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. (shrink)

The essay addresses the well‐known idea that there has to be a place for intuition, thought of as a kind of non‐inferential rational insight, in the epistemology of basic logic if our knowledge of its principles is non‐empirical and is to allow of any finite, non‐circular reconstruction. It is argued that the error in this idea consists in its overlooking the possibility that there is, properly speaking, no knowledge of the validity of principles of basic logic. When certain important distinctions (...) are observed, for instance that between recognising that Modus Ponenes is sound and recognising that it is proof against the competent discovery of basic counterexamples, the case for thinking that there is indeed no space for genuine recognition of the validity of Modus Ponens becomes increasingly impressive. It is argued however that, the impossibility of knowledge notwithstanding, we are, in an important sense, entitled to take it that Modus Ponens is sound and that this notion of entitlement can help break the trichotomy ‐ intuition, inference, experience ‐ which imprisons our ordinary thinking about logical knowledge. (shrink)

Kit Fine’s book is a study of abstraction in a quite precise sense which derives from Frege. In his Grundlagen, Frege contemplates defining the concept of number by means of what has come to be called Hume’s principle—the principle that the number of Fs is the same as the number of Gs just in case there is a one-to-one correspondence between the Fs and the Gs. Frege’s discussion is largely conducted in terms of another, similar but in some respects simpler, (...) proposal—to define the concept of direction by laying down the principle that the direction of a line a is the same as that of line b if and only if lines a and b are parallel. Such principles have come to be known as abstraction principles. More generally, abstraction principles are ones of the shape: ∀α∀β ↔ α ≈ β), where ≈ is an equivalence relation on entities of the type over which α and β vary and, if the principle is acceptable, § is a function from entities of that type to objects. Since ‘the direction of’ is intended to stand for a function from objects of a certain sort to other objects, the Direction Equivalence is a first-order, or in Fine’s terminology objectual, abstraction; since ‘the number of’ is intended to stand for a function from concepts to objects, Hume’s principle is a higher-order, or as Fine says conceptual, abstraction. As is well-known, Frege himself abandoned the idea of defining number implicitly or contextually by means of Hume’s principle—adopting instead an explicit definition of the number of Fs as the extension of the concept concept equinumerous to the concept F— because he thought that an adequate definition should settle the question whether Julius Caesar, for example, is or is not the number of some concept, but that the proposed implicit definition cannot do so. But interest in abstraction principles has revived in the last couple of decades, largely as a result of Crispin Wright’s attempt to show that in spite of the many difficulties confronting it—including the notorious Julius Caesar problem— Hume’s principle can after all serve as a foundation for arithmetic. (shrink)

In his paper in the same volume, Sider argues that, of maximalism and quantifier variance, the latter promises to let us make better sense of neo-Fregeanism. I argue that neo-Fregeans should, and seemingly do, reject quantifier variance. If they must choose between these two options, they should choose maximalism.

NeoFregeanism is an intriguing but elusive philosophy of mathematical existence. At crucial points, it goes cryptic and metaphorical. I want to put forward an interpretation of neoFregeanism—perhaps not one that actual neoFregeans will embrace—that makes sense of much of what they say. NeoFregeans should embrace quantifier variance.

This article is a critical notice of Bob Hale and Crispin Wright's *The Reason's Proper Study* (OUP). It focuses particularly on their attempts (crucial to their neo-logicist project) to say what a singular term is. I identify problems for their account but include some constructive suggestions about how it might be improved.

The neo-logicist argues tliat standard mathematics can be derived by purely logical means from abstraction principles—such as Hume's Principle— which are held to lie 'epistcmically innocent'. We show that the second-order axiom of comprehension applied to non-instantiated properties and the standard first-order existential instantiation and universal elimination principles are essential for the derivation of key results, specifically a theorem of infinity, but have not been shown to be epistemically innocent. We conclude that the epistemic innocence of mathematics has not been (...) established by the neo-logicist. (shrink)