# Hypothesis

## consequences, geometry and logical

HYPOTHESIS, in logic, the antecedent of a conditional proposition. An extensive mathematical or scientific investigation often as sumes the form of a vast conditional prop osition. In such a case the hypothesis is usually stated once for all at the beginning and taken for granted in the rest of the investigation.

Mathematics is neither more nor less than the study of hypotheses of this type and the de duction of their logical consequences. Thus the axioms of Euclidean geometry form the hypothesis on the basis of whioh the proposi tions of geometry are proved. In natural science the truth of the hypothesis, which is a matter of indifference to the mathematician, is of fundamental importance. As this is but rarely, if ever, subject to a direct inspection, and cannot always be deduced from known gen eralizations, a hypothesis is generally established or destroyed by reference to its logical conse quences. While a single unfulfilled consequence of a hypothesis is sufficient to refute it, no num ber of verified consequences, however large, is adequate to its complete demonstration. How ever, hypotheses have certain properties, by virtue of which their degrees of difference or similarity may be compared. It is, moreover,

a principle generally valid that the fewer and less important are the details in which a hypoth esis breaks down, the slighter will be the dif ference beween it and the true hypothesis. Thus a scientific hypothesis, which at first furnishes only a very crude approximation to the ob served facts, by the gradual remodeling of de tail after detail, comes as close to the truth of the matter as may be desired. The Darwinian hypothesis of natural selection is a good ex ample of this; slow variations have been found an insufficient basis for evolutionary changes, and have been replaced by mutations; the theory of the inheritance of acquired characteristics has been discarded; and the whole notion of the survival of the fittest has been accommodated to these changes in opinion. The modern theory of evolution is thus more firmly established than its predecessor, though it still remains a hy pothesis capable of further rectification. See INDUCTION.