The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.

Although the case for the judgment-dependence of many other domains has been pored over, surprisingly little attention has been paid to mathematics and logic. This paper presents two dilemmas for a judgment-dependent account of these areas. First, the extensionality-substantiality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the substantiality condition (roughly: non-vacuous specification). Second, the extensionality-extremality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the extremality condition (...) (roughly: absence of independent explanation). The paper concludes with a moral concerning the judgment-dependence of a posteriori areas of discourse that emerges from consideration of these two a priori cases. (shrink)

This article discusses the argument we cannot have knowledge of abstract entities because they are not part of the causal order. The claim of this article is that the argument fails because of equivocation. Assume that the “causal order” is concerned with contingent facts involving time and space. Even if the existence of abstract entities is not contingent and does not involve time or space it does not follow that no truths about abstract entities are contingent or involve time or (...) space. I argue that it is the latter which is required to obtain the desired conclusion. (shrink)

According to the species of neo-logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo-fregeanism-a general conception of the relation between language and reality; (2) the method of abstraction-a particular method for introducing concepts into language; (3) the scope of logic-second-order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) The issues discussed include reductionism, rejectionism, the Julius Caesar problem, the Bad Company objections, and the charge that second-order logic is set theory in disguise. (shrink)

According to the species of neo‐logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo‐fregeanism—a general conception of the relation between language and reality; (2) the method of abstraction—a particular method for introducing concepts into language; (3) the scope of logic—second‐order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) The issues discussed include reductionism, rejectionism, the Julius Caesar problem, the Bad Company objections, and the charge that second‐order logic is set theory in disguise.The irresistible metaphor is that pure abstract objects […] are no more than shadows cast by the syntax of our discourse. And the aptness of the metaphor is enhanced by the reflection that shadows are, after their own fashion, real. (Crispin Wright [1992], p. 181–2)But I feel conscious that many a reader will scarcely recognise in the shadowy forms which I bring before him his numbers which all his life long have accompanied him as faithful and familiar friends; (Richard Dedekind [1963], p. 33)1 Introduction2 Logical‐historical preliminaries3 The linguistic turn4 Reductionism5 Rejectionism6 The ‘Julius Caesar’ problem7 Second‐order logic8 Bad company objections9 Conclusion. (shrink)

Platonism in the philosophy of mathematics is the doctrine that there are mathematical objects such as numbers. John Burgess and Gideon Rosen have argued that that there is no good epistemological argument against platonism. They propose a dilemma, claiming that epistemological arguments against platonism either rely on a dubious epistemology, or resemble a dubious sceptical argument concerning perceptual knowledge. Against Burgess and Rosen, I show that an epistemological anti- platonist argument proposed by Hartry Field avoids both horns of their dilemma.

Debunking arguments—also known as etiological arguments, genealogical arguments, access problems, isolation objec- tions, and reliability challenges—arise in philosophical debates about a diverse range of topics, including causation, chance, color, consciousness, epistemic reasons, free will, grounding, laws of nature, logic, mathematics, modality, morality, natural kinds, ordinary objects, religion, and time. What unifies the arguments is the transition from a premise about what does or doesn't explain why we have certain mental states to a negative assessment of their epistemic status. I examine (...) the common, underlying structure of the arguments and the different strategies for motivating and resisting the premises of debunking arguments. (shrink)

A modest solution to the problem(s) of rule-following is defended against Kripkensteinian scepticism about meaning. Even though parts of it generalise to other concepts, the theory as a whole applies to response-dependent concepts only. It is argued that the finiteness problem is not nearly as pressing for such concepts as it may be for some other kinds of concepts. Furthermore, the modest theory uses a notion of justification as sensitivity to countervailing conditions in order to solve the justification problem. Finally, (...) in order to solve the normativity problem, it relies on substantial specifications of normal conditions such as those that have been proposed by Crispin Wright and Mark Johnston, rather than on Philip Pettit's functionalist specification. This theory is modest in that it does not meet the demands of Kripke's sceptic in full. Arguments are provided as to why this is not needed. (shrink)

Our thesis in this paper is that consideration of Wright’s account of what it is to be judgement-dependent leads us to the conclusion that teaching is judgement dependent. We begin with a consideration of Wright’s account of what it is to be judgement-dependent. We then make the case that teaching satisfies the conditions on what it is to be judgement-dependent. Our intention is not to delve into the independent plausibility of such a view. Our focus is simply on showing the (...) connection between Wright’s account of judgement-dependence and what it is to teach. (shrink)

Response‐dispositional (RD) properties are standardly defined as those that involve an object's appearing thus or thus to some perceptually well‐equipped observer under specified epistemic conditions. The paradigm instance is that of colour or other such Lockean “secondary qualities”, as distinct from those—like shape and size—that pertain to the object itself, quite apart from anyone's perception. This idea has lately been thought to offer a promising alternative to the deadlocked dispute between hard‐line ‘metaphysical’ realists and subjectivists, projectivists, social constructivists, or hard‐line (...) anti‐realists. A chief source text is Plato's Euthyphro, where the issue is posed in ethical terms: do the gods infallibly approve virtuous acts on account of their divine moral omniscience or are virtuous acts just those the gods approve? Among the areas proposed as amenable to an RD approach are epistemology, ethics, political theory, and philosophy of mathematics. It is claimed that by making due allowance for the involvement of normalised or optimised human responses one can steer a course between the twin poles of an objectivist realism that places truth beyond our cognitive grasp and an epistemic conception that confines truth within the limits of humanly attainable proof, knowledge, or verification. Here I argue—on the contrary—that RD approaches can be shown to offer nothing more than a variant of the same old realist versus anti‐realist dilemma. That is, they work out either as a trivial (tautologous) claim that ‘truth’ simply equates with ‘best judgement’ in the ideal (quasi‐objective) limit or as the claim—advanced by anti‐realists like Michael Dummett—that we cannot form any adequate conception of objective (recognition‐transcendent) truths. After looking at this issue in various contexts of debate, I conclude that one useful (if pyrrhic) outcome is to demonstrate the non‐availability of any middle‐ground stance. We are left with the strictly unavoidable choice between a realist or objectivist approach and one that assimilates truth to the consensus of accredited best opinion. This latter amounts to a roundabout, elaborately qualified version of the anti‐realist case. (shrink)

Error-theories and non-cognitivism about ethical discourse face tremendous obstacles, often stemming from their rejection of the truth of ethical assertions. However, I argue in that generic realism about ethical discourse is equally unattractive. Crispin Wright’s judgement-dependence allows for the rejection of generic realism without implausibly rejecting the truth of the distinctive assertions of a discourse. I show how Wright speedily dismisses the judgement-dependence of truth in ethics, but suggest that he has been too quick, ignoring some ways in which a (...) stronger case could have been made in its favour. However, these suggestions do not address the fact that ethical discourse violates one of Wright’s fundamental conditions on judgement-dependent accounts. Wright is able to argue that the ‘grammar’ of intention discourse allows a form of judgement-dependence to be salvaged despite its violation of the conditions on judgement-dependence. The major project of this essay is to investigate the application of this strategy to ethical discourse, although I argue that the strategy must ultimately fail. I suggest why Wright’s strategy worked for intention but not ethics, and conclude that the failure of judgement-dependence for ethics encourages us to seek a plausible judgement-independent account of the discourse. (shrink)