[The Hardy-Weinberg law] seems trivially obvious, a routine application of the binomial theorem.
Hardy, Weinberg and Language Impediments, James Crow (1999)
We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.
A Mathematician's Apology, G.H. Hardy (1940)
I. Mendelism Is Challenged
In 1908, at Cambridge, the Mendelian geneticist Reginald Punnett gave to the Pure Mathematician Godfrey Harold Hardy a stumper of a real-world problem: Considering the simple mathematical rules of Mendelian genetics, why does the observed phenotype in a population, under a dominant and a recessive allele, not converge to a 3:1 ratio? Consider what Wikipedia says today, about Punnet’s crossing squares, and whether this example could lead to a mistaken expectation that it should:
The probability of an individual offspring's having the genotype BB is 25%, Bb is 50%, and bb is 25%. The ratio of the phenotypes is 3:1, typical for a monohybrid cross.
Punnett Square, Wikipedia (2018)
Eugenicist fears are lurking here. Brachydactyly, for example, is a rare and dominant trait. If nothing were done about it, an argument from eugenics went, would not the English eventually become a short-fingered race, at a ratio of three to one?
A wider debate among evolutionary biologists over Mendelism is here too. Today, we see Darwinism and Mendelism as unified. At the time, the two schools of thought were in opposition. Punnett had been confronted with the ratio problem by the biometric statistician Udny Yule, and the confrontation on the detail might be taken as a confrontation between biometricians and Mendelians, over Mendel, as a whole.
II. The Entire History of Scientific Controversy Teaches a Lesson
There was a history of personal animosity behind the confrontation.
In 1902, for example, two years after the rediscovery of Mendel’s paper, Yule wrote about Punnett’s colleague, the Mendelian geneticist William Bateson. Bateson’s use of inverted sentences, among other irritations, had exceeded the bounds of Yule’s patience. If a Mendelian should model their prose after Carlyle, or reference stories from the bible, or praise a biometrician in the wrong way, this is more than a biometrician should have to bear.
It is difficult to speak with patience either of the turgid and bombastic preface to "Mendel's Principles," with its reference to Scribes and Pharisees, and its Carlylean inversions of sentence, or of the grossly and gratuitously offensive reply to Professor Weldon and the almost equally offensive adulation of Mr. Galton and Professor Pearson. A writer who indulges himself in displays of this kind loses his right to be treated either as an impartial critic or as a sober spectator.
Mendel’s Laws and Their Probable Relations to Intra-Racial Heredity, G. Udny Yule (1902)
Yule goes on for two pages in this style. He was attacking Bateson for attacking Weldon for attacking Bateson. He was deploying bombast against bombast. An attack of this kind is the very worst method of arriving at truth, Yule points out, in the middle of his making this kind of attack:
Apart altogether from the question of good manners, the entire history of scientific and philosophical controversy would have taught a more judicious disputant that personal polemic is the very worst method of arriving at truth; an attack of this kind can do nothing but distract attention from the scientific question and concentrate it on the ephemeral personalities.
Yule and Weldon were from the more mathematical side of the dispute; Punnet and Bateson from the more experimental. Punnet therefore enlisted mathematical help from Hardy, whose still-respected book A Course of Pure Mathematics was first published that year.
III. The Binomial Theorem Is Routinely Applied
Hardy published his solution to the 3:1 problem in a short letter to Science. He was reluctant to intrude into matters of which he had no expert knowledge, he began.
(G.H. Hardy was a careful stylist. The words surrounding the equations in his mathematics course have a model clarity. The prose of his Apology has been called poetic. But he has a socially difficult situation to navigate in his opening sentence here, perhaps leading to his long-winded way of getting to the point.)
He was reluctant to intrude into matters of which he had no expert knowledge, he began, but he wished to inform biologists of a very simple point, which he should have expected them to already know. A routine application of basic algebra shows that in the absence of outside force, the distribution of traits in a population will remain constant after the second generation.
Hardy provided a worked example, using brachydactyly, to demonstrate this, and a pointed refutation of the eugenicist fears:
In a word, there is not the slightest foundation for the idea that a dominant character should show a tendency to spread over a whole population, or that a recessive should tend to die out.
Mendelian Proportions in a Mixed Population, G.H. Hardy (1908)
IV. They Have an Impediment to Hearing Each Other
Various versions of this story have been told, to various spin. Why Hardy published in Science is a detail that will change according to teller.
Hardy published his letter in the American journal Science, not the English journal Nature, because reluctant or not, and expert or not, he was taking a side in the most heated dispute in biology ever. Nature, at the time, was in the hands of biometricians, and off limits to the side that had enlisted him.
Standard accounts of the biometrician-Mendelian dispute, in trying to explain why biometricians and Mendelians alike had missed the simple algebra, will say that if the two sides had been talking to rather than insulting each other, they probably both would have seen it.
V. The Impediment Is More Than Personal Animosities
The only way in which we may hope to get at the truth is by the organization of systematic experiments in breeding[.]
Materials for the Study of Variation, William Bateson (1894)
The questions raised by the Darwinian hypothesis are purely statistical, and the statistical method is the only one at present obvious by which that hypothesis can be experimentially checked.
Remarks on Variation in Animals and Plants, W. F. R. Weldon (1894)
Other accounts will stress the methodological differences between the sides, with experimentalism versus statistical naturalism as the most fundamental. The mutual accusations of bad faith in published results would come from different ideas about how science should be done, resulting in frequent misunderstandings of each other, and with the personal animosities then becoming intertwined. The two quotes on how best to investigate variation and evolution, above, both from the same year, directly express the opposing views. This dispute about methodology predates the dispute about Mendel.
VI. A Nonmathematical Expression Was Known
In a 1902 paper, Bateson and the Mendelian geneticist Edith Rebecca Saunders expressed the principle of genetic equilibrium in nonmathematical form. The “zero-force” model described here, where necessary conditions are stated, and where departures from equilibrium expose the presence of outside force, directly corresponds to one important way that Hardy-Weinberg is still used today.
If the degree of dominance can be experimentally determined, or the heterozygote recognised, and we can suppose that all forms mate together with equal freedom and fertility, and that there is no natural selection in respect of the allelomorphs, it should be possible to predict the proportions of the several components of the population with some accuracy. Conversely, departures from the calculated result would then throw no little light on the influence of disturbing factors, selection, and the like.
The Facts of Heredity in the Light of Mendel’s Discovery, William Bateson and E.R. Saunders (1902)
The Mendelian geneticist William Castle, in 1903, and the biometric statistician Karl Pearson, in 1904, had expressed the idea of gene frequencies reaching a stable equilibrium as well.
VII. Meanwhile, In Germany
”I have never done anything useful,” the Pure Mathematician G.H. Hardy, at Cambridge, bragged in his Apology. In Stuttgart, the obstetrician Wilhelm Weinberg had delivered 3500 babies.
Compared to Hardy’s publication on the principle now bearing both their names, Weinberg’s was a) earlier by six months, and b) not noticed in the English-speaking world until 35 years later.
The language barrier was one directional. Weinberg saw Hardy’s publication in Science, and reviewed it. He thought Hardy’s derivation was a bit clumsy.
The one directional language barrier continued after 1908. Weinberg knew of English developments in the new discipline of population genetics, and would have contributed to it, if his papers had been read there.
VIII. The Routine Equation Is Theoretically Applied
Nothing in evolutionary biology makes sense except in the light of population genetics.
The Frailty of Adaptive Hypotheses for the Origins of Organismal Complexity, Michael Lynch (2007)
In the paper quoted above, the population geneticist Michael Lynch poses a stumper of a real-world problem. The multicellular eukaryotic genome is full of complexities, such as mobile elements, introns, regulatory structures, and such. This complexity is a mutation hazard. And what adaptive advantage multicellular organisms might have is at any rate hard to explain. How, then, has multicellular life come about?
Investigation of this fundamental question is guided, at core, by the assumptions to the Hardy-Weinberg equilibrium: Population sizes must be infinite; Mating must* be random; Generations cannot overlap; No inflow or outflow of genes can exist, which includes that there cannot be natural selection or mutation.
Evolution, in population genetics, is defined as a change in gene frequencies, over time. This definition is very tightly tied to the Hardy-Weinberg equilibrium, which is a lack of change in gene frequencies over time. Evolution, and the Hardy-Weinberg equilibrium state, are two sides of one coin. Or, the genetic stability of Hardy-Weinberg is a limit, which because evolution is always present, can only be approached.
Where evolution exists, there must be a violation of the Hardy-Weinberg assumptions. The assumptions behind the math correspond to the drivers of evolutionary change.
As part of a proposed explanation for the existence of multicellular life, Lynch is focusing on how strongly multicellular organisms violate the assumption of infinite population size. Compared to bacteria, there are so very few elephants, say, and so very few humans. Compared to bacteria, there are so very few ants. That is to say, with math as a guide where to look, an explanation for the origin of multicellular life might be found not in natural selection, but in genetic drift.
The paper makes a political argument, calling us to task. That we feel entitled to speculate about evolution. That we feel we have sufficient grounding, and that we understand it. And that when we speculate about evolution, we tend to consider only natural selection as cause.
Evolutionary biology is treated unlike any science by both academics and the general public. For the average person, evolution is equivalent to natural selection, and because the concept of selection is easy to grasp, a reasonable understanding of comparative biology is often taken to be a license for evolutionary speculation.
The math of population genetics can be complex. It is not an easy field. But, having an adequate grasp of evolution requires some formal population-genetic grounding.
However, the details do matter in the field of evolutionary biology. As discussed above, many aspects of biology that superficially appear to have adaptive roots almost certainly owe their existence in part to nonadaptive processes. Such conclusions would be difficult to reach without a formal population-genetic framework, but they equally rely on observations from molecular, genomic, and cell biology.
There is a saving grace here, though. As math goes, Hardy-Weinberg, which is at the base of population genetics, which is necessary to understand evolution, is not all that hard.
Although the principle is trivially simple, it is nevertheless the foundation for theoretical population genetics.
Population Genetics History: A Personal View, James Crow (1987)
IX. The Routine Equation Is Practically Applied
HWE is the fundamental starting point for all population–genetical investigation[.]
A Century of Hardy–Weinberg Equilibrium, Oliver Mayo (2008)
A Google Scholar search on “Hardy Weinberg” finds more than 2300 results, since just the beginning of this year. There are roughly twenty results from genetic research on goats, alone. Population geneticists, routinely, will test whether a gene they are studying is in equilibrium. And where a paper in population genetics does not explicitly refer to Hardy-Weinberg, it is still standing on its shoulders.
Today, the H-W Law stands as a kind of Newton’s First Law (bodies remain in their state of rest or uniform motion in a straight line, except insofar as acted upon by external forces) for evolution[.]
Uses and Abuses of Mathematics in Biology, Robert May (2004)
The Hardy-Weinberg law would be more like Newton’s First Law for evolution, though, if the academic field of Newtonian physics was thought to be founded by the first level of scientists standing on the shoulders of Newton, rather than by Newton himself.
X. G.H. Hardy Founds the Discipline of Population Genetics
Judged by all practical standards, the value of my mathematical life has been nil; and outside mathematics it is trivial anyhow.
A Mathematician's Apology, G.H. Hardy (1940)
In a standard telling, the academic discipline of population genetics starts with a 1918 paper by Ronald Fisher, which gives a mathematical treatment of the idea, paraphrasing Fisher’s opening sentence, that the results of biometry could be interpreted in accordance with Mendelian genetics. That is, continuous variation, as seen by the biometricians, could be explained by discrete genetics, as seen by the Mendelians, if a number of genes combined to produce one phenotype. And additionally, as corollary, to say that continuous variation is compatible with Mendelian genetics, is to say that Charles Darwin’s version of the theory of natural selection, which stresses infinitesimally small difference, is compatible with Mendelian genetics as well.
The founders of population genetics in this telling, the modern synthetic one, will be listed as Fisher, Sewall Wright, and J. B. S. Haldane.
A course in population genetics, however, will perhaps first review Mendelian genetics, and then introduce Hardy-Weinberg. Here is the first line of a text by Warren Ewens, who says that beginning with Hardy-Weinberg is common practice.
It does not often happen that the most important theorem in any subject is the easiest and most readily derived theorem for that subject, and the one which is first taught to students.
Population Genetics, W.J. Ewens (1969)
There is an inconsistency, then, in how population genetics is taught, and who it will be claimed founded it.
First, the idea that continuous variation can be explained as the combined effect of multiple Mendelian genes might be needed for a particular paper, but very well might not. Hardy-Weinberg is essential to Fisher’s 1918 paper — as well as say Haldane’s Mathematical Theory series starting in 1924, and Wright’s 1921 series on the mathematics of mating systems — and not the other way around.
Second, if population genetics is tightly tied to the idea of a synthesis of Darwinian natural selection and Mendelism, this would go back to Hardy-Weinberg at any rate.
It is the "quantal" nature of the gene that leads to the stability described by the Hardy-Weinberg law…. Thus the Hardy-Weinberg law shows that far from being incompatible, Darwinism and Mendelism are almost inseparable.
Mathematical Population Genetics, Warren Ewens (2012)
And, third, G.H. Hardy, a braggart about his own uselessness, is asking for it by his own rhetoric. He has made a scientific contribution that was trivial at a mathematical level, but of entirely nontrivial importance and consequence and use.
In his apology, Hardy made a prediction of notable wrongness: That the real math in Einstein’s theory of relativity would long go unsullied by any usefulness in war. This was said in 1940.
Second after this, though, Hardy’s claim that his contribution outside mathematics is trivial and useless, could not have been more wrong.