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Some ages ago, there was no difference between chemistry, physics, mathematics, and perhaps even philosophy. These were not further distinguished and largely practiced by the same people.

Obviously, this is no longer the case. The different disciplines have differentiated and themselves have become split into subdisciplines.

I am interested at what point the distinction between pure and applied mathematics first appeared within the mathematical community (if a mathematical community even existed at that point). That includes, in particular, when terminology "pure" and "applied" first appeared.

(edit) On top of that, it is of interest whether this discourse occurred in non-Western traditions in a similar manner. Mathematicians at the Indian Kerala school or the Chinese Imperial Court may have had different point of views.

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    $\begingroup$ GPT4 gives a convincing answer: the term appeared in the late 19th century when specialization and the math community began to be large enough. The last universal mathematicians were probably Hilbert and Poincaré, who both worked somewhat in applied maths. Hilbert in continuum physics, eg with the Riemann-Hilbert shock problem, Poincaré in celestial mechanics. It is also the time when rigor demands increased, with the push to formalize maths - Frege, Dedekind, Peano, Hilbert,... So there was a growing gap between math and physics, where applied maths sprang. WW1 then promoted this trend. $\endgroup$
    – plm
    Commented Oct 5 at 10:44
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    $\begingroup$ Let me mention that such a distinction appears in F. Klein's Development of Mathematics in the 19th Century in the context of Gauß, thus it should be at least earlier than when F. Klein wrote this. $\endgroup$
    – Z. M
    Commented Oct 5 at 10:46
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    $\begingroup$ I’m voting to close this question because it is more appropriate for HSMSE. $\endgroup$
    – LSpice
    Commented Oct 5 at 13:42
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    $\begingroup$ @pm ChatGPT is designed to be convincing. Go and find some primary sources to back up its claims and then tell people. Anything ChatGPT generates is hardly admissible evidence in court... :-) $\endgroup$
    – David Roberts ♦
    Commented Oct 5 at 22:15
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    $\begingroup$ The correct spelling is ‘Kerala’. The word ‘Karela’ means a bitter gourd (vegetable) in a certain language in India. $\endgroup$
    – Cranium Clamp
    Commented Oct 7 at 4:49

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The distinction between pure and applied mathematics goes back to the ancient Greeks, where it was referred to as the study of the world of ideas (pure) versus the world of the senses (applied).

Plutarch makes the distinction in his Life of Marcellus, citing Plato:

For the art of mechanics, now so celebrated and admired, was first originated by Eudoxus and Archytas, who embellished geometry with its subtleties, and gave to problems incapable of proof by word and diagram, a support derived from mechanical illustrations that were patent to the senses. For instance, in solving the problem of finding two mean proportional lines, a necessary requisite for many geometrical figures, both mathematicians had recourse to mechanical arrangements, adapting to their purposes certain intermediate portions of curved lines and sections.

But Plato was incensed at this, and inveighed against them as corrupters and destroyers of the pure excellence of geometry, which thus turned her back upon the incorporeal things of abstract thought and descended to the things of sense, making use, moreover, of objects which required much mean and manual labour. For this reason mechanics was made entirely distinct from geometry, and being for a long time ignored by philosophers, came to be regarded as one of the military arts.

More Greek references are in the Wikipedia entry.

The terms "pure" versus "applied" came in use in the 19th century, with journals such as Journal fur die reine und angewandte Mathematik (founded 1826), and Journal de mathematiques pures et appliquee (founded 1836).

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    $\begingroup$ I was reading just before your edit on Crelle's journal. I am not sure to find in the Plutarch quote a distinction between pure and applied maths as we understand them nowadays. $\endgroup$
    – plm
    Commented Oct 5 at 10:50
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    $\begingroup$ @plm: independently on what Plutarch says, the distinction is quite clear. Euclid and Apollonius were pure mathematicians, while Ptolemy applied. Archimedes did both. You never see such things as trigonometry, for example in the work on pure mathematics. $\endgroup$
    – Alexandre Eremenko
    Commented Oct 5 at 11:17
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    $\begingroup$ @plm There is even no absolute consensus today. For example, in France, PDE is usually classified as applied, including more theoretic parts, while it does not seem to be so in the US, say. $\endgroup$
    – Z. M
    Commented Oct 5 at 13:25
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    $\begingroup$ J. reine angew. Math. and J. math. pures appl. were predated by Magazin für reine und angewandte Mathematik and Archiv der reinen und angewandten Mathematik (Leipzig 1786–1800), resp. Annales de mathématiques pures et appliquées (Nîmes 1810–1832). $\endgroup$
    – Francois Ziegler
    Commented Oct 6 at 18:23
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    $\begingroup$ @FrancoisZiegler It's remarkable that we're still using the same terms for the fundamental division of the subject almost 250 years later. $\endgroup$
    – Will Sawin
    Commented Oct 8 at 20:09
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The ancient Greeks were quite strict in their separation of pure mathematics (mathematics) and applied mathematics (logistics). Euclid in his elements covered the basics of pure mathematics: line segments were not measured (there are no units, neither for lengths, nor for areas or angles) but compared to each other, and proportions were described in terms of integers; the unit was not divisible (there is not a single fraction in Euclid). Everything related to measures of lengths (which is pretty much anything you would find in Heron) was called logistics; this includes even the Arithmetica of Diophantus since he did not regard the unit as not divisible: there are fractions everywhere in his books.

The distinction between mathematics and logistics is not quite the same as the modern one between pure and applied mathematics, and this is reflected in the generalization of the notion of "number" that took place long after Euclid, when mathematicians freely used the decimal system and fractions, and later quadratic and higher irrationals.

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  • $\begingroup$ On the other hand, Greek pure mathematics was the result of an abstraction process always starting from concrete problems and aimed to concrete applications. For instance, the interest on conics comes from the geometry of projectors, and acustics of theaters. According to Lucio Russo, Euclid's Elements is an abstract model for technical drawing. $\endgroup$
    – Pietro Majer
    Commented Oct 16 at 8:20

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