By Mark Balaguer
During this deft and lively e-book, Mark Balaguer demonstrates that there are not any reliable arguments for or opposed to mathematical platonism (ie., the view that summary, or non-spatio-temporal, mathematical items exist, and that mathematical theories are descriptions of such objects). Balaguer does this by way of setting up that either platonism and anti-platonism are defensible positions. partly I, he exhibits that the previous is defensible by way of introducing a unique model of platonism, which he calls full-blooded platonism, or FBP. He argues that if platonists propose FBP, they could then clear up all the difficulties often linked to their view, such a lot particularly the 2 Benacerrafian difficulties (that is, the epistemological challenge and the non-uniqueness problem).In half II, Balaguer defends anti-platonism (in specific, mathematical fictionalism) opposed to a variety of assaults, leader between them the Quine-Putnam indispensability argument. Balaguer's model of fictionalism bears similarities to Hartry Field's, however the arguments Balaguer makes use of to guard this view are very diverse. components I and II of this ebook taken jointly in actual fact identify that we don't have any reliable argument for or opposed to platonism.In half III, Balaguer extends his conclusions, arguing that it isn't easily that we don't at present have any stable argument for or opposed to platonism, yet that shall we by no means have such an issue, and certainly, that there's no truth of the problem as to if platonism is right (ie., no matter if there exist any summary objects).This lucid and accessibly written ebook breaks new floor in its zone of engagement and makes important interpreting for either experts and somebody else attracted to the philosophy of arithmetic or metaphysics commonly.
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Extra resources for Platonism and Anti-Platonism in Mathematics
Example text
The three central claims of the non-spatiotemporalcontact view—the claims that distinguish it from the no-contact view—are (i) that mathematical intuition is analogous to sense perception, (ii) that mathematical intuition involves a sort of information transfer between abstract mathematical objects and human beings, and (iii) that (1) is false. Here is some evidence for thinking that Godel believed all three of these claims. As for (i), I think it is fair to say that the central theme running through all of Godel's remarks on mathematical intuition is the analogy to sense perception.
20 Maddy is thinking mainly of sets here. 21 Thus, her two central claims are that sets are spatiotemporally located —a set of eggs, for instance, is located right where the eggs are —and that sets are perceptible, that is, that they can be seen, heard, smelled, felt, and tasted in the usual ways. The Epistemological Argument Against Platonism 29 I will argue here that this view, which we can call naturalized platonism, doesn't work. I will argue not just that Maddy's view doesn't work, but that no version of naturalized platonism works.
Thus, all of this is irrelevant to my argument. I conclude, then, that Maddy does not have a response to my argument. Moreover, I do not see any way for naturalized platonists to respond. The reason, in a nutshell, is that there seems to be an incompatibility between abstractness and perceptibility. Naturalized platonists need some sort of abstractness to account for the truth of theories like ZF and, hence, to avoid the semantic horn of Benacerraf's dilemma. But as soon as they let in any sort of abstractness, it becomes impossible for them to use an appeal to sense perception to avoid the epistemic horn of the dilemma.