Sunday, March 3, 2013

Three paths to becoming a mathematical anti-platonist

It may be the case that "no good arguments exist either for or against mathematical platonism" (Platonism and Anti-Platonism in Mathematics by Mark Balaguer). Mathematical anti-platonists may come from a certain naturalistic belief that there's nothing outside nature, so if it's the case that there are no infinities in nature, how can one believe in platonist mathematical objects like infinite sets and the real number continuum?

The three paths below are not arguments against mathematical platonism. They are just ways mathematical anti-platonists can travel. (The first path is complete in the sense that it actually gets to a goal of a truly non-platonistic alternative to interpreting standard mathematics. The other two are incomplete in the sense that they result either with a somewhat restrictive or a non-standard mathematics.)

1. Finite mathematics (of indefinitely large size sets)*

In Understanding the Infinite, Shaughan Lavine describes the mathematics of Jan Mycielski ("The meaning of pure mathematics", "Locally finite theories"). In this approach, the quantifiers (∀, ∃) within the sentences of standard mathematics are replaced with indexed quantifiers (∀i, ∃j), and the interpretation of these quantifiers is that the variables they govern range over finite sets (Ωi, Ωj) with the same index. The key to this approach is that the finite sets can be of different sizes (unlike in the standard interpretation where the variables range over the same set). This indexing of quantifiers in sentences of a standard-mathematical theory T is done by process called relativization, and the result of applying this process to a sentence φ of standard mathematics is called a regular relativization φ' of φ. Beginning with a standard theory T, the result is the corresponding finitary theory Fin(T). The key theorem of this approach is:

"If φ is a sentence in the language of T and φ' is a regular relativization of φ, then φ is a theorem of T if and only if φ' is a theorem of Fin(T)."

Thus every theorem of T (interpreted with possibly infinite sets) has a corresponding theorem of Fin(T) (interpreted with only finite sets).

Note: Mycielski calls this interpretation intentionalism (which is different from intuitionism), in contrast with formalism and platonism. For some examples, see:

Mathematica materialis, or How not to be lured into Plato's cave
Persons without infinities
Plato's cave is closed

* or MIFS: Mathematics of Indefinitely-large Finite Sets

The other two paths I mention briefly.

2. Computable analysis

Can computable numbers be used instead of the reals?
Computable analysis
Constructive mathematics

E.g., only consider numbers and methods of analysis that can be represented by computer programs. It would interesting to link path 2 to path 1.

3. Paraconsistent mathematics (with finite models)

Inconsistent mathematics
Paraconsistent Logic

"One interesting implication of the existence of inconsistent models of arithmetic is that some of them are finite (unlike the classical non-standard models)."

Inconsistent models of arithmetic:
Part I: Finite models, Part II: The general case