Platonism and Anti-Platonism in Mathematics

by Mark Balaguer

In this highly absorbing work, Balaguer demonstrates that no good arguments exist either for or against mathematical platonism-for example, the view that abstract mathematical objects do exist and that mathematical theories are descriptions of such objects. Balaguer does this by establishing that both platonism and anti-platonism are justifiable views. Introducing a form of platonism, called "full-blooded platonism," that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks-most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good arguments for or against platonism but that we could never have such an argument. This lucid and accessible book breaks new ground in its area of engagement and makes vital reading for both specialists and all those intrigued by the philosophy of mathematics, or metaphysics in general.

Genres: Philosophy, Mathematics

222 pages, Kindle Edition

First published January 1, 1998

About the author

Mark Balaguer is Professor in the Department of Philosophy at California State University, Los Angeles. He is the author of Platonism and Anti-Platonism in Mathematics and Free Will as an Open Scientific Question (MIT Press).

Reviews

[Yumeko (blushes) · Rating 4 out of 5]

Don't you love when the author presents you with two carefully constructed and well developed views (ah well more or less, there were bits I really can't judge because who knows tf nominalized (??) quantum mechanics is) and then in the literal final paragraph claims:
"Yeah ok that's great and all but I personally believe none of it✌."

[Joao]

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. This is mostly because he means to accomplish it through a nominalization of Quantum Mechanics which I find not just blatantly flawed but ultimately indefensible. Still Balaguer's notion of Full-Bloodied Platonism, the peculiar point-of-view he develops and embraces in this work is extremely interesting and challenging: 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 (among such objetcs)? 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 (but NOT all mathematical objects need exist)! That is 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?)

[Bryon Wilson · Rating 5 out of 5]

A really concise explanation of the metaphysical problems and solutions involving mathematics.

[Mr James · Rating 1 out of 5]

In common with much of his other writings, while Balaguer claims to be objective regarding Platonism in mathematics, the reality is that his entire output has an underlying Platonist ethos. This is demonstrated by the fact that much of his discussions rest on his Platonist assumption that there is a such a thing as an independent mathematical truth, that a mathematical statement has a 'true' or 'false' value that is completely independent of any defined mathematical system. But the reality is that a mathematical statement only has a meaning and an evaluation within a specific mathematical system. And any such system has a set of rules from which a statement of that system might be proved, or the negation of that statement might be proved. And the same statement might be provable in one mathematical system, but its negation might be provable in another mathematical system. Hence the assumption of an independently existing mathematical truth value for mathematical statements is inherently a Platonist notion that has no logical nor evidential basis.

[Arthur Drury · Rating 5 out of 5]

Excellent analysis. Balaguer makes his signature move at the end: On the available evidence, we don't know. Cf. his Free Will (The MIT Press Essential Knowledge 978-0253352378)

Apparently Balaguer studied at CUNY. But, in this book he doesn't engage much with the metaphysical realism or the semantics of Jerrold J. Katz. It seems relevant. So, I'm curious about that.

[Sarah]

This was an interesting book, especially from the "hermeneutical perspective" of analyzing and categorizing the various arguments brought by mathematicians and philosophers, which the author said was part of his aim with this book. His other aim, his "metaphysical aim" to actually make progress answering whether platonism or anti-platonism are the "correct" conceptualization of mathematics, had a disappointingly short punchline; which kind of stunk, since that is what I was most anticipating :P.

Basically it amounted to this: both those that believe in platonism and those who don't accept that when they are arguing for/against the existence of abstract mathematical objects, by "abstract" they mean things that do not exist spatiotemporally, and are causally isolated from the physical world we exist in. But if such objects are causally isolated from us, they have no relevance to us, for their existence or absence cannot change anything about our world without violating this causal isolation. Therefore, it doesn't really matter what the answer to the question is, so go ahead and believe what you want but there's no point arguing about it :P.

Obviously there is far more to the book than that, and his support of this thesis is built up over the entire thing as he anticipates objections and defines terms and so on, but in the end it boiled down to this sort of "if you both conceive of abstract objects in this particular way, you aren't gonna find an answer" argument. Now what stuck with me from this book isn't so much this conclusion as its implication that the relationship between abstract objects and physical reality needs to be conceived of differently than that if you're going to keep on thinking about it. And he mentions several arguments that DO do this, though he ends up discarding them as implausible. But what brought me to this book in the first place was seeing it in Neal Stephenson's list of resources used while writing Anathem, and I just love that book even more for running with the implications of the punchline from this book. One of the arguments from the platonism-side that doesn't rely on the assumption the author says makes the argument irrelevant is that abstract objects are NOT causally isolated from us, and information does in fact pass from platonic objects to our minds. Alternatively from the anti-platonism side is the argument that not only do mathematical truths not "exist," neither do physical truths, and it's symbol games all the way down. And it impresses me so much that, even knowing as he must have that basically no one that read his book would be familiar with this debate, let alone the all the arguments made on each side and one guy's argument that the mainstream shared assumptions nullified the argument, Stephenson actually chose to give his two opposing schools of thought in the concents portrayed in the book two worldviews that legitimately didn't share this assumption and therefore had something that could be argued about for millennia. The Halikaarnans, the platonists, take as a key assumption of their metaphysics that information does flow from abstract to concrete objects, and the Procians, the anti-platonists, take the radical anti-realist view that logic and analysis aren't advancing knowledge at all.

The author of this book attributed the platonic argument of causal interaction between abstract and concrete to Godel, though he said that was arguable. I think his argument that taking as given the complete separation of abstract and concrete makes this debate irrelevant is convincing, but this just makes me even more interested to follow this line of thought and think about what it might mean to let the two interact, and I hope the author is right that Godel, or anyone really, had something to say about this, that there is somewhere where this idea isn't dismissed in a single paragraph as being an argument about Cartesian dualism and not what Balaguer wants to talk about :P.

[Mark · Rating 1 out of 5]

Balaguer is very knowledgeable of this s ubject. It is a worthy attempt to analyze the realist-antirealist debate for mathematical universal truths as aspects of reality or merely human.
I disagree strongly with his position and the entire thesis of the book. It is literally out of this world of spacetime. He contrasts one version of anti-Platonism (fictionalism) with the "full blooded Platonist" view of universals in mathematics that seems popular now. This is the view that all logically possible mathematical objects exist outside of spacetime. Numbers, geometric shapes. This throwback theory originated with Plato and is rather fanciful and too speculative. The author presents Platonist realism more favorably than non-Platonism without analyzing nominalist theories.
From what I understand, fictionalism (a bad name) is an atypical untenable anti-Platonist position held by very few philosophers. It is easy to criticize. I think it is indefensible and I am an anti-Platonist myself. . I imagine that writing this book was quite laborious and I am glad he uses the first person style.

[Paige McLoughlin · Rating 5 out of 5]

Again whether numbers exist as eternal abstract objects outside time and space or whether they are merely instrumental fiction depends on how you like your metaphysics. Like the Wavefunction of QM, some people believe in things not seen as really existing and some find them convenient descriptive fictions we can use. The author argues for both positions being equally defensible or indefensible and thinks we should suspend judgment on the status of mathematical objects and allow for both ideas to hold sway over people with one or the other Platonist or Fictionalist interpretation. I don't know what my hunch is for platonism but fictionalism has its merits.

update 2/14/2023 I think Baleguer is being clever Platonists have an epistemological problem with how we know the platonic realm which exists outside time and space really exists, and nominalists or fictionalists have an indispensability problem with why is something made up like fiction so damn effective. Baleguer has a solution that is Godelian to make Platonism like the Axiom of choice an undecidable assumption that you can take "platonism" or leave "fictionalism".