By the second century, in the Almagest, Ptolemy provides a modern conception of "mathematics" as a "science":

'Mathematics' ... is an attribute of all existing things, without exception, both mortal and immortal: for those things which are perpetually changing ... it changes with them, while for eternal things ... it keeps their unchanging form unchanged.

When was the term "mathematics" first used in this way?


It seems the term "mathematics" has been used to express different things at different times, historically. The name itself - "mathematics" - is Greek in origin, and at that point in time, mathematics encompassed much more than it does today in terms of its breadth.

See the entry on Wikipedia: Etymology of term "mathematics".

See also the entry, History of Mathematics, which includes the following assertion:

The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction".
- (Reference given: Heath. A Manual of Greek Mathematics. p. 5.)

This may provide a start, and provides further resources, if you scroll down. (The link is to a subsection of the entry "Mathematics".

Also of interest is Earliest Known Uses of Mathematical Words, a site maintained by Jeff Miller, where you can find the origins of the use of many mathematical terms. Click on "M", then scroll down to "mathematics". (This addresses the question in your post's title.) I'll quote the start of that entry:

Words of the form math- derive ultimately from the Greek mathematike tekhne meaning "mathematical science," itself derived from manthanein, the ordinary word meaning “to learn.” How the association with a special form of learning came about is considered by T. L. Heath (A History of Greek Mathematics, vol. 1 pp. 10-11). Heath describes how the school of Pythagoras distinguished between those who had learned the theory of knowledge in its most complete form, the mathematicians, and those who knew only the practical rules of conduct. He infers that, “seeing that the Pythagorean philosophy was mainly mathematics, the term might easily become identified with the mathematical subjects as distinct from others.”

  • $\begingroup$ Regarding the differing meanings of mathematics through time, remember that Augustine wrote somewhere: "The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell." $\endgroup$ – Miha Habič Nov 17 '12 at 19:53
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    $\begingroup$ @MihaHabič I think many of the OP's here at math.se would agree with Augustine! $\endgroup$ – Namaste Nov 17 '12 at 19:56
  • $\begingroup$ @MihaHabič: Ptolemy would certainly have disagreed with Augustine. He goes on to say: "This science, above all things, could make men see clearly; from the constancy, order, symmetry and calm which are associated with the divine, it makes its followers lovers of this divine beauty, accustoming them and reforming their natures, as it were, to a similar spiritual state." And though "this science" is often assumed to be astronomy, it's clear (to me at least) from the context (in translation) that he means mathematics. Which raises the question of to what extent he saw the two fields as distinct. $\endgroup$ – orome Nov 17 '12 at 20:07

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