Fisher’s Fundamental Theorem (Part 1)

There are various ‘fundamental theorems’ in mathematics. The fundamental theorem of arithmetic, the fundamental theorem of algebra, and the fundamental theorem of calculus are three of the most famous. These are gems of mathematics.

The statistician, biologist and eugenicist Ronald Fisher had his own fundamental theorem: the ‘fundamental theorem of natural selection’. But this one is different—it’s a mess! It was unclearly stated, the first few proofs were packed with mistakes, and people don’t agree on what it says, whether it’s correct, and whether it’s nontrivial. Thus, people keep trying to clarify and strengthen it.

This paper analyzes Fisher’s work:

• George R. Price, Fisher’s ‘fundamental theorem’ made clear, Annals of Human Genetics 32 (1972), 129–140.

Price writes:

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. He stated the theorem in these words (1930, p. 35; 1958, p. 37): ‘The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.’ And also in these words (1930, p. 46; 1958, p. 50): ‘The rate of increase of fitness of any species is equal to the genetic variance in fitness.’ He compared this result to the second law of thermodynamics, and described it as holding ‘the supreme position among the biological sciences’. Also, he spoke of the ‘rigour’ of his derivation of the theorem and of ‘the ease of its interpretation’. But others have variously described his derivation as ‘recondite’ (Crow & Kimura, 1970), ‘very difficult’ (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 reviewed here) have maintained that the theorem holds only under very special conditions, while only a few (e.g. Edwards, 1967) have thought that Fisher may have been correct—if only we could understand what he meant!

It will be shown here that this latter view is correct. Fisher’s theorem does indeed hold with the generality that he claimed for it. The mystery and the controversy result from incomprehensibility rather than error.

I won’t try to explain the result here—or the various improved versions that people have invented, which may actually be more interesting than the original. I’ll try to do this in a series of posts. Right now I just want to quote Price’s analysis of why Fisher’s result is so difficult to understand! It’s amusing:

In addition to the central confusion resulting from the use of the word fitness in two highly different senses, Fisher’s three publications on his theorem contain an astonishing number of lesser obscurities, infelicities of expression, typographical errors, omissions of crucial explanations, and contradictions between different passages about the same point. It is necessary to clarify some of this confusion before explaining the derivation of the theorem.

He analyzes the problems in detail, calling one passage “most confusing published scientific writing I know of”.

Part of the problem, though only part, is that Fisher rewrote part of his paper while not remembering to change the rest to make the terminology match. It reminds me a bit of how the typesetter accidentally omitted a line from one of Bohr’s papers on quantum mechanics, creating a sentence that made absolutely no sense—though in Bohr’s case, his writing was so obscure that nobody even noticed until many years later.

Given its legendary obscurity, I will not try to fight my way through Fisher’s original paper. I will start with some later work. Next time!