An Instrument of Analysis
Armed with the view of language as a species of logic, and mindful of the power of language to deceive or mislead, writers on language in the latter part of the 18th century looked for ways to harness language in the service of enlightenment. For many,
mathematics, the embodiment of enlightened values, served as a guide. Mathematics was eminently rational to 18th-century eyes; its symbols and results were truly international; its reasoning was at once powerful and certain. More than that, at the heart of 18th-century mathematics stood analysis, and the method of analysis enjoyed privileged status among the philosophes . Small wonder that, in an age that prized the rational and the universal, mathematics—that sharp and sure instrument of reason—offered inspiration and example to the reformers of language.
The privileged status accorded to the philosophical method of analysis during the Enlightenment helped to focus attention upon its manifestation in mathematics, and in particular upon algebra as a powerful instrument of analysis, especially in the hands of a master. Few could rival the mastery of Leonhard Euler. In Condorcet's view, Euler "sensed that algebraic analysis was the most comprehensive and certain instrument one can employ in all sciences, and he sought to render its usage general. This revolution. . .earned [Euler] the unique honor of having as many disciples as Europe has mathematicians." The power of the Eulerian formulation of infinitestimal analysis provided dramatic confirmation of the mathematical potential of algebra and lent credence to the view of algebra as the fundamental language of mathematics.
Algebraic analysis found a vigorous advocate in Etienne Bonnot de Condillac, whose treatises on philosophical analysis, language, and mathematics explicitly invoked mathematics as a model for the perfection of language. Condillac's treatise on logic rejected prevailing approaches to logic and deprecated the grand philosophical systèmes of the 17th century. What was needed for building knowledge, Condillac argued, was not "to imagine ourselves a system," but instead to
"attend to what nature teaches us." And what nature teaches us is to analyze: "analysis is the only method by which accurate knowledge is to be acquired."
Condillac offered graphic examples of analysis at work. When you order a copy of a sample dress, seamstresses "will naturally perceive that it is necessary to take [the sample] apart and remake the pattern of each part." When you throw open a window and look at a landscape, you take the scene apart by focusing on first one object, then another. "We make this decomposition only because an instant is not sufficient for us to study all those objects. But we only decompose in order to recompose." Analysis for Condillac is thus nothing more than observing "in a successive order the qualities of an object, so as to give them in the mind the simultaneous order in which they exist."
Condillac threw open a window on our thought processes themselves. "To see distinctly all that at once offers itself in my mind, I must decompose it, as I have decomposed what offered itself to my sight; I must analyse my thoughts." In his enthusiasm, he contended that analysis, which some called the method of invention, "has made all discoveries": "by the medium of analysis we become capable of creating arts and sciences." He thus echoed Voltaire's dictum—"Let us make an exact analysis of things"—which served as a motto for the Enlightenment. It was clear to Condillac that seamstresses knew more than philosophers, who used synthesis, not analysis, to wrap up the truth "in a heap of vague notions, opinions, errors" and who "strayed prodigiously, when they forsook analysis" for the "tenebrous method" of synthesis.
In his depiction of language as an instrument of reason, Condillac also renewed the critique of existing languages and their unhappy effects on thought. In an oft-quoted passage, he attacked philoso-
phers as "subtle, singular, mysterious, visionary, unintelligible, [who] often seemed to be afraid of not being obscure enough, and affected to cover with a veil their real or pretended knowledge. Therefore, the language of philosophy has been nothing else but a jargon of gibberish for many centuries past." The tirade continued, "The art of misapplying words was for us the art of reasoning: arbitrary, frivolous, ridiculous, absurd, it had all the vices of disordered imaginations."
Analysis, that lever of the mind, offered Condillac a solution to the problem of language: "To speak so that we may be understood, we must conceive and express our ideas in the analytical order which decomposes and recomposes each thought." The result will necessarily be "clearness and precision." Condillac then took the argument one step further: "The art of reasoning is in truth only a well constructed language." He rejected the remedy, often proposed, of "seeking in words their essential qualities"; instead, he exhorted us to seek in words only "what we placed in them, the relation of things to us, and those relations which they bear to each other." Nature displays to us a natural order; by our analysis of nature we perceive and recreate that order in our mind. Language should reflect a natural taxonomy that corresponds to the "order of our ideas." Condillac concluded that a well-constructed language is nothing other than an analytical method.
Condillac was captivated by the power and achievement of the 18th-century brand of mathematical analysis and set out to facilitate analytical thought in metaphysics by studying algebra—the fundamental language of mathematical analysis. In his treatise on logic, Condillac insisted that no distinction need be drawn between mathematical analysis and logical or metaphysical analysis, "because in all of them [analysis] leads from the known to the unknown, by reasoning; that is, by a series of judgments which are included in each other." For Condillac, the advantage of mathematical analysis consists strictly in
the fact "that it speaks there the most simple language"—and that most simple language is algebra.
For Condillac the mathematical advances of the late 17th and especially the 18th century offered a powerful argument in favor of algebra as the language of mathematics. Algebra constituted "very striking proof that the progress of the sciences depends solely on the progress of languages; and that correct languages alone could give analysis that degree of simplicity and precision of which it is susceptible." Further, he called attention to the central role played by Euler and Joseph Louis Lagrange in putting mathematical analysis in algebraic form: they "are great mathematicians, because they are great analysists [sic ]. They excellently write algebra, a language in which good writers are most scarce, because it is the most correct." Condillac put the point forcefully. Algebra "is a language, and cannot be any thing else." It is "a language which could not be badly constructed"; this in turn means that mathematicians who couched their analysis in algebraic language are able thereby to "speak with precision."
Condillac maintained that languages are in and of themselves analytical instruments, which ought to lay bare the relations and analogies of our ideas. Most languages, in Condillac's view, were too blunt, too imprecise. But in algebra, conceived as the rules by which equations and their components could be manipulated and transformed, the language (and the analytical method) corresponded exactly to the analogies existing among mathematical ideas. As he explained it, "One goes from identity to identity until the conclusion that resolves the question, in other words, from identical equation to identical equation until the final equation." Condillac's explanation of the chain of identities sounds a good deal like a passage from d'Alembert's Preliminary discourse to the Encyclopédie : "Thus, the chain of connection of several geometrical [mathematical] truths can be regarded as more or less different and more or less complicated
translations of the same proposition and often of the same hypothesis." D'Alembert was describing a series of mathematical propositions deduced one from the other: "It is almost as if one were trying to express this proposition by means of a language whose nature was being imperceptibly altered, so that the proposition was successively expressed in different ways representing the different states through which the language had passed. Each of these states would be recognized in the one immediately neighboring it; but in a more remote state we would no longer make it out, although it would still be dependent upon those states which preceded it and designed to transmit the same ideas."
Condillac's enthusiasm for algebra went beyond a recommendation to study its method: he provided a sort of elementary algebra textbook in the work La langue des calculs . In line with the interest, common during the second half of the 18th century, in the origin of language, Condillac attempted in this work to trace the origin of algebra, that most excellent of languages. He traced the development of calculating from finger-counting to words to letters and symbols. Familiar numerical expressions like 1, 10, and 100 were not introduced until well into the book, since, according to Condillac, the use of numerals followed the use of letters and symbols in the historical development of calculating. The remainder of La langue des calculs explained the basic operations of calculating with letters and numbers. Condillac confined his exposition to the canons by which algebraic and arithemtic quantities were to be expressed and manipulated. In stopping short of the solution of polynomial equations or their application in solving problems, he showed himself to be concerned primarily with algebra as the expression of relations among mathematical entities and as the means to make those relations reveal new facts.
Condillac's intent in writing La langue des calculs was not to add another volume to the lengthening shelf of elementary algebra textbooks in the 18th century, nor can his rational reconstruction of the
birth of calculating be considered a serious addition to the historiography of mathematics. Nor should his work be read as a recommendation that all thought be "mathematicized" or "algebraicized." In our attempts to reform languages, we should not wield recklessly the instrument of algebra, but instead should attend to the lessons of mathematics as we devise new tools. Condillac called attention to algebra precisely because the chain of reasoning in a series of identities "is more easily perceived, when we express ourselves with algebraical signs." He issued a call to action: "If, therefore, any sciences are not very exact, it is not because we do not talk algebra in them; it is, because their languages are not correct. . . . All the sciences would be exact, if we knew how to speak the language of every one." His work thus represents an important example of the confluence of two deep philosophical currents in the 18th century: advocacy of the method of analysis in all domains of thought and the desire to perfect language. In both aspects Condillac was strongly influenced by what he knew of mathematics.