Editing Gottfried Wilhelm Leibniz (section)

===Symbolic thought and rational resolution of disputes===
Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion:

{{blockquote|The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, to see who is right.<ref>{{Cite SEP|url-id=leibniz-mind|title=Leibniz's Philosophy of Mind |date=June 29, 2020|edition=Winter 2020 |author-last1=Kulstad|author-first1= Mark |author-last2=Carlin |author-first2=Laurence}}</ref><ref>{{Cite web |last=Gray |first=Jonathan |title="Let us Calculate!": Leibniz, Llull, and the Computational Imagination |url=https://publicdomainreview.org/essay/let-us-calculate-leibniz-llull-and-the-computational-imagination/ |access-date=2023-06-22 |website=The Public Domain Review |language=en}}</ref><ref>''The Art of Discovery'' 1685, Wiener 51</ref>}}

Leibniz's [[calculus ratiocinator]], which resembles [[Mathematical logic|symbolic logic]], can be viewed as a way of making such calculations feasible. Leibniz wrote memoranda<ref>Many of his memoranda are translated in [[George Henry Radcliffe Parkinson|Parkinson]] 1966.</ref> that can now be read as groping attempts to get symbolic logic—and thus his ''calculus''—off the ground. These writings remained unpublished until the appearance of a selection edited by Carl Immanuel Gerhardt (1859). [[Louis Couturat]] published a selection in 1901; by this time the main developments of modern logic had been created by [[Charles Sanders Peirce]] and by [[Gottlob Frege]].

Leibniz thought [[symbol]]s 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 [[calculus]] is an example of his skill in this regard. Leibniz's passion for symbols and notation, as well as his belief that these are essential to a well-running logic and mathematics, made him a precursor of [[semiotics]].<ref>Marcelo Dascal, ''Leibniz. Language, Signs and Thought: A Collection of Essays'' (''Foundations of Semiotics'' series), John Benjamins Publishing Company, 1987, p. 42.</ref>

But Leibniz took his speculations much further. Defining a [[Grapheme|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 character]]s, and the symbols of [[astronomy]] and [[chemistry]], he deemed not real.<ref>Loemker, however, who translated some of Leibniz's works into English, said that the symbols of chemistry were real characters, so there is disagreement among Leibniz scholars on this point.</ref><!--is this paragraph correct up to this point?--> 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:

{{blockquote|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.<ref>''Preface to the General Science'', 1677. Revision of Rutherford's translation in Jolley 1995: 234. Also Wiener I.4</ref>}}

Complex thoughts would be represented by combining characters for simpler thoughts. Leibniz saw that the uniqueness of [[prime factorization]] suggests a central role for [[prime numbers]] in the universal characteristic, a striking anticipation of [[Gödel numbering]]. Granted, there is no intuitive or [[mnemonic]] way to number any set of elementary concepts using the prime numbers.

Because Leibniz was a mathematical novice when he first wrote about the ''characteristic'', at first he did not conceive it as an [[algebra]] but rather as a [[universal characteristic|universal language]] or script. Only in 1676 did he conceive of a kind of "algebra of thought", modeled on and including conventional algebra and its notation. The resulting ''characteristic'' included a logical calculus, some combinatorics, algebra, his ''analysis situs'' (geometry of situation), a universal concept language, and more. What Leibniz actually intended by his ''characteristica universalis'' and calculus ratiocinator, and the extent to which modern formal logic does justice to calculus, may never be established.<ref>A good introductory discussion of the "characteristic" is Jolley (1995: 226–240). An early, yet still classic, discussion of the "characteristic" and "calculus" is Couturat (1901: chpts. 3, 4).</ref> Leibniz's idea of reasoning through a universal language of symbols and calculations remarkably foreshadows great 20th-century developments in formal systems, such as [[Turing completeness]], where computation was used to define equivalent universal languages (see [[Turing degree]]).