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Mathematical Truth

Mathematicians have not adopted a divergent approach to expanding mathematics. The dominant formalization of mathematics is the Zermelo Frankel axioms of set theory plus the Axiom of Choice (ZFC)[3,1]. ZFC seems adequate for all practical problems in science and engineering and most open mathematical questions[5]. This approach to mathematics is justified by a Platonic philosophy that assumes the existence, in some abstract sense, of a hierarchy of infinite sets which encode mathematical truth that cannot be decided by finite processes. Some mathematicians are questioning this philosophy and the objectivity of some open problems in mathematics that depend on it, most prominently the Continuum Hypothesis. This question arose from the proof that there must be more real numbers than integers because one cannot pair up every real number with a unique integer. This is obviously correct for finite sets but can be questioned for infinite ones. The Continuum Hypothesis asserts that the reals are the smallest set larger than the integers. Both the Continuum Hypothesis and its negation have been shown to be consistent with ZFC[3]. Thus the Continuum Hypothesis can never be proved true or false within ZFC.

Solomon Feferman, the editor of Gödel's collected works, expressed his skepticism as follows.

I am convinced that the Continuum Hypothesis is an inherently vague problem that no new axiom will settle in a convincingly definite way. Moreover, I think the Platonic philosophy of mathematics that is currently claimed to justify set theory and mathematics more generally is thoroughly unsatisfactory and that some other philosophy grounded in inter-subjective human conceptions will have to be sought to explain the apparent objectivity of mathematics.
[Feferman's note to the above quote] CH [Continuum Hypothesis] is just the most prominent example of many set-theoretical statements that I consider to be inherently vague. Of course, one may reason confidently within set theory (e. g., in ZFC) about such statements as if they had a definite meaning[5].

I have argued for a philosophy of mathematical truth that limits objectively meaningful mathematical questions to those relevant to ultimate destiny in a finite but potentially infinite universe. Such questions are logically determined by a list of events that an ideal computer could enumerate[2]. I am skeptical of infinite structures in the physical universe or in an ideal Platonic abstract reality. The concept of an ideal Platonic reality is firmly rooted in the physical universe. For example, we cannot construct the Platonic perfect circle, but we can compute the ratio of its circumference to its diameter, pi, to millions of decimal places with high confidence that we have done it correctly. If we cannot reach Platonic perfection, we can often approach it to an arbitrary accuracy through technology. Nothing in physical reality seems able to approach the infinite structures in the Platonic philosophy underlying ZFC.


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