Platonism and Anti-Platonism in Mathematics

Rating:
Summary: Platonism=Fictionalism=Realism?
Review: This is a remarkably forceful and ambitious book but a very
worthy read nonetheless. Balaguer is clearer in his arguments than about any other contemporary philosopher I have read on the subject! He does however make a few discernable mistakes and shows a surprising lack of depth is some of his tangential examinations as pointed by some of his reviewers (I am thinking of Colyvan and Zalta whose review of this book can be found on the web). Also, out of breath as I was, by the time I finished this book, I cannot say I feel persuaded by its thesis with respect to the indescernability between Fictionalism and Platonism. Still Balaguer's notion of Full-Bloodied Platonism, the peculiar point-of-view he develops and embraces in this work is extremely interesting and challenge: it comes down to the notion that all "broadly possible" mathematical structures exist. This happens to be, though Balaguer seems anaware of it, a thesis currently arrived at by physical cosmologists speculating about the "Multiverse" (see Mark Tegmark's recent Scientific American article on "Parallel Worlds")! When different lines of speculation arrive at the same concepts there is some hint of
historical consensus one tends to suspect a metaphysical corner
where we are all about to get stuck for a while! On the other hand I cannot help to remark how simplistic and misleading is
the language in which philosophers insist in carrying their arguments! An example from the beginning is the characterization
of an abstract object as one that exists "non-spaciotemporaly".
Though he ends up debating some of the obvious problems with this
distinction Balaguer never addresses today's scientific consensus that space-time itself is an abstract object of some sort (except if you ask Julian Barbour and his Leibnitzian crowd), either Riemann space or Multiverse, so one may naturally ask why should it be a previledged reference for existence? On this matter I take a a more radical view than Balaguer, which I would call Full-Bodied Platonism, by arguing that all that exists are abstract mathematical objects. What he
calls spaciotemporal existents (that includes us, at least the ones among us who cartesianly think they exist) whose existence is merely contingent on our participation in the true (eternal and necessary) existence of such abstractions. (But wait! Isn't that what Plato thought?)