A demon-haunted theory

Here is a piece I’ve just published in the April issue of Physics World, in pre-edited form (sort of).

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James Clerk Maxwell originally devised his demon as a thought experiment to evade the second law of thermodynamics. But some of the physicist’s contemporaries actually believed it was an intelligent being that could bridge hidden worlds and provide a scientific route to immortality of the human soul.

Maxwell’s demon represents one of the great “thought experiments” of physics. Just like Einstein riding a light wave or Schrödinger’s cat facing quantum extermination, it poses a ‘what if’ question that illuminates a deep property of nature. The Scottish physicist James Clerk Maxwell proposed his little being as a way of picking a hole in the second law of thermodynamics by enabling heat to flow from cold to hot and resisting entropy’s disruptive influence. He had no idea that he was actually posing a riddle for the theory of information, which would lead ultimately to the recent demonstration that information and energy can be interconverted.

But the popularity that Maxwell’s demon has enjoyed thanks to the current burgeoning interest in the thermodynamics of information overlooks the way this little being was regarded by Maxwell and his contemporaries. Placed in its historical context, Maxwell’s demon played a rather different role – one that is surprising and in some respects shocking. For one thing, this wasn’t exactly a thought experiment at all. Some of Maxwell’s contemporaries saw in the demon a link between science and religion, a solution to the problem of free will, a bridge to hidden worlds, even a scientific route to immortality of the human soul. In some ways, Maxwell’s original demon seems more closely linked to ancient demonology than to the future of computing and information science.

Picking holes

Maxwell’s idea was a response to the gloomy prediction of a ‘cosmic heat death’ of the universe. In 1850 the German physicist Rudolph Clausius formulated the first and second laws of thermodynamics: the conservation of energy and the irreversibility of heat flow from hot to cold. A year later William Thomson (later Lord Kelvin) pointed out that the flow of heat involves ‘dissipation’ of mechanical energy: it flows into random motions of molecules and can never be recovered. This process, he said, must eventually create a universe of uniform temperature, from which no useful work can be extracted, and in which nothing really happens.

Maxwell realised that this inexorable slide into an inert state challenged human free will. If, as the second law says, there is only one way for things to happen, we would seem to be locked into rigid determinism, with human freedom just an illusion. As a devout Christian, he could not accept that God would arrange things this way. But how could free will be rescued without violating thermodynamics?

Maxwell’s seminal work on the microscopic theory of gases gave him an escape clause. He was convinced that the second law is simply statistical. Gases contain molecules with a bell-shaped statistical distribution of speeds, the faster ones being in a sense ‘hotter’. Temperature gradients get dissipated because it is far more likely that the faster molecules will mingle with the slower, rather than by chance congregating into a ‘hot’ patch. There’s nothing in the laws of mechanics to forbid the latter; it’s just very unlikely.

But what if we could arrange for that to happen? Then the second law would be undone. We can’t manage it in practice, Maxwell recognized, because we can’t possibly find out about the velocities of all the individual molecules. But what if there were, as Maxwell put it, a “finite being”, small enough to ‘see’ each molecule and able to keep track of it, who could open and shut a trapdoor in a wall dividing a gas-filled vessel? This being could let through fast-moving molecules in one direction so as to congregate the heat in one compartment, separating hot from cold and creating a temperature gradient that could be tapped to do work.

Maxwell laid out this idea in December 1867 in response to a letter from his friend, the physicist Peter Guthrie Tait, who was drafting a book on the history of thermodynamics. Maxwell told Tait that his aim was explicitly to “pick a hole” in the second law – to show that it was “only a statistical certainty”. The thought experiment offered a loophole that might rescue free will.

Exorcising the demon

It took over a century for the problem with Maxwell’s demon to be identified. In 1929 the Hungarian physicist Leo Szilárd believed he saw a flaw: to measure the speed of molecules the demon would have to expend energy, which would dissipate enough heat – produce enough entropy – to compensate for the demon’s manipulations. But in 1961 the German-American physicist Rolf Landauer, drawing on the relationship between information processing and thermodynamics developed by Claude Shannon in the 1940s, pointed out that measurements can in principle be conducted without increasing entropy.

That may be done, however, only by retaining all the information that the demon acquires. But, said Landauer, if he is a ‘finite being’ with a finite memory, this accumulation of data can’t go on forever: eventually some information will have to be erased to make room for more. And Landauer showed that while measurement can be free of an entropic cost, erasing data can’t be. Resetting a binary digit (from 1 to 0, say) must inevitably dissipate energy of at least kTln2, where k is Boltzmann’s constant. So in effect the demon generates entropy by forgetting. Charles Bennett of IBM’s research centre in Yorktown Heights later showed that this act of ‘forgetting’ is unavoidable, since it is equivalent to resetting the measuring equipment ready for the next measurement.

Landauer’s discovery has profound implications for the theory of computation. The digital circuits in today’s computers – which are inevitably reset from one calculation to the next – will always dissipate a certain minimum amount of heat during processing, although at present they still create far more heat than this lower limit because of other sources of dissipation. The existence of this unavoidable heat output in computing was proved experimentally last year by a team at the University of Augsburg in Germany, who were able to measure the amount of energy dissipated when a microscopic silica bead was moved between two optical traps to encode a binary digit. They found that as a cycle of switching and resetting the bead’s position was made ever slower, the amount of energy dissipated fell to a minimum of kTln2: for infinitely slow switching, all of this was due solely to the resetting operation [1].

In effect Landauer’s principle implies an equivalence between information and heat: information itself can be converted to heat. This too has recently been confirmed experimentally. In 2010 a team of physicists at the University of Tokyo led by Shoichi Toyabe moved a nanoscale polystyrene bead in a particular direction, doing useful work, not by using any energy as such but by taking advantage of the information gathered about the bead’s position [2]. They put the bead on a spiral staircase of sorts, on which the bead could hop up or down one step at a time using thermal energy. Left to its own devices, the ball would, on average, move down the staircase. But if a demon knew the position of the ball, it could place a barrier to prevent any downhill motion, so that the ball only moves uphill. In the experiment, the physicists took on the role of the demon: if the bead was measured to have moved uphill by one step, the barrier was moved upwards by one step too. By taking advantage of information gathered about the bead’s position, the physicists – using no energy as such – ensured the bead’s net uphill movement and thereby caused the bead to gain potential energy. This demonstrated experimentally that information can be converted into energy.

These studies reveal that Maxwell’s thought experiment is now accessible to direct experimental probing, and that such efforts are at the forefront of information science and technology. Moreover, even if Landauer’s principle currently represents the standard doctrine, some commentators feel that it may still be too early to be sure that the demon is dead, and that ultimately it will prove to have ramifications for the foundations of quantum information theory [3,4].

Little helpers

Maxwell didn’t intend his creature to be called a demon. That label was applied by Thomson in an 1874 paper in Nature, where he defined it as “an intelligent being endowed with free will, and fine enough tactile and perceptive organization to give him the faculty of observing and influencing individual molecules of matter.” Whether he meant it or not, this seemingly trivial change connected Maxwell’s being to a long genealogy of tiny or invisible spirits acting as agents and familiars with special powers, dating back to the demon that allegedly advised Socrates. Maxwell was not pleased. “Call him no more a demon but a valve”, he grumbled to Tait.


"Who gave them this name? Thomson." Maxwell grumbles to Tait in this letter.

Maxwell’s apparent victory over the second law in the nineteenth century might seem decidedly Pyrrhic, since as he admitted, we can’t possibly do what the demon does anyway. Maxwell presumably could have argued that we might one day have the technological means, but he didn’t seem to hold out much prospect of that. There is, however, another way that his thought experiment could work: the demons might be real. Maxwell seems to have entertained this idea, for he took seriously the possibility that free will depended on it.

The notion of invisible, perhaps demonic, beings that intervene in the world was widely shared among philosophers of the Middle Ages and the Renaissance. But surely such ideas were banished by Victorian times? Not at all. Maxwell himself seems never to have stated whether he regarded his ‘demon’ as a being – his references to a “valve” and a “self-acting” device suggest he may have preferred the image of a machine, as physicists do today – albeit a ‘machine’ with intelligence and autonomy, as he once put it “a doorkeeper, very intelligent and exceedingly quick.” Yet his touchiness about Thomson’s quip seems rather puritanical even for a religious man until one realises that Maxwell might have entertained a belief in evil spirits.


Demons performing useful work for humans in a sixteenth-century illustration.

Several of his contemporaries had little doubt that these ‘demons’ were to be taken literally. Thomson himself took pains to stress that the demon was plausible, calling it “a being with no preternatural qualities, [which] differs from real animals only in extreme smallness and agility.” Tait evidently believed they might exist, and he enlisted them for an extraordinary cause. In 1875 Tait and the Scottish physicist Balfour Stewart, an expert on the theory of heat, published a book called The Unseen Universe in which they attempted to show that “the presumed incompatibility of Science and Religion does not exist.” There must be, they wrote, “an invisible order of things which will remain and possess energy when the present system has passed away.” They believed that this “invisible” or “spiritual” domain must be capable of interacting energetically with the familiar physical world, perhaps bridged by the pervasive ether that was then thought to carry Maxwell’s electromagnetic waves. Thus energy might be transferred from the physical to the invisible realm to sustain our souls after death: through living, we store up immortality.

Tait and Stewart were aware of the apparent conflict between the Christian doctrine of the immortality of the soul and the second law of thermodynamics, which seemed to enforce an eventual universe of insensate stasis. “The dissipation of energy must hold true”, they admitted, “and although the process of decay may be delayed by the storing up of energy in the invisible universe, it cannot be permanently arrested.” Maxwell’s demon gave them a way out. “Clerk-Maxwell’s demons”, they wrote, “could be made to restore energy in the present universe without spending work” – and as a result, “immortality is possible.”

Today these speculations, coming from two highly respected scientists who Maxwell esteemed, look bizarre. But in the late nineteenth century such ideas were widely held. Spiritualism interested many scientists, including William Crookes, Oliver Lodge, J. J. Thomson and Pierre Curie. Even though some, like Tait and Stewart, were sceptical of the claims of mediums, they did not object to the basic concept.

Scientific spirit

Not only did these scientists believe in a spiritual world, but they felt that science was on the threshold of proving its existence. Many regarded the ether as a mediator. Cromwell Varley, a pioneer in transatlantic telegraphy, drew analogies between the use of electromagnetic signals for long-distance communication and the invisible messages that were alleged to pass from spirits to the living. The distinguished English physicist William Barrett wrote in 1917 that “it is not a very incredible thing to suppose that in the luminiferous ether life of some kind exists.” He speculated about “four-dimensional beings” and “human-like intelligences – good or bad daimonia” that might be responsible for events at séances. The Irish physicist Edmund Fournier d’Albe proposed in 1907 that there might exist “infra-men” on the scale of atoms, and drew on the discoveries of radioactivity and the electron to present a “physical theory of immortality.”

Indeed, the discoveries of new ‘invisible rays’, such as X-rays and radioactivity (“Roentgen rays”) bolstered beliefs in unseen universes. William Crookes – one of the most notorious sympathizers of the Spiritualists, mediums and Theosophists of the age – felt that vibrations beyond X-rays might account for telepathy. He too found a striking role for Maxwell’s demon, arguing that it might in effect explain the mystery of radioactive uranium’s seemingly inexhaustible supply of energy. He suggested that uranium atoms might be like demons themselves, mining energy from the surrounding atmosphere by sifting hot gas molecules from cold. “Let uranium or polonium”, he said at the annual meeting of the British Association in 1898, “have a structure that enables them to throw off the slow moving molecules of the atmosphere, while the quick moving molecules, smashing on to the surface, have their energy reduced and that of the target correspondingly increased.” It’s not clear that Crookes thought any intelligent agency was involved here, although he certainly believed in the possibility of invisible beings that have “intelligence, thought, and will, existing without form or matter” – and Maxwell had made clear that intelligence was needed to make the selection among gas molecules.

All this reveals that Maxwell’s demons had a much more ambiguous ontological status than imaginary ‘thought-creatures’. As well as reminding us that even apparently ‘modern’ historical scientists didn’t necessarily see things was we do, it shows how sometimes science doesn’t banish mystical beliefs but offers ‘rational’ justifications for them.

References
1. A. Bérut et al. Nature 483, 187–189 (2012).
2. S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki & M. Sano, Nat. Phys. 6, 988-992 (2010).
3. J. Earman & J. D. Norton, Stud. Hist. Phil. Mod. Phys. 29, 435-471 (1998) & 30, 1-40 (1999).
4. K. Maruyama, F. Nori & V. Vedral, Rev. Mod. Phys. 81, 1-23 (2009).

Further reading
B. Stewart & P. G. Tait, The Unseen Universe. Macmillan, London, 1875.
R. H. Harman, The Natural Philosophy of James Clerk Maxwell. Cambridge University Press, 1998.
W. H. Brock, William Crookes (1832-1919) and the Commercialization of Science. Ashgate, Aldershot, 2008.
J. Canales & M Krajewski, Interdisciplinary Science Reviews 37, 314-331 (2012).

I'm not a neuroscientist, but I know what I like

Here’s my latest Muse comment for Nature news. I recommend also taking a look at this nice piece by Steven Poole on neuropseudoscience.

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Can brain scans ever tell us why we like art?

No one with even a passing interest in scientific trends will have failed to notice that the brain is the next big thing. That’s been said for at least a decade, but now it’s getting serious, for example with the recent European award of €1 bn to the Human Brain Project to “build a new infrastructure for future neuroscience” and a $1 bn US initiative endorsed by President Obama. Having failed to ‘find ourselves’ in our genome, we’re going looking in the grey matter.

It’s a reasonable objective, but only if we have a clear idea of what we hope and expect to find. Some neuroscientists have grand visions, such as that adduced by Semir Zeki of University College London: “It is only by understanding the neural laws that dictate human activity in all spheres – in law, morality, religion and even economics and politics, no less than in art – that we can ever hope to achieve a more proper understanding of the nature of man [sic].”

Zeki heads the UCL Institute of Neuroesthetics – one of many fields that attaches ‘neuro’ to some human trait with the implication that the techniques of neuroscience, such as functional MRI, will explain it. We have neurotheology, neuroethics, neurocriminology and so on – or in popular media, a rash of books and articles proclaiming (in a profoundly ugly trope) that “this is your brain on drugs/music/religion/sport”.

If anyone is going to pursue neuroaesthetics (my brain refers that spelling), I’d be glad for it to be Zeki, who has a deep and sincere appreciation of art and an awareness of the limits of a scientific approach to the way we experience it. But some of the pitfalls of neuroaesthetics are perceptively expressed by neuroscientist Bevil Conway of Wellesley College, Massachusetts, and musicologist Alexander Rehding of Harvard University in an article in PLoS Biology [1]. They point out that “it is an open question whether an analysis of artworks, no matter how celebrated, will yield universal principles of beauty”, and that “rational reductionist approaches to the neural basis for beauty… may well distill out the very thing one wants to understand.”

For one thing, to suggest that the human brain responds in a particular way to art risks creating criteria of right or wrong, either in the art itself or individual reactions to it. Although it’s a risk most researchers are likely to recognize, experience suggests that scientists studying art find it hard to resist drawing up rules for critical judgements. The Nobel laureate chemist Wilhelm Ostwald, a competent amateur painter, devised an influential theory of colour in the early twentieth century that led him to declare Titian had once used the ‘wrong’ blue. Paul Klee, whose intuitive handling of colour was impeccable, spoke for many artists in his response to such hubris:
"That which most artists have in common, an aversion to colour as a science, became understandable to me when a short time ago I read Ostwald’s theory of colours… Scientists often find art to be childish, but in this case the position is inverted… To hold that the possibility of creating harmony using a tone of equal value should become a general rule means renouncing the wealth of the soul. Thanks but no thanks."

Even if neuroaestheticists refrain from making similar value judgements, they are already close to falling prey to one. Conway and Rehding discuss this field primarily as an attempt to understand how the brain responds to beauty. As they point out, beauty is not a scientific concept – in which case it’s not clear even which questions neuroaesthetics is examining. But the problem is deeper, for equating an appreciation of art with an appreciation of beauty is misleading. A concept of beauty (not necessarily ours today) was certainly important for, say, Renaissance artists, but until recently it had almost vanished from the discourse of contemporary art. Those who like the works of Marcel Duchamp, Joseph Beuys or Robert Rauschenberg generally don’t do so for their beauty. Scientists as a whole have always had conservative artistic tastes; a quest for beauty betrays that little has changed.

Even the narrower matter of aesthetics is not just about beauty. It has traditionally also concerned taste and judgement. Egalitarian scientists have a healthy scepticism of such potentially elitist notions, and it’s true that arbiters of taste may be blinkered and dogmatic: witness the blanket dismissal of jazz by Theodor Adorno, a champion of modernism. But the point is not whether aesthetes are right or wrong, but whether they can offer us stimulating and original ways of seeing, listening, and experiencing. In this regard aesthetics is partly a question of culture and circumstance, not a fundamental quality of the brain. Reducing it to what is shared and general recalls exercises in producing the ‘perfect’ picture or song from poll averages, the results of which are (intentionally) hideous and banal.

And what will a neuroaesthetic ‘explanation’ consist of anyway? Indications so far are that it may be along these lines: “Listening to music activates reward and pleasure circuits in brain regions such as the nucleus accumbens, ventral tegmental area and amygdala”. Thanks but no thanks. And while it is worth knowing that musical ‘chills’ are neurologically akin to the responses invoked by sex or drugs, an approach that cannot distinguish Bach from barbiturates is surely limited.

There surely are generalities in art and our response to it, and they can inform our artistic understanding and experience. But they will never wholly define or explain it.

Reference
1. Conway, B. R. & Rehding, A. PLoS Biology 11, e1001504 (2013).

Buried emotions

One more, this time from the curious world of culturomics, which is also on Nature news.

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Changes in expressions of sentiment can be discerned from Google Books

There’s nothing like having stereotypes confirmed. If you associate contemporary British fiction with the cool, detached tones of Martin Amis and Julian Barnes, and American fiction with the emotional inner worlds explored by Jonathan Franzen or the sentimentality of John Irving, it seems you’ve good reason. An analysis of the digitized texts of English-language books over the past century books concludes that, since the 1980s, emotion terms have become significantly more common in American than British books.

The study [1], by anthropologist Alberto Acerbi of the University of Bristol in England and coworkers, takes advantage of Google’s database of over 5 million digitally scanned books from the past several centuries. This resource has previously been used to examine the evolution of literary styles [2] and trends in literary expressions of individualism [3].

Such mining of the cultural information made available by new technologies has been called ‘culturomics’ [4]. Its advocates believe that these approaches can unearth trends in social opinions and norms that are otherwise concealed within vast swathes of data. “Language use in books reflects what people are talking about and thinking about during a particular time, so Google Books provides a fascinating window into the past”, says psychologist Jean Twenge of San Diego State University, author of Generation Me.

The new results certainly seem to show that familiar narratives about social mood are reflected in the literature (both fiction and non-fiction) of the twentieth century. Acerbi and colleagues find that, while words connoting happy emotions show peaks of usage in the ‘roaring twenties’ and the ‘swinging sixties’, sad words come to the fore during the years of the Second World War.

But there are surprises too: the First World War doesn’t seem to register on this happy-sad index, for example. By this measure, happiness seems to be rising since the 1990s, although it is too early to see whether the global recession will reverse that.

“The relationship between historical events and collective mood is complicated”, Acerbi admits, “but just by doing a somewhat crude analysis of emotion words it is possible to find trends that resonate with what we know about history.” He hopes that further analysis might reveal, for example, whether literature is ahead of its time or only slowly reflect other changes.

“This is a fascinating look at how two cultures have changed over time, especially how world events influence the expression of emotion in media”, says Twenge.

Overall, the use of emotion-related words in English-language books declined over the twentieth century. But when the researchers distinguished books in American and British English (about 1 million and 230,000 respectively), they found that, despite the overall decline, emotion words have become relatively more frequent in the former since about 1980, whereas previously the differences were minor. Such changes were not seen for a random selection of words. “Our results support the popular notion that American authors express more emotions than the British”, they say.

A similar change is seen in the usage of ‘content-free’ words such as pronouns and prepositions (you, us, about, within). Acerbi and colleagues interpret this as indicating that the shift in emotionality is coupled to a general shift in literary style, according to which American texts are increasingly prolix. “The correlation with mood terms is not altogether surprising, as these longer constructions provide increased opportunity for expressing sentiments”, explains biologist David Krakauer of the University of Wisconsin, who has mined Google Books for changes in literary style [2].

“Authors tend to read their contemporaries and their competitors largely within their respective cultures”, he adds, “and so we might expect British English and American English to diverge somewhat”.

Do these shifts imply that the US population in general expresses more emotion than the British? Although that doesn’t necessarily follow – literary norms may sometimes invert rather than mirror tendencies in everyday life – Acerbi feels that these new findings “may reflect a genuine cultural change, because of the size of the sample, and because Google Books is not explicitly biased towards successful or influent books.”

But Krakauer cautions that differences in literary expression don’t necessarily represent differences in the emotional mindscapes behind them. “It is a rather intriguing and open question why different cultures express the same level of feeling with different numbers of words”, he says.

References
1. Acerbi, A, Lampos, V., Garnett, P. & Bentley, R. A. PLoS ONE 8, e59030 (2013).
2. Hughes, J. M.. Foti, N. J., Krakauer, D. C. & Rockmore, D. N. Proc. Natl Acad. Sci. USA 109, 7682-7686 (2012).
3. Twenge, J., Campbell, K. W. & Gentile B. PLoS ONE 7, e40181 (2012).
4. Michel, J.-P. et al., Science 331, 176-182 (2011).

Weil to go (OK, so you have to know the pronunciation)

Here’s the initial draft (sort of) of my story for Nature news on the Abel Prize. I blanched when I read the award citation, but in the end this was fun.

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Proof of a deep conjecture about algebra and geometry nets Abel Prize

It has been four decades since Belgian mathematician Pierre Deligne completed the work for which he became celebrated, but that fertile contribution to number theory has now earned him the Abel Prize, one of the most prestigious awards in mathematics.

Given annually by the Norwegian Academy of Science and Letters and named after the famous Norwegian mathematician Niels Henrik Abel, the prize is worth 6 million Norwegian krone, or about US$ 1m.

The Academy has rewarded Deligne, who works at the Institute for Advanced Study in Princeton, New Jersey, “for seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields.”

Speaking via webcast on Wednesday, Deligne said he was surprised to learn that he had won the prize. Despite having won major prizes before, he said, he did not spend much time wondering about when the next one would come. “The nice thing about mathematics is doing mathematics,” Deligne said. “The prizes come in addition.”

Deligne has made “many different contributions that have had a huge impact on mathematics for the past 40-50 years”, says Cambridge mathematician Timothy Gowers, who delivered the award address in Olso.

“Usually mathematicians are either theory builders, who develop tools, or problem-solvers, who use those tools to find solutions”, says Peter Sarnak, also at the IAS in Princeton. “Deligne is unusual in being both. He’s got a very special mind.”

Algebraic geometry explores the links between geometric objects and the algebraic equations that describe them – for example, the expression for a circle of radius r, x2+y2=r2. It has proved to have deep connections to many areas of mathematics, particularly the properties of pure integers (number theory).

This last connection is evident in the analogy between the Riemann hypothesis, which describes a relationship between prime numbers, and the so-called Weil conjectures, which were proposed by mathematician André Weil in 1949 – the subject of Deligne’s most famous result.

The Weil conjectures concern objects in algebraic geometry called algebraic varieties, which are the set of solutions of algebraic equations. The number of such solutions can be found from a function called the zeta function. While Riemann’s hypothesis concerns the nature of the Riemann zeta function, which determines how prime numbers are distributed among all the integers, the Weil conjectures specify some of the properties of zeta functions derived from algebraic varieties.

There are four of these conjectures. The first three were proved to be true in the 1960s, but the fourth and hardest – and the direct analogue of the Riemann hypothesis – was proved by Deligne in 1974. The Riemann hypothesis itself remains “the most famous unsolved problems in mathematics”, says Gowers – which is in itself an indication of the significance of Deligne’s proof.

Gowers adds that this proof “completed a long-standing programme” in mathematics. “By solving that”, says Sarnak, “he solved a whole lot of things at once”. For example, the solution also proved a long-standing, recalcitrant conjecture by the famous Indian mathematician Srinivasa Ramanujan.

In finding his proof, Deligne built on the work of his mentor, the German-born mathematician Alexander Grothendieck, who proved the second Weil conjecture in 1965. That work introduced a crucial concept called l-adic cohomology.

The general notion of cohomology, which concerns the topological properties of spaces described by algebraic equations, was itself first developed in the 1920s and 30s, and Weil recognized that it would be needed to prove his hypotheses. Grothendieck laid the foundations for finding the right cohomology, but his student Deligne found the final proof alone – and in a different way from what Grothendieck had imagined.

Deligne’s proof won him the Fields Medal, the “other maths Nobel” besides the Abel Prize, in 1978, and in 1988 he shared the Crafoord Prize with Grothendieck – making him an obvious candidate for the Abel. Since completing the work that secured his reputation, he has applied tools such as l-adic cohomology to extend algebraic geometry and to relate it to other areas of maths. For example, because much of his work is concerned with so-called finite fields – basically modulo arithmetic – it can be applied to the kind of digital logic used in computing. “People in computer science are using his results without even knowing it”, says Sarnak.

“Even if you took away his most famous result on the Weil conjectures”, says Gowers, “you would still be left with a great mathematician.”

Deligne said he had not thought yet about how he would spend the money that came with his Abel Prize, but that he would like to find a way to make it useful for mathematics. “To some extent, I feel that this money belongs to mathematics, not to me.”

Listen up

Here are a couple of podcasts I’ve been involved in recently. First, a programme for the BBC World Service all about black (it’s the 8 Feb episode). Second, a discussion about chemistry, atoms and art broadcast by the journal Leonardo in conjunction with an ebook called Art and Atoms, which is mostly a compilation of interesting papers on the intersections of art and the molecular sciences.

Chinese made easy(er)

Here’s my latest piece for BBC Future. Hmm, will Blogger permit Chinese characters? We’ll see. If you want to find out more about this interesting learning system, there’s some stuff here, but apparently more on the way as the authors figure out how to develop this tool.

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There’s no way round it: learning Chinese is tough. As far as reading goes, what most dismays native speakers of alphabetic languages is that Chinese characters offer so few clues. With virtually no Spanish, I can figure out in the right context that baño means bath, but that word in Chinese (洗澡) seems to offer no clues about pronunciation, let alone meaning.

There seems no alternative, then, but to slavishly learn the 3,500 or so characters that account for at least 99% of usage frequency in written Chinese. This is hard even for native Chinese speakers, usually demanding endless rote copying in school. And even then, it is far more common than is often admitted for Chinese people to forget even quite routine characters, such as 钥匙 (key).

Is there a better way? Jinshan Wu of Beijing Normal University, a specialist in the new mathematical science of network theory, and colleagues have investigated the relationships between these 3,500 most used characters to develop a strategy that makes optimal use of the connections to assist learning and memorization.

Chinese characters aren’t really as arbitrary and bewilderingly diverse as they seem at first encounter. For one thing, they are made up of a fairly limited number of sub-characters or radicals, which themselves are composed of a set of standard marks or ‘strokes’. What’s more, the radicals often contain clues about meaning or pronunciation, or both. In the Chinese for bath, for example (pronounced xizao in the pinyin Romanization system), both characters start with the same radical, which denotes ‘water’, and the righthand half of both indicates the pronunciation. There are general rules (called liu shu, 六书) for building characters from radicals.

These connections can be exploited in learning. Once you know that wood is 木 (mu), it’s not so hard to remember that forest is 林 (lin) – or even more pictorially, 森林 (senlin). Assisted by the liu shu rules, Wu and colleagues mapped out the structural relationships between all 3,500 of the common characters, to form a network with over 7,000 links. This shows that the roughly 224 radicals are combined in just 1,000 or so characters that form the basis of all the others.

This network is hierarchical, meaning that it is somewhat like a tree, with a few central nodes (trunks) branching into many branch tips. That’s very different from a web-like network such as a grid or street map, in which there are often many different ways to get to any particular node. The researchers figured that it could be most efficient to start learning at the lower levels of the hierarchy – the trunks, as it were – and to progress gradually out towards the branch tips.

But would that necessarily be better than a strategy which focuses on the most frequently used words first? How, indeed, can one assess the relative learning cost of different strategies? There’s no unique way to do this, but Wu and colleagues developed a logical, intuitive method of enumerating costs. They figured that it is easier to learn a multi-part character if all the components had been learnt previously. To take a simple case, it’s easier to learn 明 (ming: bright) if you have already learnt 日 (ri: sun, day) and 月 (yue: month, moon). The researchers assigned cost values to each ‘new learning’ task.

The ‘cheapest’ way to learn all the characters in the network is then to start with the ‘trunk’ characters that have the highest number of branches, and work up through the layers. But that could leave you knowing a lot of words you rarely need to use. If, on the other hand, you simply learn characters in order of use frequency (as some learning methods do), you fail to take advantage of the network connections that can aid recognition.

The idea approach is a compromise between the two. Wu and colleagues therefore adjust the relationship network by giving a certain weighting or priority to each character depending on its use frequency. Then the learning path spreads gradually through the network while picking up most of the common characters first. It’s rather like planning a shopping trip by seeking a short total path between shops while also contriving to pick up the heaviest items last.

The researchers compared the learning cost of their strategy with that for the most widely used textbook in Chinese primary schools (covering 2,475 characters) and a popular textbook for learning Chinese as a second language. For a given cost, their new strategy picked up both considerably more characters in total and a significantly greater total use frequency than the two alternatives.

What’s more, the researchers say that their approach would allow each student’s learning strategy to be tailored to his or her individual strengths – for example, to suit those who have already learnt some characters. This just isn’t possible with traditional approaches.

Of course, the ultimate test is whether students do actually learn faster. This remains to be seen. But with a debate already raging in China over whether current teaching methods are the most suitable, this new proposal shows that there may be rational ways to pursue the question.

Reference
X. Yan, Y. Fan, Z. Di, S. Havlin & J. Wu, preprint.

Smoke signals

Ah, there is after all a route to BBC Future from the UK, in which case you can read here a piece I wrote for it yesterday on ‘papal smoke’. It’s now gone white, of course. The main difference in the version below, my original draft, is in the final paragraph, which understandably was a bit near the knuckle for the BBC.

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There’s something almost poignant in the way the Vatican has had to resort to chemistry to get its archaic method of communicating the papal election results to work properly: science helping to sustain a bizarre tradition from a distant age. Before this week, “fumata nera” and “fumata bianca” meant little to most people outside Italy. But now all eyes are on the copper chimney of the Sistine Chapel, from which the release of black smoke signalled today that the 115 cardinals voting to choose the new pope have not yet reached the two-thirds majority needed to secure a decision. When they do, the smoke will turn white.

The smoke comes partly from the burning of ballot papers in a special stove in the chapel. But to colour it white or black, this smoke is mixed with that from chemical additives burnt in a second stove. Traditionally the Vatican produced the different colours by burning wet straw for white and tarry pitch for black.

Anyone who has ever made a bonfire knows that damp grass will work for the former; the less responsible of you will know that chucking old tyres or roofing felt into the flames will turn the smoke black – and what’s more, noxious, because it is then full of sooty carbon particles that can clog the lungs and are potentially carcinogenic.

It’s not concern for the environment that has led the Vatican to change its ways, however. Rather, the smoke in some previous elections came out an ambiguous grey, prompting the decision for the last conclave in 2005 to use a more reliable method based on chemical ingredients.

The Vatican has now revealed what these are. For black, it uses a mixture of potassium perchlorate, anthracene and sulphur; white comes from potassium chlorate, lactose and the conifer resin called rosin, also used to rub violin strings to give the bow purchase.

We needn’t imagine a team of Vatican chemists labouring like alchemists to devise these magic recipes, because what they really show is that the Vatican is making plain old smoke bombs. A smoke bomb – as well as fireworks designed to be particularly smoky – works by combining an easily burnt carbon-rich compound such as sugar with a so-called oxidizing agent, which provides the oxygen for the combustion reaction. Potassium perchlorate and chlorate (which differ only in precisely how much oxygen they contain) are the most common oxidizers in these applications. Lactose (milk sugar), rosin and anthracene are the sources of carbon – anthracene, found in coal tar, is particularly good for producing big black sooty particles, although it is no longer used in pyrotechnic displays because it is carcinogenic. Sulphur also burns well, and was a traditional component of gunpowder: indeed, the Vatican’s ‘fumata nera’ mix is basically that, with the traditional oxidizer of saltpetre replaced by another.

White smoke for pyrotechnics is more commonly – and reliably – made by burning zinc dust with the organic solvent hexachloroethane and zinc oxide added. It is widely used for military training exercises. But the solvent is poisonous, and the smoke itself can cause liver damage and respiratory problems – so it’s no surprise that the Vatican chose a safer recipe.

In any event, the smoke system now leaves little to chance. Electrical heating of the flue, and backup air fans make sure that the smoke will come pumping out, and the process will surely have been tested to ensure that the black smoke doesn’t turn white as its big sooty flakes break up into smaller particles – an effect sometimes to be seen as bonfire smoke rises.

But aren’t the Vatican being a bit unimaginative and literally monotone here? Why stop at a mere two-colour signalling system? The lurid rainbow smokes used in aerial displays like those of the Red Arrows are tinted with pigments and dyes such as indigo and rhodamine. Couldn’t we have beige smoke to denote a coffee break, pink smoke to tell the world that Cardinals Monteiro de Castro and Sandoval are arguing over homosexuality, or red to indicate that Cardinal Calcagno is threatening to get out his extensive collection of firearms?

Moore's law is not just for computers

Here is my latest news story for Nature.

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Mathematical laws make industrial growth and productivity predictable

Predicting the future of technology has often seemed a fool’s game: no one forgets IBM founder Thomas J. Watson’s (possibly apocryphal) prediction that the world would need five computers. But a team of researchers in the USA now says that technological progress really is predictable, and backs up the claim with evidence from 62 different technologies.

It’s not a new claim. But a group of researchers at the Santa Fe Institute in New Mexico and the Massachusetts Institute of Technology in Cambridge, Massachusetts have put to the test several hypothetical mathematical laws describing how the costs of technologies evolve. They find that one proposed as early as 1936 supplies the best fit to the data [1].

That proposal was made by aeronautical engineer Theodore Wright, who pointed out that the cost of airplanes decreased as the total number of planes manufactured (the cumulative production) increased [2]. Specifically, this cost was proportional to the inverse of the cumulative production raised to some power. This has since been put forward as a more general law governing costs of technological products, and is often explained on the basis that, the more we make, the better and more efficient we get at making.

But it’s not the only contender. Much more famous than Wright’s law is the relationship proposed in 1965 by Gordon Moore, cofounder of the microelectronics company Intel. He pointed out that the computer power was increasing exponentially over time [3] – which means, in effect, that the cost per transistor was falling exponentially.

So who is right: Wright or Moore? Perhaps neither, for several other hypothetical relationships between scale and cost of production have been suggested – that, for example, costs fall purely due to economies of scale. All of these ‘laws’ predict that costs tend to fall over time, but at slightly different rates.

“The predictive ability of these hypotheses hadn't been tested against a large dataset before,” says Trancik. She and her colleagues tested six of them by collecting data on 62 technologies, ranging from chemicals production to energy technologies (such as photovoltaic cells) and information technologies, spanning periods of between 10 and 39 years. “Assembling a large enough data set was a big challenge”, says Trancik.

The researchers evaluated the performance of each ‘law’ with hindcasts: using earlier data to make predictions about later costs, and then seeing how these compared with the actual figures. They used statistical methods to figure out which law produced the smallest predictive errors.

And the winner was… Well, in fact there wasn’t a huge difference between any of the laws. The best was Wright’s law, but Moore’s law was close behind, at least for a relatively modest time horizon of a few decades. In fact, their predictions were so similar that the researchers suspected the two laws might be related.

This seems quite likely. In 1979 political scientist Devendra Sahal pointed out that if cumulative production of an item grows exponentially, then Wright’s law and Moore’s law are equivalent [4]. The new data confirm that production does show such growth for a wide range of products. “You wouldn’t necessarily expect that”, says Trancik.

That Moore’s law applies at all to so many different industries is a surprise, since computing has often been regarded as a special case. “It’s a much more general thing”, says coauthor Doyne Farmer, currently at the University of Oxford.

Economist William Nordhaus of Yale University warns that these laws will only work for technologies that survive – which is not itself easy to predict. “History is written only about the victors”, he says. “Those technologies that didn’t make it in the market don't make it into the data set. This is one reason why it is so difficult to forecast which of many nascent energy technologies will survive.”

The future of some technologies will depend crucially on governmental policies, not just conventional market forces. For example, in climate-change technologies, in which Nordhaus specializes, he says that the evolution will depend on the future pricing policies of carbon emissions. “Some technologies, such as carbon capture and storage, won’t even get off the ground with a zero carbon price”, he says.

Estimating the potential costs of climate-change mitigation technologies is one of the prime applications the researchers envisage for their findings. A key question is whether costs will fall just as a matter of time, as Moore’s law implies, or whether stimulating growth by public policies (such as subsidies or taxes) that boost production might accelerate the fall, as Wright’s law implies.

The results seem to imply the latter, which is good news. “We have more control over these things than we might think”, says Farmer.

References
1. Nagy, B., Farmer, J. D., Bui, Q. M. & Trancik, J. E. PLoS ONE 8, e52669 (2013). [doi:10.1371/journal.pone.0052669]
2. Wright, T. P. J. Aeronaut. Sci. 10, 302-328 (1936).
3. Moore, G. E. Electr. Mag. 38 (1965).
4. Sahal, D. AIIE Trans. 11, 23-29 (1979).

Cloud control

Here’s my latest piece for BBC Future.

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It’s quite a promise. Using existing technology, we could engineer the clouds “to cancel the entire warming caused by human activity from pre-industrial times to present day”. But this, the latest of many ‘geo-engineering’ proposals to mitigate climate change, has a drawback. Get it only a bit wrong, and you make the problem worse.

That, however, has been the worry with such ‘techno-fixes’ all along. We could fire a fleet of little mirrors into an orbit around the Sun that locks them in place to deflect sunlight from the Earth. But if it goes wrong, we could be plunged into an ice age. Manipulating the clouds has been a popular idea with would-be geo-engineers, but these proposals face the fact that the climate effects of clouds are among the hardest parts of the climate system to understand and predict, so we can’t be too sure what the results will be.

The new suggestion examined by climate scientist Trude Storelvmo of Yale University and her coworkers targets one particular kind of cloud: the cirrus ice clouds that extend their wispy tendrils in the upper troposphere, at altitudes of about 5-15 kilometres. The researchers say that these thin clouds are known with confidence to have a net warming effect on our planet, since their ice crystals trap infrared radiation from the sun-warmed surface and re-emit it back down towards ground. So if we can make cirrus still thinner, we’ll let out more heat and cool the globe.

This idea was first suggested in 2009 by David Mitchell and William Finnegan of the Desert Research Institute in Nevada (D. Mitchell & W. Finnegan, Environmental Research Letters 4, 045102 (2009)). It relies on a rather counterintuitive effect: to reduce the warming influence of cirrus cloud, one should add to the upper troposphere tiny ‘seed’ particles that actually encourage the formation of the ice crystals from which the clouds are made.

So how does that work? The ice crystals of cirrus clouds generally form spontaneously in moist, cold air. But seed particles, if present in the right concentration in the air, could grab all the water vapour to produce a small number of large ice crystals, preventing the formation of lots of unseeded little ones. This would have two consequences. First, the resulting clouds would be more transparent, just as big blobs of water in oil create a more transparent mixture as salad dressing separates out, compared with the milky, opaque mixture you get when you shake it into lots of tiny droplets. The thinner clouds absorb less radiation from the warm ground, allowing more to escape into space.

Second, clouds made from larger ice particles have shorter lives, because the big crystals sink down through the atmosphere under gravity.

This wasn’t by any means the first proposal for geo-engineering climate by modifying the reflection or absorption of light in the atmosphere. For example, British meteorologist John Latham and coworkers have suggested that a fleet of solar-powered ships might spray sea salt into the air to seed the formation of stratocumulus clouds, which cool the planet by reflecting sunlight (J. Latham, Nature 347, 339 (1990)). And climate scientist Paul Crutzen has proposed injecting a sulphur-containing gas into the stratosphere, which would form a haze of sulphate particles to reflect sunlight – a process that happens naturally in volcanic eruptions and which is known to cool the earth (P. J. Crutzen, Climatic Change 77, 211 (2006)).

One of the advantages of climate engineering via clouds is that the effects are transient: if it doesn’t go to plan, the process can be stopped and all will return to normal in a matter of weeks. Mitchell and Finnegan suggested that seeding of cirrus cloud might be done by releasing the particles from aircraft. They suggested that the somewhat exotic (but not excessively costly) compound bismuth tri-iodide would be a good material for the seeds, as it is known to promote ice formation on its surface.

But will it work as planned? That’s what Storelvmo and colleagues have now studied by using a climate model that incorporates a description of cirrus cloud formation. They find that to get climate cooling, one has to use just the right concentration of seed particles. Too few, and cirrus clouds form just as they do normally. But if there are too many seeds, they generate more ice crystals than would have formed in their absence, and the clouds are actually thicker, trapping even more heat.

If we get the seed concentration right, the effect is dramatic: the cooling is enough to offset all global warming. But this ‘Goldilocks’ window is quite narrow. What’s more, the researchers say, finding the precise boundaries of the window requires more information than we have at present, for example about the ability of bismuth tri-iodide to seed ice formation and the rates at which the ice crystals will settle through the atmosphere. So attempting this sort of engineering prematurely could backfire – even if the effect would be quite short-lived, we should hold fire until we know more.

Reference
T. Storelvmo et al., Geophys. Res. Lett. 40, 178-182 (2013).

Falstaffian science

Here’s a review of the new production of Bertolt Brecht’s Life of Galileo from the Royal Shakespeare Company, published in Nature.

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It is one of the central works of drama about science, and one of the most controversial. Bertolt Brecht’s Life of Galileo has been criticised for misrepresenting history, science and Galileo himself, with some validity. The real question, however, is whether the play works, theatrically and psychologically.

Shakespeare, after all, took vast liberties with history, yet such is his way with human portraiture that no one complains. Shakespeare and Galileo were born within a few weeks of each other in 1564 – a coincidence that the Royal Shakespeare Company (RSC) understandably makes much of for its latest production of Brecht’s play. More significantly, the play shows Brecht at his most Shakespearean, with Galileo the wily, tragically compromised sensualist redeemed by self-insight that others lack — he is, as Adam Gopnik suggested in a recent article in The New Yorker, a kind of intellectual Falstaff.

The exuberance and wit of this production owe much to the new translation by current RSC writer in residence Mark Ravenhill. Ravenhill has commented on the “comic sensibility in Brecht’s language which I think [is] often overlooked”, but which he and director Roxana Sibert have found in abundance. In the title role, Ian McDiarmid is sly and wordly while pulling off the important trick of making Galileo loveable. It’s with the basic fabric of the play, not its realization, that the questions lie.

In retrospect we can see that that Brecht set himself an impossible task, because even now there is no consensus on Galileo. Many scientists still prefer the narrative that prevailed when Brecht first wrote the play in 1937-39, of a martyr persecuted by the Catholic Church for his pursuit of truth about the arrangement of the cosmos. A more measured view holds now, recognizing that a less pugnacious man might have navigated the currents of the papal court more skilfully. It is certainly not to excuse the bullying, dogmatic Vatican to point out that Galileo’s evidence for a heliocentric universe was equivocal and in some respects (his interpretation of the tides) wrong.

Galileo’s mathematical physics, rightly adored by physicists today, was not, as some older science historians had it, the right way to do science. It was the right approach for celestial and terrestrial mechanics, but useless for chemistry, medicine, botany, zoology and much else. And while Einstein celebrated Galileo’s rejection of logical deduction devoid of empirical input as essential to modern science, Galileo was not the first to do that.

Brecht must take some blame for making Galileo more original than he was. He fell for the idea of a Scientific Revolution in which Great Men begin thinking in a totally new way. Complaints about historical accuracy could seem carping in a work of art, but Brecht himself attested of the first version of the play that “I was trying here to follow history”.

Brecht was in any case disingenuous, for his original version of the play was evidently informed by, and widely interpreted as a comment on, the political climate of the time. Brecht fled Nazi Germany after the Reichstag fire in 1933, and his cunning Galileo who subverts the ideology of the authorities – recanting on his heliocentrism in order to continue his work in secret – was regarded as a symbol of anti-Nazi resistance.

That, however, is not the Galileo of the revised 1947 version – the one most often performed, and used here. Although Brecht was already reworking the play in 1944, the bombing of Hiroshima and Nagasaki transformed his view of scientists. “Overnight the biography of the founder of the new system of physics read differently,” he wrote. He felt that the Manhattan Project scientists had betrayed their moral obligations, and criticized even Einstein as a politically naïve “eternal schoolboy”. Regardless of the merits of that view, it is the play’s downfall.

Now Galileo, confronting his former student Andrea, launches into a diatribe on how, by focusing on science for science’s sake, “you might jump for joy at some new achievement, only to be answered by a world shrieking in horror.” Nothing in Galileo’s former conduct has prepared this (anachronistic) concern about the social applications of science, leaving us with a confusing portrait.

On 30 October 1947, when the new version was premiered, Brecht got a taste of Galileo’s ordeal: he testified before the House Committee on Un-American Activities (others involved in the production refused). He left for Europe the following day, accused of having compromised artistic freedom and with perhaps a keener appreciation that ideological interference in art and science was not confined to dictatorships.

Brecht’s other impossible task was to explain how real science is done. He succumbs to the view that you just need to think clearly, believe your eyes, trust in reason. He then has to skirt around the problem that your eyes tell you that the sun, not the earth, moves. What’s more, philosophers such as Paracelsus and Bernardino Telesio had been relying on experience, rather than Aristotle, for a hundred years before Galileo, but had reached rather different ‘truths’. Nor was there any ‘scientific method’ in Galileo’s time, just as there is none today: its ad hoc combination of hypothesis, assumption, experiment, theory, logic and intuition will not reduce to any formula.

The RSC’s production is spirited and visually inventive. But the play itself is pulled between too many irreconcilable poles to make a coherent whole. It is perhaps the history of the play, rather than the text itself, that reveals the most about the difficult relationship between science and the cultures in which it is embedded.

Scribblings of a scribe

My article on cursive handwriting in Prospect seems to be creating some debate, which pleases me. I volunteered in my last comment there to offer up a sample of my own handwriting, as used for taking notes at speed. Here it is; fire away.

The demon-haunted microworld

Here's a bigger and illustrated version of an article published in Aeon magazine. It is part of a larger project to be revealed soon.

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If small means invisible, we fear what it holds

When the Dutch cloth merchant Antony van Leeuwenhoek looked at a drop of pond water through his home-made microscope in the 1670s, he didn’t just see tiny ‘animals’ swimming in there. He saw a new world: too small for the eye to register even the faintest hint, teeming with invisible life. The implications were as much theological as they were scientific.

Invisibility comes in many forms, but smallness is the most concrete. Light ignores very tiny things as ocean waves ignore sand grains. During the seventeenth century, when the microscope was invented, the discovery of such objects posed a profound problem: if we humans were God’s ultimate purpose, why would he create anything that we couldn’t discern?

The microworld was puzzling, but also wondrous and frightening. There was nothing especially new about the idea of invisible worlds and creatures – belief in immaterial spirits, angels and demons was still widespread. But their purpose was well understood: they were engaged in the Manichean struggle for men’s souls. If that left one uneasy in a universe where there was more than meets the eye, at least the moral agenda was clear.

But Leeuwenhoek’s ‘animulcules’ and their ilk indulged their opaque, wriggly ways everywhere one looked: in moisture, air, body fluids. In human semen – Leeuwenhoek studied his own, transferred with jarring haste from the marital bed – there were tadpole-like ‘animulcules’ writhing like eels. In 1687 the German mathematician Johann Sturm suggested that disease is caused by breathing in such invisible animals in air. The Jesuit priest Athanasius Kircher proposed that the plague might be caused by the microscopic ‘seeds’ of virulent worms that enter the body through nose and mouth – just a step away, it seemed, from a germ theory of contagion, although the impossibility of seeing bacteria and viruses with the microscopes of the time obstructed that leap until Louis Pasteur and Robert Koch made it in the late nineteenth century.

Pestilence was everywhere, unseen and impossible to fend off – just like medieval demons. The narrator in Daniel Defoe’s Journal of the Plague Year (1722) attests he has heard that if a person with the plague breathes on glass, “there might living Creatures be seen by a Microscope of strange, monstrous and frightful Shapes, such as Dragons, Snakes, Serpents and Devils, horrible to behold.” He admits some doubts about whether this is true, but the message is clear: the invisible microworld is labelled “Here be dragons”.

Little has changed. Electron microscopes now reveal miniature viral monsters like science-fiction aliens, with arachnoid legs and crystal heads from which they inject genetic venom into cells. MRSA bacteria lurk unseen on hospital door handles and bed sheets. We sprinkle anti-bacterial fluids like holy water to fend off these invisible fiends.



The invisible world

The idea that matter might be composed of particles and processes too small to see – the atoms of Democritus, the whirling vortices of Descartes and the corpuscles of Newton – has a long history. But this fine-grained nature of matter only came to seem like an ‘invisible world’ when the advent of the microscope enabled us, first, to appreciate the intricacy with which it was wrought, and second, to identify life amidst the grains. When Galileo used one of the first microscopes to study insects, he was astonished and repelled, writing to his friend Federico Cesi in 1624 that
“I have observed many tiny animals with great admiration, among which the flea is quite horrible, the mosquito and the moth very beautiful… In short, the greatness of nature, and the subtle and unspeakable care with which she works is a source of unending contemplation.”

This wonder at nature’s invisible intricacy was echoed by Robert Hooke, whose 1665 book Micrographia put microscopy on the map. Crucially, Hooke’s volume was not merely descriptive: he included large, gorgeous engravings of what he saw through the lens, skilfully prepared by his own hand. The rhetorical power of the illustrations was impossible to resist. Here were fantastical gardens discovered in mould, snowflakes like fronds of living ice, and most shockingly, insects such as fleas got up in articulated armour like lobsters, and a fly that gazes into the lens with 14,000 little eyes, arranged in perfect order on two hemispheres.



This was surely a demonstration of the infinite scope of God’s creative power. “There may be as much curiosity of contrivance in every one of these Pearls”, Hooke wrote, “as in the eye of a Whale or Elephant, and the almighty’s Fiat could as easily cause the existence of the one as the other; and as one day and a thousand years are the same with him, so may one eye and ten thousand.”
In comparison, the finest contrivances of man – a needle’s tip, a razor’s edge, a printed full stop – looked crude and clumsy under the microscope.

What excited Hooke and his contemporaries most was that the microscope seemed to offer the possibility of uncovering not just the invisible structures of nature but, in consequence, its hidden mechanisms. Where in previous ages natural philosophers had attributed the cause of processes to invisible, occult forces and emanations – vague and insensible agencies – the mechanistic philosophers of the seventeenth century argued that nature worked like a machine, filled with levers, hooks, mills, pins and other familiar devices too small to be seen. Now at last these structures might be revealed. Henry Power, whose Experimental Philosophy advertised the virtues of the microscope a year before Hooke’s, wrote that we could expect to see at last “the magnetical effluviums of the loadstone [magnet], the solary atoms of light, the springy particles of air.” Hooke too insisted that “‘Those effects of Bodies, which have been commonly attributed to Qualities, and those confesse’d to be occult, are performed by the small Machines of Nature.” He never quite found them; but there was no shortage of other marvels.

Life writ small

Micrographia recorded life in this microscopic realm too, but none that could not be discerned, with effort, by the eye alone: “eels” in vinegar and mites in cheese. Leeuwenhoek’s discoveries, reported in 1676 and verified by Hooke a year later, brought home the full force of a teeming, invisible microworld. The anxieties about scales of perception that run through Swift’s Gulliver’s Travels make it clear how unsettling this was. In the land of the gigantic Brobdingnagians, Gulliver is disgusted by their bodies when seen so close up: “Their skins appeared so coarse and uneven, so variously coloured when I saw them near, with a mole here and there as broad as a trencher, and hairs hanging from it thicker than pack-threads.” Among the common folk he is repelled by the immense lice crawling on their clothes, possessing “snouts with which they rooted like swine.”

Even with refinements of the microscope in the nineteenth century that enabled scientists to peer into the invisible world with unprecedented resolution, there remained questions about what might be happening down there. In 1896 the pioneering British psychiatrist Henry Maudsley proclaimed that
“The universe, as it is within [man’s] experience, may be unlike the universe as it is within other living experience, and no more like the universe outside his experience, which he cannot think, than the universe of a mite is like his universe.”

Maudsley’s avowal of ignorance was an attack on the ready assumptions of some scientists, such as the chemist William Crookes, that invisible realms were peopled with beings like us. But this lack of knowledge could equally supply licence for the most exotic of speculations. The beginnings of molecular science engendered an appreciation that life as it was known could have a minimal possible size. But when the ‘indivisible’ atom began to display a finer-grained structure of subatomic particles, and light waves proved to have much finer oscillations in the form of X-rays, no one could rule out the possibility of an entire hierarchy of material existences on smaller scales. The physicist George Johnstone Stoney, who gave the name to subatomic electrons discovered in 1897, declared that the physical universe is really an infinite series of worlds within worlds. Another physicist, the Irishman Edmund Fournier d’Albe, developed these ideas in Two New Worlds (1907), where he envisaged an “infra-world” at a scale below that which microscopes could register, peopled like Leeuwenhoek’s drop of water with creatures (“infra-men”) that “eat, fight, and love, and die, and whose span of life, to judge from their intense activity, is probably filled with as many events as our own.” The human body, he estimated, could play host to around 10**40 of these infra-men, experiencing joys and woes “without the slightest net effect on our own consciousness”.

As ever is the case with scientific advance, the new and unfamiliar are popularly interpreted by reference to the old and prosaic. Littleness has been a consistent theme in the folklore and traditions of demons and faeries. Mischievous imps and fairies that interfere in domestic matters were a stock of folk tradition, and if these beings were not necessarily invisibly small, their diminutive stature enabled them to pass unseen. One might be tempted to imagine that by the late nineteenth century such beliefs reached no further than rural backwaters – but that would be to underestimate the grip of the invisible world on the imagination. Nowhere is this better illustrated than in the ‘demon’ of James Clerk Maxwell, perhaps the most profound physicist of the nineteenth century.

Maxwell’s idea was a response to the gloomy prediction of a ‘cosmic heat death’ of the universe. In 1851 William Thomson (later Lord Kelvin) pointed out that the second law of thermodynamics, which can be expressed as the condition that heat energy must always flow from hot to cold, must eventually create a universe of uniform temperature, from which no useful work can be extracted and in which nothing really happens.

As a devout Christian, Maxwell could not accept that God would let this happen. He believed that the second law is statistical rather than fundamental: temperature gradients get dissipated because it is far more likely that faster, ‘hotter’ molecules will mingle with slower ones, rather than by chance congregating into a ‘hot’ patch. But what if there were, as Maxwell put it in 1867, a “finite being”, small enough to ‘see’ each molecule and able to keep track of it, who could open and shut a trapdoor in a wall dividing a gas-filled vessel? This being could let through fast-moving molecules in one direction so as to congregate the heat in one compartment, separating hot from cold and creating a temperature gradient that could be tapped to do work.

Maxwell didn’t intend his creature to be called a demon. That label was applied by Thomson, where he defined it as “an intelligent being endowed with free will, and fine enough tactile and perceptive organization to give him the faculty of observing and influencing individual molecules of matter.” Maxwell was not pleased. “Call him no more a demon but a valve”, he grumbled – albeit a ‘valve’ with intelligence and autonomy, or as Maxwell once put it “a doorkeeper, very intelligent and exceedingly quick.”

Several of his contemporaries had little doubt that these ‘demons’ were to be taken literally. Thomson himself took pains to stress that the demon was plausible, calling it “a being with no preternatural qualities, [which] differs from real animals only in extreme smallness and agility.” Maxwell’s friend, the Scottish physicist Peter Guthrie Tait, evidently believed they might exist, and he enlisted them for an extraordinary cause. In 1875 Tait and fellow Scot Balfour Stewart, an expert on the theory of heat, published a book called The Unseen Universe in which they attempted to show that “the presumed incompatibility of Science and Religion does not exist.” There must be, they wrote, “an invisible order of things which will remain and possess energy when the present system has passed away.” Tait and Stewart were aware of the apparent conflict between the Christian doctrine of the immortality of the soul and the second law of thermodynamics, which seemed to enforce an eventual universe of insensate stasis. “The dissipation of energy must hold true”, they admitted, “and although the process of decay may be delayed by the storing up of energy in the invisible universe, it cannot be permanently arrested.” Maxwell’s demon gave them a way out. “Clerk-Maxwell’s demons”, they wrote, “could be made to restore energy in the present universe without spending work” – and as a result, “immortality is possible.”

Modern studies have shown that Maxwell’s demon cannot after all evade the second law, since even it has to dissipate heat as part of the process of gathering information about molecular speeds. The conceit is now generally regarded as an amusing thought experiment: it is forgotten that, in Maxwell’s day, invisibly small demons going about their micro-business seemed possible, even likely.

Nano nightmares

The demonization of invisible beings is as strong as ever, now adapted to the fantasies of our age: viruses are “alien invaders”, we go to “war” on “superbugs” with super-powers, repelling them like vampires with “magic bullets”. Children are taught that invisible “germs” are the omnipresent enemy, and they are enlisted, as imps and demons once were, to instil safe behaviour. It is a case, microbiologist Abraham Baron declared in 1959, of “man against germs”. When he explained that “we share the world with an incredible vast host of invisible things”, it was a warning and not an expression of wonder. In his 1912 study of the hazards of dust, physician Robert Hessler cautioned that “It is the invisible we have to guard against.”



This fear of the malevolent designs of imperceptibly small entities was evident in the early reception of nanotechnology, which seemed to be supplementing this gallery of invisible horrors. Among scientists, nanotechnology was a loosely defined collection of attempts to visualize and manipulate matter on scales ranging from ångstrøms (the size of atoms) to hundreds of nanometres (the size of small bacteria). But in public discourse it became dominated by a single entity, which nanotechnologists were allegedly aiming to construct: the nanoscale robot or nanobot. This, it was said, would be an autonomous device that would patrol the bloodstream for pathogenic invaders, or construct materials and molecules from the atoms up. It was, in other words, a human avatar on an invisible scale.

What if nanobots ran amok, as robots are (in fiction) almost predestined to do? A rogue robot might be a menace, but it is a comprehensible one, a kind of superhuman being. A rogue nanobot, capable of replicating like bacteria and of pulling matter apart atom by atom, would be an unthinkable threat. Hidden from sight, it could reduce anything in seconds to a formless mass of atoms, which would then be reconstituted into replica nanobots: an amorphous ‘grey goo’. The terror of this imagery was crudely but effectively exploited by Michael Crichton in his novel Prey (2002).

If the image is frightful, it is also familiar. Invisible powers have long been held capable of animating clay, creating the fearsome Golem, or of disintegrating and deliquescing matter and flesh (think now of the Ebola virus). What’s more, the nanobot connects with long-standing images of the exploration of new worlds, most notably the submarine Nautilus in which Captain Nemo explores the hidden deep sea in Jules Verne’s 20,000 Leagues Under the Sea. Once again, it seems we must remake the invisible microworld in our own image before we can explore its promise and peril. This was most explicit in the 1966 movie Fantastic Voyage (based on a short story by Isaac Asimov) and the parodic 1987 remake Inner Space, in which humans are shrunk to a scale that allows them to navigate through the human body.





The extreme miniaturization that has its ultimate expression in nanotechnology has not yet given birth to an invisible nemesis, and shows no sign of doing so. What it, in conjunction with the manipulation of invisible rays such as Marconi’s ‘wireless’ emanations, has done is create an age of technological invisibility, in which things happen with no mechanism in sight, indeed even without our volition, embedded in an omnipresent field of information. Items in stores speak to barriers and computers; miniaturized sensors control our cars and refine our household environment; libraries leap into our pockets. Dust, a metaphor for worthless matter while it was the smallest thing that could (just) be seen with the unaided eye, has become “smart dust”, a nanotechnological promise of particles laced with invisible circuitry, programmed with the intelligence to self-assemble as we will them: to make a Golem, perhaps, rebranded now as a ‘reconfigurable robot’.

It has become a commonplace that these advances would have seemed in earlier times to be magical. Less often acknowledged is how traditional reactions to invisibility can help us comprehend and negotiate the cultural changes that ensue. The boundaries between rationality and insanity can no longer be policed in behavioural terms. Is the person gesticulating and talking out loud in the street communing with demons of the mind, or with a friend? Is the person fretting over the invisible threats of nearby radio masts succumbing to some modern version of the mal aria theory of contagion, or do they have a point? We entrust our digital secrets to the intangible Cloud, assume that this nebulous entity can be summoned to regurgitate them at will. With invisibly small technology harnessed to the invisible ether, we have in a real sense animated the world.

The curse of cursive

Here’s a piece published in Prospect this month. I’m not holding my breath about whether it is going to make the slightest impact on the blinkered way this aspect of education is approached – this is one of those issues so ingrained that one rarely gets much more than a dumb stare if it is raised with teachers. I’d love someone to tell me just why cursive is so important in educational terms. They haven’t yet.

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There’s something deeply peculiar about the way we teach children to play the violin. It’s a very difficult skill for them to master – getting their fingers under control, holding the bow properly, learning how to move it over the strings without scratching and slipping. But just as they are finally getting there, are beginning to feel confident, to hit the right notes, to sound a little bit like the musicians they hear, we break the news to them: we’ve taught them to play left-handed, but now it’s time to do it like grown-ups do, the other way around.

All right, I’m fibbing. Of course we don’t teach violin that way. It would be absurd. What would be the point of making it so hard? We wouldn’t do anything so eccentric for something as important as learning a musical instrument, would we? No – but that’s how we teach children to write.

It’s best not to examine the analogy too deeply, but you see the point. The odd thing is that, when most parents watch their child’s hard-earned gains in forming letters like those printed in their story books crumble under the demand that they now relearn the art of writing ‘joined up’ (“and don’t forget the joining tail!”), leaving their calligraphy a confused scrawl of extraneous cusps and wiggles desperately seeking a home, they don’t ask what on earth the school thinks it is doing. They smile, comforted that their child is starting to write like them.

As he or she probably will. The child may develop the same abominable scribble that gets letters misdirected and medical prescriptions perilously misread. In his impassioned plea for the art of good handwriting, Philip Hensher puts his finger on the issue (while apparently oblivious to it):
“You longed to do ‘joined-up writing’, as we used to call the cursive hand when we were young. I looked forward to the ability to join one letter to another as a mark of huge sophistication. Adult handwriting was unreadable, true, but perhaps that was the point.”

The real point is, of course, not that illegibility but that sophistication. When I questioned my friend, a primary teacher, about the value of teaching cursive, she was horrified. “But otherwise they’d have baby writing!” she exclaimed. I pointed out that my handwriting is printed (the so-called ‘manuscript’ form). “Oh no, yours is fine”, she – not the placatory sort – allowed. I didn’t ask whether all the books on my shelves were printed in ‘baby writing’ too.

I did also once ask my daughter’s teachers what they thought they were doing by teaching her cursive. When they realised this was not a rhetorical question but a literal one, there was bemusement and panic. “It’s just what we do”, one said. “We always have”. Another ventured the answer I’d anticipated – the children will be able to write faster – and then added that she thought she’d seen some research somewhere showing that some children find that the flowing movements help to imprint the shape of whole words more clearly in their mind. This was evidently not a question they had faced before.

We tend to forget, unless we have small children, that learning to write isn’t easy. It would make sense, then, to keep it as simple as possible. If we are going to teach our children two different ways of writing in their early years (quite apart from distinguishing capitals and lower-case), you’d think we would ensure we have a very good reason for doing so. I suspect that most primary teachers could not adduce one.

It’s not just about writing, but reading too. “As a reading specialist, it seems odd to me that early readers, just getting used to decoding manuscript, would be asked to learn another writing style,” says Randall Wallace, a specialist in reading and writing skills at Missouri State University.

There are, from time to time, proposals to stop teaching cursive handwriting – but these are usually motivated by the conviction that handwriting is passé in the digital age. The outraged response is that handwriting is an art, that there is an intrinsic value in beautifully formed script, and that to lose it would be a step towards barbarism.

Here I’m with Hensher: we should value skill with a pen. Our handwriting is an expression of our personality and humanity – not in some pseudoscientific graphological sense, but in the same way as is our clothing, our voice, our conversation. Yet these arguments are never really about cursive per se: they are about the good versus the indifferent in handwriting. It is implicitly assumed that the acme of good handwriting is beautiful cursive.

Now, I admire the elegant copperplate of the Victorians as much as anyone. But no one writes like that any more, since no one is taught to. How can we insist that to drop cursive will be to drop beauty and elegance, given that most people’s cursive handwriting is so abysmal? “It has always seemed ironic that, even after we sign a document, we have to print our signature underneath it for clarity”, says Wallace.

Surely, though, in something as fundamental to education as writing, there must be scientific evidence that will settle this matter? Let’s dispatch the most obvious red herring straight away: you will not write faster in cursive than in print. Once you need to write fast (which you don’t at primary school), you’ll join up anyhow if and when that helps. I know this to be so, because I missed the school years in which cursive was ground into my peers, and yet I never suffered from lack of speed. But don’t take my word for it – research shows that there is no speed advantage to cursive [1].

Are there any other advantages, then? Champions of cursive will always unearth tenuous arguments from dusty corners of the literature. Cursive makes it easier to learn how to write words; in cursive, b and d are not confused, and children don’t write backwards letters; the blending of sounds is made more apparent by the joining of letters; cursive helps the left-handed. None of these claims counts for very much. (There is equal reason, for example, to think that the continuous movement of the pen from left to right makes cursive especially hard for left-handers.) On the merits of learning cursive versus manuscript, Steve Graham, a leading expert in writing development at Arizona State University, avers that “I don't think the research suggests an advantage for one over the other.”

A survey in the US in 1960 found that the decision to teach cursive in elementary schools was “based mainly on tradition and wide usage, not on research findings” [2]. One school director said that public expectancy and teachers’ training were the main reasons, and that “we doubt that there is any significant advantage in cursive writing.” According to Wallace, nothing has changed. “The reasons to reject cursive handwriting as a formal part of the curriculum far outweigh the reasons to keep it”, he says.

It’s not necessarily cursive per se that’s the problem, but the practice of teaching children two different systems, perhaps in the space of so many years, without good reason. Research seems to show that it may not much matter how children learn to write, so long as it is consistent. Wallace argues that any style will do if it is “flexible enough to be perceived as similar to printed text and simple enough to last through the school years” [3].

Were there to be a choice between cursive and manuscript, one can’t help wondering why we would demand that five-year-olds master all those curlicues and tails, and why we would want to make them form letters so different from those in their reading books. But that’s a smaller matter than forcing them to struggle though one of their hardest early-learning tasks twice, with two different sets of rules, apparently because of nothing more than the arbitrary and tautological belief that only the kind of writing you had to (re)learn can be ‘grown-up’ and ‘beautiful’. After all, what’s the point of conducting research on educational methods if in the end you’re going to say “But this is how we’ve always done it”?

References
1. S. Graham et al., J. Educ. Res. 91(5), 290 (1998).
2. P. J. Groff, Elementary School J. 61(2), 97 (1960).
3. R. R. Wallace & J. H. Schomer, Education 114(3) (1994).

Folk tales show how culture spreads

Here’s another Nature News piece – there’s evidently a lot of language about (and more in the pipeline…).

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It’s harder to transmit stories than genes across linguistic barriers

Have you heard the story of the good and bad sisters? They leave home, the good sister is kind to the people and animals she meets, and gets rewarded in gold. The bad sister is haughty and greedy, and is rewarded with a box of snakes.

This is a familiar folk tale in European culture. But how similar your version is to mine depends on how far apart we live and how ethnically and linguistically different our cultures are, according to a study by a team of researchers in Australia and New Zealand published in the Proceedings of the Royal Society B [1]. They have identified what makes the transmission of cultural traits and artefacts, such as folk tales, similar to and different from the transmission of genes.

Like genes, say psychologist Quentin Atkinson of the University of Auckland and his collaborators, folk tales get passed from group to group – and the more distant two groups are, the less similarity their genes and stories possess.

“The geographic gradients we found are similar in scale to what we see in genetics, suggesting that there may be parallel processes responsible for mixing genetic and cultural information”, says Atkinson.

“But the mechanisms aren’t identical”, he adds. “The effect of ethnolinguistic boundaries is much stronger for the folktales than for genes.” This fits with recent studies looking at other aspects of culture, such as song [2]. “Our findings support predictions that cultural variation should be more pronounced between groups than genetic variation”, says Atkinson.

“This supports the view that our cultures act almost like distinct biological species”, says evolutionary biologist Mark Pagel of the University of Reading, a specialist in cultural transmission. “Our cultural groups draw pretty tight boundaries around themselves, and can absorb genetic immigrants without absorbing their cultures.”

Atkinson and his colleagues figured that the ubiquity of folk tales would make them a good proxy for cultural exchange. “Folktales can be transmitted over the world”, says folklore specialist Hans-Jörg Uther of the University of Göttingen in Germany. “The plot can stay the same while characters and other attributes change to match the cultural traits of the region.”

The researchers used the statistical tools of population genetics to investigate variations between versions of ‘The kind and the unkind girls’ across many European cultures, from Armenian to Scottish, Basque and Icelandic.

“This tale is widely known, and we were able to locate a large, well-documented collection that spanned all of Europe”, says Atkinson: about 700 variants in all. “For example, some stories involve two cousins or brothers rather than daughters, in others it is a daughter and servant girl.” The researchers built on well established methods of enumerating these differences.

If folk tales simply spread by diffusion, like ink blots in paper, one would expect to see smooth gradients in these variations as a function of distance. But instead the team found that ethnolinguistic differences between cultures create significant barriers.

These barriers are greater than those for gene flow. You could say that the attitude is “I’ll sleep with you, but I prefer my stories to yours.”

Uther finds the work interesting, but he is “a little bit sceptical about comparing variants while neglecting their historical context and mode of performance.” He suspects that, as digital archives of folk tales become increasingly available, they will provide a valuable tool for making comparative and evolutionary studies of culture more quantitative.

References
1. Ross, R. M., Greenhill, S. J & Atkinson, Q. D. Proc. R. Soc. B doi:10.1098/rspb.2012.3065 (2013).
2. Rzeszutek, T., Savage, P. E. & Brown, S. Proc. R. Soc. B 279, 1606-1612 (2011).

Rooting out the mother tongue

Catching up after winter bugs… here is an article for Nature News from a week or so back.

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Computer algorithm reconstructs ancient languages

On Fiji, a star is a kalokalo. For Pazeh Taiwanese aboriginals it is mintol, and for the Melanau people of Borneo, biten. All these words are thought to come from the same root – but what was it?

An automated computer algorithm devised by researchers in Canada and California now offers an answer – in this case, bituqen. The program can reconstruct extinct ‘root’ languages from modern ones, a process that has previously been done painstakingly ‘by hand’ using rules of how linguistic sounds tend to change over time.

Statistician Alexandre Bouchard-Côté of the University of British Columbia and his coworkers say that, by making the reconstruction of ancestral languages much simpler, their method should make it easier to test hypotheses about how languages evolve. They report their technique in the Proceedings of the National Academy of Sciences USA [1].

Automated computer methods like this have been attempted before, but the authors say these were rather intractable and prescriptive. The method of Bouchard-Côté and colleagues can factor in a large number of languages to improve the quality of reconstruction, and it uses rules that handle possible sound changes in flexible, probabilistic ways.

The method requires a list of words in each language, with their meanings, and a map of the phylogenetic ‘language tree’ showing how each language is related to the others. These trees are routinely constructed today by linguists using techniques borrowed from evolutionary biology.

The algorithm can automatically identify cognate words (ones with the same root) in the lexicons. It then applies rules known to govern sound changes to deduce the likely root of each set of cognates. For example, phonemes that are always paired will tend to get condensed into one if this doesn’t lose any semantic information.

The algorithm involves millions of parameters, whose values are found by an automated shuffling process that seeks the simplest fit to the data. It’s a little like cracking a code, based on a series of encoded phrases, by trying out possible bits of cipher and working your way towards one that gives a plausible solution to all the phrases.

The researchers tested their approach on 637 Austronesian languages spoken primarily on islands in Southeast Asia and the Pacific, including Malaysia, the Philippines and Indonesia. Manual methods have previously been used to reconstruct the protolanguage of this large group, thought to have come originally from Taiwan.

Bouchard-Côté and colleagues found that the predictions of their algorithm matched those of the manual method in 85 percent of cases (including bituqen). “Our system only uses a subset of the factors taken into consideration by a linguist, so we feel most of the discrepancies reflect things to be improved in our method”, admits Bouchard-Côté.

“It looks as though this method could be a very useful labor-saving device in some cases”, says linguist Don Ringe of the University of Pennsylvania. But he cautions that methods which are “correct or nearly correct in about 85% of the cases will never be good enough. Our reconstructions might be no better than an approximation, and if we settle for what look like approximations even to us, we might be plain wrong.”

Bouchard-Côté and colleagues used the method to test a hypothesis about language evolution first proposed in 1955 [2], which states that sounds that are particularly important for distinguishing words from each other are more resistant to change. Any such pattern is almost impossible to spot for just a few languages, but it emerged clearly from the data set of 637 languages.

There had previously been some scepticism about this so-called ‘functional load hypothesis’, and Ringe says that “the demonstration that there might be something to it after all is interesting.”

He adds that “it’s refreshing to find colleagues in other disciplines tackling a problem that historical linguists actually care about.”

References
1. Bouchard-Côté, A., Hall, D., Griffiths, T. L. & Klein, D. Proc. Natl Acad. Sci. USA doi/10.1073/pnas.1204678110 (2013).
2. Martinet, A., Économie des Changements Phonétiques (Maisonneuve & Larose, Paris, 1955).