# Borel's Law and the Origin of Many Creationist Probability Assertions

by John Stockwell
[Last update: March 13, 2002]

ften on talk.origins we have seen assertions to the effect that there exists a law that is well known to physicists and/or mathematicians (possibly implying that it is a mathematical theorem) that there is a particular order of probability below which any event is considered to be "essentially impossible". This statement usually preceeds a calculation based on some unrealistic model of the formation of complicated organic molecules via the random assembly of atoms as "proof" that abiogenesis is impossible. At the end of this article, references are given to several creationist sources that refer to this probability assertion as "Borel's Law".

### Conclusions of this FAQ

The "law" in question does not exist as a mathematical theorem, nor is there a universally decided upon "minimum probability" among the physical sciences community. Rather, Borel's Law originated in a discussion in a book written by Emil Borel for non-scientists. Borel shows examples of the kind of logic that any scientist might use to generate estimates of the minimum probability below which events of a particular type are considered negligible. It is important to stress that each of these estimates are created for specific physical problems, not as a universal law.

### A Discussion of Karl Crawford's Original Post

A post by t.o regular, creationist Karl Crawford (aka ksjj), shed some light on the possible origin of this "law".

Talk.origins regulars, of course, will recognize that all of the models that are used to generate the tremendously tiny odds are based on faulty assumptions. However, the point in question is the reference to mathematician Emil Borel:

...Mathematicians generally agree that, statistically, any odds beyond 1 in 1050 have a zero probability of ever happening.... This is Borel's law in action which was derived by mathematician Emil Borel....

I was intrigued by the reference to Emil Borel. While Borel is famous in mathematical circles, he is hardly a household name, so I wanted to see if there was such a thing as "Borel's law" in the subject of probability and statistics. After searching a number of probability and statistics textbooks, technical treatises, and other scholarly works on the subject without finding any reference to such a thing, I happened quite by chance (no pun intended) on two books by Borel, himself.

### A Discussion of Borel's Law

The first is Probability and Life, a 1962 Dover English translation of the French version published in 1943 as Le Probabilites et la Vie. The second is Probability and Certainty, a 1963 Dover English translation of the French version published in 1950 as Probabilite et Certitude. Both of these books are "science for the non-scientist" type books rather than scholarly treatments of the theory of probability.

In Probability and Life, Borel states a "single law of chance" as the principle that "Phenomena with very small probabilities do not occur". At the beginning of Chapter Three of this book, he states:

When we stated the single law of chance, "events whose probability is sufficiently small never occur," we did not conceal the lack of precision of the statement. There are cases where no doubt is possible; such is that of the complete works of Goethe being reproduced by a typist who does not know German and is typing at random. Between this somewhat extreme case and ones in which the probabilities are very small but nevertheless such that the occurrence of the corresponding event is not incredible, there are many intermediate cases. We shall attempt to determine as precisely as possible which values of probability must be regarded as negligible under certain circumstances.

It is evident that the requirements with respect to the degree of certainty imposed on the single law of chance will vary depending on whether we deal with scientific certainty or with the certainty which suffices in a given circumstance of everyday life.

The point being, that Borel's Law is a "rule of thumb" that exists on a sliding scale, depending on the phenomenon in question. It is not a mathematical theorem, nor is there any hard number that draws a line in the statistical sand saying that all events of a given probability and smaller are impossible for all types of events.

Borel continues by giving examples of how to choose such cutoff probabilities. For example, by reasoning from the traffic death rate of 1 per million in Paris (pre-World War II statistics) that an event of probability of 10-6 (one in a million) is negligible on a "human scale". Multiplying this by 10-9 (1 over the population of the world in the 1940s), he obtains 10-15 as an estimate of negligible probabilities on a "terrestrial scale".

To evaluate the chance that physical laws such as Newtonian mechanics or laws related to the propagation of light could be wrong, Borel discusses probabilities that are negligible on a "cosmic scale", Borel asserts that 10-50 represents a negligible event on the cosmic scale as it is well below one over the product of the number of observable stars (109) times the number of observations that humans could make on those stars (1020).

To compute the odds against a container containing a mixture of oxygen and nitrogen spontaneously segregating into pure nitrogen on the top half and pure oxygen on the bottom half, Borel states that for equal volumes of oxygen and nitrogen the odds would be 2-n where n is the number of atoms, which Borel states as being smaller than the negligible probability of 10-(10(-10)), which he assigns as the negligible probability on a "supercosmic" scale. Borel creates this supercosmos by nesting our universe U1 inside successive supercosmoses, each with the same number of elements identical to the preceding cosmos as that cosmos has its own elements, so that U2 would be composed of the same number of U1's as U1 has atoms, and U3 would be composed of the same number of U2's as U2 has U1's, and so forth on up to UN where N=1 million. He then creates a similar nested time scale with the base time of our universe being a billion years (T2 would contain a billion, billion years) on up to TN, N=1 million. Under such conditions of the number of atoms and the amount of time, the probability of separating the nitrogen and oxygen by a random process is still so small as to be negligible.

Ultimately, the point is that the user must design his or her "negligible probability" estimate based on a given set of assumed conditions.

Curiously, in spite of the suggestive title of the book Probability and Life, Borel has no discussion of evolution or abiogenesis-related issues. However, in Probability and Certainty, the last section of the main text is devoted to this question.

From Probability and Certainty, p. 124-126:

The Problem of Life.

In conclusion, I feel it is necessary to say a few words regarding a question that does not really come within the scope of this book, but that certain readers might nevertheless reproach me for having entirely neglected. I mean the problem of the appearance of life on our planet (and eventually on other planets in the universe) and the probability that this appearance may have been due to chance. If this problem seems to me to lie outside our subject, this is because the probability in question is too complex for us to be able to calculate its order of magnitude. It is on this point that I wish to make several explanatory comments.

When we calculated the probability of reproducing by mere chance a work of literature, in one or more volumes, we certainly observed that, if this work was printed, it must have emanated from a human brain. Now the complexity of that brain must therefore have been even richer than the particular work to which it gave birth. Is it not possible to infer that the probability that this brain may have been produced by the blind forces of chance is even slighter than the probability of the typewriting miracle?

It is obviously the same as if we asked ourselves whether we could know if it was possible actually to create a human being by combining at random a certain number of simple bodies. But this is not the way that the problem of the origin of life presents itself: it is generally held that living beings are the result of a slow process of evolution, beginning with elementary organisms, and that this process of evolution involves certain properties of living matter that prevent us from asserting that the process was accomplished in accordance with the laws of chance.

Moreover, certain of these properties of living matter also belong to inanimate matter, when it takes certain forms, such as that of crystals. It does not seem possible to apply the laws of probability calculus to the phenomenon of the formation of a crystal in a more or less supersaturated solution. At least, it would not be possible to treat this as a problem of probability without taking account of certain properties of matter, properties that facilitate the formation of crystals and that we are certainly obliged to verify. We ought, it seems to me, to consider it likely that the formation of elementary living organisms, and the evolution of those organisms, are also governed by elementary properties of matter that we do not understand perfectly but whose existence we ought nevertheless admit.

Similar observations could be made regarding possible attempts to apply the probability calculus to cosmogonical problems. In this field, too, it does not seem that the conclusions we have could really be of great assistance.

In short, Borel says what many a talk.origins poster has said time and time again when confronted with such creationist arguments: namely, that probability estimates that ignore the non-random elements predetermined by physics and chemistry are meaningless.

### References

Borel, Emil (1962), Probability and Life, Dover, translated from the original, Les Probabilite et la Vie, 1943, Presses Universitaire de France.

Borel, Emil (1963), Probability and Certainty, Dover, translated from the original, Probabilite et Certitude, 1950, Presses Universitaire de France.

### Creationist Sources That Refer to Borel's Law

1. Origins Answer Book, Paul S. Taylor. p.22.

2. In The Beginning, Walter T. Brown. p.8.

3. ibid., p.44.

4. Origins: Creation or Evolution, Richard B. Bliss. p.21.

5. Creation and Evolution, Alan Hayward. p.35.

6. It Couldn't Just Happen. Lawrence Richards. p.70-71.

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