Darwinism Intelligent Design Mathematics

Fisher’s Proof of Darwinism Flipped: William Basener replies to Bob O’Hara

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The paper, by William Basener and John Sanford, shows that the continuous flow of new mutations that would continuously replenish a population’s genetic variability and enable Darwinian evolution does not really happen.  (Paper.)

Much discussion has followed here and here.

William Basener

Basener has replied to Bob O’H, and for reader convenience, we are reproducing the comments here:

First, Bob O’H:

tjguy – The maths isn’t troubling (except that I’ sure they could have gone further). The simulation section shows that fitness can decrease, but we already knew this. Basener & Sanford don’t say what mutation rate they use though.

It’s obvious, I think, that the paper will be used to claim that mutations mean that evolution can’t work, so it’s a shame they don’t provide such an important parameter.

William Basener replies:

Bob O’H, RE 17: Your question regarding the mutation rate we used in the paper is pertinent. As you stated it has long been known that a high mutation rate can lead to decline in fitness while a low mutation rate allows for adaptation, observed in biological populations and mutation-selection models in the literature discussed in Section 2.2 (differential equations with an infinite-population). Thank you for the good question.

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.

This is in the mathematics in the paper, but you are certainly correct that we did not state this as the mutation rate and that would have been helpful to readers. In the math, Equation 5.1 for $f_{i,j}$ is the probability distribution provided by Kimura to estimate the effect of a single mutation on fitness. We modify this distribution to add beneficial mutations (at a rate and magnitude that are greater than observed estimates), and this is used as the net effect of all mutations for $f_{i,j}$ in Equation 3.2.

Anyone who wishes to explore parameters can use the JavaScript simulation I posted at my RIT web page (referenced in the paper):
This was used to create the figures in the paper, and thus provides full transparency and opportunity to validate/reproduce/modify our results.

I agree with you that it would have been nice to go further with the math, and done an exploration of the parameters space. But the paper is at 34 pages, and we wanted to provide proper support for the model (hence the beefy literature review in Section 2) and give proper context with regards to Fisher. I think you and I agree that determining behavior for parameters spanning the space of realistic values is an important next step. I think it would beneficial to have multiple different groups explore results for varying parameters.

We’ll keep readers posted re further discussions.

profileimageThe discussion on this point continues in the comments below, with Bob O’H at 1.

This is Bob O’Hara:

My career has been dedicated to discovering whether I am a biologist or a statistician – I intend to retire once I come to a definite answer. Most of my work has been on ecology and evolution, using statistical methods to put data and models together to try to learnmore about the real world.

My main interest at the moment is the development of methods to model the distributions and dynamics of species an communities, where we have a variety of data sources that need to be respected, and models of the unobserved true distributions and dynamics from which we want to infer what drives the distributions of species, and oredict how this will change in the near future.

See also: “Fisher’s Proof of Darwinism Has Been Flipped” paper is making waves – Twitter displeased


Fisher’s proof of Darwinian evolution has been flipped?

13 Replies to “Fisher’s Proof of Darwinism Flipped: William Basener replies to Bob O’Hara

  1. 1
    Bob O'H says:

    Bob O’Hara spotted?

    That’s an old me (photo taken on “Birdshit Island”). The new me is here. I have yet to examine the islands around Trondheim for avian excrement.

    Anyway, to the main point of the post. First, it’s not clear if the mutation rate is per individual per generation or per population. I suspect the former: if it’s the latter, then the expected reduction in mean fitness is (using their assumed population size of 10^9) is 10^-12 if we ignore selection. This is clearly too low: after 3000 generations it would still be almost the initial population size.

    If the mutation rate is per individual, then the result make more sense: the reduction in mean fitness due to mutation would be 10^-3. We know that in the short term, fitness cannot compensate for mutation if -U d > s, where U is the mutation rate per genome, d is the average effect of mutation on fitness (so the left hand side is the mean change in fitness due to mutation, and s is the genetic variance in fitness (I assume the Basener & Sandford model will reduce down to this too: it looks like it should).

    In the simulation in §5.4 of the B & S paper, their initial genetic variance in 0.000025, and d is -10^-3 (as near as makes no difference). So clearly the inequality is true, i.e. selection can’t compensate unless the genetic variance is increased 40 times.

    So how realistic is this mutation rate of one mutant per individual per generation? The simulation is for an asexual species, so we can look at bacteria. For E. coli the mutation rate per genome has been estimated as about 10^-3. So it seems to be too high.

  2. 2
    News says:

    Bob O’H at 2, update posted.

  3. 3
    Mung says:

    Bob O’H is my brother!

  4. 4
    bornagain77 says:

    Glad to see Bob O’Hara applying his expertise in statistics, trying to help finally develop a proper mathematical model for Darwinian evolution.

    “It is our contention that if ‘random’ is given a serious and crucial interpretation from a probabilistic point of view, the randomness postulate is highly implausible and that an adequate scientific theory of evolution must await the discovery and elucidation of new natural laws—physical, physico-chemical, and biological.”
    Murray Eden, “Inadequacies of Neo-Darwinian Evolution as a Scientific Theory,” Mathematical Challenges to the Neo-Darwinian Interpretation of Evolution, editors Paul S. Moorhead and Martin M. Kaplan, June 1967, p. 109.

    Hopefully, he is helping to develop a proper mathematical model for Darwinian evolution that finally realistically reflects reality. Such as the fact that biological form is not even reducible to mutations in DNA in the first place:

    Darwinism vs Biological Form – video

    Or perhaps he will incorporate the fact that quantum biology also now shows us that Darwinists, with their reductive materialistic framework, are not even on the correct theoretical framework to begin with in order to properly understand and/or ‘mathematically model’ biology?

    Darwinian Materialism vs Quantum Biology – video

    Of course one is always hopeful that those who unquestionably accept Darwinism will finally make mathematical models that properly reflect reality and thus finally reject Darwinism, but, having been down this road for a number of years, I am not holding my breath for that hope.

    Darwinian Evolution: A Pseudoscience based on Unrestrained Imagination and Bad Liberal Theology – video

    Verse and video:

    2 Corinthians 10:5
    Casting down imaginations, and every high thing that exalteth itself against the knowledge of God, and bringing into captivity every thought to the obedience of Christ;

  5. 5
    kairosfocus says:

    BA77, a very happy new year, we have been concerned, noted your vids with interest. God’s richest. KF

    PS: Could you email me?

  6. 6
    Origenes says:

    Happy new year bornagain77.

  7. 7
    Erasmus Wiffball says:

    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.)

  8. 8

    BA77 is back. That’s the best news I’ve heard so far today!

  9. 9
    Bill B says:


    You are correct that the mutation rate is one mutation per individual.

    I am honestly somewhat surprised you find the numerical simulation in Section 5.4 so interesting. I expected that in this paper there would be a large number of things that people would say, “yeah well we already knew that…”, but things in this “yeah we already knew that” bag would be known only to active researchers in certain communities, and would not be part of the general Neo-Darwinian discussion. A finite population with realistic parameters that has fitness decreasing to negative values should not be surprising, given the work of Lynch etc. which we review in Section 2 of the paper.

    The simulation in Section 5.4 is not presented in the paper as a model of any specific organism, or of what is generally to be expected behavior. We do not claim that most species are driven to negative reproduction in 1000 years. I will quote the last line of Section 5.4:
    “We observe that it problematic to define parameter settings that are biologically realistic yet result in continuous fitness increase, supporting the modeled buildup of very slightly deleterious mutations described in Kondrashov’s paradox.”
    or from the abstract:
    “The expanded theorem has biological implications significantly different from what Fisher had envisioned.”
    I contend that the simulation in Section 5.4, in combination with the rest of the paper, supports those conclusions.

    We could go back and forth over mutation rates and parameters for biological populations, but I find such exchanges unfruitful. As I said earlier, I would like see a proper evaluation of our model across a comprehensive span of realistic parameters, including some various estimates of specific organisms. (Since you mentioned it, I’ll include a simulation using estimates for E. Coli if I do a parameter evaluation, which is likely.) I could point out that the human mutation rate is 100 per individual (we could argue over whether they should all count against fitness given possible junk DNA, and we could argue if Kimura’s Gamma distribution for mutational effect considered mutations in junk DNA and if it matters) but in any case, 1 mutation per individual is well below human mutation rates. You have pointed out that our system is set up for asexual reproduction, which I agree although 1) Fisher did not restrict his FTNS only to asexual reproduction and 2) Lynch’s work (Section 2 of our paper) indicates that sexual reproduction slows genetic collapse rather than preventing it. I am sure we find more to go back and forth over, maybe debate what it means to be ‘realistic’, but it would be far more productive (and less wasteful of both our efforts) to just put estimated parameter values into a mathematical model and check the results. If you are concerned that the single simulation will be taken out of context to say that species generally reach negative reproduction rates within 1000 years (or something like that), I’ll agree up front that nobody should draw that conclusion.

    The point of that simulation is to show that using Fisher’s equations with realistic mutations can result in continual fitness decline; combined with the rest of the paper we can say continual up is really hard to find. This might be surprising to some because Fisher’s work originally was thought to be a rigorous proof of continual up as universal as entropy. Fisher’s view on how his theorem works in combination with mutations (which we label “Fisher’s Corollary”) is logically false because his assumptions about mutations wrong, and it is also contrary to what we see in mutation-selection models.

    The thing that I would expect to be more troubling (for someone who believes that mutations plus selection has created the diversity of life from a very simple start) would be the substantial amount of research into the mutation-selection process (See Section 2 of our paper, but also papers you have cited) that does not show an intrinsic upward push; to the contrary the main question seems to be how do we avoid fitness decline and maintain equilibrium.

    There are other more interesting things in our paper. You cited a paper with an error threshold on the mutation rate and effect separating fitness decline from staying at equilibrium – our main theorem gives a formula separating fitness decrease from increase based on the mutation distribution (which includes rate and magnitude) and the population distribution. Our model is more comprehensive than assuming every mutation has the same fixed affect (as with Lynch’s model), which could provide more comprehensive information on fitness decline in small populations – i.e. endangered species.

    Anyhow, I am going to try to drop out of this debate. I have work I have to get to, and blog debates consume my focus too much. I prefer the pace of publishing papers. Best wishes to all, especially Bob.

  10. 10
    Erasmus Wiffball says:

    It’s a lot easier to publish a bad paper than a good paper showing that a bad paper is bad, particularly if the bad paper is long and elaborate.

  11. 11

    BA77 — welcome back … you’ve been missed.

  12. 12
    Bill B says:

    For reader convenience, this comment is now also available as a post, here.

    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 (https://en.wikipedia.org/wiki/Fisher%27s_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 (https://archive.org/stream/geneticaltheoryo031631mbp#page/n57/mode/2up, 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 theorey was in popular decline, mainly due to a perceived conflict with Mendel’s work. This period has been called the eclipse of Darwinism (https://en.wikipedia.org/wiki/The_eclipse_of_Darwinism#CITEREFBowler2003). 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 (https://pdfs.semanticscholar.org/62d8/dfdbfb9d5a582c99493aefb145a066ee356f.pdf, 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 (See https://en.wikipedia.org/wiki/Modern_synthesis_(20th_century), 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” from (https://en.wikipedia.org/wiki/The_eclipse_of_Darwinism#CITEREFBowler2003): “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 (https://en.wikipedia.org/wiki/Fisher%27s_fundamental_theorem_of_natural_selection), 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 (https://en.wikipedia.org/wiki/Fisher%27s_fundamental_theorem_of_natural_selection), “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 (https://www.ncbi.nlm.nih.gov/pubmed/2814903 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” http://www.journals.uchicago.e.....086/285812, which has been cited 770 times (list provided at the link.) What John and I showed in our paper that these mutation 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 import is the very substantial amount of pervious 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 know 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 – hypothesis 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.”

  13. 13
    Bill B says:

    Adding to the above conversation – I believe there is no better view of the universe from which to make hypothesis and assemble rocks than that of a universe designed by an biblical Creator.

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