Pure vs. Applied

Why did humans first start contemplating the natural numbers and plane and solid geometry (see Thurston's definition)? Because these branches of math have applications, the natural numbers to trade and accouting, plane geometry to land surveying, and solid geometry to astronomy, astrology, and navigation. In a certain sense, all modern mathematics is similarly applied.

V. I. Arnold wrote, with tongue partly in cheek, in Polymathematics: Is Mathematics a Single Science or a Set of Arts?, 2000:

All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA.).

Cryptography has generated number theory, algebraic geometry over finite fields, algebra [footnote: The creator of modern algebra, Viete, was the cryptographer of King Henry IV of France.], combinatorics and computers.

Hydrodynamics procreated complex analysis, partial differential equations, Lie groups and algebra theory, cohomology theory and scientific computing.

Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.

So we could be doing ostensibly pure mathematics, but Arnold points out that our subfield only exists (i.e. has funding) because it is relevant to (something that is relevant to )ⁿ an application with funding. We usually use pure and applied to describe the relative distance (n) from such an application.

Real vs. Corrupt?

Pure and applied mathematics also can be distinguished by their motivations. Applied mathematics is often motivated by its use-value for the domain it is being applied to. Pure mathematics is motivated by either aesthetic concerns — it is a form of art — or fun — it is a game.

In a short article, Mathematics in War-Time, published in Eureka 1940, G. H. Hardy uses the term real mathematics to mean the whole body of mathematical knowledge which has permanent aesthetic value, as for example the best Greek mathematics has, the mathematics which is eternal because the best of it may, like the best literature, continue to cause intense emotional satisfaction to thousands of people after thousands of years. This includes both pure mathematics and some applied mathematics, such as Maxwell and Einstein and Eddington and Dirac. According to Hardy, no mathematics specially devised for war meets this aesthetic criterion, rather it is repulsively ugly and intolerably dull. And conversely, the real mathematics...has no direct utility in war. Nobody has yet found any war-like purpose to be served by the theory of numbers or relativity or quantum mechanics, and it seems very unlikely that anybody will do so for many years.

Unfortunately for Hardy, his bid to wash his hands of complicity in war by keeping to real mathematics did not work out: the theory of numbers is now essential to cryptography, relativity is necessary for the GPS satellites that drones rely on, and quantum mechanics furnished the crucial insights needed for the atomic bomb.

Due to the interconnected nature of mathematics, even the most seemingly pure mathematics is susceptible to even the most wordly applications. If we understand Arnold's point above, this is not that surprising: all math is really applied, from the get-go. Either it is funded because of an application, or because mathematicians care about it, and they care because it is relevant to (something that is relevant to )ⁿ an application with funding.