The Mathematical Philosophy
By Tore Frängsmyr
There are two distinct opinions regarding the status of science in the 18th century. One dismisses the period as an uninteresting interval between the breakthroughs of the 17th century and the expanding industrialism of the 19th. The opposing view holds that the 18th century was the time when all the important work of the previous century bore fruit.
The dismissers argue that nothing really new emerged in the 18th century. "By many historians the century has been deplored for allegedly producing science that was boring, unoriginal, lacking in rigour and overspeculative." It has been castigated as "comparatively undistinguished in its science"; if its science is included in its reputation, the siècle des lumières cannot be freed from "an element of dullness."
In direct opposition to these nay-sayers, other historians have identified much of the intellectual life of the 18th century with the fostering of its heritage from the scientific revolution. The Enlightenment thus climaxes the development of early modern science. The most recent general assessment of the role of science in the Enlightenment gives particular weight to the century's efforts to consolidate and organize what it had received. "The creation of the new scientific disciplines was probably the most important contribution of the Enlightenment to the modernization of science, and one that we might easily overlook. It marks the Enlightenment as a period of
transition between the old and the new." That careful and informed students of the 18th century have reached diametrically opposed evaluations of its science is largely a consequence of their differing ideas about what counts as science. In this volume we sidestep the difficulty by restricting our attention to the working out of a particular method, or, as it might better be called in many instances, of an appeal to a slogan. That slogan, as Galileo formulated it, holds that "the book of nature is written in the language of mathematics."
In the 18th century, this proposition inspired the work of the most productive mathematicians and underlay the program of the most ambitious philosophers, who proposed to extend it beyond the book of nature to the books of man—that is, to all exercises of human reason. The languages of reason and nature were thereby to agree. We see two phases in the 18th-century elaboration of this proposition. Until about 1760, "mathematics" was applied to the general exercise of reason (as opposed to mathematics itself and the quantified parts of astronomy and physics) more as a symbol and slogan than as a useful tool. In this first phase the philosophy of Leibniz's epigone, Christian Wolff, is exemplary. In the last third of the century, however, the equation of reason with mathematics inspired instrumentalist and quantitative approaches to a wide range of problems in various sciences, arts, and technologies. We take the activities of this quantifying spirit to have been progressive, in the sense that under its inspiration the achievements of the scientific revolution were extended and reinterpreted in a way that made possible rapid advance in mathematics, physical sciences, and statistics (conceived as the sciences of the state) in the early 19th century.
In what follows, Wolff's method is seen as representative of the process by which reason was "naturalized" during the first half of the 18th century. The purpose is to provide both an introduction and a contrast to the applications of the quantifying spirit later in the century that form the subject matter of the other essays in this volume.
Leibniz wanted not only to see the world as a machine with a "static structure of matter in geometrical and mechanical relation"; he also wanted to give exact mathematical expression to all the processes and changes that took place in the world. Through the infinitesimal calculus he found a constant law that remained valid for an infinite number of processes and changes. The elements constituting the universe, nature, and human life were not particles of matter, but forces and motions. Each element or individual substance—Leibniz called them "monads"—was complete in itself, a microworld or a mirror of the universe, and yet intimately related to all other individuals in a system of pre-established harmony. The mathematical system came by its harmony naturally.
Given this structure of the world it was natural for Leibniz to take another step: to organize human knowledge in a mathematical pattern. Drawing both on the Lullian tradition that sought a universal language and on his own mathematical work, Leibniz wanted to create a mathesis universalis or ars characteristica , which would represent, combine, and produce all possible knowledge. This language would serve not only as a philosophical method but also as the key to a true encyclopedia. In the mathesis universalis , each human concept or idea corresponded to a different symbol or combination of symbols. By combining these symbols according to mathematically defined rules, Wolff's method could produce long lists of new truths and confirm old ones. The new language became for Leibniz both a means of communication and a new kind of logic, a way to formulate statements and an instrument of reason.
The most influential philosopher in the Leibnizian tradition was Christian Wolff. Wolff was not an original thinker. Instead, he took the foundation of his system from his teacher Leibniz, systematized it, and put it within the grasp of any educated person. Because of his
firm insistence on the rationalism that was characteristic of his "mathematical method," Wolff came to be regarded as a philosopher of the Enlightenment. Mathematics held symbolic value as the driving force behind the progress of science, and also the progress of reason and free thought. As Newton had given the universe a mathematical structure, so Wolff aimed to subsume human and society within a vast mathematical system. He pushed his mathematical method into every possible area: philosophy and science, theology and ethics, politics and public finance.
Wolff's "Rational Thoughts"
Many historians of philosophy have described Wolff as one of the most influential philosophers of his time, without, however, paying much attention to his ideas. He has been called "the master of the Enlightenment in Germany," and even outside Germany; Diderot sang Wolff's praises in his Encyclopédie . Paul Hazard made Wolff "the intellectual leader of Germany" and a general-purpose sage admired by all nations. Lewis White Beck portrays Wolff as a rationalist in the Enlightenment spirit: "Wolff is the best German representative of a general movement of thought towards deism, utilitarianism and free thought that was sweeping over Europe as a whole."
These characterizations of Wolff and his influence claim both too much and too little. Too much, because terms like "rationalism" and "mathematics" did not necessarily imply Enlightenment, nor did Wolff's philosophy. It was the form of mathematics that interested him, its way of pursuing an argument to its proof. The characterizations claim too little because the association of Wolff's brand of rationalization with the Enlightenment neglects the vast effort at systematization of all knowledge inherited from Leibniz and the 17th century.
Wolff turned to the connection between mathematics and philosophy early in his career. He obtained his doctorate in 1703 with a dissertation on ethics based on mathematical foundations. He soon came into contact with Leibniz, who nominated him for election to the Academy of Sciences in Berlin. In 1707 Wolff became professor of mathematics at Halle and later assumed the chair of philosophy. In Wolff's work the one subject cannot be distinguished from the other. As a mathematician he hardly made an original contribution, but his diligence yielded bulky volumes summarizing the contemporary state of knowledge—for instance, his Anfangsgründe aller mathematischen Wissenschaften (6 vols., 1710) and Elementa matheseos universae (5 vols., 1713–5).
Wolff announced his own particular specialty—the application of mathematics to philosophy—in his Ratio praelectionum Wolfianarum in mathesin et philosophiam universam (1718). This line of work culminated in a book in German with the monumental title Vernünfftige Gedancken von Gott, der Welt und der Seele des Menschen, auch allen Dingen überhaupt (1720). The book, usually known as his "German metaphysics," sets out Wolff's philosophy virtually in its entirety; it brought him wider renown and prefigured a whole literary genre during the 18th century. Wolff himself published a series of Vernünfftige Gedancken on various topics; elsewhere philosophers followed his lead and published "rational thoughts" on one subject after another.
Wolff's stature as a liberal theologian has also contributed to his reputation as a rationalist philosopher of the Enlightenment. In Halle, the dominant theology was the conservative strain of Pietism developed by August Hermann Francke, which sought renewal within the Church. Competition came from influential orthodox
pastors and from a group of "transitional theologians," who attempted the feat of mixing orthodoxy, Pietism, and rationalism.
Wolff found himself at odds with this eclectic combination. In 1721 he delivered a formal address on the teaching of Confucius, Oratio de sinarum philosophia practica , which emphasized its strict morality. By studying human nature, but without invoking divine revelation, Confucius had attained a lofty moral position. From this Wolff argued that morality could be comprehended by reason alone. Here he linked up with the deism that had been spreading steadily since the mid-17th century and that often saw in Confucianism the confirmation of its ideas. Wolff's address contained nothing new or sensational, but under the circumstances it was taken as a challenge. The Pietists retaliated by accusing Wolff of atheism; condemnatory sermons echoed in the churches and protests ascended to higher levels; the highly regarded theologian Johann Franz Buddeus, orthodox but favorably disposed to the Pietists, wrote an indictment of Wolff's views. The campaign finally succeeded in 1723, when Frederik Wilhelm I banished Wolff from Halle. Wolff retreated to Marburg, a little university town in the landgraviate of Hessen, an environment more conducive to his reputation. Within a few years he was being hailed as the greatest mathematician and philosopher of his age; students on their grand tours included a stop in Marburg to attend his lectures; scientific academies and universities throughout Europe wooed him. He began to write in Latin, which permitted access to a wider international audience. The 1730s saw the appearance of the great classical works, Philosophia prima sive ontologia (1730), Cosmologia generalis (1731), Psychologia empirica (1732), Psychologia rationalis (1734), Theologia naturalis (2 vols., 1736–7), and Philosophia practica universalis (2 vols., 1738–9). Again, they contained little new, but developed and refined Wolff's ideas in different philosophical disciplines and generated popularity, renown, and followers for their author.
Wolff's reputation as a rationalist and freethinker derived more from his banishment than from his writings. His position was enhanced when Frederick the Great ascended the throne of Prussia and decreed that salvation was everyone's responsibility. One of Frederick's first acts was to recall Wolff to Halle and to shower him with honors. Wolff assumed positions as professor of law, vice-chancellor of the university, privy councillor, and, in three years, chancellor. In 1745 he was ennobled as baron von Wolff. The summons from Frederick the Great, prince of Enlightenment, confirmed Wolff's status as an Enlightenment philosopher.
The Mathematical Method
The characteristic aspect of Wolff's philosophical teaching was its strictly logical construction. He had a manic capacity for arranging and systematizing everything, step by step, clearly and paradigmatically, accurately and in detail. He worked from the principles of contradiction (principium contradictionis ) and of sufficient reason (principium rationis sufficientis ). He regarded the principle of contradiction as the first law of philosophy from which all other propositions could be derived. With the principle one could decide whether something was possible; that is, everything that might be held to have sufficient reason. Philosophy was the science of the possible, with the task of showing how and why things are possible.
In working out his science, the philosopher had to observe certain rules. No principles could be employed unless adequately proved, and no new ones allowed unless derived from proven principles. No departure could be made from the meaning that words had generally acquired, and if new words or concepts were required, they had to be accurately defined. Intrinsically different objects and phenomena had to be given different names. Starting with axioms, clear definitions, and distinctions, deductive methods would link truths with one another and thus reach irrefutable conclusions. In this way results
obtained in philosophy would be as reliable as those in mathematics. In none of this did Wolff go beyond Leibniz, or, for that matter, Aristotle, Thomas Aquinas (whose influence he acknowledged), Descartes, Spinoza, or Tschirnhausen, whose Medicina mentis he had studied in detail. The mathematical method nevertheless became known as Wolffian, because Wolff systematized it as never before. In his hands it became an all-purpose means for establishing truth in any sphere.
Wolff's epistemology is quite simple. Human knowledge outside Christian revelation can be acquired in three ways: by experience (historical knowledge), by reason (mathematical knowledge), or by a combination of the two (philosophical knowledge). The last of these three methods is preferable; the other two have value only to the extent that they can be of use to philosophy. Our senses awaken our most general concepts, which are innate in our consciousness, and provide many fresh ideas; reason clarifies them and puts them into context. Thus every philosophical discipline has both empirical and rational components. For Wolff, philosophy came to mean the same thing as science, as the German term for philosophy, "Welt-Weis-sheit," suggests.
Mathematics is not a part of Wolff's philosophy. Instead, it serves as an instrument of knowledge in its own right—a method, a means rather than an end. Mathematics, in both form and inherent logic, provides a model for all areas of human knowledge. From this point of view, the mathematical method was identified with the philosophical one, and assumed a central role in the scheme of all Wolffian philosophers, even if they did not all see mathematics in the same subtle light as Wolff did. For the imitators, in fact, mathematics at times meant little more than computation.
Wolff held dear the idea of a system of knowledge. Knowledge was not merely a sudden insight or an idea, but arose from step-by-step deduction leading to reliable conclusions. The complex, reliable system therefore became an emblem of Wolff's entire philosophy—
both theoretical (logic, metaphysics, psychology, natural theology, physics, and technology) and practical (law, ethics, economics, and politics).
For Wolff, mathematics was the method and philosophy the content . This distinction is essential to an appreciation of his philosophy and its place in 18th-century thought.
For and Against Enlightenment
Wolff's working life falls into two periods. In the first he was active at Halle, wrote in his native German, formulated a more popular philosophy, and spoke so freely on religious matters that he was accused of atheism and banished. In the second period he lived in Marburg, wrote books in Latin, put forward a more theoretical philosophy explicitly addressed to the world of learning, and appeared in religious matters as an orthodox and unrelenting apologist. The watershed came somewhere around 1730. In the earlier phase he followed the precepts of the Enlightenment; in the latter he opposed them.
Wolff's German books aimed at a larger public are characteristic of his enlightened period. They have a pronounced practical and utilitarian tone and allot philosophy a "liberating" role, fully comparable to later manifestos of the French Enlightenment. Good examples are plentiful in his Vernünfftige Gedancken von der Menschen Thun und Lassen, zu Beförderung ihrer Glückseligkeit (1720). A critical indication of his popular, liberal position was his vacillation between physicotheology and teleology, two doctrines not easily distinguished and therefore often confused. Physicotheology rested on the ingenious organization of the natural world in general, but it often singled out particular features in nature as evidence of a creator behind the order. This search for an all-ordering God behind the exquisite subtlety of nature could justify, and even motivate, all scientific research. Where order served as a fundamental principle in physi-
cotheology, so intention emerged as essential in teleology. If everything fulfills a definite purpose, there must be at least an implicit ordering principle—a God who intended the order. Many physicotheologists came to confuse proofs based on general order and those based on intent, and to subsume teleology under physicotheology. In either case, a "popular" proof of the existence of a divine creator was secured.
To see Wolff as a great representative of physicotheology, however, is to overplay his hand. Wolff attacked both the physicotheological and teleological approaches. He recognized the legitimacy of demonstrating divine intent in nature, but demanded a prior proof of God's existence and of the creation of the world as a free divine act. Wolff's method required that the theoretical foundation be established first. Yet a few years later, Wolff appeared as a full-fledged physicotheologist in his great work Vernünfftige Gedancken von der Absichten der natürlichen Dinge (1723). Had his views suddenly changed? No, he had distinguished between two sets of readers—academic and popular. Wolff's major works in Latin do not feature physicotheology. In a foreword to a translation of Nieuwentyt's well-known book on physicotheology (1731), he referred to his own work in German and underlined that it was intended for readers who did not speak the languages of the erudite. Whatever he said, in practice he inferred God's perfection from the usual inventory of the natural scientist. He described the cosmos and all its bodies, the sun and the fixed stars, the earth and the planets; he dealt with air and winds, different kinds of precipitation, rainbows, lightning, and thunder. The quantitative philosopher will want to know that his evidence includes "the number of heavenly bodies."
In sum, although Wolff demanded strict standards of logical proof in his mathematical method, he allowed himself to break the rules when addressing an unlearned audience. The end justified the means,
for the religious objective was all-important. He made an even more significant adjustment in his major work Theologia naturalis , an extended exercise in apologetics, rapidly becoming his specialty. He devoted more than 350 pages to demonstrating the false steps and absurd notions of atheism. He did not confine his attack to unbelievers but also took up cudgels against every conceivable kind of philosophical freethinker: atheists, fatalists, deists, naturalists, anthropomorphists, materialists, paganists, Manichæans, Spinozists, and Epicureans. Just as he had previously used the mathematical method to lend positive support to philosophy or theology, he now used it to expose and exclude those who were not orthodox. Fourteen years earlier he had been banished and accused of atheism; now Wolff attacked those who adopted the viewpoint he had then espoused. Yet he held fast throughout these sea-changes to his mathematical method and philosophical rationalism.
Weighing in the Balance
Which side of Wolff attracted his contemporaries—the mathematical-rationalist or the teleological-apologetic? The answer must be that many accepted both sides, viewing them as two parts of the same entity. The mathematical method tied them together, and became the guiding principle for Wolff's pupils, even for those most inclined to apologetics.
One of the first of these pupils was Ludwig Philipp Thümmig, appointed professor in Kassel on the master's recommendation. His extensive Institutiones philosophiae Wolfianae (2 vols., 1725–6) differed little from Wolff's own account, and functioned as an authoritative source for later Wolffians. Another pupil, regarded as even more significant, was Georg Bernhard Bilfinger, professor of philosophy first at Tübingen, then at St. Petersburg, and again at Tübingen. His
Dilucidationes philosophiae de Deo, anima humana, mundo et generalibus rerum affectionibus (1725) followed up, in title and contents, Wolff's Vernünfftige Gedancken von Gott, der Welt und der Seele des Menschen . Bilfinger emphasized the twin foundation stones of reason and experience, and stood ready to point out correspondences between natural science and Christian faith. Alexander Gottlieb Baumgarten, first in Halle, then at Frankfurt, wished above all to apply Wolff's philosophy to aesthetics. His well-balanced Metaphysics (1739), extremely faithful to Wolffian systematics, amounted to a reference book or compendium of Wolffian doctrine. In Leipzig, Johann Christian Gottsched and Carl Günther Ludovici wrote historical works on Wolff but were not themselves leading philosophers. Israel Theophil Canz at Tübingen did much to inject the doctrine into theology, as did Jacob Carpov at Jena, who sought to prove divine revelation using Wolff's teleological method. Important expositions of the mathematical method were also offered by Friedrich Christian Baumeister at Gölitz and Georg Heinrich Ribov at Göttingen.
Outside Germany Wolffianism found adherents in England, France, the Netherlands, Switzerland, and Scandinavia. In Denmark, Wolff's doctrines exerted influence through a number of mathematicians and philosophers: Christian Hee, who stayed for a time with Wolff at Marburg; Friedrich Christian Eilschow; and, above all, Jens Kraft, eminent as both a mathematician and a philosopher, who published a series of textbooks in imitation of Wolff. In Sweden, too, mathematicians introduced Wolff's ideas, which spread to the Swedish university at Åbo (Turku) in Finland.
The tendency of Wolff's disciples to tread confidently in their master's footsteps, to accept both his philosophical and his theological views, and to tie them together via the mathematical method, was muted in Sweden by resistance to the casual combination of philosophy and theology. Sweden's true introduction to Wolff came through
the efforts of the mathematicians Anders Celsius and Samuel Klingenstierna, whose reputation at Uppsala University recommended their opinions to many students. In a manual of arithmetic (1727), Celsius spoke warmly of the existence of the mathematical method and also of its importance to other disciplines. By the following year, in a pro gradu dissertation on the existence of the soul or the intelligence, Celsius completely embraced the philosophy of Wolff. He praised not only Wolff's method but also followed the master in setting out his dissertation in traditional Euclidean form, with short propositions under such headings as axioma, theorema, definitio, observatio, demonstratio , and scholion . The dissertation argues that the soul can be proved to exist by means of Wolff's philosophical laws, and Wolff himself is called "the greatest philosopher of our time," philosophus nostra aetate summus .
Several theses reminiscent of Celsius' approach appeared over the next few years, all of them loud in praise of the mathematical method and bearing the imprint of the mathematical form favored by Wolff. Early in June 1729, the brothers Erik, Nils, and Johan Gottschalk Wallerius and their close friend Olof Hammaræus performed as respondents to four Wolffian dissertations. Presented first pro exercitio , two years later all four dissertations were defended pro gradu , under Celsius and Klingenstierna as tutors. Several of the dissertations describe the mathematical method without employing it; one stresses that certain philosophical knowledge is impossible without it. Jacob Friedrich Müller, professor of philosophy at Giessen, was quoted in support of this statement, although, unknown to the author, Müller had just defected to the camp of Wolff's denigrators.
Other dissertations under Celsius follow the same pattern. One deals with the subject of "incomprehensible books," which provided
the opportunity to laud Wolff as an example of lucidity and intelligibility. Evidence suggests that Celsius was behind the most detailed presentation of Wolff's philosophy, a dissertation in two parts (1731–2) on the subject of "how to attain worldly happiness through philosophy." The subject may appear novel, but the theme is familiar. The philosophy that can promote profit and happiness is contained in logic and mathematics. Hence the mathematical method is superior to earlier instruments and Wolff is the greatest of philosophers, outshining lesser lights like Plato, Aristotle, and Descartes. With Wolff's method, the author "brushes aside the weapons of atheism and defends the truths of the Christian religion." Ad pleniorem scientiam , the dissertation demonstrates that mathematics and physics are of fundamental importance to all other sciences, technology and mechanics, medicine, law, economics, and military subjects such as fortification and pyrotechnics.
Samuel Klingenstierna had already earned a name for himself as a mathematical genius when he set off to study abroad in 1727. He went first to Marburg to hear the much admired Wolff. It is recorded that the pupil much impressed the teacher. When the chair of mathematics at Uppsala became vacant, Klingenstierna applied for it. Because Marburg was within the native state of King Fredrik I of Sweden, Wolff had access to the king and recommended Klingenstierna in the strongest terms. Appointed to the post in 1728, Klingenstierna did not take up his professorship until 1731, when he returned to Sweden with newly purchased books and fresh reports on the fashionable new philosophy. His were popular lectures, and students sought him as tutor for their doctoral theses. These theses are filled with quite simple Wolffian propositions about the excellence of the mathematical method or the role of contingency in the creation of the world, and frequently make reference to Wolff and Bilfinger. Mathematics thus linked up with philosophy, notably in Klingenstierna's seminars in philosophiam naturalem .
But enthusiasm for Wolff and the use of mathematics in philosophy went too far. Alert theologians, worried by the excessive spread of rationalism, castigated Wolff as "heathen and atheistic." In 1732 the chancellor of Uppsala, Gustaf Cronhielm, warned professors against dealing too casually with the new philosophy inspired by Wolff. The professors responded with promises to be careful and to protect their young students. Two years later the tone of admonition grew sharper. The chancellor decreed that professors should not preside at theses outside their own disciplines, a rebuke directed particularly at the professors of mathematics, who had been so ready to interpret the Wolffian philosophy. A year later, students were required to give a declaration of faith when enrolling at the university, to guard the purity of the doctrine. Such censorship and official criticism encouraged attacks on Wolff. His opponents asserted that mathematics and philosophy did not belong among the fundamental sciences; they saw all questions as ultimately teleological in nature. Philosophy, they insisted, should stick to its time-honored role as the handmaiden of faith.
The situation soon changed radically. Cronhielm was succeeded as chancellor by Gustaf Bonde, known for his deep interest in mysticism, alchemy, and Platonic mathematics. An admirer of physicotheology and of Wolff, he later published three volumes of "reflections on the wonders of God in nature." On a visit to Uppsala University in 1738, Bonde stressed the desirability of teaching the younger generation theologia naturalis as a timely defense against "atheists and indifferentists." The new chancellor thus recommended what the old one had forbidden. Bonde went even further when he engineered an offer to Wolff of the most highly regarded professorship in the university (which Wolff declined). Bonde's permissiveness sanctioned latent interests among the faculty. Petrus Ullén, professor of philosophy, became the first important figure in this new phase. By the time of his death in 1747, he had presided over a hundred theses, a third of which were Wolffian through and through. Ullén was no original
thinker. He praised mathematics for its ability to clarify and present problems in easily grasped diagrams and figures, and he insisted on the importance of the Wolffian philosophy to theology. He vehemently attacked all tendencies toward deism or "indifferentism"; true to the later Wolff, he used rationalism as a defense of orthodoxy and against the ideas of the Enlightenment.
An even louder champion of Wolffianism was Nils Wallerius, who started out as a mathematician and physicist, continued as a philosopher, and ended up as a theologian. He succeeded Ullén as professor of philosophy in 1746. Within the compass of philosophy he included logic, metaphysics, psychology, and natural theology, all slavishly arranged in accordance with Wolff's system. The mathematical method was fundamental in all philosophy, but mathematics must yield to theology, lest it lead to materialism and atheism. Wallerius shared the concern of both philosophers and theologians over theologia polemica —the struggle against Enlightenment philosophy. In 1755 Wallerius received a new chair in theology devoted to uncovering and combatting heretics. The new professor was to repudiate all freethinkers, "such as atheists, naturalists, deists, anti-scripturalists, indifferentists and other unbelievers." Wallerius warmed to the task: Moravian Brethren, Socinians, pantheists, and mystics, too—here he named Jakob Böhme, Paracelsus, Robert Fludd, and Johann Conrad Dippel—fell under his flail. In an essay on the repulsiveness and wickedness of materialism, Wallerius ranted at the ancient atomists Democritus, Epicurus, and Lucretius and their modern successors Thomas Hobbes and Pierre Bayle. "O stupida ingenia, sive mente ac ratione ," you are so blind that you cannot imagine anything beyond the bounds of the material. In 1756, Julien Offray de La Mettrie, the leading contemporary materialist, came in for particularly severe criticism. Wallerius decried the deists from Locke to Hume as
"naturalists"—Wallerius' favorite epithet for his opponents—and Voltaire as "the greatest fraud of the day."
Wallerius' guiding ambition was to reconcile mathematics, philosophy, and theology. At hand were all the necessary tools: a professional graps of mathematics, philosophy, and theology; a passion for system; and Wolff's method. His eminent elogist in the Royal Swedish Academy of Sciences, Torbern Bergman, showed restrained appreciation of Wallerius' contribution, and a like opinion was expressed in another contemporary biography: "Had he lived fifty years earlier and in a more scholastic era, his memory would have been even more illustrious."
Wolff's use of mathematics usually made the best impression on those who knew the least mathematics. During the late 1740s, in a celebrated dispute between the supporters of Newton and those of Wolff, Pierre Louis Moreau de Maupertuis was the main adversary of the Wolffians. The most telling attack on Wolff came from Kant, however, who as usual went right to the core and challenged the mathematical method itself.
The Berlin Academy had posed the question of whether metaphysical truth could be equated with mathematical truth and, if it could not, what sort of truth it then was. Kant replied with the treatise, Untersuchung über die Deutlichkeit der Gründsätze der natürlichen Theologie und der Moral (1762, published in 1764), which struck a decisive blow to Wolff's philosophical teachings.
Kant had been influenced by Wolff's philosophy, but had never
followed it slavishly. The treatise of 1762 marked his rejection of the mathematical ideal in philosophy. In it Kant draws his well-known distinction between analytic and synthetic propositions, and concludes that metaphysics is not synthetic like mathematics but rather analytic. The synthetic structures of mathematics cannot be transferred to philosophy and its "mathematical method," for philosophy, unlike mathematics, does not have at its disposal definitions and axioms from which to proceed methodically. Application of the mathematical method to the field of philosophy had thus given rise to errors and mistakes: no dogma in philosophy could be likened to the definitions and axioms of mathematics.
Kant's criticism swept away the very foundations of Wolff's influence. Wolff's mathematical method depended on the possibility of applying procedures of mathematical proof to all philosophical and theological questions. Now Kant argued that the basic structures of mathematics and philosophy were different. Wolff's system crumbled.
Several conclusions emerge. Wolffian thinking became a fashionable philosophy in the first half of the 18th century, and its exponents and detractors both saw the mathematical method as the essential element of his doctrine. It is important to distinguish between form and content in our analysis of the Wolffian mathematical method, and to recognize that Wolff's philosophy fulfilled different functions in different situations. While "rationalism" and "mathematical method" were popular rhetorical flourishes in the 18th century, they also stood for a well-defined means of attaining certain intellectual objectives. Both friends and enemies of the Enlightenment appropriated that method, since mathematics as a method of proof promised results in all areas of human knowledge. That it collapsed so readily under Kant's attack reflected that Wolff's mathematical method had already proved inadequate: it failed to provide tools for revamping critical philosophy, or for creating instrumentalist science, or for solving practical problems facing the bureaucratic states of the late 18th century. The form and content of mathematics itself, however, would continue to speak to these needs.