A Mathematician’s View Of Darwinian Evolution

A Mathematician’s View of Darwinian Evolution: How Natural Selection Fails to Design

The theory of evolution, originally based on the ideas of Darwin, proposes an explanation for life on earth. In this theory there is no place for an intelligent Designer. The sole mechanism of advancing from inorganic matter to the first life form and from there to the diversity of life found on earth relies on chance-based mutations and natural selection (the elimination of ‘unfit’ offspring). In his writings to Asa Gray, a devout American scientist, Darwin says that because he does not believe that this universe and life on earth could have been designed by a beneficent and omnipotent God, for theological rather than scientific reasons, he has to take the position that their origin is ‘left to the working out of what we may call chance’, [Darwin, 93]. While recent versions of the theory of evolution have drifted from Darwin’s original ideas, one aspect remains the same. There is a complete absence of any reference to an intelligent Designer/Creator that is beyond His creation.

For any observed phenomenon, there can be many possible explanations. A chance-based mechanism could be one explanation. We usually associate chance as being indifferent to purpose, and intelligent design as being with a purpose. When a mechanism appears to be unconcerned with or incognizant of any possible outcome, we term it a chance-based or random mechanism. In other words, chance is invoked when there appears to be no preference for one particular outcome over another.

While a chance-based mechanism can be one of the possible explanations for a phenomenon, we only consider it seriously when the probability of the proposed mechanism is significant. When the probability is insignificant, we must eliminate this possibility and consider other options. Otherwise, we may fall into what the mathematician/philosopher Dembski calls the ‘chance-of-the-gaps’ fallacy:

‘Statistical reasoning must be capable of eliminating chance when the probability of events gets too small. If not, chance can be invoked to explain anything. Scientists rightly resist invoking the supernatural in scientific explanations for fear of committing a god-of-the-gaps fallacy (the fallacy of using God as a stop-gap for ignorance). Yet, without some restriction on the use of chance, scientists are in danger of committing a logically equivalent fallacy’ one we may call the ‘chance-of-the-gaps fallacy’. Chance, like God, can become a stop-gap for ignorance.’ [Demski, 98]

To illustrate this principle let us consider three examples. The first is from a movie, related by the same author. The event related concerns the supposed spontaneous combustion of a person.

‘In the movie ‘This is Spinal Tap’, the lead singer remarks that a former drummer in the band died by spontaneously combusting. Any one of us could this instant spontaneously combust if all the most rapidly moving air molecules in our vicinity suddenly converged on us. Such an event, however, is highly improbable, and we don’t give it a second thought.’ [Dembski, 98]

Let us consider two other examples: footprints in the desert and a novel placed next to a typewriter.

Desert Landscape Highland Mountain Sky View

Footprints in the Desert

Footprints in the Desert

When a bedouin sees camel footprints in the desert, he does not attribute these to chance. He does not think that the wind, by chance, formed those patterns on the sand. Instead, he interprets them as signs that a camel has recently walked along this way. The probability of the chance-based scenario is simply too small. Having seen the patterns characteristic of the wind, the bedouin can be almost sure that the camel explanation is the correct one, even if he has not seen the camel itself.

Novel Placed Next to a Typewriter

Imagine you find a novel left next to a typewriter, next to which sits a monkey. One explanation for who might have written the novel could be that the monkey typed the novel, making words and sentences that make sense, solely by chance. We can view this as a chance-based explanation; up to now we have not witnessed any monkey that can appreciate human literature. To make this explanation more plausible, let us assume that when the monkey completes a page, a human checks the page and if it does not make any sense the page is thrown away. If there were an unlimited supply of paper and ink, and if the monkey were replaced by another monkey when it died, you might expect to see a few lines of meaningful literary work after thousands of generations. In the process you would expect to see mountains of thrown-out pages, containing meaningless sequences of letters. If you knew that the monkey could not live long enough to produce a novel, if you knew that it did not have access to enough paper, or if you could not find any thrown-out pages, you would simply have to eliminate the monkey hypothesis. You would consider other explanations, such as that there was a person who was capable of producing literary work, and this person typed the novel and left it next to the typewriter. The idea that a monkey could produce a novel by typing randomly, without regard to content (i.e. by chance) is simply too small a probability. It is not worthy of serious consideration.

These examples illustrate a general principle; highly improbable and ‘preferred’ (independently specified) patterns cannot be generated by chance-based mechanisms. The footprints in the first example illustrate highly improbable events, and shows us how one arrives at a conclusion by preference or purpose. Since Darwinian evolution relies on chance, there is only one way it can produce the diversity of life found on earth. This can be done by exhausting a significant proportion of the possibilities, producing useless organisms in the process, leaving the remains of the unsuccessful organisms on the way and by eventually producing a useful organism after using up a considerable amount of time, matter and space.

science without religion is lame, religion without science is blind.

“Science without religion is lame, religion without science is blind.”
Albert Einstein Quotes

Blind or Not Blind?

Some evolutionists argue that evolutionary algorithms are not the same as a blind search, because they include a fitness function which favors certain outcomes over others. In a way, this imagined ‘fitness function’ evaluates which organisms are the most promising in each generation. To understand the concept of a fitness function, consider the selective breeding of animals. The breeders select the members that have the most desirable properties and continue to breed them. They might breed from only the most fertile chickens or the woolliest sheep and succeed in altering the characteristics of these animals. So, the selection criteria of the breeders can be regarded as a fitness function. However, in the case of selective breeding, the human breeder is the one who defines and applies this function. In the case of Darwinian evolution, there is no room for an intelligent being. So, the imagined fitness function must be a result of the physical laws of the universe. It must be a result of the conditions on earth

What the evolutionary algorithms do is to exploit the information already encoded in the fitness function. So, evolutionary algorithms simply ‘shift’ the problem into a different space. If we assume that this fitness function is a result of the conditions on earth, we have to remember that the conditions on earth are highly improbable and specified. So, if this fitness function is assumed to be capable of design, one must explain how it came to be in the first place. Secondly, this function is ill-defined. Even evolu- tionists themselves have difficulty defining what this function is or how it operates. Various attempts to define this function by evolutionists amount to a tautology in order to justify it. ‘In this formulation, the theory predicts that the fittest organisms will produce the most offspring, and it defines the fittest organisms as the ones which produce the most offspring.’ [Johnson, 91]. Noting this trend, the famous philosopher of science Karl Popper once wrote, ‘some of the greatest contemporary Darwinists themselves formulate the theory in such a way that it amounts to the tautology that those organisms that leave the most offspring leave the most offspring.’ Since the fitness function is not well defined, it is also not possible to demonstrate how this function is capable of producing this diversity of life, neither scientifically nor mathematically. In experiments where evolutionary mechanisms were tested, they tend to favor simplicity, whereas in life we see increasing complexity. Limits of the Universe The famous physicist Carl Sagan once said, in reference to evolutionary processes, given enough time, chance will work miracles. While this statement can be true theoretically, one should also consider that chance, given enough time, will produce a disproportionately high ratio of useless outcomes before coming up with a miracle. Furthermore, the universe, as we have observed, has a limited age and a limited amount of matter. So the amount of time that we can assume chance has is limited. The amount of material to be used by the chance-based pro- cesses is also limited. So the next question is, given the age of the universe, the amount of matter in the universe and all the possible sequences of changes that can take place, what is the likelihood of chance producing a single living cell? The answer to this question is important because it will determine whether the chance hypothesis is worthy of our attention. Dembski explains that within the observed universe any probability below a universal probability bound remains improbable, even if it is assumed that all the resources available were exhausted in order to try out all the possibilities. He calculates this number as 10-150, that is 1 over 10 to the 150th power. The details of this calculation and what it means are explained by him as follows:

In the observable universe, probabilistic resources come in very limited supplies. Within the known physical universe there are estimated to be around 1080 elementary particles. Moreover, the properties of matter are such that transitions from one physical state to another cannot occur at a rate faster than 1045 times per second. This frequency corresponds to Planck time, which constitutes the smallest physically meaningful unit of time. Finally, the universe itself is about a billion times younger than 1025 seconds (assuming the universe is between ten and twenty billion years old). If we now assume that any specification of an event within the known physical universe requires at least one elementary particle to specify it and cannot be generated any faster than the Planck time, then these cosmological constraints imply that the total number of specified events throughout cosmic history cannot exceed 1080 x 1045 x 1025 = 10150. It follows that any specified event of probability less than 1 in 10150 will remain improbable even after all conceivable probabilistic resources from the observable universe have been factored in. A probability of 1 in 10150 is therefore a universal probability bound. Implicit in a universal probability bound such as 10-150 is that the universe is too small a place to generate specified complexity by sheer exhaustion of possibilities. Stuart Kauffman develops this theme at length in his book Investigations. In one of his examples (and there are many like it throughout the book), he considers the number of possible proteins of length 200 (i.e., 20200 or approximately 10260) and the maximum number of pairwise collisions of particles throughout the history of the universe (he estimates 10193 total collisions supposing the reaction rate for collisions can be measured in femto seconds). Kauffman concludes: The known universe has not had time since the big bang to create all possible proteins of length 200 [even] once. To emphasize this point, he notes: It would take at least 1067 times the current lifetime of the universe for the universe to manage to make all possible proteins of length 200 at least once. [Dembski, 98] It should be noted that the precise value of this universal probability bound is not critical. Even an approximate value is enough to judge a proposed chance-based explanation for an observed phenomenon. Carl Sagan himself estimated the probability of humans evolving from a single living cell as one chance in 102,000,000,000, [Sagan]. Combining this estimate with the universal probability bound discussed above and using common sense, one can easily dismiss the chance hypothesis for the origin of life or for the diversity of life on earth. The Fossil Record Problem Associated with the chance hypothesis, there is also the problem of fossil record which we have not discussed in detail yet in this article. If the chance hypothesis is correct, the failed attempts of the blind processes should vastly outnumber those that are successful. This would imply that in the fossil record we should have found vast numbers of fossils of dysfunctional species compared with a tiny minority of successful species. Just as failed attempts would vastly outnumber successful ones, the fossils of such attempts should reflect the same ratio. While there are signs of extinct species in the fossil record, we do not find a huge record of the fossils of the kind of wild variation we would expect from blind mutations. While the reasons for the extinction of species like dinosaurs are debated, there is agreement that they were successful living organisms during their lifetimes. Conclusion To summarize, when a mathematician calculates the probability of chance producing life on earth, he or she can easily dismiss this explanation because of the aforementioned problems. Life is too complex and intricate. The lifetime of the universe and the amount of matter in it are insufficient to blindly exhaust all possibilities and arrive at the diverse life forms we see on earth. The fossil record does not reflect the ratio of unsuccessful attempts to successful ones we would expect from chance hypothesis. When presented with the chance-based theory of evolution, an objective mathematician would thus feel obliged to say, Either you have to shut off my intellect or I cannot accept this hypothesis. Life must be the product of an intelligent Designer who is All-Wise, Omniscient, Omnipotent and cognizant of what He is doing. The mathematician would thus come to the same conclusion as an unlettered bedouin, who is nevertheless a careful observer and a solid thinker: Camel droppings point to the existence of a camel. Footprints on the sand tell of a traveler. The heaven with its stars, the earth with its mountains and valleys, and the sea with its waves – do they not all point to the Maker, All-Powerful, Knowing, Wise and Caring?

By Alphonse Dougan

This article is borrowed from The Fountain Magazine.


  • [Darwin, 93] Charles Darwin, The Correspondence of Charles Darwin 8, 1860 Cambridge University Press, 1993.
  • [Dembski, 98] William A. Dembski, The Design Inference, Cambridge University Press, New York, 1998.
  • [Dembski, 99] William A. Dembski, Intelligent Design, InterVarsity, Downers Grove, Ill., 1999.
  • [Johnson, 91] Philip E. Johnson, Darwin on Trial, Regnery Gateway, Washington DC, 1991.
  • [Khan] Khan, M., trans. Sahih Al-Bukhari: The Translation of the Meanings. Darussalam Publishers: 1997.
  • [Sagan] Carl Sagan, F. H. C. Crick, L. M. Muchin in Carl Sagan, ed., Communication with Extraterrestrial Intelligence (CETI) (Cambridge, MA: MIT Press), pp. 45-46.
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