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  Abstraction, the action of divorcing properties of physical objects from the objects themselves, is a fundamental concept, perhaps the fundamental concept, in mathematics. It marks off the beginning of mathematics from what went before. The discoverer of abstraction was the person who first realized that numbers are independent of the objects being counted, that two oranges and two apples (for instance) share a property, ‘twoness’, which is independent of what kinds of fruit they are. Ever since this discovery, abstraction has been a major theme in the development of mathematics, as those interested in the field have come up with ideas further and further divorced from their basis in the real world, and then sought ways to bring them back to tell us things about the real world which we might otherwise not have known.

Using abstraction, it is possible to reason directly about properties that hold in general; without abstraction, thought is limited to the particular. Intelligence itself has been linked by many to the ability to perform abstractions; the power of reasoning about the general by abstraction from the specific is the foundation of intellectual progress. Once, for example, the property of ‘twoness’ is isolated from the particular objects having that property, it becomes possible to determine properties of all objects having ‘twoness’—such as the fact that two pairs taken together always give a quartet—and to have such a thought, though infants can do it, is a highly sophisticated mental process.

Mathematics takes such thought-processes to lengths unimagined in ordinary life: indeed, mathematicians are often more interested in the abstraction itself than in any applications. The power of mathematics to abstract essential properties from the real world gives it its generality, and its power in its application to scientific problems. Without it, scientists would be unable to use mathematics to examine properties of the universe around them. (It is, of course, questionable whether a mathematics which was totally unrelated to the real world could ever have developed.) Each branch of the discipline is itself the product of abstraction: geometry, for example, is an abstraction of the calculations used in building and surveying. There are even branches of mathematics which are abstractions of other branches of mathematics: category theory, for example, is an abstraction of general properties of the objects studied in algebra.

Abstraction of a slightly different kind is a fundamental process in both fine art and folk art. It takes two main forms. In the first, the artist makes images which refer not to the visible world but to his or her own fancy. Typical examples are the whorls or zig-zags which adorn early pottery, or the rows and geometrical shapes made from beads which decorate Amerindian clothing. The appeal of such patterns lies in symmetry and repetition on the one hand, and on the other of interruptions in such symmetry. In fine art, main styles (of painting in particular) which use similar techniques are Abstract Expressionism, Geometric Abstraction, Neoplasticism, Suprematism and Tachism; the style is also a major feature of functional art and design throughout the world.

In the second kind of abstraction in art, the creator ‘abstracts’ images from the visible world, making (for example) the shapes of crosses, flower heads, hooks, leaves, pebbles and so on the basis for abstract patterns. The appeal here is two-fold: intellectual (appreciation of the pattern-making itself, as with the first kind of abstraction) and emotional (pleasure or challenge derived from the ‘abstraction’ of objects which still have a recognizable correlative, and non-artistic ‘meaning’, in the visible world.) This style is once again common in folk art—totemic art is a notable example—and in fine art is characteristic of those modernist movements which seek to break with the pictorial conventions of earlier art: examples are Cubism, Expressionism, Fauvism and Futurism. PD MG AJ KMcL SMcL

See also applicability of mathematics; artificial intelligence; scientific method; thought.



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