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The Calculus Ratiocinator is a concept appearing in the writings of Gottfried Leibniz, usually paired with his characteristica universalis, which he mentioned much more frequently. Gottfried Wilhelm Leibniz (also Leibnitz or von Leibniz)[1] (July 1 (June 21 Old Style) 1646, Leipzig – November 14, 1716, Hanover) was a German polymath of Sorbian origin, deemed a universal genius in his day and since. ...
The Characteristica Universalis is a recurring concept in the writings of Gottfried Leibniz. ...
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. A screenshot of computer software in action. ...
Computer hardware is the physical part of a computer, including the digital circuitry, as distinguished from the computer software that executes within the 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" (Fearnley-Sander 1982: p.164). 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 Charles 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". This
Analytic philosophy is the dominant philosophical movement in University philosophy departments in English-speaking countries and in Scandinavia, although one of its founders, Gottlob Frege, was German, and many of its leading proponents, such as Ludwig Wittgenstein, Rudolf Carnap, Kurt Gödel, Karl Popper, Hans Reichenbach, Herbert Feigl, Otto Neurath...
Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...
Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...
An inference engine tries to derive answers from a knowledge base. ...
The terms computer program, software program, applications program, system software, or just program are used to refer to either an executable program by both lay people and computer programmers or the collection of source code from which an executable program is created (eg, compiled). ...
Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ...
Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. ...
Charles Sanders Peirce Charles Sanders Peirce (pronounced purse), (September 10, 1839 – April 19, 1914) was an American polymath, born in Cambridge, Massachusetts. ...
Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ...
Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ...
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." Leibniz constructed just such a machine (picture) for mathematical calculations which was also called a Stepped Reckoner. 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) Generally, synthetic means pertaining to synthesis, i. ...
Again, in a ship, if a man were at liberty to do what he chose, but were devoid of mind and excellence in navigation (αÏετης κυβεÏνητικης), do you perceive what must happen to him and his fellow sailors? (Plato, Alcibiades, 135A). ...
Electronic engineering is a professional engineering discipline that deals with the emission, behavior, and effects of electrons (as in electron tubes and transistors) and with electronic devices, systems or equipment. ...
Systems theory or general systems theory or systemics is an interdisciplinary field which studies systems as a whole. ...
In the 1670s, a German Baron called Gottfried von Leibniz (sometimes von Leibnitz) took mechanical calculation a step beyond his predecessors. ...
Norbert Wiener Norbert Wiener (November 26, 1894 - March 18, 1964) was a U.S. mathematician and applied mathematician, especially in the field of electronics engineering. ...
- 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.
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 Weiner'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 pentium chips (for example). Hence giving electronic circuit language significance for systems and process philosophy. This article is not about combinatory logic, a topic in mathematical logic. ...
A logic gate is an arrangement of electronically-controlled switches used to calculate operations in Boolean algebra. ...
Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...
In electronics, an adder is a device which will perform the addition, S, of two numbers. ...
The process of circuit design can cover systems ranging from national power grids all the way down to the individual transistors within an integrated circuit. ...
Pentium logo, with MMX enhancement The Pentium is a fifth-generation x86 architecture microprocessor by Intel. ...
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. The Unified Science is an integral system of knowledge developed by Edward Haskell, Harold Cassidy, Willard V. Quine, Arthur Jensen, and Jere Clark. ...
In electrical engineering, Impedance is a measure of opposition to a sinusoidal electric current. ...
The adjective Boolean (sometimes boolean), coined in honor of George Boole, is used in many contexts: An evaluation that results in either TRUE or FALSE. A boolean value is a truth value, either true or false, often coded 1 and 0, respectively. ...
In electricity, current is the rate of flow of charges, usually through a metal wire or some other electrical conductor. ...
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.
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.
- Desmond Fearnley-Sander, 1982. Hermann Grassmann and the Prehistory of Universal Algebra, The American Mathematical Monthly, Vol. 89, No. 3, pp. 161-166.
Louis Couturat (January 17, 1868 - August 3, 1914) was a French logician, mathematician, philosopher, and linguist. ...
Hartley Rogers, Jr. ...
Norbert Wiener Norbert Wiener (November 26, 1894 - March 18, 1964) was a U.S. mathematician and applied mathematician, especially in the field of electronics engineering. ...
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