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.
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.
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.
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"'
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