December 22, 2017 | David F. Coppedge

Mathematician and Geneticist Team Up to Correct Fisher’s Theorem

A new paper corrects errors in Fisher’s Theorem, a mathematical “proof” of Darwinism. Rather than supporting evolution, the corrected theorem inverts it.

Flipping Fisher’s Famous Theorem

by William F. Basener and John C. Sanford

A recent paper in the Journal of Mathematical Biology  ( has uncovered major problems with the historically pivotal Fundamental Theorem of Natural Selection. That theorem was proven by Ronald Fisher – one the great scientists of the last century. Fisher’s theorem was published in 1930, and was the foundational work that gave rise to neo-Darwinian theory and the field of population genetics.

Fisher’s theorem was … the foundational work that gave rise to neo-Darwinian theory and the field of population genetics.

Fisher described his theorem as “fundamental,” because he believed he had discovered a mathematical proof for Darwinian evolution. He described his theorem as equivalent to a universal natural law – on the same level as the second law of thermodynamics.  Fisher’s self-proclaimed new law of nature was that populations will always increase in fitness – without limit, as long as there is any genetic variation in the population. Therefore evolution is like gravity – a simple mathematical certainly. Over the years, a vast number of students of biology have been taught this mantra – Fisher’s Theorem proves that evolution is a mathematical certainty.

The authors of the new paper describe the fundamental problems with Fisher’s theorem. They then use Fisher’s first principles, and reformulate and correct the theorem. They have named the corrected theorem The Fundamental Theorem of Natural Selection with Mutations. The correction of the theorem is not a trivial change – it literally flips the theorem on its head. The resulting conclusions are clearly in direct opposition to what Fisher had originally intended to prove.

In the early 1900s, Darwinian theory was in trouble scientifically. Darwin’s writings were primarily conceptual in nature, containing a great deal of philosophy and a great deal of speculation. Beyond simple observations of nature, Darwin’s books generally lacked genuine science (experimentation, data analysis, the formulation of testable hypotheses). Darwin had no understanding of genetics, and so he had no conception of how traits might be passed from one generation to the next. He only had a very vague notion of what natural selection might actually be acting upon. He simply pictured life as being inherently plastic and malleable, so evolution was inherently fluid and continuous (think Claymation). When Mendel’s genetic discoveries were eventually brought out of the closet, it could be seen that inheritance was largely based upon discrete and stable packets of information. That indicated that life and inheritance were not like Claymation, and that biological change over time was not based upon unlimited plasticity or fluidity. Mendel’s discrete units of information (later called genes), were clearly specific and finite, and so they only enabled specific and limited changes. At that time it was being said; “Mendelism has killed Darwinism”.

Ronald A. Fisher (1890-1962) believed he had proved evolution mathematically.

Fisher was the first to reconcile the apparent conflict between the ideas of Darwin and the experimental observations of Mendel. Fisher accomplished this by showing mathematically how natural selection could improve fitness by selecting for desirable genetic units (beneficial alleles), and simultaneously selecting against undesirable genetic units (deleterious alleles).  He showed that given zero mutations, the more there are good/bad alleles in the population, the more natural selection can improve the fitness of the population. This is the essence of Fisher’s Theorem. This was foundational for neo-Darwinian theory – which now reigns supreme in modern academia.

The resulting conclusions are clearly in direct opposition to what Fisher had originally intended to prove.

Remarkably, Fisher’s theorem by itself illustrates a self-limiting process – once all the bad alleles are eliminated, and once all the individuals carry only good alleles, then there is nothing left to select, and so selective progress must stop. The end result is that the population improves slightly and then becomes locked in stasis (no further change). It is astounding that Fisher’s Theorem does not explicitly address this profound problem! Newly arising mutations are not even part of Fisher’s mathematical formulation. Instead, Fisher simply added an informal corollary (which was never proven), which involved extrapolation from his simple proof. He assumed that a continuous flow of new mutations would continuously replenish the population’s genetic variability, thereby allowing continuous and unlimited fitness increase.

The authors of the new paper realized that one of Fisher’s pivotal assumptions was clearly false, and in fact was falsified many decades ago. In his informal corollary, Fisher essentially assumed that new mutations arose with a nearly normal distribution – with an equal proportion of good and bad mutations (so mutations would have a net fitness effect of zero). We now know that the vast majority of mutations in the functional genome are harmful, and that beneficial mutations are vanishingly rare. The simple fact that Fisher’s premise was wrong, falsifies Fisher’s corollary. Without Fisher’s corollary – Fisher’s Theorem proves only that selection improves a population’s fitness until selection exhausts the initial genetic variation, at which point selective progress ceases. Apart from his corollary, Fisher’s Theorem only shows that within an initial population with variant genetic alleles, there is limited selective progress followed by terminal stasis.

Since we now know that the vast majority of mutations are deleterious, therefore we can no longer assume that the mutations and natural selection will lead to increasing fitness. For example, if all mutations were deleterious, it should be obvious that fitness would always decline, and the rate of decline would be proportional to the severity and rate of the deleterious mutations.

To correct Fisher’s Theorem, the authors of the new paper needed to reformulate Fisher’s mathematical model. The problems with Fisher’s theorem were that; 1) it was initially formulated in a way that did not allow for any type of dynamical analysis; 2) it did not account for new mutations; and 3) it consequently did not consider the net fitness effect of new mutations. The newly formulated version of Fisher’s theorem has now been mathematically proven. It is shown to yield identical results as the original formulation, when using the original formulation’s assumptions (no mutations). The new theorem incorporates two competing factors: a) the effect of natural selection, which consistently drives fitness upward); and b) the effect of new mutations, which consistently drive fitness downward).  It is shown that the actual efficiency of natural selection and the actual rate and distribution of new mutations determines whether a population’s fitness will increase or decrease over time. Further analysis indicates that realistic rates and distributions of mutations make sustained fitness gain extremely problematic, while fitness decline becomes more probable. The authors observe that the more realistic the parameters, the more likely fitness decline becomes. The new paper seems to have turned Fisher’s Theorem upside down, and with it, the entire neo-Darwinian paradigm.

Sanford’s book examines the impact of mutations that are invisible to selection.

Supplemental Information – Fisher’s informal corollary (really just a thought experiment), was convoluted. The essence of Fisher’s corollary was that the effect of both good and bad mutations should be more or less equal – so their net effect should be more-or less neutral. However, the actual evidence available to Fisher at that time already indicated that mutations were overwhelmingly deleterious. Fisher acknowledged that most observed mutations were clearly deleterious – but he imagined that this special class of highly deleterious mutations would easily be selected away, and so could be ignored.  He reasoned that this might leave behind a different class of invisible mutations that all had a very low-impact on fitness – which would have a nearly equal chance of being either good or bad. This line of reasoning was entirely speculative and is contrary to what we now know.  Ironically, such “nearly-neutral” mutations are now known to also be nearly-invisible to natural selection – precluding their role in any possible fitness increase. Moreover, mutations are overwhelmingly deleterious – even the low impact mutations. This means that the net effect of such “nearly-neutral” mutations, which are all invisible to selection, must be negative, and must contribute significantly to genetic decline. Furthermore, it is now known that the mutations that contribute most to genetic decline are the deleterious mutations that are intermediate in effect – not easily selected away, yet impactful enough to cause serious decline.

Update 1/05/18: This article was referenced by World Magazine and by the Discovery Institute’s Evolution News and Science Today. 

Ed. note: We are honored to have Bill Basener and Dr. John Sanford writing in Creation-Evolution Headlines to report this important new paper. Dr Sanford is a renowned geneticist from Cornell University, inventor of the gene gun and author of Genetic Entropy. Dr. Sanford was an evolutionist who turned creationist based on the genetic evidence.

In our 9/02/2004 entry, we relayed what Steve Jones (an ardent evolutionist) thought about R. A. Fisher’s personal life and beliefs. Jones was reviewing a book about six ‘Darwin bulldogs’ or defenders, including Fisher. Calling him a “bearded bigot,” Jones made these uncomplimentary remarks about Ronald Fisher, based on what he had read in Marek Kohn’s book A Reason for Everything: Natural Selection and the British Imagination (Faber and Faber, 2004):

Fisher claimed that his fundamental theory of natural selection occupied the supreme position among the biological sciences, although others dismissed it as a verbal trick…. His Genetical Theory of Natural Selection became evolution’s equivalent of The Lord of the Rings: full of gnomic and portentous truths with rather a nasty social agenda lurking beneath (Fisher felt it his biological duty to beget eight children).  As Kohn points out, Fisher’s followers, like those of Wagner – composer of a musical on the same theme – are obsessed with the fine detail of what the great begetter meant and are still far from sure.

Fisher was also an open racist and eugenicist. About this facet of his life, Wikipedia says:

In 1910 Fisher joined the Eugenics Society (UK) at University of Cambridge, whose members included John Maynard Keynes, R. C. Punnett, and Horace Darwin. He saw eugenics as addressing pressing social and scientific issues that encompassed and drove his interest in both genetics and statistics. During World War I Fisher started writing book reviews for the Eugenic Review and volunteered to undertake all such reviews for the journal, being hired for a part-time position. The last third of The Genetical Theory of Natural Selection focussed on eugenics, attributing the fall of civilizations to the fertility of their upper classes being diminished, and used British 1911 census data to show an inverse relationship between fertility and social class, partly due, he claimed, to the lower financial costs and hence increasing social status of families with fewer children. He proposed the abolition of extra allowances to large families, with the allowances proportional to the earnings of the father. He served in several official committees to promote Eugenics. In 1934, he resigned from the Eugenics Society over a dispute about increasing the power of scientists within the movement.

These quotes illustrate that theories bearing on nature, mankind and reality do not emerge in a vacuum, but are filtered through the passions and worldview assumptions of their creators.


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