Tag: evolution

Evolutionary Innumeracy

In which Dennis McCarthy calls me out for failing to address his challenge to my PZ argument. And he’s right, I didn’t even bother to address the one he pointed out because it was so obviously wrong and I had already pointed out in the book that genetic drift is far too slow to ever fill in the gap that natural selection couldn’t. Hence my point about having addressed “every single relevant point”. What Dennis did on the challenge that I ignored was to apply Kimura’s substitution equation to the question of fixations over time while omitting Kimura’s equation for mean time to fixation for a neutral mutation.

So, I will rectify that failure now. To put it in terms everyone should be able to understand, he pointed to the speed limit sign to determine how fast the car was going instead of looking at the speedometer. I didn’t see any reason to respond to that; no biologist or population geneticist would ever suggest that genetic drift is actually faster than natural or sexual selection.

VD: Economists, mathematicians, and physicists are inordinately skeptical about evolution, because they all understand the math that the biologists don’t. Every single one who bothers to looks into it ends up dismissing it pretty quickly. Darwin was innumerate, by his own admission. Dawkins is observably innumerate. And the Wistar Symposium demonstrated that the very best and most influential evolutionists simply can’t comprehend the math involved.

DM: I am in fact “numerate” and would be happy to compare my peer-reviewed articles employing math with yours. Regardless, my challenge of your PZ arguments are correct–and you haven’t addressed them. You have just moved onto other errors. I will be responding to those new errors shortly.

VD: That’s nonsense, Dennis. I believe I have addressed every single relevant point you raised. Feel free to identify any one I failed to address and I will certainly do so.

DM: Vox, your original argument was that Kimura’s equation could not account for the roughly 35 million single-nucleotide differences that have accumulated since the human–chimpanzee lineages diverged. Below is my rebuttal, which remains correct. I have seen no direct response to it. If you have written one, please link to it here (And please don’t just link to a book and claim it’s in there somewhere–or note the “entire book” is a rebuttal–but link to the explicit argument that discusses these numbers and equations below.)

Quoting my original rebuttal: “Here is the problem: What Vox Day calculated—(1/20,000)^20,000,000 —are the odds that a particular group or a pre-specified list of 20 million mutations (or 20 million mutations in a row) would all become fixed. In other words, his calculation would only be accurate if the human race experienced only 20 million mutations in total over the last 9 million years—and every one of them then became fixed. And, yes, hitting a lottery (with odds 1/20,000) 20 million times in a row would indeed essentially be probability zero.

“But our evolutionary history does not require that an exact group of 20 million mutations become fixed—only that some 20 million out of an enormous pool of candidate mutations become fixed.

Here’s the correct analysis. Using Vox Day’s numbers, in a population of 10,000 humans, we would expect, on average, 50,000 new mutations per year. And over the course of 9 million years, this means we would expect:

50,000 x 9 million = 450 billion new mutations altogether.

So out of 450 billion mutations, how many mutations may we expect to achieve fixation? Well, as Vox Day noted, each mutation has a probability of 1/20,000 in becoming fixed.

450 billion x 1/20,000 = 22.5 million fixed mutations.

And that is a pretty close approximation to the 20 million fixed mutations that have been observed.

VD: I dealt with this in PZ before you even raised the question in response to it. The problem is that you left Kimura’s time component out of your response. This is Kimura’s standard equation for how long it takes a neutral mutation to fixate: t̄ = 4Nₑ generations — mean time to fixation for a neutral mutation (Kimura & Ohta 1969). It’s mainstream population genetics.

With the standard human effective population size, Nₑ = 10,000. t̄ = 4 × 10,000 = 40,000 generations. At 25 years per generation, that’s 1,000,000 years per neutral fixation.

Neutral substitution is much, much slower than natural selection, which is why, in the book, I point out that genetic drift is not a defense of evolution and it can’t explain the observable genetic differences. And it’s also why I knew your response was irrelevant from the start.

Now, it’s actually a lot worse for the neutral substitution case than that, because there is a hard limit to genetic drift even being able to affect a population. In any population over 10,000, neutral substitutions will not fixate at all. And we have actually found empirical evidence for this mathematical barrier in the ancient DNA database.

So this is what I mean by evolutionary innumeracy. I’m not saying that evolutionists can’t punch the numbers into the calculator and get a correct result. I’m saying they don’t understand what the numbers represent or how the various mathematical equations necessarily interact.

UPDATE: DM’s response:

That response does not deal with your first incorrect improbability equation, but introduces yet another error in which you are incorrectly claiming only one mutation can fix at a time. Do you seriously think that? Do you seriously think that only one mutation out of many millions can sweep through a population at a time? Anway using your numbers:

Ne =10,000

Expected fixation time: 4N_e=40,000 generations

At 25 years per generation: 40,000 times 25 =1 million years

Nine million years divided by 25=360,000 generations

So neutral mutations that have occurred in the last 40,000 years have not had time to fix yet. But mutations arising during the first:

360,000-40,000=320,000 generations

did have the average 40,000 generations available to fix. That is eight million years of mutation production:

50,000 x 8,000,000=400 billion mutations

Applying the fixation probability:

400 billion x 1/20,000 =20 million fixations. So when we apply the time available to the numbers we land on the exact correct result.

Obviously, mutations, whether neutral or not, do not have to increase in frequency throughout a population one at a time.

My improbability equation is irrelevant, as it even says in the book. It’s not “wrong” because it doesn’t exist. It has absolutely nothing to do with the proven fact of the mathematical impossibility of evolution by natural selection.

Here is where DM went wrong: “400 billion x 1/20,000 =20 million fixations. So when we apply the time available to the numbers we land on the exact correct result.”

No, because that requires assuming that every mutation is neutral. It isn’t. Which doesn’t matter when I’m just pointing out the scale of the improbability, but when you’re trying to prove that yes, it is actually possible, then you have to run the real numbers.

1/2Nₑ is the fixation probability for a neutral mutation. It is not a general fixation probability that applies to all mutations. There are 3x more harmful mutations than neutral ones.

Then DM said I’d “introduced yet another error in which you are incorrectly claiming only one mutation can fix at a time.”

No, what I was citing there is what standard population genetics has known since 1969, which is that every single neutral substitution requires ~one million years apiece to fixate. What you’re referring to now is the “parallel drift” defense, which is so obvious that I devoted Chapter 10 to it.

“Drift, on the other hand, filters nothing. It preserves nothing. It adapts to nothing. It is purely random. But these defenders of non-Darwinian random evolution have nevertheless resorted to insisting that mutations can spread through populations by random chance alone, fixating 1,017 times per generation, without needing natural selection at all. This is what they mean when they invoke “neutral” or “mostly neutral” processes operating “in parallel.” Without realizing it, they’re appealing to the fixation model produced by Kimura and its subsequent revision by Tomoko Ohta, even though neither model is capable of even coming close to accomplishing what is claimed of them…. Faced with this obvious problem, some of the less intellectually gifted defenders of evolution have retreated to an imaginary mechanism of genetic change: neutral processes running in parallel.”

And from my paper on the Bernoulli Barrier, published around the time the book came out: “The parallel fixation mechanism is self-defeating: it invokes a process that eliminates the conditions necessary for its own operation.”

And drift is obviously not a viable retreat when you think through the basic logic of it:

  1. For neutral substitution to work to fixation at its maximum speed, natural selection cannot be operating.
  2. There are 3x more harmful substitutions than neutral substitutions.

So if genetic drift were capable of producing actual changes to the genome, the human species would have been rendered infertile and gone extinct within centuries. We know drift didn’t account for the 410 million base-pair differences, because both chimps and humans survived.

DISCUSS ON SG

Biologists are Retarded

Mr. Matsumoto clearly groks the fullness of the fundamental problem with biology: biologists are functionally innumerate. They simply aren’t capable, and they have never been capable, of understanding the math that is required for them to make any sense of their own science.

Most of the apparent reasonableness of The Modern Creation Myth rests on people’s inability to understand Size and the scale of the things Science is actually talking about. Once you actually grok it… once numbers stop being simply “Big” and you start to understand how small some “Big” numbers can be in comparison to others…

The whole notion that we evolved from random processes just completely falls apart.

Completely.

Biologists are retarded you see? They don’t understand numbers.

The most unbelievable scene in Project Hail Mary. More so than the talking rock people. I went to grad school. You can’t fool me. There ain’t one microbiologist in the universe who could work out how long it would take a spaceship to get back home from another star. Not one. Biology is for people who want the prestige of calling themselves Scientists but who also found Calculus 1 too hard.

Here. I will state my thesis plainly: Evolution only feels believable to people because, again, above a certain threshold, all large numbers just get filed under “Big” in people’s heads.

That’s it.

Mankind’s inability to conceptualize large numbers is the only reason anyone has ever taken Darwin’s theories seriously. Only reason. You point out to somebody how improbable it is that Life would spontaneously arise from the random fallout of a random explosion and they go “Yes, well, but there was an infinite amount of time for it to happen so…”

And they shrug.

You know, because to them all large numbers are just… “Big”.

But, see, there wasn’t an infinite amount of time for Life to happen.

There actually wasn’t anywhere even close to it.

Current best estimates put the age of the universe at around 14 billion years, which, compared to infinity, might as well be a microsecond. If you understand the actual statistics behind the claim, you understand that even the most generous cosmology gives you nowhere near the amount of time or space necessary to make the formation of even one cell by random chance remotely plausible.

That is a fact.

I’m telling you a mathematical fact.

It can’t happen. The creation of life by random chance could simply never occur.

There is a reason that economists are inordinately inclined to be dubious about evolution. And that reason is this: we comprehend large numbers. For all the flaws of our field, and believe me, they are many, there isn’t a single economist who is impressed by a biologist waving his hand and summoning “millions and billions of years” or dramatically declaring that 600,000 generations is but “an eyeblink” in the context of geological time.

We work in trillions, bitch.

So waving your hand and imagining that the mere evocation of ONE MILLION YEARS allows sufficient time for anything and everything to happen not only doesn’t impress us, it confirms our original impression that you are actually retarded. I mean, when your core argument is, quite literally, an Austin Powers joke about inflation in a 1997 movie, you should probably consider the very high degree of probability that your entire hypothesis is not only ill-founded, but obviously wrong.

Evolution by natural selection never happened because it is mathematically impossible. And if you don’t accept that, it is only because you don’t understand the relevant math.

DISCUSS ON SG

Probability Weasel

An evolutionist’s summary of Richard Dawkins’s famous Weasel program, which is still very popular among the innumerate for its explicative proof of the viability of evolution:

In 1986, evolutionary biologist Richard Dawkins set out to dispel such misunderstandings by implementing a computer program that works—in Dawkins’ own words—like this:

“It […] begins by choosing a random sequence of 28 letters, […] it duplicates it repeatedly, but with a certain chance of random error – ‘mutation’ – in the copying. The computer examines the mutant nonsense phrases, the ‘progeny’ of the original phrase, and chooses the one which, however slightly, most resembles the target phrase, METHINKS IT IS LIKE A WEASEL.”

Here’s a more detailed explanation of his weasel program: it starts with a string of 28 random letters. Next, it creates N offspring strings by copying the original 28 random letters N times. When being copied, the chance of a letter being changed into another (random) letter is P. Now that we have a set of N new strings, we compare each string letter by letter against “METHINKS IT IS LIKE A WEASEL” (a quote from Shakespeare’s Hamlet, by the way) and pick the one with the most character matches (the highest match score). This one is deemed the “fittest” and kept as the survivor of the first generation; the other N-1 strings are discarded. We repeat the whole process by creating again N offspring from the survivor, then pick a new survivor and so on until the survivor finally matches “METHINKS IT IS LIKE A WEASEL”.

Let’s choose N to be 100 (one hundred offspring per generation) and P to be 5%, as in Dawkins original experiment. How many generations would it take until we finally reach “METHINKS IT IS LIKE A WEASEL”? Just guess a number, I’ll wait here…

Answer: about 50 generations!

In other words, the program theoretically explains how evolution can achieve fixation within 50 generations. Therefore, the transformation from proto-chimp to human is… well, the math still doesn’t work given the 410 million base pair gap, but we can’t expect a famous biologist to understand long division. The important point that the concept makes sense to the innumerate. Although, obviously, it doesn’t make sense, since as the usual critique correctly points out, the target phrase is known from the start, which is not the case in the evolutionary context.

But never mind that. I am not interested in the usual critique, since the distinction between known and unknown ends is far too abstract for the evolutionists, who have completely failed to comprehend the significance of that distinction for the last 40 years. Let’s try something else, something more simple and basic that even the most determined Neo-Darwinist should be able to grasp: the limits of reproduction.

First, note that Dawkins’s “population of 100” is really just one weasel that has 100 kits a year, where you keep those with the desired letter and kill the others. That’s not a species. A real weasel has about six kits, up to twice a year. So plug in the real animal, with a minimum population of 10,000 so it’s not going extinct, and here’s what changes for the Weasel program.

A phrase letter initially appears in one weasel out of 10,000. For that letter to count toward completing the phrase, all 10,000 weasels have to end up carrying it. And the only way that happens is the slow way: that one weasel’s descendants slightly out-breeding everyone else’s, litter after litter after litter, until the whole population has it. That process is called fixation.

If we apply Dawkins’s proposed 5% advantage to the population in the correct manner, we assume a selection advantage of five percent. Or s=.05. With the appropriate reproductive limit of 12 rather than 100, that means it will take 400 generations for the initial letter M to propagate throughout the species, and 11,000 generations for all 27 letters required to complete the phrase to do so.

However, Dawkins’s 5% probability was referring to the initial mutation, not the actual selection advantage, which is usually much more subtle, around s=0.001. And this gives us 20,000 generations for the initial letter M and 540,000 generations to complete the full phrase. So Dawkins’s little program which was meant to demonstrate the viability of evolution by natural selection actually demonstrates the precise opposite, since his estimate of 50 generations was off by four orders of magnitude, or about 11,000x.

Which, to be fair, is actually pretty close for a biologist attempting to math. It’s certainly much closer than any of the biologists insisting that evolution by natural selection can account for the 410 million base pairs that distinguish the modern human from the modern chimpanzee.

DISCUSS ON SG

The Uncomfortable Conclusion

Genuinely curious: does vox day suggest any alternative explanation for the phenomena that Darwin’s theory explains ?

Intelligent Genetic Manipulation. We have no idea by whom. But I suspect all the recent alien disclosure stuff is because it is fairly obvious that the genetic record is going to prove that the human species, at the very least, was constructed. With a few more advancements in genetic science, we may even be able to figure out more or less when it happened.

The uncomfortable thing for everyone, creationist and evolutionist alike, is that the range of genetic variance is also likely to show that humanity is more than one species, constructed at different times. Because neither natural selection nor genetic drift can be responsible for it.

That sounds absolutely crazy, I know. But that’s what the evidence is currently suggesting. And the bigger problem is that natural selection hasn’t been performing its real function, keeping the genome from degrading, since around 1800. This is probably the real reason for the observed decline in human fertility, not microplastics or climate change.

DISCUSS ON SG

Do Try to Keep Up, Dennis

Dennis McCarthy has absolutely no idea how absolutely hopeless the case has gotten for evolution by natural selection. Believing in evolution is now much more far-fetched and implausible than believing the Earth is flat or that it is held up by a turtle supporting four elephants.

[This was one of my more popular efforts, and I think it provides a nice intro to the evidence supporting evolution:]

I believe this is one of my more important posts—not only because it explains evolution in simple, intuitive terms, making clear why it must be true, but because it directly refutes the core claims of Vox Day’s best-selling book Probability Zero: The Mathematical Possibility of Evolution by Natural Selection. Day’s adherents are now aggressively pushing its claims across the internet, declaring evolution falsified. As far as I am aware, this post is the only thorough and effective rebuttal to its mathematical analyses currently available….

Here is the problem: What Vox Day calculated—(1/20,000)20,000,000 —are the odds that a particular group or a pre-specified list of 20 million mutations (or 20 million mutations in a row) would all become fixed. In other words, his calculation would only be accurate if the human race experienced only 20 million mutations in total over the last 9 million years—and every one of them then became fixed. And, yes, hitting a lottery (with odds 1/20,000) 20 million times in a row would indeed essentially be probability zero.

But our evolutionary history does not require that an exact group of 20 million mutations become fixed—only that some 20 million out of an enormous pool of candidate mutations become fixed.

Here’s the correct analysis. Using Vox Day’s numbers, in a population of 10,000 humans, we would expect, on average, 50,000 new mutations per year. And over the course of 9 million years, this means we would expect:

50,000 x 9 million = 450 billion new mutations altogether.

So out of 450 billion mutations, how many mutations may we expect to achieve fixation? Well, as Vox Day noted, each mutation has a probability of 1/20,000 in becoming fixed.
450 billion x 1/20,000 = 22.5 million fixed mutations.

And that is a pretty close approximation to the 20 million fixed mutations that have been observed. Change the assumptions¹ and the estimate moves. But under Vox’s own assumptions, the result is the opposite of “probability zero”: it’s basically what you’d predict.

First, Dennis was completely, absolutely, and hopelessly wrong about literally everything he said in that original post, because he doesn’t understand big numbers, genetic drift, or how probability works. That’s not his fault, not very many people do. Including, as it turns out, the population geneticists.

But what is his fault is that he didn’t even notice that I had to release a new edition of Probability Zero because it turned out that the “complete mapping of the human and chimpanzee genomes” completed in 2005, and upon which all of the numbers in the book were based, was not, in fact, the complete mapping of the human genome or the chimpanzee genome, neither of which were published until 2025.

And when Yoo et al published the genuinely complete mappings of all the great apes, that 35 million difference between the two species was considerably expanded. This is from the Introduction to the Second Edition.

In fact, the genetic difference between chimps and humans turned out to be 13.8 percent, with 410 million base pairs separating the two lineages since the Chimpanzee-Human Last Common Ancestor. This 10x increase in the number of observed differences between the two genomes has had, as you might expect, a tremendous impact on the arguments I presented in the first edition of this book. In fact, it made them approximately ten times more conclusive.

Not only that, but subsequent work on Kimura’s substitution equation rendered McCarthy’s attempt to escape to Neutral Theory and genetic drift not only highly improbable to the point of effective impossibility, but flatly impossible. I was wrong. Genetic drift in humanity didn’t stop relatively recently, as I calculated in The Frozen Gene. In fact, it hasn’t occurred in humans for millions of years, because no genetic drift can take place in any population that has exceeded 10,000 individuals at any point in its existence.

You can read more about that in my paper The Hard Limits of Fixation Through Genetic Drift, which was written a few months ago after discovering a Portuguese mathematician’s analysis of Kimura’s equation and his identification of the way in which that equation has been systematically misunderstood and misapplied by Kimura and others for the last sixty years.

Kimura’s neutral substitution rate, k = μ, is a steady-state identity: it holds only after a population has held one size for the roughly 4Nₑ generations a neutral allele needs to drift from a single copy to fixation. That transit has never been checked against the number of generations that species lineages have actually had. Checking them yields a hard ceiling on population size, X = (Vₖ + 2)·G/16 — reproductive variance and lineage generations alone, with no mutation rate, no coalescent quantity, and no fitted constant. For a large, long-lived vertebrate the effective ceiling falls to about ten thousand individuals. Above it, a neutral allele’s chance of fixing within the generations its lineage will ever have is not merely small but exponentially small, of order exp(−π²Nₑ/G) — for humans at current size, about one in ten to the seventy-eight-millionth. Humans number more than eight billion, the better part of a million times over the line; the African elephant, endangered at four hundred thousand and falling, is still forty times too abundant for drift to complete to fixation. No vertebrate population is simultaneously large enough to persist as a species and small enough to fix a neutral allele by drift. What substitution these abundant populations show is not produced at their current size; it is residual drainage from the smaller populations they descend from. The domain of k = μ is confined to demographic conditions that no non-endangered species is capable of meeting.

Evolution isn’t just dead, it is absolutely, hopelessly, and completely mathematically impossible. It’s absolutely and observably retarded to continue to believe in evolution, as that is tantamount to admitting that you can’t comprehend basic multiplication or grasp the concept of a number larger than ten thousand. It is more reasonable by far to believe that a black pygmy woman wrote the entire Shakespeare canon in Africa than to believe that evolution by natural selection or genetic drift ever produced one single vertebrate species in the entire history of the planet Earth.

What part of “mathematically impossible” is so hard to understand?

DISCUSS ON SG

The Response to the Retraction

Keruru retracted his previous endorsement of the N/Nₑ argument, and much to my chagrin, he was right to do so: the derivation of Kimura’s substitution identity never needed Nₑ on either side of the algebraic equation. Supply is 2Nμ in the census N, the fixation probability of a new copy is 1/(2N) in the same N, the two correctly cancel, and the fact that the effective population Nₑ was regularly written into the supply side was nothing more than careless notation by the authors of the biology textbooks over the years, it was not a hidden move by Kimura back in 1968. Thanks to this additional evidence that biologists can’t do math, this error means that it will be necessary to produce a 3rd Edition of Probability Zero to correct my mistake in this regard. Mea maxima culpa, dammit.

The second part of his post is not correct. He says zero fixations across the ancient-DNA window is exactly what neutral theory predicts, so the data cannot discriminate. The arithmetic holds given one number: Nₑ ≈ 10,000 — which comes from θ = 4Nₑμ, which presupposes k = μ, the identity under test. He has retracted an empirical falsification of the clock by feeding in a parameter that the clock manufactures; Charlesworth flagged this exact circularity in 2009, and Keruru himself, two sections later, calls Nₑ “estimated from the quantity it is then used to explain.” He diagnosed the disease in Section 3 and caught it in Section 2. His chains carry it from the other side: they differ in offspring-number variance, never the contested axis, while his sweepstakes parent is drawn uniformly at random — setting the one parameter in dispute, the covariance between who breeds and what they carry, to zero by fiat.

And most crucially, his work never contains a census population. His own supply term at Nₑ = 10,000 gives 740,000 new mutations per generation; Kong et al. (2012) times the global birth rate gives 2.5 × 10¹¹, a factor of 330,000 — which does not touch his repaired algebra but demolishes the literature that algebra abandons. He rescued three lines and orphaned fifty-seven years of applying them. Then he names frequency drift as “the analysis that should have been run.” As it happens, I ran it while writing the first edition of PZ: the Allen Ancient DNA Resource, 1,372 Neolithic against 680 modern Europeans, has rs35619459 moving 29.3% → 91.3%, a twelve-sigma excursion against his own stated expectation, and a drift-variance Nₑ near 2 rather than ten thousand. The signal is the size of a population turnover — structure, non-exchangeability, the very thing his model pinned at zero.

Nevertheless, I owe him a considerable debt of gratitude and not just for pointing out my mistake. For, as we’ve repeatedly seen over the course of this project, the identification of one particular flaw, and the subsequent process of correcting it, has led to a new discovery. The mechanism he reached for in the second part— that the fixation time is so enormous almost nothing destined to fix finishes inside the window — is exactly right, and it turned out to be the most productive thing in the exchange. He relied upon it to argue the data cannot test the clock; but when it is reversed, that same mechanism does not protect the clock, it provides a time limit. If the chance a destined allele completes within T generations scales as exp(−π²Nₑ/T), then k = μ cannot be realized at all in any population large enough for that transit to run 4Nₑ generations — and at census population scale that exponent becomes a ceiling on population size, about ten thousand for a large vertebrate, above which gennetic drift is not slow but entirely shut down.

That is the argument of my new paper entitled The Hard Limits of Fixation Through Genetic Drift which you can read at the preprint site, and it exists only because Keruru identified my mistaken identification of Kimura’s algebraic non-error.


ABSTRACT: The Hard Limits of Fixation Through Genetic Drift,

Kimura’s neutral substitution rate, k = μ, is a steady-state identity: it holds only after a population has held one size for the roughly 4Nₑ generations a neutral allele needs to drift from a single copy to fixation. That transit has never been checked against the number of generations that species lineages have actually had. Checking them yields a hard ceiling on population size, X = (Vₖ + 2)·G/16 — reproductive variance and lineage generations alone, with no mutation rate, no coalescent quantity, and no fitted constant. For a large, long-lived vertebrate the effective ceiling falls to about ten thousand individuals. Above it, a neutral allele’s chance of fixing within the generations its lineage will ever have is not merely small but exponentially small, of order exp(−π²Nₑ/G) — for humans at current size, about one in ten to the seventy-eight-millionth. Humans number more than eight billion, the better part of a million times over the line; the African elephant, endangered at four hundred thousand and falling, is still forty times too abundant for drift to complete to fixation. No vertebrate population is simultaneously large enough to persist as a species and small enough to fix a neutral allele by drift. What substitution these abundant populations show is not produced at their current size; it is residual drainage from the smaller populations they descend from. The domain of k = μ is confined to demographic conditions that no non-endangered species is capable of meeting.

DISCUSS ON SG

The Irrelevance of k = μ 

Keruru has uncovered some interesting things in his review of the mathematics underlying Kimura’s cancellation, which I have shown is a) incorrect for most species and b) a Portuguese mathematician has shown to be mathematically incorrect:

In February I argued that the mathematics underwriting neutral theory was broken. The argument had two legs. The first, drawn from a preprint I had read approvingly, held that Kimura’s substitution-rate identity k = μ rests on an equivocation: that the effective population size N<sub>e</sub> is made to stand for two incompatible quantities — a mutation-supply term and a drift term — which are then cancelled against each other. The second was empirical: an ancient-DNA analysis reporting essentially zero allele fixations across a million loci in the 240 generations separating Early Neolithic Europe from the present.

Both legs have given way. I want to set out how, because the manner of their failure is more interesting than the original claim, and because the instinct behind the essay turns out to have been sound while its aim was off by about thirty degrees.

The equivocation that isn’t

Written as k = 2N_e μ × 1/(2N_e) = μ, the objection looks devastating. Mutation supply is a biochemical fact about replicating cells and scales with the number of individuals actually reproducing. Drift is a statistical fact about sampling variance and scales with something much smaller and considerably slipperier. Two different numbers wearing the same symbol, cancelled against each other, yielding a result that has calibrated half of molecular anthropology.

The trouble is that the derivation does not require N<sub>e</sub> in either position, and the textbook notation is simply careless.

Mutation supply is 2Nμ with N the census count — mutations occur in gametes, and every reproducing individual contributes gametes. The fixation probability of a single new neutral mutant is its initial frequency, and one new copy among 2N copies has frequency 1/(2N) — census again. The two census terms cancel. N<sub>e</sub> never enters, and so cannot be equivocated upon.

Why is the fixation probability equal to the initial frequency? Not by assumption. Under neutrality the expected change in allele frequency each generation is zero, which makes the frequency a bounded martingale; a bounded martingale that must eventually be absorbed at 0 or 1 has absorption probability at 1 equal to its starting value. That is optional stopping, and it is a theorem rather than a modelling convention.

The claim is testable without any diffusion approximation at all, using exact finite Markov chains. I built two. The first is ordinary Wright–Fisher. The second is a sweepstakes model: half the time, half the population is replaced wholesale by clones of a single randomly chosen parent — reproductive skew of the kind documented in Atlantic cod and Pacific oysters, wildly outside anything Wright and Fisher had in mind. Measured from heterozygosity decay, the sweepstakes chain has an effective size 5.6 times smaller than the Wright–Fisher chain at identical census size.

The fixation probability of a single new mutant in both chains: exactly 1/(2N_census), to fifteen decimal places.

N<sub>e</sub> can be dragged around by a factor of six and the fixation probability does not move. It follows that my own demographic work on the collapse in variance of reproductive success — a real finding, and one I still stand behind — does not touch the substitution rate. It changes coalescent depth, standing diversity, and every estimate of N<sub>e</sub> derived from them. It leaves k = μ exactly where it was.

First, the problem with Kimura turns out to be even more significant than I originally believed. I was pretty sure that Keruru had been led off track the moment that he mentioned the word “martingale” because that’s the retreat that every AI, including both Claude Athos and the entire Red Team, initially made. You cannot use AI to provide you the answer; you have to know at least the correct shape of the answer before you ask the AI.

I’ll post my full response to Keruru later this week. His work here is very far from valueless, as I will demonstrate, but it is nevertheless incorrect due to his failure to anchor the abstract mathematical elements in material reality. This also, rather fortuitously, demonstrates the importance of the Triveritas, as any one link in the triad is capable of serious error when allowed to operate outside the bounds of the other two.

But the epicycle wasn’t elsewhere. There are multiple epicyles.

k = μ is a correct baseline for asking whether something other than drift is happening. But it is not a rate at which anything can be observed to occur in a sexual, age-structured, demographically non-stationary population — which is every organism anyone cares about.

DISCUSS ON SG

Chalub and the Kimura Cancellation

In 2022 a Portuguese mathematician named Fabio Chalub published a paper solving the neutral Kimura equation called Solution of the Neutral Kimura equation with two integral constraints. It is not a critique of neutral theory. Chalub gives no sign that he is even aware that anyone is contesting neutral theory. He is a specialist in degenerate partial differential equations doing what specialists do, which is testing the assumption that a well-known equation actually has the solution everyone has assumed it has.

Chalub opens by pointing out something that seventy years of science textbooks have failed to notice. The classical solution to the neutral Kimura equation — the one Kimura wrote down in 1955, the one reproduced in Crow and Kimura and in every graduate course since — decays to zero. All of the probability drains out of the interior of the interval and nothing is left. As a description of a population, it is nonsense: the alleles have to go somewhere, which is either fixation or extinction. The standard solution does not track them. It simply stops accounting for them. To fix this, Chalub adds two point masses at the boundaries, one representing the fraction of populations in which the allele has fixed and one representing the fraction in which it has been lost. Fine. Now comes the interesting part.

Nothing in the Kimura equation determines how the escaping probability splits between those two boundaries. The equation degenerates at both endpoints — the diffusion term goes to zero there — and there is no condition inside the mathematics that says how much goes to fixation and how much goes to extinction. The equation upon which the entire molecular clock is built does not provide the one number that it needs. So Chalub supplies the number from outside, in the form of two constraints he imposes by hand before solving anything. The first is that total probability is conserved. The second is that the average allele frequency never changes, which is not a fact about diffusion but an inherited assumption from the Wright-Fisher model, the very assumption that has been successfully disputed by me and three scientists before me. Impose that assumption and the fixation probability comes out equal to the starting frequency. Of course it does. Chalub writes in the constraint before he solves anything, recovers it as a result two pages later, and he says plainly that the constraints are inherited from the discrete process.

He has no reason to be aware of what he has demonstrated. The diffusion machinery is invoked throughout the literature as independent confirmation that a neutral allele fixes with probability equal to its frequency. Chalub shows the machinery cannot confirm any such thing, because it was given the answer in advance. That is the circular reasoning. Chalub shows that the 1/(2N) is an assumption imported from the Wright-Fisher model. It is not a result produced by the mathematics. It was never tested against, or derived from, an actual population.

This changes the shape of the Neutral theory critique even further in my favor, not that I needed any more ammunition given the N/Ne switcheroo as well as Frankham’s empirical evidence that conclusively demonstrates the inapplicability of Kimura to sexually reproducing populations. But until now, the defense of Kimura’s cancellation still had one retreat available: the critic does not understand the diffusion approximation, the result is a theorem of the continuum limit, the mathematics is above his pay grade.

That retreat is now closed, and it was eliminated by a mathematician who was not even paying any attention to its manifold implications for evolution, neutral theory, or the molecular clock. The initial-frequency theorem is not a theorem of the diffusion. The diffusion cannot produce it. The diffusion has to be provided as an assumed condition that is supplied in order to solve the equation. Which means that when we say Kimura’s cancellation rests on an assumption about how populations reproduce rather than on the math, we are no longer making an assertion that requires the reader to side with us against the scientific consensus on the basis of the evidence and the logic. We are simply repeating an observation that has been independently established in the mathematical literature and can be confirmed by anyone with the math to do so.

Kimura’s assumption about populations were always the whole of his argument. Chalub proved this, published it, and no one in population genetics appears to have either read it or realized its implications.

And if you don’t grasp what this means, here is the short version: every molecular single clock has been eliminated. All of the scientific estimates about deep time are gone. Even my own attempts at recalibration are hopeless, irrelevant, and wrong. It’s not just the CHLCA estimate. It’s also the Cambrian “molecular dates predate the fossils” literature, which was used for two decades to argue the fossil record was systematically incomplete rather than that the dates were all systematically wrong. This includes the mammalian radiation and whether it preceded or followed the K-Pg. The out-of-Africa chronology. Every single eukaryotic node in TimeTree. And the dating of angiosperm origins that produced Darwin’s abominable mystery in its modern form.

All of these time frames were calculated with a rate constant that has no derivation and no non-circular calibration. And all of it is now worthless and mathematically invalid nonsense.

For those who still have no idea what I’m talking about, let me make it even more straightforward for you. And rest assured there isn’t an evolutionary biologist on the planet who can even begin to successfully argue against any of this.

Dear Biologists,

Evolution is mathematically impossible. Your historical time estimates are not only wrong, they cannot even be calculated on the basis of the information you have. And your most important equations from your best guys are mathematically nonsensical.

Love,

The Mathematicians

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Another Failed Critique

The Kurgan discovers that AI critiques of Probability Zero cannot be relied upon.

In going over the summaries I did of your work and getting AI to check I have fairly represented what you say, I got the following, which I believe to be a narrative/adjacent/hallucination type of error AI is very prone to, as I have now become very familiar with. But I have not had time to review your latest updates, so I thought it best to get it direct from you.

The issue I am having is the text the AI spat out below when I basically said its objections didn’t make sense. It tried to say that not ALL models assume 100% replacement per generation as if that somehow invalidate your conclusions, (it had to concede almost all do use 100% replacement) So, I told it : “But in any case, even if a model does NOT assume 100% replacement that just makes Vox’s point even more valid.” This was its reply:

Does an incomplete-turnover model make Vox’s point more valid? No — and this is the part I have to be straight with you about, because it’s a vulnerability sitting in plain sight.

There are two claims tangled together:

1. Turnover is incomplete — generations overlap, not everyone is replaced each generation.
2. Incomplete turnover slows evolution, by something like a factor of d.

Claim 1 is trivially true and nobody disputes it — and yes, the Moran model “agrees.” But the entire force of the Selective Turnover Coefficient rests on Claim 2, and Claim 2 is precisely what the overlapping-generation models refute. Work evolution out in the Moran model — the rigorous overlapping-generations treatment — and you do not get a 1/dslowdown. You get essentially Wright–Fisher dynamics, and for drift, if anything, slightly faster. The unambiguous version: the substitution rate of neutral mutations equals the mutation rate whether or not generations overlap (Kimura’s result is model-independent), and a beneficial allele still fixes with probability ≈ 2s (Haldane). The engine runs at the same speed regardless of what fraction turns over per nominal “generation.”

What the d ≈ 0.45 factor actually is, is a units problem, not a biological brake. If only 45% of the population is replaced in what you’re calling a “generation,” then your “generation” is just shorter than the real mean generation time — you’ve relabelled the clock, not slowed it. The real biology (births, deaths, selection per year) is untouched, and mainstream models already handle it by using the actual generation time. There’s no hidden 1/d penalty waiting to be recovered.

So the strategic position: the moment you correctly narrow the claim to “Wright–Fisher,” you’ve conceded overlapping-generation models exist — and a population geneticist’s very next sentence is “right, and the Moran model does overlapping generations and produces no turnover slowdown, so where does the Bio-Cycle Fixation Model’s 1/d actually come from?” The burden then falls on Vox’s model to explain why it diverges from the standard overlapping-generation result, and I don’t think it can.

Error 1: “Kimura’s result is model-independent.”

This is false. The book addresses this explicitly. The derivation of k = μ goes:

k = 2N · μ · 1/(2N) = μ

The cancellation depends entirely on the initial-frequency theorem: the fixation probability of a new neutral allele equals its initial frequency, which is 1/(2N). And that result depends on exchangeability — every gene copy in the population must have the same probability of being the ancestor of the entire future population. In a Wright-Fisher model with discrete generations, exchangeability holds by construction. In a real sexual population with overlapping generations, it fails, because gene copies carried by a 20-year-old with forty years of reproduction ahead of her are not equivalent to gene copies carried by a 55-year-old with two years left. Their probabilities of fixation differ because their expected reproductive contributions differ. Once exchangeability fails, the fixation probability is no longer 1/(2N), the cancellation doesn’t go through, and k ≠ μ.

The critic asserts model-independence without engaging the assumption on which the derivation depends. That’s not a refutation. It’s a restatement of the claim being challenged.

Error 2: “The Moran model does overlapping generations and produces no turnover slowdown.”

The Moran model replaces one individual per time step — one birth, one death, chosen uniformly at random. It is “overlapping” in the trivial sense that not everyone dies at once. But it preserves exchangeability by construction: every individual is equally likely to be chosen for reproduction and equally likely to be chosen for death. There is no age structure, no differential reproductive value, no biological reality in which a grandmother and a teenager have different expected future contributions to the gene pool.

The whole point of the Selective Turnover Coefficient is that real overlapping generations are not Moran-style overlapping generations. In a real human population, individuals who were already adults in generation N are still reproducing in generation N+1, and they carry their existing allele frequencies forward, diluting the effect of selection on the new cohort. The Moran model abstracts this away by making every individual interchangeable at every time step. Citing it as evidence against d is citing a model that assumes exchangeability to refute an argument that exchangeability fails. That’s circular.

Furthermore, the Moran model isn’t the one that is relied upon by any biologists anyhow. The Moran model is a less-effective attempt to correct for the very Kimura-Wright-Fisher model that is the standard in population genetics.

Error 3: “d is just a units problem — you’ve relabeled the clock, not slowed it.”

This is wrong, and the book explains exactly why.

If you redefine the “generation” to be longer (so that 100% turnover occurs per redefined generation), you get fewer generations over the divergence interval. The math doesn’t change because d enters the calculation twice and in the same direction: it reduces both the effective selection per generation (Δp ≈ d · s · p(1-p)) and the number of effective generations (G_eff = G · d). You can’t escape this by rescaling one without rescaling the other. The total selective work done over the divergence period is d² times what the discrete model predicts, not d times, and no unit conversion eliminates a squared factor.

More importantly, the “units problem” claim is empirically falsified. If d were merely a relabeling, then the standard Kimura model and the Bio-Cycle model would produce the same predictions for allele frequency trajectories. They don’t. The book tests both models against three independent ancient DNA time series — LCT, SLC45A2, and TYR — using published selection coefficients. Kimura systematically overpredicts, driving alleles to near-fixation when observed frequencies are substantially lower. The Bio-Cycle model with d ≈ 0.45 for the Neolithic reduces prediction error by an average of 69% across all three loci. Three independent loci, different selection pressures, different time periods, different geographic regions, all converging on the same correction factor. A “units problem” doesn’t produce systematic overprediction in one model and accurate prediction in the corrected model. A real biological constraint does.

The AI critic’s objection follows a familiar pattern. It defends k = μ by citing models (Wright-Fisher, Moran) that assume the very thing being contested (exchangeability), calls the correction a relabeling rather than a physical constraint, and never engages with the empirical validation that distinguishes the two models. It is, in short, exactly the kind of narrative objection that sounds rigorous until you check whether it actually addresses the math — at which point you discover it doesn’t.

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Taleb, Anti-Fragility, and Evolution

A reader writes in about anti-fragility and evolution:

I’m reading “Anti-Fragile” and, like Taleb’s other books, appreciate his insight. I do find it interesting how much he argues for anti-fragility using evolutionary and enlightenment examples. I agree with Taleb’s main point on anti-fragility. However, having read “Probability Zero” and your recent posts on Kant, I’m thinking you’ve refuted all his arguments to illustrate anti-fragility via evolution and enlightenment thinking. Taleb spends quite a bit of time writing about how evolution “directs” certain outcomes or how enlightenment thinking supports his humanism position.

Do you know if Taleb has read your recent works? His math background would hopefully allow him to fully engage in the math you’ve revealed.

My limited understanding is you’ve shown evolution can’t direct anything or even happen. The changes we see are either random or directed by some intelligence. Secondly, by refuting Kant and enlightenment thinking, would that impact Taleb’s thinking on how he approaches so many things being “unknowable”?

Taleb was familiar with SJWs Always Lie. I doubt he is familiar with any of my newer work. While we had some contact on Twitter beating up on Mary Beard and her ahistorical nonsense together, I have had no contact with him since getting banned from there in 2017 or whenever it was.

This is where I think it’s always vital to distinguish between the What and the Why. Anti-fragility is a sound concept and an important strategy that does not rely in any way, shape, or form on Taleb’s attempt to explain it in terms of evolution by natural selection or Enlightenment illogic.

One minor correction: the changes we see are not random. Kimura-style neutral drift has also been disproven in Probability Zero and The Frozen Gene, although the disproof was totally unnecessary because anyone who actually understood the math would never have pretended it could even begin to fill in the gaps produced by the insufficient rate of evolution natural selection; the drift equation is 40x slower to fixate than the already-too-slow rates examined in my books.

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