Signs and Thought
The larger problem of the connection between signs and thought once again claimed the attention of philosophers and linguists in the last third of the 18th century. In his history of universal language schemes, Knowlson has identified some of the reasons for this renewed interest. Studies of the origin of languages and of general grammar helped to stimulate thinking on the subject by focusing on the evolution of nomenclature and the logical significance of syntactical structure. What Knowlson calls a "general climate of rationalization," as demonstrated by the design of new systems of schools and of weights and measures, promoted the search for a new and rational language. Political events and trends also played their part. Hopes that a rationally constructed language would break down political and cultural barriers were fed by republican aspirations to extend the realm of liberté, égalité , and fraternité . The reconstruction of chemical language, and its happy consequences for the science of chemistry, lent an optimistic note. Condillac's views of language also attracted more attention at the end of the century as editors prepared a complete edition of his writings. It would include the first appearance in print of La langue des calculs , left unfinished at Condillac's death in 1780.
A series of essays in the memoirs of the Berlin Academy evinces the widespread interest in "the reciprocal influence of reason on language and language on reason" (the phrase comes from the title of Sulzer's memoir of 1767). In his essay of 1774, for example, Dieudonné Thiébault argued in favor of a universal language, rejecting Johann Gottfried Herder's contention that such a language would be nearly useless. Thiébault envisioned a language with a limited number
of base words or concepts, a uniform system of derivation, and a regularization of grammatical relations. Such a language would provide, he claimed, "at once the best book of logic and metaphysics." Castillon took up a similar question at the end of the century, in his "Mémoire touchant l'influence des signes sur la formation des idées."
Anne-Robert-Jacques Turgot, well acquainted with attempts to rationalize activities of the French state, also concerned himself with the connections of language, logic, and mathematics. He suggested that "The study of language, if well done, would perhaps be the best of logics," since language served as both the "expression" and the "measure" of thought. Language exemplified Turgot's view of progress as passage through various states of perfection, each state accompanied by new symbols. For Turgot, as his biographer Frank Manuel has explained, the language of mathematics could serve as "an impregnable barrier against retrogression"—"the armor of numbers and equations" in defense of reason. Turgot's protégé Condorcet also envisioned a philosophical language, clad in algebraic notation, applicable to all of knowledge and assured, by virtue of its mathematical foundation, of precision.
Among those who thought that the whole matter of language and its relation to thought bore reexamination were the idéologues , who, building upon the ideas of Condillac, Turgot, and Condorcet, launched a thorough review of the question of signs and thought. Idéologie (the term was introduced by Destutt de Tracy) tackled the analysis of sensations and ideas, and drew upon the 18th-century discussion of general grammar and logic. The Institut national de France, and in particular its Classe des sciences morales et politiques,
proved to be a hotbed of Idéologie , and its members debated vigorously the connection between signs and thought. To encourage consideration of this issue, the Classe set as its first prize question, "To determine the influence of signs on the formation of ideas."
Among the entries was an essay by P.F. Lancelin. His work received honorable mention, and in expanded form was subsequently published under the title Introduction à l'analyse des sciences, ou de la génération, des fondemens, et des instruments de nos connoissances. As the title page announced, Lancelin aimed at nothing less than the perfection of human reason. He saw language as the lever of the mind and the perfection of language as the best mechanism for assuring the reign of reason and for avoiding error and prejudice. The mechanical metaphor is worth noting. As an Ingénieur-constructeur de la Marine française, Lancelin was a product of the technical training establishment in France, and his "analysis of the sciences" bears the stamp of his training, which saluted the military virtues of mathematics and mechanics. Lancelin's intent was to take apart the human mind in the same way an engineer takes apart a machine.
The influence of Condillac is also evident. Like Condillac, Lancelin reduced the art of thinking to the use and analysis of a well-constructed language—"a language based on good observations, correct ideas, and exact facts"—and sought to reveal the development of the human mind by examining "the exact formation of mathematical languages." Language was an analytical method; properly used, it would enable the mind to soar like "an eagle over the whole globe of human knowledge." For Lancelin, as for Condillac, algebra—the language of mathematics—was just such an analytic method and as such deserved careful attention, since a method that delivers exactitude in one sphere of human knowledge will do so in all.
Lancelin shared in the general Enlightenment enthusiasm for mathematics and professed the common faith in the correctness and certainty of mathematical reasoning. But in order to conclude that the method that gives us exact knowledge in the realm of mathematics can do so in other realms of thought, Lancelin had to consider why mathematics works at all. Mathematics deals with what Lancelin called "measurable ideas"—those formed by successive addition or repetition of identical elements. The restriction to measurable ideas clearly enhances the certainty of mathematical reasoning, Lancelin argued, but it is not the sole cause for that certainty. At least as important as a guarantee of correct reasoning is the art of mathematical signs. Or, turning the argument around, differences of opinion outside mathematics are to be blamed as much on the inexactitude of signs as on the inexact formulation of the ideas they represent.
In an attempt to improve signs and thus thought, Lancelin built upon Condillac's contention that propositions, judgments, and equations are all essentially the same—that is, expressions of the formation of complex ideas out of simple elements. He concluded that ordinary sentences can usefully be translated into a sort of algebra. He offered as example the statement "gold is yellow, heavy, meltable, and malleable," which he transformed into the "equation" O = a + b + c + d , where O denotes gold, a denotes yellow, b denotes heavy, and so on. By a liberal use of algebra, the language of mathematical analysis, Lancelin thought he could facilitate the application of analysis outside the normal realm of mathematics.
Nor did Lancelin stop there. He advanced an ambitious plan for representing "in a series of synoptic tables the ideas and languages of all peoples." His idea for tables of knowledge and nature was scarcely novel. What is noteworthy, however, is the way the suggestion flowed directly from his ideas about the perfectibility of language and the privileged nature of mathematical reasoning. The whole discussion was embedded in an extravagant reform program that also
included the perfection of naval engineering, a complete overhaul of the French educational system, a new approach to solar astronomy, and an "experimental physics of the soul," in which the "thinking force" could be measured and expressed in an analytic formula. However ambitious and impractical his plans, it is evident that they were shaped by his faith in mathematics.
A more sober and critical approach was taken by Baron Joseph Marie de Gerando in his prize-winning essay for the competition sponsored by the Institut national. In considering the relation of signs and thought, de Gerando compared analysis and synthesis in both mathematics and metaphysics, argued for the privileged nature of mathematical knowledge, and presented a careful critique of proposals for a philosophical language and an algebra of thought. His essay, published in four thick volumes, constituted a thorough reexamination of ideas on language and mathematics from the mid-17th century to the end of the 18th century.
Like so many of the authors whom he discussed, de Gerando deplored the abuse of words and urged that we hasten to discover a remedy. He thus granted the desirability of reforming language, but did not see flaws in language as the root of all wrong thinking. Instead, he blamed the imperfections of language on imperfect thought.
He then compared the methods of analysis and synthesis in both philosophical and mathematical contexts and decided in favor of analysis (although not with the same determined enthusiasm as had Condillac). De Gerando was especially interested in the "prodigious successes" of algebra, which seemed to offer the greatest hope to those who sought to perfect all abstract sciences, because the use of algebra permits us both to display the relations between quantities and to eliminate extraneous information. His analysis focused on the advantage of algebra over ordinary calculation or over reasoning using words, the praiseworthy character of the signs used in algebra, and
the connection between algebraic signs and the nature of the ideas they represent.
De Gerando also drew attention to a particular feature of algebraic calculations: there is no need at the outset, as there is in arithmetic, to determine an idea before establishing its relations to other quantities. By using signs for unknown and known quantities alike, "we are not stopped by our own ignorance." In citing the virtues of algebraic analysis, he quoted Laplace: "Such is the fecundity of analysis, that it suffices to translate particular truths into this universal language, in order to see how to proceed from these expressions to a host of new and unexpected truths. . . . Thus the geometers of this century, convinced of its superiority, have applied themselves to extending its domain and pushing back its boundaries."
De Gerando's enthusiasm for mathematics in general, and for algebraic analysis in particular, was qualified by his recognition of the privileged character of mathematical ideas. He divided all ideas into four categories, of which the most straightforward and least problematic was the class of "ideas of simple modes." This class is roughly equivalent to Lancelin's category of measurable ideas. The properties and advantages of "la langue du calcul," recalling Condillac's phrase, were to be seen as a "privilege" of such simple ideas: the essential identity of the elements of such ideas means that both the nature of the elements and their mode of combination are already determined. Thus mathematicians reason only on clearly conceived and unequivocal notions.
De Gerando then undertook an evaluation of schemes proposed for philosophical or universal languages in the hopes of demonstrating which reforms are desirable and which are feasible. He defined the desiderata for an artificial language—analogy of signs to ideas, analogies of signs among themselves, simplicity of the whole system as well as of its details, neat and precise distinctions between signs, and "a sufficient abundance" of signs—and described the general categories into which all philosophical languages can be divided.
Using these categories, and working from a thoughtful analysis of mathematical reasoning itself, he set up criteria for the evaluation of universal and philosophical languages. The conclusion was a gloomy one. He found—sometimes by use of numerical assessments of the set of base concepts or the complexity of the character systems—that such language proposals necessarily fell far short of the desired ends. Thus, for each category, including the pasigraphies of the 1790s, he found it impossible to satisfy the necessary conditions with which he began. (De Maimieux's pasigraphy came in for special criticism because it forced the distribution of ideas to conform to a scheme requiring equal numbers of division or subdivisions, regardless of content.) Hence the goal of a truly philosophical language, however desirable, was an impossible dream.
De Gerando also traced the attempts to create an algebra of thought from Ramon Lull and Athanasius Kircher through Leibniz to Lambert and other 18th-century proponents of symbolic logic. Here, too, de Gerando offered a careful and ultimately pessimistic critique, warning that a true algebra of thought will require a large number of primitive ideas to be capable of handling more than very simple and general propositions. Indeed, the number of primitive ideas and symbols required will be prohibitively large. Unlike Lancelin, with his boundless enthusiasm for mathematics, de Gerando cautioned against confusing the "method of reasoning of geometers with the mechanical processes of their calculs ." Although geometers and metaphysicians share the same method of reasoning, he argued, we cannot assume that the virtues of mathematical algebra—"the simplicity of forms and the speed of execution"—will follow from an uncritical application of its procedures to metaphysics. He thus invoked the privileged character of mathematics in order to discount the likelihood that its procedures can be applied uncritically in realms of thought outside mathematics.
Although mathematics was an instructive and inspirational guide, the path chosen by 18th-century language reformers was steep and
rocky. The efforts to construct artificial, universal languages made some progress toward rationalization, especially in replacing idiosyncracies of existing languages by means of new, regular schemes of word derivation. Most of these schemes were informed by at least the rudiments of combinatorics, and it was hoped that by pursuing the analogy to numbers language reformers would approach the precision and transparency characteristic of mathematical language. Establishment of a set of base words and categories capable of expanding to encompass all of human knowledge proved more difficult. Also elusive was the conversion of language into an instrument of discovery as effective and certain as mathematical reasoning.
Mechanicians of language, with their optical telegraph systems, were more successful. Taking inspiration from military need and encouragement from governmental coffers, they took account of combinatorics in constructing signaling apparatus, devised clear and unequivocal codes, and treated as essentially interchangeable words, numbers, and geometrical patterns. Using such systems, trained operators were able to transmit messages rapidly and without distortion over long distances. However, military exigencies, which called for the codebooks to remain secret, blocked hopes that telegraphic cipher-tables might serve as a universal language, at least in times of war.
Whether they met with success, as in the optical telegraph, or failure, as in the universal language schemes, efforts in the 18th century to invoke mathematics in the consideration of language were frequently self-conscious. Condillac, Lancelin, de Gerando, Gamble, Meyer, and Edelcrantz all took seriously the task of evaluating the possibility of achieving their objectives. Condillac, although he did not recommend the algebraicization of all fields of knowledge, still thought it wise to study algebra as the best available example of a well-made language. Lancelin's faith in mathematics, which exceeded Condillac's, led him to argue that it was practical to remake language in the image of mathematics. De Gerando's assessment was more careful and conservative. Edelcrantz and Gamble also looked for mathematical measures of the functioning of telegraph systems
and applied considerations drawn from mechanics and experimental physics in their appraisals.
Of the 18th-century attempts to employ mathematics in the construction of language, only the optical telegraph achieved practical success. The quest for a universal or philosophical language modeled on mathematics was an uphill struggle; indeed, in de Gerando's opinion, it was assured of failure. That language reformers were so determined to ground their projects in mathematical terms, whatever the outlook for success, is yet another measure of the pervasive appeal of mathematics in the late Enlightenment.