Mathematical Implications of Mach’s Ideas: Positivistic Geometry, the Clarification of Functional Connections

Abstract

The highest tribute to a scientist on an anniversary of his death is the demonstration that his ideas are still alive. Fifty years after Mach, some thoughts of this great Austrian philosopher-scientist have an unspent vitality. Two ideas of his on which I wish to concentrate lead to implications for mathematics that have not yet been fully explored: Mach’s general philosophical views suggest the development of a positivistic geometry, which has hardly begun; and his maxim that causal explanations should be replaced by functional connections calls for a clarification of the method of applying the function concept and other ideas of mathematical analysis to science — a clarification which is still in its initial stage.