Measure of Ideas, Rule of Language: Mathematics and Language in the 18th Century
By Robin E. Rider
In the 18th century mathematics both shone as an example and served as an instrument of enlightenment: it was at once rational, universal, certain, precise, unambiguous. Many 18th-century authors hoped that the same could shortly be said of common language. Toward that end they sought to infuse language with rationality and to render it universal; to guarantee its certainty, increase its precision, strip it of ambiguity. Some sought to convert into mechanical processes both the derivation of words and their transmission. Throughout their struggles with the inadequacy of existing languages and the complexity of creating new ones, language reformers were guided by mathematics.
By the second half of the 18th century the close connection between the content of thought and its form of expression was a familiar philosophical refrain. In his Diversions of Purley , for example, John Horne Tooke carefully re-examined a century's worth of views on the intimate connection between language and thought, then pushed the philosophical commonplace one step further. For Tooke, the operations of the mind were none other than the operations of language—hence, language is thought.
Growing appreciation of the link between language and thought and renewed concern for the quality of thought gave rise to new laments about the inadequacies of existing languages and their unhappy consequences. Authors in the Age of Enlightenment followed a century-old lead in condemning the arbitrary assignment of words to things. Tooke, for instance, recalled Francis Bacon, who had likened words to a Tartar's bow: they "shoot back upon the understanding of the wisest, and mightily entangle and pervert the judgment."
Language studies in the 18th century also addressed questions of syntax—the structure of phrases and sentences—and what was called universal or general grammar. The latter was defined in the Encyclopédie as the "invariable and universal rule that must serve as the foundation for the particular construction of any given language." General grammar thus rested on the premise that all languages follow "the laws of logical analysis of thought; and these laws are invariably the same everywhere and always." Or, more pointedly, "a sane logic is the foundation of grammar."
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.
In the last third of the 18th century, perceived flaws in existing languages called forth a spate of proposals for artificial ones free from flaws—and the very profusion of existing languages, like antiquated social structures and inconsistent systems of weights and measures, cried out for rationalization. To answer this call, several authors proposed a "pasigraphy," from the Greek terms for "universal" and "writing"—a set of rational, universal symbols each person could read in his or her own language. Consider the artificial language proposed by György Kalmár in his Praecepta grammatica atque specimina linguae philosophicae sive universalis, ad omnevitae genus adcommodatae . The fact that Kalmár thought it necessary to publish his proposal in Latin, German, and Italian editions in the space of just two
years testified to the need, as he saw it, for a universal mode of communication. Clearly, his native Hungarian would not suffice; nor, he thought, would other existing languages. All were crippled by grammatical irregularities and orthographic confusion. For his new language, Kalmár constructed both a general, rational grammar and a new set of 400 primitive characters. Kalmár's familiarity with Hungarian as well as other languages helped him to construct a general language capable of accommodating "the details, and even the anomalies, of all existing languages."
Two more examples, taken from opposing camps in the political turmoil of the 1790s, illustrate further the broad appeal of the rationalization of grammar and the invention of a language all nations might share. In 1795 Jean Delormel presented to the National Convention in Paris his project for a new and universal language. Shortly thereafter Joseph de Maimieux, a nobleman who had fled to Germany and no friend of the Convention, published his own pasigraphy. Both Delormel and de Maimieux intended their schemes to further the objectives of rationalization and universal communication; both drew inspiration and justification from the esprit géometrique .
Delormel recognized that "extraordinary epochs" offer the opportunity, impossible in normal circumstances, to realize "interesting projects." He presented his own proposal during just such an epoch, at a moment ripe for disseminating "the principles of equality." Delormel's proposal included a binomial classification of substantives by genera and species. In tune with the call for rationalization, Delormel coupled the taxonomy with a system for the regular formation of derivative words; no longer would language need to submit to the tyranny and caprice of usage. As a reviewer of Delormel's scheme commented in 1797, "changes on words are to be rung with all the regularity of a multiplication-table." Such comments doubtless
pleased Delormel, who considered the analogy to numeration to be a prime reason for the simplicity of his scheme.
His system prescribed ten vowels, according to the spirit of the day, and twice that number of consonants. The thicket of synonyms was cleared away, replaced by seven degrees of comparison. Delormel measured the advantages to be gained: where ordinary dictionaries contained 30,000 words, he claimed that one-tenth that number would suffice in his project. More than that, he proclaimed that his new language would promote the "central unity" of the Republic and, by uniting savants of different nations, would spur the progress of science. "Enlightenment brings together men of all sorts, and this language, by facilitating communication, will propagate enlightenment."
Joseph de Maimieux also billed his pasigraphy as offering all the advantages of a rational scheme of knowledge expressed in a universal form. And, he claimed, an enthusiastic audience awaited. One exuberant disciple would later honor the all-encompassing nature of de Maimieux's scheme by dubbing him a second Leibniz. De Maimieux likened pasigraphy to numerals in arithmetic, lines of music, and "characters of chemistry" — "equally intelligible from Petersburg to Malta, Madrid to Peru, London and Paris to Philadelphia or the isle de Bourbon." He devised twelve characters, some of which were mirror images of one another; twelve grammatical rules universally applicable and permitting no exceptions; and three sets of tables. These sets of tables corresponded to the three species of pasigraphic words: those of three, four, or five characters respectively. Words of three characters constituted what de Maimieux called the Indicule . In the Indicule , the first character specified the relevant column (out of twelve columns); the second specified the tranche (six for each column); and the third, the line (one of six) within a given tranche . The Indicule , together with the Petit
Nomenclateur (for words of four characters) and the Grand Nomenclateur (for words of five characters), made up the second part of the Maimieux's pasigraphy. Each page of the tables contained scores of French words, arrayed according to de Maimieux's outline of knowledge.
He claimed much for this scheme. Unlike the "alphabetic chaos" of standard dictionaries, "the pasigraphic order is a natural order." Every word in the system could express "thought, state, action, or passion by means of a progressive analytic development, but without any analytic appareil ." The twelve grammatical rules governed declension, modification, conjugation, and enunciation, and, de Maimieux puffed, yielded up great logical and grammatical riches. The tables of the Indicule, Petit Nomenclateur , and Grand Nomenclateur also provided a mappemonde intellectuel of visual, analytic, and mnemonic convenience.
Although the printer might have complained about having to cast a new font of type (which might never be used again), de Maimieux was fortified by enthusiasm. The relative complexity of concepts would be evident at a glance, measured by the number of characters used in a given word; nature and knowledge could be surveyed with ease from an armchair. De Maimieux proclaimed his lowered expectations, at least by comparison to those held by Wilkins and others in the creation of universal characters in the late 17th century. De Maimieux aimed, not at a representation of truth, but at a handy chart to facilitate communication across linguistic boundaries.
The coordinates in this mappemonde covered varied lexical terrain. Column 9 of the Indicule concerned simple aspects of "science, grammaire, calcul." At the 6th tranche , line 1, de Maimieux put "Plus, au plus, de plus"; four lines down he placed "Beaucoup, bien, très, fort." Cadre 6, column 6 of the Petit Nomenclateur listed civil acts, including privilège, procès-verbal , and confronter ; more lively were the inhabitants of cadre 3, column 6, tranche 4: "Rhinoceros, girafe, onagre, zèbre, buffle, cerf, daim, rène, chamois, gazelle,
grisbock, chevreuil, cabri, vigogne, musc, élan, original." In the Grand Nomenclateur de Maimieux mapped out more complex concepts, with columns for such diverse categories as Dieu, etre, esprit; astres, signes, élémens; insouciance; actes religieux; meubles; arts chymiques . He paid tribute to the reigning confusion of measures and money by supplying an additional four-page alphabetic, multilingual list of available units.
De Maimieux thus borrowed from the encyclopedic spirit of the age, reckoned with the profusion of measures and tongues, marshaled concepts of universal grammar, and hammered his system into a numerical matrix. By analogy to latitude and longitude, de Maimieux's characters could lead a reader straight to the location of the idea in the "topographic map of the domain of thought." De Maimieux followed this with a Carte générale pasigraphique in 1808. Containing some 8,000 words, the tableau of 1808 was nearly as complete as the original version, but multiplied pasigraphic confusion by its incompatibility with its predecessor.
Both de Maimieux and his defenders resorted to mathematical terms in describing the merits of his pasigraphies. De Maimieux saw his system as "a sort of general glossomètre which will give the measure of the richness of each language [and] rectify the inexactitudes of translations, in applying to languages a common scale." One of his disciples went so far as to describe algebra as "that pasigraphy of quantities."
De Maimieux's pasigraphy is a multidimensional variant on proposals in the 17th and 18th centuries for numerical dictionaries—polyglot dictionaries with numbered entries. All these proposals built upon what was seen as the universal intelligibility of Arabic numerals. As Robert Boyle had written in the mid-17th century,
"Since our arithmetical characters are understood by all the nations of Europe the same way, though every several people express that comprehension with its own particular language I conceive no impossibility that opposes the doing that in words, that we see already done in numbers." The idea that mathematical concepts, especially the concepts of number and their representation by Arabic numerals, were universally comprehensible, held persistent appeal to those who aimed at universal, rational languages. In 1801, for example, Zalkind Hourwitz proposed a Polygraphie that relied on the assignment of a number to each word in a basic polyglot dictionary. The same number thus served to designate words with the same meaning in several languages. As the subtitle of his book indicates, Hourwitz intended polygraphy to facilitate the art of communication across national boundaries. Not long afterward Jacques de Cambry did likewise, explicitly invoking the blessings of numeracy in his Manuel interprète de correspondance, on Vocabulaires polyglottes alphabétiques et numériques en tableaux . His mappemonde encompassed not merely Francophones, but also speakers and writers of Italian, Spanish, German, English, Dutch, and the ever-vexing "Celto-breton."
In the final decade of the 18th century, French revolutionary expansionism lent special urgency to the problem of long-distance communication. The logistic necessity of swift and sure communi-
cation with allied military units on distant fronts called for a language simple enough to be transmitted over many miles. Here was an obvious and practical instance of the importance, as proclaimed by philosophers, for a simple, clear, expressive, unequivocal language. In most of the late 18th-century proposals for an optical or military telegraph, the preferred means of transmission were mechanical; the elements of the message were geometrical or numerical.
The first telegraph to attract international attention was the invention of Claude Chappe. Chappe began by exploring the possibility of electrical telegraphy. In 1790 he abandoned the electrical for the mechanical as a means of transmitting language and built a prototype optical telegraph. By 1791 he had extended its range to 15 kilometers. Revolutionary crowds, however, feared telegraphy as a royalist instrument and tore down Chappe's Parisian stations.
The apparatus consisted of a 1-foot pole atop a tower or tall building, a 14-foot crossbar pivoting at the center, a 6-foot arm at each end of the crossbar, control wires by which an operator could manipulate the arms, and lanterns on the arms for nighttime operation. The crossbar could be positioned either horizontally or vertically; the arms then pivoted around the ends of the crossbar. Each arm could assume seven different positions with respect to the crossbar. Each possible position was assigned a number.
With the help of a friend familiar with diplomatic codes, Chappe formulated a manual of nearly 10,000 words. As in the numerical dictionaries, each word corresponded to a number. The positions of
the arms signaled first the page number and then the numbered word on that page. Eventually Chappe added two more code books—one containing phrases and another containing place names. Operators would then transmit sets of three signals: the first indicated which code book to use; the second, the page number; and the third, the number of the word on the page. The scheme thus resembled Hourwitz's numerical dictionary and de Maimieux's matrix of knowledge. As a reviewer commented, Chappe had coupled his mechanical apparatus with an analysis of language: "The table of characters . . . is a tachygraphic method. . . fruit of [Chappe's] long and laborious meditations." In the interest of enhancing its tachygraphic possibilities, a member of the Chappe organization later suggested the use of two-person teams, one to dictate in a sort of numerical shorthand what he saw on the distant tower, the other to record the information.
Through the good offices of his brother, a deputy in the Legislative Assembly, Chappe's system came to the attention of the Assembly's committee on public instruction. After the demise of the Assembly, its successor, the National Convention, was persuaded in April 1793 to order a test "in order to determine the utility of the telegraph" and to consider the use in war of such a "rapid messenger of thought." The commissaires appointed by the Convention—Lakanal, Arbogast, and Daunou—were joined by "several celebrated savants and artists" on the day of the test. They were apparently impressed: Lakanal's official report on the merit of Chappe's telegraph envisioned its "great utility . . . especially in wars on land and at sea, where prompt communication and rapid awareness of maneuvers can have a great influence on success." The positive report led to Chappe's appointment as official ingénieur-télégraphe and, more telling, a high-priority claim on scarce supplies with which to construct a full-scale telegraph system. In September 1794 public enthusiasm was
buoyed (and suspicions about Chappe's loyalties presumably quelled) when the newly constructed telegraph line carried to the capital news of the capture of Condé. A year later the Directory authorized continued support of Chappe's venture, and in 1798 a second line (connecting Paris and Strasbourg) was completed. Eventually the system consisted of eight principal lines and covered some 3,000 miles.
News of the successful deployment of Chappe's system spread quickly. The first printed version of the "alphabet Chappe" appeared anonymously in Leipzig in 1794; periodicals carried accounts of the bullétin télégraphique reporting the capture of Condé a second edition of Chappe's Beschreibung und Abbildung des Telegraphen appeared in 1795. The need for an optical telegraph was widely felt. From 1794, for example, committees of the Patriotic Society of Hamburg had discussed the development of a telegraph system to replace the cumbersome apparatus of messenger boats, observers, and towers used to regulate shipping traffic between Hamburg and Cuxhaven. The Society's secretary, Friedrich Johann Lorenz Meyer, also noted the military potential of a telegraph system. On a visit to Paris in 1796, Meyer witnessed a demonstration of Chappe's optical telegraph, and carried the news of its success back to Hamburg, where a select committee was convened to study its merits. Their cost-benefit analysis argued against establishing a Chappe-style system in Hamburg, on grounds both of cost (estimated at 20,000 marks) and of poor visibility in the northern German climate.
The issue of the optical telegraph also engaged the attention of savants and royalty elsewhere in the German states. In Berlin, the academician and chemist Franz Karl Achard demonstrated his version of the telegraph to Friedrich Wilhelm II in 1795, successfully sending such meaningful messages as "The King is loved by his subjects just as much as he is feared by his enemies." As reward for his efforts, Achard received a substantial bonus of 500 Reichstaler, equal to one-third of his annual salary at the Berlin Academy.
Another savant inspired to imitate or improve upon Chappe's invention was the Swedish academician and natural philosopher Abraham Niklas Edelcrantz. He seized on its military possibilities: "In case of war," a telegraph would afford the possibility of "quick communication for discussion between several armies or divisions of the same army." Edelcrantz began his investigations in September 1794 with variants on Chappe's design of crossbars and pivoting arms. He then switched to a lattice of ten "holes" or windows with shades visible in daylight and lanterns behind them at night. This arrangement permitted 1,024 different signals. The first trials of the Swedish telegraph took place between Stockholm and Drotningholm on 30 October and 1 November 1794. Demonstrations of the system the following year were conducted in "the presence of the king, the regent, and the whole royal court."
Like Chappe, Edelcrantz compiled a codebook, which he called a telegraphische Chiffren-Tabelle . It contained short words, syllables, and a few phrases, each corresponding to a three-digit number. Each
number designated one of the 1024 signals possible with the ten-window lattice. Edelcrantz saw his codebook as the basis for a universal language: "A cipher-table or dictionary explaining signs for all languages should be made to go along with this language-instrument, which can be completely portable."
Thomas Northmore, inspired by what he knew of Polybius' telegraph, also linked his proposal for a Nocturnal or Diurnal Telegraph (which used reflecting lamps moved by a winch) with an account of his new universal character. He proclaimed the "ground-work of the whole super-structure" to be "that if the same numerical figure be made to represent the same word in all languages, an universal medium is immediately obtained." "Diversity of idioms" would prove no obstacle in Northmore's simplified language, which required as a "constant companion" only a "small pocket numerical dictionary," containing some 5,000 to 6,000 "select words."
One of de Maimieux's royalist disciples likewise saw the connection between universal language and telegraphy. The comte de Firmas-Périès, who served in the army of the prince de Condé, recognized the versatility of de Maimieux's pasigraphy and reformulated it for use in telegraphy. Firmas-Périès' mechanical semaphore system, which he labeled "pasitelegraphy," converted de Maimieux's latitude and longitude indicators to arrangements of wooden "hands" on a large clock face operated by chains and pulleys à la Vaucanson. To decipher the message, the operator turned to de Maimieux's tables of words (and knowledge).
Reports concerning the French telegraph, including one retrieved from the pocket of a French prisoner in 1794, also inspired English inventors, including John Gamble, chaplain of the staff of H.R.H.
Frederick, Duke of York and Albany. As would Edelcrantz, Gamble analyzed "the different modes which have been, or may be adopted for the purpose of distant communication"—this at the request of the Duke. Gamble recognized the need for a "figurative language" adequate for communication and capable of quick transmission, and settled on a portable apparatus of five shutters ("lever boards"), which in various combinations denoted the letters of the alphabet. The use of lamps behind the shutters looked promising for nighttime transmissions. The Admiralty implemented Gamble's suggestion on a series of "telegraph hills" between London and Deal as early as 1796 and found that a short message could be relayed in a minute's time. Like the other systems of the late 18th century, the Admiralty telegraph was quick (though labor-intensive) and promptly proved its value for the transmission of military intelligence.
The efficiency of mechanical-optical telegraph systems itself came under scrutiny in Edelcrantz' study. He treated the visibility of signals at a distance like a problem in exact experimental physics, and assessed the accuracy of contemporary telescopes, threshold values for the angular diameter of an object visible at a given distance, and differential effects of color and humidity on visibility. The analysis was informed by Edelcrantz' familiarity with contemporary work on experimental optics, meteorology, and other branches of 18th-century physics. Indeed, Edelcrantz saw the possibility that a network of telegraph operators and stations, suitably equipped with an array of scientific instruments, might also advance the cause of science: "Every telegraph is a real Observatory, which can become an
astronomical one, with the appropriate instruments and the understanding of the director, and science would be enriched with new discoveries."
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.