TWODEPEP (now PDE/PROTRAN ), I am intrigued by the analogy between the 11 year evolution of this computer code and the multi-billion year history of the genetic code of life, which contains a blueprint for a species encoded into billions of bits of information. Like the code of life, TWODEPEP began with primitive features, being capable of solving only a single linear elliptic equation in polygonal regions, with simple boundary conditions. It passed through many useful stages as it adapted to non-linear and time dependent problems, systems of PDEs, eigenvalue problems, and as it evolved cubic and quartic elements and isoparametric elements for curved boundaries. It grew a preprocessor and a graphical output package, and out-of-core frontal and conjugate gradient methods were added to solve the linear systems.

Each of these changes represented major evolutionary steps--new orders, classes or phyla, if you will. The conjugate gradient method, in turn, also passed through several less major variations as the basic method was modified to precondition the matrix, to handle nonsymmetric systems, and as stopping criteria were altered, etc. Some of these variations might be considered new families, some new genera, and some only special changes.

I see one flaw in the analogy, however. While I am told that the DNA code was designed by a natural process capable of recognizing improvements but incapable of planning beyond the next random mutation, I find it difficult to believe that TWODEPEP could have been designed by a programmer incapable of thinking ahead more than a few characters at a time.

But perhaps, it might be suggested, a programmer capable of making only random changes, but quite skilled at recognizing improvements could, given 4.5 billion years to work on it, evolve such a program. A few simple calculations would convince him that this programmer would have to rely on very tiny improvements. For example, if he could produce a billion random "mutations" per second (or, for a better analogy, suppose a billion programmers could produce one "mutation" per second each), he could not, statistically, hope to produce any predetermined 20 character improvement during this time period. Could such a programmer, with no programming or mathematical skills other than the ability to recognize and select out very small improvements through testing, design a sophisticated finite element program?

The Darwinist would presumably say, yes, but to anyone who has had minimal programming experience such an idea is preposterous. The major changes to TWODEPEP, such as the addition of a new linear equation solver or new element, required the addition or modification of hundreds of lines of code before the new feature was functional. None of the changes made during this period were of any use whatever until all were in place.

Even the smallest modifications to that new feature, once it was functional, required adding several lines, no one of which made any sense, or provided any "selective advantage", when added by itself.

Consider, by way of analogy, the airtight trap of the carnivorous bladderwort plant, which has a double sealed, valve-like door which is opened when a trigger hair is activated, causing the victim to be sucked into the vacuum of the trap (described by R.F.Daubenmire in "Plants and Environment," John Wiley and Sons, N.Y. 1947). It is difficult to see what selective advantage this trap provided until it was almost perfect.

This, then, is the fallacy of Darwin's explanation for the causes of evolution--the idea that major (complex) improvements can be broken down into many minor improvements. French biologist Jean Rostand, in "A Biologist's View" (William Heinemann Ltd., London, 1956) recognized this:

"It does not seem strictly impossible that mutations should have introduced into the animal kingdom the differences which exist between one species and the next...hence it is very tempting to lay also at their door the differences between classes, families and orders, and, in short, the whole of evolution. But it is obvious that such an extrapolation involves the gratuitous attribution to the mutations of the past of a magnitude and power of innovation much greater than is shown by those of today."
The famous "problem of novelties" is another formulation of the objection raised here. How can natural selection cause new organs to arise and guide their development through the initial stages during which they present no selective advantage, the argument goes. The Darwinist is forced to argue that there are no useless stages. He believes that new organs and new systems of organs arose gradually, through many small improvements. But this is like saying that TWODEPEP could have made the transition from a single PDE to systems of PDEs through many five or six character improvements, each of which made it work slightly better on systems.

It is interesting to note that this belief is not supported even by the fossil evidence. Harvard paleontologist George Gaylord Simpson, for example, in "The History of Life," Volume II of "Evolution after Darwin," (University of Chicago Press, 1960) points out:

"It is a feature of the known fossil record that most taxa appear abruptly. They are not, as a rule, led up to by a sequence of almost imperceptibly changing forerunners such as Darwin believed should be usual in evolution...This phenomenon becomes more universal and more intense as the hierarchy of categories is ascended. Gaps among known species are sporadic and often small. Gaps among known orders, classes and phyla are systematic and almost always large. These peculiarities of the record pose one of the most important theoretical problems in the whole history of life: Is the sudden appearance of higher categories a phenomenon of evolution or of the record only, due to sampling bias and other inadequacies?"
Another way of describing this same structure is expressed in a recent Life magazine article (Francis Hitching, "Was Darwin Wrong on Evolution?", April 1982, which concludes that "natural selection has been tested and found wanting") which focuses on the "curious consistency" of the fossil gaps:

"These are not negligible gaps. They are periods, in all the major evolutionary transitions, when immense physiological changes had to take place."
Unless we are willing to believe that useless, "developing" organs (and insect traps which could almost catch insects) abounded in the past, we should have expected the fossil structure outlined above, with large gaps between the higher categories, where new organs and new systems of organs appeared.

Nevertheless, despite the fact that the structure of the fossil record is the only argument against Darwin which has received much attention lately, this is not the real issue. The "problem of novelties" correctly states the real argument, but too weakly. Consider, for example, the human eye, with an aperture whose size varies automatically according to the light intensity, controlled by reflex signals from the brain; with a lens whose curvature varies automatically according to the distance to the object in view; and with a retina which receives the picture on color sensitive cells and transmits it, complete with coded intensity and frequency information, through the optic nerve to the brain. The brain superimposes the pictures from the two eyes and stores this 3D picture somehow in memory, and it will be able to search for and recall this image later and use it to recognize an older but familiar face in a different picture. Like TWODEPEP, the eye has passed through various useful stages in its development, but it contains a large number of features which could not reach usefulness in a single random mutation and which provided no selective advantage until useful (e.g. the nerves and arteries which service it), and many groups of features which are useless individually. The Darwinist may bridge the gaps between taxa with a long chain of tiny improvements in his imagination, but the analogy with software puts his ideas into perspective. The idea that all the magnificent species in the living world, or the human brain with its human consciousness, could have arisen from simple organic molecules guided by a natural process unable to plan beyond the next tiny mutation, is entirely comparable to the idea that a programmer incapable of thinking ahead more than a few characters at a time could, given a lot of time, design any sophisticated computer program.

I suggest that, with Jean Rostand, "we must have the courage to recognize that we know nothing of the mechanism" of evolution.


Related links:

PDE2D (current version of PDE/PROTRAN) web page

A Mathematician's View of Evolution , The Mathematical Intelligencer, 2000.