At this point in time, it’s like beating a dead horse to attack Fisher’s Fundamental Theorem of Natural Selection.

This thread surveys problems outside of Basener Sanford 2018. There are at least 3 problems, and the 4th is the deal breaker for me.

The first problem is that there was no clear and stable statement of the theorem. Not good for a theorem that was hailed as so highly for a long time.

Fisher’s Fundamental Theorem of Natural Selection was hailed as “biology’s central theorem” by Richard Dawkins. The theorem was first stated in the book, *The Genetical Theory of Natural Selection* , which WD Hamilton said was “one of the greatest books of the present century” and “second in importance in evolution theory to Darwin’s *Origin* ”. Price described how Fisher spoke of his own theorem:

He [Fisher] compared this result to the second law of thermodynamics, and described it as holding ‘the supreme position among the biological sciences’

But even as the theorem was given such accolades, even by Fisher himself who called it “fundamental”, the theorem was “view by most population geneticists as out-of-date and of modest historical interest” (Basener and Sanford 2018). Ewens and Lessard give a “negative assessment” to “the long term relevance of the theorem.” And Joe Felsenstein referred to it almost derogatory terms as the “not so fundamental Fundamental Theorem of Natural Selection."

Regarding the statement of the theorem, Price notes Edward’s commentary:

only a few (eg. Edwards, 1967) have thought that Fisher may have been correct – if only we could understand what he meant!

So in fairness, how did Fisher himself state his theorem?

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

My reading of this is that this statement is the most general continuous-generation model statement of the theorem.

Ewens and Lessard 2015 give what they call a discrete-generation version of Fisher’s Theorem in equation 13 here:

On the interpretation and relevance of the Fundamental Theorem of Natural Selection - ScienceDirect

They said:

the correct statement of the FTNS in the discrete generation case

My reading is that this is a multi-loci discrete generation case, not a single loci haploid case. So, I’ll state the single loci haploid case of FTNS (which Joe Felsenstein doesn’t consider THE statement of Fisher’s theorem, so we can call it Fisher like). But in my defense, Queller 2017 and Barton 2009 describe it as THE statement of the theorem. I think Ewens and Lessard give the more accurate description that this is merely a special case application of FTNS. So this is the special case statement of FTNS:

\LARGE {\bar{w}}^\prime-\bar{w}=\frac{Var\left(w\right)}{\bar{w}}

just to clarify, in the discrete generation case, \bar{w} is a function of the generation K, so to emphasize this I like to say, \bar{w}(K).

So another way of stating FTNS in the simplest case is:

\LARGE \bar{w}\left(K+1\right)-\bar{w}\left(K\right)=\frac{Var\left(w,K\right)}{\bar{w\left(K\right)}}

But this leads to the 2nd problem of FTNS, namely, it’s a nothing burger!

Consider the example where we have a set of 3 numbers:

{1, 2, 3}

The mean is 2.0 and the variance is 2.0. But there are many sets of 3 numbers that posses the same variance and mean. For example, the following sets have the same mean and variance as the above set (within a few significant digits):

{ 0.845299462, 2.577350269, 2.577350269}

and

{ 0.941699476, 2.129150262, 2.929150262 }

Fisher’s theorem relates the variance in fitness at one point in time to the increase in fitness. But given that many sets of values of W_i might result in the same variance in relative fitness w_i, any statement of Fisher’s theorem in terms of variance is insufficient to determine the exact trajectory of the population sizes at any given time. If anything, information is lost by appealing to summary statistics such as variance of numbers rather than the numbers themselves!!!

So even if one rejects the simplified version of FTNS above as THE statement of FTNS, the inadequacy remains for FTNS to improve the computation of the system state at any time compared to simply using the fitness values themselves!!!

Why not just use the fitness values themselves to compute the state of the population than the variance of the fitness values, which doesn’t work for all time any way??? FTNS is an interesting mathematical relationship, but it actually degrades and blurs the evoluton of the system relative to the clarity the fitness values themselves provide.

The 3rd problem is that although the idealization of infinite sized populations make many models tractable, for small finite-sized real populations of complex multi-locus organisms, the relevance and value of describing traits in terms of fitness become increasingly suspect.

The 4th problem, perhaps the worst problem is that W_i for allele i or whatever i is – that 99% of the “fit” types might be function compromising. Framing fitness in term of differential reproductive success doesn’t really speak to the long term evolution of complexity that Darwin envisioned for Natural Selection. In fact it suggests, even on the assumption that the mean fitness is always increasing where:

\bar{w} \equiv \text{adaptedness}

functional compromise is always increasing!

So even if one argues fitness and adaptedness are increasing, it’s only an equivocation of common sense understanding of what it means to be improved or more fit.

NOTES:

here was my derivation of the simple case of FTNS:

\bar{w}\left(K\right)\equiv\sum{p_i\left(K\right)w_i}

and

\mathrm{Var}\left(w,K\right)\equiv\sum{\{p_i\left(K\right)w_i^2\}}-{\bar{w}\left(K\right)}^2

but it can be shown

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

With

\LARGE \bar{w}(K+1) \equiv\sum{p_i\left(K+1\right)w_i}

then

\LARGE \bar{w}\left(K+1\right)=\sum{\frac{w_ip_i\left(K\right)}{\bar{w}}w_i}=\sum\frac{p_i\left(K\right)w_i^2}{\bar{w}(K)}

We can state the above relations as:

\LARGE \bar{w}\left(K+1\right)-\bar{w}\left(K\right)=\left\{\sum\frac{p_i\left(K\right)w_i^2}{\bar{w}(K)}\right\}-\bar{w}(K)

but since:

\LARGE \bar{w}=\frac{{\bar{w}}^2}{\bar{w}}

we can say:

\LARGE \bar{w}\left(K+1\right)-\bar{w}\left(K\right)=\frac{\left\{\sum{p_i\left(K\right)w_i^2}\right\}-{\bar{w\left(K\right)}}^2}{\bar{w\left(K\right)}}

which reduces to:

\LARGE \bar{w}\left(K+1\right)-\bar{w}\left(K\right)=\frac{Var\left(w,K\right)}{\bar{w}\left(K\right)}

QED