Sunday, July 29, 2012

Reflections on realism

Today, the set theorist Hugh Woodin is probably the strongest advocate for an objective mathematical realism akin to Platonism, However, among set theorists in general, there is nothing close to consensus on such philosopical matters. (E.g. Joel Hamkins prominently advocates his "multiverse" conception of set theory.) Probably most of them don't even want to think about such questions; they just want to do set theory, whatever it "is." Most mathematicians don't even care about the set theory beyond its role in providing a common syntax and semantics that, in principle if not in practice, unify mathematics. (In reality, today there is no such thing as a true expert in three major fields of mathematics; experts in two are rare and valuable.) Mostly, each of us just wants to do mathematics in his field. Unless one's field is set theory/foundations, set-theoretic language is taken for granted, and intersections of one's field with non-trivial mathematical phenomena from set theory are mostly considered pathologies to be avoided.

Woodin's case starts with an empirical fact: set theorists have discovered a natural linear heirarchy of stronger and stronger "large cardinal" axioms that appear to absorb all other mathematical axioms in that they can "naturally" instantiate models of them. This empirical (but not yet mathematical formalized) regularity leads to the hypothesis, which I share, that there is one way up, meaning that we can converge towards (but, as Godel demonstrated, never arrive at) an "objectively best" or "true" concept of set. (More precisely, I conjecture that all the "best" concepts of set will be mutually interpretable in some very strong way.)

This is analogous to the hypothesis that the regularities of the physical universe are objective and that our laws of physics are converging towards a true description of these regularities. However, strictly speaking, the "one way up" hypothesis carries no ontological baggage. One could interpret it as a mere property of human mathematical thought. Nevertheless, my own thinking is that everything has an explanation. E.g., the Christian God is the most parsimonious explanation for the conjunction of our universe's observed regularities and the testimonies we have of the miraculous, and it is not satisfactory to say that our universe simply is. Therefore, I find an objective transfinite realm a very appealing explanation of the large cardinal hierarchy.

I cannot logically exclude the possibility that someday our concept of set will "split" into mutually uninterpretable concepts that each have their own merits and cannot all be naturally interpreted by a single "better" concept of set. However, I have faith that any such split would be at most a temporary state of affairs, just as most physicists have faith the someday general relativity and quantum field theory will be unified (perhaps by loop quantum gravity; perhaps by string theory). The physicist's faith is that nature is a unity, and can be described as such. I believe that mathematics is a unity, if only in God's mind.

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