Flowery Regions of Algebra

There were fears of revolutionary violence in Paris in the spring of 1773. The police tried to quell the disturbances and make those responsible account for their actions, but they had no success. The trouble spread, first downriver to Normandy, then elsewhere. Statements in the newspapers designed to assuage fears and explain the source of the trouble had little effect, and the crisis lasted well into the summer. The cause of all the bother was some arcane speculation on the chances of comets crashing into the Earth. Journalists grumbled that future Frenchmen would be amused by the tale of ignorant citizens disordered by a mere astronomical calculation. Voltaire joined in, mocking Parisians who’d fled the city to escape the cometary apocalypse: they ‘aren’t philosophers and, if you believe what they believe, don’t have enough time left to turn into philosophers’. The Académie Royale des Sciences had inadvertently started the panic by announcing a talk on comet collisions and their possible hazards. To restore calm, a committee was organised, the evidence reviewed, and an announcement put out to the effect that the chance of a crash was 1:64,000. The sums did not soothe Parisian nerves. ‘The vulgar, ignorant and timid, having no other reason to reassure themselves about rather unusual phenomena but the example and authority of enlightened people, become alarmed very easily,’ the committee concluded.

When foreigners grope for some way of making sense of what they see as the extreme rationalism of the French, they use the word ‘Cartesian’. But they ought rather to say ‘Laplacian’. Pierre Simon Laplace was born into a prosperous rural family in 1749; he trained for the priesthood; and in 1773 was elected to the Académie. He had recently composed a series of astonishingly assured calculations about the application of probability calculus to the orbits of the planets and the distribution of comets in space and he was chosen to lead the Académie committee on comets’ orbits. More than most other Frenchmen Laplace helped define the ways in which precisely calculable laws seemingly govern both nature and society. He was interior minister under the Consulate, senator, head of the Parisian scientific community, and survived the Bourbon restoration as a newly ennobled marquis and grand old man of the establishment. Roger Hahn has been studying this career for half a century. He has located letters and papers thought to be lost, written on the tough problems of Laplace’s religious beliefs and his relation with Newtonian cosmology, and at last written a new biography, first released under the title Le Système du monde in France a couple of years ago.

Though Laplace’s life spanned Enlightenment, Revolution, Empire and Restoration, it was rarely disturbed and seems to have been devoid of much private crisis. An exception was the public embarrassment the family suffered a couple of years after the comet scare, when Laplace’s father was sued for supplying Carmelite nuns in Dieppe with adulterated cider. This was happily resolved when Laplace’s ally, the chemist Antoine Lavoisier, joined a commission of inquiry into the affair. Hahn gives much detail of the episode, and indicates how academic inquiries increasingly governed public affairs at the end of the Ancien Régime.

Hahn’s aim here is to give the stern mechanisms of Laplacian science a human face. The challenge is considerable, as he acknowledges: ‘He was a consistently rational and deliberate man, almost as steady in his behaviour as in his vision of the stable solar system.’ The vision included a grandiose account of the origins of the sun and the planets, tracing the current order back to its primitive vortex. Hahn does the same with Laplace: his opening chapters dwell at length on the Norman upbringing, the initial training for the priesthood, the fierce doctrinal controversies between his professors at the university of Caen. Even Laplace’s deliberation can somehow be found in his Norman roots, or so it’s claimed here: a ‘stereotypical Norman villager’, whether cider grower or devout student, was ‘openly cautious, but inwardly stubborn’. His decisive move to Paris in 1769, when he set himself on the path to the Académie and mathematical sciences, looks like a radical change of heart, a move from provincial piety to urban reason.

From then on, any life of Laplace must also be a study of collective enterprise, shifting between the salons of the high Enlightenment, the academic committees and Revolutionary tribunals, and the more relaxed world of imperial administration. Cash was never a problem. When he married the teenage Marie Courty de Romange in 1788, Laplace’s annual income was almost ten thousand livres; his wife brought a dowry ten times that. The Directory and Empire brought more wealth, servants, coaches, horses and a stately house near the Luxembourg Gardens. Every Sunday afternoon, during the imperial years of his greatest prestige, Laplace hosted a group of studious disciples and scientific tourists near Paris at Arcueil, in a pastoral retreat chosen by Marie: an entire Laplacian school of physical sciences was forged there.

His acolytes defined their work as a search for the physical forces that governed nature and which human ingenuity could then exploit. The sect’s doctrines were laid down, secured by measurement and promulgated as undeniable truths. All events had determinate causes, to be analysed and known by sufficiently precise measurements. There was no need to appeal to a meddlesome deity to explain anything in the course of nature. If there were a supreme intelligence that knew the positions and velocities of all particles in nature, it could then predict all future states. The task was admittedly impossible for humans, even Laplace: but his celestial mechanics allegedly gave a shadowy intimation of what this knowledge would be like.

For this astronomy, gravity was the fundamental force in the universe. It alone explained the long-term stability and orbital dynamics of all bodies in the solar system. Light was a stream of fast particles that moved under the influence of strong, short-range forces. Heat was a material fluid that could be captured and accurately measured in ingenious ice machines. Electricity, weighed and numbered, obeyed the same kind of mathematical dynamics as gravitation.

The ambition was global. Laplace’s group claimed Napoleon, ‘this triumphant hero’, as their presiding genius. Acknowledging that it was as hard to predict political revolutions as it was the state of the weather, Laplace dedicated his Analytical Theory of Probabilities to ‘Napoleon the Great’, then mired in the swamps of Russia. For Laplace, the Revolution was best seen as ‘a great experiment’. He had been Napoleon’s tough-minded examiner at military school just before the Revolution (he ranked the future emperor 42nd in a class of 58) and a decade later welcomed his former student as a mathematics fellow in the new Institut de France: ‘We expected everything from you, general, except geometry lectures!’ Laplace was in many ways the Bonaparte of physical sciences: their inspirational commander, conscientious planner and fearsome legislator.

Both men’s relations with the British were complex and somewhat fraught, though the distressing story that Laplace’s manuscripts were destroyed in the British bombardment of Caen after D-Day turns out to be false, as Hahn showed some years ago. Admittedly, Laplace’s initial reactions to the Revolution were astonishingly anglophile. He considered moving to England to escape the turmoil; he called the British constitution ‘the most perfect one can imagine’; he compared Cartesian antipathy to Newtonianism with the Ancien Régime’s hostility to constitutional monarchy. But this warmth was private and transient. A cottage industry started in Regency Britain to find respectable British precedents for each of the advances Laplace claimed. The leading London science lecturer and administrator Thomas Young denounced Laplace’s over-reliance on abstraction and his ignorance, or perhaps theft, of British achievements. ‘We complain also, on national grounds, of an unjustifiable want of candour . . . Mr Laplace may walk about and even dance, as much as he pleases, in the flowery regions of algebra, without exciting our smiles.’ Laplace’s cunning analysis of probability allowed reliable calculations to be made about the outcome of games of chance, and was then extended to the management of errors in astronomical observations. Laplace himself never made any professional astronomical observations, Hahn writes, but he was keen to tell observers how to cope with their slip-ups. Yet all these masterly techniques were traced by enthusiastic London mathematicians back to the work of Thomas Bayes. Surely, it was argued, there was little in the new French celestial mechanics not already present in the home-grown Newtonian tradition?

In fact, Laplace was so successful at turning Newton’s tantalising analyses into a coherent and authoritative world-system that many all too easily read the rationalist cosmos of the Laplacian Enlightenment back into Newton’s decidedly spookier and more creationist universe. Britain’s boast was thus retrospectively given a French accent. This is the comfortingly Newtonian Laplace who figures, for example, in Stephen Hawking’s Brief History of Time, as the celebrated author of a version of determinism which nevertheless left open the choice of physical laws and of initial configuration, and whose optimism about the possibility of deducing all future states of the universe from perfect knowledge of its current state has now, we’re told, been undermined by the uncertainty principle of quantum mechanics. For Hawking, as for many other commentators, Laplace’s philosophy represents the most ambitious and thus decisively flawed version of classical physics.

So for more than two centuries, experts of various stripes have been trying to explain to anglophone audiences what Laplace did and why it matters. Laplace had his own answer: his Exposition of the System of the World, an attempt to popularise the most up-to-date cosmology, astronomy and mechanics that started in 1795 as lectures at the short-lived first Ecole Normale and went through five increasingly lengthy editions in his lifetime. Hahn reckons this ‘the most elegant and readable work Laplace ever composed’, and points out that throughout the 19th century engineers at the Ecole Polytechnique were assigned it for their French language courses.

The final sections of Laplace’s exercise in public science set out a tediously triumphalist history of astronomy, from stupid error and mystical reverie through keener observation to analytical triumph, with Laplace himself as its culmination. This was the story soon seized on by Auguste Comte for Positivism, the version of Catholicism without Christianity that dominated much 19th-century French philosophising. Disciples also learned Laplace’s story of the origin of the solar system: just as humanity had gradually and inexorably developed through reason from error to astronomical truth, the world system evolved under gravity from a primordial spinning gas cloud into the planets and Sun. The typically Laplacian argument was probabilist: the chance of such an orderly alignment of all the orbits of the planets and moons was so slim that some unsurprising primitive cause must be responsible. Hence the inference of an original gas vortex.

There has been a string of translations of his work, notably of the Treatise of Celestial Mechanics, a programmatic part-work issued over decades from 1799, incorporating both the most complete versions of Laplace’s attempt to derive all the phenomena of astronomy from gravitational dynamics and his application of that programme to earthly physics too: to the behaviour of light, heat and air. Staying for a couple of months in late 1802 in Lima, Alexander von Humboldt grumbled about the city’s tedium, selfishness and indifference to suffering. Though he’d just climbed Chimborazo, then the highest recorded ascent, and happily surveyed the coastal guano deposits, by far the most significant event during his time in Peru’s capital was the belated delivery of a package from Paris. After three years’ wait, the second volume of Laplace’s Celestial Mechanics had arrived: ‘I set upon it with an unbounded avidity.’ For Humboldt and his colleagues, the book became ‘an invaluable code of law of which I only understand a few words here and there’. This telling mix of astonished respect and somewhat puzzled incomprehension defined what Laplace’s works meant in the age of Enlightenment and its newly global sciences.

Perhaps the most important – certainly the most adept – English-language version was the one made by the wealthy Massachusetts insurance manager Nathaniel Bowditch in the 1810s; the most celebrated was a rival version commissioned a decade later by the Society for the Diffusion of Useful Knowledge and made by the polymathic Mary Somerville. One radical MP grumbled that Somerville’s government pension was being wasted on this work: none of the great unwashed, it was predicted, would benefit from a bluestocking’s English version of a French physics textbook. Laplacian materialism and determinism played exceptionally badly in anti-Jacobin Britain. Professors at Cambridge and Edinburgh conceded that his mechanics and astronomy were indispensable tools in the sciences, but complained that his doctrines would undermine the Church and seduce men of science into regicide atheism. Unlike the great Sir Isaac, they insisted, Laplace and his kind had strayed from the path of faith into the snares of unbelief. Bonapartists responded that the honours heaped by the French state on Laplace showed how well a truly visionary government treated its intellectual elite. It became fashionable for bright young sparks to make the pilgrimage to Arcueil to lay tribute at the feet of the founder of modern physical sciences.

Hahn’s book often reads like a guide for such pilgrims. It is too brief to allow Laplace’s programmes to be seen in the wider context of revolution and world war, where dramatic changes in the status and direction of the physical and social sciences were taking place. Some of Laplace’s failings are acknowledged. Hahn oddly judges that ‘Laplace never reached the top tier of creative mathematicians of his day.’ He notes, more plausibly, the persistent troubles that Laplace encountered as a result of his somewhat lackadaisical, sometimes self-serving failure to cite the work of contemporaries and immediate predecessors adequately. Hahn has seemingly followed the advice once offered to Hawking: mathematical equations lower book sales. This restriction is severe. Previous books, notably Pierre-Simon Laplace 1749-1827: A Life in Exact Science by Charles Gillispie and his collaborators in 1997, were less timorous and so more effective in communicating the motifs of Laplacian sciences. Hahn’s study is presented as a complement to that work. Without it, it is a bit hard to see the deep and radical effect of Laplace’s memoirs on the methods through which true causes can be inferred from events that appear to be randomly distributed. The Enlightenment had not previously known a systematic science of probability. It had a doctrine of chances, focused on the hopes and rewards of players at their gaming tables. With Laplace and his allies, however, this set of techniques began to acquire truly cosmic meaning. There was no such thing as real chance in the world: anyone who thought otherwise was a victim of superstition and ignorance.

Hahn is at his best in his exposition of the materials with which he has been working since the 1950s: Laplace’s reflections on probability and religion, which include a striking group of manuscripts, preserved in relative secrecy in a black envelope in the library of the Académie des Sciences, where the great analyst set down his views on power, causation and the authority of scripture. These papers are reproduced here as an appendix, under the factitious and perverse label of ‘Non-Scientific Manuscripts’. In very private jottings, which Hahn plausibly dates to the period of the Consulate, Laplace traced the origin of the spurious Christian cult to astronomical myths. ‘The greatest truth is universal gravitation. Transubstantiation seems to me the greatest of all absurdities.’ The private notes insist on the immutable laws that govern human will, thought and society, and urge that what looks like chance is simply a sign of ignorance of the true causes governing all phenomena. This ignorance, manifest, he implies, in the turmoil of revolution and war, is the key source of popular stupidity, its superstitious fears and passions so evident in the comet scare of 1773.

Laplace’s acquaintances sometimes revealed that the establishment of order was as much a domestic passion with him as it was with Revolutionary France. As Hahn’s book only rarely hints, Laplace’s social life did not always display the virtues of balance, order, reason and measure. When François Arago, the astronomer and political radical from Perpignan, first visited Laplace’s home in 1804, he was astonished by his immoderate wrath and absurdly petty tyrannies. ‘Ah, I said to myself, the ancients were really inspired when they attributed these kinds of failing to those who nevertheless could make Olympus tremble even with the slightest frown.’ Ambitious to weld social order to scientific standards, Laplace’s Olympian world-view demanded political and scientific assessment simultaneously. Hahn well shows that successive versions of his magisterial text on probability, written during the years between Borodino and Saint Helena, drew strikingly apt lessons about life and liberty, war and peace: ‘Every time a great power intoxicated by the love of conquest aspires to universal domination, the sense of liberty among the unjustly threatened nations breeds a coalition to which it always succumbs.’

Laplace reckoned this was far more than an ethical maxim tragically ignored by his former student and patron, Napoleon. Rather, it was the inevitable consequence of the iron law according to which ‘natural limits operating as constant causes’ must prevail. His sciences showed that nature was a secure empire governed by Laplacian forces, but that an empire of Napoleonic scope and violence could not itself be secure. Such a principle chimed with Laplacian scientism, with the intemperate confidence that quantitative analysis has long possessed about its ability to master human life. The continuing authority of this kind of scientism is an important reason why we are in need of better accounts of how such faiths began and the effects they had.