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Anything worth regarding as logicism about number theory holds that its fundamental laws – in effect, the Dedekind-Peano axioms – may be known on the basis of logic and definitions alone. For Frege, the logic in question was that of the Begriffschrift – effectively, full impredicative second order logic - together with the resources for dealing with the putatively “logical objects” provided by Basic Law V of Grundgesetze. With this machinery in place, and with the course-of-values operator governed by Basic Law V counting as logical, it is possible for all the definitions involved in the logicist reconstruction of arithmetic and analysis to be fully explicit, abbreviative definitions. Had Frege’s project succeeded, he would therefore have been in position – by his own lights – to regard the axioms of number theory simply as definitional abbreviations of certain theorems of his pure logic. Basic Law V, as every interested party knows, is inconsistent. But twentieth century orthodoxy would have scorned its description as a law of logic in any case, purely on the grounds of its existential fecundity. Contemporary Neo-Fregeanism in the foundations of mathematics does not, in intention at least, pick any quarrel with the idea that pure logic should be ontologically austere. It does however maintain that the existence of the natural numbers and the real numbers as classically conceived, and thereby the truth of the traditional axioms of arithmetic and analysis, may still be known a priori on the basis of logic and definitions. For the purposes of this claim, logic is once again conceived as essentially the system of Begriffschrift. But Basic Law V is superseded by a variety of abstraction principles, of which Hume's Principle is the best known example, which we are regarded as free to lay down as true by way of determination of the meaning of the non-logical vocabulary that they contain. Thus — the idea is — the Dedekind-Peano axioms, for example, may be known, a priori, to be true by virtue of their derivation in pure logic from a principle which may be regarded as stipulatively true, and whose very stipulation may be regarded as conferring content upon the sole item of non-logical vocabulary – the cardinality operator – which it contains and thereby as conferring content upon Hume's Principle itself..

This paper takes the form of a critical discussion of Crispin Wright’s notion of entitlement of cognitive project. I examine various strategies for defending the claim that entitlement can make acceptance of a proposition epistemically rational, including one which appeals to epistemic consequentialism. Ultimately, I argue, none of these strategies is successful, but the attempt to isolate points of disagreement with Wright issues in some positive proposals as to how an epistemic consequentialist should characterize epistemic rationality.

Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the most powerful presentation yet of a neo-Fregean program.

Since Carol Gilligan's analysis of the “Heinz dilemma,” many philosophers working on care have articulated critiques of abstraction and principles in ethics. Their objections to abstraction and principles have not always been systematically set out. In this paper, I try to clarify the debate. I begin by distinguishing several aspects of the care critique. I then consider the strengths of each from a Kantian perspective. I conclude that, although some of these objections point out potential misuses of abstraction and principle, and in doing so, suggest strategies and cautions for their correct and careful use in ethics, they do not present a successful challenge to abstraction or principles as such.

A central element in neo-Fregean philosophy of mathematics is the focus on abstraction principles, and the use of abstraction principles to ground various areas of mathematics. But as is well known, not all abstraction principles are in good standing. Various proposals for singling out the acceptable abstraction principles have been presented. Here I investigate what philosophical underpinnings can be provided for these proposals; specifically, underpinnings that fit the neo-Fregean's general outlook. Among the philosophical ideas I consider are: general views on a priori justification; the idea of abstraction as reconceptualization, the idea that truth is prior to reference in the sense associated with Frege's context principle; and various broadly relativistic views. The conclusions are by and large negative.

Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these counterexamples can be avoided only by major departures from the existing neo-Fregean programme.

On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to explain one way in which it may be extended to encompass the theory of real numbers, introducing the reals, by means of suitable further abstraction principles, as ratios of quantities.

This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (of rational numbers) and r a rational number. Say that r is an upper bound of P, written P≤r, if for any rational number s, if Ps then either s<r or s=r. In other words, P≤r if r is greater than or equal to any rational number that P applies to. Consider the Cut Abstraction Principle: (CP) ∀P∀Q(C(P)=C(Q) ≡ ∀r(P≤r ≡ Q≤r)). In other words, the cut of P is identical to the cut of Q if and only if P and Q share all of their upper bounds. The axioms of second-order real analysis can be derived from (CP), just as the axioms of second-order Peano arithmetic can be derived from Hume’s principle. The paper raises some of the philosophical issues connected with the neo-Fregean program, using the above abstraction principles as case studies.

In “Double Vision Two Questions about the Neo-Fregean Programme”, John MacFarlane’s raises two main questions: (1) Why is it so important to neo-Fregeans to treat expressions of the form ‘the number of Fs’ as a species of singular term? What would be lost, if anything, if they were analysed instead as a type of quantifier-phrase, as on Russell’s Theory of Definite Descriptions? and (2) Granting—at least for the sake of argument—that Hume’s Principle may be used as a means of implicitly defining the number operator, what advantage, if any, does adopting this course possess over a direct stipulation of the Dedekind-Peano axioms? This paper attempts to answer them. In response to the first, we spell out the links between the recognition of numerical terms as vehicles of singular reference and the conception of numbers as possible objects of singular, or object-directed, thought, and the role of the acknowledgement of numbers as objects in the neo-Fregean attempt to justify the basic laws of arithmetic. In response to the second, we argue that the crucial issue concerns the capacity of either stipulation—of Hume’s Principle, or of the Dedekind-Peano axioms—to found knowledge of the principles involved, and that in this regard there are crucial differences which explain why the former stipulation can, but the latter cannot, play the required foundational role.

Much of The Reason’s Proper Study is devoted to defending the claim that simply by stipulating an abstraction principle for the “number-of” functor, we can simultaneously fix a meaning for this functor and acquire epistemic entitlement to the stipulated principle. In this paper, I argue that the semantic and epistemological principles Hale and Wright offer in defense of this claim may be too strong for their purposes. For if these principles are correct, it is hard to see why they do not justify platonist strategies that are not in any way “neo-Fregean,” e.g. strategies that treat “the number of Fs” as a Russellian definite description rather than a singular term, or employ axioms that do not have the form of abstraction principles.