Meaning in Mathematics by Polkinghorne John;

Author:Polkinghorne, John;
Language: eng
Format: epub
Publisher: Oxford University Press
Published: 2011-08-11T16:00:00+00:00

Modern views

Mathematicians and philosophers of the modern era held similar views, as the following statement by Leibniz illustrates:

…the concept of circle put forward by Euclid—namely, that it is the figure described by the motion of a straight line in a plane about one fixed end— affords a real definition, for it is clear that such a figure is possible. It is useful …to have [such] definitions…beforehand…[For] we cannot safely devise demonstrations (secure texere demonstrationes) about any concept, unless we know that it is possible; for of what is impossible or involves a contradiction (impossibilibus seu contradictionem involventibus), contradictories can also be demonstrated. This is the a priori reason why real definition is required for possibility.

Leibniz (1683:1973, pp. 12–13 (294), brackets added)16

Leibniz thus held that genetic or constructive definitions are valuable because they make known a priori the possibility of the concepts they define (see Leibniz, 1764: 1916, Bk. II, ch. II, §18). This in turn, he suggested, makes demonstration ‘safe’ in a certain way.

He made a confusing variety of claims concerning impossible concepts and their existence, however. Sometimes he suggested that there are impossible concepts (i.e., concepts that imply a contradiction).

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