A Mathematician's View of Evolution

In 1996, Lehigh University biochemist Michael Behe published a book entitled "Darwin's Black Box" [Free Press], whose central theme is that every living cell is loaded with features and biochemical processes which are "irreducibly complex"--that is, they require the existence of numerous complex components, each essential for function. Thus, these features and processes cannot be explained by gradual Darwinian improvements, because until all the components are in place, these assemblages are completely useless, and thus provide no selective advantage. Behe spends over 100 pages describing some of these irreducibly complex biochemical systems in detail, then summarizes the results of an exhaustive search of the biochemical literature for Darwinian explanations. He concludes that while biochemistry texts often pay lip-service to the idea that natural selection of random mutations can explain everything in the cell, such claims are pure "bluster", because "there is no publication in the scientific literature that describes how molecular evolution of any real, complex, biochemical system either did occur or even might have occurred."

When Dr. Behe was at the University of Texas El Paso in May of 1997 to give an invited talk, I told him that I thought he would find more support for his ideas in mathematics, physics and computer science departments than in his own field. I know a good many mathematicians, physicists and computer scientists who, like me, are appalled that Darwin's explanation for the development of life is so widely accepted in the life sciences. Few of them ever speak out or write on this issue, however--perhaps because they feel the question is simply out of their domain. However, I believe there are two central arguments against Darwinism, and both seem to be most readily appreciated by those in the more mathematical sciences.

1. The cornerstone of Darwinism is the idea that major (complex) improvements can be built up through many minor improvements; that the new organs and new systems of organs which gave rise to new orders, classes and phyla developed gradually, through many very minor improvements. We should first note that the fossil record does not support this idea, for example, Harvard paleontologist George Gaylord Simpson ["The History of Life," in Volume I of "Evolution after Darwin," University of Chicago Press, 1960] writes:

"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?"

An April, 1982, Life Magazine article (excerpted from Francis Hitching's book, "The Neck of the Giraffe: Where Darwin Went Wrong") contains the following report:

"When you look for links between major groups of animals, they simply aren't there...'Instead of finding the gradual unfolding of life', writes David M. Raup, a curator of Chicago's Field Museum of Natural History, 'what geologists of Darwin's time and geologists of the present day actually find is a highly uneven or jerky record; that is, species appear in the fossil sequence very suddenly, show little or no change during their existence, then abruptly disappear.' These are not negligible gaps. They are periods, in all the major evolutionary transitions, when immense physiological changes had to take place."

Even among biologists, the idea that new organs, and thus higher categories, could develop gradually through tiny improvements has often been challenged. How could the "survival of the fittest" guide the development of new organs through their initial useless stages, during which they obviously present no selective advantage? (This is often referred to as the "problem of novelties".) Or guide the development of entire new systems, such as nervous, circulatory, digestive, respiratory and reproductive systems, which would require the simultaneous development of several new interdependent organs, none of which is useful, or provides any selective advantage, by itself? French biologist Jean Rostand, for example, wrote ["A Biologist's View," Wm. Heinemann Ltd. 1956]:

"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."

Behe's book is primarily a challenge to this cornerstone of Darwinism at the microscopic level. Although we may not be familiar with the complex biochemical systems discussed in this book, I believe mathematicians are well qualified to appreciate the general ideas involved. And although an analogy is only an analogy, perhaps the best way to understand Behe's argument is by comparing the development of the genetic code of life with the development of a computer program. Suppose an engineer attempts to design a structural analysis computer program, writing it in a machine language that is totally unknown to him. He simply types out random characters at his keyboard, and periodically runs tests on the program to recognize and select out chance improvements when they occur. The improvements are permanently incorporated into the program while the other changes are discarded. If our engineer continues this process of random changes and testing for a long enough time, could he eventually develop a sophisticated structural analysis program? (Of course, when intelligent humans decide what constitutes an "improvement", this is really artificial selection, so the analogy is far too generous.)

If a billion engineers were to type at the rate of one random character per second, there is virtually no chance that any one of them would, given the 4.5 billion year age of the Earth to work on it, accidentally duplicate a given 20-character improvement. Thus our engineer cannot count on making any major improvements through chance alone. But could he not perhaps make progress through the accumulation of very small improvements? The Darwinist would presumably say, yes, but to anyone who has had minimal programming experience this idea is equally implausible.

Major improvements to a computer program often require the addition or modification of hundreds of interdependent lines, no one of which makes any sense, or results in any improvement, when added by itself. Even the smallest improvements usually require adding several new lines. It is conceivable that a programmer unable to look ahead more than 5 or 6 characters at a time might be able to make some very slight improvements to a computer program, but it is inconceivable that he could design anything sophisticated without the ability to plan far ahead and to guide his changes toward that plan.

If archeologists of some future society were to unearth the many versions of my PDE solver, PDE2D , which I have produced over the last 20 years, they would certainly note a steady increase in complexity over time, and they would see many obvious similarities between each new version and the previous one. In the beginning it was only able to solve a single linear, steady-state, 2D equation in a polygonal region. Since then, PDE2D has developed many new abilities: it now solves nonlinear problems, time-dependent and eigenvalue problems, systems of simultaneous equations, and it now handles general curved 2D regions.

Over the years, many new types of graphical output capabilities have evolved, and in 1991 it developed an interactive preprocessor, and more recently PDE2D has adapted to 3D and 1D problems. An archeologist attempting to explain the evolution of this computer program in terms of many tiny improvements might be puzzled to find that each of these major advances (new classes or phyla??) appeared suddenly in new versions; for example, the ability to solve 3D problems first appeared in version 4.0. Less major improvements (new families or orders??) appeared suddenly in new subversions, for example, the ability to solve 3D problems with periodic boundary conditions first appeared in version 5.6. In fact, the record of PDE2D's development would be similar to the fossil record, with large gaps where major new features appeared, and smaller gaps where minor ones appeared. That is because the multitude of intermediate programs between versions or subversions which the archeologist might expect to find never existed, because-- for example--none of the changes I made for edition 4.0 made any sense, or provided PDE2D any advantage whatever in solving 3D problems (or anything else) until hundreds of lines had been added.

Whether at the microscopic or macroscopic level, major, complex, evolutionary advances, involving new features (as opposed to minor, quantitative changes such as an increase in the length of the giraffe's neck*, or the darkening of the wings of a moth, which clearly could occur gradually) also involve the addition of many interrelated and interdependent pieces. These complex advances, like those made to computer programs, are not always "irreducibly complex"--sometimes there are intermediate useful stages. But just as major improvements to a computer program cannot be made 5 or 6 characters at a time, certainly no major evolutionary advance is reducible to a chain of tiny improvements, each small enough to be bridged by a single random mutation.

2. The other point is very simple, but also seems to be appreciated only by more mathematically-oriented people. It is that to attribute the development of life on Earth to natural selection is to assign to it--and to it alone, of all known natural "forces"--the ability to violate the second law of thermodynamics and to cause order to arise from disorder. It is often argued that since the Earth is not a closed system--it receives energy from the Sun, for example-- the second law is not applicable in this case. It is true that order can increase locally, if the local increase is compensated by a decrease elsewhere, ie, an open system can be taken to a less probable state by importing order from outside. For example, we could transport a truckload of encyclopedias and computers to the moon, thereby increasing the order on the moon, without violating the second law. But the second law of thermodynamics--at least the underlying principle behind this law--simply says that natural forces do not cause extremely improbable things to happen**, and it is absurd to argue that because the Earth receives energy from the Sun, this principle was not violated here when the original rearrangement of atoms into encyclopedias and computers occurred.

The biologist studies the details of natural history, and when he looks at the similarities between two species of butterflies, he is understandably reluctant to attribute the small differences to the supernatural. But the mathematician or physicist is likely to take the broader view. I imagine visiting the Earth when it was young and returning now to find highways with automobiles on them, airports with jet airplanes, and tall buildings full of complicated equipment, such as televisions, telephones and computers. Then I imagine the construction of a gigantic computer model which starts with the initial conditions on Earth 4 billion years ago and tries to simulate the effects that the four known forces of physics (the gravitational, electromagnetic and strong and weak nuclear forces) would have on every atom and every subatomic particle on our planet (perhaps using random number generators to model quantum uncertainties!). If we ran such a simulation out to the present day, would it predict that the basic forces of Nature would reorganize the basic particles of Nature into libraries full of encyclopedias, science texts and novels, nuclear power plants, aircraft carriers with supersonic jets parked on deck, and computers connected to laser printers, CRTs and keyboards? If we graphically displayed the positions of the atoms at the end of the simulation, would we find that cars and trucks had formed, or that supercomputers had arisen? Certainly we would not, and I do not believe that adding sunlight to the model would help much. Clearly something extremely improbable has happened here on our planet, with the origin and development of life, and especially with the development of human consciousness and creativity.

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*Ironically, W.E.Loennig's article "The Evolution of the Long-necked Giraffe," has since convinced me that even this feature could not, and did not, arise gradually.

**An unfortunate choice of words, for which I was severely chastised. I should have said, the underlying principle behind the second law is that natural forces do not do macroscopically describable things which are extremely improbable from the microscopic point of view. See "A Second Look at the Second Law," for a more thorough treatment of this point.

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Granville Sewell completed his PhD at Purdue University. He has subsequently been employed by (in chronological order) Universidad Simon Bolivar (Caracas), Oak Ridge National Laboratory, Purdue University, IMSL (Houston), The University of Texas Center for High Performance Computing (Austin), and the University of Texas El Paso; he spent Fall 1999 at Universidad Nacional de Tucuman in Argentina on a Fulbright grant. He has written three books on numerical analysis.

Why do you assume it was pejorative?

I just want to understand where those with whom I am having a discussion are coming from.

If it's a debate about science -- especially one in which the stakes are not really that high -- I will assume it is OK to challenge thier view without expecting to see their feelings hurt.

In a matter of religion, one would be expected to be more careful.

If someone is claiming to be debating science but would be more accurately described as defending a religious view, then that would present a problem wouldn't it?

I am not misrepresenting gradualism or Darwin's view or the fossil record. When you imply, however, that saltation is gradualism you are most certainly misrepresenting it.

Darwin considered all the possible implications of the fossil record, so when you omit this fact, you misrepresent his position. The fact that his final position -- made without benefit of the subsequent 150 years of evidence -- was wrong, is no more damning than Copernicus' inclusion of epicycles in the orbits of planets. Science progresses.

Saltation happens in plants. That is a fact. Nearly everything we eat is the result of a polyploidy event in recent history. Nearly everything we eat is either a grain that cannot procreate without cultivation, or an animal that feeds on these grains. It is not inconsistent with gradualism; neither is it the same thing as gradualism.

In general, every individual is the same species as its parents, with variations. Plants have modes of evolution that are extremely rare in animals.

Darwin considered all the possible implications of the fossil record, so when you omit this fact, you misrepresent his position.

He considered all the possible implications and came to the wrong conclusion and pointing that out in defense of someone else is not a misrepresentation.

Or, if you will, Darwin understood that the fossil record did not back his view but had faith that it would eventually.

Regardless, he was a gradualist and the fossil record does not back up gradualism.

I just want to understand where those with whom I am having a discussion are coming from.

You demonstrate no evidence of this.

When you say evolution is a religion, you have a motive. It is not a random string of words. Do you routinely call physics a religion? Chemistry? Geology? Astronomy? Biology?

I'm guessing you call evolution a religion because it implies that evolution is counter to evidence, or not based on evidence. In other words, a fable or fairy tale.

Otherwis, I don't see why you and countless other anti-evolutionists insist on calling evolution a religion. If I'm wrong, tell we what you do imply when you call evolution a religion.

Regardless, he was a gradualist and the fossil record does not back up gradualism.

Define gradualism.

Be specific. Give me a specific "gap" in the fossil record that is inconsistent with evolution as currently understood. Describe the length of the gap. Name the two species that comprise the bookends of the gap.

V-A & s: Some "Catholic" parochial schools lazily hitchhike along with public skewel incompetence by requiring teachers to be certified Horace Mannite secular humanist robots. Some of the more textured reformed Christians around here can confirm that I am no Protestant but I admire the Protestant Christian Academies for teaching their children the truth rather than teaching them darwinian fantasies and lies. A growing number of actually Catholic privately owned (non-diocesan) schools are doing the same. That is where my kids attend and my wife teaches. Liberal agnostic feminazi school administratrices (collectively known as Sister Gloria Pantsuit) need not seek to influence our schools. Likewise Brother Steve Socialist, Father Lance Lavender or Mr. Computerized Photo Swapper. Science will be taught by those who can distinguish darwinianism from the Truth.

Whoops, I had said I was outta here but, in the words of Don Michael Corleone, onetime Dartmouth math professor until his careers as war hero and,ummmm, underground business whiz, "They keep pulling me back in!!!!" (Godfather III)

Did you read the post to which you are responding?

Rarely do I see the hostility in discussions about chemistry, geology etc.

Even with YECers concerning the age of the earth, which I suppose shows a confidence held by geologists and physicists that evolutionists don't have.

These posts?

Vade, this is a religion with you. Not with me. 384 posted on 09/23/2006 9:52:02 AM EDT by Tribune7

The mistake your are making is equating your worldview with "science". It's not. It's a faith based on emotion not objective reason. 410 posted on 09/23/2006 1:16:26 PM EDT by Tribune7

Rarely do I see the hostility in discussions about chemistry, geology etc.

Then you are blessed with blinders. The freeper anti-evolutionists include a geocentrist. Not one anti-evolutionist has called him out, even though he is a frequent poster.

can't even count the number of freepers who do not accept physics, or which have a physics that includes variable gravity, variable speeds of light, and so forth.

Rarely do I see the hostility in discussions about chemistry, geology etc.

That's because all the really good hostility is saved for evolutionists. We are routinely called "evol doers," "evilutionists," "evofreaks," and a few other choice terms, and often compared to Hitler, Stalin and who know what other scum. We tend to resent it and fight back.

It is your side that is starting this fight. Just look back on some of the threads.

Gradualism is the belief that evolution occurred via the accumulation of slight modifications over generations

How about I just quote Stephen Gould? You accept him as an authority right?

All paleontologists know that the fossil record contains precious little in the way of intermediate forms; transitions between major groups are characteristically abrupt. Gradualists usually extract themselves from this dilemma by invoking the extreme imperfection of the fossil record--if only one step in a thousand survives as a fossil, geology will not record continuous change.

So when someone points out that Stalin's authoritzed 1940 biography published in Moscow has him crediting Darwin with giving him the confidence to be an atheist, or that history professor has written a book showing Darwin's influence on the zeitgeist of Nazi Germany, you feel justified in attacking?

These attacks are on the theory of evolution by stretching the truth and attempting to link either evolutionists or the theory to some of history's nastiest folks.

It is the equivalent of tying your beliefs into Jim Jones in order to discredit you and your beliefs.

Read the threads.

Out of context quotes are a form of lying, and lying, we know, is a sin. It certainly alienates one from any honest attempt at understanding.

Gould argued for variable rates of evolution as seen in the fossil record. If we saw wolves and chihuahuas separated by a million years in the fossil record, it would qualify as an abrupt change.

But we know from actual experimentation that such changes in body form can be accomplished in a few hundred years, simply by selection of naturally occurring variants.

So again, I'm going to have to ask you for a specific gap in the fossil record that is inconsistent with evolution by stepwise variation and selection, each individual of the same species as its parents.

These attacks are on the theory of evolution by stretching the truth and attempting to link either evolutionists or the theory to some of history's nastiest folks.

To a degree I see your point. I personally have no theological problem with theistic evolution, and I have no personal problem with skeptics who accept and keep Christian values.

However, it appears undeniable that the evolution was used as an excuse to reject our foundational values leading to some of the most horroific events in the history of man, so regardless as to the extent of its influence on biodiversity it is important to keep in mind that there is a value system which far transcends it and someone to which we will be held accountable for our acts.

It is the equivalent of tying your beliefs into Jim Jones in order to discredit you and your beliefs.

Hitler gets thrown at us a lot, although less and less it seems with Weikert's book and the release of the OSS papers showing his plans for Christianity.

" Rarely do I see the hostility in discussions about chemistry, geology etc."

Evolution is about chemistry, biochemistry in particular. Biochemistry is what provides for life. It just so happens that all the nonscientific attacks come, because the chemistry doesn't fit the preconcieved outcomes desired by some men. The same chemistry isn't attacked when it doesn't pose a threat to some preconceived notion. when it does, then it's attacked.

Name an idea that cannot be misused or corrupted.

Or alternately, give a reason a potential tyrant would promise hell on earth for his followers.

The archetypal evil totalitarian is Big Brother. Think about that.

Science was called upon by 20th century tyrants because science was associated with good things -- medicine, labor saving devices, bounteous crops, and so forth.

Religion has been misused in the same way.

Are you saying Gould did not believe that the fossil record had large gaps?

Yes I am. and Gould directly addressed the use you are making of his writings and directly called it a lie.

Gould acknowledged gaps at the species level, but not at higher levels -- the levels now being talked about as "kinds."

Go back to my post on wolves and chihuahuas. That represents a tremendous change in body form, but it can happen in a few human lifetimes. That violates nothing in our understanding of evolution -- every individual is the same species as its parents. But such a change occurring over a few thousand generations would be completely invisible in the fossil record.

What Gould argued is that such "rapid" changes are typical, and the kinds of evolution visible in fossils is not spread evenly over time.

No, not those posts. But you really, really think that was name calling?

I have asked you to provide an alternative explanation. Were you flattering Vade when you said his worldview was faith based? Was that a compliment? Did you intend to mean that he was obviously correct in his worldview?

And when you said, "Vade, this is a religion with you" Were you implying that he had the TRUTH?

You tell me what you intended.

Suppose I said your worldview was Nazi-based? Would that be name-calling? I suppose it depends on the meaning of "is."

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