Ancient Greece Mathematical Platonism 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." 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." The Greek mathematician
Apollonius of Perga, asked about the usefulness of some of his theorems in Book IV of
Conics, asserted that 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. 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.". This duality of
platonism vs
Aristotle is still present in modern mathematics . Different historical records of the school of
Pythagoras show this contradiction too from elements of
mysticism similar to
Plato to the proof
irrationality which leads to the
paradox that an infinite non repetitive set of digits cannot belong to the "finite" and "well defined" world of
Euclidean geometry. Last but not least
Zeno paradox shows again this duality between the pure logical reasoning (splitting distance by two is always possible) and the applied realm (the fact that the traveler never arrives).
19th century Pure mathematics invented The term "pure mathematics" itself is enshrined in the full title of the
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 (1777 to 1855) 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.
The problem of infinities After
Weierstrass, by the end of 19th century, an important discussion about the role of infinities came from authors like
Gregor Cantor and early examples of
fractals and
chaos.
Ludwig Wittgenstein considered Cantor's approach with
uncountable sets the Cancer of mathematics.
Henri Poincare seems to have compared set theory to a temporary disease Infinities in general are difficult to treat axiomatically, and therefore have been always considered on the fringe of pure mathematics, and this complexity became evident in the works of
Bertrand Russell and
Gödel on
paradoxes in the early 20th century.
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 structure of
propositions 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 was made pure mathematics is useful in
engineering education: :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. Major advances in the beginning of 20th century was the formalization of
abstract algebra and
topology; these two fields were deeply influenced by the pure mathematics philosophy.
Finiteness, Number theory and Algebraic geometry Historically areas often considered attached to pure mathematics are
number theory, where these infinities are typically
countable and
algebraic geometry where functions are typically tamed functions (i.e.
piecewise polynomial or
rational functions). The success in the proof of
Weil conjectures and the unification of these two fields ultimately in the
geometric Langlands program gave momentum to the concept of pure mathematics as a research activity that can self sustain independently. On the other hand, the fields of
functional analysis,
partial differential equations,
statistics and
dynamical systems were often considered in the
applied mathematics camp, and this is reflected in the typical organization of mathematics curriculum.
The advance of computers By the end of the century numerical proofs of the
four color theorem were primary examples of the advance of computers . Famous mathematicians such as
Paul Cohen already in the 1970s challenged the concept of pure mathematics, stating that there will be a moment in the future that most mathematicians will be replaced by computers.
French vs Russian schools The French school of mathematics and the western authors in general were deeply influenced by the Bourbaki group and therefore by the philosophical idea to detach mathematics from natural sciences. The Russian school of mathematics instead (e.g.,
Kolmogorov,
Gelfand and
Vladimir Arnold) believed that mathematics is actually fully grounded in experimental sciences such as physics and biology, this was reflected in the separation of the two schools during the
Cold War era due to geopolitical motivations and by the works of authors like
Robert Langlands which started to piece wise reconnect work from both schools .
21st century At the beginning of the 21st century, the application of
artificial intelligence to pure mathematics has attracted the attention of such luminaries as
Ken Ono and
François Charton. Looking at recent research fields such as the Analytic Langlands conjecture one can see that the distinction between pure and applied blurs again. Another example is the emergence of
chaos in
number theory and authors like Robert Langlands advocate for the unification of mathematics with Physics through the
Langlands program. ==Generality and abstraction==