Online research in philosophy

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.

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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.

Are there, in addition to the various actual objects that make up the world, various possible objects? Are there merely possible people, for example, or merely possible electrons, or even merely possible kinds? We certainly talk as if there were such things. Given a particular sperm and egg, I may wonder whether that particular child which would result from their union would have blue eyes. But if the sperm and egg are never in fact brought together, then there is no (...) actual object that my thought is about.1 Or again, in the semanti cs for modal logic we presuppose an ontology of possibilia twice over.2 For first, we coutenance various possible worlds, in addition to the actual world; and second, each of these worlds is taken to be endowed with its own domai n of objects. These will be the actual objects of the world in question, but they need not be actual simpliciter, i.e., actual objects of our world. W ha t a r e w e t o m a k e o f such discourse? There are four options: (i) the discourse is taken to be unintelligible; (ii) it is taken to be intelligible but nonfactual, i.e. as not in the business of stating facts; (iii) it is taken to be factual but reducible to discourse involving no reference to possibilia; (iv) it is taken to be both factual and irreducible.3 These options range from a fullblooded form of actualism at one extreme to a full-blooded form of possibilism at the other. The two intermediate positions are possibilist in that they accept the intelligibility of possibilist discourse but actualist in that they attempt to dispense with its prima facie commitment to possibilia. All four positions have found advocates in the literature. Quine, in his less irenic moments, favours option (i); Forbes ([85], p. 94) advocates option (ii), at least for certain parts of possibilist discourse; many philosophers, including Adams [74] and myself, opt for (iii); while Lewis [86] and Stalnaker [75] have endorsed versions of (iv), that differ in how full-blooded they take the possible objects to be.. (shrink)

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. (shrink)