(from el, μάθησις, "science or learning", and la, universalis "universal") is a hypothetical

universal science Universal science (german: Universalwissenschaft; la, scientia generalis, scientia universalis) is a branch of metaphysics. In the work of Gottfried Wilhelm Leibniz, the universal science is the true logic.Stanley Burris"Leibniz's Influence on 19th ...

modelled on

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

envisaged by Descartes and

Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...

, among a number of other 16th- and 17th-century philosophers and mathematicians. For Leibniz, it would be supported by a ''

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 conce ...

''.

John Wallis John Wallis (; la, Wallisius; ) 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 ...

invokes the name as title in his ''Opera Mathematica'', a textbook on

arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...

, and

Cartesian geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...

History

Roomen - Universae mathesis idea, 1602 - 151336

Descartes' most explicit description of ''mathesis universalis'' occurs in ''Rule Four'' of the ''

Rules for the Direction of the Mind In 1628 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''. The work was eventually pub ...

'', written before 1628. Leibniz attempted to work out the possible connections between

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...

infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 ari ...

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...

, and universal characteristics in an incomplete treatise titled "''Mathesis Universalis''" in 1695.

Predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

could be seen as a modern system with some of these ''universal'' qualities, at least as far as mathematics and

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

are concerned. More generally, ''mathesis universalis'', along with perhaps

François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...

, represents one of the earliest attempts to construct a

formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A fo ...

. One of the perhaps most prominent critics of the idea of ''mathesis universalis'' was Ludwig Wittgenstein and his philosophy of mathematics. As Anthropologist Emily Martin notes:

René Descartes

In Descartes' corpus the term ''mathesis universalis'' appears only in the ''

''. In the discussion of ''Rule Four'', Descartes' provides his clearest description of ''mathesis universalis'':

Gottfried Leibniz

In his account of ''mathesis universalis'', Leibniz proposed a dual method of universal synthesis and analysis for the ascertaining

truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as belief ...

, described in ''De Synthesi et Analysi universale seu Arte inveniendi et judicandi'' (1890).

''Ars inveniendi''

''Ars inveniendi'' (

Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...

for "art of invention") is the constituent part of ''mathesis universalis'' corresponding to the method of synthesis. Leibniz also identified synthesis with the '' ars combinatoria'', viewing it in terms of the recombination of symbols or human thoughts.

''Ars judicandi''

''Ars judicandi'' (

for "art of judgement") is the constituent part of ''mathesis universalis'' corresponding to the method of analysis.

Michel Foucault

In ''

The Order of Things ''The Order of Things: An Archaeology of the Human Sciences'' (Les mots et les choses: Une archéologie des sciences humaines, 1966) by French philosopher Michel Foucault proposes that every historical period has underlying epistemic assumptions ...

'',

Michel Foucault Paul-Michel Foucault (, ; ; 15 October 192625 June 1984) was a French philosopher, historian of ideas, writer, political activist, and literary critic. Foucault's theories primarily address the relationship between power and knowledge, and ho ...

discuses ''mathesis'' as the conjunction point in the ordering of simple natures and algebra, paralleling his concept of ''taxinomia''. Though omitting explicit references to universality, Foucault uses the term to organise and interpret all of human science, as is evident in the full title of his book: "''The Order of Things: An Archaeology of the Human Sciences''".

References

Bibliography

* * * * * *

External links

* Raul Corazzon's Ontology web page
''Mathesis Universalis'' with a bibliography
{{History of science Mathematical logic Gottfried Wilhelm Leibniz René Descartes Intellectual history History of logic History of mathematics Michel Foucault Philosophy of mathematics Philosophy of science Latin words and phrases