Population Increases in a Geometrical Ratio?

[This is part of my series on Thomas Malthus’ “Essay on the Principle of Population,” first published in 1798. You can find an overview of all my posts here that I will keep updated: “Synopsis: What’s Wrong with the Malthusian Argument?”]

In my second to last post in this series, I discussed Thomas Malthus’ argument that “subsistence increases in an arithmetical ratio,” or rather the lack thereof. In the last post, I supplied some independent arguments why exponential growth is not the appropriate model for what human populations actually do. And in this post, I would now like to look into how Malthus makes his case for exponential population growth.

Unfortunately, the exact reasoning is hard to pin down. Thomas Malthus makes various statements that allude to a larger argument in the background and which are strewn over his essay. However, he never assembles them into a coherent proof. One reason for this might be that this would lay the problems bare, another that Malthus was convinced that the point was obvious, and that’s why he did not bother to work it out explicitly. In any event, I have to reconstruct the underlying argument, which opens me up to a critique that I misrepresent Malthus and only battle with a strawman. To avoid this, I will supply detailed quotes.

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Here is what Malthus’ needs to prove (cf. I.18):

“Population, when unchecked, increases in a geometrical ratio.”

An increase “in a geometrical ratio” is exponential growth on a continuous time scale. Although this is by now called the “Malthusian growth model,” Malthus himself never used the concept because he could think of developments only as a sequence of events. That is curious because the exponential function had been studied for at least a century before Malthus published his essay. But then his level of mathematical sophistication is quite low even by the standards of the time.

Exponential growth means that something increases at a fixed rate, ie. the percentage increase — which is the slope divided by the value of the function — is a constant. This is actually the definition of the exponential function. So to prove that something grows exponentially because it grows at a fixed rate is just a tautology without any force.

— — —

As Malthus insinuates the claim follows from his two postulata where only the second can be relevant (cf. I.14):

That the passion between the sexes is necessary and will remain nearly in its present state.

The problem here is to understand what this postulatum is supposed to mean. I will skip the claim about the necessity of the “passion between the sexes.” You can give it a meaning, but it is inessential to the argument. My guess is that this part is only there for rhetorical reasons: It creates a parallelism with the first postulatum “That food is necessary to the existence of man.” This is indeed a truism, and the feeling of self-evidence might carry over to the second postulatum. It is then easy to miss that the second part about the “state” of the “passion between the sexes” is not obviously true, but contentious.

So, let’s focus on the important part: Malthus does not explain his terminology and just treats it as intuitive. There are hints strewn over his essay, though, that can help with the interpretation. However, both the “passion between the sexes” and its “state” are ambiguous. And in addition, there is also the qualification by “nearly,” which adds some unspecified wiggle-room.

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As far as I can see, there are three different readings for what the “passion between the sexes” may mean.

The first would be very literal: Men and women are interested in each other, which is accompanied by strong feelings. And there is supporting evidence for such an interpretation. For example, Malthus writes about “so delightful a passion as virtuous love” (cf. IV.11) and “the pleasures of pure love” (cf. XI.1). When he attacks the “detractors” of the “passion between the sexes,” he faults them because they “have never felt what love is” (cf. XI.1).

Still, in the context of population dynamics, this would perhaps be too platonic an interpretation. So the second reading of the “passion between the sexes” is more down-to-earth. Malthus might mean an interest of men and women in having sexual intercourse. Malthus seems to think of something like this when he discusses parallels with plants and animals (cf. II.20):

They are all impelled by a powerful instinct to the increase of their species; and this instinct is interrupted by no reasoning, or doubts about providing for their offspring.

As for humans, he sees the same principle at work, only moderated by the rational side of human nature (cf. II.22):

Impelled to the increase of his species by an equally powerful instinct, reason interrupts his career, and asks him whether he may not bring beings into the world, for whom he cannot provide the means of subsistence.

But there is also evidence for a third interpretation, namely that the “passion between the sexes” is just a coy way of referring not only to an interest in sexual intercourse, but to the act itself. Malthus speaks of “the power of this passion, to contribute to the sum of pleasurable sensations in life” (cf. XI.1), and makes this favorable comparison (cf. XI.1):

The superiority of intellectual, to sensual pleasures, consists rather in their filling up more time, in their having a larger range, and in their being less liable to satiety, than in their being more real and essential.

The physical side is also apparent when Malthus remarks (cf. XI.1):

Those who have spent their youth in criminal excesses, and have prepared for themselves, as the comforts of their age, corporeal debility, and mental remorse, may well inveigh against such pleasures as vain and futile, and unproductive of lasting satisfaction.

We now have three different readings for the “passion between the sexes” as (1) longing love, (2) interest in sexual intercourse, and (3) its realization. The last directly matters for population dynamics, the former only indirectly as preconditions. It also seems as if Malthus thinks of (1) and (2) as only two aspects of the same thing. They are the cause, while (3) is the effect.

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Let’s now turn to the question what Malthus means by the “state” of the “passion between the sexes.” There is again very little information apart from a few remarks that show he has two aspects in mind. The first is that the “passion between the sexes” is ubiquitous. In his attack on Godwin, Malthus rejects the possibility that it could vanish completely (cf. I.16):

There are individual exceptions now as there always have been. But, as these exceptions do not appear to increase in number, it would surely be a very unphilosophical mode of arguing, to infer merely from the existence of an exception, that the exception would, in time, become the rule, and the rule the exception.

But Malthus also speaks about the “force” or “vigour” of the “passion between the sexes” (cf. I.16 and IV.17, respectively). This would be an intensity, which is also supported by his assertion that the “passion between the sexes” is a “given quantity” (cf. VII.11). A plausible conclusion here would be that Malthus thinks of the “state” of the “passion between the sexes” as a combination of a share of the population and an average intensity for those who experience it.

In the third sense of actual sex, it is easy to merge both aspects into one, namely by looking at the average level of actual sex over a population. As for the first two senses, as longing love and interest in sexual intercourse, it is a little hard to see how to measure them even in principle. But we could do it indirectly. They are a potential for actual sexual intercourse. If they lead to the same average level of sex under certain circumstances, we can call this their “state.” In this way, we can get around the problem of comparing “many people who have a tepid experience” with “few people who have an intense one.”

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Let’s now try to understand the qualification by “nearly” for the constancy of the “state” of the “passion between the sexes.” There is only very little we can learn from Malthus’ explanations. On the one hand he repeatedly stresses constancy in the past (cf. VII.11):

The passion between the sexes has appeared in every age to be so nearly the same that it may always be considered, in algebraic language, as a given quantity.

On the other hand, he grants a certain variability. When Malthus discusses various examples, he notes that “the passion between the sexes is less ardent among the North American Indians, than among any other race of men” (cf. III.1). He describes their attitude as “apathy” and points out that (cf. III.1):

[…] in the savage state, it rarely happens that above one or two in a family grow up to maturity. The same observation has been made with regard to the Hottentots near the Cape.

This would imply rather little actual sex, so little that the population should actually be shrinking. But it depends on the interpretation of what the “passion between the sexes” means. Native Americans in the “savage state” could still have a stronger appetite for sexual intercourse than they act out. It is hard to tell. So we have almost no hint from Malthus how much leeway he is willing to grant. The only thing he forcefully rules out is that the “passion between the sexes” could vanish altogether. He faults William Godwin for conjecturing (cf. I.16):

[…] that the passion between the sexes may in time be extinguished. […] But towards the extinction of the passion between the sexes, no progress whatever has hitherto been made. It appears to exist in as much force at present as it did two thousand or four thousand years ago.

This complaint is hammered in also at other points (cf. XI.1, XI.5 and XIII.1). All we can learn from this, though, is that the qualification “nearly in its present state” in the postulatum rules out that the “passion between the sexes” could go to zero. But that leaves us with a wide range of possibilities.

— — —

Curiously, Thomas Malthus completely abandoned this line of argument in the later editions. The two postulata are gone and do not form the basis of his deduction. There are still some references to the “passion between the sexes,” but they are only marginal. Apparently, Malthus had misgivings about what had been so self-evident to him in the first edition. Why?

One reason might be that Malthus became aware of even more examples where populations showed little “passion between the sexes.” This would have made it harder to view its “state” as a constant of human nature up to some unimportant wiggle-room. And once you concede this, also a constant rate for population growth becomes shaky.

The other reason may have been that already the data Malthus used in the first edition cause a problem for his theory. If the “passion between the sexes” is some constant of human nature, one should expect that populations under similar circumstances grow at the same rate. However, when Malthus discusses various cases where populations were presumably close to being “unchecked,” he finds rather different rates. But then again, he could blame this on different conditions.

— — —

Let me sum up what we can conclude from the second postulatum as a plausible interpretation of what Malthus means:

Human populations have a certain average level of sexual activity under ideal conditions “when unchecked.” The level is constant up to some inessential leeway as a matter of human nature which supplies the necessary appetite. This is just a claim on Malthus’ part, though. As he concedes, the ideal conditions have never applied, so the average level cannot be observed. But then he interprets actual populations as imperfect realizations of the ideal.

Now, this does not yet imply that there will be population growth at a fixed rate. Two more ingredients would be needed: (1) A population “when unchecked” also pursues a fixed level of birth control. Note here that already in Malthus’ time some methods were known although it is unclear how frequently they were used. (2) A population also has to have fixed mortality “when unchecked.”

If both assumptions apply, then a constant average level of sexual activity will lead to a constant level of fertility. Combined with a constant level of mortality, this yields a fixed rate of growth for the population. There are some minor quibbles here because that would only apply in a steady state, and should be slightly different if there is a change initially. However, that is a minor qualification we can ignore here.

— — —

The next step in understanding Malthus’ hidden proof is to make sense of the term “when unchecked.” I write it in scare quotes for a reason. In a literal sense, a population is “unchecked” when there are no “checks.” So we might take this as a definition. However, that in turn depends on what the word “check” means. As becomes clear in later posts, Malthus’ implicit definition of a “check” is any factor or constraint that can result in a deviation from a state of society “when unchecked.” But then the definition becomes circular because we only know what a “check” is after we know what a state of society “when unchecked” is. So it cannot work this way.

We have to look at what Malthus tells us directly about a state of society “when unchecked.” We can glean some information from remarks as in this paragraph where he almost supplies a definition as it seems (cf. II.5):

[…] a state therefore of great equality and virtue, where pure and simple manners prevailed, and where the means of subsistence were so abundant, that no part of the society could have any fears about providing amply for a family, […]

Note that the assumption that there is “great equality and virtue” in this society is a constraint on the behavior of the population, and hence what should reasonably be called a “check.” The population does not do what it could do: Stray away from equality and virtue. The same goes for the “pure and simple manners.” So, paradoxically, Malthus builds constraints into his definition of a state of society “when unchecked” contrary to an intuitive understanding of the term.

The connection between abundant means of subsistence, the “pure manners” and population growth become clear in the following paragraph which turns around the situation in the United States of America. The conditions there lead to fewer “checks to early marriages” (cf. II.6) as Malthus notes. He does not think that the United States of America are in a state “when unchecked,” but that they come reasonably close. The idea, therefore, seems to be that a population has higher fertility if people marry earlier.

As for the impact of “great inequality,” there are some hints in another paragraph (cf. II.3):

I think it will be allowed, that no state has hitherto existed (at least that we have any account of) where the manners were so pure and simple, and the means of subsistence so abundant, that no check whatever has existed to early marriages; among the lower classes, from a fear of not providing well for their families; or among the higher classes, from a fear of lowering their condition in life. Consequently in no state that we have yet known has the power of population been left to exert itself with perfect freedom.

Again, earlier marriages lead to higher fertility. They are inhibited by material considerations for the lower classes and status anxiety for the higher classes. With “great equality” and hence no higher and lower classes, both should be minimal. There is also a remarkable concession here that I have already alluded to above: While the assertion about population growth “when unchecked” may appear at first glance as very general, it is actually extremely narrow, so narrow that it applies to “no state that we have yet known.”

There is another aspect of a population that is “unchecked.” In his critique of William Godwin, Malthus writes about the most favorable conditions imaginable, which seem to be those “when unchecked” (cf. X.10):

Let us suppose all the causes of misery and vice […] removed. War and contention cease. Unwholesome trades and manufactories do not exist. Crowds no longer collect together in great and pestilent cities for purposes of court intrigue, of commerce, and vicious gratifications. Simple, healthy, and rational amusements take place of drinking, gaming, and debauchery. There are no towns sufficiently large to have any prejudicial effects on the human constitution. The greater part of the happy inhabitants of this terrestrial paradise live in hamlets and farm-houses scattered over the face of the country. Every house is clean, airy, sufficiently roomy, and in a healthy situation. All men are equal. The labours of luxury are at end. And the necessary labours of agriculture are shared amicably among all. […] The spirit of benevolence, guided by impartial justice, will divide this produce among all the members of the society according to their wants. Though it would be impossible that they should all have animal food every day, yet vegetable food, with meat occasionally, would satisfy the desires of a frugal people, and would be sufficient to preserve them in health, strength, and spirits.

If this is a description of a society “when unchecked,” then a laundry list of constraints on behavior has to apply, which is at odds with the intuitive meaning of the term to repeat myself. Apart from the above conclusions, all these assumptions mean that there is no extraordinary mortality beyond a minimum that is presumably a constant of human nature. But then there are further constraints that have an impact on fertility (cf. X.11):

Let us suppose the commerce of the sexes established upon principles of the most perfect freedom. Mr. Godwin does not think himself that this freedom would lead to a promiscuous intercourse; and in this I perfectly agree with him. […] It would be of little consequence, according to Mr. Godwin, how many children a woman had, or to whom they belonged. Provisions and assistance would spontaneously flow from the quarter in which they abounded, to the quarter that was deficient.

Malthus sums all this up as “extraordinary encouragements to population” and the removal of “every cause of depopulation” (cf. X.13). The main difference with his earlier characterization is not only that nothing constrains fertility or pushes it down, but also that no conceivable causes of extraordinary mortality are present beyond a minimum base-rate.

These remarks are all we can find in the first edition, which does not differ much from later editions in this regard. Hence we now have basically three different interpretations of what the qualification “when unchecked” means:

(1) A list of enumerated conditions applies. Some are constraints that a population behaves in certain ways and not others, eg. they have “pure and simple manners.” Others are the absence of constraints on fertility and also of factors that push it down, and presumably also the absence of constraints that keep mortality above a minimum and of factors that lead to extraordinary mortality.

(2) An exhaustive list of conditions applies. There are again certain constraints on the behavior of the population, and otherwise neither any conceivable constraints which keep fertility away from maximum fertility or factors that push fertility down, nor any conceivable constraints that keep mortality above a minimum or factors that result in extraordinary mortality.

However, this implies only that the population can have maximum fertility and minimum mortality. The absence of constraints and factors establishes an upper bound for fertility and a lower bound for mortality. However, Malthus appears to be after an even stronger reading:

(3) Conditions apply where a population indeed has maximum fertility and minimum mortality. The claim is not that there is an upper bound for fertility and a lower bound for mortality, but that these are the exact outcomes for each.

Why do I insist on maximum fertility and minimum mortality here? Malthus does not explicitly say so. But then as becomes clear when I discuss the role of the “checks” in Malthus’ theory, those are either factors that push fertility down or mortality up, or constraints that bound fertility from above or mortality from below. There is no indication, in the first edition at least, that there could also be factors that raise fertility above the level or lower mortality below the level “when unchecked.” Because of this asymmetry, Malthus must think of a maximum and a minimum, respectively. Fertility can only go down, mortality only up.

The problem now is that the three different interpretations of “when unchecked” are not equivalent per se. If the list of conditions is not exhaustive, then (1) and (2) do not coincide. Any list of enumerated conditions, though, might be viewed as a shorthand for an exhaustive list, which would then collapse (1) into (2). Malthus’ explanations seem to support such a conclusion. But then, if a population can have maximum fertility and minimum mortality, it is not necessarily so that it will have them. Therefore, (2) and (3) do not coincide per se either. Maybe there is an equivalence here, but the conclusion would have to rely on additional assumptions.

Let’s now try to reconstruct the missing argument. If we interpret (1) only as a shorthand for (2), then by assumption there are no factors or constraints that keep a population away from maximum fertility and minimum mortality. The only way we could conclude from this potential to an actual outcome would be the constraints that there is “great equality and virtue” and that the population has “pure and simple manners.”

Equality would not only imply that people do not have concerns about the social status of their children, it would also ensure that everybody partakes in the abundant means of subsistence. Otherwise, there could still be segments of the population who have to fear for their survival. However, equality is then only a supporting assumption that the absence of factors and constraints really works out as expected.

This leaves as with the “pure and simple manners,” which probably also subsume “great virtue.” One effect would be on mortality. As already noted, if a population can have minimum mortality, this does not have to imply that the population will have it. People could voluntarily behave in reckless ways although nothing forces them to do this.

But if the population has “pure and simple manners,” or what appears be a part of it: if it practices “[s]imple, healthy, and rational amusements” instead of “drinking, gaming, and debauchery” (cf. X.10), mortality might indeed be at a minimum, and a lower bound would turn into an exact outcome.

However, then Malthus needs the constraint that a population behaves in a way that leads to minimum mortality. If that is the assumption, of course, the tautological conclusion follows that the population has minimum mortality. But then this only begs the question.

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As for fertility, the implicit argument seems similar, but must be somewhat more complicated. We first have to understand how the “pure and simple manners” affect fertility. The opposite appear to be “vicious customs” (cf. II.28). When speaking about reasons that keep people from getting married, Malthus states that “in all societies, even those that are most vicious, the tendency to a virtuous attachment [ie. monogamous marriage] is so strong that there is a constant effort towards an increase of population” (cf. II.23).

Since Malthus draws a straight line from “virtuous attachment” to population growth, “vicious customs” appears to lower fertility. One interpretation here might be that Malthus means practices that do not lead to procreative sex. But there are some clues to another interpretation in Malthus’ attack on Condorcet later on. The complaint is that “[…] he alludes, either to a promiscuous concubinage, which would prevent breeding, or to something else as unnatural” (cf. VIII.10).

My best guess as to “something else as unnatural” is that Malthus refers to contraception and abortion here. At first glance, however, the idea that “promiscuous concubinage” would “prevent breeding” seems odd. As becomes apparent in other statements (eg. Malthus’ discussion of Otaheite, ie. Tahiti, in chapter I.V of the sixth edition), he indeed believes that promiscuous sex lowers fertility.

But that cannot be true. Compare a society with perfect monogamy with a society that is promiscuous. Some couples will find that one partner is infertile. That means they will have no children even if the other partner could have them. In a promiscuous society, there should be hence more children all else equal.

One possibility is that Malthus thinks of “promiscuous concubinage” as equivalent to polygamy. Although from the male side it may seem as if this leads to higher fertility, only the female side is important for population dynamics. If one man takes turns at having sex with several women, the probability that each of them becomes pregnant should be lower than with only monogamous relationships.

But then again, it seems as if Malthus only means promiscuity (cf. X.11). In that case, there might only be an ever changing pattern of effectively monogamous relationships, which could lead to the same level of sexual intercourse in a population as with ubiquitous “virtuous attachment.” It seems even plausible that more opportunities for sex would arise in this way. Or as per the above argument, that fertile women otherwise married to an infertile man would find a partner who is also fertile.

Another possibility is that Malthus starts from the observation that “promiscuous concubinage” in his time resulted in astonishingly few children being born or none at all. There are several possible reasons why this could have been so: Contraception and abortion might have been in the background, which Malthus does not have on the radar. Sexually transmitted diseases and abortions could lead to premature infertility. Or those who are infertile were drawn to “promiscuous concubinage.”

Whatever the explanation, Malthus assumes that “vicious customs” lower fertility versus a state where everybody is committed to “pure and simple manners.” In addition, Malthus assumes a “dictate of nature in an early attachment to one woman” (cf. II.23). He confidently states that under the most favorable conditions: “I do not conceive that there would be one woman in a hundred, of twenty three, without a family.” (Cf. X.12).

If he wants to arrive at maximum fertility, though, Malthus has to go even further and assume that people marry as early as possible, basically when they reach puberty or even before. Malthus, though, avoids such a delicate point. This is similar to his claims about monogamy leading to higher fertility. Concessions in these regards would probably be awkward for Malthus because one of the main goals in his essay is to make an argument for monogamous marriage and also as customary in his time. That’s why he presumably avoids the obvious conclusions for extraneous reasons.

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However, even if we grant these rather minor problems for the argument, all this only implies that a population can pursue maximum fertility, not that it will. The probable argument now seems to be the second postulatum that the “state” of the “passion between the sexes” is approximately constant. If you view this as a driving force for how much sex people will have, the consequence is that, at least on average, all those married couples will have sexual intercourse at a regular pace. Since “pure and simple manners” appear to rule contraception and abortion out, that would then also lead to a certain level of fertility, which is constant.

This still does not establish maximum fertility, but only approximately constant fertility at some level. The implicit argument, as evidenced by Malthus’ view of the “checks” as one-sided, can only be that he assumes that the regular pace of sex within marriage has to be the maximum, and as a potential for actual sexual intercourse also the “state” of the “passion between the sexes.”

— — —

Basically, what this means is: People always want to have as much sex as they can. The “state” of the “passion between the sexes” is not just any constant, but the maximum. That makes it also plausible why it should be constant at all because there is some hard constraint imposed by human nature. For any other constant level, the constancy is not as obvious. But then the unassuming second postulatum was (cf. I.14):

That the passion between the sexes is necessary and will remain nearly in its present state.

As it now turns out, it has a hidden meaning that only becomes apparent if we look backwards from how Malthus wants to use it.

It is not about some level as it sounds, but a very specific one: People will try to live their “passion between the sexes” out to the max if given a chance, and that means the average level of sexual intercourse will also be at a maximum under ideal conditions. Because of Malthus’ assumptions about monogamy, people will try to marry as soon as possible to have maximum sex, logically before or at puberty. This Malthus assumes is a “dictate of nature.”

To obtain also maximum fertility from maximum sex, Malthus assumes that such a population “when unchecked” is constrained in its behavior and will abstain from anything that might lower fertility: contraception, abortions, and — due to his understanding — also promiscuity. This assumption is hidden in the casual introduction of the “pure and simple manners” and “great virtue.”

And then Malthus assumes that a population “when unchecked” is also constrained in its behavior regarding reckless behaviors. They will do everything to have minimum mortality. That this and also maximum fertility are feasible comes from Malthus’ further assumptions that food is abundant, that there is “great equality,” which does away also with status anxiety as an inhibition, and that there is nothing that leads to extraordinary mortality.

— — —

But then all this means is the following:

A population “when unchecked” cannot only have maximum fertility and minimum mortality, but must have them. In as much as the maximum and the minimum are constants of human nature, there is a constant rate at which further children are born and a constant rate for how many people die. Together this means there will be the maximum rate possible for population growth, which is nothing but another word for exponential growth at the maximum rate.

However, the conclusion that a population “when unchecked” must have maximum fertility and minimum mortality is mostly so by definition. Malthus first assumes their possibility via abundant food and the absence of extraordinary mortality. And then there is a whole series of properties that define such a state of society according to Malthus: People have “pure and simple manners,” there are “great equality and virtue,” “simple, healthy, and rational amusements” instead of “drinking, gaming, and debauchery,” no “vicious customs,” etc. Ironically, he calls this constrained state of society “unchecked.” And all this is not a conclusion, but an assumption.

The only part that is not assumed right away is the part that follows from the almost constant “state” of the “passion between the sexes” in the second postulatum. At first glance, this sounds rather innocuous as if it only meant that Malthus claims that people will always feel longing love and and interest in sex, which is intuitively plausible.

Quite cleverly, he contrasts this with a claim that the “passion between the sexes” could be “extinguished,” which sounds weird. He insinuates that Godwin and Condorcet have to make this assumption, which is not even true. But then this is another huge red herring in Malthus’ essay: It creates the impression as if you had to prove this implausible claim to refute the general argument. But you don’t. All it takes is to show that fertility can go down to or below the replacement level, not to zero.

As it turns out, the actual claim that Malthus has to use in his argument is far stronger, namely that the “passion between the sexes” is at a maximum, not just some level. People will always want as much sex as they can and will do everything to have it. But then this is another strong assumption that comes out of nowhere. Malthus never even tries to show it, and he even eliminated it completely from his argument in the later editions.

The only reason you have to believe this is that Malthus states his postulatum and then swiftly moves on without argument and begs his readers to concede his premises (cf. I.17):

Assuming then, my postulata as granted, I say, that the power of population is indefinitely greater than the power in the earth to produce subsistence for man.

This and and all the other assumptions taken together are just a complicated way of defining a population “when unchecked” as one that has maximum fertility and minimum mortality. And that is again only a reformulation of maximum population growth at a fixed rate or equivalently exponential growth.

The whole argument that Malthus never makes in the open is just another case of “begging the question,” assuming what he wants to conclude. You can write it down thusly:

Grant me all the conditions that mean a population “when unchecked” grows exponentially, and I will conclude that it indeed grows exponentially. That is tautologically true, but also meaningless as a proof. It is because you never see the whole argument, but only the bits and pieces strewn over the essay. You simply lose track. And that’s why you as a reader miss what is going on.

— — —

There is another twist here: Instead of developing the relevant argument, Malthus shifts the focus to a different question, just as with his “proof” for linear growth of the food supply, namely to the question what a lower bound for population growth might be “when unchecked.”

This is only tangentially related with the real question and mostly irrelevant for his further arguments. Any positive rate would do in principle, and that is extremely easy to establish: Populations have often grown, so maximum growth has to be with a positive rate. At most, Malthus would have to show that it is not excruciatingly slow. But that is again obvious from the evidence as populations have often grown very fast. So, to a certain extent, this is all another red herring.

Readers are nonetheless blindsided with this issue, which takes center stage in the argument: Malthus points to the experience in the United States of America that come at least close to being “unchecked” in his view (cf. II.6). Population seems to double every twenty-five years there. As he explains in the sixth chapter, even higher population growth has sometimes occurred (cf. VI.5):

In New Jersey the period of doubling appeared to be 22 years; and in Rhode Island still less. In the back settlements, where the inhabitants applied themselves solely to agriculture, and luxury was not known, they were found to double their own number in 15 years, a most extraordinary instance of increase.

That is not all. In the tenth chapter, Malthus lets on that a doubling every fifteen years is not yet the maximum for population growth (cf. X.13):

England is certainly a more healthy country than the back settlements of America; and as we have supposed every house in the island to be airy and wholesome, and the encouragements to have a family greater even than with the back settlers, no probable reason can be assigned, why the population should not double itself in less, if possible, than fifteen years.

What is funny here is how Malthus inserts the qualification “if possible” at the last moment, which makes his claim tautologically true. If it is possible, of course, there is “no probable reason […] why population should not double itself in less.” But then this just begs the question. We can ignore this unforced error. Yet, it is still notable how often Malthus uses this logical fallacy.

The conclusion of Malthus’ discussion should now be that a population “when unchecked” doubles in less than fifteen years. In the sixth edition, Malthus goes even further and eyes a period of only 10 years for a doubling where he relies on an estimate by Sir William Petty (cf. I.II.14). So, the lower bound could even be a doubling in less than ten years, whatever the exact number may be. Instead, Malthus makes this logical leap (cf. II.7):

This ratio of increase, though short of the utmost power of population, yet as the result of actual experience, we will take as our rule; and say,

That population, when unchecked, goes on doubling itself every twenty-five years or increases in a geometrical ratio.

What we have here is no proof at all. Malthus just assumes his conclusion by defining it as a “rule,” or in other words: he begs the question again. This is not the consequence of the argument he has developed. He has only derived that a population doubles faster than every 25, 15, or 10 years with maximum population growth, not that a population “when unchecked” does that. Only if it does, can we also obtain this lower bound, which is rather irrelevant for his further arguments anyway. And this is not even an exact value, but a lower bound if you take his arguments seriously. As it stands, Malthus’ “proof” fails on several counts.

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To sum up: Malthus spends an inordinate amount of time on establishing a lower bound for maximum population growth although any moderately positive lower bound for the rate is sufficient for his further arguments. The real argument, namely whether a population “when unchecked” indeed grows exponentially is strewn over the essay and never developed. Readers have to reconstruct it from various hints and almost necessarily have to lose track.

If you make the effort and reconstruct the hidden argument, it turns out that it is only another exercise in “begging the question.” A state of society “when unchecked” is not what it seems: one where there are no constraints. Quite to the contrary, such a population has to abide by very stringent constraints on its behavior.

Malthus assumes that “when unchecked” means that a population can have maximum fertility and minimum fertility. This does not imply that it must have them. That is then another assumption: Such a population does everything to have minimum mortality, and it does everything to have the maximum level of sex. Via another assumption, the population refrains from all activities that would lead to less than maximum fertility. And so this is the result by definition.

In as much as minimum mortality and maximum fertility are constants of human nature, assuming them is equivalent to assuming that a population “when unchecked” has population growth at the maximum rate, which is again only another word for: it has exponential growth at the maximum rate. All this follows because it is so by definition or because Malthus states it and asks his readers to grant him the assumption.

Nowhere does he make an argument that the “state” of the “passion between the sexes” is constant, or how much leeway you have to grant. He leaves all the ambiguities of the very terms open and never tries to explain them. Initially, you may concede that the “passion between the sexes” is rather stable. But only later does it turn out that the almost constant “state” is the maximum level.

You are distracted from this not only by the side discussion on a lower estimate for maximum population growth, but also by the strawman that to refute the whole theory you would have to prove that human population lose all interest in sex. In addition, there is the confusion that examples of population growth that come close to the maximum for some time show that a population “when unchecked” must have it.

Subtle as all this is, it is nothing but a series of logical fallacies, astute rhetorical tricks, and misdirections. Malthus has no argument why the food supply increases linearly, and he also has no argument why population growth has to be exponential. Even worse: The supposed proof is only for a population “when unchecked,” which Malthus concedes has never been the case. So at first glance, even if he had shown the result, it would be irrelevant for actual human populations. In a further post, I will look into the reasons why Malthus thinks he can also draw conclusions in the general case, which lays another deep confusion bare.

As I have tried to show in my previous post: “Human population do NOT grow exponentially,” bthere is also no theoretical reason to think they should or empirical evidence that the do or at least almost. All this is only an idée fixe of the Malthusian worldview where this behavior is assumed as self-evident because “Malthus has proved it.” But he hasn’t. Not even close.