Return to Haldane's Dilemma area
by Walter ReMine
March 9, 2006
Fred Hoyle and J.B.S. Haldane were both excellent mathematicians. Concerning Haldane's Dilemma, their mistakes were not mathematical (I can testify), but due instead to something else.
Haldane's Cost of Substitution concept is a brilliant insight; an important breakthrough in our understanding. His 1957 paper focused repeatedly on "reproductive capacity," and his math was correct. Unfortunately, he didn't explain his concept well, and this source of confusion has even been expanded upon by succeeding generations of commentators. Fred Hoyle2 bought into the confusion, and thereby calculated a different mathematical result from Haldane. This was an unfortunate mistake for Hoyle, but understandable given the general confusion surrounding the topic, even to the present day.
I reconstruct Haldane's thinking as follows. Haldane was a preeminent mathematician, and he saw something lurking within his math equations, something previously unrecognized – something important. For it accurately predicted the "reproductive capacity" required by a given scenario. The concept worked, and was mathematically accurate. Unfortunately, Haldane did not derive his concept from first principles. Rather, he 'plucked it' from amidst his equations. (This is illuminated in my forthcoming paper.) His idea worked, and he gave it a name – the Cost of Substitution – he also called it "selective death", soon more widely known as "genetic death". His paper defined it mathematically via his equations, not physically in terms of concrete first principles. His physical description of 'genetic death' was rather poor, and led to much confusion for his successors.
If you doubt it, then I challenge you to explain Haldane's cost equations in a straightforward physical way, in terms of elimination of the old-type. Conversely, I challenge you to start with a simple physical definition of "genetic death" (in terms of elimination of the old-type), and thereafter derive Haldane's cost equations. (You will find this an interesting exercise in back bending. Hard on the back, but good for the mind.) In other words, there is a disconnect between Haldane’s mathematical equations for the Cost of Substitution, and any reasonable interpretation of it in terms of elimination of the old-type (i.e. "genetic death"). A connection between genetic death and Haldane's math is awkward for simple cases, and quickly becomes incomprehensible or impossible for more complicated cases.
As it happens, Hoyle bought into the confusion surrounding the physical interpretation of genetic death. Physically, just what is this thing called "genetic death"? Hoyle (like Feller and Moran) focused on the obscure physical meaning of "genetic death", and clarified it; and thereby derived mathematical equations at odds with Haldane's. Hoyle's mistake was not in the math, nor was it in the population genetics (both of which he knew thoroughly). Rather, it was solely in interpreting this new thing called "genetic death."1 (This is the sole technical mistake in Hoyle's otherwise excellent book, Mathematics of Evolution.)
So who is correct, Hoyle or Haldane? They contradict, yet evolutionists have not pursued a resolution. Instead, some evolutionists use Hoyle's position in an attempt to brush aside Haldane's Dilemma – while neither resolving, nor acknowledging, the contradictions. The truth is that evolutionary literature is manifestly confused and self-contradicted concerning the fundamentals of the cost of substitution.
Hoyle asked the wrong question. He asked: What does 'genetic death' really mean? He should have asked: What does the Cost of Substitution really mean? The physical interpretation of genetic death used by Hoyle (and Feller, and Moran) is useless. No one uses it today. It is dead as a door-nail (except as an erroneous pretense that Haldane's Dilemma is "solved" or an "illusion"). On the other hand, Haldane's mathematical definition of the cost of substitution is still potent, needing only a clearer illumination of its physical meaning, which is given in my book and my paper.
Let me conclude with what I've been saying for years. If you focus on genetic death, then you focused on the wrong thing. You will be led in confusion down a blind dead end. The issue is not the death or elimination of the old-trait, but rather the issue is the growth of the new-trait, and the reproduction rate required to make it happen. This limitation is calculated by the Cost of Substitution.
1 Hoyle defined genetic death in a given generation as the actual reduction of the old-type individuals, in other words, the difference in the actual number of old-type individuals between the beginning and end of the generation. That definition is clear and physically concrete. However, it is substantially different from what Haldane called "genetic death", which he defined mathematically.
2 Hoyle, 1987, 1999 Mathematics of Evolution, Acorn Enterprises, Memphis, Tennessee
Hoyle acknowledged there is ambiguity in the concept of genetic death.
"[T]here is no absolute fiat as to how one must define the concept of 'genetic deaths' quantitatively, ..." (Hoyle, p 119)
Hoyle then attempts to clarify genetic death in a way that "seems sensible" to him:
"[I]t is sensible it seems to me to adopt [Hoyle's equation 7.20], which then expresses the necessary condition that [when the population size remains constant] the gain of individuals due to A [the favored allele] is equal to the loss of individuals due to a [the disfavored allele]." (Hoyle, p 119)
Using that definition, he derives the following point, (though the derivation is scarcely needed, since the result is already obvious from his definition): In order to replace the original population with the newly favored allele A, almost exactly one entire population must die. One population (according to Hoyle), not 30 populations (as seemed implied by Haldane's figure of 30). Under Hoyle's reckoning, Haldane's figure seems 30 times greater than it should be. This occurred because Hoyle focused on 'genetic death', rather than understanding the physical meaning of Haldane's mathematical cost equations.
Hoyle further pursues his interpretation of 'genetic death' by combining it with another confusion factor – the 'environmental-change' scenario. He successfully shows the genetic death concept is useless, and the "principle is an illusion." (Hoyle, p 128) But he was incorrect to think the cost concept is useless.
When discussing Haldane's Dilemma, Hoyle was correct on two points that other evolutionists usually overlook:
"It would not be unreasonable to suppose that 40 percent of juveniles fail to reach maturity for accidental nongenetic reasons ..." (Hoyle, p 112)
Hoyle was correct that those two items are tied together into Haldane's Dilemma. My paper discusses how the various types of cost are interrelated.