Fisher’s theorem, reportedly proving Darwinism, is currently disputed in mathematical literature by William Basener and John Sanford. (Paper.) The controversy is attracting quite some attention. Dr. Basener has kindly offered an explanation for one of the questions raised in a comment and, for reader convenience, we reproduce both the question and the response as a post:
Question: Erasmus Wiffball at 7:
William Basener (Bill B):
Do you agree that ID proponents commonly mistake mathematical models of evolution as attempts to prove that evolution works?
Would you please tell everyone here what Fisher’s objective was in formulating his model? What was he attempting to model? To what degree did he succeed or fail in what he was attempting to do? (Surely he did not fail categorically.)
Response: William Basener at 12:
Erasmus Wiffball RE 7
The short (and thus incomplete) answer is this: Fisher attempted to prove that Mendelian genetics logically must lead to a Darwinist evolution. He believed that he was successful and along the way he (co)invented population genetics and modern statistics. However, took 40-80 years for people to realize he did not achieve his original goal. His attempt to “prove Darwin from Mendel” however did give a framework for connecting Mendelian discrete genetics with gradual change in observed traits across a population.
Now for the (apologetically) long answer…
We can see what Fisher wanted to prove and what he actually proved on the Wikipedia site for his Fundamental Theorem of Natural Selection. First, here is what Fisher claimed he proved:
“The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.”
The idea is conceptually simple: if you have a population, then the organisms with higher fitness (=reproduction rate) will reproduce more quickly, and thus over time become a large proportion of the population. Fisher made the simple observation that all populations have genetic variation, and thus all populations must be undergoing perpetual increase in mean fitness. From his book Genetical Theory of Natural Selection,, p.36): “As in the physical world we can conceive of theoretical systems in which dissipative forces are wholly absent, and in which the entropy consequently remains constant, so we can conceive, though we need not expect to find, biological populations in which the genetic variance is absolutely zero, and in which fitness does not increase.”
At the time of Fisher’s work, Darwin’s theory was in popular decline, mainly due to a perceived conflict with Mendel’s work. This period has been called the eclipse of Darwinism. Fisher’s goal was to revive Darwinism by showing that Mendelian discrete genetics leads to Darwinism as a required consequence, as we can see in the following quote from the very good historical paper Plutynski (2006): “His aim was to vindicate Darwinism and demonstrate its compatibility with Mendelism —indeed, its necessity given a Mendelian system of inheritance”
Fisher was fundamental in providing a sound theoretical justification for what has become known as the Modern Synthesis, especially the nice figure showing the relation between Mendelian Genetics and Natural Selection which fits the quote from Plutynski 2006 above).
For another explanation of what Fisher did, here is a quote from “The end of the eclipse”: “During the period 1916–1932, the discipline of population genetics developed largely through the work of the geneticists Ronald Fisher, J.B.S. Haldane, and Sewall Wright. Their work recognized that the vast majority of mutations produced small effects that served to increase the genetic variability of a population rather than creating new species in a single step as the mutationists assumed. They were able to produce statistical models of population genetics that included Darwin’s concept of natural selection as the driving force of evolution.”
BUT, that is not what Fisher actually proved, and his belief that his theorem gives ongoing fitness increase (what we labeled as “Fisher’s Corollary”) is not actually true, and Fisher’s theorem probably does not actually give a theoretical support for the Modern Synthesis. From the Wikipedia article on Fisher’s theorem , people have determine, over 70 years of re-re-evaluation of his theorem, that what Fisher actually proved is: “The rate of increase in the mean fitness of any organism at any time ascribable to natural selection acting through changes in gene frequencies is exactly equal to its genetic variance in fitness at that time.”
The new part, (ascribable to natural selection acting through changes in gene frequencies), is there because people have realized that there are a lot of things that affect fitness other than competition between pre-existing gene frequencies. One of the leading people sorting out what Fisher did and did not prove was George Price, who according to Wikipedia, “did not find it to be of great significance.” This is mainly because there are phenomena such as epistasis and dominance that make it technically not true. See, also, for example, the abstract for Ewens 1989: “Fisher’s “Fundamental Theorem of Natural Selection” has long caused controversy in population genetics theory. Viewed as a statement about the increase, or rate of increase, of mean fitness over time, it encounters difficulties with cases arising in a multi-locus system for which mean fitness can decrease. An interpretation of the theorem is put forward here which implies that it is correct as a mathematical statement, but of less biological value than was claimed by Fisher.”
So, while Fisher’s FTNS has been known to be an incomplete model of the effects of natural selection, is often still treated as if the general idea supports the Modern Synthesis – observe the wording by Ewens, “for which mean fitness can decrease”, as if this is an exception to the norm. It is often perceived that the upward pressure of natural selection examined by Fisher still exists, if not provable in the manner attempted by Fisher.
In our paper, John and I provided a model which allows the inclusion of general mutations (using a realistic distribution for the effect of mutation on fitness) and showed that mutations have the potential to drive fitness down and cause extinction. This has been known to be possible before (See: Lynch 1995, “Mutation Accumulation and the Extinction of Small Populations,” which has been cited 770 times. What John and I showed in our paper that these mutations can be incorporated in Fisher’s original framework and provided a way to measure the downward pressure of (realistically modeled) mutations against the upward pressure of selection (…in the model at least – it is important to always remember that models are only a partial representation of reality…)
Something that is important is the very substantial amount of previous research, which we reviewed in Section 2 of the paper, showing that in models of populations and in real populations, mutations can and sometimes do accumulate to the eventual “mutational meltdown” of the population. The vast majority of mutations are deleterious (decrease fitness). Here is a quote from Section 2.3 of our paper:
The predominance of deleterious mutations over beneficial ones is well established. James Crow in (1997) stated, “Since most mutations, if they have any effect at all, are harmful, the overall impact of the mutation process must be deleterious”. Keightley and Lynch (2003) given an excellent overview of mutation accumulation experiments and conclude that “…the vast majority of mutations are deleterious. This is one of the most well-established principles of evolutionary genetics, supported by both molecular and quantitative-genetic data. This provides an explanation for many key genetic properties of natural and laboratory populations”. In (1995), Lande concluded that 90% of new mutations are deleterious and, the rest are “quasineutral” (Also see Franklin and Frankham (1998)). Gerrish and Lenski estimate the ratio of deleterious to beneficial mutations at a million to one (Gerrish and Lenski 1998b), while other estimates indicate that the number of beneficial mutations is too low to be measured statistically (Ohta 1977; Kimura 1979; Elena et al. 1998; Gerrish and Lenski 1998a). Studies across different species estimate that apart from selection, the decrease in fitness from mutations is 0.2–2% per generation, with human fitness decline estimated at 1% (See Lynch 2016; Lynch et al. 1999). Estimates suggest that the average human newborn has approximately 100 de novo mutations (Lynch 2016). Research using finite population models has been driven by the need to understand the impact of the buildup of deleterious mutations (called mutational load) in small populations of endangered species (See Lande 1995; Franklin and Frankham 1998). Of special interest is the mutational load in the human species given the relaxed selection due to social and medical advances (Kondrashov 1995; Crow 1997; Lynch 2016).
Moreover, the deleterious mutations can have a very significant effect on fitness, potentially accumulating and leading to a negative growth rate and “mutational meltdown” extinction. Also from Section 2.3 of our paper:
Of critical importance are deleterious mutations that are small enough in effect to accumulate, which Kondrashov calls “very slightly deleterious mutations” (VSDMs) (Kondrashov 1995). He states, “The study of VSDMs constitutes one of the pillars of population genetics” and attempts to quantify the most dangerous range of VSDMs as follows: “deleterious mutations with an effect less than 1/G (where G is the length of the genome) have little effect no fitness even in large numbers, and that deleterious mutations with an effect greater than 1/4Ne (where Ne is the effective population size) will be eliminated via selection”. He then observes, “In many vertebrates Ne ? 10^4, while G ? 10^9, so this dangerous range includes more than four orders of magnitude” (Kondrashov 1995). Other authors (e.g. Butcher 1995) have described this dangerous range in terms of Muller´s ratchet; deleterious mutations with a larger effect give a larger turn of the ratchet at each click but have a slower rate of clicks (because they are more susceptible to selection), while mutations with smaller effects give a smaller rotation at each click but have a higher click-rate. The mutations with the greatest long term impact on fitness are in the middle range with the greatest net rotation rate of the ratchet. These are the mutations, like those in the range of values observed by Lynch et al that minimize time to extinction (Lynch et al. 1993), which can accumulate over time and have significant net impact over time on fitness. In (1997), Crow describes the effect of these mutations as follows: ‘…diverse experiments in various species, especially Drosophila, show that the typical mutation is very mild. It usually has no overt effect, but shows up as a small decrease in viability or fertility, usually detected only statistically. … that the effect may be minor does not mean that it is unimportant. A dominant mutation producing a very large effect, perhaps lethal, affects only a small number of individuals before it is eliminated from the population by death or failure to reproduce. If it has a mild effect, it persists longer and affects a correspondingly greater number. So, because they are more numerous, mild mutations in the long run can have as great an effect on fitness as drastic ones.’
In mutational meltdown, each individual mutation is like a speck of rust on a car – too small to be selected out. But as they accumulate, eventually they build up to cause big problems as parts fall off the car. This is a fine analogy (as analogies go) for the very important mutation accumulation in small populations. (Browse the 770 papers in the “cited by” list for the Lynch paper above, and you will see a lot of papers working with real populations of endangered species.)
Anyhow, in Section 2 we review literature on mutation-selection models, and there is a theme of trying to maintain equilibrium against downward mutational effects. A goal of our paper is to give a mechanism whereby research can get some quantitative evaluation of the upward pressure of selection and the downward selection of mutations.
It’s worth noting that Fisher considered mutations in his FTNS, but he believed they could be ignored because the extremely deleterious ones were lethal (or nearly so) and could be ignored, and the mutations with mild effects were 50/50 beneficial to deleterious. This view of mutations is now known to be untrue, as described in the research quoted above.
So, Fisher was an unquestioned genius, one of the top scientists of the 20th century. He (co)-founded population genetics and modern statistics. Yet, his desire to justify Darwinism led him to make logical flaws and overstate the certainty and applicability of his work.
There are good lessons in that, and I don’t mean only for those with a Darwinist view of the world. Scientists are people trying to form a big picture out of a collection of observations/measurements/facts. I think it was Henri Poincare who said that a collection of facts are not more a theory than a pile of rocks is a house. We are trying to assemble the rocks into a form that makes sense to us on a larger scale. It is human insight and some level of belief in something beyond the rocks themselves that makes us believe they form a house. All scientists have a faith that there is *some* overall house, some order to the universe.
The greater house is constructed as scientists argue of how the rocks go together – hypotheses are posed and refuted – but it is critical to distinguish between the hard observations of rocks and our view of how they should go together. Everybody has a philosophy or worldview that informs the lines of investigations and hypothesis they pose, but we have to check that view when it comes to validating experimental results, otherwise, we are just validating preconceptions instead of doing science. Properly done science works.
There is a long discussion that can take place about what the best worldview for building hypothesis is, but for the interested reader, I’ll defer to Pearcey and Thaxton’s book The Soul of Science.
See also: Fisher’s Proof of Darwinism Flipped: William Basener replies to Bob O’Hara. The mutation rate used in the paper is 1 mutation per generation. As with all the parameters in the paper we chose this parameter so that if there is any bias, the parameter selection favors selection and increasing fitness.
“Fisher’s Proof of Darwinism Has Been Flipped” paper is making waves – Twitter displeased
Fisher’s proof of Darwinian evolution has been flipped?