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.

The top speed of animals is irrelevant to whether a man could outrun them. A human can run continuously for thirty miles. Any fur covered animal will overheat. On the plains of Africa, a human can run down almost anything he is likely to want to catch.

This may seem like a pedant debate - but I really have an issue with this statement. Maybe I am wrong but with my current data this statement doesn't sound correct. I am not trying to be a pain - I was concede if you show me I am wrong.

Wolves stalk their prey - the first time the prey realizes the wolf is there, the wolf is somewhat near - no more than 300 feet (guessing). A wolf can sprint at 40 MPH for say two minutes and then it can run at 7 MPH for as much 7 hours (I got that from a link). Man can run say 18 MPH for a long time. The wolf is going to over take the man in the first few minutes. (math is not my strong subject so maybe I am miscalculating - or not calculating at all)

"Given the incredible improbability of life arriving from non-life.....do you really believe that life as we observe it today came to be by chance and natural process, and you are willing to base your life on that belief?

I'm not willing to limit my definition to a couple of phrases, but basically, yes. I find Pascal's wager morally loathsome. Any deity that operated on that principle would be morally indistinguishable from Satan."

Although Pascal's wager is relevant, asking whether or not one will make a life choices based the probability of something is true is not equivalent to Pascal's wager. Would you walk across a bridge that has a 99% chance (probability of random generation of even a single protein is much much lower even given your suggested ridiculously long three billion years.....much less spontaneous generation....one method of calculation below) of failing down is not equivalent to:

God is, or He is not. But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up...Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.

We make choices based on the probability of something being likely or likely true all the time....engineers, actuaries, scientists etc. This is not equivalent to Pascals wager: Your only chance of winning eternal happiness is believing, and your only chance of losing is not believing. Pascal's wager is not asking the question whether or not something is likely or likely true, merely the consequences of being wrong.

Given your belief that matter is all there is Darwinian worldview, what is your source of ethics that makes you 'find Pascal's wager morally loathsome' (other then your presuppositions are hostile to 'God')?

"Your criticism of Avida and evolutionary algorithms is rather beside the point."

The criticism was not directed at evolutionary algorithms-- a particular stochastic method of machine learning. In fact, I use stochastic methods frequently. The criticism was directed at the fallacy of a carefully designed algorithm somehow proving that that the world has come to be by chance and natural process.

Calculations of Bradley and Thaxton for random production of a single protein.

Walter L. Bradley and Charles B. Thaxton calculated the probability of a random formation of amino acids into a protein to be 4.9 x 10-191. They began with the assumption that the probability of starting with an L-amino acid was .5, and the probability of starting with an L-amino acid was .5, and the probability of two L-amino acids joining with a peptide bond was also .5. They assumed that the twenty necessary amino acids existed in equal concentration in the prebiotic soup so that the probability of the right amino acid in the required position was .05.
Bradley and Thaxton were also generous towards the proponents of random processes when they also assumed that all of the chemical reactions would be with amino acids, ignoring the high probability of reactions with non-amino acid chemicals. They calculated the probability of the necessary placement of one amino acid to be .5 x .5 x .05 or .125. This, of coarse, meant that the probability of assembling N such amino acids would be .0125 x .0125 for N terms. Assuming a protein with 100 amino acids (.0125 x .0125 for 100 terms ), the mathematically impossible probability would be 4.9 x 10-191.
Bradley and Thaxton noted their agreement with Hubert P. Yockey and concluded that even assuming that all the carbon on earth existed in the form of amino acids and reacted at the greatest possible rate of 1012/s for one billion years (when actually only 130 million years were available), the mathematically impossible probability for the formation of one functional protein would be 10^-65.

Walter L. Bradley and Charles B. Thaxton, “Information and the Origin of Life” in The Creation Hypothesis, ed. J. P. Moreland (Downers Grove, Il : InterVarsity Press, 1994), p. 190

"All the calculations in the world are useless if you're not asking the right question."

The question: What is the probability of life arriving solely by chance and natural process? is the right question. It is the crux of the issue.

a) Calculations of Sir Fred Hoyle and Chandra Wickramasinghe for random generation of a simple enzyme and calculations for a single celled bacterium.

Although he is an evolutionist, and an atheist, Hoyle sees the mathematical statistical difficulty in producing a single bacterium like E. coli. In his calculations of the probability of life emerging from chance interactions with chemicals, Hoyle assumed that the first living cell was much simpler than today’s bacteria. However, his calculation for the likelihood of even one very simple enzyme arising at the right time in the right place was only chance in 10^(20). Because there are thousands of different enzymes with different functions, to produce the simplest living cell, Hoyle calculated that about 2,000 enzymes were needed with each one performing a specific task to form a single bacterium lie E coli.

No matter how large the environment one considers, life cannot have a random beginning….there are about two thousand enzymes, and the chance of obtaining them all in a random trial is only one part in (10^20)^2000 = 10^40,000, an outrageously small probability that could not be faced even if the whole universe consisted of organic soup. If one is not prejudiced either by social beliefs or by a scientific training into the conviction that life originated on the Earth, this simple calculation wipes the idea entirely out of court….the enormous information content of even the simplest living systems….cannot in out view be generated by what are often called “natural” processes, …For life to have originated on the Earth it would be necessary that quite explicit instruction should have been provided for its assembly…There is no way in which we can expect to avoid the need for information, no way in which we can simply get by with a bigger and better organic soup, as we ourselves hoped might be possible a year or two ago.
-Hoyle & Wickramasinghe, Evolution from Space (London: J.M. Dent & Sons, 1981).

Other calculation models posted here:

http://www.freerepublic.com/focus/f-news/1689062/posts?page=185#185

"$1,000,000 reward to the first evolutionist to get life to evolve from any sort of primordial soup in a reproducible fashion."

This is actually real.....Evolutionists have a $1,000,000 reward for anyone who can explain spontaneous generation by chance and natural processes...The Orgin-of-Life prize. Spontaneous generation is monumental problem for the evolutionist. The evolutionist has to accept it by faith, despite its incredibly low probability.

http://www.us.net/life/index.htm

The website is actually fairly interesting, and provides an interesting list of issues references, judges, etc...

One part that is particularly funny....

"Other than announcements in scientific journals, The Prize will not be publicly advertised in lay media. The Origin-of-Life Foundation, Inc. wishes to keep the project as quiet as possible within the scientific community. No media interviews will be granted until after the Prize is won."

"'As far as you know, is there a compelling mathematical argument in support of macro-evolution?'

Yes. (1 + µ)n ~ 1 only if µ=0 (micorevolution does not exist) and/or n is small (young Earth). If you accept microevolution and an old Earth, macroevolution is inescapable."

Random formation of even a single protein is mathematically highly improbable even if n is ridiculously large (one billion years). Spontaneous generation is much, much less probable. Faith in the highly improbable is inescapable for those with the presupposition of "matter is all there is".

c) Calculations of Bradley and Thaxton for random production of a single protein.

Walter L. Bradley and Charles B. Thaxton calculated the probability of a random formation of amino acids into a protein to be 4.9 x 10-191. They began with the assumption that the probability of starting with an L-amino acid was .5, and the probability of starting with an L-amino acid was .5, and the probability of two L-amino acids joining with a peptide bond was also .5. They assumed that the twenty necessary amino acids existed in equal concentration in the prebiotic soup so that the probability of the right amino acid in the required position was .05.
Bradley and Thaxton were also generous towards the proponents of random processes when they also assumed that all of the chemical reactions would be with amino acids, ignoring the high probability of reactions with non-amino acid chemicals. They calculated the probability of the necessary placement of one amino acid to be .5 x .5 x .05 or .125. This, of coarse, meant that the probability of assembling N such amino acids would be .0125 x .0125 for N terms. Assuming a protein with 100 amino acids (.0125 x .0125 for 100 terms ), the mathematically impossible probability would be 4.9 x 10-191.
Bradley and Thaxton noted their agreement with Hubert P. Yockey and concluded that even assuming that all the carbon on earth existed in the form of amino acids and reacted at the greatest possible rate of 1012/s for one billion years (when actually only 130 million years were available), the mathematically impossible probability for the formation of one functional protein would be 10^-65.

other probability models posted here:

http://www.freerepublic.com/focus/f-news/1689062/posts?page=185#185

a) Calculations of Sir Fred Hoyle and Chandra Wickramasinghe for random generation of a simple enzyme and calculations for a single celled bacterium
b) Calculations of Hubert Yockey for random generation of a single molecule of iso-1-cytochrome c protein.
c) Calculations of Bradley and Thaxton for random production of a single protein.
d) Calculations of Harold Morowitz for single celled bacterium developing from accidental or chance processes.
e) Calculations of Bernd-Olaf Kuppers for the random generation of the sequence of a bacterium.

"Now if you can get the Sun to shine on the fieldstone pile in my backyard and assemble it into that wall I've been meaning to build, you will have thoroughly destroyed Sewell's hypothesis."

..gotta add the random changes and evalution via a stochastic fitness algorithm to this....: )

...that will save me so much work and sore muscles.... I will go and build a fire in my backyard tonight. I have a pile of concrete blocks that I have been meaning to assemble into a wall for some time. The light and heat from the fire will surely assemble the blocks into a wall given enough time. Oh wait..if I get some dynamite and make a big bang in my backyard it will assemble the wall even faster......To wall to build even faster I will iteratively make random changes via a sledge hammer to the blocks and evaluate the blocks based on fitness via a (designed) stochastic algorithm. I will reproduce the blocks most fit for building a wall via (designed) molds, then apply energy via fire, sunlight, and dynamite so the blocks will assemble themselves into a wall. Given enough time, enough fire, and enough random changes via dynamite and sledge hammers, I will have assembled the wall....you saved me so much work I thought that I was going to have to actually design the wall by stacking the blocks in order....

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