Tag Archives: science

The forgotten revolution

257-132 BCE


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 thrust.

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.


Rama’s ape

12.3-11.7 million years ago

Ramapithecus (Rama’s ape) is no more. Another Hindu god has taken over the franchise; Ramapithecus is now subsumed under Sivapithecus, an earlier discovery, and is no longer a valid taxon name.

The story is interesting from a history-of-science point of view. Ramapithecus used to be presented as the very first ape on the human line, postdating the split between humans and great apes, maybe even a biped. This was given in textbooks not so long ago as established fact. Then geneticists (Sarich and Wilson) came along, and declared that the genetic divergence between chimps and humans is so low that the split had to be way later than Ramapithecus. There was a lot of fuss over this. Paleoanthropologists didn’t like geneticists telling them their job. Eventually, though, the paleoanthropologists found some new fossils. These showed in particular that the line of Ramapithecus‘s jaw was not arch-shaped, like a human’s, but more U-shaped, like a non-human ape’s. So after thinking it over a while, paleoanthropologists decided that Ramapithecus (now part of Sivapithecus) looked more like an orangutan relative: likely ancestor of a great radiation of orangutan kin that left just one surviving species in the present.

rama jaw

There are plenty of examples of experts in different fields coming up with different answers. For example, paleontologists didn’t like physicists telling them why dinosaurs went extinct. And we’ll see other examples in days to come: geneticists, physical anthropologists, and archeologists arguing over modern human origins. And very recently geneticists coming in on the side of old-fashioned historical linguists, and against recent generations of archeologists, in the matter of Indo-European origins.

It would be nice if there were a simple rule of thumb to decide who’s right in these cases. Maybe experts know what they’re talking about (except that experts were telling us recently that low fat diets were the key to losing weight and eggs would kill us with cholesterol). Or maybe harder science experts know better than softer science experts (except that physicists like Kelvin were telling geologists that the Sun couldn’t possibly have produced enough energy to support life on Earth for hundreds of millions of years – then along came Einstein and E=mc2). So the best we can do maybe is realize people, scientists included, are prone to overconfidence and group think – and not just those other people, either, but you and me.

Evolution and broken symmetries

9.3-8.8 Billion years ago.

No big news in the universe today. Some evolutionary thoughts: Species evolve. Do planets? stars? galaxies?

Charles Darwin didn’t use the word “evolution” often. He mostly wrote about “descent with modification.” But this is pretty much the same as what biologists mean by evolution. For example, the usual definition of genetic evolution is “change in gene frequency,” i.e. descent with (genetic) modification.

However, people sometimes talk about evolution that doesn’t involve descent with modification, in contexts that have nothing much to do with biological evolution – cosmic evolution or stellar evolution in the history of the universe, for example, or mineral evolution in the history of the earth. Another Victorian writer, the sociologist and philosopher Herbert Spencer, offered a definition of evolution that might cover these cases.

Evolution is an integration of matter and concomitant dissipation of motion; during which the matter passes from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity.

It’s easy to make fun of this definition. It’s the sort of abstract word pile that style manuals tell you to avoid, and that gives sociology a bad name. For that matter, it’s easy to make fun of Herbert Spencer. He may be some of the inspiration for the character of Mr. Casaubon, the dried up, impotent pedant in George Eliot’s “Middlemarch.” (Spencer probably turned down a chance to marry George Eliot = Mary Ann Evans.) But it may be that Spencer was groping toward the important modern concepts of symmetry and symmetry breaking.

A simple example: imagine you’re holding a bicycle exactly upright. The bicycle is pretty much bilaterally (mirror image) symmetrical. (OK, not really, the gears are on the right side, so it’s not a perfect mirror image. But just pretend …) Now let go of the bike. It will fall to one side or the other. The symmetry is broken, and you need one extra “bit” of information to tell you which side the bicycle is on.

Symmetry breaking is a fundamental concept in physics. In the very early history of the universe, the four forces of nature — gravitational, strong, weak, and electromagnetic – were united, but then as the universe cooled, one by one, these forces broke the symmetry and turned into separate forces. More symmetry breaking generated elementary particles, and nuclei, and atoms. When atoms first formed, they were distributed symmetrically through the universe as a diffuse gas. But gravitation pulled atoms and other particles together into clumps, leaving other parts of space emptier, and the spatial symmetry was broken. (A “translational” symmetry in this case.)

Symmetry breaking will keep showing up throughout the history of the universe. Consider sexual reproduction. A simple early form of sex involved two equal sized gametes (sex cells) joining to produce a new organism. Some species still do it this way. But more commonly the symmetry is broken – some organs or organisms produce little gametes that move around easily (sperm or pollen), others produce big gametes that don’t move around so easily (eggs or ovules). We call the first sort of organs or organisms males and the second sort females. Sex in most large organisms is a broken symmetry.

Or consider the rise of political stratification, the move from small-scale societies where “every man is a chief” to large-scale societies of chiefs and commoners, rulers and ruled. Another broken symmetry. It may be more or less an accident (good or bad luck, Game of Thrones style) who ends up being king, but it’s not an accident that somebody is, past a certain social scale.

We don’t attach much moral significance to broken symmetries where the physical world is concerned. You’re being way too emotional if you feel sorry for the poor weak nuclear force that missed its chance to be the strong nuclear force, or for the dwarf galaxies that got cruelly tossed around and cannibalized by the Milky Way. Broken symmetries in social life – males and females, kings and commoners – are another matter …


Some years are famous in their own right: 1848, 1492, 1066, 622 (= 1 Anno Hegira), 4004 BC. Will 2016 be another famous year? Interesting as this year’s political developments have been, I doubt they’ll make it famous in the long run. But there’s a science story from this year that might make a bigger impression on the future. Astronomers announced that they have discovered a planet close to Earth’s mass, orbiting in the habitable zone around Proxima Centauri b. This star is part of the Alpha Centauri multiple star system, the closest stars to Earth (visible to observers in the Southern hemisphere, but not in most of the Northern hemisphere).

Proxima Centauri b is a red dwarf, far less luminous than our sun. The new planet must be very different from Earth, possibly tidally locked. with one face permanently facing its sun. But it may be habitable, or at least terraformable. In the next few years and decades we will learn more about it. In the next centuries we might visit or even settle it. So 2016 might be remembered a thousand years from now as the year we found our first home outside the solar system.

Steam engine time

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 Wooton’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 – 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.

The last of the magicians


If I have seen further than others, it is by standing upon the shoulders of giants.

I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

These familiar quotations from Newton are sometimes presented as expressions of his humility (a quality not otherwise much in evidence). In fact, though, it became clear in the course of the twentieth century that they are really an expression of Newton being not just a great scientist, but a kook. Here’s a less familiar quotation from Newton about the intellectual origins of heliocentrism

It was the opinion of not a few, in the earliest ages of philosophy … that under the fixed stars the planets were carried about the sun; that the earth, as one of the planets, described an annual course about the sun, while by a diurnal motion it was in time revolved about its own axis. … This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato, … and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world, erected a round temple … and ordained perpetual fire to be kept in the middle of it.

The Egyptians were early observers of the heavens; and from them, probably, this philosophy was spread abroad among other nations … It was their way to deliver their mysteries, that is, their philosophy of things above the common way of thinking, under the veil of religious rites and hieroglyphic symbols.

The Renaissance was committed to recovering the ancient past. This led not only to the recovery of ancient art, literature and science, but also to the recovery of a whole body of ancient magic and pseudoscience. For example, many Renaissance thinkers, and Newton, were fascinated by the mystical writings of “Thrice Great Hermes,” an alleged Egyptian sage. By Newton’s time it had already been demonstrated that these writings were “pseudepigraphia,” a nice scholarly term for “fakes”; Hermes Trismegistus never existed. Nevertheless, Newton was convinced that there was an esoteric tradition preserved by the ancient Egyptians, passed on to Moses, Pythagoras, and Plato, and hidden away in the Bible. Read the Bible or stories of Pythagoras closely enough and you could recover the inverse square law of gravitation. Newton spent huge amounts of time throughout his life trying to recover further scientific secrets from the Bible.

John Maynard Keynes, who got hold of some of Newton’s papers wrote

In the eighteenth century and since, Newton came to be thought of as the first and greatest of the modern age of scientists, a rationalist, one who taught us to think on the lines of cold and untinctured reason.

I do not see him in this light. I do not think that any one who has pored over the contents of that box which he packed up when he finally left Cambridge in 1696 and which, though partly dispersed, have come down to us, can see him like that. Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago.

After Newton, it became clear that modern science had surpassed anything achieved by the ancient world, and natural science and the humanities went their separate ways. Reverence for the secret wisdom of the ancients would survive in the novels of Dan Brown and in the weird Masonic pyramid on the back of the US one dollar bill.dollar-bill-eye

Copernicus versus the scientific method


I have to confess that for a long time I didn’t really “get” Copernicus. That is to say, while I knew that Copernicus is right and Ptolemy is wrong, I wasn’t clear on just why Copernicus had a better scientific theory, partly because I didn’t bother understanding Ptolemy. So here’s a brief summary of the two. (Howard Margolis’s book helped me out.) There’s a larger point here: what makes Copernicus’s theory better doesn’t quite fit with a lot of pronouncements about “the Scientific Method.”

First, a diagram that illustrates Ptolemy’s model.


To account for the motion of the planets, Ptolemy needed to assume that the five planets (not counting the sun and moon) have both cycles (the big circles) and epicycles (the little circles). If you’re going to put Earth at the center of the system, then you have to have epicycles to account for the motions of the planets, like the retrograde motions where planets seem to go backwards.

Here’s something to note about this diagram: some of the cycles and epicycles vary independently, while others are exactly tied to the motions of the sun. For Mercury and Venus, the epicycles vary independently, taking different periods of time (88 days, 225 days) to complete a circuit. Their cycles, by contrast, take exactly one Earth year to complete a circuit. Furthermore, the deferent, the point at center of each epicycle, is always exactly in line with the sun. For Mars, Jupiter and Saturn on the other hand, the cycles vary independently (1.88, 11.86, and 29.46 years to make a complete circuit). But the epicycles take exactly one Earth year to complete a circuit. Furthermore, in each case the line from deferent to planet is exactly parallel to the line from Earth to Sun. Note that it’s hard to see any reason why the epicycle for Jupiter, say, couldn’t take 3.14 years to complete a circuit. But instead somehow, mysteriously, it’s connected with the sun’s motion around the earth.

(A large fraction of diagrams on the web purporting to illustrate Ptolemaic astronomy get this crucial point wrong! They show higgledy-piggledy non-parallel deferent/planet lines, pointing any which way. This makes it impossible to understand why Copernicus had a better theory. So I’m not the only person not to get Ptolemy.)

Copernicus’s model, by contrast, doesn’t just replace five circles (the cycles for Mercury and Venus, and the epicycles for Mars, Jupiter and Saturn) with one (for the Earth going around the Sun). It also automatically explains why the five superfluous cycles show an otherwise unexplained synchronic parallelism.


People who read Copernicus 1543 book carefully (not many at first) could see he had a real explanation for something that’s just a mysterious coincidence in Ptolemy. But contrary to what you may have heard, and what students get taught, about the Scientific Method, Copernicus did not formulate a hypothesis and then collect data to test his hypothesis, and show that it made the right predictions. Ptolemy and Copernicus make the same predictions about where the planets will appear in the sky. (Both are slightly off because they assume circular rather than elliptical orbits.) Eventually other scientists would gather data in support of Copernicus – Galileo’s observation of the phases of Venus was the real clincher – but the explanatory economy of his theory was a very strong reason for believing in it even before that.

Fortunately, there is a modern theory of how induction works – Solomonoff induction – that can explain why Copernicanism is a better theory. According Ray Solomonoff, induction has two parts. First there is Bayes’ Rule. Bayes’ Rule is an application of probability theory that tells you how you should revise probability estimates in the face of new evidence. Eliezer Yudkowsky gives one of the best introductions around to a counter-intuitive approach that has become enormously influential in recent years. (It’s fun to read too).

But Bayes’ Rule is only part of the story. The rule assumes that you have already assigned some prior probabilities to events before you look at the evidence. And where do scientists get their prior probabilities? Yudkowsky gives one answer: “There’s a small cluttered antique shop in a back alley of San Francisco’s Chinatown. Don’t ask about the bronze rat,” i.e. “Don’t ask.” Solomonoff offers a different answer. He argues that we can use the theory of algorithmic complexity, as developed by Kolmogorov, to assign prior probabilities. Roughly, if your theory were turned into a computer program, how long would the program be? The longer the program, the lower the prior probability. (Probabilities fall off exponentially with length of program, and are weighted to sum to one.) Suppose I give you a sequence of numbers corresponding to the first 1000 decimal digits of π. A computer program to calculate the first 1000 digits of π is going to be a lot shorter than just a list of the first 1000 digits, so the theory that I generated the list by calculating π is astronomically more likely than the theory that I generated the list at random. This is a formalization of Occam’s Rule, that simple explanations with fewer working parts are better.

So collecting evidence in support of a theory is part of good induction. But proposing more economical theories, accounting for more data with fewer working parts, is another part. Sometimes a new theory is so much better than the alternatives that we can assign it a much higher likelihood even before we collect more evidence. With Copernicus, explaining some striking coincidences which were otherwise unexplained, this was the first act in the modern Scientific Revolution.

This isn’t just a matter of historical interest. Physicists now, as they close in (hopefully) on a final theory of fundamental physics find themselves in a similar situation to Copernicus. Our best looking theory – some version of string theory – provides immense explanatory economy (or so we’re told), but just now is difficult or impossible to test. Here’s an article on the conundrums this poses for philosophy of science and the “scientific method.”