# Fisher's fundamental theorem of natural selection

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Creationists have reason to doubt the theory based on **Fisher’s fundamental theorem of natural selection** published in 1930. In modern terminology Fisher’s theorem has been stated as: “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”.

It relies on the premise that a gene (allele) may have a fitness of its own being a unit of selection. Historically this way of thinking has also influenced the view of egoism as the most important force in evolution; see for instance references in Gaussian adaptation about Hamilton's kin selection, 1963, or Dawkins about the selfish gene, 1976.

A proof as given by Maynard Smith, 1998, shows the theorem to be formally correct. Its formal validity may even be extended to the mean fitness and variance of individual fitness values.

A drawback is that it does not tell us the increase in mean fitness from the offspring in one generation to the offspring in the next (which would be expected), but only from offspring to parents in the same generation. Another drawback is that the variance is a genic variance in fitness (in vertical direction) and not a variance in phenotypes (horizontal direction). Therefore, the structure of a phenotypic landscape – which is of considerable importance to a possible increase in mean fitness - can’t be considered.

And there is something dubious in the premise, so let us have a look at the definition of fitness as for instance given by Maynard Smith 1998, in the following way: Fitness is a property, not of an individual, but of a class of individuals – for example homozygous for allele A at a particular locus. Thus the phrase “expected number of offspring” means the average number, not the number produced by some one individual.”

Even if the definition is useful in breeding programs, it can hardly be of any use as a basis of a theory of an evolution selecting individuals. It seems to me that this definition denies the fitness of the individual. Nevertheless, the individual fitness is needed, because otherwise the expected average number of offspring from a certain class of individuals can’t be determined. In addition, if it possible to define the fitness of an allele as an average of individual fitness values, then, it must be possible to define the mean fitness of a whole population from such values.

Gaussian adaptation is based on the selection of individuals using definition given by Hartl, 1981. The fitness of the individual is the probability s(x) that the individual having the n characteristic parameters x^{T} = (x_{1}, x_{2}, …, x_{n}) – where x^{T} is the transpose of x – will survive, i. e. become selected as a parent of new individuals in the progeny. This definition is perhaps less useful in breeding programs, but may be useful in certain philosophical discussions about evolution. If the selection of individuals rules the enrichment of genes, then Gaussian adaptation will perhaps give a more reliable view of evolution.

The image below shows two different cases (upper and lower) of individual selection, where the green points with fitness = 1 - between the two lines - will be selected, while the red points outside with fitness = 0 will not. The centre of gravity, m, of the offspring is fat black and ditto of the parents and offspring in the new generation, m* (according to the Hardy-Weinberg law), is fat red.

Because the fraction of green feasible points is the same in both cases, Fisher’s theorem states that the increase in mean fitness in vertical direction is equal in both upper and lower case. But the phenotypic variance (not considered by Fisher) in the horizontal direction is larger in the lower case, causing m* to considerably move away from the point of intersection of the lines. Thus, if the lines are pushed towards each other (due to arms races between different species), the risk of getting stuck decreases. This represents a considerable increase in mean fitness (assuming phenotypic variances almost constant). Because this gives room for more phenotypic disorder/entropy/diversity, we may expect diversity to increase according to the entropy law, provided that the mutation is sufficiently high.