On Mathematical and Logical Realism and Contingency

Abstract

This study presents the claim that mathematics and logic are merely highly formalized reflections, grounded in the physical laws of conservation. The claim generally correlates with John Stuart Mill’s known stance, but unlike his general view, it specifies which elements of the natural kingdom are reflected by mathematical objects and statements. As the study claims one version of physicalism, it raises the question of the necessity vs. contingency of mathematics and concludes that the necessity of mathematical judgments depends on the necessity of the conservation laws themselves. Since the conservation laws are only certain, it follows that there is no basis to claim the necessity of mathematical statements themselves, and that it is only possible to speak of a conditional necessity in the sense that mathematics is necessarily such as it is only in a world governed by conservation laws. Such conditional necessity does not possess the being of absolute necessity. Mathematics can only be considered necessary to the extent that the reflected world described by it is necessary, which further implies the claim that mathematics is necessarily a posteriori and synthetic. The entire series of mathematical proof types, including the most commonly utilized reduction ad absurdum, ultimately derives its strength from experience.