Leaving my introduction for a later time, as my first post I'd like to present the case that the correct framework in Philosophy of Mathematics is not Aristotelianism but plenitudinous theistic Platonism, where the meaning of each word is:

- plenitudinous: all self-consistent formal systems are on an equal standing.
- theistic: formal systems do not exist in a real world of their own, only virtually in God.
- Platonism (as oposed to Aristotelianism) = the existence (real or virtual) of a formal system is independent of its correspondence with a feature of the physical world.

The term "plenitudinous" was, AFAIK, coined by philosopher of Mathematics Mark Balaguer [1], who BTW does not choose between plenitudinous Platonism (whether strict or theistic) or plenitudinous fictionalism, but rather dismisses the issue. The choice clearly depends on the philosophical worldview of the chooser:

- A strict Platonist, who believes that there is a real world of forms, will obviously hold strict Platonism: self-consistent formal systems exist really from all eternity, "floating" in a world of pure forms.

- A classical theist, who believes that there is one absolutely simple Subsistent Being, will hold theistic Platonism: self-consistent formal systems exist virtually in God from all eternity, and temporally in the minds of people who discover them.

- An atheist will obviously hold fictionalism [3]: self-consistent formal systems exist virtually only in the minds of the people who build them or learn of them, just as the plot of a novel exists virtually only in the minds of its author and its readers.

"Discover" in Platonism and "build" in fictionalism is just the same activity viewed in different perspectives. Everyone agrees that mathematicians are restricted to build only those self-consistent formal systems that can exist, and in that sense they may be said to "discover" them. However, for an atheist those formal systems existed nowhere in any way before being "discovered", and therefore they are actually "built".

Back to the "plenitudinous" aspect, it means that ALL self-consistent formal systems exist virtually from all eternity, independently of whether there is a feature of the physical world that corresponds to any of them. Thus, all self-consistent formal systems are on an equal standing, so that

- Euclidean geometry is no less or more "real", or "true", than elliptic or hyperbolic geometry as formal systems, and
- (ZFC + CH) is no less or more "real", or "true", than (ZFC + ¬CH) as formal systems [2].

[1] Balaguer, M., 1998. "Platonism and Anti-Platonism in Mathematics".

To note, Balaguer's plenitudinous Platonism is equivalent to Resnik's and Shapiro's ante rem structuralism:
Resnik, M., 1997. "Mathematics as a Science of Patterns".
Shapiro, S., 1997. "Philosophy of Mathematics: Structure and Ontology".

[2] ZFC = Zermelo–Fraenkel set theory with the axiom of choice
CH = continuum hypothesis

Kurt Gödel showed in 1940 that CH cannot be disproved from ZFC.
Paul Cohen showed in 1963 that CH cannot be proved from ZFC.
Therefore, if ZFC is consistent, then (ZFC + CH) and (ZFC + ¬CH) are also consistent.

[3] As e.g. Mario Bunge, who also calls it "fictionism".
 

Last edited by Johannes (7/10/2015 9:27 pm)