Mark Balaguer has written a provocative and original book. The book is as ambitious as a work of philosophy of mathematics could be. It defends both of the dominant views concerning the ontology of mathematics, Platonism and Anti-Platonism, and then closes with an argument that there is no fact of the matter which is right.

In his précis of a recent book, Richard Joyce writes, “My contention…is that…any epistemological benefit-of-the-doubt that might have been extended to moral beliefs…will be neutralized by the availability of an empirically confirmed moral genealogy that nowhere…presupposes their truth.” Such reasoning – falling under the heading “Genealogical Debunking Arguments” – is now commonplace. But how might “the availability of an empirically confirmed moral genealogy that nowhere… presupposes” the truth of our moral beliefs “neutralize” whatever “epistemological benefit-of-the-doubt that might have been extended (...) to” them? In this article, I argue that there appears to be no satisfactory answer to this question. The problem is quite general, applying to all arguments with the structure of Genealogical Debunking Arguments aimed at realism about a domain meeting two conditions. The Benacerraf-Field Challenge for mathematical realism affords an important special case. (shrink)

It is commonly suggested that evolutionary considerations generate an epistemological challenge for moral realism. At first approximation, the challenge for the moral realist is to explain our having many true moral beliefs, given that those beliefs are the products of evolutionary forces that would be indifferent to the moral truth. An important question surrounding this challenge is the extent to which it generalizes. In particular, it is of interest whether the Evolutionary Challenge for moral realism is equally a challenge for (...) mathematical realism. It is widely thought not to be. In this paper, I argue that the Evolutionary Challenge for moral realism is equally a challenge for mathematical realism. Along the way, I substantially clarify the Evolutionary Challenge, discuss its relation to more familiar epistemological challenges, and broach the problem of moral disagreement. The paper should be of interest to ethicists because it places pressure on anyone who rejects moral realism on the basis of the Evolutionary Challenge to reject mathematical realism as well. And the paper should be of interest to philosophers of mathematics because it presents a new epistemological challenge for mathematical realism that bears, I argue, no simple relation to Paul Benacerraf's familiar challenge. (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)

What is the Moral Problem? NORMATIVE ETHICS VS. META-ETHICS It is a common fact of everyday life that we appraise each others' behaviour and attitudes from ...

Metaethical—or, more generally, metanormative— realism faces a serious epistemological challenge. Realists owe us—very roughly speaking—an account of how it is that we can have epistemic access to the normative truths about which they are realists. This much is, it seems, uncontroversial among metaethicists, myself included. But this is as far as the agreement goes, for it is not clear—nor uncontroversial—how best to understand the challenge, what the best realist way of coping with it is, and how successful this attempt is. (...) In this paper I try, first, to present the challenge in its strongest version, and second, to show how realists—indeed, robust realists—can cope with it. The strongest version of the challenge is, I argue, that of explaining the correlation between our normative beliefs and the independent normative truths. And I suggest an evolutionary explanation as a way of solving it. (shrink)

Contemporary realist theories of value claim to be compatible with natural science. In this paper, I call this claim into question by arguing that Darwinian considerations pose a dilemma for these theories. The main thrust of my argument is this. Evolutionary forces have played a tremendous role in shaping the content of human evaluative attitudes. The challenge for realist theories of value is to explain the relation between these evolutionary influences on our evaluative attitudes, on the one hand, and the (...) independent evaluative truths that realism posits, on the other. Realism, I argue, can give no satisfactory account of this relation. On the one hand, the realist may claim that there is no relation between evolutionary influences on our evaluative attitudes and independent evaluative truths. But this claim leads to the implausible skeptical result that most of our evaluative judgements are off track due to the distorting pressure of Darwinian forces. The realist’s other option is to claim that there is a relation between evolutionary influences and independent evaluative truths, namely that natural selection favored ancestors who were able to grasp those truths. But this account, I argue, is unacceptable on scientific grounds. Either way, then, realist theories of value prove unable to accommodate the fact that Darwinian forces have deeply influenced the content of human values. After responding to three objections, the third of which leads me to argue against a realist understanding of the disvalue of pain, I conclude by sketching how antirealism is able to sidestep the dilemma I have presented. Antirealist theories of value are able to offer an alternative account of the relation between evolutionary forces and evaluative facts —
an account that allows us to reconcile our understanding of evaluative truth with our understanding of the many non-rational causes that have played a role in shaping our evaluative judgements. (shrink)

This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether platonism or anti-platonism is true.

According to many, that the normative supervenes on the non-normative is a truism of normative discourse. This chapter argues that those committed to more specific moral, aesthetic, and epistemic supervenience theses should also hold : As a matter of conceptual necessity, whenever something has a normative property, it has a base property or collection of base properties that metaphysically necessitates the normative one. The main aim in this chapter is to show that none of the available arguments establish, or indeed (...) the relevant epistemic, aesthetic, and moral supervenience theses. is not a conceptual truth. This has considerable dialectical importance. One interesting upshot is that it affords non-reductivists and non-naturalists a novel way of resisting certain prominent supervenience-based objections to their views, including objections that formulate supervenience as a purely metaphysical thesis. (shrink)

Particularism's model of explanation is challenged on the ground that a sensible intelligibility principle requires that there must be an explanation for the difference between a good and a bad action. Raz is concerned with what it is to be guided by reason, as well as with the results of the fact that reason can often undermine particular outcomes. What determines the moral status of an action must extend beyond what the agent's reason for acting is. It is argued that (...) there is a clear distinction between the evaluative and guiding functions of reasons. (shrink)

Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, an evaluation of arguments (...) and counterarguments is presented. Results: The main result is that strict finitism is indeed a viable option, next to other constructive approaches, in mathematics. Implications: Arguing for strict finitism is more complex than is usually thought. For future research, strict finitist mathematics itself needs to be written out in more detail to increase its credibility. In as far as strict finitism is a viable option, it will change our views on such “classics” as the platonist-constructivist discussion, the discovery-construction debate and the mysterious applicability problem . Constructivist content: Strict finitism starts from the idea that counting is an act of labeling, hence the mathematician is an active subject right from the start. It differs from other constructivist views in that the finite limitations of the human subject are taken into account. (shrink)

This entry introduces the reader to the main ideas in Frege's philosophy of logic, mathematics, and language.

In this paper I examine a strategy which aims to bypass the technicalities of the indispensability debate and to offer a direct route to nominalism. The starting-point for this alternative nominalist strategy is the claim that--according to the platonist picture--the existence of mathematical objects makes no difference to the concrete, physical world. My principal goal is to show that the 'Makes No Difference' (MND) Argument does not succeed in undermining platonism. The basic reason why not is that the makes-no-difference claim (...) which the argument is based on is problematic. Arguments both for and against this claim can be found in the literature; I examine three such arguments, uncovering flaws in each one. In the second half of the paper, I take a more direct approach and present an analysis of the counterfactual which underpins the makes-no-difference claim. What this analysis reveals is that indispensability considerations are in fact crucial to the proper evaluation of the MND Argument, contrary to the claims of its supporters. (shrink)

According to many, that the normative supervenes on the non-normative is a truism of normative discourse. This chapter argues that those committed to more specific moral, aesthetic, and epistemic supervenience theses should also hold (NS*): As a matter of conceptual necessity, whenever something has a normative property, it has a base property or collection of base properties that metaphysically necessitates the normative one. My main aim is to show that none of the available arguments establish (NS*), or indeed the relevant (...) epistemic, aesthetic, and moral supervenience theses. (NS*) is not a conceptual truth. This has considerable dialectical importance. One interesting upshot is that it affords non-reductivists and non-naturalists a novel way of resisting certain prominent supervenience-based objections to their views, including objections that formulate supervenience as a purely metaphysical thesis. (shrink)