Autor: Lonergan, Bernard J.F.

Buch: Understanding and Being

Titel: Understanding and Being

Stichwort: 3 Arten von Definition: Wortdefinition, erklärende u. implizite; Hilbert

Kurzinhalt: Nominal, Explanatory, and Implicit Definition, Hilbert's Geometrie: Punkte und Linien; die implizite (implicit) Definition sieht völlig von der materie ab -> Relation

Textausschnitt: () three types of definition: nominal, explanatory, and implicit. Nominal definition supposes insight into the use of words.
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An explanatory definition adds a further element which, if not added in the definition, would have to be added by way of a postulate.
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Finally, the postulational element of the explanatory definition can be used alone, and then the definition is implicit. Implicit definition is of far greater generality. In Hilbert's Foundations of Geometry, points and straight lines are defined by the postulate, A straight line is determined by two and only two points. Two points and a straight line are set in correlation; if we have two points, that will determine what we mean by a straight line, and if we have a straight line, that will determine what is meant by a point. The only thing that is settled is the relation between the two points and a straight line.
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By 'point' Hilbert does not mean position without magnitude, something that you approximate by a dot, nor does he mean by 'line' something with length but without breadth or thickness. That is one case of his definition, but his definition is satisfied equally directly if by a straight line you mean ...
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Implicit definition, then, prescinds from the matter; it gets entirely away from the matter. It is just the expression of the relational element, and it picks out what is of scientific significance, introducing us to complete generality.
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When you base your geometry on implicit definitions, you are dealing purely with relations; you have moved on to a further stage of abstraction. Implicit definitions are simply relational structures, and the terms of the relations are left indeterminate. When definitions are implicit, the concrete meaning can be anything that will satisfy them.