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In the Foundations of Arithmetic, Frege famously developed a theory which today goes by the name of logicism - that it is possible to prove the truths of arithmetic using only logical principles and definitions. Logicism fell out of favor for various reasons, most spectacular of which was that the system, which Frege thought would definitively prove his thesis, turned out to be inconsistent. In the early 1980s a movement called neo-logicism was begun by Crispin Wright. Neo-logicism holds that Frege was almost right, in that arithmetic can be proven in second-order logic using only definitions and one quasi-logical proposition, called Hume's Principle, which says that the number of Ps equals the number of Qs if and only if they can be put into one-to-one correspondence. There has been some controversy about the status of Hume’s Principle - for instance, whether it counts as a logical or analytic proposition. (See e.g. the similarly titled, “Is Hume’s Principle Analytic?, by Crispin Wright and George Boolos.) In this paper a different tack will be tried. Indeed Frege is almost right. He is almost right because a large part of arithmetic and number theory, or at the least a large part of something which looks like them, can indeed be generated using only logical principles and definitions, without the assumption of any quasi-logical assertion and in particular without Hume’s Principle. Specifically, logic will be taken as second-order logic with full comprehension and the addition of one distinguished 2-ary predicate “!”. A large amount of arithmetic and number theory will then be developed, using only (second-order) logical principles and definitions. It can thus be seen that the epistemological status of this large part of arithmetic is independent of the question of the status of Hume’s Principle.

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.

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..

The neo-Fregean account of arithmetical knowledge is centered around the abstraction principle known as Hume’s Principle: for any concepts X and Y , the number of X ’s is the same as the number of Y ’s just in case there is a 1–1 correspondence between X and Y . The Caesar Problem, originally raised by Frege in §56 of Die Grundlagen der Arithmetik , emerges in the context of the neo-Fregean programme, because, though Hume’s Principle provides a criterion of identity for objects falling under the concept of Number–namely, 1–1 correspondence—the principle fails to deliver a criterion of application. That is, it fails to deliver a criterion that will tell us which objects fall under the concept Number, and so, leaves unanswered the question whether Caesar could be a number. Hale and Wright have recently offered a neo-Fregean solution to this problem. The solution appeals to the notion of a categorical sortal. This paper offers a reconstruction of their solution, which has the advantage over Hale and Wright’s original proposal of making clear what the structure of the background ontology is. In addition, it is shown that the Caesar Problem can be solved in a framework more minimal than that of Hale and Wright, viz . one that dispenses with categorical sortals. The paper ends by discussing an objection to the proposed neo-Fregean solutions, based on the idea that Leibniz’s Law gives a universal criterion of identity. This is an idea that Hale and Wright reject. However, it is shown that a solution very much in keeping with their own proposal is available, even if it is granted that Leibniz’s Law provides a universal criterion of identity.

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.

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.

Hale proposes a neo-logicist definition of real numbers by abstraction as ratios defined on a complete ordered domain of quantities (magnitudes). I argue that Hale's definition faces insuperable epistemological and ontological difficulties. On the epistemological side, Hale is committed to an explanation of measurement applications of reals which conflicts with several theorems in measurement theory. On the ontological side, Hale commits himself to the necessary and a priori existence of at least one complete ordered domain of quantities, which is extremely implausible because science treats the logical structure of quantities as subject to experimentally and theoretically motivated refinements and revisions.

Defining the real numbers by abstraction as ratios of quantities gives prominence to then- applications in just the way that Frege thought we should. But if all the reals are to be obtained in this way, it is necessary to presuppose a rich domain of quantities of a land we cannot reasonably assume to be exemplified by any physical or other empirically measurable quantities. In consequence, an explanation of the applications of the reals, defined in this way, must proceed indirectly. This paper explains the main complications involved and answers the main objections advanced in Batitsky's paper in this issue.

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 this paper I examine the prospects for a successful neo–logicist reconstruction of the real numbers, focusing on Bob Hale's use of a cut-abstraction principle. There is a serious problem plaguing Hale's project. Natural generalizations of this principle imply that there are far more objects than one would expect from a position that stresses its epistemological conservativeness. In other words, the sort of abstraction needed to obtain a theory of the reals is rampantly inflationary. I also indicate briefly why this problem is likely to reappear in any neo–logicist reconstruction of real analysis.