Mathesis universalis

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Mathesis universalis (Greek μάθησις, mathesis "science or learning", Latin universalis "universal") is a hypothetical universal science modeled on mathematics envisaged by Descartes and Leibniz, among a number of more minor 16th and 17th century philosophers and mathematicians. John Wallis invokes the name as the title to a textbook on Cartesian geometry. For Leibniz, it would be supported by a calculus ratiocinator.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

Gottfried Wilhelm Leibniz German mathematician and philosopher

Gottfried Wilhelm (von) Leibniz was a prominent German polymath and philosopher in the history of mathematics and the history of philosophy. His most notable accomplishment was conceiving the ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments. Mathematical works have always favored Leibniz's notation as the conventional expression of calculus, while Newton's notation became unused. It was only in the 20th century that Leibniz's law of continuity and transcendental law of homogeneity found mathematical implementation. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of all digital computers.

John Wallis English mathematician

John Wallis was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is credited with introducing the symbol ∞ to represent the concept of infinity. He similarly used 1/∞ for an infinitesimal. John Wallis was a contemporary of Newton and one of the greatest intellectuals of the early renaissance of mathematics.


Descartes' clearest description of the mathesis universalis occurs in Rule IV of the Rules for the Direction of the Mind, written before 1628. The desire for a language more perfect than any natural language had been expressed before Leibniz by John Wilkins in his An Essay towards a Real Character and a Philosophical Language in 1668. Leibniz attempts to work out the possible connections between algebra, infinitesimal calculus, and universal character in an incomplete treatise titled "Mathesis Universalis" in 1695.

<i>Rules for the Direction of the Mind</i> book

In 1628 or a few years earlier, René Descartes began work on an unfinished treatise regarding the proper method for scientific and philosophical thinking entitled Regulae ad directionem ingenii, or Rules for the Direction of the Mind. This work outlined the basis for his later work on complex problems of mathematics, science, and philosophy. 36 rules were planned in total, although only 21 were actually written. This work was not published during the author's lifetime. A Dutch translation appeared in 1684, and the first Latin edition in 1701.

In neuropsychology, linguistics, and the philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages can take different forms, such as speech or signing. They are distinguished from constructed and formal languages such as those used to program computers or to study logic.

John Wilkins Secretary of the Royal Society and Bishop of Chester

John Wilkins, (1614–1672) was an Anglican clergyman, natural philosopher and author, and was one of the founders of the Royal Society. He was Bishop of Chester from 1668 until his death.

Predicate logic could be seen as a modern system with some of these universal characteristics, at least as far as mathematics and computer science are concerned. More generally, mathesis universalis, along with perhaps François Viète's algebra, represents one of the earliest attempts to construct a formal system.

Computer science Study of the theoretical foundations of information and computation

Computer science is the study of processes that interact with data and that can be represented as data in the form of programs. It enables the use of algorithms to manipulate, store, and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems.

François Viète French mathematician

François Viète, Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry III and Henry IV of France.

Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

One of the perhaps most prominent critics of the idea of mathesis universalis was Ludwig Wittgenstein and his philosophy of mathematics. As Anthropologist Prof. Emily Martin notes: 'Tackling mathematics, the realm of symbolic life perhaps most difficult to regard as contingent on social norms, Wittgenstein commented that people found the idea that numbers rested on conventional social understandings "unbearable"' [1] [2]

Ludwig Wittgensteins philosophy of mathematics

Ludwig Wittgenstein considered his chief contribution to be in the philosophy of mathematics, a topic to which he devoted much of his work between 1929 and 1944. As with his philosophy of language, Wittgenstein's views on mathematics evolved from the period of the Tractatus Logico-Philosophicus: with him changing from logicism towards a general anti-foundationalism and constructivism that was not readily accepted by the mathematical community. The success of Wittgenstein's general philosophy has tended to displace the real debates on more technical issues.

See also

Ars inveniendi is a chief notion of mathesis universalis and implies ascertaining truth through the use of mathematics.

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.

Universal language may refer to a hypothetical or historical language spoken and understood by all or most of the world's population. In some contexts, it refers to a means of communication said to be understood by all living things, beings, and objects alike. 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.

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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

History of mathematics aspect of history

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, together with Ancient Egypt and Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy and to formulate calendars and record time.

Classical logic is the intensively studied and most widely used class of logics. Classical logic has had much influence on analytic philosophy, the type of philosophy most often found in the English-speaking world.

<i>La Géométrie</i> mathematical appendix to Descartes Discourse on Method, published in 1637

La Géométrie was published in 1637 as an appendix to Discours de la méthode, written by René Descartes. In the Discourse, he presents his method for obtaining clarity on any subject. La Géométrie and two other appendices, also by Descartes, La Dioptrique (Optics) and Les Météores (Meteorology), were published with the Discourse to give examples of the kinds of successes he had achieved following his method.

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as the numbers 0, 1 and 2; function symbols such as sin; operator symbols such as "+"; conceptual symbols such as lim and dy/dx; equations and variables; and complex diagrammatic notations such as Penrose graphical notation and Coxeter–Dynkin diagrams.

The alphabet of human thought is a concept originally proposed by Gottfried Leibniz that provides a universal way to represent and analyze ideas and relationships by breaking down their component pieces. All ideas are compounded from a very small number of simple ideas which can be represented by a unique character.

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Wilhelm Leibniz independently discovered calculus in the mid-17th century. However, both inventors claimed that the other had stolen his work, and the Leibniz-Newton calculus controversy continued until the end of their lives.

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.

A philosophical language is any constructed language that is constructed from first principles, like a logical language, but may entail a strong claim of absolute perfection or transcendent or even mystical truth rather than satisfaction of pragmatic goals. Philosophical languages were popular in Early Modern times, partly motivated by the goal of recovering the lost Adamic or Divine language. The term ideal language is sometimes used near-synonymously, though more modern philosophical languages such as Toki Pona are less likely to involve such an exalted claim of perfection. It may be known as a language of pure ideology. The axioms and grammars of the languages together differ from commonly spoken languages today.

In elementary mathematics, a variable is a symbol, commonly a single letter, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown. Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation. A typical example is the quadratic formula, which allows one to solve every quadratic equation by simply substituting the numeric values of the coefficients of the given equation for the variables that represent them.

Arithmetica Universalis is a mathematics text by Isaac Newton. Written in Latin, it was edited and published by William Whiston, Newton's successor as Lucasian Professor of Mathematics at the University of Cambridge. The Arithmetica was based on Newton's lecture notes.

In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.

Diagrammatic reasoning

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.

Martin Knutzen was a German philosopher, a follower of Christian Wolff and teacher of Immanuel Kant, to whom he introduced the physics of Isaac Newton.

Mathesis may refer to


  1. Martin, Emily (October 2013). "The Potentiality of Ethnography and the Limits of Affect Theory". Current Anthropology. 54 (S7): 156. doi:10.1086/670388.
  2. Rhees, Rush (1970). Discussions of Wittgenstein. New York: Schocken.