Calculus ratiocinator

The Calculus Ratiocinator is a theoretical universal logical calculation framework, a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently mentioned characteristica universalis, a universal conceptual language.

Two views

There are two contrasting points of view on what Leibniz meant by calculus ratiocinator. The first is associated with computer software, the second is associated with computer hardware.

The analytic view

• The received point of view in analytic philosophy and formal logic, is that the calculus ratiocinator anticipates mathematical logic — an "algebra of logic". The analytic point of view understands that the calculus ratiocinator is a formal inference engine or computer program which can be designed so as to grant primacy to calculations. That logic began with Frege's 1879 Begriffsschrift and C.S. Peirce's writings on logic in the 1880s. Frege intended his "concept script" to be a calculus ratiocinator as well as a lingua characteristica. That part of formal logic relevant to the calculus comes under the heading of proof theory. From this perspective the calculus ratiocinator is only a part (or a subset) of the universal characteristic, and a complete universal characteristic includes a "logical calculus".

The synthetic view

• A contrasting point of view stems from synthetic philosophy and fields such as cybernetics, electronic engineering and general systems theory. It is little appreciated in analytic philosophy. The synthetic view understands the calculus ratiocinator as referring to a "calculating machine". The cybernetician Norbert Wiener considered Leibniz's calculus ratiocinator a forerunner to the modern day digital computer:

The history of the modern computing machine goes back to Leibniz and Pascal. Indeed, the general idea of a computing machine is nothing but a mechanization of Leibniz's calculus ratiocinator. (Wiener 1948: 214)

...like his predecessor Pascal, [Leibniz] was interested in the construction of computing machines in the Metal. ... just as the calculus of arithmetic lends itself to a mechanization progressing through the abacus and the desk computing machine to the utra-rapid computing machines of the present day, so the calculus ratiocinator of Leibniz contains the germs of the machina ratiocinatrix, the reasoning machine (Wiener 1965: 12)

Leibniz constructed just such a machine for mathematical calculations which was also called a Stepped Reckoner. As a computing machine, the ideal calculus ratiocinator would perform Leibniz's integral and differential calculus. In this way the meaning of the word, "ratiocinator" is clarified and can be understood as a mechanical instrument that combines and compares ratios.

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  Stepped Reckoner
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Implications

The full implications of these views are yet to be explored in philosophy and the history of computing. Philosophers familiar with the foundations of microprocessors will appreciate that Wiener's view implies a link between Leibniz's On the Art of Combinations and the combinational logic, itself an art of combining logic gates (and truth tables) to form the adder circuits used for circuit design in Intel Pentium chips (for example). Hence giving electronic circuit language significance for systems and process philosophy.

Moreover any unified natural philosophy (i.e. unified science) must be able to explain the difference between the two views above. R.Hartley saw a link between the two, defining the calculus ratiocinator as "an algorithm which, when applied to the symbols of any formula of the characteristica universalis, would determine whether or not that formula were true as a statement of science" (Hartley Rogers, Jr. 1963; p. 934). Some clarification might be found in the fact that electronic components need to have very high electrical impedance ratings in order to perform Boolean logic. This results in very small amounts of electrical current energy flow. However macroscopic ecological, engineering and general systems sciences that employ synthetic reasoning, are characteristically concerned with larger energy flows.

A classic discussion of the calculus ratiocinator is Couturat (1901: chpts. 3,4), who maintained that the characteristica universalis --and thus the calculus ratiocinator--were inseparable from Leibniz's encyclopedic project (chpt. 5). Hence the characteristic, calculus ratiocinator, and encyclopedia form three pillars of Leibniz's Enlightenment project.

Whether one understands the calculus ratiocinator as a formal logic or as a calculating machine, therefore has both metaphysical implications and affects how one views its relation to Leibniz's characteristica universalis, encyclopedia, and unfinished Enlightenment project. If one understands the calculus ratiocinator as a forerunner of the digital computer, then the coupling of digital computer and encyclopedia, i.e. Wikipedia or the like, is a necessary outcome of the Enlightenment.

See also

• Gottfried Leibniz
• characteristica universalis

References

• Louis Couturat, 1901. La Logique de Leibniz. Paris: Felix Alcan. Donald Rutherford's English translation in progress.
• Hartley Rogers, Jr. 1963, An Example in Mathematical Logic, The American Mathematical Monthly, Vol. 70, No. 9., pp. 929–945.
• Norbert Wiener, 1948, "Time, communication, and the nervous system," Teleological mechanisms. Annals of the N.Y. Acad. Sci. 50 (4): pp. 197–219.
• -- 1965, Cybernetics, Second Edition: or the Control and Communication in the Animal and the Machine, The MIT Press.
• Desmond Fearnley-Sander, 1982. Hermann Grassmann and the Prehistory of Universal Algebra, The American Mathematical Monthly, Vol. 89, No. 3, pp. 161–166.

External links

• Language as Calculus versus Language as Universal Medium