"mathematical science," late 14c. as singular noun, mathematik (replaced since early 17c. by mathematics, q.v.), from Old French mathematique and directly from Latin mathematica (plural), from Greek mathēmatike tekhnē "mathematical science," feminine singular of mathēmatikos (adj.) "relating to mathematics, scientific, astronomical; pertaining to learning, disposed to learn," from mathēma (genitive mathēmatos) "science, knowledge, mathematical knowledge; a lesson," literally "that which is learnt;" from manthanein "to learn," from PIE root *mendh- "to learn."
As an adjective, "pertaining to mathematics," from c. 1400, from French mathématique or directly from Latin mathematicus.
"one skilled or learned in mathematics," early 15c., mathematicion, from Old French mathematicien, from mathematique, from Latin mathematicus "of or belonging to mathematics," from Latin mathematica (see mathematic).
"the science of quantity; the abstract science which investigates the concepts of numerical and spatial relations," 1580s; see mathematic (the older form of the word in English, attested from late 14c.) + -ics. Originally one of three branches of Aristotelian theoretical science, along with first philosophy (or metaphysics) and physics (or natural philosophy).
"of, pertaining to, or of the nature of mathematics," early 15c., from Medieval Latin mathematicus "of or belonging to mathematics," from Latin mathematica (see mathematic) + -al (1). Also, by 1765, "pertaining to the quadrivium," comprising arithmetic, geometry, astronomy, and music. It also could include optics. Related: Mathematically.
The four mathematical arts are arithmetic, geometry, music, and astronomy; these anciently were termed the quadrivium, or fourfold way of knowledge. [Sir John Hawkins, "A General History of the Science and Practice of Music," Sir John Hawkins, 1776]
It is the hypothetical source of/evidence for its existence is provided by: Greek menthere "to care," manthanein "to learn," mathēma "science, knowledge, mathematical knowledge;" Lithuanian mandras "wide-awake;" Old Church Slavonic madru "wise, sage;" Gothic mundonsis "to look at," German munter "awake, lively."