What is "pure mathematics"

Pure \Pure\, a. [Compar. Purer; superl. Purest.] [OE. pur, F. pur, fr. L. purus; akin to putus pure, clear, putare to clean, trim, prune, set in order, settle, reckon, consider, think, Skr. p? to clean, and perh. E. fire. Cf. Putative.]

1. Separate from all heterogeneous or extraneous matter; free from mixture or combination; clean; mere; simple; unmixed; as, pure water; pure clay; pure air; pure compassion.

   The pure fetters on his shins great.
   --Chaucer.

   A guinea is pure gold if it has in it no alloy.
   --I. Watts.

2. Free from moral defilement or quilt; hence, innocent; guileless; chaste; -- applied to persons. ``Keep thyself pure.''
   --1 Tim. v. 22.

   Now the end of the commandment is charity out of a pure heart, and of a good conscience.
   --1 Tim. i. 5.

3. Free from that which harms, vitiates, weakens, or pollutes; genuine; real; perfect; -- applied to things and actions. ``Pure religion and impartial laws.''
   --Tickell. ``The pure, fine talk of Rome.''
   --Ascham.

   Such was the origin of a friendship as warm and pure as any that ancient or modern history records.
   --Macaulay.

4. (Script.) Ritually clean; fitted for holy services.

   Thou shalt set them in two rows, six on a row, upon the pure table before the Lord.
   --Lev. xxiv. 6.

5. (Phonetics) Of a single, simple sound or tone; -- said of some vowels and the unaspirated consonants.

   Pure-impure, completely or totally impure. ``The inhabitants were pure-impure pagans.''
   --Fuller.

   Pure blue. (Chem.) See Methylene blue, under Methylene.

   Pure chemistry. See under Chemistry.

   Pure mathematics, that portion of mathematics which treats of the principles of the science, or contradistinction to applied mathematics, which treats of the application of the principles to the investigation of other branches of knowledge, or to the practical wants of life. See Mathematics.
   --Davies & Peck (Math. Dict. )

   Pure villenage (Feudal Law), a tenure of lands by uncertain services at the will of the lord.
   --Blackstone.

   Syn: Unmixed; clear; simple; real; true; genuine; unadulterated; uncorrupted; unsullied; untarnished; unstained; stainless; clean; fair; unspotted; spotless; incorrupt; chaste; unpolluted; undefiled; immaculate; innocent; guiltless; guileless; holy.

Abstract \Ab"stract`\ (#; 277), a. [L. abstractus, p. p. of abstrahere to draw from, separate; ab, abs + trahere to draw. See Trace.]

1. Withdraw; separate. [Obs.]

   The more abstract . . . we are from the body.
   --Norris.

2. Considered apart from any application to a particular object; separated from matter; existing in the mind only; as, abstract truth, abstract numbers. Hence: ideal; abstruse; difficult.

3. (Logic)

   1. Expressing a particular property of an object viewed apart from the other properties which constitute it; -- opposed to concrete; as, honesty is an abstract word.
      --J. S. Mill.

   2. Resulting from the mental faculty of abstraction; general as opposed to particular; as, ``reptile'' is an abstract or general name.
      --Locke.

      A concrete name is a name which stands for a thing; an abstract name which stands for an attribute of a thing. A practice has grown up in more modern times, which, if not introduced by Locke, has gained currency from his example, of applying the expression ``abstract name'' to all names which are the result of abstraction and generalization, and consequently to all general names, instead of confining it to the names of attributes.
      --J. S. Mill.

4. Abstracted; absent in mind. ``Abstract, as in a trance.''
   --Milton.

   An abstract idea (Metaph.), an idea separated from a complex object, or from other ideas which naturally accompany it; as the solidity of marble when contemplated apart from its color or figure.

   Abstract terms, those which express abstract ideas, as beauty, whiteness, roundness, without regarding any object in which they exist; or abstract terms are the names of orders, genera or species of things, in which there is a combination of similar qualities.

   Abstract numbers (Math.), numbers used without application to things, as 6, 8, 10; but when applied to any thing, as 6 feet, 10 men, they become concrete.

   Abstract mathematics or Pure mathematics. See Mathematics.

n. mathematics which is done for its own sake rather than being motivated by other sciences.

n. the branches of mathematics that study and develop the principles of mathematics for their own sake rather than for their immediate usefulness

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.

Another insightful view put forth is that pure mathematics is not necessarily applied mathematics: it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians.

To develop accurate models for describing the real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research.