The Missing Link in the Darwinian Argument
[In another series of posts, I develop a critique of the Malthusian argument. Since it forms the basis for Darwin’s original theory of natural selection, this also leads to a critique thereof. In an independent series of posts, I will explore this and also other criticisms. To make sure noone gets the wrong idea where I am going: I am an atheist and have no stake in “creationism,” which is in my view silly.]
The tautological definition of “fitness” for a heritable feature is that those specimens of a species with it have more descendants than those without it. If that is so, then there will be relatively more with the feature in the next generation. I call this definition tautological because it begs the question. It assumes what is then presented as a conclusion: the share with the feature will rise over time.
If you also assume that “fitness” in this sense is constant, then you can conclude that the share of those with the feature goes to 100% over time. Again, this begs the question because the conclusion is the same as the assumption. You can see it in this way:
If those with the feature have (1+R) descendants per capita and those without the feature (1+r) descendants in the next generation where R and r are positive constants with R > r, then we have for respective sizes of C and c for the two populations at the start (and assuming exactly equal generation lengths for all specimens and all in parallel, which would only hold perhaps for annual plants):
There are C * (1+R)^N + c * (1+r)^N specimens in the total population after N generations.
Every generation yields another factor of (1+R) or (1+r), respectively, which leads to the powers. The first term stands for those with the feature and the second for those without. There is no constraint here, the population can grow to infinity. You can build that into the formula and scale the total population down in each generation, but I leave it out because it does not change the conclusion materially.
Now, the first term goes to infinity faster than the second term. That means that the ratio of the second to the first term, c * (1+r)^N / C * (1+R)^N, goes to zero as N goes to infinity. To see this, we can ignore the two constants because they only introduce a fixed constant c / C. With each generation the term is multiplied by (1+r) / (1+R) which is less than 1 by assumption because R > r. We can call the ratio q, and then we have q^N here, which goes to zero as N goes to infinity. (On a continuous time scale this is exponential decay.)
We can now rewrite the size of the total population as
C * (1+R)^N + c * (1+r)^N = C * (1+R)^N * (1+o(N))
where o(N) is a function that goes to zero as N goes to infinity. This is just the quotient before. The last factor goes to 1 from above as N goes to infinity. The share of those with the feature is: 1 / (1+o(N)), which then goes to 1 (ie. 100%) from below.
All this is only a long-wound way to state an intuitive fact: In each generation, those with the feature get a bigger boost than those without. The former keep gaining on the latter, and hence their share grows. (The intuition is not enough, though, because without the assumption that “fitness” is constant, the gain could slow down and then convergence need not be to 100%. A function that keeps rising can also level off below 100%.)
You can relax the assumption of constant “fitness” somewhat. For example, you could think that R can change over time, but still remains strictly away from r, ie. that there is some positive constant s, that R > r + s always holds. If so, you can argue that the factor by which each generations boosts those with the feature is at least (r+s), which is greater than r. If that already implies that the share goes to 100%, then the conclusion for R follows a forteriori (it is even stronger) because the share has to be at least as high.
Or you could also make r variable over time, again with a safe distance to R. Or you could add some noise and think of R and r as a mean value, etc. That all makes the derivation more complicated and changes the conclusions perhaps somewhat, but the principle basically remains the same.
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There is nothing wrong with the tautological definition of “fitness.” All of mathematics is about tautologies. However, that also means that we cannot arrive at empirical results that are not tautological. Many who like to operate with the tautological definition of “fitness” are fascinated with the logical stringency here. You assume a higher constant “fitness,” and then the share must go to 100%. Yet, in a way this is only another way to state the assumption although that may not be obvious from the start. If that is warranted, then the conclusion is inescapable. The tricky part is to establish the assumption in a concrete situation. It does not follow just because the rest of the conclusion is necessarily true, which is easy to miss if you focus only on that.
Now, the tautological definition of “fitness” was not what Charles Darwin had in mind in his original theory. First of all, he thought of a concrete concept (and Herbert Spencer helped him along when he coined the term “survival of the fittest,” which was then picked up by Darwin). “Fitness” is that a specimen somehow fits into its environment or is fitted to it, ie. that it is adapted to it. “Nature” as an outside entity selects those that have higher “fitness,” and that is also why Darwin calls all this “natural selection.” Noone would have ever called “having more descendants in the next generation” “fitness” otherwise because there is no meaningful connection between the two words. And “natural selection” makes only sense if “Nature” really “selects” those who survive because of some feature and those who do not.
Darwin’s intuition is that if a specimen fits better into the environment, eg. because it can handle it better or beat other competitors in some way, then it has a higher chance of survival. The phrase is hence also “survival of the fittest.” Not only “fittest,” but also “survival” are meant in a very concrete sense. Darwin thinks of survival not only as survival for some time, but over the longer run, ie. long enough for a specimen to have descendants at all, or long enough that it can even have the maximum number possible. That is the assumption that comes from the underlying Malthusian argument that Darwin took for granted. Thomas Malthus claims that animals or plants always have the maximum number of descendants possible although he makes no argument to prove the claim. Darwin accepted that on faith.
If you can assume that all those who survive long enough have the maximum number of descendants, then this concrete definition of “fitness” collapses into the tautological one. Darwin also implicitly takes if for granted that the chance of survival is constant over time, which is not obvious. For example, it might depend on the size of the population. Or it might change with the environment. If constancy or some weaker variant that still keeps the conclusions intact does not apply, then it need not be true that the share of those with a feature goes to 100%. A simple case would be that better chances of survival end at some point. Then the share of those with a feature would remain stuck at the last level, or begin to float around if there is some noise that randomly changes it from one generation to the next.
The boost each generation gets from the feature might also be very small. So, although the share of those with the feature goes to 100%, it may take very long. The conclusion only says that this happens as time goes to infinity. It is a naive, but common fallacy to interpret this as: If “fitness” is higher, then the share must go to 100% over any period of time. In this explicit form, the confusion is obvious. If the advantage in “fitness” evaporates fast enough, this need not happen.
Still, on an intuitive level, ie. in a worldview, it is easy to make this mistake. And I would say that that happens very often. You can find many examples where someone jumps from the assertion “a feature leads to higher “fitness”” to “hence its share must be 100% in no time.” This is similar and even related to another intuitive fallacy that already Malthus subscribed to: The exponential functions (the powers in the formula are just a discrete version of this) feels extremely “strong.” Rightly understood, that is true, but only in the sense of “eventually as time goes to infinity,” not in the sense of “it can achieve anything over an arbitrary interval of time.”
More as an aside: An even funnier fallacy along these lines is to claim that also exponential decay is extremely fast, ie. when the rate is negative, or in the form with powers, the base is less than 1. People who make such a claim only look at the “exponential,” equate it with “very strong,” and then jump to the conclusion. But here it is not true even in the asymptotic sense, ie. as time goes to infinity. Exponential decay is extremely slow. The exponential function with a negative rate, exp(-t), takes infinitely long to go from 1 at time t=0 to zero. Although the claim is total nonsense, I assure you that you will find many examples where someone falls for this fallacy. I have seen examples of: “Fertility in Germany is below the replacement level, so the population shrinks exponentially, and that is very fast!”
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The main problem for Darwin’s orginal theory is that it need not be true that a higher chance of survival with the feature leads to more descendants. If surviving to fertile age immediately means that a specimen has the maximum number of descendants, and that is a constant for the species, then you have a direct correspondence. Chance of survival then translates one-to-one to the number of descendants. A higher chance means more descendants, and a strictly positive difference for the chances of survival means also a strictly positive difference for the number of descendants. If that is so, the conclusion above goes through that the share has to go to 100%. But that depends on the “if.”
Not to be understood: My point is not that natural selection is an implausible mechanism how a feature becomes ubiquitous. But the critical question is whether it is the only way, and hence natural selection is always the reason why a feature it common. If that were so — and that is Darwin’s claim in his original theory — then you can conclude also backwards that any ubiquitous feature must be the result of natural selection. Once you concede also other ways how this can happen, and there are many, it is not longer correct to draw this conclusion.
That does not mean that there is no natural selection in Darwin’s original sense. Here is a stark example: If a specimen of a species has some feature that means that it will never survive to fertile age, it will also never have descendants. In that case, any other feature that leads to survival to fertile age will win out by default. If there is none, then the species is not there in the first place.
A common claim especially popular with eugenicists, is that natural selection, understood in the original sense has come to a close in modern industrial societies. But that is not true in this sweeping sense. If at conception a fertilized egg has the misfortune to have a gene that codes for a dysfunctional form of, for example, hemoglobine, which carries oxygen in the blood, then no matter what, this fertilized egg will never grow even far enough to result in a both, let alone to someone who will have children.
Actually, this is very frequent. It is hard to estimate the share of those fertilized eggs that never make it too birth. As far as I understand it, it is at least 50% of even more. Not all of it may be due to heritable features, but for some it probably is the case. And then we have extremely strong natural selection.
This is not “natural selection” in the sense that a specimen is adapted to an outside environment, but in the sense that it can only survive if some necessary conditions apply, which are dictated by natural laws. If your blood cannot carry oxygen, then tissues cannot live long, and if the can’t the whole organism will die. In this sense, natural selection can never go away, even with good conditions in modern industrial societies as long as not every random genetic sequence will lead to an organism that can survive and procreate, ie. never.
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The challenge to his original theory came already in Darwin’s lifetime. Either someone pointed it out to him or he found the counter-example himself:
The male peacock has feathers that make it stick out. That cannot improve his chances of survival if there are also predators around. He would be much better off with less outrageous feathers. So if there is a heritable alternative here, which is obviously so because other birds don’t have such extravagent plumage, then according to Darwin’s original theory of natural selection, there could be no male peacocks and hence also no peacocks. In this case, natural selection should work against this feature very fast. Still, there are peacocks.
Darwin fixed this refutation of his original theory with another theory he tacked on: “sexual selection.” Not only does “Nature” select those who are to survive and those that are not, but also the other sex in a species (or both) select those that will have descendants. First you have to survive, but then you also have to find a mate. If you don’t, survival alone does not do the trick.
Sexual selection can now explain why there are peacocks. Female peacocks (or peahens?) go nuts for flashy feathers, so the explanation goes. Only a male peacock that has them will find a mate. And female peacocks discriminate against those who fail in this regard. We now have two probabilities instead of only one: the probability to survive to fertile age and the probability to find a mate after doing that. If we assume that these are independent (which is not obvious), then the probability to have descendants is just the product of the two.
Darwin as a Malthusian thinks this is the end of the story. Once you have a mate, you will have the maximum number of descendants which is something of a constant for a species. Hence the combined probability is equivalent to “fitness” in the tautological sense. If it is constant and higher than for those without the feature, then the argument above goes through and the share of those with the feature goes to 100% over time.
The explanation for the peacock via sexual selection is that in the product of the probability to survive to fertile age and the probability to find a mate, indeed the former goes down with fancier feathers, but the latter goes up, and it goes up enough to undo the disadvantage for natural selection (in the original sense). That is entirely possible. And if it is so, then also a feature that works against natural selection can go to 100%.
As I have already noted in another post: Once you concede that both natural and sexual selection play a role, Darwin’s initial confidence that everything can be explained via natural selection collapses. It is not obvious, which of the two sides plays the main role in general (apart from the question whether these two possibilities are really exhaustive). In actual cases, sexual selection might be more common than natural selection (perhaps in an even narrower sense of being adapted to something in the environment excluding the species itself).
However, Darwin started with a worldview where is seems “obvious” that the explanation was always natural selection. And so he backtracked from there to “it is almost always natural selection, only in some fancy cases it can also be sexual selection.” That may be so (again assuming that there are only two possibilities), but it is not a conclusion, only a bias that stems from how Darwin’s thinking evolved. It is easily conceivable that with a different course, his view would have been “it is almost always sexual selection, only in some fancy cases it can also be natural selection.” Still, this has become part of what I would call the Darwinian worldview where natural selection in a narrow sense is always assumed as the go-to explanation unless it becomes too absurd.
There are many attempts to get around this awkward point. The main route is to try and reduce sexual selection to natural selection. For example, one such theory is that the male peacock actually has higher “fitness” because it shows that it can survive even with the fancy feathers. But that is a play with words. If “fitness” means the probability of survival, then the male peacock obviously has lower chances of survival because of his plumage. It is not possible to claim that he also has higher chances of survival at the same time. Surely, you can confuse this issue by redefining “fitness” as “higher chances of survival than those peacocks without fancy feathers if they had also fancy feathers.” However, that is just an equivocation.
The reason why such a reduction cannot work is that it is conceivable that there can be sexual selection that works strongly enough that natural selection may go in the other direction, and “fitness” in the tautological sense still goes up along with it. That’s why it cannot be necessarily so that all sexual selection is also natural selection at the same time. Maybe you can show that this is the case for the male peacock, although I find that implausible from the start, but then this does not show that it can never be the case. Once you concede this, sexual selection remains on the table as an independent mechanism that does not coincide with natural selection.
Modern Malthusians get around this by calling both natural selection and sexual selection in the original selection “natural selection.” But renaming it and equivocating between different meanings does not do away with the problem either that there can be two principles that might align or also work against each other with one winning out despite of the other. I find this approach easy to explain, though. The challenge is not for the argument, but for a worldview where it can only be natural selection in a narrow sense. The concession that it might not always be so creates a tension, and so there is a demand for spurious explanations that make the worldview whole again.
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I come now to the missing link in the Darwinian argument that I have announced in the title. The conclusion that all you need to know is the probability of survival to fertile age and the probability of finding a mate to arrive at “fitness” in the tautological sense of having more descendants only holds if surving and finding a mate implies the latter. If you accept the underlying Malthusian argument (which is actually an unfounded claim that Malthus made) that all species of animals and plants have the maximum number of descendants once they have survived and found a mate, then this works.
But what if that is not true? As I argue in my series on the Malthusian argument, it looks like this assumption is false for humans. And I think it is also false for many other species. Basically, what this means is that the probability to survive to fertile age and the probability to find a mate does not yet determine the number of descendants for those with or without some feature. The number of descendants might not be a constant for those with a feature who survive and find a mate as Darwin, chanelling Malthus, assumes. If that is so, then it is not sufficient to look only at natural and sexual selection.
What we have is that “fitness” in the tautological sense can be written as the product of three terms:
Probability to survive to fertile age * probability to find a mate * number of descendants once you have made it so far.
If the number of descendants is a constant, we only have to look at the first two terms. But if it is variable and that depends on a certain feature, that is not enough. We then have some rule how those who survive and with a mate have descendants. That may depend on things like population density or also the amount of food available or its volatility and many other things. It may also be intertwined with natural or sexual selection. For example, if a specimen takes more time to find a mate, it might have fewer descendants even if they come at a constant rate. Or the decision to have fewer or more descendants might depend on their chance of survival at the moment, ie. how abundant or scarce food is or how volatile, whether there are diseases around, or whether there is shelter, and so forth.
Once you have a rule in the equation which makes things variable and which may depend on lots of things, it becomes much more complicated because this is not a number like the probabilities. Note also that there is no reason to exclude interactions between the three terms, even less so than for the first two terms alone.
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If you cannot think of a feature that might work through the third term, take as an example altruism, which totally stomps Darwinians. It does not obviously raise the chances of survival for the feature (actually it looks like it lowers them versus a very narrow egoism) or the chances of finding a mate (again that looks not very plausible prima facie).
Darwinians have spent a lot of time trying to explain how natural and sexual selection could produce altruism anyway. If you are altruistic towards those closely related to you that might promote the feature even if it does not work via your own personal “fitness.” This may be so, but altruism in humans is not that narrowly directed only at close relatives.
It looks somewhat like the attempt to do away with sexual selection by reducing it to natural selection. You have a contradiction in your worldview and then you accept any explanation that seems to fix it. There is also the equivocation technique where people try to sell you that altruism is just egoism in disguise. That may sometimes be so, but there are far too many cases where this doesn’t make sense. I have to think of Arnaud Beltrame, the French police officer who swapped his position with that of a hostage and was killed by the terrorist. At best it is silly to call this an example of egoism disguising as altruism. But actually, such a claim shows a deep incomprehension of what happened here and even callousness.
However, if a feature leads to more descendants once you have survived and found a mate, then it can even lower your chances regarding natural and/or sexual selection somewhat and still lead to higher “fitness” in the tautological sense, ie. it going to 100% in the species.
What if those who show more altruism (not necessarily to the total exclusion of all egoism) might perhaps have a married life that leads to more children than those who narrowly hunt after the next chance to mate also with some else? If so, it is at least conceivable that they end up with more descendants although their altruism might lower both their chances for natural and sexual selection. If they don’t overdo it, altruism goes to 100% in the species.
I have no idea whether this is so for altruism, this is “just so”-story. But then my point is not to prove anything in this way. What I only want to point out is that if you have a third variable term in the formula for “fitness,” conclusions based only on the first two and a constant third term no longer follow. Once this is no longer necessarily so, it is also not necessarily so that you have to show that altruism improves the chances of survival and/or the chances of finding a mate to remove the objection that it cannot exist in a species over the longer run.
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Here is another conclusion that is kind of a reductio ad absurdum: Darwin’s assumption following Malthus is that specimens of a species always have the maximum number of descendants possible once they survive to fertile age and find a mate. This is the rule that the third term above is constant and the biological maximum. Now you can argue that it cannot be otherwise. If a specimen has a feature that leads to less than maximum fertility after survival and with a mate, it will be edged out of the population because it has fewer descendants. However, that only holds if you assume that the species has this rule. Only then can you conclude that nothing apart from natural and sexual selection can play a role. But that is somewhat circular because you assume a Malthusian behavior to show that it must be so.
You can now wonder how any other rule could ever have come about in a species that first pursued constant maximum fertility with a mate. Here is how: If a rule leads to less natural selection over the longer run although it does not lead to maximum fertility with a mate at all times, and if we hold the effect for sexual selection constant, then it is possible that the rule will lead to higher “fitness” in a tautological sense, ie. more descendants and the feature going to 100% in the species.
Here is why this is plausible for a rule that precludes constant maximum fertility (not fertility that can also be so high at times). As I have explained in another post: Exponential growth is not feasible for beings that essentially live on a two-dimensional plane, even if there were no constraint for the food supply. And exponential growth would be the consequence if the first two terms are fixed and the third term corresponds to a Malthusian rule. The reason is that beings in two dimensions can at most expand quadratically with a finite speed of expansion. Exponential growth would imply that the population density goes to infinity and/or that the speed of expansion goes to infinity, where neither is obviously possible.
That’s also why a species with constant maximum fertility with a mate must run into the Malthusian endgame because mortality has to pick up because of some constraint. Mostly a constraint for the food supply, and if not that, a constraint on sheer space must stop the species. However, the Malthusian endgame for the descendants with a Malthusian rule should be really bad for natural selection. It puts them into a precarious position almost all the time. Any slight change in conditions can wipe them out in droves. Fine that those with the feature have so many descendants over the short term when there is still leeway, but it is not a winning strategy if they finally land on the brink of starvation and die out all at once.
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An example where this happens is a cancer. Cells lose control over their growth and divide as soon as they can. What that leads to is the Malthusian endgame. The cells divide so fast that a solid cancer usually has a dead core because the cancer cells on the interior cannot get their hands on sufficient nutrients to keep dividing. Only on the frontier can the cancer grow where it draws in more an more resources from the surrounding body. Once it stops growing with even that little structure, it infiltrates other tissues, metastasizes, and tries to force the body to supply it with nutrients to feed its insatiable growth.
However, the endgame is not some stabilization at the brink of starvation for the cancer, but death of the whole body. Actually, natural selection works here. Features that lead to cancer right away will not go to the next generation. Only those features that lead to cancer later on may remain in the population.
Healthy tissues do not behave like this. Their cells react to pressure from around. When it gets too high, they stop their growth. When nutrients become scarcer, they also stop growing. And in healthy tissues this happens long before a constraint for the food supply is reached. Even bacteria don’t behave like a cancer. It is quite funny how Malthusians look at them. You can show them a Petri dish with a colony of bacteria that has grown from a single cell. And a Malthusian will look at it and exclaim: That is exponential growth!
However, that is false. The colony expands at a regular speed, and that means it grows quadratically in two dimensions (cubically in three dimensions if the bacteria grow in colonies in a suspension or the soil), not exponentially. The confusion is that Malthusians know that a cell in the colony can divide after a short time. They then conclude that they actually do that all the time, which would indeed mean exponential growth. You can fault bacteria for being dumb. But they are at least not as dumb as Malthusians. They do the same thing as the cells in a tissue. They perceive the pressure, they perceive what the supply of nutrients is, and then decide to divide. Or not.
Actually, if you think about it, it is very implausible that a species would pursue a Malthusian strategy. I am unsure whether there are some exceptions, perhaps viruses who are really stupid. But not to pursue a Malthusian strategy seems like a winner hands down. And the reason why rules that are not Malthusian can exist in a species is that it raises the chance of survival over the longer run although it also means that those with the feature do not have the maximum number of descendants all the time.
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To sum up:
The tautological definition of “fitness” as more descendants is the same thing as the conclusion that those with higher “fitness” will see their share in the population rise, and with constant fitness it goes to 100%. This is no proof, but only a reformulation of the assumption, a tautology.
Darwin’s original theory relies on the Malthusian claim that specimens will always have maximum fertility. Even if you concede this, the argument is incomplete because survival alone does not determine “fitness” in the tautological sense.
Darwin tried to fix this with an additional theory: sexual selection. If you concede the Malthusian assumption of constant maximum fertility, then natural and sexual selection determine “fitness” in the tautological sense. If natural and sexual selection are independent, their product determines the number of descendants.
It is now no longer true that natural selection explains everything as Darwin orginally contended. Sexual and natural selection can work against each other, and features that lead to lower chances of survival can still go to a share of 100% in a species. Attempts to reduce sexual selection to natural selection are doomed for logical reasons. They can only work via equivocations.
The missing link is that there is a third term that determines “fitness” in a tautological sense (disregarding interactions and assuming stability for all three components, which is anything but obvious). “Fitness” can then be written as:
Probability to survive to fertile age * probability to find a mate * number of descendants once you have made it so far.
The third term stands for a rule how many descendants those have that survive and find a mate. It is possible that a rule lowers the chances of survival, the chances of finding a mate or even both, but still leads to higher “fitness” in the tautological sense, and hence it goes to a share of 100% in a species.
The Malthusian assumption is that there can only be one rule: The number of descendants after survival and finding a mate is constant at the biological maximum (in principle any other constant would do, but that would plausibly be edged out by those rules that increase it until it cannot go any further).
A Malthusian strategy is that of a cancer. It is plausible that that will lead to lower chances of survival for those that pursue it. In the short term, they may have an edge, but over the longer run, they end up in the Malthusian endgame and are pushovers for any slight move against them.
Other rules for the third term are certainly conceivable and also ones that lead to higher “fitness” in a tautological sense versus a Malthusian strategy, even if they lower the probabilities in the first two terms. If that is so, their share in a species will go to 100%, and that is stable.
Bacteria and healthy tissues are examples on a very low level that there are such rules that lead to a variable number of descendants that may also sometimes go to the biological maximum, but will not do so on a regular basis. Humans are another example where it is plausible that there is such a rule. And it is also plausible that many, if not practically all species have such rules as well.
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Note
I have edited my remarks on altruism, which were too speculative and which I had not thought through well enough. Since this was not an essential part, it does not change the thrust of my argument. More generally, my argument needs more work and is perhaps not easy to understand in all parts. I will try to make up for this in further posts, especially regarding what I mean by a “rule” that effects the third term which determines “fitness” in a tautological sense.
