Sunday, January 8, 2012

Closing Plato's Cave

Jan Mycielski

February 7, 2012 will be Jan Mycielski's 80th birthday. He is known perhaps to topologists and graph theorists for his work in those fields, but I know about him for his work in mathematical logic and his interpretation of mathematics, intentionalism (Mathematics is "a description of finite structures consisting of finitely many individually imagined objects"), and a series of papers ("Locally Finite Theories" <>, ...) as covered by Shaughan Lavine's Understanding the Infinite. I wrote about that in Mathematica materialis, or How not to be lured into Plato's cave.

What is this all about?

Mathematics as generally taught in schools is thought to commit one to the existence of sets with an infinite number of elements.

Take one of the axioms of arithmetic of natural numbers:

(forall (m) (if (natural m) (exists (n) (and (natural n) (succ m n)))))

Here I'm using a formulation of the axioms with S-expressions for the object language. (succ m n) says n is the successor of m.

(forall list-of-variables expression)
(exists list-of-variables expression)

are interpreted in the standard way as the universal and existential quantifiers.

What Jan Mycielski does is to add a domain of quantification:

(forall P list-of-variables expression)
(exists Q list-of-variables expression)

where P and Q are the domains over which the variables range.

The above axiom becomes

(forall P (m) (if (natural m) (exists Q (n) (and (natural n) (succ m n)))))

and the interpretation is that when domains of quantification are nested, then the domain of the inner quantification is a superset of the domain that encloses it. (The inner-more you go, the bigger the sets get.) And not only that, but all domains of quantification are finite! (When the domains are the same as in standard logic one is committed to an infinite set of natural numbers, but this is not the case for nesting domains.)

If one has an entire mathematical theory (expressed as a collection of S-expressions) then one is talking about variables only ranging over finite sets, but one gets bigger and bigger finite sets as needed in any practical application of the theory. Lavine's book (discussed in the previous post) likens this to getting bigger and bigger bags of beans when needed. (I guess it helps to be a bean counter.)

So whatever Platonic infinities mathematicians hold dear can be dispensed with (with the Mycielskian change of quantification, of course).

Plato's Cave is closed.

2012/02/29: I came across a more recent paper, "A System of Axioms for Set Theory for the Rationalists", Notices of the AMS, February 2006 (the infinite sets and universes of pure mathematics are not actually but only potentially infinite). Again, infinite is replaced by only as large as needed.

Note: There is a comment in the above AMS paper of Mycielski that "it seems actual infinity exists in physical reality" (the supposed space-time continuum of Penrose's The Road to Reality), but I don't think there is a sufficient basis for that.