Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

by Alexander, Amir

  Under the king’s patronage, the Royal Society became a trend-setting scientific organization and, along with the French Royal Academy of Sciences, a model for scientific institutions in Europe and beyond. Its regular meetings in these early years were devoted to public experiments in optics, the structure of matter, the reality of a vacuum, and telescopic observations, among other topics, executed by the Society’s curator of experiments, Robert Hooke. Most famously, Robert Boyle’s experiments with the air pump, in which he investigated the structure and composition of air, were conducted in public laboratories of the Royal Society in front of numerous witnesses. In 1665 the Society’s secretary, Henry Oldenburg, launched Philosophical Transactions of the Royal Society of London, one of the first scientific journals, and certainly the longest-running in the world. Philosophical Transactions reported not only the investigations of the Society’s fellows but also studies conducted by others, making the Society a world center of scientific research.

  Some of the practices of the early Royal Society might seem peculiar to a modern scientist. For example, there was little distinction between what today would be considered amateurs and professionals, and the pages of early Philosophical Transactions were filled with reports of unusual weather phenomena and the births of monstrous or malformed livestock. Social rank also mattered a great deal in the Society, and quite a few prominent gentlemen owed their fellowship to their illustrious lineage rather than to any scientific distinction. Also puzzling from our perspective is that the experiments were performed in public, that is, in front of an audience of Society fellows and sometimes other notable guests. All those present would then discuss what they had seen, examining its meaning and significance. To a modern scientist this would seem more like a circus performance than a proper scientific experiment.

  Some of the differences between the early Royal Society and modern scientific procedures can be attributed to the fact that science in the seventeenth century was still young, and its practices still very much in flux. The professional career scientist is a creation of the nineteenth, not the seventeenth, century. Other differences are due to the Royal Society’s viewing itself as much more than a scientific institution of the kind we know today. A modern scientific institute or university department is concerned exclusively with scientific research and education, its success measured by the number and quality of its publications and innovations. The Royal Society also emphasized its research and innovation, always insisting on the usefulness of its discoveries. But in addition to this, it took upon itself a mission unlike that of any of its modern counterparts: to provide a model for the operation of the state as a whole.

  This mission has its roots in the meetings of the London group, back in the 1640s. Outside their meeting hall the group’s members might be radicals or moderates, Presbyterians or Independents, Parliamentarians or even Royalists, all engaged in a life-and-death struggle for domination. But inside their meetings, none of that mattered: irrespective of their religious and political affiliations, they could all pursue the investigations of nature in peace and civility. “It was Nature alone,” wrote Sprat of those early meetings, that “… draws our minds off from past, or present misfortunes … that never separates us into mortal Factions; that gives us room to differ, without animosity; and permits us to raise contrary imaginations upon it, without any danger of Civil War.” In pursuing nature, Wallis, Boyle, and their associates created a safe space where even disagreements could be managed in peace and civility. It was a welcome relief from the cutthroat politics of the Interregnum.

  But what started as a simple refuge ultimately developed into an ideal: if reasonable men of different backgrounds and convictions could meet to discuss the workings of nature, why could they not do the same in matters that concerned the state? Why could Parliamentarians and Royalists not resolve their differences in peace and civility rather than murdering each other on the battlefields of northern England? Why could Independents, Presbyterians, and Anglicans not come to a reasonable agreement on Church government instead of each trying to impose their own system and suppressing all others? The harmony that prevailed in the natural philosophers’ meetings, even among men who disagreed sharply, seemed to hold an important lesson for the entire English body politic. For as Sprat put it, in those meetings, “we behold an unusual sight to the English Nation, that men of disagreeing parties, and ways of life, have forgotten to hate, and have met in the unanimous advancement of the same Works … For here they do not only endure each other’s presence without violence and fear; but they work and think in company, and confer their help to each other’s Inventions.”

  In the harsh climate of Interregnum politics, Wallis and his fellows reveled in their ability to conduct their business in peace, cooperate despite disagreements, and together advance the cause they all cherished. By the time they emerged from the shadows and were officially chartered by Charles II, they were ready to spread the word, and use their experience to reconstitute the entire body politic. The dogmatism of the preceding decades would be replaced by the moderation and open-mindedness that characterized their meetings and their science. The hubris of the fanatics would be replaced by the modesty of the experimenter, passions by rational debate, and the intolerance of sects by the tolerance of different but reasonable men working together for a shared cause.

  Presenting itself as a model for the state, the Royal Society tried to be as inclusive as possible. There was nothing democratic about it, to be sure, and members of the lower classes were unwelcome in its halls just as they were unwelcome in the political class. Wallis, Boyle, and their fellows feared and distrusted the common people, convinced that the only way to achieve peace and order was to restore the authority of the propertied classes. But when it came to gentlemen, the Society sought to set an example of openness, and this meant accepting into their ranks men of even modest accomplishment. For Society fellows to have presented themselves as “professionals,” to the exclusion of “amateurs,” would have smacked too much of the sectarianism of the past, in which one group sets itself up as the judge of all others.

  The public experiments of the early days also played a role in the Royal Society’s political mission, serving as an example of how reasonable men of good faith could discuss difficult issues and come to an agreement. The model was to be the private meetings of the Society’s founders during the Interregnum, in which they experimented and debated, offering different interpretations of what they were observing. In the end, though, they would arrive at some interpretation they could agree on—even if it left many questions unanswered. But for such discussions to take place, now that the Society was an official institution, it would not do to conduct the experiments in the privacy of a secluded laboratory. If members were to form an opinion, they would have to observe the proceedings themselves. It was therefore essential that the experiments be conducted before witnesses of unimpeachable character, most often other fellows, who could then discuss what they had seen and come to an agreement on what had occurred. A modern-day laboratory, in contrast, does not carry the ideological burden of the early Royal Society. It relies exclusively on the testimony of experts, safely assuming that laymen would not comprehend the proceedings anyway.

  Not all forms of natural philosophy were equally suited for the Society’s goals of promoting peace, tolerance, and public order. Particularly suspect were grand philosophical systems that claimed to arrive at indisputable truths through the power of pure reason. One such system, which was very much on the minds of the Society’s founders, was Cartesian philosophy (named after its originator, René Descartes), which was sweeping the Continent at that very time. In his writings, Descartes purported to dismantle all unsubstantiated presuppositions, reducing all knowledge to a single unshakeable truth: “I think, therefore I am.” From this rock of certainty he then recreated the world through rigorous step-by-step reasoning, accepting the validity of only clear and distinct ideas. And since his reasoning was flawless, Descartes
(and his followers) argued, his conclusions must inevitably be true.

  Boyle, Wallis, Oldenburg, and the other leaders of the early Society were deeply impressed by Descartes, but also very critical of his approach and conclusions. They were even more concerned with another system anchored in pure reasoning that was lurking in their own backyard, and that of course was Hobbes’s philosophy. Hobbes and Descartes differed radically on many critical issues, but this much they had in common: both believed that their system was structured like Euclidean geometry, founded on self-evident assumptions and proceeding through rigorous reasoning to truths. And it was precisely this unquestioning confidence in the validity of their systematic reasoning and the absolute truth of their conclusions that the founders of the Royal Society found particularly dangerous.

  The problem with dogmatic philosophy, Sprat explained in his History of the Royal Society, “is that it commonly inclines such men, who think themselves already resolv’d, and immovable in their opinions, to be more imperious, and impatient of contradiction.” Such an attitude is detrimental to science because “it makes them prone to undervalue other men’s labours, and to neglect the real advantage that may be gotten by their assistance. Least they should seem to darken their own glory.” It “is a Temper of mind, of all others the most pernicious,” Sprat continued, and one to which he attributes the “slowness of the increase of knowledge amongst men.” Even worse, this kind of arrogance easily leads to the subversion of the state: “The reason of men’s contemning all Jurisdiction and Power proceeds from their idolizing of their own Wit … they suppose themselves infallible.” This leads inevitably to sedition, because “the most fruitful parent of Sedition is Pride, and a lofty conceit of men’s own wisdom; whereby they presently imagine themselves sufficient to direct and censure all the Actions of their Governors.”

  Sprat was only twenty-eight when he was elected fellow in 1663, a young and not particularly distinguished man who was probably recruited for the express purpose of writing the Royal Society’s history. But if Sprat was a relative nonentity at the time, the men who commissioned him to write were the Society’s greatest men. These included the Society’s president, Lord Brouncker; its secretary, Henry Oldenburg; and its leading scientist, Robert Boyle, all of whom reviewed and corrected Sprat’s text to make sure it accurately presented their views. As a result, History of the Royal Society is not just a summary of Sprat’s private reflections, but a public statement of the goals and purpose of the Royal Society as understood by its leaders at the time. And when it came to their views on dogmatic philosophies, their verdict was clear: dogmatism leads to sedition and subversion of the state, and was not the kind of approach that would be practiced in the Royal Society.

  The alternative to the dogmatic rationalism of Descartes and Hobbes, the founders of the Royal Society believed, was experimental philosophy. Instead of pride, experimentalism bred humility, and whereas the rationalist philosophies led to pettiness and envy of rival philosophers, experimentalism fostered cooperation and mutual trust. Most important, instead of sedition and subversion, “the influence of experiments is Obedience to the Civil Government.” Unlike the rationalist philosopher, the experimentalist never claims he has discovered the only true system or that his results are absolutely and irrefutably true. Instead, making no assumptions about what he will find, he humbly proceeds from experiment to experiment, trying to make sense of what he finds. His conclusions are always the best that he can supply at the moment, but can always be overturned by the next experiment. Not for him are Hobbes’s bold pronouncements about matter, human nature, and the only viable commonwealth. To the contrary, he proceeds slowly, conducting many different experiments many times over, and only then will he venture, carefully and somewhat reluctantly, to provide a provisional interpretation of the results.

  Experimentalism is a humbling pursuit, very different from the brilliance and dash of systematic philosophers such as Descartes and Hobbes. It is, wrote Sprat, “a laborious philosophy … that teaches men humility and acquaints them with their own errors.” And that is precisely what the founders of the Royal Society liked about it. Experimentalism, as Sprat noted, “removes all haughtiness of mind and swelling imaginations,” teaching men to work hard, to acknowledge their own failures, and to recognize the contributions of others. This is precisely the attitude the founders of the Royal Society hoped to engender in the body politic as a whole. In place of the intolerant fanaticism of the parties and sects that had thrown the commonwealth into violence and chaos, experimentalism would breed moderation, cooperation, respect for differing opinions, and ultimately civic peace.

  When members of the Royal Society celebrated the glories of the experimental method, they also celebrated the man whom they considered the founder of it all, the “one great man who had the true imagination of the whole extent of this Enterprize.” He was Francis Bacon, Lord Chancellor to James I, who in his retirement had authored some of the most influential works ever on proper scientific method. In contrast to his younger contemporary Descartes, who had argued that true knowledge must be based on clear and rigorous reasoning, Bacon had insisted that true knowledge of nature could be acquired only by observation, experimentation, and the careful gathering of facts. To the Royal Society, Bacon was the prophet of the experimental method, and the spiritual father of the Society itself, though he died many years before its founding. In fact, the Society considered itself the true incarnation of Bacon’s “Salomon’s House,” a state institution for the study of nature that he proposed in his utopian work New Atlantis.

  There is irony here, because Bacon’s secretary in his final years was none other than Thomas Hobbes. As an avowed rationalist, Hobbes had derided the value of experiments in his dispute with Robert Boyle, and his thinking was not much influenced by his distinguished employer (except perhaps in his abiding interest in the natural sciences). But there was no getting around the fact that for all their idolization of Bacon, none of the Society grandees had actually known the Lord Chancellor, whereas their enemy Hobbes had been his intimate companion.

  The brilliance of Bacon’s reputation has hardly diminished, even to this day. Though not a creative scientist himself, he is nevertheless considered one of the crucial figures in the scientific revolution, whose writings made possible the growth and expansion of science. Bacon provided a brilliant defense of the experimental method, which had been viewed as suspect during the centuries in which Scholastic dispute and reliance on ancient authority were considered the proper path to true knowledge. He provided a road map for the development of experimental science, advocating for the systematic collection of data by a multitude of field-workers, and its concentration in a centralized institution for systematic evaluation. More than anything, perhaps, he made the experimental method respectable.

  Long before Bacon’s time, there were always those who tried to extract the secrets of nature through the rough method of trial and error. Sometimes they succeeded brilliantly, as in the inventions of gunpowder and the compass, other times less so, as in the case of the alchemists, who built sophisticated laboratories equipped with chemicals and furnaces in their search for the elusive philosopher’s stone. But any knowledge gained by these methods, even when it proved useful, was not considered appropriate for teaching in institutions of higher learning. It was “rude” and “mechanic,” associated with the lower classes, who dirtied their hands and worked for a living. No self-respecting gentleman would ever stoop to engage in such work, for fear of being tainted by its plebeian association. True knowledge, worthy of academic study, was to be found in the writings of the great masters of the past, or derived from them through exacting logical reasoning. Experimental results were not considered knowledge at all, since they relied on the notoriously unreliable senses and did not therefore rise to the required level of certainty. Bacon, almost single-handedly, demolished this perception. Here was no less a personage than the Lord Chancellor of England promoting experimentalism as th
e proper path to true knowledge. At a stroke, the uncouth practices of “rude mechanics” became a worthy pursuit for the intellectually curious gentleman.

  There is one aspect of Bacon’s methodology, however, that has often been criticized: his belittlement of mathematics as a tool of science. It is not that he ignored mathematics completely, since he did acknowledge that objects in the world had quantity, and mathematics was the science of quantity. But Bacon thought that mathematical knowledge was far too general to be of serious use. “It is the nature of the human mind,” he wrote, “to delight in the open plains (as it were) of generalities, rather than the woods and inclosures of particulars,” and mathematics was the best field to “satisfy that appetite.” Such an approach, however, is “to the extreme prejudice of knowledge,” because all knowledge worth pursuing lies in the particulars of the tangled woods, not in the generalities of the open plains. Mathematics can be useful, Bacon concedes, but only as the handmaiden of the experimental fields, not as a science unto itself. Nothing could be worse for the growth of knowledge than “the daintiness and pride of mathematicians, who will need have this science almost domineer over Physic.”

  Bacon’s suspicion of mathematics as a tool for comprehending the world is not hard to understand. For mathematics to describe nature correctly, nature must be mathematical—that is, structured according to strict mathematical principles. If that is the case, then all one needs in order to gain insight into the workings of nature is to follow the rules of rigorous mathematics, and all observations and experiments are superfluous. But Bacon made no such assumption. There is no way of knowing how the world is structured, he believed, until one engages in careful and systematic observations. The idea that one can deduce the workings of nature by mere mathematical reasoning is a dangerous illusion based on unwarranted pride, and is bound to lead any scientist astray.