These two aspects of Poincaré’s work—the variable and the fixed—emerged together and can only be understood together. He says, in different ways over many years: Manipulate the flexible aspects of knowledge as tools; choose the form that makes the problem at hand simple. Then seize those relations that stand fast despite the choices made. Those fixed relations stand for knowledge that endures. Together the variant and invariant make scientific progress possible.
For a hundred years, scholars have struggled to get at the root of Poincaré’s conventionalism. Some, with good reason, have stressed the role of geometry. As Poincaré emphasized over and again, however, mathematical statements can be made in the language of non-Euclidean just as well as in Euclidean geometry. Other scholars have scrutinized the works of earlier figures in geometry—such as Felix Klein, the great German geometer who propagandized so forcefully for different kinds of geometry. Still others have gone back to Sophus Lie for the root of Poincaré’s picture of free choice among geometries. After all, Poincaré explicitly cited Lie as a mathematical forebear, and Lie was quite clear on the arbitrariness of many choices made by the mathematician. For example, Lie said that Descartes identifies the variables x and y by a point on the plane. But (according to Lie) Descartes could, “with equal validity” have chosen to symbolize x and y by a line, and develop geometry from that asssumption. Moreover, Descartes defined x and y according to a specific coordinate system—the x and y refer to their distances from the x- and y-axes. There too is a certain free choice: “[P]rogress made by geometry in the 19th century,” wrote Lie, “has been made possible largely because this two-fold arbitrariness . . . has been clearly recognized as such.” Lie was here contending that mathematical progress followed from the recognition that there were always many ways to represent mathematical concepts. Choosing the particular representation of mathematics, choosing this or that geometry is, Lie argued, a matter of “advantage and convenience.” It was, as one scholar has persuasively argued, “one of the grounds of Poincaré’s commitment to a view of geometry that held our choice of geometries to be one of open choice, grounded by convenience.”36 No doubt Poincaré’s emphasis on free choice in geometry can be hunted back to the German polymath Hermann Helmholtz, who struggled to disentangle factual meaning from definitions in geometry, always emphasizing the central role of mobile, rigid objects in fixing our concepts of space. It may be too that Poincaré’s ideas on mathematical conventionalism should be pursued back to the myriad geometries of Bernhard Riemann, or for that matter to the more recent work of Poincaré’s teacher, Charles Hermite.37
Propelling this sense of a freedom of choice was a pedagogical conventionalism (if you will) to be found in the stridently agnostic instructional style of the Ecole Polytechnique. That abstention from absolute commitment to any particular theory was a prominent feature of Alfred Cornu’s courses, which Poincaré had attended as a student. Alternative theories each had advantages and disadvantages; all were constrained only by the much-emphasized experimental fixed points. For Poincaré, the invariants of physics (which provided objective knowledge) were the fixed relations between experiments, relations that survived the ever-changing flux of theories. Recall that the same free choice among theories had informed Polytechnique’s even-handed hiring—some representative scientists for atomism, some against. Such an abstention from absolute theoretical commitment characterized Poincaré’s own courses; lecturing on electricity and optics in 1888, 1890, and 1899, for example, he gave each major theory its moment in the sun, displaying its virtues and vices for the students to judge. A rhyzomatic “root” of conventionalism here too.
Finally, in Poincaré’s exchanges with his brother-in-law’s (Emile Boutroux’s) philosophical circle, Poincaré would have found conventionalism in a philosophical register. This loose affiliation of philosophers and philosophically minded scholars offered Poincaré, early on, a more reflective view of the mathematical sciences. Raw empiricism was avoided as woefully inadequate to account for the generality and extent of scientific knowledge. Pure idealism (reducing reality to mental life) could not explain the concordance of ideas with the world. Drawing strongly on the Kant revival underway in Germany, Boutroux and his circle rejected both the extremes of idealism and empiricism. Taking science and the humanities to be inextricably bound, these philosophers saw both structured by an active role for the mind and a suspicion toward the purely metaphysical. In his encounters with Auguste Calinon’s work on the philosophical foundations of physics, Poincaré walked this philosophical middle line straight toward the problem of simultaneity.
Geometry, topology, pedagogy, philosophy—each of these ways of parsing Poincaré’s world tells us something about the ways in which scientific “free choice” made sense to him. Intriguingly, around 1890, Poincaré began regularly to call “free choice” by a new name, insisting (as he had in 1887) that the geometrical axioms are neither experimental facts nor (as some Kantians would have it) printed in advance on our human minds. In a curt, insistent sentence printed in 1891, he lay down a new formulation of his view of geometric axioms: “They are conventions.”
Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true, and if the old weights and measures are false; if Cartesian co-ordinates are true and polar co-ordinates false. One geometry cannot be more true than another; it can only be more convenient. Now, Euclidean geometry is, and will remain, the most convenient.38
Here the status of the axioms of geometry have been explicitly likened to terms in a language that can be freely chosen and (in this quotation) also to the freedom the mathematician or physicist always has to choose a coordinate system. The new element is that Poincaré depicts the choice of Euclidean or non-Euclidean axioms not just as a choice among groups, but as a choice between the arbitrary system of meters and kilograms and the arbitrary system of feet and pounds.
To appreciate this aspect of Poincaré’s use of “convention,” we must recognize that his reference to weights and measures contains the trace of an entire world of conventions. At the same time, as we will see, Poincaré’s concern with meters and seconds cannot be considered an external “influence,” one that determined his scientific and philosophical work the way a hidden magnet rearranges the iron filings above it. “Roots” and “influences” are terms too weak and external to capture Poincaré’s thoroughgoing engagement with the practice of setting planetwide conventions.
The world of decimalized, conventionalized time and space was anything but abstract to Poincaré. He flourished in (and contributed to) the Parisian, indeed worldwide skein of wires, meetings, and international treaties. Here was, after all, the consummate Polytechnician who moved as easily in the depths of a coal mine as in the far reaches of astronomical stability. But to make visible the mechanisms of clocks, rods, and wires—above all, to grasp the production of the late-nineteenth-century conventional understanding of simultaneity—it is necessary to pull back to a wider perspective. We need to move back and forth, in and out, between the details of work in philosophy, mathematics, and physics and the larger-scale social and technological conventionalization of time and space in which Poincaré took part.
In to the precision swing of master clock pendulums, out to the undersea telegraph cables crisscrossing the oceans. In to follow the minutae of individual train schedulers, jewelers, and astronomers; then back out to the legal recalibration of national and world-covering time zones. In this process of scrutiny, historical light necessarily plays off the very different scales utilized by technological, scientific, and philosophical activity. Between 1870 and 1910, conventions of space and time scintillated with a critical opalescence.
Chapter 3
THE ELECTRIC WORLDMAP
Standards of Space and Time
PARIS, HÔTEL DES AFFAIRES étrangères, 20 May 1875, 2:00 P.M. Represented by their decorated plenipotentiaries, seventeen names will be put to a treaty, their resplendent titles marching across the p
age: “His Majesty, the Emperor of Germany,” “His Majesty, the Emperor of Austro-Hungary,” “His Excellency, the President of the United States of America,” “His Excellency, the President of the French Republic,” “His Majesty, the Emperor of All Russia. . . .” We are at the solemn signing of the Convention of the Meter. After years of negotiation, the High Contracting Parties now called into existence an international bureau of weights and measures. The new prototypes of the meter and kilogram it was charged with certifying would supplant the myriad of competing national measures, establish the relation between these gauges and all others, and compare results with the standards used to map the earth.
Here, in the convention, diplomacy met science. When Duke Louis Decazes, the French minister of foreign affairs, sent out invitations to other countries back in 1869 for a diplomatic conference on this issue, he invited politicians, but also leading scientists like the German astronomer Wilhelm Förster, who was director both of the German Bureau of Weights and Measures and the Berlin Observatory. By March 1875 the committee had come far enough for Decazes to gently retire the scientific domain, in which the assembled held only a “relative competence,” in order to focus on “questions of a political and conventional order (ordre conventionnel),” where they had “absolute competence”: their conclusions would form the basis for binding international law. Conventions joining scientific and legal technologies had been concluded before—in 1865, for example, governing telegraphy. Indeed dozens of conventions had aimed to smooth collisions between countries in trade, post, and colonization. Now, in the vital domain of the metric system, even more than the telegraph accord, the delegates had produced an “international contract” as dear to scientists as it was to industrialists and politicians: a legal document that would rule from the spotless precision of the physics laboratory to the smoke and steam of the factory.1
If Decazes spoke for diplomacy, Jean Baptiste André Dumas, organic chemist and, since 1868, perpetual secretary of the French Academy of Sciences, spoke for French scientific enthusiasm. As head of the special (scientific) commission on the meter, Dumas had been responsible for the recommendations that now stood before his colleagues. Partially summarizing, partially lobbying, Dumas stood before his fellow delegates to advocate a permanent bureau in Paris vested with the authority to set, maintain, and distribute international standards. Above all, Dumas wanted to justify the universal meter as a standard for industry, for science, for France, for the world. As he saw it, anyone who had set foot in London’s 1851 Universal Exposition immediately recognized that “chaos” reigned between national systems. Each country’s peculiar system of weights and measures made comparison among them impossible without tedious calculation. At the same time, every subsequent exposition had demonstrated that the reach of the metric system was steadily growing. Everywhere people wanted to throw out discordant measures; they yearned to smash intellectual barriers between peoples. For Dumas, indeed for many senior French scientists, the call for international standards would be heard by all “enlightened men.” Having embraced the metric system throughout physics and chemistry laboratories, scientists now taught it widely. Factories, builders, telegraphs, and railroads had seized the meter. Now, Dumas urged, public administration should back the rational meter.
Dumas: Decimals mattered. For both sides, practical and pure science, it was the decimal character of the metric system that mattered. Twelve inches in a foot, three feet in a yard—neither plumber nor physicist could cherish such a hodgepodge. “As for the geodesic origin of the metric system,” that pride of the French Revolution, by now “it is absolutely without interest for commerce, for industry and even for science.” Upon its adoption in 1799, the meter was supposed to be exactly one ten-millionth of a quarter of the earth’s circumference. Dumas assured his listeners that modern proponents of the metric system made no such claim; the assembled knew perfectly well that the earth’s size could not be measured with the precision needed for an international standard. For Dumas, the reason for adopting the metric system was because it divided lengths into sensible units of ten. That was what pure scientists wanted and what pragmatic journeymen demanded. To spread this new rational system a center was needed. It should be “neutral, decimal, international.” It should be ça va sans dire, in Paris.2
Dumas reminded his audience that the metric standards had become international precisely because revolutionary France had designed the system to make it so. Long ago, ancient Hebrews had put their measuring prototypes in the Temple. Romans set their standard in the Capitol, Christians sequestered theirs in the Church (which was how Charlemagne’s standard kept its original purity). For eighty years, the Archives had performed this task for France, preserving the standard meters since revolutionary times. But now that the high contracting parties had decided to make the meter a truly international standard, they judged the revolutionary meter neither strong enough nor sufficiently invariable to serve as the prototype for the world’s measures.
Signing the Convention of the Meter started, rather than ended, the process of distributing the meter. Bureaucrats and scientists lobbied, bullied, and negotiated their countries toward putting the scheme into practice. Some of the great experimenters of Europe and the United States contributed to it: Armand Fizeau, who had measured the “dragging” of the ether by water, as well as the American Albert Michelson, who invented the interferometer, an instrument capable of measuring length to within a fraction of the wavelength of visible light. For fourteen years, French engineers and British metallurgists hammered and smelted their way to a tough, durable iridium-platinum alloy.
While a British firm pounded these hard, pure bars into meter sticks with an inflexible “X” cross section, the French concentrated on producing an enormous “universal comparator” (see figure 3.1), that would, by strict procedure, allow a standard length to be reproduced on another bar to within two ten-thousandths of a millimeter. It was painstaking, nerve-wracking work. When the British metal workers delivered their precious bars to the French, the operator at the conservatoire would set both the standard meter and the blank bar on the bridge of the comparator. Peering through a microscope (M), the operator would line up the one-meter mark on the standard. Then the operator would activate a lever, causing a diamond blade to inscribe a fine line precisely at the one-meter point on the blank. Carving subdivisions was just as difficult. The two microscopes would be set, say, ten centimeters apart. The operators would mark that length. Sliding the bar down, they would etch a second ten-centimeter length into the bar, and so on. To prepare the 30 standard bars that the international delegates would take home with them, the operators repeated this operation 13,000 times. The slightest slip with the diamond point meant starting over again with repolishing of the blank.3
Finally, on Saturday, 28 September 1889, two years after Poincaré was elected to the Academy of Sciences, eighteen representatives of the contracting parties gathered in Breteuil for the final sanctioning of the meter. The president of the conference canvassed their votes—unanimous—and then pronounced: “This prototype of the meter will from now forward represent, at the temperature of melting ice, the metric unit of length,” while “this prototype [kilogram] will be considered from now on the unit of mass.” All standards stood on display in the meeting room: meters sheathed by protective tubes, kilograms nested in triple glass bell jars. According to plan, each delegate ceremoniously picked a ticket from an urn, the number received assigning his country a meter stick, for which he offered a signed receipt.
Figure 3.1 Universal Comparator. This machine served to rule precise lengths for platinum-iridium copies of the standard meter, M. For engineers, physicists, politicians, and philosophers—especially in France—the international success of the standardized unit of length served as a model for what they hoped would be the decimalization and standardization of time. SOURCE: GUILLAUME, “TRAVAUX DU BUREAU INTERNATIONAL DES POIDS ET MESURES” (1890), P. 21.
Suddenly, these care
fully scripted proceedings ground to an abrupt halt. The most important act—the deposit of the meter in its underground safe—was possible only with the three keys needed to open the vault. One of those keys would be in the hands of the director of the French Archives, but he was not there. The president suggested they ask for instructions from the French minister of commerce, but the delegates vigorously objected. Swiss astronomer Adolph Hirsch insisted that the conference was international, not French. The conference would not address an ordinary French minister. Out of the question: Hirsch and his colleagues would deal with France only through its minister of foreign affairs. Diplomacy apparently produced the missing key.
Later that afternoon, at 1:30 to be precise, the commission charged with depositing the international prototypes gathered in the lower basement of the Breteuil Observatory. There the delegates certified that the international prototype M would from that moment forward be enclosed in a case covered on the interior with velvet, lodged within a hard cylinder of brass, screwed tight, locked, and placed in the vault. Alongside M the standard bearers then prepared two “witnesses” for burial (meter sticks, not delegates). These metallic observers would forever testify, by the very conditions of their bodies, to anything that might befall M. In the same ceremonial interment, convention delegates sanctioned the kilogram, K, elevating and renaming it as the universal standard of mass. It too found its eternal resting place in the underground iron vault in the company of its witnesses. With two keys, and in full view of the delegates, the director of the International Bureau of Weights and Measures locked the case, secured the inner basement door with a third key, and bolted the exterior door with a fourth and a fifth key. At the conclusion of these solemn events, the president of the conference handed these latter keys in separate, sealed envelopes: one to the director of the International Bureau, one to the general guard of the National Archives, and the last to the president of the International Committee. From that time on, all three basement keys would be needed to enter the sanctum sanctorum.4
Einstein's Clocks and Poincare's Maps Page 8