Editing Pure mathematics (section)

==History==

===Ancient Greece===
Ancient Greek mathematicians were amongst the earliest to make a distinction between pure and applied mathematics. [[Plato]] helped to create the gap between "arithmetic", now called [[number theory]], and "logistic", now called [[arithmetic]]. Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who "must learn the art of numbers or [they] will not know how to array [their] troops" and arithmetic (number theory) as appropriate for philosophers "because [they have] to arise out of the sea of change and lay hold of true being."<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0-471-54397-7 |chapter=The age of Plato and Aristotle |pages=[https://archive.org/details/historyofmathema00boye/page/86 86] |chapter-url=https://archive.org/details/historyofmathema00boye/page/86 }}</ref> In this wise [[Euclid of Alexandria]], when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must make gain of what he learns."<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0-471-54397-7 |chapter=Euclid of Alexandria |pages=[https://archive.org/details/historyofmathema00boye/page/101 101] |chapter-url=https://archive.org/details/historyofmathema00boye/page/101 }}</ref> The Greek mathematician [[Apollonius of Perga]], asked about the usefulness of some of his theorems in Book IV of ''Conics'',   asserted that<ref name="Apollonius">{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0-471-54397-7 |chapter=Apollonius of Perga |pages=[https://archive.org/details/historyofmathema00boye/page/152 152] |chapter-url=https://archive.org/details/historyofmathema00boye/page/152 }}</ref>
<blockquote>They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason.</blockquote>
And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of ''Conics'' that the subject is one of those that "...seem worthy of study for their own sake."<ref name="Apollonius" />

===19th century===
The term itself is enshrined in the full title of the [[Sadleirian Professor of Pure Mathematics|Sadleirian Chair]], "Sadleirian Professor of Pure Mathematics", founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of ''pure'' mathematics may have emerged at that time. The generation of [[Carl Friedrich Gauss|Gauss]] made no sweeping distinction of the kind between ''pure'' and ''applied''. In the following years, specialisation and professionalisation (particularly in the [[Weierstrass]] approach to [[mathematical analysis]]) started to make a rift more apparent.

===20th century===
At the start of the twentieth century mathematicians took up the [[axiomatic method]], strongly influenced by [[David Hilbert]]'s example. The logical formulation of pure mathematics suggested by [[Bertrand Russell]] in terms of a [[Quantifier (logic)|quantifier]] structure of [[Proposition (mathematics)|proposition]]s seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of ''[[rigorous proof]]''.

Pure mathematics, according to a view that can be ascribed to the [[Bourbaki group]], is what is proved. "Pure mathematician" became a recognized vocation, achievable through training. That said, the case has been made pure mathematics is useful in [[engineering education]]:<ref>[[A. S. Hathaway]] (1901) [https://www.ams.org/journals/bull/1901-07-06/S0002-9904-1901-00797-5/S0002-9904-1901-00797-5.pdf "Pure mathematics for engineering students"], [[Bulletin of the American Mathematical Society]] 7(6):266–71.</ref>

:There is a training in habits of thought, points of view, and intellectual comprehension of ordinary engineering problems, which only the study of higher mathematics can give.