Pure mathematics

An illustration of the Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.

Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts.  This was a recognizable category of mathematical activity from the nineteenth century onwards, at variance with the trend towards meeting the needs of navigation, astronomy, physics, economics, engineering, and so on.

Quotes

• ... mathematics resembles an iceberg: beneath the surface is the realm of pure mathematics hidden from the public ... Above the water is the tip, the visible part of which we call applied mathematics.
  • Armand Borel, quoted in "Opening Ceremonies". Proceedings of the International Congress of Mathematicians, 1994, Zürich. Birkhäuser, Basel. 1995. p. xxv. 

• Proof is the idol before whom the pure mathematician tortures himself.
  • Arthur Eddington, "Science and Mysticism," ch. 15 of The Nature of the Physical World (1928).

• Pure mathematics is, in its way, the poetry of logical ideas.  One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships.  In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.
  • Albert Einstein, Obituary for Emmy Noether (5 May 1935).

• There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand.
  • Peter Rowlett, "The unplanned impact of mathematics", Nature 475, 2011, pp. 166-169.

• Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true … If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.
  • Bertrand Russell, Recent Work on the Principles of Mathematics, published in International Monthly, vol. 4 (1901).

• Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.
  • Bertrand Russell, Principles of Mathematics (1903), Ch. I: Definition of Pure Mathematics, p. 3.

See also

• applied mathematics