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Haldane's Dilemma

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webpage by Walter ReMine

This page last updated:  August 21, 2007

  "In my opinion the [Haldane's Dilemma] problem 
was never solved, by Wallace or anyone else."

    George C. Williams, a highly respected evolutionary geneticist, 1992,
in his book, Natural Selection: Domains, Levels, and Challenges, p 143-144


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Responding to evolutionist critiques:

Welcome to my place for updates on Haldane's Dilemma.  This website is intended to complement my book and papers, not replace them. 

To the present date, evolutionists have not responded in ink, so there is nothing to report there. 

Things are different on the internet, where content is temporary and often anonymous; a situation that invites much mischief. On the internet, evolutionists abundantly misrepresent my material, and continue it even after I notify them. I here identify the misrepresentations, in hopes there will someday arise an honest scientific discussion of this important unsolved problem, Haldane's Dilemma. 

NEW:  Haldane's view of Haldane's Dilemma 

NEW:  Evolutionists Withhold Evidence! 


Also: My paper, "Cost Theory and the Cost of Substitution – a clarification" is now published. Leading evolutionary geneticists, James Crow and Warren Ewens, acknowledge it is correct. It is a fundamental clarification needed in this field. See it here. Also, see its tale of peer-review.  Then compare my paper with James Crow's paper, to see that my paper includes many clarifications that Crow's paper did not make. In other words, my paper is correct, new, and relevant to an important evolutionary problem – yet evolutionists refused to publish it. 

What is Haldane's Dilemma?

Briefly. Haldane's Dilemma establishes a limit of 1,667 beneficial substitutions (where a substitution is almost always one nucleotide) over the past ten million years of the lineage leading to humans.  The origin of all the uniquely human adaptations would have to be explained within that limit.1 That is a serious problem. 

All the key data, assumptions, models, and calculations are taken from evolutionists, and put a limit on the rate of beneficial evolution. I call it the "Haldane limit," or the "1,667 limit."4  Since my book came out, there has been no serious dispute that Haldane's analysis (if correct) places a 1,667 limit on human evolution. 

My book identifies many factors further reducing that figure by orders of magnitude.5 In other words, Haldane's estimate is overly optimistic in favor of evolution. Yet I focus on the 1,667 limit because it derives directly from Haldane – so evolutionists cannot evade the issue by blaming it on me. 

All those matters were known to evolutionary geneticists in 1957 when Haldane published his argument.  Yet despite it being interesting, important, and easy to communicate, they did not inform the public. No, there was no conspiracy. But it was a staggering bit of negligence. Haldane's Dilemma is not just the problem itself, but also the evolutionists' negligence for not communicating it to the public. 

Thus, there are three facets of Haldane's Dilemma:  

  1. The simple core problem. 
    • The 1,667 limit
    • How it concerns evolutionary theory.
  2. The history of Haldane's Dilemma.  
  3. The technical details

The entire issue is richly fascinating, no matter which side you are on. 

   Haldane's Dilemma puts a limit on the rate of beneficial evolution.  It does not limit the rate of neutral or harmful evolution, which can be far more rapid. However, my book also contributes a style of argument previously unheard of – a serious limit on the rate of expressed neutral substitutions.  The argument involves something routinely left out of evolutionary discussions – error catastrophe.  By seeing the connection between error catastrophe and plausible substitution rates, I was able to create a new type of argument.  

2    Haldane's calculations included the possibility of many substitutions overlapping in time. His argument did not require single substitutions tacked end-to-end. 

3    Sometimes the 'thing' being substituted into the population might be larger than a nucleotide, such as:  insertion, deletion, gene inversion, gene duplication, or the relative order of genes on a chromosome.  Each of these would count as a substitution, and the argument puts a limit on the total number of substitutions.  

4    The math is easy:  1,667 substitutions = 10,000,000 / (20 * 300) 
 Substitutions  = (Years) / [(Years / Generation) * (Generations / Substitution)]

5    As discussed in my book, several factors could reduce the 1,667 limit significantly.  For example, according to Eldredge and Gould's evolutionary theory, punctuated equilibria, species are in statis at least 99 % of the time, and  Gould claimed punc-eq applies to human evolution.  According to Gould (in his last book, The Structure of Evolutionary Theory) genetic change would typically cease during statis.  If correct, this factor alone could reduce the Haldane limit by a factor of about 100, to a limit of 17 substitutions.  I was the first to bring up this relationship between punc-eq and Haldane's Dilemma.  Evolutionists should have seen this relationship, but if they did, they did not publicize it.  

1/24/2005 - by Walter ReMine

Are the substitutions "genes"?

The thrust of my argument does not speak of a limit of "1,667 gene substitutions." Rather my argument focuses on typical substitutions, which – according to evolutionists – are almost always point mutations (a single nucleotide). Yet evolutionists traditionally discussed Haldane's Dilemma in terms of "gene substitutions," which created the false impression that large blocks of new DNA are being replaced, rather than just a single mutation (typically one nucleotide). That habit further obscured the severity of Haldane's Dilemma from public view.  In fact, a well-known evolutionary genetics professor from Cornell University expressed shocked resistance when I first explained this point to him:  The 1,667 substitutions are typically single nucleotides, not 1,667 whole genes. The substituting 'thing' is a mutation, not a gene. It's a simple concept when explained clearly, though it was habitually overlooked even by professionals at the time. The traditional focus on "gene substitutions" is one of many factors that garbled Haldane's Dilemma for so long.

1/24/2005 - by Walter ReMine

Regulatory genes?

Mutations to regulatory genes can sometimes have a large biological effect. So, are mutations to "regulatory genes" a solution to Haldane's Dilemma?1  Brief answer:  It scarcely affects Haldane's Dilemma. 

Evolutionists do not get to choose the substitutions, say, as "mutations to regulatory genes" or as "mutations with a large biological effect" or as "gene clusters". Rather, evolutionists must accept what nature doles out – and we can observe what nature doles out.  The issue is fundamentally empirical and observable, not one of telling stories about regulatory genes. Take the blend of beneficial substitutions observed in nature. The coloring of melanic moths and the beak size of Galapagos finches are noted examples.  Those represent the basic building blocks that nature has to work with, the creative 'power' of nature. Leading evolutionists acknowledge (indeed most of them insist) that substitutions with small effect predominate in evolution.  Are 1,667 substitutions (like those) sufficient to create all the human adaptations?  The tripling of brain size, fully upright posture, language, speech, hand dexterity, hair distribution, and appreciation of music, to name a few. 

It scarcely matters what kind of tall-tales one can devise about regulatory genes. What matters is the blend of the beneficial substitutions observed in nature.  Are 1,667 of those sufficient for creating all the human adaptations? 

Moreover, when Haldane calculated the total cost of a substitution (=30), he assumed selection coefficients approaching zero, which gives the absolute lowest possible total cost of substitution. The figure rises for higher selection coefficients, and rises especially rapidly when selection coefficients get above one-tenth (s>0.1). Therefore, if evolutionary scenarios invoke high selection coefficients (such as mutations for antibiotic resistance or pesticide resistance), then the total cost of substitution gets quite high – and the number of substitutions (previously at 1,667) goes to a much lower limit. 

In summary, whatever the blend of substitutions, (a) they must correspond to what we observe in nature, and (b) they must meet the cost constraints, (higher selection coefficients will lower the plausible number of substitutions). 

1    Professor Massimo Pigliucci raised regulatory genes as a "solution" to Haldane's Dilemma during our live debate at the University of Minnesota.

1/24/2005 - by Walter ReMine

Living chimps compared to living humans?

My argument is framed within a single line of inheritance.  My argument questions whether a particular number of beneficial substitutions is sufficient to create all the human adaptations starting from some alleged ape-human-like ancestor ten million years ago. This is the total number of beneficial substitutions – not the net genetic difference between the initial organism and end-point organism.1 I cast the problem this way to make it more direct, and harder for evolutionists to finagle around.  (The reasons are technical and not necessary for this essay, which is about evolutionists misrepresenting my argument.)

My argument is not about the genetic difference between divergent organisms (organisms that do not have an ancestor-descendant relationship, sometimes called "sibling species"). If you compare divergent organisms (such as living chimps and living humans) on the basis of genetic or protein sequences, then you are misrepresenting my argument. That makes my argument less direct, more confusing, and most importantly, it grants evolutionists large fudge-factors that operate in their favor. That is why evolutionists typically  (near universally) misrepresent my argument in this manner. They are attempting to grant favors to themselves, by misrepresenting my argument. 

Typically they do that by claiming I made an "invalid assumption" that they will "correct."  That is falsehood. My argument assumes humans evolved from an ape-like ancestor (as evolutionists allege), and then shows a serious problem with that. Evolutionists cannot legitimately attack that as an "invalid assumption" in need of "correction."

In an attempt to "document" their misrepresentation, some evolutionists avoided my book (where they admit they found no support for their misrepresentation), and they searched among my countless tutorial posts still available from Google (and still found no support), and apparently found from a third-party a single electronic communication alleged to be mine (but can no longer be independently authenticated), and took out-of-context a parenthetical comment (a side comment contained within parenthesis), and misrepresented it as though it were my argument.  Those several errors in scholarship are visible from what they wrote.2 Moreover, those same individuals ignored my repeated notifications of their misrepresentations. Therefore, if you see anyone misrepresenting my argument, (even the occasional creationist who inadvertently absorbed the evolutionists’ line), then correct them on it and send them here.

That is:  Comparisons between living organisms (such as living chimps and humans, on the basis of genetic or protein sequences) are interesting.3 But they are not, and never were, my argument on Haldane’s Dilemma. I cast my argument specifically to give the least latitude for evolutionary finagling, fudge-factors, and story-telling.

1    For example, the evolution of human adaptations may likely require a series of substitutions (in a particular order) involving the same nucleotide sites.  In that case, the net genetic difference between initial and final organisms would underestimate the number of substitutions required.  

2    Robert Williams is a key example of this, but there are others too. 

3    I am not saying a comparison of living chimps and living humans is useless. (On the contrary, such a comparison is useful on many grounds, and will likely grow considerably as the data from the Chimp Genome Project comes forth.) Rather I am saying such comparison – as a means of resolving Haldane's Dilemma –  inherently favors evolutionists, and greatly caters to evolutionary story-telling. Such a comparison does not address, and does not solve, Haldane's Dilemma in its most challenging form. Haldane's Dilemma cannot be considered 'solved' until it is solved in its most challenging form. 

1/24/2005 - by Walter ReMine

Is the 1,667 limit "not a problem" ???

Some internet evolutionists chastise me for raising Haldane's Dilemma.  They aggressively post on internet discussion groups, (typically under anonymous pen-names, such as: "Huxter," "SLP," "Gordy," "Art," "KC," or "Robert"). They suggest the 1,667 limit is "not a problem," and they try to shame me for raising the issue. They claim it is "unscientific" to raise something as a problem unless I can "prove" it. And so forth.

My immediate response is, "Who are they kidding."  I would feel differently about their posturing if the public could recall a serious history of leading evolutionists claiming a limit of 1,667 beneficial nucleotides is "not a problem." But no such history exists.  Even the 1,667 limit itself was the trade secret of evolutionary genetics.1  I was the first to bring it to public attention.

The public can understand Haldane's Dilemma, and readily sees it as a serious problem. Everything necessary to communicate the problem (the 1,667 limit) was available in 1957 when Haldane published his paper.  Yet evolutionists did not communicate it to the public.  They still haven't.  In this matter they were grossly negligent. This is now a fact of history, and cannot be changed. 

For many decades already, evolutionary literature claimed Haldane's Dilemma was "solved." But that would require evolutionists to solve all the crucial components of the problem — including the number of substitutions needed to create all the human adaptations.  Evolutionists have scarcely begun to do that. Haldane's Dilemma was never solved

Haldane's Dilemma cannot be considered solved until the solution exists in ink, by capable scientists willing to put their names and reputations to it.  Evolutionary operatives posting their opinions anonymously at unreadable, internet shouting-matches, do not count as solutions. 

1    I am using the same wording — "trade secret" — with the same meaning as when Stephen Gould said the widespread absence of gradualism is "the trade secret of paleontology." There is no charge of conspiracy in either case. 

1/24/2005 - by Walter ReMine

500,000 substitutions?

My book had the ambitious goal of teaching this cutting-edge subject to the ordinary person. I therefore began my discussion with an uncomplicated scenario for tutorial purposes.  I discussed an ape-human-like species, with: 

This scenario is highly unrealistic in favor of evolution, but it is useful for teaching the basic parameters of the problem and how they interact. Such as the role played by the mutation rate, the selective value of the mutation, the population size, and of course the required reproduction rate (which is a central parameter), plus how these limit the number of substitutions.  The discussion rhetorically raises questions to get my readers thinking in new terms.  Once I teach the fundamentals, I move to a more realistic scenario — Haldane's — where one substitution occurs every 300 generations, for a total of 1,667 substitutions over the same era. This teaching approach has been quite successful, and many people (who previously knew nothing of population genetics) say they are thrilled by that chapter. Considering the difficulty of the subject, I am quite pleased.

However, some of my evolutionary opponents seized that figure of "500,000 substitutions" as though it has some special role in my argument.  It has none.  It's purpose is solely tutorial, to teach, and to get the ordinary person thinking in population genetic terms — a subject most people hadn't thought about.  The example simply reveals the mechanisms behind Haldane's Dilemma as understandable to the ordinary person. That also gets readers wondering why they hadn't seen such an obviously important topic before. ............

The figure "500,000" derives solely from the example itself (10 million years / 20 year effective generation time), and my goal of dovetailing this tutorial example easily into Haldane's example. Yet evolutionists misappropriate this figure.1 

Throughout my book I avoid causing you to rely on my opinion, rather I cite leading evolutionists to support what I am saying.  Or, I assemble the facts from evolutionists, and let you make up your own mind.  This "de-emphasis of me," is key to making the book so compelling. This approach applies also to Haldane's Dilemma, where, for simplicity and authenticity, I advanced only one figure — Haldane's!  It is essentially Haldane's figure. The limit of 1,667 substitutions comes (in all its deepest respects) from Haldane.  I merely framed it so the ordinary person can understand, and so evolutionists would have the most difficulty finagling around.

If you see anyone mis-representing the "500,000 substitutions" figure, then correct them on it, and send them here. 

1    Evolutionists frequently combine two misrepresentations together:  (1) the 500,000 figure, and (2) a raw genetic comparison between living chimps and living humans  (see here). This combination doubly misrepresents me.

1/24/2005 - by Walter ReMine

A history of neglect

Everything necessary for the 1,667 limit was known to evolutionary geneticists since 1957, but they did not communicate it to the public.  The issue ought have appeared widely (in television programs, popular science magazines and popular books).  But it didn't.  It still hasn't.  I hope the sociology of science will someday examine this remarkable history. 

But the negligence also permeates the technical literature itself. We examine that next.

After Haldane published his 1957 paper, the technical journals discussed Haldane's Dilemma from time to time, in muted tones and technical jargon.  The phrase "Haldane's Dilemma" was used, though not typically.  Typically, readers might understand the oblique references to the issue, only if they already understood the problem.  Usually no revealing figure was given.  A direct figure – such as the Haldane limit of 1,667 substitutions for human evolution – is virtually non-existent.  (That these were approximately 1,667 nucleotide substitutions was not acknowledged in print, though it should have been.)  In cases where a figure was given, it was indirect, such as "300 generations per gene substitution" – a phrase that may as well be in code, rather than revealing the problem. 

Despite the general lack of forthrightness, some leading evolutionary geneticists at least took the problem seriously. Motoo Kimura and John Maynard-Smith each cited the problem as the leading reason for their revolutionary new views of evolutionary process. They saw the problem as real, (whether or not their approach was an adequate solution). 

But they were implicitly contradicted by numerous other evolutionary geneticists, who gave radically different definitions of the problem's central concept – known as the "cost of substitution." 

In other words, the technical literature become overwhelmed with contradictions and confusions – which were seldom acknowledged, nor resolved. They still exist today, virtually unchallenged by evolutionists. Yet the contradictions occur under the simplest conceivable circumstances – single, isolated substitutions in asexual haploids. In other words, the contradictions involve the most basic fundamentals

Of all the evolutionary authorities, Motoo Kimura was the most effective proponent of the problem, which he saw (legitimately) as a compelling limitation on selective evolution (and on Darwinian evolution).  Unfortunately, Kimura's causal account of the problem skipped key portions of the reasoning behind it.  Kimura, an evolutionist, was explaining a serious evolutionary problem to other evolutionists, perhaps without wanting to reveal too much, or be too explicit, or draw too much attention?  More significantly, other evolutionists scarcely bothered to expand Kimura's causal account.  Perhaps they too did not care to reveal too much?  Whatever the cause, this lamentable situation brought some experts (such as Fred Hoyle) to note the incomplete reasoning, and thereby cast the entire problem aside as nonsense.1 More commonly, evolutionists amplified various confusion factors into so-called "solutions."  

This is another notable feature of Haldane's Dilemma, namely the evolutionists' lack of pursuit of clarity and resolution.  Virtually any suggestion was given a free pass, so long as it blurred, confused, or allegedly "solved" Haldane's Dilemma, regardless of the inconsistencies and contradictions raised thereby.  These tended to remain in the literature uncorrected, and even be repeated, cited, and pressed forward.  On the other hand, challenges to the confusions, contradictions and "solutions" were rare, half-hearted, usually non-explicit, and seldom cited or repeated, so these were effectively lost in the vast ocean of evolutionary literature.  The disparity is noteworthy.  Again I hope the sociology of science will someday examine this history. 

Around the mid-1970s, twenty years after Haldane's paper, evolutionists largely set the issue aside, believing it had been solved.  I say Haldane's Dilemma was never actually solved.  Instead, it was garbled, confused, and prematurely brushed aside as "solved," when it wasn't.2  

Haldane's Dilemma was already a decades-long scandal3 when I raised it in 1993.4  I called for evolutionists to publicly address the issue, to clean-up their confusions and contradictions, and finally to resolve Haldane's Dilemma. My calls fell on deaf ears, or more precisely, on angry ears. Evolutionists responded (not in print, but on the internet) by repeatedly maligning me personally, and misrepresenting what I wrote.  To this date (January, 2005), the evolutionary literature has responded neither to me, nor to the highly respected evolutionary geneticist, George C. Williams, who wrote: 

"In my opinion the [Haldane's Dilemma] problem was never solved, by Wallace or anyone else.  It merely faded away, because people got interested in other things.  They must have assumed that the true resolution lay somewhere in the welter of suggestions made by one or more of the distinguished population geneticists who had participated in the discussion. .... I think the time has come for renewed discussion and experimental attack on Haldane's dilemma."  (George C. Williams, 1992, Natural Selection: Domains, Levels, and Challenges, p 143-148)5

One would think the Internet would speed up resolution of this mess. But that has not happened. 

For example, since 1971, evolutionist Joe Felsenstein has claimed that beneficial substitutions inherently have "zero cost" – a profoundly mistaken view also embraced by many evolutionists as a "solution" to Haldane's Dilemma.  But more importantly for our analysis of history, Felsenstein is contradicted – implicitly – by other evolutionist experts on Haldane's Dilemma.  I pointed out the contradiction on the newsgroup, and spent many months there personally pressing Felsenstein and other evolutionists to resolve the contradiction, if only among themselves.  They still haven't resolved it.  They scarcely address it. Not in the literature; not anywhere.  Either beneficial mutations have a cost of substitution, or not.  If evolutionists cannot resolve such utterly fundamental matters, then they cannot (or will not) resolve anything much about Haldane's Dilemma. Confusion reigns. 

If you advance the subject on an internet "discussion group" (a term I use with reservations), the evolutionist response tends to be exceedingly partisan, not the open-minded pursuit of truth we expect from science.  They attempt to wear readers down with many confusion factors and misrepresentations, switching from one to another and back again, until readers are exhausted – without evolutionists acknowledging (much less resolving) even one.  They will attempt to show that you, dear reader, do not understand the issue. Those tactics are cynical and insincere, when the truth is that the evolutionary literature is manifestly self-contradicted on the issue. 

Here is another example.6  Sir Fred Hoyle's book claims, "Haldane's cost principle is an illusion." So evolutionists began using Hoyle's statement to brush aside Haldane's Dilemma. The problem is, Hoyle is contradicted by most other evolutionary geneticists, including Haldane.  So an obvious question is: Who is right, Hoyle or Haldane?  It happens they used different definitions of "genetic death" (Hoyle used Definition 3 below, while Haldane used Definition 4) and that led to their different conclusions.  In other words, the contradiction is both obvious and fundamental, yet evolutionists do not pursue its resolution.  Internet evolutionists were using confusion (their own) to deflect Haldane's Dilemma. 

A profound example of negligence occurred January, 2005, at a leading journal of population genetics.  

Do not despair, there are many good and decent evolutionists.  They just happen to remain mostly silent on Haldane's Dilemma.  Someday they will make their voices heard. In the meantime, Haldane's Dilemma remains a monument to an unhealthy scientific environment.7  

1    Fred Hoyle (1987) rightly complains that the key expositors of the problem (Haldane, Wright, and Kimura) did a poor job explaining the basis for it. (However, nothing prevented other evolutionists from making clearer explanations widely known within their journals, which they didn't do.)

"To start with, Kimura gives no explanation, only statements, and then only on pages 26 and 30 of his book.  It seemed to me curious that, with 250 pages available, a clearer account of why the Darwinian theory was as devastatingly wrong as it is said to be shouldn't have been given.  A carefully reasoned argument, at whatever length was necessary, would have been worthwhile in establishing so profound a result." (Hoyle, 1987, 1999, Mathematics of Evolution, p 117)

"... if a logically sensible explanation of ([equation] 7.30) existed, Haldane's paper would have been written more clearly than it was, Sewall Wright would have explained ([equation] 7.30) instead of conjuring it like a rabbit out of a hat, and Kimura would not have based his book merely on the obscure remarks quoted above.  He would surely have devoted a whole chapter at least to a careful analysis of the precise argument itself." (Hoyle, 1987, 1999, Mathematics of Evolution, p 124) 

2    At this point in the discussion, evolutionists often charge that I am a "conspiracy theorist." But I never claimed such a thing, and on numerous occasions I explicitly disavowed any evolutionist conspiracy.  However, I do charge evolutionists with a robust history of negligence on Haldane's Dilemma, (and their "conspiracy theorist" charge is an attempt to smear my character and divert attention away from their history of negligence). 

3    I am using the same word — "scandal" — with the same meaning as when evolutionist Graham Bell wrote, "[T]he casualness of the few attempts to provide a functional account of haploidy and diploidy constitutes a major scandal." (The Masterpiece of Nature: The Evolution and Genetics of Sexuality)  In either case, there is no charge of conspiracy. 

4    At the time, I was unaware of evolutionist George C. Williams's similar (though more gently worded) assessment of Haldane's Dilemma.  (Creationist Paul Nelson brought it to my attention in March 1995.)  

5    My critique of Haldane's Dilemma and its many confusion factors is, in my opinion, considerably fuller than that of George C. Williams. 

6    From the ARN discussion group (July 2004).  

7    The brief history given here focuses mostly on the neglect, and does not attempt to cover all the history, all the contributors, or all the ideas  (good or bad). The good ideas were always attached to confusion factors, in virtually every conceivable combination, which garbled Haldane's Dilemma, rather than solved it.

1/24/2005 - by Walter ReMine

Haldane's model

Evolutionists attempt the following argument (often on the internet, but seldom if ever in print).  They claim Haldane's Dilemma "only applies to the conditions assumed by Haldane."  They suggest Haldane's Dilemma is "solved" by breaking one of Haldane's starting assumptions.  They say Haldane used an old, unenlightened model from 1957, not a model based on modern knowledge. 

Their argument is mistaken, for the following reasons:  

  1. Haldane used principles, and models of genetics and selection, that remain exceptionally predominant today in evolutionary genetics textbooks. 
  2. In addition, Haldane used assumptions that favor evolution. When you 'break' such an assumption, not only do you not solve Haldane's Dilemma, you make the problem worse. 
  3. Some of Haldane's assumptions were used (as still commonly done today) in order to simplify the math and generalize the results.  You cannot solve Haldane's Dilemma by breaking one of these particular assumptions, since that merely complicates the problem without solving it.  There is always a cost of substitution, no matter what model is used.  Evolutionists must identify a model that actually solves Haldane's Dilemma, while remaining plausible on other grounds.  They have not done that. 

 For example: 

Haldane's old model? – Haldane used a multiplicative-fitness model. (Moreover, for the parameters he used, it also approximates an additive-fitness model.)  Both of those fitness models are still predominantly used today.  Everything in "Haldane's model" is current with today's practice of evolutionary genetics (including Haldane's uses of fitness, fitness models, selection, alleles, genes, dominance, and Mendelian segregation).  So if evolutionists throw-out "Haldane's model" they must also throw-out the modern textbooks on evolutionary genetics. 

Small selection coefficients? – Haldane assumed selection coefficients approaching zero.  This gives the absolute minimum total-cost of substitution in each case.  If you break Haldane's assumption, and invoke higher selection coefficients, then the cost increases, resulting in fewer substitutions, and Haldane's Dilemma worsens.  

The environmental-change scenario? – Haldane assumed substitutions begin in a peculiar way, via an environmental-change scenario.  The scenario operates as follows.  Neutral and slightly harmful mutations (though almost always eliminated outright) sometimes drift upwards in frequency, to arrive at moderate frequencies.  Then, when the environment changes, one of these neutral or slightly harmful mutations is converted (it is alleged) into a beneficial mutation.  This elevated starting frequency is where Haldane begins to tally the total-cost of the substitution.  By giving the substitution a free head-start to an elevated frequency, it lowers the total cost of substitution.  This cost-reduction is the only impact of the environmental-change scenario that Haldane allowed into his calculations. If you break Haldane's assumption, then it raises the total-cost of substitution, and worsens Haldane's Dilemma.1 

Constant population size? – Haldane assumed the population size remains constant throughout a given substitution (though he allowed large varieties of population size, each for a different substitution).  That was done partly for mathematical simplification (in the era before computers were readily available to readers).  When evolutionists break this assumption, they do not "solve" Haldane's Dilemma.  They merely obscure it further.  There is always a cost of substitution; it is unavoidable.  It is not enough to merely object to Haldane's simplification.  Evolutionists must actually s-o-l-v-e Haldane's Dilemma. 

Infinite population size? – Evolutionists sometimes claim Haldane assumed an unrealistic "infinite population size." That is untrue.  If Haldane had done that, then the total-cost of substitution would always be infinite – when Haldane calculated its average value is 30.  So Haldane obviously did not use an infinite population size.  Rather, Haldane used something at the other end of the spectrum.  To see it, take a haploid species, and suppose there are two independent alleles, A and B (at independently segregating loci), each with a frequency of one per thousand.  By random mating, the genotype AB (containing both alleles, A and B) would have a frequency of one per million.  But if the population size is only one thousand individuals, then in a given generation, genotype AB cannot actually exist at a frequency of one in a million.  Instead, either that genotype exists as a whole individual, or it does not exist – it either has a frequency of one per thousand, or zero.  There is no 'in-between' when dealing with individuals that are quantized into whole-bodies.  This difficulty is handled by Haldane, and by virtually all textbooks today, in the same way – by using non-quantized individuals.  To greatly simplify the math, and to generalize the results, they allow a genotype to exist at its expected frequency (without having to quantize the genotype into, say, 1000 whole-bodies).  Put simply, Haldane assumed non-quantized individuals, not infinite population size.  If evolutionists want to throw-out that simplifying assumption, then they would have to throw-out virtually all of today's evolutionary genetics textbooks.  And it still would not solve Haldane's Dilemma. 

There is nothing in "Haldane's model" that evolutionists can object to, without shooting themselves in the foot. A solution to Haldane's Dilemma (if one exists at all) will require moving beyond the models, assumptions, and techniques commonly employed in evolutionary genetics today. 

1    My book shows that if the environmental-change scenario is fully accounted (which Haldane did not do), then it worsens Haldane's Dilemma.  In any case, Haldane's handling of the matter unrealistically favored evolution.  

1/24/2005 - by Walter ReMine

Artificial selection

Cost theory has wide applicability.  Cost arguments can apply to any evolutionary scenario, for any species, in any environment, and even including computer simulations.  However, Haldane's calculation – 300 generations per substitution – applies to organisms with low reproduction rates in natural environments. That would include the higher vertebrates, and especially elephants, whales, apes, and humans. 

Under artificial selection – using human intervention and contrived circumstances – the population's reproduction rate can be dramatically increased. That can, and should, be accounted using cost theory.  In addition, a larger fraction of the reproduction rate can be devoted to substitutions (by artificially reducing the fraction of the reproduction normally disposed in other ways, such as random losses). Cost theory shows that these factors can dramatically increase the substitution rate under artificial circumstances, compared to the wild. 

Moreover, cost arguments put a limit on the beneficial substitution rate. For realistic species, the neutral substitution rate, and the harmful substitution rate, are unlimited by cost, and can be rapid.  (The reasons are technical, and covered in my paper.)  Therefore, when examining the results of artificial selection – such as the various breeds of dogs or chickens – keep in mind that many of the observed "changes" are unintended neutral or harmful outcomes.  These 'changes' should not be tabulated as "beneficial" substitutions. 

When correctly tallied, the results of artificial selection do not violate cost theory.  

2/03/2005 - by Walter ReMine

Haldane's Dilemma does not test "Darwinian natural selection"

"Darwinian natural selection" is a a broad, ambiguous term with many various meanings. For some it means "Survival of the fittest" – period. For others, it means repeated cycles of mutation, variation, excess reproduction, and selection of individuals. For others it includes wild, exceedingly rare but exceedingly fortuitous meta-mutations (as in the Hopeful Monster hypothesis). For others it includes processes that are downright harmful to the population (such as harmful mutation and inbreeding) – needed in order to get past barriers in the fitness terrain. For others it includes mechanisms for warping and deforming the fitness terrain (such as the Red-Queen hypothesis, or the Stationary hypothesis) – needed in order to get past barriers in the fitness terrain.. Then there are others for whom that term includes other levels of selection (such as kin selection, group selection, deme selection, species selection, and gene selection), which (according to Stephen Jay Gould) have complex interactions between levels, with positive and negative feedback between levels. 

My point is that the term "Darwinian natural selection" can be rather ambiguous, and thereby convey unintended meanings – which is why I sometimes avoid the term, in favor of something more specific.

In the case of population genetics or evolutionary genetics, a more specific term is needed. Indeed, I would argue that these fields do not use "Darwinian natural selection", rather they use differential survival or differential reproductive success at the genotype level (where "survival" and "reproductive success" are averaged over the entire life-cycle, which makes them two terms for the same thing). These fields rarely ever (never?) deal directly with the features and functions of organisms (the phenotype, such as what makes tigers and toads). Rather, these fields take values for "differential survival" at the genotype level and calculate the genetic outcomes of it.

In this sense, Haldane's Dilemma does not assume "Darwinian natural selection", but rather something much more modest – mutation and differential survival – both of which are observed facts. Mutation exists. And different organisms "survive and reproduce" at different rates. Those are observed facts.

Haldane's Dilemma then grants assumptions that are quite favorable to evolutionists, which makes the dilemma all the more problematic. Namely, Haldane's Dilemma assumes there are no barriers in the fitness terrain. It lets the mechanisms of mutation and differential- survival free-run at their fastest conceivable rate, unimpeded by barriers in the fitness terrain.

Recall your elementary math classes, where you can prove a new mathematical theorem by assuming something is true, and then showing the assumption leads to a contradiction – thus proving the assumption is false, and thereby establishing a new theorem. Well, that same method is also valid in empirical sciences, except that the source of a contradiction is often far less clear, and takes time to unravel. Haldane's Dilemma arises from a bundle of many assumptions, (as is universal in empirical sciences). So the source of a contradiction can be awkward to pin down. The question is: Which assumption in the 'bundle' is wrong? Which one is the source of contradiction?

"Darwinian natural selection" is not assumed within Haldane's Dilemma, so it cannot be the source of contradiction. Rather, Haldane's Dilemma assumes mutation and differential survival, both of which are observed facts, and cannot be discarded nor rejected. Other assumptions favor evolution, so rejecting those will only aggravate the contradiction. So the 'source' of the contradiction resides in the assumption that humans arose from ape-like creatures in ten million years, or in the assumption that the currently used models of genetics-selection are valid, and so forth.

The point is, Haldane's Dilemma does not test "Darwinian natural selection".

2/5/2005 - by Walter ReMine

Technical material

The cost of substitution:  confusion factors

Haldane's Dilemma is based on a technical concept known as the "cost of substitution."  All evolutionary authors define the cost of substitution in terms of at least one (and often two or more) of the following concepts: 

In truth, none of those concepts is essential to the cost of substitution. They are all secondary, at best.  Those confusion factors survive to this day.  When someone teaches those are the central issue, they have confused it.  

The so-called "solutions" to Haldane's Dilemma fail, in almost every case because they are built on a mistaken view of the cost of substitution.  Those failures are fundamental, due to a basic misunderstanding.  That is why I say Haldane's Dilemma was never solved, instead it was garbled into oblivion.  

Any progress toward resolving Haldane's Dilemma will require: (a) the elimination of many traditional confusion factors about the cost of substitution, and (b) the clarification of the physical meaning of the cost of substitution. My book and papers do that. 

1/24/2005 - by Walter ReMine

Genetic death

Haldane defined the cost of substitution in terms of genetic death. But what is a genetic death? First some terminology.  For simplicity here, call the substituting mutation the new-allele, and the previous version (which is being replaced) the old-alleleGenetic death, as a physical concept, focuses on the elimination of the old-allele, by the elimination of individuals containing it. 

Genetic death - various definitions

I challenge you to defend the genetic death concept.

Evolutionary geneticists contradict each other on what genetic death is, what it means physically, and how (and whether) it limits the substitution rate.  These contradictions are an observable fact of the evolutionary literature.  

There are four basic types of definition in the literature: 

  • Definition 1 – Genetic deaths are actual deaths ("or their equivalents in reduced fertility") of individuals containing the old-allele, due to the presence of the new-allele. Of the three basic definitions, this one is the closest to that used by Kimura, Maynard-Smith, Crow, and other evolutionary geneticists who view the cost of substitution as a broad problem to be dealt with. 
  • Definition 2 – Genetic deaths are actual deaths ("or their equivalents in reduced fertility") of individuals containing the old-allele, due to a deteriorating environment. This definition is closest to that used by Felsenstein and others to brush aside Haldane's Dilemma.  This definition is the most mischievous and least correct.  

Other things being equal, a higher cost of substitution reduces the plausible substitution rate. That sentence is correct, and is also the traditional view.  Accordingly, that means a higher number of genetic deaths (of the old-allele) will reduce the substitution rate (of the new-allele).  Again that is the traditional view.  Now consider this, under the above two definitions, an individual who is never even conceived (due to reduced fertility) counts as a genetic death.  But why should these 'virtual' (or imaginary) deaths affect the substitution rate? Why should virtual (or imaginary) deaths (of the old-allele) reduce the substitution rate (of the new-allele)?  That represents the traditional position, awkward as it is.  I challenge you to make its physical basis clear.1 

The above confusion caused some researchers to avoid the notion of 'virtual' genetic death as nonsense. The following two definitions do that. 

  • Definition 3 – Genetic deaths are a net reduction in a given generation. Count the adults containing the old-allele at the beginning and end of the generation, then the difference gives the genetic deaths.  In the simplest cases, this definition has an unambiguous, concrete, physical interpretation, and is used by Feller, Hoyle, and others, who concluded that the concept is un-important.2 
  • Definition 4 – Haldane defined genetic death mathematically, that is, he gave equations for it.3  (His physical notion of genetic death was vague, poorly defined, and unclear,4  though bore some resemblance to definition 1, and less so to definition 2.) There was an explanatory disconnect between Haldane's equations for genetic death, and a useful physical meaning of genetic death.  There was a great gap in logic that was never filled-in.  In the following challenge you will attempt to fill-in that gap.  

The Genetic Death challenge

The challenges below are not trick questions. Rather, they are fundamental and designed to reveal existing confusions. They are designed to probe the physical reasoning or logic behind the evolutionists "genetic death" concept. If evolutionists cannot resolve this riddle, then they have no business claiming their genetic death concept was understood. 

Answer the following questions:

  1. What is a "genetic death" physically?  What does it mean?

  2. Why would a higher number of genetic deaths (of individuals containing the old-allele) lower the substitution rate (of the new-allele)? Plainly stated, this is the traditional view. 

  3. From the above answers, derive the cost equations (e.g. Haldane's cost equations)

Evolutionists never answered those questions with any clarity, and that is the root cause of many mistaken "solutions" to Haldane's Dilemma. The fundamentals were misunderstood. 

To help your discussion, start with the simplest conceivable scenario.5  The simpler the scenario, the more obvious will be your confusion:

  1. A haploid species (preferably asexual, for simplicity), 
  2. with a constant population size,
  3. during a single isolated substitution (Thus, there are only two alleles, A and a. The new-allele A is substituting into the population; the old-allele a is being eliminated.),
  4. with pure viability selection (e.g. selection solely by elimination of juveniles containing the old-allele), and
  5. with a tiny selection coefficient (s<<1). 

Focus strictly on a single generation. If you can answer the questions for a single generation, then the same logic can be applied over-and-over again, to each and every generation. But if you cannot answer the questions for even one generation, then you have no business going further.6  

Then answer those questions again at each step of the following ever-looser assumptions:

  1. Next break assumption #4 and allow other types of selection (such as fertility differences).  In this case, some or all of the genetic deaths are 'virtual', not real, they are organisms who are never even conceived. That adds another layer of confusion.  Answer the questions again.  Note especially the conundrum: Why would a virtual death of the old-allele – a death that is not real – count toward reducing the substitution rate (of the new-allele)? In effect that is the traditional view, but rationalizing it is another matter. 

  2. Next, also break assumption #5, and allow larger selection coefficients, which adds another layer of complication (and confusion).  Note that this disallows a simplifying assumption used by Haldane, so it will require a greater level of physical understanding.  Answer the questions again.

  3. Next, also break assumption #1, and allow a sexual diploid species.  Note especially the conundrum: For individuals containing both the old- and new-allele (i.e. a heterozygote), where the new-allele may be dominant or recessive: What is a genetic death? Answer the questions again.

  4. Next also break assumption #3, and allow multiple concurrent substitutions (overlapping in time).  Note especially the conundrum: Various individuals contain various mixes of old- and new-alleles, so What on earth is a genetic death? Remember also that when a stream of many concurrent substitutions are occurring, the optimal genotype might never actually exist in the population.  Answer the questions again. 

  5. For extra credit, also break assumption #2, and allow changes in population size.  (Note that none of the traditional definitions allow non-constant population size.) Answer the questions again.

I challenge you to defend the genetic death concept.


Correct answers are conceivable for the simplest cases, but the logic is indirect, convoluted, and back-bending.  The above exercise is helpful nonetheless, because (if done clearly) you will inevitably brush up against the real meaning of the cost of substitution (which is far simpler and more direct than "genetic death"). In other words, genetic death was an intuitively intriguing concept that (in the simplest cases) was sufficiently correct to captivate our interest. 

However, as soon as you move beyond the simplest cases, answers to the above questions lay somewhere between incomprehensible, undefined, and false.  The above challenges help you experience that. 

The correct understanding – without the genetic death concept – can give simple, compelling answers under all the above circumstances.  With proper clarifications, Haldane's equations and argument, re-emerge as potent and important. 


1    A correct answer to that question exists (if only for the simplest cases).  But it is indirect, convoluted, and back bending.  You will likely only discover the line of reasoning by beginning with the correct understanding of the cost of substitution, and then reverse-engineering that into terms of 'genetic death'.  In other words, in the simplest cases, the genetic death concept had a toe-hold on the truth.  That is what intrigued us and kept the concept alive for so long. 

2    Definition #3, when summed over the entire substitution, and divided by the population size is always very nearly 1.0, (not Haldane's figure of 30).  For this reason, Feller, Hoyle, and others felt that Haldane's "genetic death" concept did not make physical sense, or was "an illusion," or not a problem.

But definition 3 can give useful insight. That is, combined with another parameter, definition 3 can be shoehorned into a simple and correct interpretation of the cost of substitution. I offered such a derivation in the appendix of my book. At the time, I saw this as a simple additional way to show the connection between genetic death and my cost concept. I have since jettisoned any need for the genetic death concept, and re-derived all of Haldane's equations (and Crow's equations) directly using my clarified cost concept. 

3  Haldane's original paper (1957) used the term "selective deaths," though the term "genetic death" soon replaced it in common usage.  Haldane defined it mathematically, (not physically), by giving equations for each genetic situation:  haploids; and diploids with varying degrees of dominance, inbreeding, and sex-linked loci. 

4  Haldane's 1957 paper (where he establishes the cost of substitution) was not too concerned with a physical definition of a genetic death. His equations did not calculate the number of genetic deaths (a number such as 300,000). Instead his equations for genetic death were normalized (to the adult population size), and this normalized figure (or ratio) was his focus.  For example, he estimated that the total-cost of substitution is 30, which is a normalized figure, a ratio.  That gives a proper clue to Haldane's thinking, that is, he did not dwell on genetic death as a discrete, identifiable, countable or important thing. Rather, he emphasized the ratio.  Indeed, he was correct – the ratio is important, not the actual number of genetic deaths.  Haldane's equations accurately predict something useful, but the genetic death concept does a poor job  illuminating it.  There is a clearer interpretation of that ratio, and of Haldane's equations. My paper supplies the technical details. 

5   The simplifying assumptions (1 thru 5 above) are effectively identical to the ones Haldane used to first teach his cost concept. 

6  In discussing these matters with evolutionists, I find they habitually rush to add more complexity (and confusion) before resolving the fundamentals.  I claim evolutionists traditionally garbled Haldane's Dilemma at its fundamentals.  Therefore, we must demand clarity and resolution on the simplest cases first, and build with solid-footing toward the complicated cases. Otherwise, the confusion factors (in countless various combinations) just keep coming back.  


1/24/2005 - by Walter ReMine

The Genetic Load challenge

Next retake the above challenge, except replace the words "genetic death" with "genetic load" (or more precisely "substitutional load").  

You will come to the same conclusion as before.  The genetic load concept (with some back bending and convoluted reasoning) can be made to work only in the simplest cases.  Then quickly becomes incomprehensible, undefined, or false, as the scenarios get more complex.  



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