Volume 25, 2008
Philosophy of Mathematics
Wilhelm K. Essler
On Using Measuring Numbers according to Measuring Theories
It was shown by Frege that four of the five axioms of Peano can be regarded as analytical truths; and it was shown by Russell that the remaining axiom cannot be regarded as being analytically true or even as being analytically false, that this axiom thus is to be regarded as a synthetic statement. In using the concept of apriority in the sense of Reichenbach, it can be shown that this synthetic axiom is to be regarded as an apriorical truth within the usual background theory of measuring theories, which are used not as generalizations of empirical results but as— not moreover provable— preconditions of receiving measuring results and of ordering these results. Furthermore, the systems of numbers, starting with the natural numbers, are developed in a way such that the pre-rational numbers— but not the rational ones— turn out to be those ones which are used in performing measurements according to such theories, while the pre-real numbers— but not the real ones— then turn out to be those ones which are used in using such measuring theories together with their background theories for purely theoretical reasons.