Tag Archives: science

The curve of binding energy

9.86 – 9.33 billion years ago

Determinism

13.0 – 12.4 billion years ago

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
Pierre Simon Laplace 1814

Laplace gives a classic statement of determinism, which he supposed to follow from classical mechanics. And Einstein thought that relativity implied a “block universe,” with no distinction between past, present, and future. This is from a letter he wrote to the family of Michele Besso, collaborator and close friend, after Besso’s death:

Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion.

If this is true, then everything that happened in the year 2023, down to the finest details, and everything that’s going to happen in 2024, was destined to happen from the earliest moments of the Big Bang.

But it may be that this idea is more a matter of metaphysics – more precisely, of a particular mathematical framework – than a necessary implication of physical theories. This is the argument that the Swiss physicist Nicolas Gisin has been making lately, drawing on intuitionist mathematics. In contrast to the mainstream tradition in math, intuitionists argue that numbers are real only insofar as they can be constructed. The irrational numbers pi and e can be constructed, and so it makes sense to ask “Is the googlth digit of this number 7?” But for “most” irrational numbers, it is neither true nor false that their googlth digit is 7 until the number has actually been calculated. A popular summary of Gisin’s work is here.

Some possible implications:

Robert May, a physicist turned population biologist, was a pioneer in the study of chaos. In a classic article, he considered the potentially chaotic behavior of a discrete logistic equation. Such an equation might govern the behavior of an organism subject to density dependent mortality which reproduces at discrete intervals, say once a year. At low intrinsic rates of reproduction, say r=2, the equation is well behaved.

But at high rates of reproduction, say r=4, it goes through crazy looking booms and busts. These are chaotic: if you know the starting point up to n digits, you can only predict the precise behavior of the system up to some multiple of n time units. After that the only prediction you can make is probabilistic.

If Gisin is right, then the behavior of the system, contra Laplace, is not determined from the beginning.

I took Robert May’s course in population biology as an undergraduate at Princeton University long ago, and learned about chaos before most people. May was, among many other things, a really inspiring lecturer.

Gradualism

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1824 – 1836

Charles Lyell’s great work, Principles of Geology, came out between 1831 and 1833. Lyell advocated an uncompromising uniformitarianism: the same geological forces at work today, causing small changes over the course of lifetimes, were at work in the past, causing massive changes over the course of geological ages. We’ve seen over the course of this blog that uniformitarianism is not a completely reliable guide either to geology or to human history, which have been punctuated often enough by catastrophes – asteroid strikes, continent-scale floods, volcanic eruptions, and devastating wars and plagues. But the theory is nonetheless at least part of the story of history, and Lyell’s work was deservedly influential.

In 1837 Charles Darwin, a careful reader of Lyell, published a short article entitled On the Formation of Mould. This would eventually led to his last book, The Formation of Vegetable Mould through the Action of Worms. Darwin’s work on soil formation was Lyellianism in miniature. He demonstrated, through a combination of careful reasoning and experiment, that the surface layer of pasture soil is formed by earthworms. “Although the conclusion may appear at first startling, it will be difficult to deny the probability that every particle of earth forming the bed from which the turf in old pasturelands springs, has passed through the intestines of worms.” Reading Darwin on worms you get the feeling he identifies with his humble subjects, gradually remaking the world through their patient industry.

The doctrine of progress through gradual change was appealing for more than just scientific reasons. In the 1830s, English liberals (of whom Darwin was one) were attempting to reform their society gradually, without the violence of the French Revolution, and without turning over politics to a Great Man in the style of Napoleon. (Darwin was also a gradualist with regard to his own work: he came up with the theory of natural selection in 1838, but England at the time wasn’t ready for anything so radical, and he didn’t publish On The Origin of Species for another twenty years.)

George Eliot (Mary Ann Evans), a friend of Darwin’s, set her greatest novel, Middlemarch, around the time of the Reform Act of 1832, which moved England one big step closer to a genuinely representative government. The novel’s heroine, Dorothea Brooke, might in another age have been a famous saint, another Theresa of Avila. In the England of her time she has another fate. Here is the famous conclusion of the novel, a paean to gradualism and the cumulative force of small deeds:

Her full nature … spent itself in channels which had no great name on the earth. But the effect of her being on those around her was incalculably diffusive: for the growing good of the world is partly dependent on unhistoric acts; and that things are not so ill with you and me as they might have been is half owing to the number who lived faithfully a hidden life, and rest in unvisited tombs.

Steam engine time

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The steam engine was a child of seventeenth century science; the Scientific Revolution gave birth to the Industrial Revolution. That’s not at all the conventional story, but David Wootton’s recent book The Invention of Science: A New History of the Scientific Revolution makes the case.

According to the conventional story, the steam engine resulted from the work of generations of inspired tinkerers, ingenious craftsmen with no particular scientific training and no great scientific knowledge. Indeed, according to one historian, “Science owes more to the stream engines than the steam engine owes to science.” (After all, the steam engine did inspire Carnot’s thermodynamic theory.)

But Wooton traces a path from scientific theory to practical application, beginning with the seventeenth century science of vacuum, air and steam pressure. The pioneering scientists here were not just theorists. They built (or at least designed) a number of devices for making use of differences in gas pressures, including an air gun (Boyle), a steam pressure pump (della Porta), and a vacuum-powered piston (von Guericke). Huygens took up the last idea, using an explosion to empty air from a cylinder, through a valve, and then using the partial vacuum to move a piston. This in turn was taken up by Denis Papin, a French Protestant medical doctor and mathematician, who worked as an assistant to Huygens, and then to Boyle. Papin combined scientific knowledge and engineering experience to design several steam engines. None of these was very practical – sadly Papin ended his life in failure and poverty. But the first of them was very similar to the first commercially viable steam engine, produced by Newcomen in 1712 – so similar that many historians have been convinced that Newcomen must have been familiar with Papin’s design.

Up to recently there’s been no convincing account of how Newcomen could have learned of Papin. But now Wooton has discovered the likeliest link, a book by Papin with the unpromising title A Continuation of the New Digester of Bones. The book has been neglected by historians, not surprisingly, but sold well in its own day. It gives plans for a pressure cooker (hence the title). But it also contains detailed descriptions both of vacuum powered piston, and of the use of steam condensation to produce a vacuum: just what Newcomen needed to put together to build his first engine. Wooton writes:

Newcomen’s steam engine is a bit like a locked-room plot in a detective story. Here is a dead body in a locked room: How did the murderer get in and out, and what did he use as a weapon? … We cannot exclude the possibility that Newcomen went to London and met Papin in 1687 … But we do not need to imagine such a meeting. With a copy of the Continuation in his hands, Newcomen would have known almost everything that Papin knew about how to harness atmospheric pressure to build an engine. … From this unintended encounter, I believe, the steam engine was born.

He concludes:

Historians have long debated the extent to which science contributed to the Industrial Revolution. The answer is: far more than they have been prepared to acknowledge. Papin had worked with two of the greatest scientists of the day, Huygens and Boyle. He was a Fellow of the Royal Society and a professor of mathematics. … Newcomen picked up … where Papin began. In doing so he inherited some of the most advanced theories and some of the most sophisticated technology produced in the seventeenth century. … First came the science, then came the technology.

Brave New World

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Columbus’s discoveries overthrew the Medieval conception of Earth’s place in the cosmos. No, he did not discover that the Earth was round. Educated Greeks had known that two millennia earlier. But he also did more than just discover new lands.

The standard, educated medieval view of the cosmos was a synthesis of Aristotle and Christian theology. The universe consisted of larger and larger spheres of more and more rarefied elements: a sphere of earth, a sphere of water, a sphere of air, a sphere of fire (the sublunary sphere, home of meteors), and successive quintessential spheres for the planets, the fixed stars, and heaven beyond. The first two spheres were not concentric, obviously – otherwise the earthly sphere would have been entirely underwater. Instead, Providence had set the earthly sphere sufficiently off-center that some of it – including the whole inhabited world – stuck above the water (Genesis 1:9-10).

Here’s a representation of the old view, still surviving just after Columbus (from David Wooton’s fine recent book The Invention of Science: A New History of the Scientific Revolution). At the very center of the chart, inside the wavy lines representing the sphere of water, is a funny shape: a T-and-O map of the inhabited world. The East, and Asia, are the white area above the horizontal crossbar of the T. The vertical bar of the T is the Mediterranean, with two further horizontal black lines separating Iberian, Italian and Balkan peninsulas to the North (left). Africa is South (right) of the Mediterranean. Not shown on this map, at the very crux of the T, is the holy city of Jerusalem, site of the Crucifixion and Resurrection. T-and-O maps aren’t much use for navigation, but they were popular for a long time because they showed a Higher Truth about the divine order of the Cosmos.

It’s hard to square this conception of the universe with the discovery of a whole New World sticking up on the opposite side of the watery sphere. Columbus tried out various theories. At first he imagined that he had reached the (East) Indies. Later, he started thinking that the earthly “sphere” might have a pear-shaped extension sticking up out of its far side (like a woman’s breast, he put it), rather than being strictly spherical, so you could reach the site of earthly Paradise (the nipple) by sailing up the Orinoco.

The generation following Columbus, beginning with Amerigo Vespucci, abandoned the nested spheres idea, at least as far as earth and water were concerned. When Medieval writers wrote about “the Earth” they generally meant just the earthly sphere, minus the sphere of Ocean. After Columbus, “the Earth” would come to refer to the whole terraqueous globe.

The Waldseeemüller map (1507) is one of the first to show the Old World and the New. Copernicus almost certainly saw a copy of the map. It spurred him to imagine that the Earthly globe – land and water – could revolve around its own axis, and – even more radically – might revolve around the sun.

A cycle of Cathay

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1108 – 1158 CE

The innovations which make their appearance in East Asia round about the year 1000 … form such a coherent and extensive whole that we have to yield to the evidence: at this period, the Chinese world experienced a real transformation. … The analogies [with the European Renaissance] are numerous – the return to the classical tradition, the diffusion of knowledge, the upsurge of science and technology (printing, explosives, advance in seafaring techniques, the clock with escapement …), a new philosophy, and a new view of the world. … There is not a single sector of political, social or economic life in the eleventh to thirteenth centuries which does not show evidence of radical changes in comparison with earlier ages. It is not simply a matter of a change of scale (increase in population, general expansion of production, development of internal and external trade) but of a change of character. Political habits, society, the relations between town and country, and economic patterns are quite different from what they had been. … A new world had been born.

Jacques Gernet. A History of Chinese Civilization, pp. 298-300

Scholars contemplating the sweeping economic, social, and political transformation of China under the Song dynasty (960-1279) seem compelled to draw analogies with later dramatic occurrences in Europe – with the Renaissance (as in the quote above) or with the Economic Revolution in England on the eve of the Industrial Revolution.

The changes are dramatic. Population roughly doubles, from about 50 million to about 100 million. Cities grow. Both internal and external trade boom. The division of labor advances, with different households and different parts of the country specializing in “goods such as rice, wheat, lighting oil, candles, dyes, oranges, litchi nuts, vegetables, sugar and sugarcane, lumber, cattle, fish, sheep, paper, lacquer, textiles and iron.” In a number of fields of technology – iron production, shipbuilding – China reaches heights which the West will not attain for many centuries.

With changes in the economy come changes in the relation between society and state. Taxes come to be mostly collected in cash rather than kind. Eventually revenues from taxes on commerce, including excise taxes and state monopolies, will greatly exceed those from land tax. A Council of State will put constitutional checks on the power of the emperor.

Yet Imperial China will ultimately follow a different, less dramatic developmental pathway than Europe. Some reasons why:

Missing Greek science and math. The Greeks figured out the shape of the Earth (it’s a sphere) by the fourth century BCE, and Eratosthenes produced a fairly good estimate of its circumference in the third century BCE. The news spread: educated Muslims and Christians in the Middle Ages knew the earth was round. Remarkably, however, China didn’t get the message, or didn’t pay it much attention. The standard cosmological model in China was a round heavens above a flat, square Earth, until Jesuits in the seventeenth century convinced the literati otherwise. And, while China had a sophisticated mathematical tradition (including an ingenious method of solving systems of linear equations with rods on a counting board, equivalent to Gaussian elimination), the massive mathematical legacy of the Greeks didn’t get that far. In his recent history of Greek mathematics, Reviel Netz argues that this alone is enough to explain the “Needham question” of why China did not produce a scientific revolution.

Church, state, and kin. Europe and China arrived at very different bargains between an imported ascetic otherworldly religious tradition, an imperial state, and patrilineal kin groups.

“Without the towering synthesis of the Principia there would have been no Newtonianism to define the eighteenth and nineteenth centuries, arguably no Enlightenment, and a very different trajectory to modern history. But, working backward, without Galileo and Kepler, there would have been no Principia, and … both Kepler and Galileo would have been strictly impossible without conic sections. … Kepler and Galileo, and their entire generation turned to conic sections because they had Archimedes. … Conic sections … emerged exactly once in history – as the parting shot of the generation of Archytas and as the central theme of the generation of Archimedes. Take away these two generations and you take away the tools with which to make a Newton. … Europe, rather than China or India, produced the scientific revolution because, unlike the other major civilizations, Europe had the resources of Greek mathematics”
A New History of Greek Mathematics pp. 497 et. seq.

The nomad brake. By 1000, Western Europe has largely tamed its barbarians, folding them into a settled, stratified, Christian society. But the civilized folk bordering the Eurasian steppe, in Eastern Europe and continental Asia, are in for a rougher ride. During the whole Song period, China faces a threat from nomads to the north. In the Northern Song period (960-1126), the Khitan empire, founded by steppe nomads, occupies Mongolia, Manchuria, and part of northern China. In the Southern Song period (1127-1279), the Song lose all of northern China to a new barbarian dynasty, the Jin. Finally, the Song dynasty ends when all of China is conquered by the Mongols under Genghis Khan and his heirs, with the loss of about a third of the population. For all the wealth and sophistication of the Song, the succeeding native Chinese dynasty, the Ming, does not regard them as a model to be emulated.

Rice economics. Rice is the main food crop in southern China, the most populous and developed part of the country. Here’s a basic fact about rice versus wheat production (hat-tip pseudoerasmus): diminishing marginal returns to labor are less pronounced with rice than wheat. In other words, with rice, you can produce a lot more if you’re willing to put in a lot more work. With wheat, you more quickly reach a point where additional labor yields little additional production. This simple fact has far-reaching implications. Imagine an economy with two sectors, agriculture and manufacturing. And imagine that population expands up to a Malthusian limit. Under these assumptions, and given standard economic reasoning, it makes a big difference whether the principal crop is rice or wheat. With rice (diminishing marginal returns less pronounced), equilibrium population density is greater, output per capita is less, and more of the labor force is in agriculture, less in manufacturing.

So an economic model incorporating information about labor productivity of rice and wheat seems to account for some basic differences between China and the West. But rice cultivation may have more subtle implications.

Rice psychology. An older generation of humanist scholars was willing to generalize about Chinese thinking.

It is quite clear to all those who have been in contact with this world that it is quite different from the one in which we ourselves have been moulded. … China does not know the transcendent truths, the idea of good in itself, the notion of property in the strict sense of the term. She does not like the exclusion of opposition, the idea of the absolute, the positive distinction of mind and matter; she prefers the notions of complementarity, or circulation, influx, action at a distance, of a model, and the idea of order as an organic totality. … Chinese thought does not proceed from an analysis of language. It is based on the handling of signs with opposing and complementary values.

Gernet p. 29

Within the social sciences, sweeping pronouncements like this are suspect. To hard-headed materialists and quants they look hopelessly impressionistic and unscientific. To post-colonialist critical theorists, they reek of old-fashioned, condescending Orientalism. But there is now a substantial body of research demonstrating real differences in cognitive style across cultures, and between the West and China (and other East Asian societies), in line with the quotation above.

Of note here: there is also regional variation within China. Rice paddy farming requires high levels of cooperation, including joint work keeping up irrigation systems, and reciprocal labor exchanges. And research shows that there are differences in psychology as well between wheat and rice growing regions in China. Chinese from rice growing regions are more inclined to holistic, context dependent thinking. Chinese from wheat growing regions have a more independent, individualizing cognitive style. In other words, the expansion of rice cultivation in China may have reinforced some of its characteristic cognitive inclinations.

In conclusion: the history of the Song period poses in particularly clear form the “Needham puzzle” of why the Industrial Revolution did not originate in China. The answer, it seems, is complicated, combining (at least) political and social responses to external threat, the nature of agricultural economies, and more intangible (but still measurable) differences in intellectual traditions and cognitive style.

The forgotten revolution

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252 – 127 BCE

103,049

Here’s where this number comes from: Take a sequence of symbols, (a b c d e f g h i j), say. Construct as many groups as you want by sticking parentheses around any two or more symbols or groups. For example, ((a b) c d (e (f g)) h i j). Or (a (b (c (d e f g) h) i) j). Or (a (b (c d e)) f g (h i j)). There are 103,049 ways of doing this with ten symbols, so 103,049 is the tenth Schröder number, named after the man who published this result in 1870. But it turns out that the same number is given in Plutarch – attributed to Hipparchus (190-120 BCE) – as the number of “affirmative compound propositions” that can be made from ten simple propositions. It is only in 1994 that somebody connected the dots, and realized that Schröder numbers had been discovered 2000 years before Schröder.

This is just one example of the very high level of mathematical, scientific, and technical accomplishment attained within the Hellenistic world — the world of Greek culture after Alexander. Lucio Russo calls Hellenistic science The Forgotten Revolution. A couple more examples from his book:

Everybody knows that Aristotle – and thus “the Greeks” – thought that heavy objects fall faster than light ones. Supposedly it took Galileo to prove him wrong. But in fact there is a clear statement in Lucretius (De Rerum Natura II:225-239) that objects of different weight fall at the same speed, unless air resistance kicks in; Russo argues that circumstantial evidence points to Hipparchus as the source.

Russo also argues that Hellenistic thinkers understood that gravity could account for the spherical shapes of the earth and planets, and that the balance between gravity and linear velocity could account for circular orbits. He shows that some strange passages in Vitruvius and Pliny about the sun making planets go around by shooting out triangular rays make sense if you assume the authors were looking at, but not understanding, vector diagrams of successive straight line motions bent into a circle by a centripetal pull.

Russo argues that scientific progress largely came to an end by 150 BCE, and the Roman period saw an actual decline in scientific understanding. Later writers like Ptolemy and Galen, often taken to represent the height of Classical learning, were derivative, and didn’t really understand their predecessors: a stark reminder that a civilization may avoid collapse, and even maintain a decent level of prosperity, but regress intellectually.

The way and the word, continued

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Continuing yesterday’s post: What accounts for the differences between classical Greek and early Chinese intellectual traditions? Below are a few things that might be involved; this is hardly a complete list.

Non-degenerate limit random variables

Here’s a nice little puzzle involving probability:

Take a bag with two marbles in it, one red and one green. Draw a marble at random. Put it back in the bag, and add another marble of the same color. Repeat: randomly draw one of the (now three) marbles in the bag, put it back, and again add a marble of the same color. Continue, adding a marble every time. What happens to the frequency of red marbles as the number of marbles in the bag goes to infinity?

Answer: When you carry out this procedure, the frequency approaches a limit. As the number of marbles grows larger, you sooner or later get, and stay, arbitrarily close to the limit. Now carry out the same infinite procedure a second time. This time you also approach a limit. But the limit this time is different! The first time, the limiting frequency might be .23748… . The second time it might be .93334… . If you keep on doing the infinite experiment a bunch of times, you’ll approach a different limit every time, with the various limits uniformly distributed over the interval [0,1]. These are non-degenerate limits. This is different from what you get when you flip a fair coin infinitely many times. The frequency of heads will always approach the same “degenerate” limit, .50000… .

A chance element like this is probably involved in the intellectual traditions of major civilizations. The first few great thinkers to come along have a massive influence on the direction of intellectual life, just as picking a red or green ball on the first round makes a big difference to the final limit. So Pythagoras’ and Plato’s obsessions with numbers and geometry as the keys to the universe have a disproportionate influence on later Western thought (allowing that the “Pythagoras” we know is encrusted with legends). Subsequent thinkers have progressively less and less influence, just as picking a green or red ball when there are already a hundred balls in the bag doesn’t make much difference in the ultimate limiting frequency.

Temperament

But there may be more systematic things going on. Daniel Freedman was a psychologist, white, married to a Chinese-American woman. While awaiting the birth of their first child, the couple found that relatives on the two sides of the family had very different ideas about how newborns behave. Freedman was sufficiently intrigued that he carried out an investigation of assorted newborns in a San Francisco hospital, including babies of Chinese and European origin.

It was almost immediately apparent that Chinese and Caucasian babies were indeed like two different breeds. Caucasian babies started to cry more easily, and once started they were more difficult to console. Chinese babies adapted to almost any position in which they were placed.. … In a similar maneuver … we briefly pressed the baby’s nose with a cloth, forcing him to breathe with his mouth. Most Caucasian and black babies fight this … by immediately turning away or swiping at the cloth. However … the average Chinese baby in our study … simply lay on his back, breathing from his mouth. … Chinese babies were … more amenable and adaptable to the machinations of the examiners. p. 146

This might seem like a minor curiosity, but it fits neatly with later work demonstrating East-West differences in adult cognitive styles. This raises the possibility that differences in temperament evident at a very early age might influence the evolution of intellectual traditions.

Coinage

Coined money apparently initially appeared in Lydia, in Asia Minor, around 600 BCE. It was quickly taken up by the Lydians’ Ionian Greek neighbors. And it is in Ionia too that we find the earliest philosophers. In Money and the Early Greek Mind, Richard Seaford argues that these developments are connected. The monetization of the Greek economy accustomed Greeks to the idea that a common impersonal material measure of value, relatively independent of individual control, underlay the multifarious goods and services produced by the polis economy. This led in turn to the pre-Socratic philosophers, who were obsessed with finding the one impersonal natural element – water, air, number – of which the whole heterogeneous variety of the natural world was made.

In Athens, the expansion of a monetary economy led to a curious insult – opsophagos, or fish-eater. What made this an insult is that fish were sold in the marketplace. They were a mere commodity, free of the ritual and taboos that surrounded the sacrifice and distribution of animal flesh. The fish-eater was a rich man indulging the pleasures of consumption free from the constraints of tradition and decorum. And his conspicuous consumption offended not only tradition but the spirit of democracy. Better that he spend his wealth on the public good.

In traditional China, by contrast, coins, and later paper money, would challenge but never break the hold of state patriarchy. And Spartans too recognized the subversive potential of money. Sparta used iron bars for money, precisely because they were inconvenient.

The way and the word

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523 – 384 BCE

Over the course of the mid to late first millennium BCE, Greeks and Chinese developed impressive intellectual traditions that would profoundly influence later civilization. These traditions differed a lot. In content:

The fundamental concepts at play in Greece and China were strikingly dissimilar. The Greeks focused on nature and on elements, concepts that seem familiar and obvious to those educated in modern science. They invented the concept of nature to serve distinct polemical purposes – to define their sphere of competence as new-style investigators and to underline the superiority of naturalistic views to … traditional beliefs. … Chinese investigators had a very different set of fundamental concerns, not nature and the elements, but the tao, ch’i, yin–yang, and the five phases. Where Greek inquirers strove to make a reputation for themselves as new-style Masters of Truth, most Chinese Possessors of the Way, had a very different program, namely to advise and guide rulers. … To that end they … redefined existing concepts … to produce a synthesis in which heaven, earth, society, and the human body all interacted to form a single resonant universe.

Geoffrey Lloyd and Nathan Sivin. The Way and the Word: Science and Medicine in Early China and Greece, p. 241

And in style of engagement:

Ancient Greek culture encouraged disagreement in natural philosophy and science as in every other field; the Chinese emphasized consensus. Success in debate was how you made your name in Greece, in a way that has no analogue in China.

Ibid. p. 247

What accounts for these differences? A few thoughts tomorrow.

The curve of binding energy

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9.86 – 9.33 billion years ago

Logarithmic History

The history of the universe — from the Big Bang to the end of the year — day by day

Tag Archives: science

The curve of binding energy

Determinism

Gradualism

Steam engine time

Brave New World

A cycle of Cathay

The forgotten revolution

The way and the word, continued

The way and the word

The curve of binding energy