Introduction

The prodigious mental abilities exhibited by geniuses throughout history are both fascinating and awe-inspiring. Many well known (some possibly apocryphal) stories speak to the remarkable abilities of Carl Friedrich Gauss, John von Neumann, and several others. Recently I saw someone comment that they had never heard such a story about Euler. That’s a good excuse to finally write a little about the man.

Leonhard Euler (1707–1783) was the greatest mathematician and physicist of the 18th century and the most prolific mathematician of all time.1 The depth and breadth of his contributions to mathematics and physics are so great that they are difficult to convey. In his eulogy to Euler, Marquis de Condorcet (1743–1794) spoke of the “impossibility . . . of conveying in detail . . . an accurate idea of the multiplicity of discoveries, new methods of investigation, and ingenious views” contained in his work.2

This short piece is not about that work, but about the man behind it: his talents and personality. By all accounts, Euler had an extraordinary memory and capacity for mental computation which, together with an unstoppable drive, made possible this uniquely prolific career. His personality was also a refreshing contrast to the stereotype of the neurotic and isolated genius. His work, intellectual ability and personality all made a great impression on his contemporaries and he died as a revered celebrity.

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A blossoming talent

Leonhard Euler was born in 1707 in Basel, Switzerland. As a boy, he was cheerful, sociable and highly talented, though the term “prodigy” has not as routinely been applied to him as with, say, Gauss. This, however, must partly be understood in light of his lack of early dedicated mathematical training. After all, Leonhard’s father wanted his son to become a pastor; and his upbringing and early education reflected that.

Nevertheless, Euler did receive basic instruction in mathematics from his first teachers: his parents. As a boy, his father gave him an old-fashioned algebra textbook from the 1500s. This book must’ve been extremely difficult to work through for the little child. As Euler-biographer Ronald Calinger notes: “[A]t just seven years of age, only an exceptional child could have hoped to make it through such a difficult text.”3 But little Euler was exceptional.

Seeing that he had extraordinary intellectual ability, the 8-year-old Leonhard was sent by his parents to live with his maternal grandmother so he could attend the Latin grammar school (a Gymnasium) in Basel.4 The Latin school offered little to no mathematics instruction, though Leonhard’s father hired in 1715 a theologian tutor with a tolerable background in mathematics for private lessons.5 In 1720, 13-year-old Euler registered at the University of Basel for courses in the philosophical faculty.

Euler mastered all his subjects in university through hard work and an astonishing memory. His memory was a gift for acquiring languages. At age 14, the young Euler gave a speech in Latin on arithmetic and geometry. The year after, he was appointed three times to exchange views publicly about Latin papers.6

As part of the standard Latin curriculum, Euler was also exposed to Virgil’s Aeneid. He learned this epic poem entirely by heart:

He now displayed his eidetic memory by reciting long passages from Virgil’s Aeneid, which contains more than ninety-five hundred verses and which he knew entirely by heart. Even at the age of seventy, he could cite the beginning and closing words on each page of the text he had read as a young man.7

In 1723, the 16-year-old Euler passed the examination for his master of arts degree.8 At the graduation session, he gave a public lecture in Latin comparing the natural philosophy of René Descartes with that of Isaac Newton and indicating the consequences of each.9

Young Euler had a knack for solving mathematical problems with only minimal external assistance. Starting in October 1723, he had weekly meetings with the famous Basel mathematician Johann Bernoulli (1667–1748). “Private lessons, however, he [Johann Bernoulli] categorically ruled out because of his busy schedule,” says Euler in his short autobiography, however:

[H]e gave me a far more beneficial advice, which consisted in myself taking a look at some of the more difficult mathematical books and work through them with great diligence, and should I encounter some objections or difficulties, he offered me free access to him every Saturday afternoon, and he was gracious enough to comment on the collected difficulties, which was done with such a desired advantage that, when here solved one of my objections, ten others at once disappeared, which certainly is the best method of making auspicious progress in the mathematical sciences.10

“This instilled an excellent process,” said Nicolas Fuss (1755–1826), mathematician and later assistant of Euler, “but only one that can succeed with an extremely talented genius which Mr. Euler possessed.”

In line with his family’s wishes, Leonhard Euler began studying theology in 1723. It was probably Bernoulli who in 1725 helped convince Leonhard’s father that his son’s true calling was mathematics. Johann Bernoulli (who was the most highly regarded active mathematician at the time11) now considered his 18-year-old student a genius who alone was worthy to be his successor. He described Euler as possessing the highest acumen, together with the mental agility and ingenuity, to be able to penetrate the most profound secrets of higher mathematics.12

Once he received his father’s blessing, and once his full attention was devoted to mathematics and science, Euler’s genius could truly flourish. He began independent investigations at the age of eighteen and had his first paper published in 1726. The following year Euler participated in the prestigious Paris Academy science competition which proposed for 1727 the problem of the most efficient arrangement of masts on a ship. Having grown up in a landlocked country with no practical experience with ships, it was impressive that the 19-year-old Euler’s essay managed to secure an honorable mention (practically speaking a second place finish). Euler would, in time, go on to win the prestigious prize more times than any other individual.13

It was also in 1727 when Euler was hired by the St. Petersburg Academy in Russia, where he was eventually appointed an adjunct of higher mathematics. This marked the beginning of his long and prolific career in mathematics and natural philosophy.

In St. Petersburg, Euler had two early breakthroughs in the years 1735–1736 which brought him considerable fame. First he solved the famous Basel Problem, a problem concerning the sum of the reciprocals of squares that had stumped the leading mathematicians for half a century. Secondly, he also published Mechanica (1736), a widely praised book that was the first one on mechanics entirely in the language of the new calculus. His reputation was further bolstered by Paris Academy competition prize wins in both 1738 and 1740. In 1741 he began working in the Berlin Academy in Prussia, and in 1766 he returned to the St. Petersburg Academy once again where he stayed until his death in 1783.

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Euler, the person

Euler was in many respects the polar opposite of the prevalent mathematics genius stereotype. He was not neurotic, isolated or introverted. He was cheerful and sociable, generous and kind, and always humble. He also enthusiastically took on the role as teacher (unlike, say, Gauss or Newton).

Nor was Euler the stereotypical lifelong bachelor. He had a long and healthy marriage. He and his wife together had thirteen children (of which, sadly, only five survived past early childhood), and the children were a great source of happiness for Euler.14 For entertainment, he and his wife enjoyed taking their children to the zoo and marionette shows. On the rare occasion that there was any time for leisure, Euler enjoyed chess and playing the clavier. He was also an ardent pipe smoker.15

Despite giving up becoming a pastor, Euler remained a wholehearted believer and closely followed the religious practices of his upbringing until his death. Before bed, Euler would often read Biblical passages for his children.16 His unshaken Christian conviction was occasionally the target of criticism from the contemporary French Enlightenment philosophers, but thick-skinned Euler took it in stride.

It was not just his work and talents that made an impression on his contemporaries; his personality did as well. As a person, Euler was described as:

ascetic; always ready to engage in conversations, with the rare ability to shift immediately from deep discussion to a casual level and back; gentle and cheerful, enjoying harmless jokes; good-hearted and fair, refusing to hold grudges. Euler, Fuss avowed, had been an incomparable teacher; the student recalled a scholar devoutly religious, unmarked by the skepticism and the crisis of conscience that often went with the mentality of the Enlightenment. He was also hailed as a good husband, father, and friend.17

In 1738, a dangerously high fever led to total blindness in Euler’s right eye. But he was not perturbed. Some years later, Euler wrote a letter expressing joy about life in Berlin:

After having left the Petersburg Imperial Academy, I have every reason to be satisfied with my lot. . . . I can do just what I wish [in my research] and no one expects anything from me. The king calls me his professor [mon professeur] and I think I am the happiest man in the world.18

Over the years, his vision also gradually disappeared in the left eye. By the time he was 60, he was practically blind in both eyes. His good spirits persisted despite this: “One more distraction removed,” he said about the loss of his vision.19

A living computer

Euler’s blindness only made his exceptional abilities and tenacity all the more unmistakable—the condition had no apparent effect on his productivity. Aided by scribes, Euler wrote an average of one mathematical paper per week in the year 1775. As remarked by William Dunham, “Never was his remarkable memory more useful than when he could see mathematics only in his mind’s eye.”20

Contemporaries repeatedly commented on his extraordinary memory and aptitude for mental calculation: “[H]is memory did not permit him to forget anything,” said Condorcet in his eulogy dedicated to Euler. He had a “prodigious memory which loses nothing from a lecture he has heard,” according to Fuss.

As previously mentioned, Euler had learned the entirety of Virgil’s epic poem Aeneid by heart as a teenager. Even when he was seventy, he could cite the beginning and closing words on each page of the text he read when he was young. Ready on demand, Euler had access to enormous formulae and vast tables of numbers in his mind’s eye. For making the calculation of roots easier, he apparently memorized the first six powers of all natural numbers up to a hundred (i.e., n², n³, …, n⁶ for all n = 1 to 100).21

Euler clearly took pleasure in calculating. Various mathematical constants (π, e, ζ(2), etc) were calculated by him to many digits of precision. He discovered the then largest known prime which no one surpassed for nearly a century. He was also able to factorize 2³² + 1 as 641 × 6,700,417, thereby disproving Pierre de Fermat’s conjecture that it was a prime number. These are remarkable feats without access to modern computing tools.

The man was, of course, not merely a mindless computer. As a great mathematician, Euler often saw connections that no one else had. For the prime factorization, he proved that any composite “Fermat number” must have a particular structure; this drastically reduced the search space for possible factors. Similarly, on his quest to solving the Basel Problem—finding the exact sum of 1 + 1/2² + 1/3² + 1/4² + etc—he estimated the value of the sum so accurately that it would require summing the first 30,000 terms of the original series had it been done through naïve computation.22 Instead of doing brute force calculation, Euler invented a method now called Euler-Maclaurin summation which required far fewer terms to reach the desired accuracy.

Euler’s peers were frequently amazed by his ability to calculate. One notable incident happened while Euler was working in the Saint Petersburg Academy, where he was tasked with a project requiring astronomical calculations. He astonished the academicians by completing in three days a task they expected might take three months.23

Euler could perform incredible calculations entirely in his head. Two of Euler’s pupils (as we are told by Fuss, pupil himself) had one day calculated a converging series as far as the seventeenth term. Upon comparing their written results, they found a discrepancy at the fiftieth place. They communicated this difference to their master, and the blind Euler made the correct calculation in his head.24

Euler was also a man of extraordinary erudition. People were amazed by the breadth of his knowledge outside his area of expertise. According to Marquis de Condorcet, “Euler had studied nearly everything available in the branches of Physics, Anatomy, Chemistry and Botany.” Nicolas Fuss, who served as an assistant for the blind Euler, wrote:

He possessed what is called erudition to a very important degree. All that is left to us by the great writer of ancient Rome, he read; old mathematical manuscripts were perfectly known to him; the history of all the ages and nations were found in his head and he knew to quote whatever he knew without mistakes. He was familiar with medicine, botany and chemistry; he knew much more than was expected of a scientist whose science was other than the ones he knew so well.

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Respect and admiration

Not all great men receive due recognition in their lifetime, but Euler did. Near the end up his life, Euler’s reputation had extended far beyond the scientific community. Of all 18th century intellectuals, Euler’s fame reached a status that was second only to that of Voltaire.

Early in Euler’s career, Johann Bernoulli addressed Euler in a letter as the “most famous and learned man of mathematics.” When Euler returned to the Saint Petersburg Academy in 1766, Catherine the Great wrote in a letter:

I am certain that the academy will be resurrected from its ashes by such an important acquisition, and congratulate myself in advance in having restored this great man to Russia.25

In his final years, Euler’s home became something of a tourist attraction. Strangers and notable people visited and spoke with him about non-mathematical topics, experiencing the venerable old man’s erudition. Visitors left his home with “surprise mixed with admiration” that he “could have retained so many facts, much of which was unimportant and useless to the pursuit of his studies,” remarked his assistant.26 In 1782, one year before his death, Euler was made an honorary foreign member of the American Academy of Arts and Sciences.

One final heartwarming anecdote is worth telling. In 1783, the final year of Euler’s life, princess Yekaterina Romanovna Dashkova (1743–1810) was appointed as the director of the Russian Imperial Academy of Sciences by Catherine the Great. On her way to the inauguration, the princess stopped to visit at home the man she called Euler the Great. She begged Euler to let her enter the academy on his arm, and he was honored to do so. Euler, his son, and his assistant together accompanied princess Dashkova on the horse carriage to the ceremony.27 At the ceremony, after noticing that the seat of honor had been taken, Dashkova (as related in her memoirs) treated Euler with the utmost respect:

I [Princess Dashkova] therefore turned to Mr. Euler and told him to sit down wherever he thought fit, for any seat he occupied would always be the first among all. His son and grandson were not alone in showing appreciation . . . at my remark, and the professors, who all had the highest respect for the venerable old man, had tears in their eyes.

With support from Catherine the Great, the Petersburg Academy offered tribute to Euler by bestowing on him alone the grand medal of the academy and commissioned a mural in the assembly hall in his honor.28

The 76-year-old Euler died not long thereafter. On the morning of 17 September 1783, Euler instructed his brightest grandson in the sciences. Still equipped with a sharp mind, he also made some difficult mental calculations about the motion of hot-air balloons. In the afternoon, Euler started joking and playing with one of his grandsons while sitting on the couch with his pipe. Suddenly, he suffered a stroke and dropped his pipe. “My pipe!” he exclaimed, “I am dying.”29

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Introduction

People with a Western education will have been taught about the transatlantic slave trade. Those historically inclined will also be familiar with the large Arab slave trades that similarly affected Africa.

Few will need convincing that racial ideology played a key role in the transatlantic slave trade. In contrast, the important role that race played in the Arab slave trades is far less appreciated.

While people of any race could be enslaved on religious grounds (that is, for being non-Muslim), slavery in the Muslim world was still highly racialized. Black slaves were treated differently from non-black slaves. They were more often reduced to lower forms of labor, had less opportunity for social mobility, and were more likely to be victims of the horrific practice of castration. Additionally, there was widespread opposition to “contaminating” one’s lineage through intermarriage with blacks.

Medieval Muslim scholars provided specific intellectual justification for the enslavement of black people—they described them as subhuman and as natural slaves. Just as Muslims were considered superior to non-Muslims, so were pale-skinned Arabs deemed inherently superior to dark-skinned Africans. Culturally, as evident in attitudes and language, blackness and slavery gradually came to be regarded as synonymous in Muslim society.

The connection between blackness and slavery became increasingly overt with time. In the early 1600s, the West African scholar Ahmad Baba tried (and failed) to convince his Muslim contemporaries that black people shouldn’t be enslaved for their race alone. Later in 1699, the Moroccan Sultan gave orders to enslave all black Africans in Morocco. In the 19th century, an Arab historian lamented that “many common folk believe that the reason for being enslaved according to the Holy Law is merely that a man should be black in colour.” All this and more is the subject of the full piece below.

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If we travel sufficiently far back in time in Europe, we encounter homicide rates vastly exceeding what we’re used to. A classic illustration is found in the English long-run homicide rate estimates by Eisner (2003):

These estimates suggest a homicide rate of approximately 20 homicides per 100,000 inhabitants in England in the 14th century (note the logarithmic axis). While these local estimates contain substantial uncertainty, the massive downward trend cannot be doubted.

Today Western European homicide rates tend to hover around ~1 homicide per 100,000, and have thus declined by more than an order of magnitude. (For comparison, the American homicide rate has been 4-7 per 100,000 in recent years.)

It is rarely mentioned that long-run homicide trends pose a serious challenge to the belief that poverty is the mother of crime. The homicide rate decline preceded the economic growth of the Industrial Revolution. When the West eventually did grow rich, it wasn’t met with additional widespread decline in homicide.

Sweden, for instance, has experienced a ~30-fold increase in GDP per capita from 1750 to 2022, and yet the homicide rate showed no systematic decline over that period. How could this be if the absolute economic level of a country plays a central role in determining violence? A “violence isn’t monocausal”-retort is hardly convincing—the economic transformation is far too large to shrug off so casually.

These observations are all the more astonishing in light of population aging (violence is a young man’s game), and medical and emergency advances that have made it far easier to save the life of an injured person.1

In this piece, I will analyze the data on long-run homicide trends in greater detail. When did homicide rates reach modern levels? And where did homicide rates decline first? The data provide insight into the deep roots of regional disparities in violence, and conflict with popular ideas about causes of violence.

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Introduction

There is much to admire in the history of mathematics in India (see, e.g., Plofker, 2009). One can reasonably claim that, for most of the Middle Ages (after classical antiquity), the most proficient mathematicians were found in India.

Despite this perfectly admirable reality, many feel compelled to make exaggerated or downright false claims to embellish Indian mathematics. The most pervasive false claims are in the realm of calculus: that India invented calculus and/or that European calculus was derived from Indian methods.

In this piece I will first address what “calculus” is and why India didn’t invent it. Then I will refute the hypothesis that calculus was built upon a knowledge foundation transmitted from India to Europe, addressing some of the common arguments. The bulk of the article is spent on documenting the true story of how the development of calculus began in Europe.

These false claims are not only worth refuting because we should strive for accurate beliefs, but also because the story of how calculus emerged is fascinating in its own right.

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India did not invent calculus

The misconception that India invented calculus is primarily promulgated by the popular media and in popular discussions. Few relevant experts of note will make such a claim. Addressing it nevertheless provides helpful context. The misconception rests on a fundamental misunderstanding of what “calculus” (and its invention) refers to in the history of mathematics.

Calculus is a systematic theory of integration and differentiation, and nowhere did such a theory exist before it was developed in Europe in the 17th century. The development of calculus at the very least entails three components:

• Differential calculus. A systematic method for differentiating general curves. (I.e., for finding the instantaneous rate of change, or slope of the tangent, at any given point for general curves)

• Integral calculus. A systematic method for integrating general curves. (I.e., for finding the area under general curves)

• Fundamental theorem of calculus. To establish the connection between integral and differential calculus, unifying the two branches.

None of these three criteria had been previously satisfied anywhere—not in India, not in the Islamic world, nor by the ancient Greeks. Every assertion that calculus had been developed anywhere before the 17th century misses the crucial distinction between (a) applications of infinitesimal methods in specific contexts and (b) the systematic methods and theory that make up a true calculus.

It is true that methods for differentiating specific curves were known, and methods for integrating specific curves or objects were known, and the sums of specific infinite series were known. But nowhere had there been developed systematic methods for differentiating or integrating general classes of curves. Historically, infinitesimal methods had to be tailored to the specific problem at hand and didn’t generalize. Importantly, no one had connected the two branches of calculus into one (the fundamental theorem of calculus).

To appreciate what it means for calculus to be a systematic theory, consider Leibniz’s influential 1684 paper on differential calculus. In this paper, Leibniz presents a general algorithm for algebraically differentiating curves, general methods for finding minima and maxima, as well as general rules of differentiation (e.g., rules regarding constants, as well as the product and quotient rules). In the following years, he would publish similar papers on integral calculus (1686), and the fundamental theorem of calculus (1693).

Leibniz was far from the first to publish on infinitesimal analysis—he was building upon developments of the preceding century—but these papers were the first to publicly communicate a truly systematic theory of differentiation and integration. They gave the world calculus (the name is also derived from the title of Leibniz’s 1684 paper).

Beyond the development of the theory of calculus, of equal if not greater importance are its applications. After all, the reason calculus is held in so high regard is its potential to describe the world and solve problems. The Brachistochrone problem is a famous early problem in mechanics addressed by calculus in the late 17th century (including a solution by Newton). But in the 18th century, Euler would be the person to truly show the power of applied calculus.

Regarding the notion of “calculus” outside Europe, the historian of mathematics Victor Katz (1995) summarizes it well:

[S]ome of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. There were apparently only specific cases in which these ideas were needed.

There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented the calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today.

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There was no transmission from India to Europe

While calculus was clearly not invented in India, more often the weaker claim is made that European mathematicians learned certain basic ideas of calculus from India, from which Europe built calculus upon.

The hypothesis suggests that Jesuit missionaries in India in the late 16th century (1578 at the earliest) discovered Indian mathematics of the Kerala school—including trigonometric methods and basic ideas of calculus—and that this knowledge was transmitted to correspondents in Europe from which it eventually disseminated via scholarly networks (Plofker, 2009).

Of course, even if this was true, calculus would still be a European invention, as that’s where the systematic theory of integration and differentiation was developed. But this is a moot point—there is no good reason to believe that such a transmission occurred at all.

The basic problem with the transmission hypothesis is that it lacks any historical evidence. As expert on the history of Indian mathematics Kim Plofker (2009, p. 252) writes:

There are no known records of sixteenth- or seventeenth-century Latin translations or summaries of these mathematical texts from Kerala. Nor do the innovators of infinitesimal concepts in European mathematics mention deriving any of them from Indian sources.

Moreover, as Plofker also notes, the European infinitesimal techniques used in the early developments of calculus do not resemble the Indian methods, but instead resemble those of their Hellenistic forerunners. This important point will be elaborated upon in the section on how the development of calculus truly began.

Similarly, David Bressoud (2002) writes:

There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century.

The historian of mathematics Victor Katz (1992) has also stated that he remains unconvinced (not just specifically regarding an Indian transmission of calculus, but also regarding certain other claims of an Eastern impact on Western mathematics):

Although there were clearly wide-ranging mathematical developments in China, India, and the Islamic world, Joseph [in The Crest of the Peacock] was not able to convince me that the mathematics of these non-Western cultures had a significant effect on the development of European mathematics. In fact, if Europeans had been aware of this work, it would have saved them many centuries of struggle with the same ideas.

Ugo Baldini (2009) has raised many strong points against the Indian transmission hypothesis. Among the mathematicians in the Jesuit Society, practically all of them were destined to China, not India. Regardless if their perception was justified or not, the Jesuits perceived the scientific traditions of China to be more worthy of attention.

He also summarizes the implausible circumstances that are implied by the transmission hypothesis:

As evident, the ideal situation implies a number of circumstances which could hardly occur together: A Hindu – presumably a Brahmin – should have allowed a foreign person, who was also his rival in religion, to get in touch with precious and reserved texts; the other person should have been highly proficient in both Sanskrit and mathematics; he should also have been so patient and curious about those texts’ content to overcome the barrier produced by unusual expressions and demonstration procedures. In front of this, it must also be recalled that, as a matter of fact, no hint about Indian mathematical texts – perhaps some elementary ones excepted – is found in Jesuit sources before 1650 at least, and no such qualified person seems to have existed among Indian missionaries in those years.

After carefully considering the plausible candidates, noting that none of them seem to have gathered any notable mathematical information from Kerala, Baldini concludes:

Unless new evidence is found and some basically new circumstance is established, the only possible deduction seems to be that not only no information exists on a Jesuit mathematician having managed to study some advanced Indian text (not to say to transmit it, or its content, to Europe), but no serious clue appears of a scientific interchange not purely superficial and more than occasional.

Even proponents of the transmission hypothesis have been forced to admit that, despite a great deal of effort, the search for transmission evidence has left us empty-handed. As Almedia & Joseph (2009, p. 267) write:

The painstaking trawl of the mass of manuscript and other materials mentioned earlier in this Report has yielded no direct evidence of the conjectured transmission. We therefore have to report that on the basis of the evidence of the documents studied so far that the evidence supports the null hypothesis [of no transmission] formulated earlier. Thus the European Renaissance developments of the prototypical calculus may well have been independent of the developments in this subject in Kerala some centuries earlier.

In short, there is a total absence of historical evidence for an Indian transmission. And contrary to common belief, absence of evidence is evidence of absence. In a mathematical culture as heavily documented as 16th and 17th century Europe, you would expect to find plenty of evidence of such a transmission if it were true. After all, this was after the printing press and the establishment of postal services. There is an abundance of textual records from this era, and they have seemingly left us with no detectable trace of a transmission.

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On “Circumstantial evidence” for transmission

Despite the lack of direct evidence, a select few still maintain that a transmission did occur. For example, in The Imperishable Seed: How Hindu Mathematics Changed the World and Why this History was Erased, Kamble (2023) admits that “no record of the Kerala mathematics having been transmitted to Europe has been found”, yet he still argues in favor of the transmission on the basis of “circumstantial evidence.” A similar case is forwarded by George Gheverghese Joseph in the The Crest of the Peacock (3rd ed), though Joseph is more tentative in his conclusions.

The premise of Kamble’s argument is that a “sudden and unexplained flourishing of mathematics and science in Europe” occurred after Jesuits arrived in India in 1578, and that a number of suspicious ‘coincidences’ suggest a transmission occurred.

This premise is false. Many of the most important advances in mathematics, technology, and science in Europe occurred in the centuries leading up to 1578 (e.g., solutions to the cubic and quartic equations, advances in symbolic algebra, and trigonometric advances; the mechanical clock and printing press; the observation of a supernova, and advances in anatomy, and Copernicus’s heliocentric model).

When the development of infinitesimal analysis began (which largely did fall in the latter part of the 16th century), Europe was already fully accustomed to mathematical and scientific advances. The transformative advances of the 17th century were a continuation of processes already ongoing before 1578. Moreover, many of the innovations of the ‘Scientific Revolution’ had no equivalent outside Europe, and thus clearly cannot be explained by imitation of outsiders. The incredulity aimed at the invention of calculus is unwarranted.

Nevertheless, Kamble is determined to frame history through the lens of a “sudden” change caused by a transmission, resulting in a distorted history with relevant details and context omitted. He forwards three ‘coincidences’ which together supposedly show that a transmission must have occurred. Let’s see how compelling they are.

Purported “coincidences”

Trigonometric tables

The first purported major “coincidence” is that Christoph Clavius in 1608 published a table of sines accurate up to 7 decimal places, “without either explaining the method he used in obtaining the values or mentioning the source.” Indian mathematicians already had sine tables accurate to 7 decimal places, so this is suspect according to Kamble.

This argument can be dismissed because we know that Clavius’s sine tables were taken from Reinhold’s 1554 table (Roegel, 2021). That is, it was based on work prior to Jesuit contact in India. The trigonometric methods were also explained in Clavius’s previous publication Astrolabium (1593).

The author also neglects to mention that the German Johannes ‘Regiomontanus’ Müller had already calculated a sine table to 7 decimal places more than a century prior to Jesuit contact. Regiomontanus’s sine table was completed in 1468 (Roegel, 2021), and was well known to Clavius. There is no strange ‘coincidence’ in need of explanation.

To make matters worse, starting around 1540, Georg Joachim Rheticus was already working on trigonometric tables with accuracy that far exceeded tables in India, or anywhere else for that matter. These tables were calculated to an accuracy of 15 to 20 decimal places (depending on the angle). They took decades to complete, but he had begun long before the Jesuit contact.

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The Tychonic astronomical model

The second purported “coincidence” is the Tychonic astronomical system proposed in 1588 by Tycho Brahe. The author claims that it’s “exactly the same” model as one proposed by the Indian astronomer Nilakantha in 1501.

One gets the impression that the two models share remarkably intricate details that couldn’t plausibly occur by coincidence. In reality, they are simply both geoheliocentric models—that is, they both propose that all the planets except the Earth revolve around the Sun. This is relatively unremarkable.

Once you go beyond these gross similarities and look for more subtle astronomical ideas that could indicate a transmission, you find nothing. For example, in the same 1501 treatise in which his planetary model is presented, Nilakantha offers computational schemes for longitudes and latitudes of the Sun. Despite these being supposedly more accurate than contemporary European methods (Ramasubramanian et al., 1994), they were never adopted in Europe. These schemes were the primary focus of the treatise, not the implied astronomical model. If they were aware of Nilakantha’s work, why would they not adopt his superior computational methods?

The geoheliocentric (Tychonic) system (while ultimately incorrect) was the natural alternative to the geocentric (Ptolemaic) and heliocentric (Copernican) systems. A compromise between the two competing models, it attempted to address the problems of both (i.e., the overly complicated epicycles of the Ptolemaic system and the apparent lack of evidence for the moving Earth integral to the Copernican system). Proposing the geoheliocentric model is therefore completely understandable in context and requires no special explanation.

The Gregorian calendar reform

The third and final “coincidence” is that the Gregorian calendar reform was proposed in 1582, just four years after the Jesuit contact in 1578.

The calendar reform was motivated by the observation that the date of Easter was inaccurate. That is, there was an increasing divergence between the canonical date of the equinox and observed reality.

The author yet again neglects to mention that this issue had been known since the Middle Ages, and that there had been multiple earlier calls for reform prior to 1578 (including in 1515).

The issue became more pressing as the accuracy of astronomical observations increased—the divergence between the two became increasingly impossible to ignore. Much of this is due to the previously mentioned Regiomontanus, who in 1475 was invited to the Vatican in an attempt to undertake such a reform. Tycho Brahe, who had the most accurate pre-telescope observations, also remarked on the large discrepancies in his 1573 De nova stella (Dreyer, 1890, p. 52). Yet again, there is no ‘coincidence’ to explain.

Summary

The Indian transmission hypothesis is based on no historical evidence and the arguments proposed in its favor are flawed. The hypothesis rests on vague suggestions of ‘coincidences’ which, upon closer inspection, are not coincidences at all and amount to little of substance. If any uncertainty still persists, that should be eliminated in the following section in which I provide the actual account of how calculus emerged, grounded in incontrovertible historical evidence.

It is important to note that a central theme in the arguments forwarded by Kamble and Joseph is the charge of widespread Eurocentrism, which both of them believe underlies the unwillingness to accept an Indian transmission.

But is this a case of people in glass houses throwing stones? Perhaps the eagerness to adopt such a hypothesis on the basis of so flimsy evidence (at least in the case of Kamble), as well as their own refusal to accept the mainstream explanation (which boasts far more evidentiary support), is itself an indication of a bias in the opposing direction?

Contrary to Kamble or Joseph’s suggestions, the reason that the Indian transmission hypothesis is widely rejected by experts is not Eurocentrism. Rather, the hypothesis is based on no solid evidence and, just as importantly, the true origins of European calculus are already well understood and documented. The transmission hypothesis not only lacks evidentiary support, it is also unnecessary. It is being presented as an explanation to an unexplained mystery. But there is no real mystery.

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How calculus truly came to be

Calculus was not invented over night. The advances of Newton and Leibniz were not just the product of their own ingenuity, but also the culmination of countless contributions of earlier mathematicians.

Prerequisite mathematical steps first had to be taken. In the 15th and 16th centuries, Europe saw rapid advances in algebra (e.g., solutions to the cubic and quartic equations) and in trigonometry. Starting in the mid-16th century, European algebra morphed from fully verbal form to an increasingly symbolic form. In the early 17th century, geometry also became conceptualized algebraically, with curves being described with equations and visualized in a coordinate system. The symbolic and algebraic advances provided a universal language from which general methods of calculus could be developed.

There were structural factors aiding mathematical and scientific development too. Most notably, the invention of the printing press (c. 1440) and the establishment of the international postal service (1490) led to a revolution in communication and sped up the rate of innovation in all of science and mathematics.

The era’s scientific “fashion” even played a role. In the 16th and 17th centuries it was motion that became the subject of fascination and investigation, regardless if it was motion of planets or of falling bodies on earth. Necessity is the mother of invention and calculus was the tool necessary for describing motion.

However, the proximate cause of the rise of infinitesimal analysis in Europe was the reintroduction of high-quality translations of ancient Greek works. These ancient works contained the precursors of calculus which Renaissance mathematicians built upon. In contrast with the Indian transmission hypothesis, the direct evidence in favor of this assertion is overwhelming.

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The influence of ancient Greek works

Though the ancients never came close to developing a systematic theory of integration and differentiation (i.e., calculus per se), they had nevertheless studied basic aspects of calculus. Among all the mathematicians from classical antiquity, Archimedes was the one who took ‘proto-calculus’ the furthest.

In his treatise On Spirals, Archimedes was able to find the tangent of a specific spiral. In Quadrature of the Parabola, he masterfully calculated the area enclosed by a parabola and a line segment, using what has come to be known as the “method of exhaustion.” These two works illustrate ancient proto-calculus at its best, being early precursors of differential and integral calculus, respectively.

More well known, Archimedes also used the ‘exhaustion’ procedure to approximate π. He inscribed and circumscribed a circle with regular polygons. The perimeters of the polygons provided lower and upper bounds of the circumference of the circle. The greater the number of sides used for the polygons, the better the approximation.

As part of the Renaissance revival, several ancient mathematical works had already been translated. The quality of many early translations was poor, however, because those responsible were often not mathematicians themselves and they were not based on the original Greek manuscripts.

In the 16th century, a concerted effort was made to translate ancient mathematical works from original Greek, and for them to be prepared by mathematicians (Katz, 2009, 3rd ed., p. 407). This led to a new wave of high-quality translations, and their dissemination aided by the printing press. The transmission of ancient Greek knowledge, not a transmission of Indian knowledge, explains the ‘sudden’ rise of infinitesimal analysis in Europe. The earliest developers of calculus in Europe were all directly inspired by ancient works, especially those of Archimedes.

As historian of mathematics Jens Høyrup (2022) writes:

The decisively productive problems taken up by the 17th-century mathematicians came . . . from Archimedes, Apollonios, and Pappos; they were wholly different from anything circulating in or in the vicinity of the abbacus environment.

The Northern Renaissance only discovered Archimedes in the 1530s. Initially, Archimedes was mainly revered as an inventor. Gradually the appreciation for Archimedes the mathematician grew, but it took a few decades before this appreciation was converted into substantive engagement with his mathematics (Høyrup, 2017).

In De subtilitate (1550), Gerolamo Cardano placed Archimedes first in a list of great minds, and Federico Commandino (1509–1575) held the opinion that “that one can hardly call himself a mathematician who has not studied the works of Archimedes” (Høyrup, 2017).

Commandino was one of the most important translators of the Renaissance. In 1558, he published Greek-to-Latin translations of many of Archimedes’s works. Being a competent mathematician himself, Commandino’s translations and corresponding commentaries were of the highest quality and they were exceptionally influential.

In a 1586 book on statics, Simon Stevin applied an Archimedean principle to demonstrate that the center of gravity of a triangle lies on its median. He directly references the work of Archimedes and Commandino. Stevin also argued that one could fill the triangle with an infinite number of inscribed parallelograms, an early move towards a limit concept (Boyer, 1959).

But François Viète would be the first mathematician to make creative use of Archimedean mathematics (Høyrup, 2017). In his Variorum (1593), Viète became the first to discover an infinite product, drawing direct inspiration from Archimedes. Viète’s method was a variation of Archimedes’s approximation of π, using an exhaustion-like procedure of inscribing a circle with a sequence of polygons. The Variorum contains an abundance of references to the Greeks (Høyrup, 2022). Already on the first page does Viète mention Archimedes.

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Countless other mathematicians would soon follow suit in building upon ancient Greek works. Fascinated with Archimedes’s approximation of π, Ludolph van Ceulen (1540–1610) emulated his method and calculated the first 35 digits of π by doubling the sides of an inscribed square an incredible 60 times (Cipra, 2009). In 1606, Luca Valerio (1553–1618) used the methods of Archimedes to find volumes and centres of gravity of solid bodies.

The next important step towards calculus was taken by Johannes Kepler (1571–1630), the greatest mathematical astronomer of his time. In Astronomia nova (1609), he computed the time elapsed between any two points in a planet’s orbit. He was also directly inspired by the exhaustion procedure Archimedes had used to approximate π. Kepler had become acquainted with Archimedes’ work through Commandino’s 1558 translation and it made a great impression on him (Aiton, 1976).

As Kepler himself tells us (Astronomia Nova, chapter 40, translation by Donahue):

For I had remembered that Archimedes, in seeking the ratio of the circumference to the diameter, once thus divided a circle into an infinity of triangles – this being the hidden force of his reductio ad absurdum [proof by contradiction]. Accordingly, instead of dividing the circumference, as before, I now cut the plane of the eccentric into 360 parts by lines drawn from the point whence the eccentricity is reckoned.

Kepler took to heart the idea that one can approximate complicated geometrical shapes by using sufficiently many simpler shapes, and developed further upon it in his book Nova stereometria doliorum vinariorum (1615). Here he calculated the volumes of solid objects not considered by the ancients: wine barrels.

A barrel’s exterior could be approximated by infinite number of cylinder strips of varying width, he reasoned, and he used this to integrate the volume of the whole barrel. The book also contains an early step in differential calculus in the form of analyses of maxima and minima. Kepler analyzed how a barrel should be proportioned if one wants to maximize its volume, and then went on to consider similar questions for other shapes. He made the important observation that the rate of change is small where maximal value is attained (dy/dx = 0, as we would say). In the preamble of this book, Archimedes is once again explicitly referenced.

Kepler’s idea of splitting an object into infinite strips, which was inspired by the ancient method of exhaustion, was systematically expanded upon in Bonaventura Cavalieri’s Geometria indivisibilibus (written 1627, published 1635). This work introduced the “method of indivisibles”—the first major step towards integral calculus.

As is noted by Kim Plofker, an expert in Indian history of mathematics, these early European advances towards calculus bear no resemblance to Indian infinitesimal works. Indian infinitesimal techniques were applied in a trigonometric context (Plofker, 2001). Instead, the methods used by the Europeans during the early development of calculus, as well as the kinds of problems they tackle, resemble those found ancient Greek works (Plofker, 2009, p. 252):

[H]ow would Sanskrit texts on infinitesimal methods for trigonometry and the circle, transmitted no earlier than the late sixteenth century, have inspired the quite different European infinitesimal techniques used at the start of the seventeenth century? Mathematicians like Kepler and Cavalieri focused instead on mechanical questions such as centers of gravity and areas and volumes of revolution, as well as general problems of quadrature explicitly associated with the work of Hellenistic forerunners and of Renaissance geometers such as Nicholas of Cusa.

The problem with the Indian transmission hypothesis is not just that it lacks historical evidence. To maintain belief in it, you must also ignore the historical evidence altogether. Because once you look inside the early European infinitesimal works, it becomes abundantly clear who they were learning from—they explicitly tell us.

The increasing societal reverence and influence of Archimedes is also seen in Galileo’s works. In Two New Sciences (1638), Galileo praised Luca Valerio, referring to him to as the new Archimedes of his age. He also gave an Archimedean demonstration of the quadrature (integration) of the parabola and included in an appendix some work on centers of gravity (Boyer, 1959).

One could list countless other examples of ancient Greek inspiration, but just one more will do. In De dimensione parabolae (1644), Evangelista Torricelli (1608–1647) gave 21 different proofs of the quadrature (integration) of the parabola. About half of these used the ancient method of exhaustion, and the other half used the new method of indivisibles. One of the proofs was essentially the one Archimedes himself gave in his treatise On the quadrature of the parabola, extant and well-known amongst European scholars at the time (Boyer, 1968, p. 391). In 1645, Torricelli also calculated the length of a logarithmic spiral based on infinitesimal methods he had learned from Archimedes, Galileo and Cavalieri (Boyer, 1968, p. 376).

It would take a couple more decades before a fully systematic theory of calculus would emerge (Leibniz’s trio of influential papers were published in 1684, 1686, and 1693, and Newton’s unpublished calculus was completed in 1671). But from the 1620s onward, its development was proceeding at rapid pace. While the early stage built upon an ancient Greek foundation, through the cumulative efforts of European mathematicians of the 17th century, infinitesimal analysis was developing far beyond what the ancients had achieved.

In the following sections, I will describe the steps that were taken towards developing systematic infinitesimal methods and theory, and how this ultimately culminated in calculus proper as achieved by Newton and Leibniz. I will also describe how the theory of calculus became a powerful scientific tool for describing natural phenomena. Lastly, I will discuss the plausibility of an alternative reality in which India or even ancient Greece invented calculus.

For those interested, I have also included an appendix of advances in European mathematics in the 15th and 16th centuries that preceded European investigations into infinitesimal analysis.

Towards a general theory of calculus

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Foreign travellers who visited India during the sixteenth and seventeenth centuries present a picture of a small group in the ruling class living a life of great ostentation and luxury, in sharp contrast to the miserable condition of the masses — the peasants, the artisans and the domestic attendants. Indigenous sources do not disagree; they often dwell on the luxurious life of the upper classes, and occasionally refer to the privations of the ordinary people. (The Cambridge Economic History of India, p. 458)

Introduction

In popular discussions it is often suggested that India was historically a wealthy society. The standard line is that wealthy precolonial India accounted for a quarter of global GDP, a fraction that dramatically fell with ongoing British colonialism.

But the basic premise is false. Precolonial India was not rich. The living standards of commoners in 17th century India were not any better than medieval European peasants. The misconception is based on bad arithmetic, on conflating a large population with prosperity, on generalizing from the lives of elites, and on bad data.

The purpose of this piece is not to analyze the consequences of British colonialism, but to provide a more accurate account of the precolonial Indian economy. I’ll consider both the qualitative and quantitative evidence on the living standards in India. Then I will trace the origins of the developmental divergence between the West and India.

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A misguided argument

“India used to be very rich. It once had more than 25% of the global GDP. After British colonialism, it fell to just 4% of world GDP.”

This is, in essence, the most common argument for the two-part claim that (a) India used to be very rich, and (b) that British colonialism caused the decline from this height. It is often accompanied by a graph like this (using long-run GDP estimates from Maddison, 2007).

There are three major problems with this argument.

The first is basic arithmetic. Because shares have to add up to 100%, a decline in global GDP share does not tell you that the size of the economy declined. A decline in share is just as consistent with other economies experiencing growth. The share of global GDP thus cannot be used as evidence for economic decline.

The second problem is data quality. In truth, Maddison’s estimates above were largely guesswork, at least for preindustrial non-Western countries. The economic historian Gregory Clark went further saying that, “All the numbers Maddison estimates for the years before 1820 are fictions.” Luckily, we now have access to better data than Maddison did.

The third problem is a conceptual misunderstanding about economics, namely the use of GDP rather than GDP per capita as a proxy for a “wealthy society.” In an agrarian society, GDP is largely food production and is mostly a proxy for population size rather than living standards.

The real measure of prosperity is quantified on a per-capita basis as that better reflects the living standards of actual people. Even if we accepted Maddison’s estimates for the sake of argument, we find that India was substantially poorer in per-capita terms than England in 1700 AD (Maddison, 2007; Table A.7).

In a similar vein, there’s a common misconception that natural resources like spices, diamonds or gold made India rich. On the contrary, a large, populous country containing natural resources tells you little about the living standards of its general population. Indian commoners were poor and couldn’t themselves afford many spices, diamonds, or luxurious textiles. Instead, modern wealth has emerged primarily through rises in productivity, not from natural resources.

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The precolonial Indian economy

Living standards in 17th century India were not that unlike those in medieval Europe centuries earlier. It was an agrarian society, commoners lived primarily off the food they cultivated themselves, only a minority were acquainted with urban life, and the vast majority were illiterate. The average person was poor, and famine and epidemics were recurring threats.

There were, of course, wealthy kings, nobles and elites who lived in better conditions, but their rarity makes them less representative of the country’s living standards than the peasant. Slavery was also widespread in Mughal India, while serfdom predominated in medieval Europe.

A major difference was population size. India was substantially more populous than Europe prior to the Industrial Revolution, perhaps a quarter of the global population in 1500 AD, a fact often mistakenly interpreted as richness.

In a preindustrial world dominated by Malthusian constraints, a large and dense population was a liability—it reduced average living standards. Historically, a “large GDP” has been functionally equivalent to “many poor people.” Only after the Industrial Revolution could economic growth widely and persistently exceed population growth, resulting in skyrocketing living standards.

Qualitative evidence

British colonization of India happened well after the Age of Exploration, and many European travelers had already set foot in India. The British had not taken any significant control of India prior to around 1760. By 1820, India was de facto under British control, and their rule was formal between 1858 and 1947 under the British Raj.

European travelers in India encountered a society with a small ruling class living a life of great ostentation and luxury, sharply contrasted with the miserable poverty of the masses.

The Dutch merchant Francisco Pelsaert (c. 1595–1630), for instance, said of India:

The common people (live in) poverty so great and miserable that the life of the people can be depicted or accurately described only as the home of stark want and the dwelling place of bitter woe . . . their houses are built of mud with thatched roofs. Furniture there is little or none, except some earthenware pots to hold water and for cooking.

“Their bedclothes are scanty, merely a sheet, or perhaps two”, he continues, “this is sufficient in the hot weather, but the bitter cold nights are miserable indeed.”

Descriptions of this sort are repeated in numerous accounts. François Bernier (1620–1688) writes about Delhi:

for two or three who wear decent apparel, there may always be reckoned seven or eight poor, ragged and miserable beings.

Bernier compares this unfavorably with Paris in which “seven or eight out of ten individuals seen in the streets are tolerably well clad.”

William Harrison’s Description of England in 1577 provides another point of comparison. The book illustrates the rising prosperity in England for lower and upper classes alike. As he tells us, a growing number of Englishmen have chimneys instead of smoke holes; glazed windows instead of wooden lattices; regular beds instead of straw pallets; pewter plates instead of wooden platters, and tin and silver spoons instead of wooden ones. He also writes:

many farmers . . . garnish their cupboards with plate, their joined beds with tapestry and silk hangings, and their tables with carpets and fine napery

While modest in modern perspective, he is describing living standards that significantly surpass that of Indian commoners in 16th and 17th centuries. As we will see later, quantitative evidence supports this position.

Regarding food consumed in India, John Fryer (1650–1733) writes:

Boiled rice, nichany [the ragged millet], millet, and grass-roots, are the common food of the ordinary people.

Their single-room houses could be crowded, for Fryer notes that they “themselves, their family, and cattle, are all housed, and many times in no distinct partition.”

Jean-Baptiste Tavernier (1605–1689) writes about the small wages of Indian diamond miners:

These poor people only earn 3 pagodas per annum, although they must be men who thoroughly understand their work. As their wages are so small they do not show any scruple, when searching the sand, in concealing a stone for themselves when they can, and being naked, save for a small cloth which covers their private parts, they adroitly contrive to swallow it.

Pelsaert concurs about the low wages in India, saying that “for the workman there are two scourges, the first of which is low wages” and “the second [scourge] is [the oppression of] the Governor, the nobles, the Diwan, the Kotwal, the Bakhshi, and other royal officers.”

Mughal sources rarely comment on the lives of commoners, instead focusing on elite perspectives, but when they do, they do not disagree with the European characterization.

Babur (1483–1530), the founder and emperor of the Mughal Empire, said “Peasants and people of low standing go about naked.” He also lists the deficiencies of India:

[T]here are no good horses, no good dogs, no grapes, musk-melons or first-rate fruits, no ice or cold water, no good bread or cooked food in the bāzārs, no Hot-baths, no Colleges, no candles, torches or candlesticks.

The Persian historian Firishta (ca. 1570–1620), in agreement with Palsaert about the oppressive ruling class in India, writes about Delhi sultan rule:

The duties levied on the necessaries of life realised with the utmost rigour, were too great for the power of industry to cope with: the country, in consequence, became involved in poverty and distress. The farmers fled to the woods, and maintained themselves by rapine; the lands were left uncultivated; famine desolated whole provinces, and the sufferings of the people obliterated from their minds every idea of subjection.

Abul Fazl (1551–1602), the great historian of Akbar’s reign, says that the common people of Bengal “for the most part go naked wearing only a cloth about the loins.”

Rice, millets and pulses formed the stable diet of the masses. But, as noted by Habib (1963, p. 91), “generally speaking, it was the lowest varieties, out of his produce, which the peasant was able to retain for his own family.” In the 16th century Ain-i-Akbari, Abul Fazl mentions that the ‘indigent’ resorted to eating ‘pea-like grain’, which used to cause sickness (Habib, 1963, p. 91). Meat was rarely eaten by Hindus due to taboos on beef and pork, but fish was consumed in some regions.

Based on these and other accounts (see, e.g., Chandra, 1982; Habib, 1963), and acknowledging that there could be substantial variety between regions, we may summarize the living standard of a typical Indian commoner:

• Peasants were generally scantily clad, being naked save for a cloth around their waist, and usually went barefoot.

• Daily food was very simple; a typical meal might be a portion of pulse mixed with rice and sometimes ghee. Fruits were also available according to region and season. Only the more prosperous villages could afford more than one meal a day. Many spices (e.g., cumin, cardamoms and pepper) were too expensive for commoners, and capsicums or chilies were unknown.

• Peasants typically lived in single-room houses made of mud, wood or bamboo, with thatched roofs. The homes had minimal furniture and floors were traditionally coated with cow dung. Cities and houses were often crowded, and large families could reside in a single hut. In some areas, houses were so low that a person could not stand upright in them.

• Famines and epidemics were two major scourges of village life. Lack or excess rain were frequent causes of failed crops. Big famines accompanied by epidemics could disrupt entire rural economies and led to mass migrations. Entire villages or towns could be temporary and huts were often not durable or long-lasting.

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Quantitative data

The qualitative descriptions are consistent with quantitative data. Recent data solidifies the conclusion that India had substantially lower GDP per capita than West Europe before 1700 AD, as Maddison originally had surmised (Maddison, 2007, Table A.7). I will summarize this data after a little background on the agrarian economy.

Agrarian life

Major civilizations in history have all centered around agriculture. But as countries have gotten richer and more productive, fewer have needed to devote their efforts to agriculture in order to sustain the population. The share of labor in agriculture is therefore a strong (inverse) proxy for economic development.

Precolonial India was a prototypical agrarian society—the great majority lived in villages and worked in agriculture to feed themselves. As Gavin Hambly writes:

It is hardly necessary to observe that the majority of the inhabitants of the Indian sub-continent during the sixteenth and seventeenth centuries passed their entire lives in a predominantly agrarian village-oriented environment, and that only a small minority were acquainted with urban patterns of living, however loosely the term ‘urban’ is applied.

Earlier in the same book, it is noted that:

the great majority of the people mainly consumed what they themselves produced or secured from their neighbours on the basis of customary arrangements.

An exact figure is impossible to nail down, but multiple authors have estimated that 80% to 85% of the population were dependent on agriculture in Bengal (Roy, 2010).

The transition into less reliance on agriculture happened significantly earlier in north-west Europe. Already by the 15th century in Europe, a substantial minority did not do agricultural labor. By 1700 AD, less than half worked in agriculture in some European countries.

As shown in the chart below, only very recently has India caught up to figures that were achieved some 300 years earlier in north-west Europe. To this day, agricultural productivity in India lags far behind the West (Fuglie et al., 2024).

Income data

Maddison’s original GDP per capita estimates for precolonial India were largely “guesstimates.” However, recent work confirms the basic conclusion: living standards in India were substantially below that of England.

In a detailed case study of real incomes in Bengal in 1763, Tirthankar Roy demonstrates that these were much lower than those in England (Roy, 2010).

More recently, Broadberry et al. (2015) have provided new estimates. The table below shows that there was a significant difference between India and Britain by the year 1600, which grew over time.

Based on this work, Gupta (2019) analyzes the divergence in greater depth. Indian living standards modestly declined past 1600 AD, but relative to Britain the decline was great. The divergence was therefore more a product of British growth than Indian decline, though both contributed.

The author next decomposes the productivity into agricultural and industrial components. The following image conveys two important points: (a) trends in Indian GDP per capita and agricultural output per capita are nearly identical (indicative of India’s complete reliance on agriculture); and (b) Indian agricultural productivity (per-capita) decline preceded British colonization.

While colonialism happened after the productivity decline, the decline instead does coincide with a population growth starting from 1600 AD.

A straightforward Malthusian dynamic is a likely explanation (de Zwart, 2025). India was completely reliant on agriculture. Under Malthusian constraints, population growth outpaced agricultural output, thus the per-capita output declined as a result.

The conclusion of historically poor living standards in India has been greatly strengthened in recent years. This is due to work by De Zwart & Lucassen (2020), providing us with far more data—taking the number of wage observations from a few hundred to over 7500. The data primarily comes from Northern India, probably the wealthiest part of the subcontinent.

The wages of adult unskilled Indian men were very low. Since the 1600s, they have hovered around or even below subsistence level. This is shown for Bengal in the figure below.

In the words of the authors (de Zwart & Lucassen, 2020):

[I]t can be seen that the newly unearthed Indian wage data confirm the more pessimistic views of Indian early modern living standards (as put forth by Allen, and Broadberry and Gupta) rather than the more optimistic views (such as those of Parthasarathi and Sivramkrishna).

And in conclusion:

Our evidence does not support claims of comparatively high Indian real incomes in the eighteenth century nor views that suggest the decline was a post-1800 phenomenon, or even a purely colonial phenomenon

In agreement with Gupta (2019), they remark that “most of the decline in living standards preceded it,” i.e., preceded colonialism.

Finally, based on even earlier data from western India, recent work by Carvalhal et al. (2024) shows that the Great Divergence between England and India goes back even further.

Summary on economic divergence

Qualitative and quantitative evidence agree: commoners in precolonial India lived in poor conditions. Even in Northern India, wages were barely at subsistence level, if not below, and were declining in Malthusian fashion before colonization. The economic divergence between England and India was well under way in the 1600s.

In the upcoming part I will argue that the divergence, more broadly conceived, began much earlier still. There is more to societal development than income, and the foundations for the eventual economic divergence were being established centuries earlier. In particular, divergences in technology and human capital were underway significantly earlier.

Interrogating these foundations helps not only explain the economic divergence between the West and India. It also helps explain why there was more potential for an “East Asian growth miracle” than an equivalent “South Asian growth miracle.”

Understanding the divergence

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Introduction

A common belief is that U.S. prisons are filled with offenders of victimless or trivial crime—imprisoned merely for being in possession of weed for personal use, or perhaps driving under the influence without accident, et cetera. At its most extreme, this belief gets you books like “The New Jim Crow.”

The U.S. prisoner count is the usual starting point of this argument—which, indeed, is high. With 541 prisoners per 100,000 inhabitants, the American prisoner rate ranks 5th in the world, only beaten by a handful of less developed countries. Owing to its large population size and high prisoner rate, the U.S. has more prisoners than any other country. As I recently pointed out, the fraction of men who receive a prison sentence in their lifetime is non-negligible.

Yet this belief remains false. Despite the high prisoner count, American prisons are not filled with people who committed innocuous offenses. The modal prison inmate has a long criminal record and is guilty of violent crime.

I will substantiate these points, and then I’ll provide context for the high American incarceration rate.

Prisoners have committed serious offenses

Prison population composition

To address this question, we can simply look at the prison population categorized by their most serious offense (Carson, 2022):

The data dispel the notion that the prisoner rate is high because countless people are imprisoned for trivial offenses. A clear majority of prisoners have committed violent crime (62.5%). Nearly five times as many prisoners are in for murder compared to drug possession (15.0% vs 3.2%).

Even non-violent offenses tend to be serious. Burglary comprises over half of the property offenses. Drug possession only accounts for a small fraction of drug crimes and just 3.2% of all inmates—and even this figure should be interpreted in the light of possible plea bargains.

This shouldn’t really be surprising. After all, prisons (as opposed to jails) are designed for long-term incarceration. Jails are where people are held while awaiting trial or sentence, or while serving short sentences.

Prison admissions

There’s a distinction between the prisoner composition, and the composition of prison admissions. People enter and exit prisons every day, and those with shorter sentences leave more quickly.

The composition of the prison population is therefore a function of both admissions and sentence length—it will be skewed towards more serious crimes with longer sentences.

Yet, even when we look at prison admissions, a plurality (but not a majority) have committed violent offenses:

Unfortunately the available admission data do not allow me to break down drug or public order offenses into more detailed categories. However, recall that drug possession and DUIs account for only small proportions of their respective categories (drug crimes, public order crimes) in the prison population.1

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Prisoners are repeat offenders

Prisoners have not only typically committed serious offenses, they have also typically committed many offenses.

The figure below shows the number of prior arrests for people admitted to state prison (including the arrest that led to the prison sentence).

The median number of prior arrests was nine. More than three quarters have at least 5 prior arrests. Having 30+ prior arrests was more common than having no arrest other than the arrest that led to the prison sentence (i.e., 1 prior arrest).

It should go without saying that those imprisoned with no prior arrests have usually committed highly serious offenses that would warrant imprisonment with no prior criminal record.

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Prisoners have committed more crime than records suggest

As striking as these figures are, they still understate prisoners’ criminal histories. This is because of the dark figure of crime—the amount of unreported, undetected, or undiscovered crime (Biderman & Reiss, 1967).

A highly replicated finding is that criminals self-report having committed many offenses for each police contact. That is, they readily admit to having “gotten away with” many offenses.

One study of 411 males found that the self-reported number of offenses was over 30 times greater than convictions (Farrington et al., 2014). For sexual offending, studies have estimated the dark figure to be anywhere from 6.5 to 20 times the official figure (Drury et al., 2020). In a recent study of American delinquent youths, the self-reported number of delinquent offenses was 25 for every police contact (Minkler et al., 2022).

However large the dark figure of crime exactly is, it is undoubtedly practically significant. Prisoners’ criminal histories are therefore substantially more extensive than their criminal records would suggest.

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Why is the U.S. prisoner rate so high?

It is true that the U.S. prisoner rate is very high. But why?

As we’ve just seen, this is not because prisons are filled with people guilty of one or two insignificant offenses. Prisoners are overwhelmingly repeat offenders, and the bulk of them have committed violent offenses. In fact, violent offenses are so predominant that, if all non-violent offenders were released from American prisons, the U.S. would still have a higher prisoner rate than most highly developed countries. Even the Prison Policy Initiative—an advocacy group against mass incarceration—recognizes this.

The real answer lies in a combination unique to the United States: high development and capacity, high rate of serious crime for its development, and longer sentences.

Compared with other highly developed nations, the United States has a much higher rate of serious crime (e.g., homicide). The high prisoner rate is a direct result of that. If we benchmark the prisoner numbers on a per-homicide basis, rather than a usual per-capita basis, the U.S. has a typical number of prisoners and an exceptionally low number of police.2

Globally there are obviously many countries with much higher homicide rates than the United States. But these countries tend to be underdeveloped and do not have the same capacity to fight crime—thus they have lower imprisonment rates. In many third world countries, crime runs rampant and law enforcement lacks the means to properly combat it. The U.S. is rather unusual in its combination of high crime and high capacity.

Lastly, the U.S. has longer prison sentences than other highly developed countries. For example, in a previous post I noted that:

Whereas prison sentences are typically more than one year in the United States, the modal length of an unsuspended prison sentence is one to two months in Denmark (StatBank, STRAF47). This leads to a much larger share of American people being imprisoned at any given time.

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Many will naturally have one of two follow-up questions: (1) but why is the U.S. rate of serious violent crime so high? (2) Does America’s co-occurrence of high prison rate and high violent crime rate not disprove the notion that prisons reduces crime? To keep this post short and focused, I will address these only superficially.

National crime disparities exist for many reasons, only some of which are directly related to law enforcement: deterrence from police presence (Chalfin & McCrary, 2017), demographics, and likely many others (e.g., culture, gun access).

Even if prisons reduce crime (and they do, mainly through incapacitation), it is not surprising for high crime rates and high incarceration rates to coexist. The question is based on flawed causal thinking. To see the fallacy, we might draw an analogy between incarceration and medication. Incarceration is a direct response to crime, like medication is a treatment against a disease or condition.

People on medication tend to be more ill than those who aren’t on medication, yet this obviously doesn’t disprove that the medication helps alleviate symptoms. Cross-sectional correlations are highly misleading in contexts where the treatment is directly responding to a symptom.

Evidence with better causal identification shows that incapacitation reduces crime (e.g., Buonanno & Raphael, 2013; Vollaard, 2013; Johnson & Raphael, 2012; Levitt, 1996). Crime would therefore be more prevalent in a counterfactual without prisons.

Conclusion

The U.S. prisoner rate is not high because prisons are filled with people who have committed trivial or victimless offenses. The modal prisoner is both a violent and a repeat offender. Due to unrecorded crime, prisoners’ true criminal histories are more extensive than criminal records would suggest.

The high U.S. prisoner rate can primarily be understood by a combination of factors: its high serious crime rate (relative to its development), its high capacity to fight crime, and its long sentences.

“African time is the perceived cultural tendency […] toward a more relaxed attitude to time.” (Wikipedia)

“African time (noun). Unpunctuality.” (Collins Dictionary)

“African Time is when one says they will arrive at a certain time, but arrive much later.” (Urban Dictionary)

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Introduction

A week ago or so, a video went viral on Twitter/X. A young African woman in the video asserts that, “for many African societies, the concept of the future doesn’t exist.” This is a strange claim to the Western ear, and the video’s virality is unsurprising.

This was not her own theory; she was referencing the work of an influential African philosopher and writer. African societies have traditionally conceptualized time very differently from the West. To this day, African attitudes towards time can be a bit of a culture shock for Western visitors. This phenomenon is sometimes called African time.

When used today, African time refers to a more loose and relaxed attitude towards time, or is even sometimes used synonymously with ‘unpunctuality.’

The following anecdote illustrates how ‘African time’ can manifest in practice (Onyeocha, 2010):

There was once a gathering of all the Rectors of Polytechnics in Nigeria and the Governor was scheduled to come and address them at 10 o’clock before their two-day business meeting which was slated to commence at 11 o’clock. Invitations were sent out and invitees were directed to be seated by 9.30 before the governor’s arrival at 10 o’clock. Everyone complied. At ten, there was no governor. Then ten-thirty, eleven, eleven thirty, twelve, twelve-thirty there was no governor and no message. Everyone began to murmur ‘African Time’! ‘African Time!’ Then half an hour later, the sirens were heard and in scampered a handful of security operatives officiously brandishing antiquated walkie-talkies. The long-awaited governor trotted in wearing a stern, no-nonsense look. That was ‘African Time’—perhaps at its worst.

This piece will explore this phenomenon. Did traditional African societies really have no concept of the future? I will consider the source cited in the viral video together with observations from early explorers. Then I will look at ‘African time’ in the modern context, which has given rise to self-commentary and self-criticism within Africa.

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How many are criminals? It’s a simple question to state, but upon moment’s thought, it becomes clear that there is no simple answer. Nearly everyone has technically committed a minor inconsequential offense at some point, yet “nearly everyone” would be a misleading answer. Obviously, people have more serious offenders in mind when they think of “criminals”, so the question must be answered according to a stricter operationalization. For instance, in the United States, we might consider those people who, at some point in their life, are convicted of a felony.

If you have read an earlier article of mine, or you have a grasp of criminology, you may know that a large portion of crime is due to a small number of repeat offenders. From this you might infer that there are very few criminals — but that would be a mistake. Yes, a few repeat offenders account for a large share of crime, but there are many more criminals who each have a small number of offenses under their belt.

To address the question head on, consider the proportion of men in the United States who eventually end up in prison (Robey et al., 2023).

People don’t end up in prison for trivialities. Prison, as opposed to jail, is designed for long-term incarceration—prison sentences in the United States are rarely shorter than a year. Jails are where people are held while awaiting trial or sentence, or while serving short sentences.

As the figure above shows, one in four to one in three black men eventually end up in prison. The same is true for one in seven Hispanic men, and about one in fifteen (non-Hispanic) white men.

These high figures left many in disbelief when I posted them a little while back. Surely this included jail, or something else was off, right? But no, it really is that many.1

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If you think this is because the United States is some huge outlier, you might be in for a surprise. In Sweden, a study found that 1% of the population was responsible for 63% of violent crime convictions. But the same study also showed that ~7% of men were convicted of a violent crime in their lifetime (Falk et al., 2013). One in fourteen men in Sweden, convicted of a violent crime.2

In Denmark, depending on birth cohort, about 6–7% of men receive an unsuspended prison sentence at some point in their lifetime (StatBank, STRAFFO2). In the same birth cohorts, about 7–9% of men are eventually convicted of a violent crime (StatBank, STRAFFO1).

This also provides a clue as to why the American prison population is so comparatively large (per capita). It is not just because they imprison more people but, probably more importantly, because American prison sentences are much longer. Whereas prison sentences are typically more than one year in the United States, the modal length of an unsuspended prison sentence is one to two months in Denmark (StatBank, STRAF47). This leads to a much larger share of American people being imprisoned at any given time.

If we consider all crimes under the Danish criminal code, which includes a larger number of offense types but still excludes minor traffic offenses, the number of criminals increases markedly. We also see large ethnic disparities.

As shown in the figure above, already by age 24, about 9% of men of Danish origin and 27% of men of non-Western origin have been convicted under the criminal code.

Thus the common refrain that only a tiny, insignificant share of immigrants commit crime is clearly false. It is not true for native Danes, and it is even less true for many immigrant groups.

In summary, while it is certainly true that a large share of crime is committed by a small number of persistent offenders, there are still more criminals than you might expect.

This is part of a series. See also:

• Immigration and crime: Sweden

• Immigration and crime: Denmark

• Immigration and crime: Norway

Introduction

A major source of contemporary Western political debate are questions related to immigration and crime. This post consolidates data from the four major Nordic countries: Denmark, Finland, Norway and Sweden.1 I will first provide a brief review of evidence and sources of data from each country, and afterwards go into greater detail.

Beyond the basic question of whether immigrants are overrepresented in crime or not, I will address several related points:

• Immigrant heterogeneity. How do crime rates vary between different immigrant groups?

• Etiology. Which factors explain (or don’t explain) criminal disparities?

• Immigrant persistence/assimilation. Do criminal disparities persist to the next generation, or do groups assimilate?

• Specific types of crime. Additional data regarding specific types of crime.

Brief literature summary

There is no doubt that people with foreign background are overrepresented in crime in the four Nordic countries. I will briefly summarize some of the data that establishes this conclusion, but first I need to address terminology used throughout this article.

Definitions

In line with typical convention in Nordic statistics, “immigrants” refer to people born outside the country in question (i.e., first-generation immigrants). “Descendants” refer to those born inside the country, but have two immigrant parents (roughly speaking, second-generation immigrants). People with “foreign background” are those who are either immigrants or descendants. “Native background” is simply the absence of foreign background.

Sweden

In a review of Swedish data, criminologist Amber Beckley (2015) writes: “Studies unequivocally show that people with a foreign background have about double the rate of crime compared to people with a Swedish background … The crime rate disparity persists after standardizing for socioeconomic factors.”

This overrepresentation is observed regardless if suspect data or conviction data is used (Brå, 2021; Bäckman et al., 2021). Studies show that individuals with foreign background are overrepresented in offenses of violent crime, deadly violence, and rape (Frisell et al., 2011, Khoshnood et al., 2020; Khoshnood et al., 2025).

Denmark

Unlike in Sweden, annual reports are issued by the Danish government and published by Statistics Denmark. They consistently find that people with foreign background are overrepresented in crime. Individuals with foreign background are also overrepresented in prisons (Kriminalforsorgen, 2024).

After adjusting for sex and age, non-Western immigrants and (especially) descendants are overrepresented in crime (DST, 2024, p. 115). According to the same report, this remains true if income is also adjusted for (table 6.7).

Norway

Over the years, there have been several studies about immigration and crime in Norway using register data, most of which are reviewed by Skardhamar (2017). These studies confirm that individuals with foreign background are overrepresented in criminality in Norway.

Notable examples include Andersen et al. (2017); Skardhamar et al. (2014); Skardhamar et al. (2011); Evensen (2009); and Kardell & Carlsson (2009). More recently, Statistics Norway (SSB) have published relevant data in 2020 and 2024, though without any associated detailed reports or statistical analyses.

Finland

Statistics Finland directly provide crime rates by national origin which can be publicly accessed by query. These statistical tables show that many nationalities are overrepresented in crime (see, e.g., StatFin, Table 13jg). Several detailed reports and articles confirm this (Danielsson et al., 2020; Kotiniemi, 2015; Lehti et al., 2014; Skardhamar et al., 2014).

With this brief overview established, I will now analyze and visualize Nordic data in greater detail. I will consider how crime rates vary between immigrant groups and by offense type; whether differences in age, sex and socioeconomic factors can explain criminal disparities; and to what extent criminal disparities lessen with assimilation; and more.

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Immigrant heterogeneity

Crime rates differ markedly between immigrant groups—to such an extent that Skardhamar et al. (2014) conclude that “it is questionable whether one should treat immigrants as one single group at all.” I concur.

When approaching this topic, the better framework is to view it as part of a broader analysis of group disparities in crime. Whether immigrants are overrepresented in crime depends crucially on the group in question (or the composition of groups), and on the people they’re being compared to.

While people with foreign background are overrepresented in crime in the Nordic countries, when looked upon with finer granularity, this evidently doesn’t apply to all groups. In the Nordic countries, the foreign criminal overrepresentation is driven mainly by higher crime rates among non-Western immigrants and descendants—in particular those who descend from MENAPT (Middle East, North Africa, Pakistan and Turkey) and Sub-Saharan African countries.

Using conviction data provided by Statistics Denmark, I have illustrated this below.

The differences are large. Crime rates of Western immigrants and descendants are similar to those of people of Danish origin, and even slightly lower for violent crimes. Non-Western immigrants and descendants, however, have substantially higher crime rates across offense categories. This is especially the case for people with MENAPT background, who have more than five times the violent crime rate as Western immigrants.

These findings are not adjusted for age, sex or socioeconomic factors. As I’ll show later, such factors contribute to the gaps, but criminal disparities remain substantial after accounting for them.

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Data provided by Statistics Denmark allow for an even more fine-grained analysis, namely by country-of-origin. I’ve illustrated violent crime rates by country-of-origin below, aggregated across 2010 to 2021.

The highest crime rates are observed for MENAPT and Sub-Saharan African countries. Immigrants from some East European countries also have higher crime rates than Danish natives. Many Western countries have very low violent crime rates, as do several Asian countries (Japan, Philippines, India, Indonesia and China).

The official reports published by Statistics Denmark conduct a similar nation-of-origin analysis (see, e.g., DST, 2024; Table 6.10). The main difference between the analyses is that I aggregate the data across multiple years to increase reliability, which also allows me to provide results from a larger set of countries. Other than that, my results and the results of the government reports coincide closely.

It should be noted that for both Kuwait and Lebanon, a large share of the immigrants are actually Palestinians who, due to being considered stateless, were categorized by the country they traveled from rather than by birth certificate.

In the rest of the article, I’ll show that similar patterns are observed in Finland, Norway and Sweden. We’ll see that these criminal disparities persist substantially into the second generation. We’ll also see that the criminal disparities cannot be explained by differences in population structure and socioeconomic conditions.

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Consistency across the Nordic countries

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Introduction

Life expectancy has dramatically increased across the West. In Sweden1, life expectancy at birth was 30-some years in the 18th century, and today it’s above 80 years. A Swedish person, on average, lives half a century longer today.

Though widely acknowledged, there are two opposing pitfalls of interpretation that people often fall into.

The first camp believes that, since life expectancy used to be 30-some years, people fell dead when they reached their thirties. According to this camp, it would be historically unusual to see a 70-year-old person. This is wrong. Even the Bible tells us, “The days of our years are threescore years and ten.”

The second camp rightly points out that life expectancy is an average that is greatly dragged down by infant mortality. They then go further and argue that if a person survives infancy (or perhaps young childhood), life expectancy has remained largely unchanged. But this is a major over-correction.

Yes, the infant mortality rate has dramatically declined. But the reality is that mortality rates have dramatically declined—at all ages. While it wouldn’t be unusual to see a 70-year-old in the past, the majority did not survive to that age, even among those who escaped the deadly first few years. Today, it is more unusual to not survive to age 70.

In this piece, I will systematically analyze the nature and extent of the rise of life expectancy.

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A glance at history

People have never lived as long as they do today. Despite idealistic notions about the simple past, hunter-gatherers lived substantially shorter lives than us. This is not to say that people never grew old, it was just much less common.

In a cross-cultural review, Gurven & Kaplan (2007) analyze longevity data among several hunter-gatherer societies. The average life expectancy at birth ranged from 21 to 37 years, compared to 84 in modern Sweden. Much of this is certainly driven by infant mortality; however, even those who lived to age 45 were not expected to live to age 70—much lower than life expectancy at birth in modern Sweden. More precisely, the (additional) life expectancy at age 45 varied from 14 to 24 years, compared to 40 in today’s Sweden.

True prehistoric times may have been even worse. Trinkaus (2011) analyzed Late Pleistocene specimens of human remains for which age could reliably be assessed. The results confirm a dearth of older adults, consistent with short life expectancy.

Below is a table of life expectancy estimates for various periods or populations, compared with the Swedish 2024 equivalent. What’s clear is that modern Swedes not only have much higher life expectancy at birth, but also much higher life expectancy conditional on surviving past infancy and childhood.

Individuals in history

As someone interested in the history of mathematics, I often read biographies of notable mathematicians in history. In doing that, it’s difficult to not take note of early deaths. While many notable historical figures lived to the 70s, and some even to the 80s, it is striking how many died in the early-60s, 50s, 40s, and sometimes even younger. John Graunt (1620-1674), often regarded as the founder of demography, and who analyzed mortality data, himself died at 53.

Did any live very long lives? Certainly. Isaac Newton (1643-1727) lived to an impressive age of 84. The polymath Christopher Wren (1632-1723) died at 90. I was amazed to discover that the Danish 16th century mathematician Thomas Fincke (1561-1656) died at 95 years of age.

These cases are exceptional, however. Many of the great mathematicians of history died before their 50s. Bonaventura Cavalieri (1598-1647) and Evangelista Torricelli (1608-1647), both very important figures in the development of calculus, died at 49 and 39, respectively. Blaise Pascal (1623-1662) died of disease when he was 39 years old.

Young deaths remain relatively common well into the 19th century. Arguably Norway's greatest mathematician ever, Niels Abel (1802–1829), died of tuberculosis when he was just 26 years old. Bernhard Riemann (1826-1866), one of the greatest mathematicians of all time, died at 39 of the same cause. Even Frank Ramsey (1903-1930) died as a 26-year-old, possibly due to an infection caught while swimming. Lastly, it would be remiss of me not to mention Évariste Galois (1811-1832), who famously died at just 20 years old. Though he died in a duel—not the usual public health matter.

The point is not to suggest that these young deaths represent the norm, but rather that dying as a young adult was a very real concern. These are all major figures in the history of mathematics; it’s not like I had to dig deep to find obscure examples of mathematicians who died young. And, of course, many people in history would have made important advances and become notable had they lived longer lives.

The conclusion is clear. Even if you survived young childhood, you were by no means guaranteed a long life. The next sections are devoted to illustrating this in a systematic manner.

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Data analysis

In the following sections I will document the rise of life expectancy, visualizing it from multiple perspectives. Owing to its long time series, I will focus on data from Sweden. The establishment of Tabellverket in 1749 gave Sweden a head start with population statistics (Sköld, 2004).

I will also focus primarily on females to minimize the effects of violent deaths (less relevant from a public health standpoint). With that said, the broad trends uncovered here will apply to both sexes and to other Western countries, even if they would differ slightly in detail.

1 — Conditional Life Expectancy

To show that the rise of life expectancy isn’t merely a matter of infant mortality, we can simply track life expectancy for people who live beyond infancy. Below I show that, conditional on reaching age 15, the life expectancy has nevertheless massively grown.

As the figure shows, even when restricted to the subset of people who live to age 15, life expectancy has increased by a staggering ~26 years.

Though I focus on Sweden due to its long time series, it’s helpful to illustrate other countries at least once to verify that Sweden is not some exceptional outlier. Using all countries where data is available since before 1900 (in the Human Mortality Database), I have illustrated life expectancy conditional on reaching age 15 below.

In the 19th century, across these countries, a 15-year-old female on average lived to approximately age 59. In 2024, the life expectancy is 85 years, or 26 years higher.

We can go further and illustrate the equivalent for many conditional ages at once. As the figure below shows, even women who managed to survive to an age of 60 years old were historically only expected to live to their early-to-mid 70s. That’s much lower than today’s life expectancy at birth. Today, 60-year-olds can expect to live to age 87. That’s twice as many additional years of life for a 60-year-old woman.

2 — Age-specific mortality rates

A second visualization approach is to look at age-specific mortality rates. As the illustration below shows, the mortality rate has dramatically declined at all ages. It is not merely mortality rates in infancy or childhood that have decreased.

And don’t be deceived by the logarithmic axis—the differences are much larger than what they might appear visually at first glance. For example, a 30-year-old Swedish woman born in 1750-1759 was 33 times more likely to die before their next birthday than a 30-year-old woman born in 1990-1994.

The next chart illustrates the same phenomenon. But instead I’ve shown the entire time series for 5 different age groups.

The mortality rates gradually decline over the entire period, but the decline becomes especially pronounced around the turn of the 20th century.

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3 — Survivorship curves

We can also analyze the proportion who survive until a given age. I’ve shown this below, comparing the 18th century with recent data.

The two curves could hardly be any more different. In the 18th century, there was a much lower chance of surviving to any given age. Clearly much of this is driven by infant and childhood mortality, as indicated by the plunge in survival in the first few years of life for the 18th century period.

But is it just a matter of childhood mortality? Just as before, we can simply exclude people who don’t live beyond the first few years. I’ve taken the same data, but restricted the analysis to females that survive to at least age 10.

The survival curves still look very different. In the 18th century, if you survived to age 10, there was still roughly 1 in 5 chance you’d die before age 40. Only about half of 10-year-olds would survive to age 60, and 1 in 3 survived to age 70. Today, 90% of 10-year-olds survive to age 70.

How have these survival rates evolved over time? Below I have shown the proportion of Swedish females who survived to age 60. One of the curves represent the probability of a newborn surviving to age 60, the others represent the survival probabilities for those who managed to survive to age 10 and 20, respectively.

The chance of surviving to old age has enormously improved. Historically, fewer than half survived to age 60 and, even if you managed to survive to age 10 or 20, your chance was still only slightly better than a coin toss. Today, it is rare not to survive to age 60.

We can also ask, what age do X% of the population survive to? This is illustrated for various percentiles below. For example, the median line tells us the age which 50% of females lived to. As we can see, the median lifetime was a little less than 40 in the 18th century. Today, the median woman survives to age 87.

In the 18th century, the 25th percentile is just above the age of zero, reflecting that a little less than a quarter died in infancy on average. Infant mortality rate dropped rapidly over the years. By 1900, fewer than 10% die in infancy (10th percentile lifts off from the horizontal axis). And by 1930, fewer than 5%.

If we consider the other end of the spectrum, we can see that the top 5% of females (95th percentile) historically survived to the age of ~81. Even the median female lives longer than that today. In modern Sweden, 5% of women reach the age of 99. The adult lifetime has clearly greatly increased, when even the 95th percentile has grown by almost two decades.

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4 — Age distribution of deaths

The change in life expectancy can also be illustrated by showing the distribution of ages at people’s deaths. The figure below shows that, in the 18th century, if you survived the first few years, the most common age of death was about 70, echoing the Bible’s assertion, “The days of our years are threescore years and ten.”

Yet we must also recognize how many used to die between the ages 10 and 60. The frequency of young adult death stands in stark contrast with modernity, where nearly all deaths are clustered at ages 60 and above. It is not just that the average has increased, the variance or “inequality” in lifetime has greatly diminished.

If we ignore infancy and young childhood, the most common age at death (“mode”) for women used to around ~70 in 18th century Sweden. Today, it is 90 years old. The mode has thus increased by 20 years, even if we discount childhood deaths.

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How much is due to infant mortality reduction?

Clearly the secular rise in life expectancy cannot simply be attributed to a reduction in infant mortality. However, is it mostly a matter of infant mortality?

This is actually possible to quantify. Due to a method of Arriaga (1984), we can decompose the total change in life expectancy into individual age-specific contributions. The method properly takes into account that surviving infancy (or any other age) not only allows you to survive to your first birthday, but (most likely) let’s you live many more years.

Arriaga decomposition results are shown for the change in life expectancy in Sweden between 1751 and 2024. Specifically, I’ve shown the cumulative relative age-specific effects.

The results tell us that reductions in infant mortality (age <1) account for 34% of the female change in life expectancy between 1751 and 2024. So not only is infant mortality insufficient to explain the change, it cannot even explain the majority of the change. However, 53% of the change is explained by reductions in mortality accumulated over the first five years of life. Therefore, ages 0–4 are just barely responsible for the majority of the change.

The importance of reduction in adult mortality shouldn’t be overlooked. About 32%-35% of the life expectancy increase is due to reduced mortality at ages 20 and up. That is similar to the contribution of infant mortality.

A simple summary is that the total increase in life expectancy between 1751 and 2024 can in roughly equal parts be explained by the contributions of (a) the first five years of life, and (b) ages 5 and above.

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Causes and prevention of death

The causes of mortality vary in time and space. In the 14th century, Europe was ravaged by bubonic plague (“The Black Death”). Sweden too was struck between 1350 and 1351, and also by recurring outbreaks of bubonic plague in the following centuries. The last major outbreak in Sweden was in 1710, representing the largest spike in mortality rate between 1650 and 1950 (see below).

With the establishment of Tabellverket in 1749, we can make more reliable assertions about the dominant causes of death in the latter half of the 18th century in Sweden.

These were smallpox, dysentery, fevers (e.g., typhoid), tuberculosis (historically, “consumption”), as well as other infectious diseases such as measles and whooping cough. Some of these causes of death were exacerbated in times of poor harvests and famine. In the 1830s, Sweden was struck by epidemics of cholera, a new disease originating from India in 1817.

Some causes of death were more sex-specific. Victims of war were mainly men, however war deaths represent only a small share of all deaths in the period 1750-1800. As for women, childbirth was a serious risk. Even still, in 1751, maternal deaths accounted for “just” 10% of female deaths in the 15-49 age group (Högberg & Wall, 1986), and a much smaller share of all female deaths.

The vast majority of deaths (including adult deaths) were therefore from general causes that were not highly differentiated by sex.2

Prevention of death

Clean water, proper sewage systems, medication and vaccines all contribute to modern public health. Waterborne diseases like cholera, dysentery, and typhoid were primarily reduced by better sanitation systems. Vaccines have helped tackle respiratory and person-to-person transmitted diseases, such as smallpox, measles, and whooping cough.

I’ll briefly describe some of the major public health breakthroughs, ranging from sanitation, to vaccines and antibiotics. Out of brevity, I will not discuss each in-depth, nor address any potential controversies here.

To help combat maternal deaths, Sweden implemented midwife training in the early 18th century. Additional midwifery programs led to reduced maternal mortality in the 19th century (Lorentzon & Pettersson-Lidbom, 2021). In 1847, Ignaz Semmelweis famously proposed that doctors and midwives should wash their hands with chlorinated lime solutions. In 1879, Swedish midwives were required to wash their hands before performing procedures.

Variolation (deliberate infection to develop resistance) had been used in Sweden to address smallpox since the 1750s, though it wasn’t widespread (Sköld, 1996). Edward Jenner’s smallpox vaccine of 1796 was the major advance; it was safer and could be more widely applied. In 1816, vaccination of children became mandatory by law in Sweden (Gustavsson, 2024). Sweden was the first major country to eliminate indigenous smallpox, which it achieved in 1895. By 1980, smallpox had been globally eradicated.

In 1854, John Snow’s work in London linked cholera to contaminated water, which influences urban planning. Water supply and sewer systems were soon established across Finland and Sweden. In addition to public health justifications, water supplies were also necessary for efficient firefighting, and sewers are important for drainage to protect buildings (Juuti et al., 2009). Repeated improvements in sanitation systems helped reduce waterborne diseases (cholera, dysentery, typhoid).

In the 1930s, sulfa antibiotics were introduced, and penicillin in the 1940s. Antibiotics are used to prevent or treat infections, and have been instrumental in addressing tuberculosis—the last of the diseases that was a major killer in the preceding century.

Less obviously, antibiotics have also been responsible for a reduction in stillbirths. The major turning point in stillbirth rates coincided with the 1930s (Woods, 2008), as I’ve shown for Nordic countries below.

The decline not only perfectly matches the timing of antibiotics (and not other obvious candidates, such as frequency of c-section use), but Løkke (2012) analyzes detailed Danish cause-of-death data and finds strong support for the importance of antibiotics.

Antibiotics not only helped cure non-obstetric infections before and during pregnancy, they also allowed more invasive interventions to save the fetus without wanton hazard to the life of the mother. This reduced the number of stillbirths.

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Conclusion

In this post, I have illustrated the rise of life expectancy in Sweden in several ways. The basic conclusion is that, though infant mortality is a major part of the story, life expectancy has greatly increased even beyond infancy and childhood.

Arriaga decomposition shows that reduction in infant mortality accounts for a third of the increase in life expectancy in Sweden. This is similar to the cumulative effects of reductions in mortality rates at ages 20 and above.

While war and maternal deaths were serious risks and highly differentiated by sex, the vast majority of deaths were due to more generic causes. In 18th century Sweden, these were particularly smallpox, dysentery, typhoid fever, tuberculosis, and other infectious diseases such as measles and whooping cough. In the 19th century, cholera arrived and was responsible for a large share of deaths.

These deadliness and prevalence of these diseases have all been greatly reduced, due to a combination of public health measures, including improved sanitation, vaccines, medication and sufficient nutrition.

Introduction

It is no secret that Sub-Saharan Africa is poor. According to World Bank figures seen below, the difference between Sub-Saharan Africa and the United States is huge, even after accounting for the costs of living. On a per-capita basis, Africa is by far the poorest continent.

This piece is about a related problem, namely Africa’s poor numbers—the lack of data and low data quality that pervades Africa.

Critical commentary about Africa’s numbers is not a new thing. In the 1981 book Social and Economic Statistics for Africa, Kpedekpo and Arya wrote that African national statistical offices were simply ill-prepared to publish adequate data, “resulting in figures that are little better than random numbers.”

Roger Riddel once wrote: “perhaps the most fundamental problem with the available Africa data is that these are widely known to be inaccurate but the degree of inaccuracy cannot easily be judged—itself a sign of the underdevelopment of the region.”

More recently, Devarajan (2013), then chief economist for Africa at the World Bank, described the state of affairs as “Africa's Statistical Tragedy”. Others call it Africa’s “data gap” (Adebisi & Lucero-Prisno, 2022). But perhaps the most well-known critical appraisal of Africa’s developmental statistics is Morten Jerven’s 2013 book Poor Numbers—the inspiration of the title of this piece.

As an economist, Jerven’s interest pertained to the production of GDP figures, the sort of numbers that are illustrated in the earlier graph. What methodology and data were used to produce those numbers? How reliable were the numbers?

In 2007, Jerven visited the statistical office responsible for Zambia’s national accounts. It was only with personal experience that the gravity of the situation truly revealed itself.

There were only three employees, and just one regularly in the office while he was visiting. No employee could account for how income estimates had been generated more than a decade ago. They lacked resources and staff. They lacked basic data. Many sectors of the economy had no usable data, or estimations were made based on a very small subset of the sector’s economy. Many numbers were functionally guesstimates with no underlying rigorous and reproducible methodology.

When he revisited in 2010, the situation was even worse. As he explains (Jerven, 2013):

“What happens if I disappear?” In 2010, I returned to Zambia and found that the national accounts now were prepared by one man alone. His question was not hypothetical, but one of real concern. Until very recently he had had one colleague, but that man was removed from the National Accounts Division to work on the 2010 population census. To make matters worse, lack of personnel in the section for industrial statistics and public finances meant that the only statistician left in the National Accounts Division was responsible for these data as well.

In total, he visited eight Sub-Saharan African countries, and had extensive e-mail correspondence with statistical offices in several others.1 The problems were by no means unique to Zambia. “Similar anecdotes could be told from research visits I have made to other countries in sub-Saharan Africa since 2007,” he writes. Nor is this problem contained to economic statistics.

Africa’s poor numbers are the result of several interrelated factors: limited funding, inadequate infrastructure for data collection and management, lack of expertise and skilled workers, political instability impeding data collection, political incentives leading to misreported numbers, and corrupt politicians opposed to data transparency.

The majority of what we “know” about African state of affairs can be traced to estimates made by external organizations and actors (e.g., WHO, UN). Though the situation is slowly improving, crucial data expected from African governments are either missing or are too unreliable to be used.

In this article, I critically analyze Africa’s numbers. I begin with a summary of Africa’s general statistical capacity. Afterwards, I separately analyze three broad classes of data: economic data, crime data, and population data.

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Statistical capacity in Africa

While Jerven’s Poor Numbers focuses on the gross domestic product, issues of poor data quality or missing data go well beyond economic figures. These problems are pervasive—usually especially so—in Africa. Examples include:

• Lack of birth registration.

• Less up-to-date censuses.

• Lack of death registration, and poor cause-of-death data quality (Karlinsky, 2024; GBD 2016 Causes of Death Collaborators, 2017).

• Poor civil registration and vital statistics data quality (Mikkelsen et al., 2015).

• Poor anthropometry data quality (Perumal et al., 2020).

• Poor homicide data quality (UNODC, 2023).

• Misreported answers in surveys (e.g., Larsen et al., 2019; Seidler et al., 2025).

A nation’s ability to collect, analyze and disseminate high-quality data about its population and economy is referred to as statistical capacity. One can measure national statistical capacity by using a composite of several indicators like those just mentioned. This is what Dang et al. (2023) have done. The overall scores are illustrated below.

While poor statistical capacity is by no means exclusive to Africa, it is a near-universal problem in Africa. Of all major regions, Sub-Saharan Africa has the lowest average statistical capacity score.

The capacity of an African statistical system is strongly related to the country’s overall governance. This is according to the latest 2023 African Governance Report by the Mo Ibrahim Foundation, finding a strong correlation between the two (r = 0.78). A nation’s statistical capacity is therefore also roughly proxied by its overall ability to govern.

The same report acknowledges that the statistical capacity of Africa “remains low compared to other world regions”. In fact, only 33.5% of data sources on Sustainable Development Goals (SDGs) indicators on Africa are produced by the country’s own statistical system (p. 29). To put it bluntly, this tells us that African governments still cannot be relied upon for providing comprehensive and reliable data about the state of affairs.

While many factors contribute to Africa’s low statistical capacity, the most commonly cited contributors can be organized into four categories:

• Lack of resources. Statistical capacity is expensive. A large personnel is necessary to collect, produce, analyze and disseminate high quality, comprehensive and reliable statistics. Tracking accurate statistics of births, deaths (and their causes), crimes, economic data, et cetera, for a nation’s entire population is just not a trivial task.
  To take a concrete example, consider the task of conducting a census. The U.S. 2020 census alone reportedly had a price tag of approximately $15.6 billion, involving hundreds of thousands of temporary workers. Labor is cheaper in Africa, but the total costs of a census can still easily amount to the equivalent of 100’s of millions of U.S. dollars. Many African countries do not have the necessary funds, and have often relied heavily on external donations for assisting in large statistical projects such as censuses.

• Low human capital. Comprehensive statistical systems are reliant on many skilled workers with technical expertise. Tracking, say, health data alone requires trained doctors handling the individual cases (births, autopsies, etc), but also skilled workers to handle, analyze and disseminate the data. Furthermore, low numeracy and literacy among the general public also results in less reliable survey data.

• Political conflicts and instability. Many African countries have experienced armed conflicts and general political instability in the past few decades. Examples include Boko Haram terrorism in Nigeria and civil war in Ethiopia. Collecting comprehensive and reliable data is obviously much more difficult in conflict zones.

• Political incentives, corruption and aversion to transparency. It is no secret that political motivations can affect statistics. For example, there’s a political pressure to demonstrate that reforms are successful to domestic political leaders and external stakeholders. This creates a push towards reporting overly optimistic numbers in Africa (see, e.g., Desiere et al., 2016; Sandefur & Glassman, 2015; Jerven, 2014). In Nigeria, resources were allocated to states in proportion to population size, leading to fabricated and grossly inflated population counts. Lastly, political leaders may simply have an aversion to full transparency, if facts would be in conflict with the political image they want to portray.

In the rest of this article, I will consider three types of data and analyze them more in-depth separately, namely: (1) economic statistics, (2) crime statistics, and (3) population count statistics.

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Economic statistics

In 2010, Ghana rebased and reviewed the basic data underlying their GDP calculation. This revision resulted in Ghana’s GDP shooting up by 62%. Newspapers reported on Ghana’s remarkable growth (Devarajan, 2013), but this wasn’t “growth”—Ghana hadn’t suddenly gotten much richer. Rather, it highlighted the unreliability of African income estimates.

African GDPs are difficult to estimate for a number of reasons. The economic data is outdated. Basic data for various economic sectors are missing. A large swathe of economic activity is unaccounted for, a problem especially pervasive in low-income countries where labor is more informal (Medina & Schneider, 2018). Newborns aren’t registered, and population censuses aren’t up-to-date (discussed more in the section on population statistics). The list of meaningful sources of error is long.

Devarajan (2013) described this as “Africa’s Statistical Tragedy,” writing: “The ‘tragedy’ is that the development community was happily publishing GDP statistics and growth figures for Ghana over the last decades, pointing out how well the country had been doing. Now we have to revise those figures. So in fact we did not know how well Ghana was doing.”

It wasn’t unique to Ghana, either. Malawi revised their GDP by +30% (Devarajan, 2013). In 2014, Nigeria’s GDP nearly doubled, being revised upward by a shocking 89%. Senegal’s grew by about 30%. Tanzania’s GDP grew by a third, Kenya’s and Zambia’s increased by a quarter, and Uganda’s rose by 13 percent (Sy, 2015).

On the one hand, the revisions told us that African countries were doing better than previously thought. On the other hand, such major revisions speak to how outdated and potentially unreliable African GDP estimates had been.

This raises an important question: how meaningfully can we really compare African GDPs? To address this question, Jerven (2010) compared African per-capita GDP estimates from the three main sources: World Bank data, Penn World Tables, and Angus Maddison estimates.2 He found a surprising amount of disagreement in country rankings between the sources.

The fundamental problem was that African GDPs were estimated from a sparse and unreliable data foundation. GDP estimates were thus highly sensitive to minor variations in assumptions and methodology.

The dearth of comprehensive and reliable economic data in Africa continues to this day. Luken et al. (2020) analyzed the industry-related data quality and conclude: “We find data quality issues least severe for two out of the five conventional economic indicators but extremely severe for the other three economic indicators.”

The organization World Economics have a data quality rating scheme designed to evaluate the usefulness of official GDP data of individual countries. African countries tend to receive the lowest scores, a fact they explicitly label as “Africa’s data quality problem.”

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How reliable are national accounts data?

The central problem for African economic figures is the lack of basic data, making it impossible for national accounts to accurately track the economy. This is made even more challenging by the large “shadow economy” of informal labor present in low-income countries (Medina & Schneider, 2018).

After having visited many statistical offices of several African countries, and being in contact with even more, Jerven (2013, p. 3) points out that, in practice, African national accounts involve an unhealthy amount of guesswork, “For large shares of the economy we have little or no information and the figures involve a great deal of guesswork.”

This is verified by relevant informants. As Marivoet & Herdt (2014) mention, an informant at the World Bank office in Kinshasa acknowledged that national accounts figures of the DRC often had little basis in hard data, and were rather based on consensus of opinion:

The same informant also mentioned in passing that most national accounts of the DRC are simply subject to consensus, or, it appears, experts and consultants who sit together and decide on the size of the Congolese economy over a pint of beer. At the same time, such a consensus strategy should not come as a surprise given the large amount of pure guesswork that seems to characterise the national accounting in the DRC. (Marivoet & Herdt, 2014)

But can we measure how accurate national accounts data is? And are errors like white noise or, worse, systematic?

In a recent analysis of economic data from Mozambique in south-east Africa, Jones & Mucocana (2025) compare the official national account of agricultural output with micro-survey and other data. Their results confirm that African “official” national accounts data cannot naïvely be taken as factual.

They triangulate different sources of evidence: micro-surveys, household budget surveys, market price, and Food and Agriculture Organization (FAO) production data. These alternative data sources all roughly agree with each other, but disagree with the national accounts data.

The error appears to be substantial: “agricultural production may be one-third lower than the value stated in national accounts and long-run real growth rates in the sector have barely exceeded rural population growth.” As shown in the figure below, the alternative data sources suggest that there has hardly been any growth between 2002 and 2020.

As agriculture remains a large fraction of the overall economy in developing countries, this implies that Mozambique’s GDP growth has been overstated to a practically significant extent.

The authors then expanded their analysis to other countries. They conclude that this problem may not be unique to Mozambique. Rather, the evidence suggests that “national accounts estimates of agricultural production may be more prone to being up-ward biased in lower-income economies.” A different analysis of agricultural data from six African countries also found no increase in productivity between 2008 and 2019 (Wollburg et al., 2024).

So not only has Mozambique’s agricultural productivity been overstated in national accounts data. This is possibly a general tendency in low-income countries. African economies may be performing systematically worse than official data suggests.

Mozambique is not the only African country where unreliable agricultural statistics have been uncovered.3 In Rwanda, data provided by the agriculture ministry showed a dramatic increase in yields by 60 percent following agricultural reforms. But Desiere et al. (2016) argue that these statistics are dubious, showing that are not consistent with other data (which show much less growth). Further, they argue that there’s a political incentive to inflate numbers to demonstrate the efficacy of the reforms.

There are also other examples of poor numbers in Rwanda. Many have argued that poverty figures have been manipulated in Rwanda. Briefly summarized, the accusation is that the basket of goods used to calculate the poverty line had been changed in a systematic way, effectively shifting the poverty line to make the situation look more favorable.

Though the World Bank have denied the accusations of manipulation, they have also acknowledged discrepancies between survey and national accounts estimates, writing in their Rwanda systematic country diagnosis: “The results of the three latest surveys show that per capita consumption has grown by 0.8 percent annually between 2011 and 2017, but national accounts show 4 percent growth.”

Survey data

Since official data supplied by the government is often unavailable or unreliable in low-income countries, economists have sought to measure development with alternative methods.

Household surveys have been the most widely used alternative approach. By asking people about their educational level, incomes, consumption, demographics, and so on, in principle one should be able to measure living standards without relying on government data.

In fact, household surveys have been so widely conducted to allow estimates of global variation in economic development (Smits & Permanyer, 2019). A map of sub-national income estimates is shown below:

In many African countries, where state capacity is low or where the government may even be corrupt, there’s little doubt that household surveys tend to be more reliable than data provided by the government.

With that said, like any other data source, household survey data can suffer from reliability issues. In a recent assessment of household survey data quality across 35 Sub-Saharan African countries, Seidler et al. (2025) focus on three simple indicators of data quality:

• Incomplete age: birth month or year missing.

• Age heaping: ages ending with the digits 0 or 5. Excess age heaping is the result of rounding due to low numeracy or lack of proper documentation.

• Flagged HAZ: missing or biologically implausible values for the measured heights of children.

The results show that African household survey data is often poor quality, but that unreliability is not evenly distributed across the continent. Nigeria, Niger, Ethiopia and Chad, for example, have less reliable household survey data than, say, South Africa.

Within-country local variation in data quality is also substantial. In general, data quality tends to be worse in more remote rural areas. Regional differences can be large. For example in Nigeria, the observed amount of age heaping ranged from 25% to over 60% across districts (with 20% being the expected amount if digits are approximately uniformly distributed, since 0 and 5 are two out of possible digits). In Chad, incomplete age averaged at 67.3%. But it ranged from 8.1% in Dar Tama (bordering Sudan) to 91.6% in Loug Chari in the south of Chad.

One potential issue with surveys is that people have imperfect memory. This factor has been analyzed in the context of agricultural surveys in Africa. Wollburg et al. (2021) find that systematic error increases with recall period—with longer recall periods, farmers report higher quantities of harvest, labor and fertilizer inputs.

These findings remind us that statistical capacity is not just about having well-funded statistical agencies, but also about the general public’s ability and willingness to answer questions accurately. This is partly a matter of human capital: literacy, numeracy, memory, et cetera (see, e.g., Larsen et al., 2019; Rosenzweig, 2021). Relatedly, Kinyondo & Pelizzo (2018) hypothesize that the problem is due to a “lack of research culture” among the general public.

The verdict

I’ve argued that African national accounts data should be taken with a grain of salt, to put it mildly. However, this does not mean we have no reliable information about Africa’s economic development.

I have already shown that household surveys have been widely conducted across the world, and though they have their own reliability issues, we can reasonably conclude that they paint a more realistic picture of African development than relying solely on “official” numbers supplied by the respective governments.

Even still, to satisfy skeptics of both national accounts data and household survey data, economists have sought yet other ways of estimating economic activity.

The pioneering work of Henderson, Storeygard & Weil (2012) showed that nighttime light, using satellite data, could serve as a strong proxy for economic activity. This approach has since been further worked upon to produce fine-grained development estimates (Sherman et al., 2023). Other proxies, such as mobile phone or social media data have also been used (Chi et al., 2022). Recently, Rossi-Hansberg & Zhang (2025) have combined diverse predictors and machine learning methods to produce subnational economic estimates.4

Together, all this evidence help us form a more reliable picture of African economic development, despite the low statistical capacity in the region.

Crime statistics

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Introduction

A society’s fertility level can be measured in different ways. By far the most commonly used measure is the “total fertility rate.” To calculate it for a given year, you take the number of births by women in an age group divided by the total number of women in the same age group.1 These ratios are added up for all individual childbearing years (convention being 15 to 49 years old):

The measure attempts to “simulate” the number of children a woman will have over the course of all her childbearing years. A fertility rate of just over 2 children per woman is the level at which a population is sustained.

A different measure is the “completed cohort fertility rate.” Instead of adding up births in a given year, you follow all women born in a given year (a cohort) and track how many children they have had by the end of their childbearing age.

A major strength of TFR as a measure is its convenience. Being a period measure, it requires only data from the given year to calculate. Such period data is much more commonly available than cohort data (which requires tracking people for decades). Based on period data, some countries have fertility time series available for multiple centuries. Many countries do not have any cohort data.

The difference between the total fertility rate (TFR) and the completed cohort fertility rate (CCFR) is well-illustrated by Our World in Data:

In the figure, we see that the total fertility rate in a given year (say, 1990) approximately coincides with the cohort fertility of women born 30 years earlier (in 1960). However, the large fluctuations in total fertility rate between 1970 and 2010 are mainly artifacts of so-called “tempo” effects. That is, trends in birth timings rather than the actual number of children women have over their lifetime.2

In truth, Sweden’s underlying fertility was much more stable than the total fertility rate suggested. Motivated by this problem, a period measure that attempts to correct for tempo effects has also been developed (Bongaarts & Feeney, 1998).

Completed cohort fertility (unlike the TFR) is a “true” fertility measure in the sense that it actually tracks how many children women have over their lifetime. It isn’t affected by the push and pull of birth timings, and being not subject to tempo effects, it is much more stable—and arguably more suited for revealing underlying long-term trends.

There is, however, one major weakness of the completed cohort fertility measure: it is always out-of-date. As of 2025, women born in 1980 still haven’t fully passed their childbearing years. This cohort’s completed fertility value is therefore only available after they pass a certain age (a typical cut-off might be 50).

In this post, I partially address this issue. I present a simple method for projecting completed cohort fertility a number of years forward (up to ~15 years). The results of 33 countries will be made available for download.

While these projections are still not fully up-to-date to period measures, they lag substantially less behind than the conventional cohort fertility measure.

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Forecasting completed cohort fertility

For the sake of illustration, I will consider the 1985 cohort in Sweden. These women are now about 40 years old—their childbearing years are largely but not entirely over.

I start off by considering this cohort’s age-specific fertility rates. In the data I use, age-specific fertility rates are available up to age 38.

Age-specific fertility rates are often modeled by scaled normal distributions or a mixture of scaled normal distributions (Peristera & Kostaki, 2007). The age-specific fertility rate (ASFR) in the right tail can thus be approximated as:

where a, b, and c are parameters to be estimated. I estimate the missing age-specific fertility rates by finding the curve that best fits for the ages beyond peak fertility age (the shape of the curve to the the left of the peak is not relevant, as data is not missing there).

The sum of these age-specific fertility values provide the total fertility attained over this cohort’s childbearing years. This is shown accumulated by age below.

In the real observed data, the woman had 1.72 children on average by age 38. The projected value for completed cohort fertility is only modestly higher at 1.85 children per woman. That’s not surprising—women typically do not have many children after the age of 38.

And thus the 1985 cohort’s fertility has been forecasted: 1.85 children per woman. These steps can then be repeated for other cohorts.

This is what I’ve done for Swedish data in the chart below. It illustrates the true cohort fertility for older cohorts, and projected fertility for cohorts with incomplete data. Specifically, 1973 is the latest cohort for which there is full data. These projections extend 15 additional years to the 1988 cohort. As seen in the chart, the Swedish cohort fertility is expected to drop modestly between the 1973 and 1988 cohorts.

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Results

The methodology described above has been repeated for 33 countries’ worth of data. The results can be downloaded below. The data used for the projections is supplied by the Human Fertility Database.3

Projectedcohortfertility V2

58.7KB ∙ XLSX file

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Updates:

• May 30, 2025: slight improvements to the stability and prediction interval calculation. The relevant changes are to the prediction intervals. Changes to the point estimates are negligible.

The time series are of varying length and quality, and some are more up-to-date than others. All this depends on the data that the HFD have access to and have supplied.

Below I will briefly discuss the results of three nation groups: the Nordics, East Asian and Southern Europe. I’ve put figures for some additional country groups in the appendix. The results of all countries can be accessed in the file.

Nordic fertility

Apart from Iceland, the Nordic countries have historically had fairly stable cohort fertility rates, mostly hovering in the 1.8-2.1 range for several decades. These forecasts suggest that cohort fertility will gradually decline between the 1975 and 1989 cohorts.

Denmark, Sweden and Norway all follow very similar trajectories and the 1987-1989 cohorts are expected to have fertility rates of about 1.7-1.8. Finland appears to have the lowest fertility with a more dramatic decline in the latest cohorts, but it is also estimated with a fair bit of uncertainty.

East Asian fertility

East Asia is the region with the lowest fertility, and this remains true for completed cohort fertility. Using HFD data, I could construct time series for Taiwan and Japan. For the criteria I used, there was only a single cohort (1988) with adequate data to generate an estimate for South Korea.

For both Japan and Taiwan, the trend is clearly indicating decreasing fertility for more recent cohorts. The decline is quite rapid (compared to, say, the Nordics). The projections suggest that the most recent cohorts will have lifetime fertility of less than 1.25 per woman. For Taiwan and South Korea, the estimate is only about 1.06.

South European fertility

South European countries like Italy and Spain have had very low total fertility rates for a while, not unlike East Asian countries. This is also reflected in their low completed cohort fertility rates.

However, the projections suggest that South European fertility is stabilizing for many cohorts around 1.4-1.5. And while this is a low level, these countries do not appear to experience the same continued decline as the East Asian countries mentioned above.

France is included here just for completion’s sake. But it should be noted that France is very unlike Spain, Italy and Portugal with respect to fertility. In fact, France is notable for having one of the higher fertility rates in Western Europe.

Conclusion

In this post, I’ve constructed estimates of completed cohort fertility for cohorts with incomplete data. There are estimates for 33 countries, and the results are available for download.

Using these projection estimates, completed cohort fertility lags much less behind period fertility measures than the typical cohort measure. How much does it lag behind? Several countries have estimates for cohorts as recent as 1988-1990.

While this sounds old, keep in mind that cohort fertility is roughly equivalent to period fertility 30 years later (i.e., in 2018-2020), and that cohort fertility is better suited for capturing underlying trends.

With more up-to-date data, discussions about the causes of fertility trends can hopefully be more fruitful. To understand a phenomenon, it is usually best to start with a solid foundation of descriptive data.

Appendix

North and South American fertility

Central European fertility

Baltic fertility

British Isles fertility

• The American Assimilation Myth

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Introduction

In a previous piece, I argued that commonly held notions about assimilation in America—“the melting pot”—are filled with exaggeration and fiction. Rather than quickly vanishing, ethnic disparities were often remarkably persistent, and their consequences can still be observed to this day.

This is by no means unique to America. Using international data, this piece evaluates the strength of assimilation with respect to several socially important domains. I focus mainly on three dimensions of socioeconomic performance: human capital, economic performance and crime. Lastly, I also briefly consider some other dimensions of cultural inheritance.

What exactly is meant by “assimilation”? According to the Encyclopedia Britannica: “The process of assimilating involves taking on the traits of the dominant culture to such a degree that the assimilating group becomes socially indistinguishable from other members of the society.”

The fundamental consequence of assimilation is that group disparities disappear, as one group converges to the other in the aggregate. This, as I’ll argue, is a process that is typically slow and/or incomplete.

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Human capital

Human capital is arguably the central component of the modern knowledge economy. We can measure human capital with various proxies such as educational attainment and educational performance, or performance on standardized cognitive or aptitude tests.

Immigrants and natives often differ substantially in human capital. This in itself isn’t terribly surprising. Immigrants typically grow up with a different language and attending highly different schools.

The question then is, do immigrants generally rapidly assimilate in human capital? The answer is a clear no. This observation is well-captured by the title of a 2011 article analyzing PISA data: “Why do the results of immigrant students depend so much on their country of origin and so little on their country of destination?”

But it’s not just first-generation immigrants. It’s also their descendants.

As shown below, there is a very strong relationship between parents’ native country-of-origin scores and second-generation immigrant scores (De Philippis & Rossi, 2020). That is, second-generation immigrants tend to score much more similar to people in their parents’ country-of-origin, and not like the country they were actually born and raised in. In short, human capital persists substantially across borders.

In another systematic evaluation of immigrant competences using international PISA, TIMMS and PIRLS data, Rindermann & Thompson (2016) found that the native-immigrant gap, on average, is reduced only by the equivalent of ~1 IQ point from first and second generation (though there was significant heterogeneity by origin and receiving country).

The findings of De Philippis & Rossi (2020) go even further. They show that, even among students who attend the same school and controlling for socioeconomic factors, students whose parents come from high-scoring countries tend to perform better.1

Where large deviations from the trend are observed — like India — it is indicative of strong immigration selection, not of assimilation. That is, those who migrated were unrepresentative of their country fellows prior to migration.2

The fact that immigrant performance is strongly related to immigrant selectivity or, more broadly, to pre-arrival characteristics is well-documented (Duncan & Trejo, 2015; Werfhorst & Heath, 2018; Ichou, 2014; Hanson & Liu, 2023). Note, for instance, the high educational level of Indian first-generation immigrants illustrated below (which is far higher than the average person in India). These pre-arrival educational disparities also predict immigrants’ earnings.

In Denmark, non-Western immigrants attain significantly lower grades than native Danes (and Western immigrants for that matter). There is a significant improvement from first- to second generation (especially in Danish language tests), but substantial differences remain.3

Interestingly, Statistics Denmark have also published results on the children of descendants — that is, third-generation immigrants. The results suggest there is little to no further convergence from second to third generation.4

This is consistent with findings by Dutch researchers (van de Beek et al., 2023). They found that there are large differences in human capital by immigrant regional origin, and there is strong persistence at least until third generation.5

Similarly, based on more than four generations of historical data, Telles & Ortiz (2008) found no support for the assimilation theory with respect to educational attainment among Mexican immigrants in the United States.

Human capital also doesn’t just persist with willing migration. Toews & Vézina (2025) show that the enemies of the people who were forcibly sent to Gulag labor camps nevertheless had persistent human capital across generations. The loss of human capital, such as the dismissal of scientists in Germany under World War 2, also resulted in long-run costs (Waldinger, 2016).

In a classic paper, Borjas (1992) argues that persistent disparities in human capital (or SES) must be understood in terms of what he calls “ethnic capital.” The idea is that intergenerational mobility depends not only on the parental input (nature and nurture), but “also on the average skills of the ethnic group in the parent’s generation.” He suggests that other ethnic group members represent externalities that influence (for better or for worse) the environment in which the child accumulates human capital.

Regardless if Borjas is right about the underlying mechanism, it’s an important observation that individuals from different groups regress to different means. This makes the between-group convergence slower than what is observed for individual differences within-group.

In the following sections, I evaluate the evidence regarding assimilation with respect to economic performance, crime and culture.

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Economic

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• The Assimilation Myth

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Introduction

When it comes to the topic of immigration, it is widely believed that assimilation is both important (normatively) and rapid and widespread (descriptively). Rapid assimilation is often simply taken for granted. For example, in economic analyses of the impact of immigration, it is not uncommon to see it assumed that second-generation immigrants have similar economic behavior as natives.1

But nowhere is the story of assimilation more important than in America, the so-called “Melting Pot.” In this piece, I will argue that the notion of widespread and rapid assimilation in America is largely a fiction. In reality, assimilation in America has been slow and incomplete. In a second part, I argue that this is by no means specific to America.

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Persistence is the norm

It’s worth first addressing more generally why rapid and widespread assimilation shouldn’t be expected. Outside the context of immigration, the reality of widespread long-run persistence has already been well-documented. Two growing scientific literatures are of particular note here.

The first of these literatures is the deep roots branch of economics. I won’t review the literature here, but the gist is that modern economic prosperity is surprisingly strongly associated with factors that long predate the Industrial Revolution (see, e.g., Spolaore & Wacziarg, 2013).

One illustrative example is that a country’s level of technological adoption in 1500 AD is strongly associated with modern national economic prosperity (Comin et al., 2010). Importantly, the migration-adjusted association is much stronger than the non-adjusted association. In simple terms, this means that development follows people, not places.

What exactly people pass down through so many generations — whether it’s culture, institutions, genes or something else — is a separate and important question, but, whatever the case, it speaks strongly to the importance of persistence beyond geographical borders.

The second is the literature on the long-run multi-generational social persistence (or mobility).2 Briefly summarized, the literature shows that social status is highly persistent, with convergence to the mean happening very slowly across generations.

As one illustrative example, in a study using rare surnames, it was shown that people overrepresented in the Oxbridge universities had descendants that were still still overrepresented many centuries later (Clark & Cummins, 2014).

Many other studies, using a variety of methods, measures and from various countries, confirm that high persistence of social status is the norm (e.g., Barone & Mocetti, 2021; Bukowski et al., 2022; Clark, 2012; Clark & Ishii, 2012; Clark et al., 2020).

An important discovery is that naïve measures of social mobility have overestimated how mobile societies truly are. Using multi-generational data, the degree of persistence is significantly stronger than you’d predict based on simple parent-offspring correlations (Braun & Stuhler, 2018; Colagrossi et al., 2019; Adermon et al., 2021; Engzell et al., 2020).

The problem is also exacerbated by measurement error. Worse measurements, imperfect intergenerational linkage and bad data all artificially make the perceived mobility much higher than the true underlying mobility (Clark et al., 2022; Ward, 2023; Zhu, 2024). A related finding is that siblings have higher income-similarity when measured in terms of lifetime, accumulated incomes rather than income measured over a few years (Grätz & Kolk, 2022).3

We can extend these observations to the study of assimilation — measurement error and imperfect linkage across generations will lead us to overestimate the effect of intergenerational assimilation and underestimate persistence.

Further, it has been found that the inheritance of social status is highly stable in the face of external shocks and perturbations. A handful of illustrative studies include:

• The enemies of the people who were sent to Gulag camps under the Soviet regime had surprisingly persistent human capital across generations (Toews & Vézina, 2025).

• Inequality that existed before the Chinese communist revolutions persisted through said revolutions (Alesina et al., 2020).

• Though slave wealth was nullified after the US Civil War, their sons almost entirely recovered in wealth ranking, and grandsons fully jumped back (Ager et al., 2021). A similar finding comes from the Danish West Indies (Galli et al., 2024).

• The Cherokee Land Lottery of 1832 provided a large windfall to the winners. Despite that, the sons of the winners fared no better in terms of wealth, income or literacy (Bleakley & Ferrie, 2016).

• The children of lottery winners in Sweden see no improvement in terms of drug consumption, scholastic performance, or skills (Cesarini et al., 2016).

In short, long-term persistence is the norm, and is surprisingly resistant to external shocks. The obvious question is: why should immigration be any different? As I’ll argue—it’s not.

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The Not-so Melting Pot

Immigration and assimilation are intimately connected to the American history and identity. It is a “nation of immigrants” and part of the “American dream” was the freedom and opportunities that America could provide over other countries.

Most importantly, America is also the “The melting pot.” This metaphor speaks directly to the American assimilation story — disparate ethnic groups coming together, melting together into a singular homogeneous American culture.

We can hardly expect a metaphor to perfectly capture the complicated messiness of reality. But how much truth is there to this story? Is the American story really one of widespread and rapid assimilation? The evidence suggests no.

European immigrants

When it comes to the Age of Mass Migration (1850 to 1920s), it is often ignored that large shares of European immigrants did not “assimilate”—they returned home because they couldn’t assimilate. At least 25-40% of European immigrants returned to their home, but serious estimates as high as 60-75% exist for some decades (Bandiera et al., 2013). Return migration was also negatively selected (Abramitzky et al., 2019). Large shares of migrants could not successfully integrate, and those were typically the ones who returned. What at the surface appears to be widespread successful integration is thus to some extent a manifestation of survivorship bias.

Contrary to common belief, economic disparities were also not quick to disappear. Using an impressive three-generation dataset linking immigrant grandfathers in 1880 to grandsons in 1940, Ward (2020) shows that European ethnic disparities in occupational income were strongly persistent. Ward’s study is the first to use actual linked grandparent-grandson data, and it confirms once again that persistence is stronger when high-quality data is used. As the author notes: “This result may be surprising since it is commonly believed that ethnic differentials faded quickly in the early twentieth century as in the melting pot metaphor”.

The assimilation story also ignores the fact that European immigrants did not simply arrive and passively absorb the local culture. They brought their own values and characteristics, and changed the local culture in their own image.

We all intuitively understand this when it comes to the things we are fond of — like ethnic cuisines. When Italians came to America, not only did they continue to make and eat Italian food, other Americans also began to eat more Italian food. But of course, it doesn’t just stop at food. Many studies have shown that cultural behaviors and beliefs persist across generations to some extent (e.g., Richwine, 2023; Giuliano & Tabellini, 2020).

The European differences have had long-term economic consequences which can be observed to this day. In an analysis of county ancestry compositions since 1850, Fulford et al. (2018) show that counties that received higher shares of people from more well-off countries tend to have higher GDP per-capita. Berger & Engzell (2019) have also shown that county variation in economic inequality and mobility mirrors that of their European origins.

Internal migration within the country also supports the notion that the arrival of new people can influence local customs and institutions. The Great Northward Migration of African Americans in the 20th century provides ample examples. But even the migration of Southern whites had an impact on their destinations, specifically by making them more politically conservative (Bazzi et al., 2023a; Bazzi et al., 2023b).

The successful immigrant

It is well known that many immigrant groups perform well economically and academically in America. Indian and East Asian immigrants, for example, significantly outperform whites in America. As a result, they are often used as examples of successful assimilation.

This argument misses the fact that those immigrants are also exceptionally positively selected on economically relevant traits. Their relative success is chiefly explained by pre-arrival characteristics (such as human capital and other correlated traits), not by traits acquired through assimilation after arrival (see, e.g., Duncan & Trejo, 2015; Hanson & Liu, 2021). Assimilation is thus given undue credit for their success.

If assimilation was really operating at great force, you’d expect rapid convergence to the average from both directions. Just as underperforming groups should improve over time, overperforming groups would expect to decline over time. But nothing indicates that Asian Americans will lose their academic and economic edge in the near future. From a normative perspective, one can certainly argue that lack of assimilation is good in this scenario. Descriptively, however, it’s yet another instance of persistence rather than assimilation.

Mexican immigrants

There is one major immigrant group that is not highly positively selected in modern times, namely Mexican immigrants (and Latin American immigrants more broadly). Consequently, their economic and academic performance is well below that of American whites.

It’s a common trope to compare Mexican immigrants with European immigrants, usually the Irish or Italians. The idea is that, while the Irish or Italian used to be dysfunctional or have poor economic performance, they rapidly assimilated and integrated into American society. A similar path should be expected for Mexicans, the proponents suggest.

This analogy does not withstand much serious quantitative scrutiny. As I’ve already shown, European economic disparities did not rapidly vanish. To expect the same of Hispanics would be to expect them not to rapidly assimilate economically.

With respect to crime, the argument also doesn’t hold up. In one study, Moehling & Piehl (2009) compare the crime rates for various immigrant groups in 1904. They show that, while the Irish were overrepresented in crime, this was entirely due to minor offenses (e.g., drunkenness). In major offenses, there was no meaningful Irish overrepresentation. Italians were overrepresented in major offenses, but this overrepresentation was very modest when compared to the distinct outlier of Mexican immigrants (overrepresentation of 67% versus 432%; see Table 7). This is also consistent with more recent findings. Abramitzky et al. (2024) demonstrate that Mexicans were a clear outlier in incarceration rates among early immigrant groups in 1900-1930, and that differences between European groups were relatively small.

Looking at their data below, it becomes clear that the modern tale of massive Irish or Italian assimilation is an exaggerated fiction. European differences were real but comparatively modest—it’s not an example of great assimilation success if the differences are already slight in the first place. The disparity between non-Hispanic whites and Hispanics, whether then or today, is much more substantial than this.4

But we should still address the pertinent question: do Mexicans rapidly assimilate to their non-Hispanic white counterparts?

It can be challenging to study Mexican immigration. Many are undocumented, and repeated waves of new immigrants make it difficult to distinguish between assimilation and compositional changes.

In terms of crime, it is well known that second-generation Hispanic immigrants have somewhat higher crime rates than first-generation immigrants. In that sense, they move further away from non-Hispanic whites. But how about socioeconomic outcomes?

Chetty et al. (2018) are optimistic about Hispanic economic mobility. They predict that: “Hispanic Americans are moving up significantly in the income distribution across generations. For example, a model of intergenerational mobility analogous to Becker and Tomes (1979) predicts that the gap will shrink from the 22 percentile difference between Hispanic and white parents observed in our sample to 10 percentiles for their children and to 6 percentiles in steady state.”

But can we trust this long-run extrapolation? Extrapolations are only as valid as the model used and its associated assumptions. After all, the steady state result is not something observed in the outcomes of real, breathing human beings.

One of the most important lessons of the social persistence literature is that multi-generational persistence is much greater than what you’d predict based on correlations between parents and children, such as Becker-Tomes style models (Braun & Stuhler, 2018; Colagrossi et al., 2019; Adermon et al., 2021; Engzell et al., 2020). More precisely, the Becker-Tomes model predicts faster-than-geometric decay over generations, whereas the true (empirically observed) decay is slower-than-geometric (Solon, 2018).5

It would be a great mistake to optimistically extrapolate from the jump between first- and second-generation to subsequent generations. Common sense tells us that the first jump is distinct from all subsequent generations, as only first-generation immigrants are born outside the country.6

Evidence also supports that there is stalling beyond the first generation for Latino immigrants. Villarreal & Tamborini (2023) show that incomes for second-generation Latino immigrants grow less with age, stalling relative to other immigrant group descendants.7 Moreover, third-generation Hispanic men seemingly perform worse than second-generation Hispanic men.

The findings of a unique and impressive study are particularly illuminating regarding historical Hispanic assimilation. In their 2008 book, Telles & Ortiz were able to follow Mexican immigrants and their actual descendants across several generations.8 Contrary to the authors’ expectations, there was little support for the assimilation narrative: “the educational progress of Mexican Americans does not improve over the generations […] our data show no improvement in education over the generation-since-immigration and in some cases even suggest a decline.” Concerning college completion rates, they write: “Sadly and directly in contradistinction to assimilation theory, the fourth generation differs the most from whites.”

All taken together, there is little reason to expect that economic, educational and criminal disparities between (non-Hispanic) whites and Hispanics will vanish any time soon. Yes, there is a significant boost from first- to second generation, but any subsequent generational improvements are much more modest and less systematic.

The elephant in the room

Setting aside European, Asian and Hispanic immigrants, there is a much more glaring problem with the American assimilation narrative—the persistent American underclasses, black Americans and Native Americans.

It is well known that large black-white disparities exist in important domains such as economics, education and crime. And, as the example of household income below shows, there has been little convergence with respect to some important measures for over 70 years.9

Yet people rarely appreciate this relevant fact in discussions of assimilation. Some will surely object to this argument on account of African Americans not being immigrant groups. But this makes the example even more striking, not less. Having lived together so long — and yet so large disparities persist — clearly speaks to America’s inability to integrate and assimilate disparate groups of people.

Conclusion

Evidence shows that social status persists strongly across multiple generations. Much the same applies in the context of immigration assimilation: multi-generational persistence is the norm, with only very gradual convergence that can easily take well over 100 years. The better the data and measurements, the stronger the persistence. While there is often a significant jump from first- to second-generation, subsequent generations do not experience large systematic boosts.

Contrary to popular opinion, the Age of Mass Migration was not a period of widespread and rapid assimilation. Large shares of European immigrants left the country because they couldn’t successfully integrate, and, among those who stayed, disparities were not quick to fade. Immigrants also played an important role in shaping America’s trajectory, rather than merely passively assimilating.

Today, many immigrant groups are highly successful in the United States — much more successful than US-born natives. This, however, is not explained by some assimilation process after arrival, but by their pre-arrival characteristics due to national origin and extreme selectivity.

The prediction that Mexican immigrants will rapidly assimilate to their non-Hispanic white peers is also unlikely to come to fruition. The analogy between Mexican immigrants and former European immigrants is flawed for two reasons. First, it exaggerates how different European ethnic groups were in the past. In reality, their differences were comparatively modest. Second, it exaggerates how quickly European immigrant groups assimilated. These two exaggerations together create a false image of massive and rapid convergence which simply isn’t historically accurate. Historical multi-generational data on Mexican immigrants and their descendants is also consistent with persistence rather than rapid assimilation.

Lastly, the elephant in the room is that of the persistent underclasses in America, namely African Americans and Native Americans. Despite living in the same country for many generations, large disparities persist between black and white Americans. These disparities will also not disappear any time soon.

While the melting pot metaphor sounds appealing, it is largely a fiction. America is not a place where widespread and rapid assimilation occurs. This should not be taken as a slight, however, for I am not claiming this is particular to America.

The second part addresses exactly this. Using data from a variety of countries, I will argue that lack of quick and widespread assimilation is the norm, not the exception.

Introduction

It is well known that fertility rates are falling across the globe, and that most of the developed world has long had fertility below the replacement rate. Less known is the fact that much of the Western world had already reached sub-replacement fertility before the Second World War, only to be interrupted by the “baby boom.” This dramatic decline in fertility sparked much public discussion, concern and even led to pro-natalist legislation.

This post is about the contemporary reaction to declining and even sub-replacement fertility prior to World War 2. First I will document the relevant fertility statistics. Afterwards, we will see what contemporary demographers had to say about it and compare with modern explanations.

Trends in European effective fertility

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Introduction

Jigsaw puzzles have a long history. In the 1760s, the cartographer John Spilsbury sold “dissected maps,” which some credit as being the earliest jigsaw puzzles. But apparently there has been a recent surge in jigsaw popularity. Some YouTube creators have amassed hundreds of thousands of subscribers, and the World Jigsaw Puzzle Championship was held for the first time in 2019.

Upon observing this little community, you’ll immediately notice one thing: there are lots of women. The content creators, their followers, the contestants in the tournaments, are all overwhelmingly women.

There is nothing wrong with this, but it does invite obvious questions. Are women just more interested in jigsaw puzzles? Are they better at solving them?

These questions are relevant to a broader literature about sex differences in cognitive abilities. It is well known, for example, that men perform better on 3d mental rotation tasks on average. One might intuitively have expected the same to be true for jigsaw puzzles.1

There is very little research on the topic of sex differences in jigsaw puzzle solving ability. McGuinness & Morley (1991) found no significant sex difference among small kids. More recently, Kocijan et al. (2017) and Ramirez et al. (2024) found that females performed better among children and young adults, respectively.

This is not a large collection of studies — and the samples are small (recent studies, n = 46 and n = 79, respectively). With this little data, any confident conclusions would be premature.

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Enter... The World Jigsaw Puzzle Championship!

The World Jigsaw Puzzle Championship (WJPC) has been held annually since 2019 (skipping 2020 and 2021 due to the epidemic). Participants compete to show who is the fastest at solving jigsaw puzzles.

The tournament is divided into three rounds: qualification rounds (6 groups), semifinals (2 groups) and finals (1 group). Based on their performance, a select number within each group can advance to the next round. The person with the quickest solution in the finalist round ultimately wins the championship.

Data analysis

I gathered the relevant data for participants in the most recent 2024 championship. Unfortunately, participant sex is not officially registered anywhere. Therefore, I manually went through the participant list and identified sex from their names. This process is no doubt prone to error, but I tried (and hopefully managed) to keep the number of coding errors low.

Results

As expected, participants in the WJPC 2024 were overwhelmingly female (79%). Did male competitors underperform? At first glance, it appears so! While the qualifying rounds were 21% male, just 11% of competitors who made it to the finals were male. This difference is highly statistically significant (p < 0.001).

The following figure shows men’s actual performance (blue) compared to expected performance (shaded band) under the null hypothesis of no sex difference. More precisely, it shows the fraction of male participants that reached a given placement in the overall tournament. I labelled it such that 0 to 1 refers to the qualification rounds, and those with rankings just below 1 are those who fell barely short of qualifying for the semifinals. Similarly, placements between 1 and 2 are the semifinals, and between 2 and 3 the finals.

In the qualifying rounds, male participants very clearly (and significantly) underperform relative to expectations. The immediate drop at the start suggests there is an excess of males among the absolute worst performers. This is even easier to see if we plot the frequency distributions of placements attained by sex.

I even noticed this while preparing the raw data. In most qualification groups there was a handful of men clustered near the bottom, but otherwise men’s rankings looked roughly evenly distributed. It seemed odd and looked like a distinct phenomenon to me.

We should keep in mind that, despite it being named the “World Championship”, the WJPC is not just an event for the highly competitive. It is also a casual event for amateurs and other puzzle enthusiasts alike. The 1st place prize is also just €1,000. For all we know, there may be friends, families and/or couples who attend together, even if not all parties participate with serious competition in mind. It’s worth considering if a larger share of the men are casual participants.

There is no perfect way to distinguish between serious competitors and those who aren’t. Still, I settled on a simple criterion: let’s consider only participants who managed to finish their jigsaw in the qualification round in the allotted time of 90 minutes. This is not a very strict criterion — 69% of participants managed to do so.

What happens to our results if we exclude those who did not finish their puzzle in the qualification round? That’s shown below. In this reduced sample, men perform exactly as you’d expect.

One might object to this analysis in that it artificially removes the worst performers who are disproportionately male, thereby restricting the range in ability. This objection has some merit, but given that the event is partly casual and partly competitive, and the seemingly distinct phenomenon among worst performers, it seemed reasonable to do. Furthermore, 7 out of 10 participants finished their puzzle in time, so it’s not as restrictive as it might sound.

One possible (but speculative) interpretation is that men’s underperformance in the WJPC might be driven a small group of men who aren’t participating with serious competition in mind, who tend to be among the worst performers.

The best of the best

Men are overrepresented at the bottom. Are they therefore also underrepresented among the absolute top performers? Relative to the number of male participants, not really.

The WJPC has been held four times. In two (half) of the championships, the gold medal was won by a man. Out of twelve medals in total, four (a third) were won by men — precisely one each year. This exceeds the male share of participants.

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Discussion and conclusion

The overwhelming majority of participants in the World Jigsaw Puzzle Championship are women. The purpose of this post was to analyze whether male participants also perform worse than female participants.

Before addressing that, a short comment must be made. It’s possible that the reason there are fewer male participants is partly because they are worse at jigsaw puzzles. After all, talent inspires interest. If so, comparing male and female participants is slightly unfair towards the women, as the (fewer) male participants would be more selectively drawn from the underlying distribution in ability. Unfortunately, there is no way to account for this.

Do men underperform relative to their number of participants? On average, yes. I find that men are highly overrepresented among the absolute worst performers, and thus attain worse placements on average. However, I do find some suggestive (though admittedly speculative) evidence that this might be driven by a number of male amateur participants. How would the results look if male and female participants were equally determined competitors? We don’t know.

Interestingly, men are far from absent among the absolute top performers. Across the four championships held so far, two of four gold medals and four out of twelve medals have been won by men — greater than the male share of participants.2

So… are women better at speed solving jigsaw puzzles? I lean towards yes. However, rather than providing a clear and definite answer, my analysis has left me with a rather tentative conclusion instead.

What we can say is that, whatever advantage women have, it must be relatively modest. For comparison, consider a game like chess that is instead male-dominated. I think most would say that sex differences in chess ability — to the extent they exist — are relatively modest. Yet in the game’s significantly longer history, no woman has ever been world champion (or even gotten particularly close).3 Men are not equivalently absent from the top in these jigsaw championships.

This analysis obviously cannot say why a difference in jigsaw puzzle solving ability might exist — whether it’s biology, culture, or socialization. Stereotypes suggest that women are better than men at perceiving more subtle color distinctions.4 If this is true, that sounds like a plausible mechanism of a female advantage to me.

Ultimately, this analysis has probably raised more questions than it has resolved. I will leave it to the reader to speculate and form their own conclusions about the questions raised in here.

This is part of a series. See also:

• Immigration and crime: Denmark

• Immigration and crime: Sweden

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Introduction and overview

In previous posts, I looked at the topic of immigration and crime in Denmark and Sweden. Here I continue by reviewing the data from Norway. As we will see, things that apply in Sweden and Denmark generally do in Norway as well.

Over the years, there have been several studies about immigration and crime in Norway using register data, most of which is reviewed by Skardhamar (2017). Notable examples include Andersen et al. (2017); Skardhamar et al. (2014); Skardhamar et al. (2011); Evensen (2009); and Kardell & Carlsson (2009). More recently, Statistics Norway (SSB) have published relevant data in 2020 and 2024, though without any associated detailed reports or statistical analyses.

Based on recent and prior findings, a few conclusions follow. Overall, immigrants are just moderately overrepresented in crime in Norway. This assertion, however, masks significant heterogeneity. Western immigrants are underrepresented in criminality and non-Western immigrants are substantially overrepresented. This is not specific to the Norwegian context — there is considerable consistency in relative crime rates of immigrant groups across all the Nordic countries.

Why do crime rates differ between immigrant groups, or between immigrants and natives? The usual answers given tend to be population structure and socioeconomic status. Of these, immigrants’ lower age is an important contributor. But population structure and socioeconomic status together are insufficient to fully account for the disparities.

Content

• Prisoners in Norway

• Violent crime rates by national origin

• Offense type

• Causes of immigrant crime disparities

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Prisoners in Norway

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As for the upcoming year, I am currently working on new articles and have many other ideas for posts—so consider subscribing! If there are any topics in particular you’d like to see, you’re free to make suggestions in the comments. I’ll take them into consideration, but of course I can’t promise anything.

About half of my posts are open to everyone. Paid subscribers have access to otherwise restricted posts, future ones and those already written.

Finally, thanks for all the support and for tagging along.

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Crime

Many of my articles in 2024 have addressed various aspects about crime.

Perhaps most noteworthy is my investigation into the Nordic rehabilitation model. In a 2-part article, I challenge the widely held idea that the Nordic countries are so much better at rehabilitating criminals.

The myth of the Nordic rehabilitative paradise

In this first part, I look at the challenges with comparing recidivism rates across countries due to varying definitions. Equivalent defined, Nordic recidivism rates are more alike the United States than often thought. Furthermore, many factors contribute to varying recidivism rates, not just rehabilitation effectiveness.

Recidivism in Norway (paid subscribers)

In the second part, I find that low Norwegian recidivism rates predate the Norwegian rehabilitation reforms, so attributing low Norwegian recidivism rates to their rehabilitation model is highly dubious. I then discuss what other factors may contribute to low recidivism rates.

Following that, I also wrote a short post discussing the effectiveness of prisons and what the purpose of prisons is.

The case for prisons

In this short post, I argue that prisons reduce crime through incapacitation and deterrence, and that rehabilitation is not the main goal of prisons.

In a series of 3 articles, I discussed the relationship between poverty and violent crime.

Does poverty cause violent crime? (paid subscribers)

In this post, I argue that the causal effect of poverty/SES on violent crime is rather minor, contrary to common belief. There is a strong correlation between poverty and violence, but it’s mostly explained by selection and reverse causality.

Race, economics and homicide

Based on a county-analysis, I show that economic disparities cannot account for racial/ethnic disparities in homicide in the United States.

Race, economics and homicide in the United States: a summary

This is a short write-up, combining the points of the two previous articles. If economic disparities cannot statistically account for much of the racial disparities in homicide, and what is “accounted” isn’t causal, American racial homicide disparities must have little (causally) to do with economic disparities.

I have also written two posts comparing crime rates across countries. The first discusses all crime variation across the entire globe; the second discusses America’s high homicide rate compared to the rest of the developed world.

Global crime (paid subscribers)

In this post, I look at how crime rates vary across countries and, further, the quality and reliability of the evidence. Homicide evidence can most reliably be compared. Beyond that there is survey evidence. I also discuss why rape rates are exceptionally difficult to compare across countries.

Demographics and American Homicides (paid subscribers)

I look at whether America’s high homicide rate can be explained by racial/ethnic demographic differences between the United States and the rest of the West. I conclude that the majority of the disparity can be, but that the (non-Hispanic) white homicide rate in the United States remains substantially higher than similar developed countries.

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Immigration

In a series of articles, I have looked at immigration in the Nordic context. In one article, I wrote about how taboo the subject is in Sweden.

Sweden's immigration taboo

In this post, I discuss how the Swedish government has been unwilling to publish information about immigration in the country, and how Sweden systematically makes researching immigration more difficult.

In two other articles, I have looked at immigration and crime. I started with Denmark and Sweden, but I still plan to write about Finland and Norway, too.

Immigration and crime: Denmark (paid subscribers)

I use government data to analyze whether immigrants are overrepresented in crime in Denmark, and to what extent. I also briefly consider possible explanations for the disparities.

Immigration and crime: Sweden (paid subscribers)

Like the article on Denmark, I analyze patterns in immigrants’ participation in crime in Sweden. I also look at the role in organized crime, which has become a substantial problem in Sweden.

Lastly, in a short article I looked at how immigrants assimilate in Denmark in terms of school performance.

Immigrant assimilation in Denmark

Looking at school grade performance, I find that there is significant assimilation from first to second generation among immigrants, but there appears to be no measurable improvement from second to third generation.

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Nature, nurture and parenting

In 2024, I wrote two posts discussing the importance of nature and nurture for social outcomes. The first is dedicated to the role of nurture, the other to the role of nature.

Where parents make a difference

In this post, I argue that behavioral genetics results do not support the notion that parenting is unimpactful. Specifically, I argue that parenting nontrivially affects (socially important) outcomes like educational attainment, income and criminality.

Genes and social stratification (paid subscribers)

In this post I review a large body of evidence showing that genes influence socioeconomic mobility and standing. This has important implications for the distribution of genetic variants in society. In particular, socioeconomic classes differ systematically in traits such as intelligence, personality and mental disorder frequency partly for genetic reasons.

Related to parenting from a different angle, my most recent post discusses the relationship between age and fecundity — that is, the ability to become a parent in the first place.

Age and infertility

In this post, I discuss how age is related to fecundity and other birth outcomes, the relative effects of male and female age, and what it is about the female reproductive system that is affected by aging.

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History

I have a great interest in the history of science and technology, particularly the period spanning medieval Europe up to the Industrial Revolution. I have previously written about the West’s ascent between 1000 and 1500 AD. As a follow-up, I wrote about the period that preceded, roughly 500 to 1000 AD, popularly known as “The Dark Ages.”

The Dark Ages (paid subscribers)

In this post, I analyze quantitative evidence regarding the decline of the Roman Empire. Then I look at how “Dark” the Dark Ages were, in terms of written works, scholars, technology and more.

Historical Accounts of African Cannibalism

In this post, I document various historical accounts of cannibalism in Africa. These accounts come from varying sources including medieval Islamic scholars, early European explorers, later European anthropologists, and also modern accounts.

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Statistics and social science

Why controlling for variables is insufficient

In this post, I argue that “controlling” for confounding variables is rarely sufficient in observational social science, and explain the reasons why (omitted variables, measurement error, etc). I also discuss more broadly how to think about confounding, and some potential methods to address it.

The shape of the age-fecundity curve

Age is the greatest cause of infertility. But the exact shape of the age-relationship remains contested. It’s widely believed that fecundity begins to rapidly decline in the mid-30s, while any decline in the 20s is comparatively minor (e.g., Menken et al., 1986). This is roughly what I believed until I read an excellent paper by Geruso et al. (2023).

Part of the challenge is what to measure. We typically measure the fertility rate by age. That is, the number of children women have. But the true variable of interest is fecundity, i.e., the potential for childbearing, which is more difficult to measure. If 21-year-old women do not attempt to conceive at the same rate as 31-year-old women, the fertility rate of the latter group may be higher, even if fecundity of the former group is higher.

An additional issue of prior analyses is that months where women are already pregnant (or months of postpartum recovery) have inappropriately been included in the denominator to estimate age-specific fertility rates. The magnitude of this bias is not constant with age.

These issues produce an artificial relationship that is biased relative to the true age-fecundity relationship. In fact, Geruso et al. show that even if the true underlying relationship between age and fecundability is a linear decline, the methods that have been used would artificially produce this concave, accelerating shape.

With the aforementioned issues addressed, the authors produce a new set of estimates for the age-fecundity curve, using an impressive dataset totalling 2.8 million women from nationally representative data across 62 countries. Their central result is illustrated below.

As expected, there is a strong (non-monotonic) relationship between age and fecundity. According to their analysis, women’s fecundity peaks when they are aged 19-20 years old.

More importantly, fecundity declines roughly linearly between 20 and 40. Contrary to common belief, there is no sudden accelerating decline in the mid-30s. The drop in fecundity between age 20 and 25 appears no less meaningful than the one between age 35 and 40.

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Birth outcomes

Beyond the risk of miscarriage, other health outcomes are also affected by advanced maternal age.

One potential health issue that can arise are genetic disorders resulting from chromosomal abnormalities. While these often result in miscarriage, that is certainly not always the case. The relationship between maternal age and chromosomal abnormalities has been studied in Denmark by Frederiksen et al. (2024). They find that the frequencies of trisomy 21 (Down syndrome), trisomy 18 (Edwards syndrome) and trisomy 13 (Patau syndrome) all increase greatly with maternal age. Down syndrome, in particular, is surprisingly common when maternal age exceeds 40 years old.

Selection bias?

Maternal age is also a known risk factor for infant low birth weight or pre-term birth. In cross-sectional analyses, one generally observes a “U-shaped” association between maternal age and gestational age or birth weight. That is, increased risk of low birth weight in older mothers and in younger mothers.

However, when we deal with human subjects, we should always ask whether selection bias might affect our results. It is possible that older mothers differ from younger mothers in relevant characteristics other than age itself.

One advantage of the previously cited Danish study on chromosomal abnormalities is that, in Denmark, a large fraction of fetuses are cytogenetically examined, even in case of a late miscarriage or stillbirth. In many other studies, only live-born children are examined (Frederiksen et al., 2024). Thus, this study should at least be less contaminated by selection/survivorship bias.

But other research designs have also been used to help address selection bias, such as the sibling design. In this research design, they compare siblings born to the same parents, where parental age at childbearing (obviously) differs. By comparing siblings with the same parents, any parental factors that are stable from one birth to the next are automatically controlled for.

At least three sibling studies have been used to further interrogate the association between maternal age and the risk of low birth weight (Hvide et al., 2021; Sujan et al., 2016; Lawlor et al., 2011). Their findings suggest that the two sides of the U-shaped curve have different causes.

The relationship between advanced maternal age and low birth weight persists within families, consistent with a causal interpretation. In fact, if anything, these studies suggest that the cross-sectional associations slightly underestimate the effect of advanced age. This might be because those who have children later otherwise tend to be healthier for other reasons, partly masking the deleterious aging effect.

On the other hand, the studies suggest that the relationship between young maternal age and low birth weight is due to confounding. This would suggest that — contrary to advanced age — young maternal age itself is not an issue. Rather, women who become mothers at young ages are different in relevant respects other than just age.

Hvide et al. (2021) also use the sibling design to ascertain the effect of parental aging on other birth defects. The relationship persists within families, consistent with the hypothesis that age causally affects risk of birth defects.

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Men’s or women’s age?

So far I have focused exclusively on maternal age. But partner ages are strongly correlated. If the mother is old, the father typically also is.

It’s therefore worth investigating to what extent this is driven by the father’s age rather than the mother’s. Geruso et al. (2023) show that fecundity also declines with paternal age, albeit at a slower rate.

The fact that maternal age has greater influence is more clearly illustrated in the following graph. The decline in fecundity is much more steep for women.

Hvide et al. (2021) also show that the risks of congenital malformations and infant mortality are greater when the mother is older than the father, supporting the notion that maternal age has a greater influence.

Bad wombs or bad eggs?

What is it about the female reproductive system that is so affected by age?

If we compare people who have received donor eggs and those who use their own eggs, the answer becomes immediately clear. As shown by CDC (2020) and Johnson & Tough (2012), when women use donor eggs, their own age has very little effect. It’s not the age of the mother that matters per se — it’s the age of the eggs.

In fact, there are two major determinants of success in assisted reproductive technology: (1) age of retrieved eggs, and (2) number of eggs retrieved. As long as many eggs are retrieved, and they are not retrieved too late, the success rate is high — as illustrated below.

These findings suggest that women who freeze eggs can easily become pregnant later in life, as long as they freeze their eggs when they are sufficiently young.

Unfortunately, this latter point has not been sufficiently acknowledged, and poor results of fertility treatments can mostly be chalked up to women starting the process too late. If egg freezing is here to stay, it has to be communicated to women that they should do it when they are still young.

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Conclusion

We all know that infertility becomes more likely with advanced age. But it is widely believed that fecundity only marginally declines in a woman’s twenties, and then rapidly declines in her mid-thirties.

Yet a new study has shown that this appears to be a misconception. The shape of this age-fecundity curve is a statistical artifact, the result of various statistical issues related to how fecundity has been improperly estimated. In fact, the decline in fecundity that women experience in their twenties is as real as the decline in their thirties.

It is not only women’s age that matters. Though the effect is smaller, paternal age also affects the likelihood of a successful pregnancy. And, in regards to the importance of maternal age, we can be more specific about what really matters: mainly the age of the eggs. Evidence shows that, if a woman uses donor eggs, her own age plays little role in determining the success rate.

As the practice of egg freezing becomes more common, it is important that women have an understanding of the relevant facts. There is a high chance of success, provided that she doesn’t wait too long to do it.

For any woman who considers a future with children, and isn’t attempting to become pregnant in the short term, it might be a good idea to undergo the process sooner rather than later.

Introduction

Personal safety is a major determinant of quality of life. But where in the world are people most at risk for criminal victimization? And where are people safest?

Though simple to ask, these questions are more difficult to answer than one might think. Countries differ in the data they report and track, and, more importantly, in the reliability of their data.

Still, this article attempts to answer these central questions. In doing so, we will also discuss the various sources of evidence that are used, and how reliable they are.

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