I consider how procedural postulationism might be combined with the view that postulational possibility should itself be open to expansion.

The article presents the Augustinian Vocabulary on prayer, as a personal prayer (Soliloquiorum and Confessionum), as a prayer of the Christian people (Enarrationes in Psalmos), as a prayer of the monks (Sancta Virginitate), as a need for grace (a topic which is present in the Works of the Pelagian Polemic). The article discusses the Augustine’s explanation of the Vocabulary on prayer in 1 Tim 2: 1-9 obsecrationes, orationes, postulationes, gratiarum actiones, as it is explained in Augustine’s ep. 149, giving the (...) exegetical background of the texts, and the Augustinian interpretation. Finally it presents the prayer of gratiarum actio, as a clue to understand Christian prayer. (shrink)

In recent work, Kit Fine proposes a new approach to the philosophy of mathematics, which he calls procedural postulationism: the postulates from which a mathematical theory is derived are imperatival, rather than indicative, in character. According to procedural postulationism, what is postulated in mathematics are not propositions true in a given mathematical domain, but rather procedures for the construction of that domain. Fine claims some very significant advantages for procedural postulationism over other approaches. This chapter raises some (...) questions for the view and its promised advantages. One crucial set of questions concerns how exactly the commands of procedural postulationism are to be understood. And in particular, how literally are we to take talk of construction? (shrink)

In a recent paper, the mathematician Harold Edwards claimed that Euler’s alleged proof, that Fermat’s last theorem is true for the case n = 3, is flawed. Fermat’s last theorem is the conjecture that there are no positive integers x, y, z, or n, such that n is greater than two and such that xn + yn = zn. In this paper we shall first briefly explain the specific flaw to which Edwards called attention. After that we briefly explain the (...) nature of mathematical proofs and with reference to such proofs explain the nature of gaps in proofs. Then we critically discuss an alternative view concerning the nature of proofs in mathematics and discuss the alternative view critically. Specifically we argue that advocates of this alternative view, which we call mathematical postulationism, have not provided a satisfactory account of the nature of gaps in proofs. The unsatisfactoriness of postulationist accounts of gaps in proofs is revealed through reflection concerning appropriate repairs to proofs with gaps. (shrink)

This chapter develops a metaphysics-first view of natural numbers and real numbers. The account gives a philosophical priority to applications: to the application of natural numbers as numbering property-instances, and to the application of real numbers as ratios of extensive magnitudes. Each natural number is individuated by the condition for it to be the number of a property. The account is contrasted with the neo-Fregean approach to natural numbers advocated by Wright; but it does have a natural marriage with the (...) postulationist approach of Fine. This metaphysics of numbers can then be deployed in combination with the principle that, for these ontologies, Individuation Precedes Representation. To be capable of representing numbers of these kinds is to have tacit knowledge of the principles that individuate them. The resulting account has both differences from and affinities with the views of Carnap. (shrink)

If her view is to diffuse charges of mystical censorship, the relativist needs a well-motivated account of what prevents our quantifying over an absolutely comprehensive domain. But relativists may seek to meet this challenge in different ways. One option is to draw on more familiar cases of quantifier domain restriction in order to motivate the thesis that a quantifier’s domain is always subject to restriction. An alternative is to permit unrestricted quantifiers but maintain that even these fail to attain absolute (...) generality on the grounds that the universe of discourse is always open to expansion. This chapter outlines restrictionist and expansionist variants of relativism and argues that the importance of the distinction comes out in two influential objections that have been levelled against relativism. (shrink)

We analyse Kit Fine’s proposal of a procedural Postulationism for mathematics. From a linguistic perspective, we argue that Postulationism is better understood in terms of declarative speech acts. Based on this observation, we argue in favor of a form of Declarationism able to account for both the objectivity of mathematics and its creative dimension.

Relativism is distinguished from pluralism; mathematical theses which are true in some systems but not others are held to express different propositions which different truth values in each. Comparisons are drawn with if-thenism, postulationism, and deductivism. A number of objections are tackled: that proofs need not be formal, that mathematicians believe theses without having proofs, that axiom systems can be inadequate and incomplete, that any consistent sentence counts as a mathematical truth, since it is provable from some system. In (...) response, further comparisons are drawn with neo-logicism, and the role of abstraction principles such as Hume's principle discussed. The superiority of neo-formalism is urged, in the treatment for example of the ‘Julius Caesar problem’ and the problem of pairwise inconsistent abstraction principles. Finally Gödel's incompleteness theorem and the problem of true, but unprovable, Gödel sentences, is raised. A quick response by the neo-formalist is rejected, and the difficulty set aside until the last chapters. (shrink)