 | This article includes a list of general references, but it lacks sufficient corresponding inline citations .(May 2010) |
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.
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 received point of view in analytic philosophy and formal logic, is that the calculus ratiocinator anticipates mathematical logic —an "algebra of logic". [1] 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 universal characteristics. 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 characteristics, and a complete universal characteristics includes a "logical calculus".
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."
"...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 ultra-rapid computing machines of the present day, so the calculus ratiocinator of Leibniz contains the germs of the machina ratiocinatrix, the reasoning machine."
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.
Hartley Rogers 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". [2]
A classic discussion of the calculus ratiocinator is that of Louis Couturat, [3] who maintained that the characteristica universalis — and thus the calculus ratiocinator — were inseparable from Leibniz's encyclopedic project. [4] Hence the characteristics, calculus ratiocinator, and encyclopedia form three pillars of Leibniz's project.
A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computations are mathematical equations and computer algorithms.
Gottfried Wilhelm (von) Leibniz was a German polymath active as a mathematician, philosopher, scientist and diplomat. Leibniz is also called "The Last Universal Genius" due to his knowledge and skills in different fields and because such people became less common during the Industrial Revolution and spread of specialized labor after his lifetime. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science by devising a cataloguing system whilst working at Wolfenbüttel library in Germany that would have served as a guide for many of Europe's largest libraries. Leibniz's contributions to a wide range of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, French and occasionally in German.
Classical logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic.
Norbert Wiener was an American mathematician, computer scientist and philosopher. He became a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher in stochastic and mathematical noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.
Walter Harry Pitts, Jr. was an American logician who worked in the field of computational neuroscience. He proposed landmark theoretical formulations of neural activity and generative processes that influenced diverse fields such as cognitive sciences and psychology, philosophy, neurosciences, computer science, artificial neural networks, cybernetics and artificial intelligence, together with what has come to be known as the generative sciences. He is best remembered for having written along with Warren McCulloch, a seminal paper in scientific history, titled "A Logical Calculus of Ideas Immanent in Nervous Activity" (1943). This paper proposed the first mathematical model of a neural network. The unit of this model, a simple formalized neuron, is still the standard of reference in the field of neural networks. It is often called a McCulloch–Pitts neuron. Prior to that paper, he formalized his ideas regarding the fundamental steps to building a Turing machine in "The Bulletin of Mathematical Biophysics" in an essay titled "Some observations on the simple neuron circuit".
Mathesis universalis is a hypothetical universal science modelled on mathematics envisaged by Descartes and Leibniz, among a number of other 16th- and 17th-century philosophers and mathematicians. For Leibniz, it would be supported by a calculus ratiocinator. John Wallis invokes the name as title in his Opera Mathematica, a textbook on arithmetic, algebra, and Cartesian geometry.
Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.
Louis Couturat was a French logician, mathematician, philosopher, and linguist. Couturat was a pioneer of the constructed language Ido.
Universal language may refer to a hypothetical or historical language spoken and understood by all or most of the world's people. In some contexts, it refers to a means of communication said to be understood by all humans. It may be the idea of an international auxiliary language for communication between groups speaking different primary languages. In other conceptions, it may be the primary language of all speakers, or the only existing language. Some religious and mythological traditions state that there was once a single universal language among all people, or shared by humans and supernatural beings.
Universal science is a branch of metaphysics. In the work of Gottfried Wilhelm Leibniz, the universal science is the true logic. The idea of establishing a universal science originated in the seventeenth century with philosophers Francis Bacon and Rene Descartes. Bacon and Descartes conceptualized universal science as a unified approach to collect scientific information similar to encyclopedias of universal knowledge but were unsuccessful. Leibniz extended their ideas to use logic as an "index" to order universal scientific and mathematical information as an operational system with a universal language. Plato's system of idealism, formulated using the teachings of Socrates, is a predecessor to the concept of universal science and influenced Leibniz' s views against materialism in favor of logic. It emphasizes on the first principles which appear to be the reasoning behind everything, emerging and being in state with everything. This mode of reasoning had a supporting influence on great scientists such as Boole, Frege, Cantor, Hilbert, Gödel, and Turing. All of these great minds shared a similar dream, vision or belief in a future where universal computing would eventually change everything.
The Latin term characteristica universalis, commonly interpreted as universal characteristic, or universal character in English, is a universal and formal language imagined by Gottfried Leibniz able to express mathematical, scientific, and metaphysical concepts. Leibniz thus hoped to create a language usable within the framework of a universal logical calculation or calculus ratiocinator.
The history of computer science began long before the modern discipline of computer science, usually appearing in forms like mathematics or physics. Developments in previous centuries alluded to the discipline that we now know as computer science. This progression, from mechanical inventions and mathematical theories towards modern computer concepts and machines, led to the development of a major academic field, massive technological advancement across the Western world, and the basis of a massive worldwide trade and culture.
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.
Diagrammatic reasoning is reasoning by means of visual representations. The study of diagrammatic reasoning is about the understanding of concepts and ideas, visualized with the use of diagrams and imagery instead of by linguistic or algebraic means.
The mathematical concept of a function dates from the 17th century in connection with the development of the calculus; for example, the slope of a graph at a point was regarded as a function of the x-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme.
Cybernetics: Or Control and Communication in the Animal and the Machine is a book written by Norbert Wiener and published in 1948. It is the first public usage of the term "cybernetics" to refer to self-regulating mechanisms. The book laid the theoretical foundation for servomechanisms, automatic navigation, analog computing, artificial intelligence, neuroscience, and reliable communications.
Mathematicism is 'the effort to employ the formal structure and rigorous method of mathematics as a model for the conduct of philosophy'. or else it is the epistemological view that reality is fundamentally mathematical. The term has been applied to a number of philosophers, including Pythagoras and René Descartes although the term is not used by themselves.
The following outline is provided as an overview of and topical guide to Gottfried Wilhelm Leibniz:
