Leibniz - On symbol systems
Type of Spiritual Experience
Leibniz thought symbols were important for human understanding. He attached so much importance to the development of good notations that he attributed all his discoveries in mathematics to this. His notation for the calculus is an example of his skill in this regard. C.S. Peirce, a 19th-century pioneer of semiotics, shared Leibniz's passion for symbols and notation, and his belief that these are essential to a well-running logic and mathematics.
But Leibniz took his speculations much further. Defining a character as any written sign, he then defined a "real" character as one that represents an idea directly and not simply as the word embodying the idea. Some real characters, such as the notation of logic, serve only to facilitate reasoning. Many characters well known in his day, including Egyptian hieroglyphics, Chinese characters, and the symbols of astronomy and chemistry, he deemed not real. Instead, he proposed the creation of a characteristica universalis or "universal characteristic", built on an alphabet of human thought in which each fundamental concept would be represented by a unique "real" character:
A description of the experience
It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters insofar as they are subject to reasoning all that we can do in arithmetic and geometry. For all investigations which depend on reasoning would be carried out by transposing these characters and by a species of calculus.
In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.
Quoted in G Simmons Calculus Gems (New York 1992).