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This chapter discusses Hartry Field's attempt to respond to the indispensability argument by showing how to dispense with mathematics in formulating our best scientific theories. It is pointed out that Field does not want to stop us from using mathematics in our scientific theorizing, but rather, that he wishes to explain the use of mathematics as a ‘theoretical juice extractor’, which allows us to draw out the consequences of our non‐mathematical assumptions. Objections to Field's programme are considered, including objections to the logical assumptions made by his account of applications, and objections to his claim that mathematics can always be dispensed with. While these objections are not conclusive, it is noted that mathematical assumptions may be valuable enough to remain present in even our best formulations of our scientific theories. Hence the book's project, to consider the case for anti‐platonism on the assumption that mathematics is indispensable to empirical science.

Hartry Field has attacked the indispensability argument by arguing that mathematics is in fact dispensable to science. He does this by showing a way to dispense with quantification over mathematical entities in Newtonian gravitation theory. He, thus, advances a kind of fictionalism about mathematics. In this chapter, Field's program is outlined and the prospects for its success are evaluated. An important part of the task of assessing what is required of Field's program involves understanding how we should read the crucial term “indispensable” in Quine's argument.

Hartry Field argues that meaning‐based approaches to explaining the apriority of certain propositions fail to succeed in their endeavour. He suggests that adopting what one calls a ‘non‐factualist’ view of justification itself removes the mystery of the apriority of such propositions, and sketches what such a view of justification involves.

This paper argues that the epistemological challenge against a Platonist conception of mathematics can be met. The challenge is expounded in a general version and does not rely on the specific adoption of a causal theory of knowledge, but rather it disputes the ability to explain our reliability of our beliefs concerning abstract objects. It is then argued that for a Platonist, the above challenge can be met by giving a satisfactory epistemology of necessary truths in general. That is to reduce the explanation of our reliability in mathematical beliefs to beliefs of necessary statements as such. In the following, the worry is raised that such a transition is illicit, as it might still be the case that albeit logical beliefs are necessary, our mathematical beliefs, even if they were true are never necessarily so.

This paper discusses the idea, labelled as “the traditional connection” that implicit definitions aim to found a priori knowledge of logic and mathematics. In the first part, it discusses and rejects a specific understanding of certain constraints (existence, uniqueness, possession problem, and explanation problem) on the theory of implicit definitions, as suggested by Horwich, on the basis of it being committing to some robust Platonist version of meaning facts. In contrast, it motivates further new constraints on the success of implicit definitions, such as arrogance, conservativeness, Evan's generality constraint, and harmony. Then, the standard view of implicit definitions for scientific terms, which appeals to so called Carnap Conditionals is discussed and an alternative model, i.e. the inverse Carnap Conditional is proffered. Lastly, this latter model is then applied to Hume's Principle and the conditionalized version of Hume's Principle as offered by Field is rejected. Furthermore, the problem of the ontological commitments of Hume's Principle and its status as a meaning––conferring successful stipulation are further discussed.

This paper starts by offering a brief reconstruction of the Neo‐Fregean approach as suggested in Frege's Conception of Numbers as Objects and distinguishes various challenges against the method of Abstraction. It then focuses on one line of criticism––Rejectionism––which is endorsed by Field in his Review of the previously mentioned book. The thought is to grant that the method of abstraction provides singular terms, however questions its ability to produce true statements. Furthermore, Field draws an analogy between the stipulation of Hume's Principle, which commits one to the existence of numbers and the ontological argument, which commits one to the existence of God. It is then shown that this analogy is amiss and that there is no real point of affinity with the Fregean platonist's ontological strategy and the ontological arguments. A further objection concerning the tacit ontological commitments on the right hand side of Abstraction Principle is discussed. The paper concludes considering ‘the onus of proof’ – issue for Nominalism‐Platonism debates.

This chapter homes in on Aristotle and recent Aristotelianism, which have mainly assumed that the virtues are unified, that in order to have one virtue one must have them all. What can be said about partial values and the inevitability of imperfection stands diametrically opposed to the Aristotelian picture of human good and virtue, and our examples—together with the complexities and richness of modern life that they illustrate—offer us some reason to finally reject the simpler Aristotelian ethical picture. The notion of partial values on which so much in the more complex account depends also turns out to have interesting parallels in what Freud said about “partial instincts” and what Hartry Field has more recently said about “partial signification.”

This chapter discusses the Benacerraf–Field Challenge – i.e., the reliability challenge. It argues that neither Benacerraf’s formulation of the challenge, nor any simple variations on it, satisfies key constraints which have been placed on it. It then turns to more promising analyses, in terms of sensitivity and safety. The challenge to show that our beliefs are sensitive is widely supposed to admit of an evolutionary answer in the mathematical case, but not in the moral. The chapter argues that it does not, but that, even if it did, this is an inadequate formulation of the challenge. But understanding the reliability challenge as the challenge to show that our beliefs are safe is more promising. The chapter shows that whether this challenge is equally pressing in the moral and mathematical cases depends on whether “realist pluralism” is equally viable in the two areas.

A common objection to realism (robust or otherwise) is that 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 chapter first distinguishes between many different ways of understanding this epistemological challenge to Robust Realism, then focusing on the strongest version of the challenge, namely, the need to explain the correlation between our normative beliefs and the independent normative truths (or else accept that there is no such correlation, and that skepticism about the normative is the way to do). After the challenge is clearly stated, a way of coping it is suggested. The way to explain the correlation is by resorting to a (godless) pre-established-harmony kind of explanation, one that utilizes some plausible evolutionary speculations. In a final section there is a preliminary discussion of the somewhat related problem of accommodating semantic access.

This chapter argues against the view that the time-series is made up of metaphysically basic time points. It shows that the only plausible version of such a temporal substantivalism is an ontologically costly variant of a modal account of time. Unlike metaphysically basic spatial points, metaphysically basic time points would also not perform any theoretical work in accounting for inertial forces or free fields.

Scientism is expounded. Then its two major challenges are stated and responses to them sketched. The first challenge is to its epistemology of mathematics-how we know the necessary truths of mathematics. The second challenge is to the very coherence of its eliminativist account of cognition. The first of these problems is likely to be taken more seriously by philosophers than by other advocates of scientism. It is a problem that has absorbed philosophers since Plato and on which little progress has been made. The second is often unnoticed, even among those who endorse scientism, since they don’t recognize their own commitment to eliminativism and so do not appreciate the threat of incoherence it poses. It is important for scientism to acknowledge these challenges.