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Platonism and Anti-Platonism in Mathematics, Mark Balaguer attempts to show that there is (1) one and only one defensible version of platonism, (2) one and only one defensible version of anti-platonism, and (3) no fact of the matter as to which is true. His arguments depend essentially on the notion of supervenience, yet he rejects metaphysical necessity. I argue that he cannot use logical, conceptual, or nomological necessity to explicate supervenience. Balaguer must either give up the arguments that make use of supervenience or accept metaphysical necessity. I also consider and reject a possible response to my arguments.

In connection with John Searle's denial that computers genuinely act, Hauser considers Searle's attempt to distinguish full-blooded acts of agents (e.g., my raising my arm) from mere physical movements (my arm going up) on the basis of intent. The difference between me raising my arm and my arm's just going up (e.g., if you forcibly raise it), on Searle's account, is the causal involvement of my intention to raise my arm in the former, but not the latter, case. Yet, we distinguish a similar difference between a robot's raising its arm and its robot arm just going up (e.g., if you manually raise it). Either robots are rightly credited with intentions, or it is not intention that distinguishes action from mere movement. In either case full-blooded acts under "aspects" are attributable to robots and computers. Since the truth of such attributions depends on "intrinsic" features of the things not on the speaker's "intentional stance," they are not merely figurative "as if" attributions.

I argue that it is a main theme of Davidson's theory of interpretation that interpretive charity implies the impossibility of massive disagreement. There is clear textual support for that. I then argue that from the first-person point of view of a full-blooded interpreter, the theme must be accepted; and that is precisely why Davidson accepts it. If massive disagreement between speaker and interpreter seems to us easy to imagine, it is only because the imagination involved is third-personal and not full-blooded.

I argue that it is a main theme of Davidson's theory of interpretation that interpretive charity implies the impossibility of massive disagreement. There is clear textual support for that. I then argue that from the first-person point of view of a full-blooded interpreter, the theme must be accepted; and that is precisely why Davidson accepts it. If massive disagreement between speaker and interpreter seems to us easy to imagine, it is only because the imagination involved is third-personal and not full-blooded.

This is an extended, critical review of Mark Balaguer's book *Platonism and Anti-Platonism in Mathematics* (New York: Oxford University Press, 1998). After describing his theory ("full-blooded Platonism"), we raise two criticisms. The first concerns the fact that Balaguer's theory offers no way to uniquely identify the denotations of the terms appearing in mathematical theories. The second concerns the fact that Balaguer overlooks the possibility that the fact, that Platonism and anti-Platonism agree on numerous points but differ only on whether mathematical objects exist, can be explained if both views turn out to be two different interpretations of the same formal theory.

A response is given here to Benacerraf's 1973 argument that mathematical platonism is incompatible with a naturalistic epistemology. Unlike almost all previous platonist responses to Benacerraf, the response given here is positive rather than negative; that is, rather than trying to find a problem with Benacerraf's argument, I accept his challenge and meet it head on by constructing an epistemology of abstract (i.e., aspatial and atemporal) mathematical objects. Thus, I show that spatio-temporal creatures like ourselves can attain knowledge about mathematical objects by simply explaininghow they can do this. My argument is based upon the adoption of a particular version of platonism — full-blooded platonism — which asserts that any mathematical object which possiblycould exist actuallydoes exist.

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("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 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.

Just what is full-blooded platonism?’ Greg Restall outlines several objections to Mark Balaguer's theory of full-blooded platonism. I reply to these objections by explicating the semantic framework for the reference of mathematical terms that full-blooded platonism requires. Expanding upon these replies, I then explain how the full-blooded platonist, in light of the explicated semantic framework, should treat mathematical terms and statements in order to avoid certain pitfalls. I want to thank Mark Balaguer, Phillip Bricker, and Greg Restall for helpful comments on earlier drafts of this paper.

Mark Balaguer's Platonism and Anti-Platonism in Mathematics presents an intriguing new brand of platonism, which he calls plenitudinous platonism, or more colourfully, full-blooded platonism. In this paper, I argue that Balaguer's attempts to characterise full-blooded platonism fail. They are either too strong, with untoward consequences we all reject, or too weak, not providing a distinctive brand of platonism strong enough to do the work Balaguer requires of it.