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! The theorem was based on confusing definitions, Fisher’s proofs were packed with typos and downright errors, and people don’t agree on what the theorem 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!

The whole series:

Part 1: the obscurity of Fisher’s original paper.

Part 2: a precise statement of Fisher’s fundamental theorem of natural selection, and conditions under which it holds.

Part 3: a modified version of the fundamental theorem of natural selection, which holds much more generally.

Part 4: my paper on the fundamental theorem of natural selection.

5 Responses to Fisher’s Fundamental Theorem (Part 1)

  1. Roice Nelson says:

    Btw, I much recommend Price’s biography The Price of Altruism: George Price and the Search for the Origins of Kindness.

    Ironically (given this content of your post), the writing felt a bit confounding at times. But Price’s life story is an amazing one, and I found the ideas woven into the book deep. It changed my understanding of evolutionary effects.

    • John Baez says:

      Interesting! I don’t know anything about George Price except his paper on Fisher’s fundamental theorem, which is really fun to read—first taking apart Fisher’s three papers on this subject and then trying to figure out what Fisher must have meant.

      • Roice Nelson says:

        I should have mentioned I heard about George Price on the Radiolab episode An Equation for Good. They interview the author of his biography, so his story is available there in short form. In a sense, George’s life played out as a metaphor of his own equation.

  2. Ishi Crew says:

    Fisher’s FTNS and G Price’s later version (with interludes by Hamilton) are central to the issue of ‘group selection’ (GS) –a sort of controversial topic since 1960’s though i view it as ‘resolved’ (i think a few outliers or ‘hold outs’ such as S Pinker and J Coyne (and probably more–D Sperber…) still are against it (The 2 Wilson’s–E O and DS –are on board DS always was and E O changed his opinion), but they both have GS theories which i don’t really think even get ‘price’s equation’.

    Although i read as much of the literature on GS as i could –probably mostly in late 80’s-90’s–2000’s —Price equation was never mentioned until fairly recently. I view Price equation as a generalization of Fisher’s FTNS (its sort of like the difference between the continuous logisitic equation and its discrete version—both are correct as are Fisher and Price—but one has a stable equilbirium distirbution and the other is chaotic. (I learned this in a paper by E Montroll.)

    Price deals with ‘extended or inclusive fitness’ –Fisher deals with a more limited form (in fact i would say Fisher’s theorem deals with ‘haploids’ or ‘clones’. The most fit clones survive. (One can look at some influential papers in theoretical population genetics and they make this assumption for ‘mathematical conveniance’ –ie they are studying th genetics of some mating (diploid, sexually reproducing ) species but for ‘conveniance ‘ they assume its haploid (clones). This way they only have to follow one gene’s trajectory. (In many cases it actually works—this approach is called Fisher’s ‘beanbag genetics’).

    ( Fisher dids deal with diploids (eg sex, epistasis, dominance)—under special cases (but Fisher just linearizes out all interactions.
    i view this as conceptually similar to trying to model phase transitions in condensed matter physics by ‘linearizing out the interactions’ –eg they say ‘as a first approximation a crystal, or computer, is essentially an ideal gas’. It actually works in many cases.)

    tThere is a famous paper by astrophysicist Chandresekhar called ‘stochastic problems in physics and astronomy’ in Rev Modern Physics in 1943–i just read the physics part. its last section has an error in it ( on diffsuion in a potential) which confused me for a long time–until i read some article on WW saying it was an error. ) is one more paper—i think I agree with this paper although i think the authors also make the same ‘mistake’ Fisher made –at least elsewhere —they linearize out interactions and end up with ‘beanbag genetics’. In a sense one could interpret this as aiming to legitimize GWAS (and discount Landes LP-model –Landes of MIT sort of was one of creators of GWAS but he revised it the way Price revised Fisher–‘optimizers dont optimize everythijng’ though Fisher’s theorem still hold–‘surivivors survive’.I think this is analgous to Schrodinger’s view in his book ‘what is life’—explains wave function collapse by saying most of the ‘other states’ ‘dissapear’ via Born’s rule. ) .

    Those quotes from Price are amusing (since i’m not sure ive ever seen his original paper ). Alex Rosenburg (philosophy of science at Duke ) has a 2005 paper/exchange i Philo and Biology called ‘Mathen’s and Ariew’s Obituary for Fitness :reports of its death has been greatly exaggerated’. (This is a sort of debate over whether Fiher’s theorem is the same as 2nd la of thermodynamics or whether its a version of newton’s law –a principlal of least action. Confusion seems to be a trait that is selected for (maybe one could ave a ne law similar to MEP (max entropy proudction) like MCPP Max confusion production pcp (tho the acronym and similar ones are in use including this area—refers to a kind of ‘boat’– a a large asian country).

    Some apparent ‘creationists’ have adopted price’s theorem to make their point (i’m not intelligent enough to understand what the or their grand design is) see the book geneticentropy(dot)org –their view seems to be that while Fisher possibly had some similar views, if Price’s revised argument prevalied over Fisher then they will use Price. This is analogous to some free market economists said if adam smith and arrow-hahn’s profoof that the ‘free market’ needs some revisions due to chaos and computability theory, then they will use chaos and computability thoery to prove that free markets are still optimal or possible—ie ‘this is the best of all possible worlds’.

    Price gave up theoretical biology, becamer a christian, and i hear worked serving the poor and died as pauper (didnt get no royalties)..

  3. Zach says:

    “Fisher just linearizes out all interactions” – is this related to “up to first order, you’re never learning anything” from the Azimuth article “Information Geometry (Part 16)”?

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