On the Influence of Signs in Mathematical Reasoning

1 p.l., 53 pp. Large 4to, orig. blue wrappers, uncut. London: J. Smith, 1826.<p> A remarkable association copy of this extremely rare offprint, linking two of the greatest scientists of the 19th century, and pre-figuring Babbage's later work on his difference and analytical engines. This copy bears the following inscription on the title-page in Babbage's hand: "To M. Gauss from the Author." Additionally, Babbage has written on the upper wrapper: "M. Gauss. Influence of Signs. 3. Ch. Babbage. On the influence of signs in mathematical reasoning." <P>Babbage here presents his views on the importance of symbolic notation in mathematical reasoning. He argues that algebraic symbolism enables one to express ideas more briefly and precisely than in ordinary language; it enables one to consider problems in great generality, rather than only in special cases; and it often enables one to consider simultaneously different cases of a problem that would otherwise be treated separately. The use of mathematical symbols is more efficient, and less prone to error, than other forms of reasoning, as he emphasized particularly when discussing the superiority of algebraic analysis over geometrical reasoning: "[T]he power which we possess by the aid of symbols in compressing into a small compass the several steps of a chain of reasoning, whilst it contributes greatly to abridge the time which our enquiries would otherwise occupy, in difficult cases influences the accuracy of our conclusions: for from the distance which is sometimes interposed [in geometrical reasoning] between the beginning and the end of a chain of reasoning, although the separate parts are sufficiently clear, the whole is often obscure" (p. 8). Babbage also emphasizes the importance of choosing the correct notation: it should remind the user of the nature of the quantity itself (so use 'v' for velocity, 't' for time, and so on); and related quantities should be denoted by similar symbols (so v, v', v'' etc. for the velocities of different bodies in the same problem, for example). Babbage singles out Lagrange's Mecanique Analitique as a model to be emulated in the correct choice of mathematical symbols. <P>The present paper may be seen as part of a chain of ideas that links not only his mathematical and scientific work, but also his views on politics and industry. While a student at Cambridge, Babbage formed, together with Herschel and Peacock, the "Analytical Society." This society was principally concerned with a matter of mathematical symbolism: its aim was famously to support "the principles of pure D'ism in opposition to the Dot-age of the University." Theological overtones apart, this was a plea to replace what Babbage and the other members of the Society viewed as the outdated and inefficient Newtonian fluxional "dot" notation still used in England with the Leibnizian dy/dx notation which was universally employed on the Continent. "For Herschel and Babbage, however, there was more to analysis than a debate about the appropriate mathematical symbols...The key to the success of analytical algebra as they saw it was its efficiency. It was a problem-solving technology that could produce answers quickly and without wasting resources...It was a way of economizing mental labor. As such it could be used to recognize what the most efficient way of proceeding in other enterprises might be too. It could provide the key, for example, to the most profitable way of deploying resources in order to maximize factory production."-Morus, When Physics became King, p. 36. <P>"Babbage's ultimate solution to the problem of how to guarantee efficiency, transparency, and accuracy in reasoning was the same as his solution to the same problem in political economy: replace humans with machinery. Babbage was a firm exponent of the division of labor in factory management and equally enthusiastic for mechanization as the ultimate realization of the principle. His primary concern throughout the 1820s and beyond was to work on his projected calculating and analytical engines...The calculating engine would replace the human drudge work for calculating mathematical tables to be used (for example) in actuarial work and in astronomy. The analytical engine would go further - it would replace the human capacity to reason as well."-ibid., pp. 37-38. When developing his difference engine Babbage again realized the importance of symbolism: in "A Method of Expressing by Signs the Action of Machinery" (Philosophical Transactions: 1826) he developed a special notation, related to Boolean algebra (which was developed later), to accurately describe the working of the engine. <P>Babbage first met Gauss in the course of a European tour he undertook starting at the end of 1827. After travelling through Holland, Belgium, Germany and Italy, Babbage arrived in Berlin in September 1828 to meet Alexander von Humboldt, then regarded as the greatest scientist of the century. At the time, Humboldt was organizing the seventh annual Congress of German scientists. Babbage was present when the Congress opened on 18 September 1828, with Gauss among the several hundred scientists and luminaries present. It was there that Babbage met Gauss. <P>Babbage's paper was read at a meeting of the Cambridge Philosophical Society on 16 December 1821, but it was not published in its Transactions until more than five years later (Vol. 2, 1827, pp. 325-377). The item offered here is a very rare pre-publication offprint, dated 1826 on the title page. <P>In very fine and fresh condition, entirely uncut. Preserved in a handsome blue morocco-backed box. With the stamps of the "Gauss-Bibliothek" (also on upper wrapper) and the Royal Observatory at Göttingen on title with release stamp date 15 October 1951. <p>❧ D.S.B., I, pp. 354-56. Dunnington, G.W., Gauss: Titan of Science, 2004. Poggendorff, I, 81.