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.

If you try to say evolution is random then evolutionists will say no, because natural selection is the acting force on those mutations.

so far, OK

... But when you try to say evolution is NOT random (IE, designed or something) ...

"not random" isn't the same as "(IE, designed or something)". It's just not random; some combinations of genes favor survival and reproduction, and some don't. "The race is not always to the swiftest, but that's the wy to bet."

... they'll say no, because the thing natural selection acts on is random mutation. ...

True enough. Evo itself has both random (genetic combinations, mutations) and nonrandom (some are better adapted to the environment than others) components.

Is a casino a random process? To the gamblers it is; to the owners it's completely predictable business. See, it can depend on the scale at which you're viewing the process.

It's a cute little catch-22!

Nope. Catch-22 was a contradiction in the Army's rules; the fact that life has both random and deterministic facets is not a contradiction, it's a fact of life.

alterkaker: Huh? Peanuts are not nuts, they are legumes. I don't believe that there is any taxonimical controversy over their classification.

sittnick: The world is a lot bigger than taxonomy, which is a man-made construct. Nutritionally they are regarded as nuts, and are nutritionally classified in the meat group. Taxonomical categorization is only ONE way of thousands to categorize things. It is often impractical and dopey to insist on taxonomical

But not in a discussion about evolution.

Thank y'all so much for the ping to this fascinating sidebar!

In the end, it seems to me that focusing only on material and efficient causes puts Neo-Darwinism in a situation where it's letting the tail [of its desire] wag the dog [of its science]: its methodological materialism precludes it from recognizing that formal and final causes actually do operate in nature. I think science -- especially physics and mathematics -- is increasingly aware that an absolutist materalist reductionism may be creating a false picture of reality.

So very true and well said. That is exactly the point which troubles me - there cannot be a complete picture when half of it has been shoved off the table (two of the four causes.) To say that it is a complete picture is a delusion, a second reality. It's not "real."

... (If an archeologist in the year 20000 AD came upon a poodle and rottweiler skeleton, not knowing of the existence of either because they died out, he would likely conclude they were different species) ...

I would argue that domestic dogs are in fact a ring species. Consider a thought experiment or two:

Procure an island with lots of game, water, shelter, etc, but no dogs. Introduce 100 male teacup poodles and 100 female great Danes. Come back in 20 years. I predict that there will be no dogs, that the difference in size prevents mating.

Same setup, only this time the the poodles are the bitches and the Danes are the males. I predict the same result.

Same setup, only this time 100 male poodles, 100 female poodles, and the same number of Danes. I redict that when you return to the island there will be two true-breeding populations, and no mutts.

If my predictions are correct, then you have to conclude that the poodles and Danes are different species.

The fact that the poodles can mate with Jack Russells, the Jacks with terriers, the terriers with beagles ... with Danes, shows that they would be a ring species.

A deer as smart as Einstein would come up with something better than running and possibly getting caught...

1) Kill a rabbit or other animal, then kick it around while scavenging. When a wolf attacks, kick the rabbit. Hopefully the wolf's prey instinct will get it to chase the rabbit, not the deer.

2) Bathe frequently to get rid of scents.

3) Find and carry a dead wolf.

4) When in a pack about to be chased by wolves, kick one of your fellow deer before running. Not friendly, but no one said our Einstein deer was a good guy.

Fantastic post and I really like this as to the 'Darwinian euphoria'

Darwinian euphoria is the notion that somehow we all emerged from the primordial soup, that a non-existent God was irrelevant to the process (come, come, Catfish, you don't think that Darwin is defended in a vacuum, do you? The enthusiasm stems from the idiot savant belief that evolution disproves God and is therefore a Promethean gift freeing man fron enslavement to mere reality.) If evolution disproves God, voila!!!! Kill whom you will and mistreat the rest all in the name of the "greatest good for the greatest number."

"Stochastic Process" is typically preferred.

It's like a process, an algorithm, that is based on randomness, but still tends to be predictable and reliable.

It's like statistical mechanics, which deals with huge numbers of particles ("events") at the same time. It's unreasonable, and not fruitful, to try and determine what each one will do precisely, but we can still tell you what the entire system (if it's big enough) will do reliably.

Evolution is less predicable since there are so many freakin factors involved. Like a stochastic process, it's unreasonable to try and determine exactly which mutations will happen and will stick around - so we look at the large system instead, typically as allele frequencies of a population, which we can make lots of predictions about without ever referring to individuals.

Ah man, you started out so strong and got me all excited.

Evolutionary algorithms are surprisingly strong... if made correctly. I have alot of experience with them. Of course they've made new things. They've redesigned computer chips by manipulating everything from the layout to the number, type, and sequence of logic gates. I've used them in simplier hypercubic minterm applications as well as condensed matter optimizations. I even worked with a guy who wanted to apply them to make new quantum computing algorithms.

I'm well aware of the problems stochastic algorithms have with local minima in the fitness landscape of the search space. I have two problems with your interpretation:

(1) The search space doesn't have to be bounded. In many applications the actual search space is so buried in abstract mathematics (A fun little section called Matroid Theory) that its unreasonable to describe changes as simply changing parameters within a space.

(1) As many have said, "New" or "New genetic information" is subjective and, as far as I can tell from the rather ambiguous and ever-changing definitions put forward by Creationists, is environment dependent.

Think of the fitness landscape (environment/time dependent) of the genetic search space (which potentially has infinite dimension, but in practice only has a couple of billion dimensions - 750B base pairs in an amoeba is the highest found). First of all, note that all known life is contained (approximately) in this search space. All evolution is doing is changing parameters... there's no "information" involved.

and then you did the turn to "But the results are impossible to construct randomly even though we're talking about a stochastic process (very different)." which was disappointing.

None of us know all of the creatures that were around three or four or thirty thousand years ago.

Of course we do. We have intact humans from longer ago than that. 3000 years ago was well within the realm of recorded history, in many places.

There you go with the logical fallacies again. I state that we do not know all of the creatures that existed thousands of years ago, and you say we have intact human fossils. I will grant that we certainly had recorded history, and that tells us a lot more than fossils alone. However, it hardly tells us all of the species that existed. We have a significant number of specialists looking for new species, and sometimes they are still found. The fossil record, even from 3,000 years ago is far from complete, much of the historical record is lost, and large parts of the world have no extant history from that era. So, of course we do not know all of the creatures walking the earth 3,000 years ago. There might be a moa or dodo type creature that died out, forgotten to all, with no found remains. There might be more than a handful of such creatures . . . or not. We just don't know.

We certainly don't know what similar looking specimens could mate with others and produce fertile offspring.

No, and without DNA from early Equines, we will likely never know. But we can make a very educated guess, judging from the fact that extant equines (like horses, zebras and asses) cannot normally produce fertile offspring.

Agreed. And educated guesses are not scientific fact.

Peanuts are different from legumes in all sorts of ways.

Name one.

I should have said "other" legumes, though I think you should have know that from the context. The root of the peanut is edible, while the fruit of other legumes are edible. Peas are green and grow in soft pods. Peanuts are light brown and grow in harder shells. In short everything that anybody from a grocer, to a botanist to a four year-old can say that separates peanuts from the other legume of your choice.

I am glad to see that you no longer pretend that the grocer is "wrong" to put the peanuts in with the other nuts in the supermarket. Should grocers pack Swedish Fish in the seafood section too?

If they do, maybe it should go by the imitation crabmeat. Of course, we call peanuts "nuts" nut primarily because the word "nut" is in it. After all, we don't think of coconuts as nuts. Of course, Swedish Fish are manmade, the base of seafood is natural (ditto for candy corn). The grocer puts peanuts in with nuts (with USDA approval) because of their size, edibility out of the shell, texture and general use. Except for a slight remblance to tiny fish, Swedish fish don't pass muster in that regard.

Taxonomical categorization is only ONE way of thousands to categorize things. It is often impractical and dopey to insist on taxonomical

But not in a discussion about evolution.

I didn't say that it was. It is certainly not the only category to refer to. In any event, the taxonomical categories are artificial and man-made. Insisting on using only taxonomical categorizations is "begging the question" (e.g. whales are mammals, and therefore evolved from land-mammals), since we are discussing the natural (not man-made) origin of species. Taxonomical categories describe species, but do not tell us whether one came from another. ( whales could have theoretically developed from fish without leaving the water leaving the water, while the other mammals happened to get their systems independent of what was going on with the whales.)

Wow.

I'm a Stanford graduate student right now. I majored in Math, Computer Science, and Physics as an undergrad. I would LOVE to watch you come to Stanford and conduct this little crusade of yours.

There's nothing mystical or contradictory in "materialism". In fact, it all fits together surprisingly well. In all the billions of ways our experiments could have gone wrong and actually contradict each other... none happen. Small discrepancies or "contradictions" are heavily sought after because that means the concepts/theories are premature and new science can be done to see what we missed.

Scientists change their theories to fit the facts.
Creationists change the facts to fit their theories.

Which is better?

if made correctly...if designed correctly...that was the point.

You seem to have a grasp of evolutionary algorithms... unless you mined that information... I'm confounded how you can understand evolutionary algorithms but not understand the natural process the idea came from.

Of coarse DNA is not an example of information theory. There is no information involved. (Pardon the sarcasm)

You said so yourself: "Evolutionary algorithms don't produce anything new, just find different parameters..

x[t+1] = s( v( x[t]) )".

ALL you need to do is let x be an open set (a "population") in the unbound genetic search space, v() be genetic variation (reproduction, mutation, etc), and s() be natural selection. Working in the space, you would not be able to tell me which genotype contains more information... they're all simply points in genetic space. Only the fitness function (which, as you know, is implicitly built into the s() function and, in this case, is completely environment dependent) can give you a sense of "good" and "bad" adaptation - the gradient (slope) of the fitness function (in a billion-dimensional space remember). No "information"... no "new"... Just a simple evolutionary algorithm in an unbound genetic search space with really really complicated v() and s() functions (which are dependent on time and position within the search space).

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