The Ontological Status of Mathematical Entities: The Necessity for Modern Physics of an Evaluation of Mathematical Systems

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FAR FROM BEING A PURELY ESOTERIC CONCERN of theoretical mathematicians, the examination of the ontological status of mathematical entities, I submit, has far-reaching implications for a very practical area of knowledge, namely, the method of science in general, and of physics in particular. Although physics and mathematics have since Newton's second derivative been inextricably wedded, modem physics has a particularly mathematical dependence. Physics has moved and continues to move further away from the possibility of direct empirical verification, primarily because of the increasingly complex logistical problems of experimentation within the parameters of the very large and of the very small. As certain areas become more and more theoretical, with developments of this century in astrophysics, cosmology, and quantum mechanics, and more specifically, with the postulation of new hypothetical elementary particles based almost exclusively upon mathematical data, physics is forced to depend increasingly upon mathematics as a method for verifying physical possibility. Typically, a mathematical formulation descriptive of an empirically established phenomenon x is manipulated and made subject to derivation on the assumption that the new formulation will continue to correspond with physical reality, and may even yield new information about the phenomenon's behavior. Why, however, should a coherence between the empirically-defined world and mathematical processes be assumed? This coherence is, above all, dependent upon a hidden metaphysically strong presupposition about the ontological status of mathematical entities and their systems.

That there is a metaphysically strong presupposition of the sort to which I refer is not immediately obvious, and I would like here to address three common refutations of this position initially given. Perhaps the most immediate is the insistence that mathematics serves a purely descriptive function in the sciences, that it acts only as a kind of language. Although this characterization is certainly applicable in some cases, it cannot possibly justify the present use of mathematics to make hypotheses and predictions in physics. It cannot explain the prescriptive use of mathematics to verify and suggest physical possibility.

Assuming the prescriptive use of mathematics, another argument can be made that mathematics is simply logic, in its most absolute, noncontroversial tautological sense. Thus, the use of mathematics in physics simply ensures the same consistency, although in a much more easily manipulatable form, that would occur by our following out the implications of theories using what amounts to common sense reason, for it is obvious that our knowledge of physical reality (physics) must be limited by, or at least not be inconsistent with, our own mental principles of logic. Unfortunately, this tautological view of mathematics too is untenable; for besides its rather narrow view of the role of mathematics, it makes the mistaken assumption that mathematics as used in physics is in fact logical, never mind tautological. One need only think of the prominent use in physics of complex numbers and common surds such as the exponential function and pi to realize how many mathematical inconsistencies have been wholeheartedly embraced without question and with success. The intuitionist school of mathematics, very much concerned with consistency and solid grounding, deems the use of infinity as unacceptably anti-intuitive, yet where would its absence leave calculus, a veritable cornerstone of the foundations of physics? A merely tautological system would severely limit the present scope of the physical and even social sciences.

The final and most common argument against inherent assumptions of mathematical Platonism in physics is simply that of cold pragmatism, which claims that we use the mathematical systems that we use not because we endow them with any real ontological status, but because they are effective. …