###### Evolution

# Peer-Reviewed Science: A “Mathematical Proof of Darwinian Evolution” Is Falsified

Due to the tradition of professional scientific writing, major developments in scientific literature often arrive muffled in language so bland or technical as to be totally missed by a general reader. This, along with the media’s habit of covering up for evolution, is how large cracks in the foundation of Darwinism spread unnoticed by the public, which goes on assuming that the science is all settled and will ever remain so.

A case in point is a recent article in the *Journal of Mathematical Biology*, a significant peer-reviewed publication from the influential publisher Springer. The title of the article announces, “The fundamental theorem of natural selection with mutations.”

Including a verb would, presumably, be too much of a concession to populist sensationalism. Yet the conclusion, if not sensational, is certainly noteworthy.

Generations of students of biology and evolution have learned of the pioneering work of Ronald A. Fisher (1890-1962). A founder of modern statistics and population genetics, he published his famous fundamental theorem of natural selection in 1930, laying one of the cornerstones of neo-Darwinism by linking Mendelian genetics with natural selection. *Wikipedia* summarizes, “[T]his contributed to the revival of Darwinism in the early 20th century revision of the theory of evolution known as the modern synthesis.”

Fisher’s theorem, offered as what amounts to a “mathematical proof that Darwinian evolution is inevitable,” now stands as falsified.

His idea is relatively easy to state. It goes:

The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.

His proof of this was not a standard mathematical one; fitness is not rigorously defined, and his argument is more intuitive than anything else. The theorem addresses only the effects of natural selection. Fisher did not directly address any other effect (mutation, genetic drift, environmental change, etc.) as he considered them to be insignificant. Later mathematicians took issue with Fisher’s lack of rigor, some at considerable length. But the omission of the effects of mutation got the most attention.

Now along come mathematician William F. Basener and geneticist John C. Sanford who propose an expansion of the fundamental theorem to include mutations. Basener is a professor at the Rochester Institute of Technology and a visiting scholar at the University of Virginia’s Data Science Institute. Sanford is a plant geneticist who was an associate professor at Cornell University for many years. He is an editor of the volume *Biological Information: New Perspectives* (World Scientific, 2013). The *Journal of Mathematical Biology* is the official publication of the European Society for Mathematical and Theoretical Biology.

Basener and Sanford expand the Fisher model to allow both beneficial and deleterious mutations, following and extending earlier work. They use zero mutation levels to test their model’s agreement with Fisher’s. They establish that there is an equilibrium fitness level where selection balances the mutational effects. However, if mutations at biologically plausible levels are used, overall fitness is compromised. In some cases this leads to “mutational meltdown,” where the effect of accumulated mutations overwhelms the population’s ability to reproduce, resulting in extinction.

Extinction is the opposite of evolution. They conclude:

We have re-examined Fisher’s fundamental theorem of natural selection, focusing on the role of new mutations and consequent implications for real biological populations. Fisher’s primary thesis was that genetic variation and natural selection work together in a fundamental way that ensures that natural populations will always increase in fitness. Fisher considered his theorem to essentially be a mathematical proof of Darwinian evolution, and he likened it to a natural law. Our analysis shows that Fisher’s primary thesis (universal and continuous fitness increase) is not correct. This is because he did not include new mutations as part of his mathematical formulation, and because his informal corollary rested upon an assumption that is now known to be false.

We have shown that Fisher’s Theorem, as formally defined by Fisher himself, is actually antithetical to his general thesis. Apart from new mutations, Fisher’s Theorem simply optimizes pre-existing allelic fitness variance leading to stasis. Fisher realized he needed newly arising mutations for his theorem to support his thesis, but he did not incorporate mutations into his mathematical model. Fisher only accounted for new mutations using informal thought experiments. In order to analyze Fisher’s Theorem we found it necessary to define the informal mutational element of his work as Fisher’s Corollary, which was never actually proven. We show that while Fisher’s Theorem is true, his Corollary is false.

In this paper we have derived an improved mutation–selection model that builds upon the foundational model of Fisher, as well as on other post-Fisher models. We have proven a new theorem that is an extension of Fisher’s fundamental theorem of natural selection. This new theorem enables the incorporation of newly arising mutations into Fisher’s Theorem. We refer to this expanded theorem as “The fundamental theorem of natural selection with mutations”.

After we re-formulated Fisher’s model, allowing for dynamical analysis and permitting the incorporation of newly arising mutations, we subsequently did a series of dynamical simulations involving large but finite populations. We tested the following variables over time: (a) populations without new mutations; (b) populations with mutations that have a symmetrical distribution of fitness effects; and (c) populations with mutations that have a more realistic distribution of mutational effects (with most mutations being deleterious). Our simulations show that; (a) apart from new mutations, the population rapidly moves toward stasis; (b) with symmetrical mutations, the population undergoes rapid and continuous fitness increase; and (c) with a more realistic distribution of mutations the population often undergoes perpetual fitness decline.

Is this unfair to a historical figure? What about models developed after Fisher?

In the light of Fisher’s work, and the problems associated with it, we also examined post-Fisher models of the mutation–selection process. In the case of infinite population models, what has commonly been observed is that populations routinely go to equilibrium or a limit set — such as a periodic orbit. They do not show perpetual increase or decline in fitness, but are restricted from either behavior because of the model structure (an infinite population with mutations only occurring between pre-existing genetic varieties). On a practical level, all biological populations are finite. In the case of finite population models, the focus has been upon measuring mutation accumulation, as affected by selection. Finite models clearly show that natural populations can either increase or decrease in fitness, depending on many variables. Not only do other finite mathematical population models show that fitness can decrease — they often show that only a narrow range of parameters can actually prevent fitness decline. This is consistent with very many numerical simulation experiments, numerous mutation accumulation experiments, and observations where biological systems have either a high mutation rate or a small population size. Even when large populations are modeled, very slightly deleterious mutations (VSDMs), can theoretically lead to continuous fitness decline.

The final blow comes wrapped in compliments:

Fisher was unquestionably one of the greatest mathematicians of the twentieth century. His fundamental theorem of natural selection was an enormous step forward, in that for the first time he linked natural selection with Mendelian genetics, which paved the way for the development of the field of population genetics. However, Fisher’s theorem was incomplete in that it did not allow for the incorporation of new mutations. In addition, Fisher’s corollary was seriously flawed in that it assumed that mutations have a net fitness effect that is essentially neutral. Our re-formulation of Fisher’s Theorem has effectively completed and corrected the theorem, such that it can now reflect biological reality.

What they mean to say is stated most bluntly earlier in the article:

Because the premise underlying Fisher’s corollary is now recognized to be entirely wrong, Fisher’s corollary is falsified. Consequently, Fisher’s belief that he had developed a mathematical proof that fitness must always increase is also falsified.

That’s the “biological reality.” Fisher’s work is generally understood to mean that natural selection leads to increased fitness. While this is true taken by itself, mutation and other factors can and do reduce the average fitness of a population. According to Basener and Sanford, at real levels of mutation, Fisher’s original theorem, understood to be a mathematical proof that Darwinian evolution is inevitable, is overthrown.

Kudos to Basener and Sanford for making this important point. Now, will the textbooks and the online encyclopedia articles take note?

*Photo: Ronald A. Fisher, via Wikicommons.*