Title: Platonism and Anti-Platonism in Mathematics. Publisher: Oxford University Press. In this deft and vigorous book, Mark Balaguer demonstrates that there are no good arguments for or against mathematical platonism ie. Balaguer does this by establishing that both platonism and anti-platonism are defensible positions. In Part I, he shows that the former is defensible by introducing a novel version of platonism, which he calls full-blooded platonism, or FBP. He argues that if platonists endorse FBP, they can then solve all of the problems traditionally associated with their view, most notably the two Benacerrafian problems that is, the epistemological problem and the non-uniqueness problem.
In Part II, Balaguer defends anti-platonism in particular, mathematical fictionalism against various attacks, chief among them the Quine-Putnam indispensability argument. Balaguer's version of fictionalism bears similarities to Hartry Field's, but the arguments Balaguer uses to defend this view are very different.
Parts I and II of this book taken together clearly establish that we do not have any good argument for or against platonism. In Part III, Balaguer extends his conclusions, arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument, and indeed, that there is no fact of the matter as to whether platonism is correct ie. This lucid and accessibly written book breaks new ground in its area of engagement and makes vital reading for both specialists and anyone else interested in the philosophy of mathematics or metaphysics in general.
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More filters. Sort order. Nov 12, Joao added it.
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 thi This is a remarkably forceful and ambitious book but a very worthy read nonetheless. 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? May 31, Bryon Wilson rated it it was amazing. A really concise explanation of the metaphysical problems and solutions involving mathematics.
Jul 24, Sarah added it.
List this Seller's Books. Although Benacerraf focuses on arithmetic, the objection naturally generalizes to most pure mathematical objects. Therefore, it logically follows that no scientific evidence will ever be forthcoming to establish their objective existence. Enrico Martino. Structuralism is a position holding that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their places in such structures, consequently having no intrinsic properties. A number of gravitons is not so much a plurality of objects as a label denoting a particular quantum state.
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.
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.
"excellent book exceptionally clear, insightful, and useful critical survey"--The Review of Modern Logic "Platonism and anti-platonism in mathematics is an. Platonism and Anti-Platonism in Mathematics. ykoketomel.mler. June , This is a painful book to read. It is not that the author is full of nonsense, airing.
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.
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