The fundamental problems with Fisher's not-so-fundamental Fundamental Theorem of Natural Selection

Frist the Royal Society Web page:

Ronald Fisher | The Royal Society

He therefore could be said to have provided researchers in biology and medicine with their most important research tools, as well as with the modern version of biology’s central theorem.

This sound hauntingly like Dawkins here:

Evolutionary Zoologist, University of Oxford. Author, The Blind Watchmaker; The Greatest Show on Earth

I’ve listened to Armand Leroi’s talk: beautifully delivered (apart from his infuriating use of the historic present), but his nomination of Aristotle is obviously just contrarian for the sake of it. He is simply bending over backwards to contrive a way of answering the question with a name other than Darwin. And that’s a hopeless cause! I prefer to answer the question straight, making no attempt to be contrarian or ‘interesting’. Darwin, of course, is the greatest biologist ever.

Who is the greatest biologist since Darwin? That’s far less obvious, and no doubt many good candidates will be put forward. My own nominee would be Ronald Fisher. Not only was he the most original and constructive of the architects of the neo-Darwinian synthesis. Fisher also was the father of modern statistics and experimental design. He therefore could be said to have provided researchers in biology and medicine with their most important research tools, as well as with the modern version of biology’s central theorem.

And Fisher himself it was a Fundamental Theorem, like the 2nd law of Thermodynamics! Who is anyone to dispute such an esteemed, honorable, Christian (albeit a neo-Darwinist).

Regarding Lewontin, Santa Fe Winter 2003 Bulletin:

The problem is that it is not entirely clear what fitness is. Darwin took the
metaphorical sense of fitness literally. The natural
properties of different types resulted in their differential
“fit” into the environment in which they lived.
The better the fit to the environment the more likely
they were to survive and the greater their rate of
reproduction. This differential rate of reproduction
would then result in a change of abundance of the different

In modern evolutionary theory, however, “fitness”
is no longer a characterization of the relation of the
organism to the environment that leads to reproductive
consequences, but is meant to be a quantitative expression
of the differential reproductive schedules themselves.
Darwin’s sense of fit has been completely

In pop gen:

W is the number of offspring associated with an allele. In simple models W is a constant. In simple discrete-generation, infinite-population-size models the number of individuals P(K) in generation K is

\LARGE P(K) = P(0)W^K

where P(0) is the initial population at generation zero.

Relating this to exponential growth functions that are solutions to simple differential equations, let:

\LARGE W = e^\alpha


\LARGE P(K) = P(0)e^{\alpha K}

So this stuff has been known for centuries.

For multiple alleles with each allele associated with an index i, for each allele:

\LARGE P_i(K) = P(0) W_i^K

If we scale things down to relative terms, we get the p_i 's and w_i 's used in the FTNS-like formula above that is stated in terms of variances and means. It’s a nothing burger.

That’s not my assessment. Ewens and Lessard gave FTNS a negative assessment as far as it’s utility and centrality. It was central in to a mythologized view of the theorem itself. Fisher had many things to be admired for. One could even admire the FTNS as a mathematical realation, but it’s hardly fundamental, it is “not-so-fundamental”.

Then how could Salvador Cordova have persisted under the misapprehension that evolutionary theory says that something which is “fit” is “necessarily good” and “adapted towards future environmental uncertainties”?

Or that the usage of the phrase “fitness increasing” becomes “dubious” when fitness increasing mutations have phenotypic effects that are “functional compromises”?


The word “FITNESS” in the biological sense from the dictionary: (search google on “fitness dictionary”).


an organism’s ability to survive and reproduce in a particular environment

Does not seem too quasi-philosophical or metaphysical to me.

There seems to be a move afoot about with the genetic decay crowd to redefine fitness to suit the narrative of an originally flawlessly “fit” creation of ideal kinds, characterized by functional perfection and unblemished by the accumulation of generations of mutation.

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I wasn’t clear as I was referring to equations II-2, II-3, II-4, II-5, II-6 which are for 2 alleles. To be properly generalized for your proof toward page 93, these have to be generalized to k alleles from the 2 allele case because you were specifically referencing those equations in your proof on page 93.

I had to prove it to myself that these equations at the beginning of Chapter II were generalizable and could be used without accidental equivocation for the proof on page 93. That was easy enough, and they also had to scale the absolute fitnesses to relative fitnesses in those equations to be strictly in line with their use on page 93. That was also easy enough. But I went the extra mile for the sake of rigor. I didn’t show that part of the proof in my OP.

I’m happy to post that part of my derivation which I didn’t earlier since I was trying to be terse.

I think a version for multiple alleles in haploids will be found in the pioneering textbook Population Genetics by C. C. Li in 1954. He may cite it to Sewall Wright, I don’t recall whether he does.

Thanks for the response on that.

The one glaring problem I see is in assuming the environment and selective pressures are completely constant and homogenous. In reality, different alleles can be fitter in different parts of a species range. For example, there are several different coat color genotypes in pocket mice that are either beneficial or deleterious, depending on where the mice are living geographically:


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Sal Cordova said

OK, let’s look on page 93 (current edition of my online text) at the passage deriving the change of mean fitness for haploids with multiple alleles. It says right there that:

Equation (II-17) that I mention there is described, when it is given on page 58, as simply a repetition of equation (II-7), with the current and next generations indicated by the gene frequencies being p and p' instead of having superscripts for the generation number. (II-7) in turn is the rewriting of equation (II-6) in terms of gene frequencies rather than numbers (in the asexual or haploid cases these are also the genotype frequencies of asexuals).

So I think that the haploid case the equations for change of gene frequencies are in fact derived. But tastes may differ – perhaps it is necessary for me to “dot the i’s and cross the t’s” a little bit more, to be fully rigorous.

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Thank you for your response.

I had no doubt you were correct, but as an exercise in trying to understand the math, I tried to work derivations out on my own.

Plus I was trying to cull the essentials into a few pages for an educational essay and lesson I’m writing on Fisher’s theorem. Also I’m trying to translate the conventions in pop gen literature to the symbolic conventions more familiar to typical math, engineering and physics students.

This is what I came up with, and even then some of my notation is sloppy partly because latex rendering on this site doesn’t work like in MS word.

This is what I came up with. Your notation uses N, I use P (for population). N tends to have connotations for other things.

Let capital P be absolute population size and W absolute Darwinian fitness:


I define:

P_{tot}\left(K\right) \equiv \sum_{i=1}^{N}{P_i\left(K\right)}

\LARGE p_i\left(K\right) \equiv \frac{P_i\left(K\right)}{P_{tot}\left(K\right)}

\LARGE \bar{W}\left(K\right)\equiv\sum_{i=1}^{N}{p_i\left(K\right)W_i}

Given the definition of W_i we can relate the population size of an allele in one generation with the population size in another generation:


Given this we can state a similar result for allele frequnces across two generations:

\LARGE p_i\left(K+1\right)=\frac{P_i\left(K+1\right)}{P_{tot}\left(K+1\right)}=\frac{P_i\left(K+1\right)}{\sum_{i=1}^{N}{P_i\left(K+1\right)}}

Combining some of the above:

\LARGE p_i\left(K+1\right)=\frac{W_iP_i\left(K\right)}{\sum_{i=1}^{N}{W_iP_i\left(K\right)}}

Dividing both numerator and denominator simultaneousl by P_{tot} (since this is basically dividing by 1):

\LARGE p_i\left(K+1\right)=\frac{W_ip_i\left(K\right)}{\sum_{i=1}^{N}{W_ip_i\left(K\right)}}

which implies

\LARGE p_i\left(K+1\right)=\frac{W_ip_i\left(K\right)}{\bar{W}\left(K\right)}

The bar symbol to indicate a mean value doen’t show up to well unfortuantely with latex.

Let W_{ref} be the absolute fitness used to scale the other absolute fitnesses to relative fitness. Then given customary notions of relative fitness, we can scale the above as follows

\LARGE p_i\left(K+1\right)=\frac{\frac{W_ip_i\left(K\right)}{W_{ref}}}{\frac{\bar{W}\left(K\right)}{W_{ref}}}=\frac{w_ip_i\left(K\right)}{\bar{w}(K)}

which then connects to the proof you stated on page 93 with slightly different notation.

I know that the prime symbol corresponds to the K+1 generation, but the prime symbol can be mistaken for the first derivative! Also, some of the the commas in the equations look like prime symbols for terms in the denominator.

Since I’m writing for non-popgen types students of math (like me), I’m writing my essay with symbols and conventions they may be more familiar with. I want math students to be able to understand your textbook so I’m trying to connect the notations in my essay. For example, this was actually very helpful notion in your book, and I merely preserved it and amended it:

\LARGE p_A^{(t)}
\LARGE p_A^{(t+1)}

I used K instead since t might be interpreted as a continuous variable.

So my notation is

\LARGE p_i(K)
\LARGE p_i(K+1)

I’m not suggesting pop gen adopt new conventions, I’m merely trying to translate the specialized conventions in pop gen for typical students of math, engineering, physics so they can more easily understand pop gen literature and concepts without having to go through an entire semester.

I’m merely pointing out one of the reasons I’m avoiding the prime notation in my essays (in addition that it is sometimes used to indicate a derivative).

Here is something from page 52 of Joe’s book. The red arrow highlights a comma, which when I was skimming the page in smallish print, I first thought was a prime symbol!


On further reading, it became clear. It’s a very minor thing.

ME (on complete no genetic diversity = MAXIMAL FITNESS):

How is this necessarily good?


Did someone claim it was?

I think that was the implicit assumption of Darwinian evolution leading to ever more fitness and complexity.

But let’s consider what this means in pop gen lingo. The most fit population of alleles is when the allele with the best growth rate finally ends up fixing at the locus and then there are no more other alleles. For a total genome, assuming a population could actually survive the journey, all the individuals become clones with homozygousity at all loci. Eh, that means when it is maximally fit, it evolves no further!!!

For it to evolve, it has to become UNfit again by having new mutations. But as I pointed out, if “beneficials” are function compromising, the collective genomes of the population doesn’t move forward in complexity.

So Dawkins Weasel evolves to “METHINK ITS LIKE A WEASEL” and never evolves to something more complex! It’s at a dead end, a fitness peak.

Natural Selection in this case PREVENTs the creature from evolving to a far more complex and memorable Shakesperean speech like this one by Henry:

No Sal. When the environment changes the population is no longer at the local fitness peak. The population doesn’t have to have mutations to become less fit. Sadly we’re seeing that right now as climate change is altering environments so quickly previously well adapted species can’t evolve quickly enough to keep up. Many are facing extinction for exactly that reason.

You never did tell us where you got the “99% of all fitness increasing mutations are function compromising”. We’ll just assume you made it up.


Then you have misunderstood Darwin, and subsequent generations of evolutionary biologists. Full stop.

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In reality, species encounter many changing and shifting environments, both geographically and temporally. The other species they compete against are also evolving. Like any scientific model, Fisher’s model is not reality and it can’t fully capture the complexity of how the real world operates. Animals at the edge of their range will experience different selective pressures than those in the middle of their range. For example, a species found in a desert will face different selective pressures at the edge of the desert where it starts to give way to grassland or forest compared to the species in the middle of the desert.

You are ignoring the possibility of exploring new niches. One of the strategies for increased fitness is to adapt to a different environment. For example, a fish well adapted to tidal regions can start to adapt to land in order to escape competition with others in its species or with other species. As the species moves onto land the species already on land will experience competition with a new species, changing the selective forces on the land species. On and on it goes.


This idealization completely misunderstands the situation.

A population does not exist in a narrow niche. It exists in a broader range of narrow niches. If the population is optimal for a narrow niche, then it will be confined to only part of that range of niches. Suboptimal might be better and allow for a larger population because it can exploit a wider range of niches.

So you should not expect a population with no other alleles. Having some genetic diversity is better for the population.


You are correct, that comma is too close to the N_a(t) term. I will put in a bit more space. And make sure that I warn the reader that primes do not indicate derivatives unless I specifically say they do.


Reminds me of something I just read

In orchid bees, these genes are expressed in their antennae, allowing them to detect airborne molecules. The researchers identified olfactory receptor gene 41 (OR41) as being different between the two species.
“That gene has accumulated a lot of changes between these two species, suggesting that those changes are responsible for the collection of different perfume compounds,” said Ramirez. “The idea here is that as these olfactory genes evolve and accumulate new mutations, they’re more sensitive to different molecules and therefore enable the bees to collect or not collect certain compounds.”

Fitness is relative to the current environment, not the future. How could it be?


Sal: first, thanks for this thread since it helps me understand some highlights of Joe’s book without having to do the hard work of actually studying it.

Philosophers of biology have opined on the concept of fitness as a term of art in biology. The competent philosophers do their work by first understanding the biology and only then analysing how fitness is used in contemporary theories, including mathematical models. If you are interested in this philosophical work, try topics “Philosophy of Biology” in IEP and “Fitness” in SEP as a starting point. There are also good intro Phil of Biology texts by Godfrey-Smith and Rosenberg.

A comment on your various posts on maximal fitness: I’m not clear from these posts whether you have a specific pop gen model in mind (other than in the Weasel post). If you do, it would be helpful to mention the assumptions and purposes underlying the model – I’m referring to material like that at the top of page 52, just before section II.2, in 2017 version of Joe’s book.

Profuse apologies for the late reply John, I actually had to track down where I did get that figure from. In the process I realized I got the 99% from a presentation by Michael Behe to a private group. I did not realize he may not have used that figure outside the group.

Michael Behe has publicly characterized Lenski’s “beneficials” as 90% function compromising.

This is an important topic in and of itself, I’m going to start a thread on it.

I’ll provide a link shortly.


here’s the link


Thanks for weighing in and the kind words.

I’m not clear from these posts whether you have a specific pop gen model in mind

The notion of W (absolute Darwinian fitness) and w (relative fitness) have roots in the single gene locus haploid model with discrete generations and INFINITE population sizes.

Much of pop gen that uses those symbols tries to specify under what conditions the symbols remain a valid model of fitness, and I would argue “GROWTH capacity” is a more appropriate term than fitness to describe the symbols W and w. Fitness is too metaphysical a word!

The simplest model of W is the number of bacterial individuals with an allele at a a single locus. Since bacteria “offspring” are the result of splitting apart the parent into two kids, W = 2. If we start the model off with only 1 bacteria, then the number absolute number of bacteria P for any given generation K is:

P(K) = W^K=2^K

It’s that simple. Everything proceeds from that. All the discussions about W and w are variations or amendments to that idea.

That’s mostly what I’m working from. There was a long discussion at TSZ over the computer science version of WEASEL vs. the pop gen version of weasel. It resulted in the alarming discovery by some (but no surprise to me) that the computer people and the pop gen people have different notions of the meaning of fitness.

OK, thanks, but I’m not sure if your comments reflect the two points I was trying to make.

  1. When scientists use a term of art like “fitness”, it’s not the everyday meanings that matter, eg those in dictionary. Instead, it is the meanings implicit in the theory and the mathematics of models capturing that theory. It’s that meaning what the philosophers try to explicate. The concept of fitness is nothing special in that regard; for example, there is similar philosophical work for the concepts of entropy, space, time in the light of modern physics.

  2. Any model has assumptions and idealizations/abstractions that limit its domain of applicability. For example, it would be wrong to try to draw conclusions about maximum fitness with a model where the role of the niche is purposely ignored.

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