# “These confounded dials…”

Reading The Seven Measures of the World by Piero Martin and Four Ways of Thinking by David Sumpter, for New Scientist, 23 October 2023

Blame the sundial. A dinner guest in a play by the Roman writer Plautus, his stomach rumbling, complains that

“The town’s so full of these confounded dials
The greatest part of the inhabitants,
Shrunk up with hunger, crawl along the streets”

We’ve been slaves to number ever since. Not that we need complain, according to two recent books. Piero Martin’s spirited and fascinating The Seven Measures of the World traces our ever-more precise grasp of physical reality, while Four Ways of Thinking, by the Uppsala-based mathematician David Sumpter, shows number illuminating human complexities.

Martin’s stories about common units of measure (candelas and moles rub shoulders here with amperes and degrees Kelvin) tip their hats to the past. The Plautus quotation is Martin’s, as is the assertion (very welcome to this amateur pianist) that the unplayable tempo Beethoven set for his “Hammerklavier” sonata (138 beats per minute!) was caused by a broken metronome.

Martin’s greater purpose is to trace, in the way we measure our metres and minutes, kilogrammes and candelas, the outline of “a true Copernican revolution”.

In the past fundamental constants were determined with reference to material prototypes. In November 2018 it was decided to define international units of measure in reference to the constants themselves. The metre is now defined indirectly using the length of a second as measured by atomic clocks, while the definition of a kilogramme is defined as a function of two physical constants, the speed of light, c, and Planck’s constant, h. The dizzying “hows” of this revolution beg not a few “whys”, but Martin is here to explain why such eye-watering accuracy is vital to the running of our world.

Sumpter’s Four Ways of Thinking is more speculative, organising reality around the four classes of phenomena defined by mathematician Stephen Wolfram’s little-read 1,192-page opus from 2002, A New Kind of Science. Sumpter is quick to reassure us that that his homage to the eccentric and polymathic Wolfram is not so much “a new kind of science” as “a new way to convince your friends to go jogging with you” or perhaps “a new way of controlling chocolate cake addiction.”

The point is, all phenomena are mathematically speaking, either stable, periodic, chaotic, or complex. Learn the differences between these phenomena, and you are half way to better understanding your own life.

Much of Four Ways is assembled semi-novelistically around a summer school in complex systems that Sumpter attended at the Santa Fe Institute in 1997. His half-remembered, half-invented mathematical conversations with fellow attendees won me over, though I have a strong aversion to exposition through dialogue.

I incline to think Sumpter’s biographical sketches are stronger. The strengths and weaknesses of statistical thinking are explored through the life of Ronald Fisher, the unlovely genius who in the 1940s a almost single-handedly created the foundations for statistical science.

That the world does not stand still to be measured, and is often best considered a dynamical system, is an insight given to Alfred Lotka, the chemist who in the first half of the 20th century came tantalisingly close to formulating systems biology.

Chaotic phenomena are caught in a sort of negative image through the work of NASA software engineer Margaret Hamilton, whose determination never to make a mistake — indeed, to make mistakes in her code impossible — landed the crew of Apollo 11 on the Moon.

Soviet mathematician Andrej Kolmogorov personifies complex thinking, as he abandons the axiom-based approach to mathematics and starts to think in terms of information and computer code.

Can mathematics really elucidate life? Do we really need mathematical thinking to realise that “each of us follows our individual rules of interaction and out of that emerges the complexity of our society”? Maybe not. But the journey was gripping.

# Soaked in ink and paint

This book, soaked, like the Dutch Republic itself, “in ink and paint”, is enchanting to the point of escapism. The author calls it “an interior journey, into a world of luxury and leisure”. It is more than that. What he says of Huygen’s milieu is true also of his book: “Like a ‘Dutch interior’ painting, it turns out to contain everything.”

Hugh Aldersey-Williams says that Huygens was the first modern scientist. This is a delicate argument to make — the word “scientist” didn’t enter the English language before 1834. And he’s right to be sparing with such rhetoric, since a little of it goes a very long way. What inadvertent baggage comes attached, for instance, to the (not unreasonable) claim that the city of Middleburg, supported by the market for spectacles, became “a hotbed of optical innovation” at the end of the 16th century? As I read about the collaboration between Christiaan’s father Constantijn (“with his trim dark beard and sharp features”) and his lens-grinder Cornelis Drebbel (“strapping, ill-read… careless of social hierarchies”) I kept getting flashbacks to the Steve Jobs and Steve Wozniak double-act in Aaron Sorkin’s film.

This is the problem of popular history, made double by the demands of explaining the science. Secretly, readers want the past to be either deeply exotic (so they don’t have to worry about it) or fundamentally familiar (so they, um, don’t have to worry about it).

Hugh Aldersey-Williams steeps us in neither fantasy for too long, and Dutch Light is, as a consequence, an oddly disturbing read: we see our present understanding of the world, and many of our current intellectual habits, emerging through the accidents and contingencies of history, through networks and relationships, friendships and fallings-out. Huygens’s world *is* distinctly modern — disturbingly so: the engine itself, the pipework and pistons, without any of the fancy fairings and decals of liberalism.

Trade begets technology begets science. The truth is out there but it costs money. Genius can only swim so far up the stream of social prejudice. Who your parents are matters.

Under Dutch light — clean, caustic, calvinistic — we see, not Enlightenment Europe emerging into the comforts of the modern, but a mirror in which we moderns are seen squatting a culture, full of flaws, that we’ve never managed to better.

One of the best things about Aldersey-Williams’s absorbing book (and how many 500-page biographies do you know feel too short when you finish them?) is the interest he shows in everyone else. Christiaan arrives in the right place, in the right time, among the right people, to achieve wonders. His father, born 1596 was a diplomat, architect, poet (he translated John Donne) and artist (he discovered Rembrandt). His longevity exasperated him: “Cease murderous years, and think no more of me” he wrote, on his 82nd birthday. He lived eight years more. But the space and energy Aldersey-Williams devotes to Constantijn and his four other children — “a network that stretched across Europe” — is anything but exasperating. It immeasurably enriches our idea of Christiaan’s work meant, and what his achievements signified.

Huygens worked at the meeting point of maths and physics, at a time when some key physical aspects of reality still resisted mathematical description. Curves provide a couple of striking examples. The cycloid is the path made by a point on the circumference of a turning wheel. The catenary is the curve made by a chain or rope hanging under gravity. Huygens was the first to explain these curves mathematically, doing more than most to embed mathematics in the physical sciences. He tackled problems in geometry and probability, and had some fun in the process (“A man of 56 years marries a woman of 16 years, how long can they live together without one or the other dying?”) Using telescopes he designed and made himself, he discovered Saturn’s ring system and its largest moon, Titan. He was the first to describe the concept of centrifugal force. He invented the pendulum clock.

Most extraordinary of all, Huygens — though a committed follower of Descartes (who was once a family friend) — came up with a model of light as a wave, wholly consistent with everything then known about the nature of light apart from colour, and streets ahead of the “corpuscular” theory promulgated by Newton, which had light consisting of a stream of tiny particles.

Huygens’s radical conception of light seems even stranger, when you consider that, as much as his conscience would let him, Huygens stayed faithful to Descartes’ vision of physics as a science of bodies in collision. Newton’s work on gravity, relying as it did on an unseen force, felt like a retreat to Huygens — a step towards occultism.

Because we turn our great thinkers into fetishes, we allow only one per generation. Newton has shut out Huygens, as Galileo shut out Kepler. Huygens became an also-ran in Anglo-Saxon eyes; ridiculous busts of Newton, meanwhile, were knocked out to adorn the salons of Britain’s country estates, “available in marble, terracotta and plaster versions to suit all pockets.”

Aldersey-Williams insists that this competition between the elder Huygens and the enfant terrible Newton was never so cheap. Set aside their notorious dispute over calculus, and we find the two men in lively and, yes, friendly correspondence. Cooperation and collaboration were on the rise: “Gone,” Aldersey-Williams writes, “is the quickness to feel insulted and take umbrage that characterised so many exchanges — domestic as well as international — in the early days of the French and English academies of science.”

When Henry Oldenburg, the prime mobile of the Royal Society, died suddenly in 1677, a link was broken between scientists everywhere, and particularly between Britain and the continent. The 20th century did not forge a culture of international scientific cooperation. It repaired the one Oldenburg and Huygens had built over decades of eager correspondence and clever diplomacy.