Mathesis universalis

Mathesis universalis (from μάθησις, mathesis "science or learning", and universalis "universal") 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.

History
Descartes' most explicit description of mathesis universalis occurs in Rule Four of the Rules for the Direction of the Mind, written before 1628. Leibniz attempted to work out the possible connections between mathematical logic, algebra, infinitesimal calculus, combinatorics, and universal characteristics in an incomplete treatise titled "Mathesis Universalis" in 1695.

Predicate logic could be seen as a modern system with some of these universal qualities, 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.

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: "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"."

René Descartes
In Descartes' corpus the term mathesis universalis appears only in the Rules for the Direction of the Mind. In the discussion of Rule Four, Descartes' provides his clearest description of mathesis universalis:"; Rule Four [...] I began my investigation by inquiring what exactly is generally meant by the term 'mathematics' and why it is that, in addition to arithmetic and geometry, sciences such as astronomy, music, optics, mechanics, among others, are called branches of mathematics. [...] This made me realize that there must be a general science which explains all the points that can be raised concerning order and measure irrespective of the subject-matter, and that this science should be termed mathesis universalis — a venerable term with a well-established meaning — for it covers everything that entitles these other sciences to be called branches of mathematics. [...]"
 * We need a method if we are to investigate the truth of things.

- René Descartes, Rules for the Direction of the Mind; translated by John Cottingham

Gottfried Leibniz
In his account of mathesis universalis, Leibniz proposed a dual method of universal synthesis and analysis for the ascertaining truth, described in De Synthesi et Analysi universale seu Arte inveniendi et judicandi (1890).

Ars inveniendi
Ars inveniendi (Latin 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 (Latin for "art of judgement") is the constituent part of mathesis universalis corresponding to the method of analysis.