

































Volume 2, Issue 2

Fisher’s Fundamental Theorem of Natural Selection
Isn’t Fundamental After All

Salvador Cordova
DOI: 10.33014/issn.2640-5652.2.2.cordova.1

Abstract

Fisher’s Fundamental Theorem of Natural Selection
(FTNS) was called “biology’s central theorem” (Fisher,
1930, pgs. 36–37; Brockman, 2011; Royal Society, 2020).
FTNS might possibly have been accorded this status for
decades because Fisher himself declared his own theorem
to be fundamental to biology (Fisher, 1930, pgs. 36–37).
However, the idea that Fisher’s theorem is biology’s cen-
tral theorem is by-and-large a myth promoted by popu-
lar science writers like Richard Dawkins (Brockman, 2011).
Joseph Felsenstein, when delivering the 2018 Fisher Memo-
rial Lecture declared that FTNS was “alas, not so funda-
mental” (Felsenstein, 2018; Felsenstein, 2017, pg. 94). One
may be hard-pressed to find a biology textbook or biology
student who can explain how FTNS helps them understand
biology. Even the meaning and proof of the FTNS have re-
mained contentious even to this day (Price, 1972; Basener
and Sanford, 2018).

Not only does FTNS do little to nothing to explain bi-
ological evolution, but like most population genetic and
evolutionary literature, FTNS relies on a definition of fit-
ness in terms of population growth rates rather than the
biophysical notions of fitness which are more in line with
the common-sense intuitions of the medical and engineering
communities.

From the perspective of the biophysical (rather than the
population growth) notion of fitness, natural selection
might be more accurately described as an agent against the
increase of complexity rather than an agent for it. Thus,
metaphorically speaking, some sort of anti-Weasel model of
natural selection might better describe how selection actu-
ally works in nature rather than Dawkins’ Weasel or other
man-made genetic algorithms.

However, the main focus of this article is to provide some
pedagogical insights through simple numerical illustrations
of Fisher’s theorem. The hope is that this will show the
general irrelevance of FTNS to the question of the evolution
of complexity by means of natural selection, and thus show
that Fisher’s theorem is not so fundamental after all.

1 Introduction: The Problem of
Defining Fitness

In order to understand the meaning of Fisher’s Fundamen-
tal Theorem of Natural Selection, it is helpful to understand
the definition of fitness which Fisher was using. This is not
so trivial a problem because in a 2009 paper on fitness, Allen
Orr quipped, “biologists have o!ered a staggering number
of definitions of fitness” (Orr, 2009), and worse Richard
Lewontin lamented, “it is not entirely clear what fitness is”
(Lewontin, 2003).

Andreas Wagner went even further:

However, fitness is hard to define rigorously and
even more di"cult to measure. . . An examination
of fitness and its robustness alone would thus not
yield much insight into the opening questions. In-
stead, it is necessary to analyze, on all levels of
organization, the systems that constitute an or-
ganism, and that sustain its life. I define such
systems loosely as assemblies of parts that carry
out well-defined biological functions.
(Wagner, 2005, pg. 1)

Ironically, “systems loosely as assemblies of parts that carry
out well-defined biological functions” sounds remarkably
similar to Michael Behe’s definition of Irreducible Complex-
ity: “A single system composed of several well-matched, in-
teracting parts that contribute to the basic function of the
system” (Behe, 1996, pgs. 39–40).

The population-genetic/evolutionary biology definition of
fitness seems disconnected, if not o!ensive, to common
sense notions of fitness used by the medical and engineering
community. For example, in the medical sense, sickle cell
anemia is a heritable trait that can lead to sickness, but
it is viewed as a “beneficial” adaptation in the population-
genetic/evolutionary sense. Natural selection could selec-
tively favor loss of optical structure in Gammarus minus
(an aquatic creature that dwells in caves) or tusks in Ele-
phants. Allen Orr used the example of Gammarus minus

https://dx.doi.org/10.33014/issn.2640-5652.2.2.cordova.1


26 Fisher’s Fundamental Theorem of Natural Selection

to chide Daniel Dennett’s naïve views of natural selection
when Orr pointed out:

In reality, selection “sees” only brute birth, death,
and reproduction, and knows nothing of Design.
Selection—sheer, cold demographics—is just as
happy to lay waste to the kind of Design we as-
sociate with engineering as to build it. Consider
the eyes of cave organisms who live in total dark-
ness. If eyes are expensive to make, selection can
wreck their exquisite engineering just as surely as
it built it. An optic nerve with little or no eye
is most assuredly not the sort of design one ex-
pects on an engineer’s blueprint, but we find it
in Gammarus minus. Whether or not this kind
of evolution is common, it betrays the fundamen-
tal error in thinking of selection as trading in the
currency of Design.
(Orr, 1996)

In fact “reductive evolution” (loss of genes and function)
rather than “constructive evolution” appears to be the
dominant mode of evolution throughout most of geologi-
cal history (Wolf and Koonin, 2013), but punctuated with
episodes of complexification that have no clear mechanistic
explanation. Reductive evolution has also been confirmed
by direct observation in the field and in the laboratory to
be the dominant mode of evolution (Behe, 2010). This
is in contrast to a widespread and long-standing historic
and present belief that natural selection generally leads to
increase in complexity, and the basis for these beliefs not
rooted in empiricism but either conjecture or computer sim-
ulation far removed from actual experiments (Darwin, 1859;
Adami, Ofria, and Collier, 2000; Vaughan, 2019).

Amusingly, the paucity of direct examples of complexity in-
crease and the abundance of complexity decrease by natural
selection was unwittingly highlighted by Sharon Moalem’s
book, Survival of the Sickest, where he argues why we need
birth defects like Tay-Sach’s disease on account of natu-
ral selection (Moalem, 2008). Furthermore, Moalem went
on to celebrate the facts of disease as evidence of Darwin’s
genius.

There are a variety of the notions of the term, “fit,” and the
population genetic/evolutionary definition (elaborated be-
low) seems idiosyncratic relative to historical and common
notions of the word. An aircraft is “fit” to fly, a subma-
rine is fit to operate underwater, a car is fit to operate on
land, a spaceship in space, etc. None of these notions of fit
need be conflated with reproductive success. Biophysicists
have established eyes are fit to sense and count individual
photons (Bialek, 2015), that bacteria can count individual

molecules (Bialek, 2012), that sharks are fit to sense elec-
tric fields in conditions at the extreme limit permissible by
physics (Angier, 2010), and birds are fit to fly and navigate
thousands of miles via compasses that leverage quantum
spin chemistry (Ritz, Adem, and Schulten, 2000; Ruben-
stein and Hack, 2013).

Although the definition and treatment of fitness in the
population-genetic/evolutionary sense may seem irrelevant
if not repugnant to common sense notions of fitness in
the medical and engineering sense, one cannot understand
Fisher’s theorem without understanding the population-
genetic/evolutionary definition of fitness. Therefore, the
next section will be devoted to elucidating the population-
genetic/evolutionary definition of fitness. The stated defi-
nitions follow from conventions used in Joseph Felsenstein’s
graduate-level textbook, Evolutionary Theoretical Genetics
(Felsenstein, 2017).

The population-genetic definition of fitness arises in the
context of a highly idealized model that approximates some
real world populations in an exponential growth stage. The
notion of fitness in this simplest of models can be extended,
with a modest amount of amendment, to definitions used
by a large amount of population genetics literature.

However, the staggering amount population genetics liter-
ature that uses such simplistic definitions does not at all
imply these population genetic models are accurate or use-
ful. Felsenstein points out:

Fisher’s and Wright’s one-locus equations turn out
to be approximations, sometimes bad ones. . . the
mathematical tools at hand have not revolution-
ized our understanding of the evolutionary pro-
cess. . . many evolutionists will fail to find the clear
and simple messages that population genetics the-
ory once seemed to promise.

(Felsenstein, 1989)

Unfortunately, many preliminaries are needed to provide a
cursory understanding of the terminology associated with
the population-genetic/evolutionary definitions of fitness
required to understand Fisher’s theorem. One must suf-
fer through some of the math in order to understand what
Fisher’s theorem means.



Volume 2, Issue 2

2. ABSOLUTE “DARWINIAN” FITNESS VS. RELATIVE FITNESS 27

2 Absolute “Darwinian” Fitness
vs. Relative Fitness

Because populations can grow exponentially, it is possible
that a single molecular change in one organism can result
in immense macroscopic consequences for the population.
For example, in the evolution of bacteria or viruses, the
emergence of a novel mutational change in their genomes
can result in their epidemic spread which has very notice-
able e!ects on the macro scale. In light of the exponential
growth of such populations, there is reasonable justification
for choosing an exponential function for modeling growth
of populations such as these, and this leads to the begin-
ning of a mathematical definition of fitness in the sense of
population genetics.

The exponential growth function arises out of one of the
most elementary di!erential equations:

! (") = d! (")
d" (1)

Where !(") is the size of the population and is a function
of time. One of the simple solutions to this di!erential
equation is:

! (") = #$! (2)

where # is a constant.

A slightly more complex di!erential equation that is tradi-
tionally related to population growth is

%! (") = d! (")
d" (3)

where % is a constant.

A solution for ! (") under the constraint of (3) is:

! (") = #$"! (4)

Where # again is also a constant. % is customarily called
the continuous growth constant and # the initial population
size. This equation can serve as an idealization of exponen-
tially growing populations.

Note that for " = 0,

! (0) = # (5)

A discrete version of (4) can be synthesized by defining the
following constant which we’ll call the absolute fitness or
Darwinian fitness:

& ! $" (6)

[It bears mentioning that calling & “Darwinian fitness”
might be a misnomer, since Lewontin argues implicitly that
describing fitness in this way violates the spirit of what Dar-
win meant by fitness. Lewontin said, “Darwin’s sense of fit
has been bypassed” by using reproductive schedules them-
selves rather than the functional aspects of an organism in
relation to its environment (Lewontin, 2003)].

We can then define the size of the population '(() as a
function of the number of generations

' (() ! ' (0)& # (7)

where ( is a non-negative integer representing a particular
generation cycle of the population, and '(0) is the initial
size of the population in generation zero. The assumption
is that in such a population, all parents have their children
simultaneously at the same time, and & represents the av-
erage number of children each parent has, and for simplicity
it is assumed & is constant for all time.

If we restrict the values of " to be only discrete points such
that " = (, and noting that ' (() is defined in terms of & # ,
we can relate ' (() to the continuous time version of popu-
lation size ! ("). In other words, if we assume " has integer
values and that the units of time are stated in generations
rather than seconds or years, so that " = 1 corresponds to
the amount of time for one generation, " = 2 is the amount of
time for two generations, etc., then under these constraints
we can then make this relation:

& # = {$"}# = {$"}! = $"! . (8)

If further we let the constant ' (0) = #, then

' (0)& # = #$"! (9)

Again, there is an implicit assumption laid out so far;
namely every parent of generation k will bear all their o!-
spring for generation k+1 simultaneously.



28 Fisher’s Fundamental Theorem of Natural Selection

If we have a population composed of ) sub populations
that each obey the above simple model, we can extend (7)
to cover multiple sub populations. With * " {1, 2, . . . )},
the size '$ (() of each the * sub populations in generation
( is:

'$ (() = '$ (0)& #
$ (10)

Note that capital “'” is used to designate absolute popu-
lation numbers. Depending on the context, '$ can describe
individuals and/or alleles. For the sake of simplicity, we
will discuss the special case where the number of individu-
als is the same as the number of alleles, namely a haploid
asexual population where only one gene locus is considered.
Aspects of this simple special case will not hold true for
diploid populations but is used primarily for pedagogical
purposes.

The total population 'tot for a given generation is straight
forwardly stated as the sum of the sizes of the individual *
sub populations:

'tot (() =
%!
$=1

'$ (() (11)

The proportion, the relative frequency, or simply frequency
+$ (() (lower case p), of alleles/individuals of a given sub-
population * that appear in the total population in genera-
tion k is:

+$ (() =
'$ (()
'tot (()

(12)

+$ (() is often called the gene frequency, although strictly
speaking it is more properly called the allele frequency.

In a somewhat analogous manner, instead of absolute fit-
ness &$ of each of the individuals of a sub population, we
can state the relative fitness ,$ of each of the individuals.
Because absolute fitness is designated with a capital & , it
is customary to designate relative fitness with a lowercase
,.

A method of defining relative fitness, though not the only
way, is to take the highest absolute fitness value found in a
total population and designate it as the reference absolute
fitness &ref. We can then define relative fitness ,$ for each
subpopulation in relation to &ref as follows:

,$ !
&$

&ref
(13)

For example, if

&1 4.0
&2 3.0
&3 2.0

then &ref = 4.0, thus:

,1 1.0
,2 0.75
,3 0.50

3 Mean and Variance of Relative
Fitnesses

For an arbitrary generation (, we can define a mean of the
relative fitnesses , (() as:

, (() !
%!
$=1

+$ (() ,$ (14)

We can also define the variance of relative fitnesses
Var (, (()) for a given generation ( as:

Var (, (()) !
%!
$=1

{+$ (() ,2
$ } # , (()2 (15)

Also let us define the change of mean relative fitnesses
!, (() as:

!, (() ! , (( + 1) # , (() (16)

This definition of !, (() may or may not be in agreement
with other texts, but it will nevertheless be su"cient to
derive a proof of Fisher’s Fundamental Theorem of Natu-
ral Selection in the discrete generation single locus haploid
case.

When it is said, “the fitness of the population is increas-
ing,” it usually means , (() is increasing. Qualitatively, it
simply means the individuals with the highest reproductive
rate are occupying a larger and larger proportion of the pop-
ulation. If there is only 1 type of individual with the high-
est fitness, a fitness increase means that the population is
asymptotically approaching a state of being homogeneous.



Volume 2, Issue 2

4. NUMERICAL EXAMPLES TO ILLUSTRATE FISHER’S THEOREM 29

With these concepts now defined, we can now examine some
numerical examples that give insights into Fisher’s Funda-
mental Theorem of Natural Selection.

4 Numerical Examples to
Illustrate Fisher’s Theorem

Repeating the example in Section 2, suppose we have the
following scenario defined by the given boundary conditions
specified by Tables 1 and 3; let us call it Scenario 1.

Table 1 lists the given absolute fitness values for the sub
populations.

&1 4.0
&2 3.0
&3 2.0

Table 1: Absolute Fitness Values of Subpopulations
(Scenario 1)

The given values in Table 1 result in the computed relative
fitness values for Table 2:

,1 1.0
,2 0.75
,3 0.50

Table 2: Relative Fitness Values (Scenario 1)

The given initial absolute individual/allele populations at
generation 0 are listed in Table 3:

'1 (0) 1
'2 (0) 2
'3 (0) 7

Table 3: Absolute Individual/Allele Populations at
Generation 0 (Scenario 1)

which results in the computed relative population frequen-
cies in Table 4:

+1 (0) 0.1
+2 (0) 0.2
+3 (0) 0.7

Table 4: Relative Population Frequencies (Sce-
nario 1)

Given the boundary conditions defined by Tables 1 and 3,
the following values were computed over 5 generations for
Scenario 1 and listed in Table 5.

[Tables 1 through 5 were from given and computed values
used in the associated supplementary spreadsheet, Supple-
ment 1.1]

One should readily observe that the last two columns of
Table 5 have identical values! This is an illustration of
Fisher’s Fundamental Theorem of Natural selection in the
simplest discrete generation, single locus, haploid model.
The statement of the Fisher’s theorem in the discrete gen-
eration model is:

!, (() = Var (, (())
, (()

(17)

and is illustrated by the fact the last two columns are iden-
tical.

Running more generations, the mean fitness ,(() will ap-
proach 1.0, meaning the population will asymptotically ap-
proach homogeneity. Ironically a condition where all mem-
bers of a population are all clones of each other might be
deemed to actually be unhealthy since diversity of alleles in
a population can help a species’ survivability since an en-
vironmental change might eliminate some individuals car-
rying a certain allele but not others.

It must be emphasized (17) is not the most general form
of Fisher’s theorem despite the fact Barton and Queller re-
fer to it as Fisher’s theorem (Barton et al., 2007; Queller,
2017). Lessard and Ewens would call this the discrete gen-
eration case (Ewens and Lessard, 2015) for only one locus
(albeit they use fairly di!erent math symbols to describe
the theorem). The most general statement Fisher’s theo-
rem was for the continuous generation case and for multiple
loci (Price, 1972).

That said, below is the proof the discrete (17) version of
Fisher’s theorem for the haploid single locus case which
follows almost exactly a proof provided in Joseph Felsen-
stein’s graduate textbook, Evolutionary Theoretical Genet-
ics (Felsenstein, 2017), but with some clarifying notation
added.

1Supplement 1 is available online at https://journals.
blythinstitute.org/ojs/index.php/cbi/article/view/67/64.

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30 Fisher’s Fundamental Theorem of Natural Selection

generation (() ,(() Var (, (()) !, (() Var(& (#))
& (#)

0 0.6000000 0.0275000 0.0458333 0.0458333
1 0.6458333 0.0360243 0.0557796 0.0557796
2 0.7016129 0.0420135 0.0598813 0.0598813
3 0.7614943 0.0429713 0.0564303 0.0564303
4 0.8179245 0.0387825 0.0474157 0.0474157
5 0.8653403 0.0316793 0.0366091 0.0366091

Table 5: Scenario 1 (5 Generations)

4.1 Proof of (17)

Starting from (10), the following must also hold by simple
substitution of ( + 1 for (:

'$ (( + 1) = '$ (0)& #+1
$ (18)

Thus, in light of (10) and (18):

'$ (( + 1) = '$ (0)& #+1
$ = '$ (0)& #

$ &$ = '$ (()&$ (19)

Using this result we can say by extending (11) and (12):

+$ (( + 1) = '$ (( + 1)
'tot (( + 1) =

'$ (( + 1)"%
$=1 '$ (( + 1)

(20)

However, in light of (19), (20) can be restated as:

+$ (( + 1) = '$ (( + 1)"%
$=1 '$ (( + 1)

=
&$'$ (()"%
$=1&$'$ (()

(21)

We can simultaneously divide the numerator and denomi-
nator of (21) by 'tot ((), and given (12), instead of relating
+$ (( + 1) in terms of the absolute sizes '$(k) of the * sub
populations, we can express it in terms of the frequency of
individuals (or alleles) +$ in total population:

+$ (( + 1) = &$ +$ (()"%
$=1&$ +$ (()

(22)

Referring back to equation (13), if we divide the numerator
and denominator of (22) simultaneously by &ref, we scale
the absolute fitnesses down to relative fitnesses:

+$ (( + 1) = ,$ +$ (()"%
$=1 ,$ +$ (()

(23)

The term in the denominator is equal to the mean of the
relative fitnesses , ((), thus

+$ (( + 1) = ,$ +$ (()
, (()

(24)

By way of extension, equation (14) must hold true also for
using ( + 1:

, (( + 1) =
%!
$=1

+$ (( + 1) ,$ (25)

Substituting the righthand side of (24) for +$ (( + 1) into
(25) yields

, (( + 1) =
%!
$=1

,$ +$ (()
, (()

,$ =
%!
$=1

+$ (() ,2
$

, (()
(26)

Subtracting , (() from the right hand side and left hand
side:

, (( + 1) # , (() =
#

%!
$=1

+$ (() ,2
$

, (()

$
# , (() (27)

Since

, (() = , (()2

, (()
(28)

We can state (27) alternatively as:



Volume 2, Issue 2

5. DISCUSSION 31

, (( + 1) # , (() =
%"%

$=1 +$ (() ,2
$

&
# , (()2

, (()
(29)

The numerator of (29) is the same as the variance
Var (, (()) from (15), thus

, (( + 1) # , (() = Var (, (())
, (()

(30)

noting equation (16) we have

!, (() = Var (, (())
, (()

(31)

this completes the proof of the discrete-generation, haploid,
one-locus case of Fisher’s theorem.

It is worth mentioning many, if not all, texts will not use
the notation here, especially the explicit statement of a vari-
able’s dependence on generation (. But the dependence on
( is emphasized here for clarity.

4.2 How Much Di!erence Does it Make?

Now suppose we have the following scenario defined by the
boundary conditions described by Tables 6 and 7; let us
call it Scenario 2. The absolute fitness and absolute popu-
lation numbers need not be stated explicitly for Scenario 2
because (23) enables the computation of values for Table 8
with only relative fitnesses and the initial allele/population
frequencies. One can simply assume there exists some abso-
lute fitness values and initial absolute population sizes that
will yield Tables 6 and 7.

The relative fitness values which are constant for every gen-
eration are listed in Table 6:

,1 1.00000
,2 0.53125
,3 0.53125

Table 6: Relative Fitness Values (Scenario 2)

The initial allele frequencies at generation 0 are listed in
Table 7:

+1 (0) 0.146666666
+2 (0) 0.426666667
+3 (0) 0.426666667

Table 7: Initial Allele Frequencies (Scenario 2)

Given the boundary conditions defined by Tables 6 and 7,
the following values were computed over 5 generations for
Scenario 2 and listed in Table 8.

[Tables 6 through 8 were from given and computed values
used in the associated supplementary spreadsheet (Supple-
ment 2).2]

What is noteworthy is that for generation 0, the entries for
both Scenario 1 and Scenario 2 are identical! However, at
generation 5, the mean fitness in Scenario 1 is 0.8653403
whereas for Scenario 2 it is 0.9073945. This illustrates the
obvious fact that di!erent sets of relative fitness values ,$

can possibly result in identical mean fitness and variance
values for some generations but not others. This shows
that Fisher’s theorem, in and of itself, cannot predict the
trajectory of population structure. This is owing to the
simple fact that infinite sets of numbers can share the same
mean and variance somewhat analogous to how several dif-
ferent curves with di!erent trajectories can share a point in
a 2D plane.

Ewens and Lessard highlighted this fact as a part of a gen-
erally negative assessment of Fisher’s theorem (Ewens and
Lessard, 2015). It shows Fisher’s theorem adds no greater
insight to population trajectory than provided by the rel-
ative fitness values ,$ themselves and the relative initial
allele frequencies +$ (0). Thus, Fisher’s Fundamental The-
orem of Natural Selection adds no additional insight into
the evolution of complexity, not only for the reasons stated
by Ewens and Lessard, but also due to the definition of
fitness used, namely fitness defined in terms of population
growth rather than biophysical complexity.

5 Discussion

The di"culty of establishing the relevance Fisher’s theorem
to real-world questions about the evolution of complexity
should be apparent because Fisher’s theorem could just as
well be applied to the increase of fitness due to loss of com-
plexity. The numerical illustrations above have no indi-
cation whether the fitness values are due to gain or loss of
complexity. It would appear Dawkins lionizing and mythol-
ogizing of Fisher’s theorem is exactly that, since Fisher’s
theorem doesn’t directly establish that biological complex-
ity must necessarily increase.

2Supplement 2 is available online at https://journals.
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32 Fisher’s Fundamental Theorem of Natural Selection

generation (() ,(() Var (, (()) !, (() Var(& (#))
& (#)

0 0.6000000 0.0275000 0.0458333 0.0458333
1 0.6458333 0.0405816 0.0628360 0.0628360
2 0.7086694 0.0516877 0.0729363 0.0729363
3 0.7816056 0.0546763 0.0699538 0.0699538
4 0.8515594 0.0475469 0.0558351 0.0558351
5 0.9073945 0.0348330 0.0383880 0.0383880

Table 8: Scenario 2 (5 Generations)

The above discrete-generation version of Fisher’s theorem
can be generalized to a continuous multi-locus diploid ver-
sion of Fisher’s theorem. Fisher described his theorem for
the continuous case where the continuous case does not
require dividing Var (, (()) by ,(() in relating “rate of
increase in fitness” to “genetic variance in fitness.” The
proof of the most general version is substantially more di"-
cult, but was demonstrated by George Price in 1972 (Price,
1972), and thus not repeated here.

An amusing historical anecdote regarding Fisher’s theorem
is that he did not use su"cient rigor to describe his own
fundamental theorem. For decades people accepted FTNS
as true even though they did not understand it. It could be
speculated this perpetuated its mythical status as biology’s
central theorem, even though population geneticists rarely,
if at all, used FTNS after Fisher first introduced it. He
stated his theorem in this way without rigorously defining
the terms mathematically:

"The rate of increase in fitness of any organism at
any time is equal to its genetic variance in fitness
at that time."
(Fisher, 1930, pgs. 36–37)

These vague words lead George Price to observe in 1972:

It has long been a mystery how Fisher (1930, 1941,
1958) derived his famous ‘fundamental theorem of
Natural Selection’ and exactly what he meant by
it. . . Also, he spoke of the ‘rigour’ of his deriva-
tion of the theorem and of ‘the ease of its inter-
pretation’. But others have variously described his
derivation as ‘recondite’ (Crow & Kimura, 1970),
‘very di"cult’ (Turner, 1970), or ‘entirely obscure’
(Kempthorne, 1957). And no one has ever found
any other way to derive the result that Fisher
seems to state. Hence, many authors (not re-
viewed here) have maintained that the theorem
holds only under very special conditions, while

only a few (eg. Edwards, 1967) have thought that
Fisher may have been correct – if only we
could understand what he meant!
(Price, 1972)

However, as with the discrete-generation version, the most
general version of Fisher’s theorem su!ers the same prob-
lems in terms of its lack of adding any further insight to the
population’s trajectory than already specified by the requi-
site boundary conditions. As with the discrete single locus
haploid case, the most generalized FTNS adds no insight to
the details of the evolution of biological systems in terms
of the common sense notions of fitness used in the medical
and engineering disciplines.

Amazingly, the original generalization of FTNS clarified by
Price does not include e!ects of the infusion of additional
mutations over time. Basener and Sanford demonstrated
that given enough mutations, mean fitness will never max-
imize (Basener and Sanford, 2018).

Further, apart from even adding mutations, Patrick Moran
and Richard Lewontin demonstrated that fitness is not
guaranteed to maximize if epistasis and gene linkage are
modeled. Grodwohl characterized these shocking mathe-
matical developments as “The Rise and Fall of Fitness Max-
imization” (Grodwohl, 2016).

In light of the fact that most “beneficial” mutations are
loss of function, then the individual with the highest fit-
ness could be the one that has lost function, rather than
gained it. This suggests that even in the existing frame-
work of population genetics, natural selection could just
as well eliminate function out of a population rather than
construct it.

Thus, Fisher’s theorem could just as well be used in models
where natural selection facilitates destruction of function,
and such destruction of function has direct observational
confirmation in the field and in experiments as the over-
whelming cause of beneficial mutations (Behe, 2010).



Volume 2, Issue 2

REFERENCES 33

It is also worth mentioning that Ronald Brady, J. G. Olla-
son, and others point out the tautologous and inadequate
definition of fitness in evolutionary biology, such that it has
no utility in helping determine the veracity of evolutionary
theory (Brady, 1979; Ollason, 1991).

Dawkins’ Weasel program (Dawkins, 1986, pgs. 46–51)
metaphorically describes how evolution is conceived in
the minds of some and has been augmented and ex-
tended to teach evolution as part of an outreach program
(Christensen-Dalsgaard and Kannewor!, 2009). Indepen-
dent of the scholarly value of Dawkins’ Weasel, metaphori-
cally speaking an anti-Weasel model of evolution by natural
selection might be a more valid description of how certain,
if not most, organisms evolve by natural selection.

6 Conclusion

Fisher may have envisioned that his Fundamental Theorem
of Natural Selection was proof that evolutionary progress
is inevitable. However, Fisher’s theorem is perfectly consis-
tent with the view that reductive evolution is the dominant
mode of evolution. And if direct observation of the ubiq-
uity of reductive evolution can be extrapolated to the past,
then complexity decrease by natural selection would be the
norm, and thus complexity increase from simple bacteria
to complex multicellular eukaryotes must come about by
a mechanism other than natural selection. An anti-Weasel
model of natural selection might more accurately describe
the operation of natural selection in the wild.

Beyond that, on many levels Fisher’s theorem has been
given a negative assessment by Ewens and Lessard, was crit-
icized by Price for not being understandable and to be in-
completely derived and proven, and declared by Felsentein
as being not-so-fundamental afterall. All of these consider-
ations are in stark contrast to Dawkins’ characterization of
Fisher’s theorem as biology’s central theorem.

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	About This Journal
	The Purpose of the Journal
	Paper Submission Policies
	Other Journal Content

	From the Editors
	Annie CrawfordAnnie CrawfordMetaphor and Meaning  in the Teleological Language of Biology
	Introduction
	The History of Teleology in the Biological Sciences
	Teleological language is Essential to Biology
	If teleological language is essential to biology, then life must be teleological
	Conclusion: Life All the Way Down

	Salvador CordovaSalvador CordovaFisher's Fundamental Theorem of Natural Selection  Isn't Fundamental After All
	Introduction: The Problem of Defining Fitness
	Absolute ``Darwinian'' Fitness vs. Relative Fitness
	Mean and Variance of Relative Fitnesses
	Numerical Examples to Illustrate Fisher's Theorem
	Discussion
	Conclusion

	Eric HollowayEric HollowayTutorial: Bioinformatics Basics
	Introduction
	Genetic Code and Sequence Translation
	Sequencing and Assembly
	Accessing Data
	Finding Things With BLAST
	Summary

	Letters and Notes
	Eric HollowayYou Cannot Get Meaning From Randomness
	Jonathan BartlettActive Information is a Specified Complexity Model
	Eric HollowayEvolution in the Valley of Illusions
	Jonathan BartlettIs Active Information Applicable to Biology?
	Eric HollowayEmpirical Active Information

	News

