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Relating mathematical questions to physical reality

Paul Budnik

Mathematics is generally thought to have its origins in counting and measuring. The abstractions that it introduces have been invaluable, but they can be carried too far as many have suggested. Where and how can boundaries be drawn? Some questionable mathematics has significant physical implications. Can that mathematics be preserved, but interpreted in a way that is fully relevant to physical reality?

One can define four categories of relationships between mathematical questions and a finite physical reality that may be potentially infinite. First are questions solvable in a finite time with a sufficiently powerful computer. Second are those that may or may not have a finite solution depending on the answer such as the consistency of a formal system which has a finite solution iff it is inconsistent. Third are questions which are logically determined, either directly or indirectly, by a recursively enumerable sequence of events. They include the hyperarithmetic hierarchy and some questions requiring quantification over the reals as discussed in The Platonistic philosophy of mathematics. In contrast to the previous two categories, this is a philosophical principle that cannot be fully defined with mathematical rigor.

Fourth are questions about "all" reals or other "absolutely" uncountable sets that are not reducible to questions about a recursively enumerable sequence of events. Questions in this fourth category include the axioms of choice and determinancy, the continuum hypothesis and large cardinal axioms.

Reals seem to be human conceptual creations that do not exist physically, but can, in some cases, be related to physical reality as, for example, the recursive reals are. Seen from this standpoint, some questions about the set of all reals, are a bit like asking if the largest integer is odd or even. There can be no largest integer. Similarly, in our universe, there can be no formalization of the set of "all" reals. This follows from Cantor's proof that the reals are uncountable combined with the Lowenheim-Skolem theorem. We can always define more reals than those provably definable in any consistent formal system just as we can always define a larger integer than any definition of a specific integer.

This is not to say that the relative versions of these questions are not important. The axioms of choice and determinancy are mutually inconsistent if they must hold simultaneously. However one might consider expanding ZF consistent with one axiom and then further expanding it consistent with the other. Neither may be the wrong way to go. It is simply a question of what works best at a given stage of development. Absolutely uncountable sets cannot be rigorously formulated as specific things. Sets can be defined as uncountable relative to the sets provably definable in a specific formal system and thus countable as seen from outside the system. Alternatively uncountable sets can be thought of as place holders for conceptual creations that can always be expanded. These possibilities are where I think the boundaries should be drawn. I do not think it invalidates any mathematics. It requires interpreting it with a realistic view of what finite human minds are capable of.

Related links:

Foundations of Mathematics (FOM) postings
What is Mathematics About?
Formalizing objective mathematics
Ordinal arithmetic calculator and research tool
What is and what will be: Integrating spirituality and science, chapters 5 and 6
Mathematical Infinity and Human Destiny Video---transcript
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