Friedrich Waismann, a little-known mathematician and onetime student of Wittgenstein's, provides answers to problems that vexed Wittgenstein in his attempt to explicate the foundations of mathematics through an analysis of its practice. Waismann argues in favor of mathematical intuition and the reality of infinity with a Wittgensteinian twist. Waismann's arguments lead toward an approach to the foundation of mathematics that takes into consideration the language and practice of experts.

Bertand's Postulate is proved in Peano Arithmetic minus the Successor Axiom.

In a short, technical note, the system of arithmetic, F, introduced in Systems for a Foundation of Arithmetic and "True" Arithmetic Can Prove Its Own Consistency and Proving Quadratic Reciprocity, is demonstrated to be equivalent to a sub-theory of Peano Arithmetic; the sub-theory is missing, most notably, the Successor Axiom.

Using an axiomatization of second-order arithmetic (essentially second-order Peano Arithmetic without the Successor Axiom), arithmetic's basic operations are defined and its fundamental laws, up to unique prime factorization, are proven. Two manners of expressing a system's consistency are presented - the "Godel" consistency, where a wff is represented by a natural number, and the "real" consistency, where a wff is represented as a second-order sequence, which is a stronger notion. It is shown that the system can prove at least its (...) Godel consistency and that closely allied systems can prove their real consistency. (shrink)

The system of arithmetic considered in Consistency, which is essentially second-order Peano Arithmetic without the Successor Axiom, is used to prove more theorems of arithmetic, up to Quadratic Reciprocity.

Neo-logicism uses definitions and Hume's Principle to derive arithmetic in second-order logic. This paper investigates how much arithmetic can be derived using definitions alone, without any additional principle such as Hume's.

Second-order Peano Arithmetic minus the Successor Axiom is developed from first principles through Quadratic Reciprocity and a proof of self-consistency. This paper combines 4 other papers of the author in a self-contained exposition.

A new second-order axiomatization of arithmetic, with Frege's definition of successor replaced, is presented and compared to other systems in the field of Frege Arithmetic. The key in proving the Peano Axioms turns out to be a proposition about infinity, which a reduced subset of the axioms proves.

The Successor Axiom asserts that every number has a successor, or in other words, that the number series goes on and on ad infinitum. The present work investigates a particular subsystem of Frege Arithmetic, called F, which turns out to be equivalent to second-order Peano Arithmetic minus the Successor Axiom, and shows how this system can develop arithmetic up through Gauss' Quadratic Reciprocity Law. It then goes on to represent questions of provability in F, and shows that F can prove (...) its own consistency and indeed the consistency of stronger systems. So, arithmetic without the Successor Axiom has an exceptional combination of three chracteristics: it is natural, it is strong, and it proves its own, as well as stronger systems’, consistency. (shrink)

General Arithmetic is the theory consisting of induction on a successor function. Normal arithmetic, say in the system called Peano Arithmetic, makes certain additional demands on the successor function. First, that it be total. Secondly, that it be one-to-one. And thirdly, that there be a first element which is not in its image. General Arithmetic abandons all of these further assumptions, yet is still able to prove many meaningful arithmetic truths, such as, most basically, Commutativity and Associativity of Addition and (...) Multiplication, but also Lagrange’s Four-Square Theorem. Adding one more axiom, the one-oneness of succession, one can prove many more theorems, such as Quadratic Reciprocity and Fermat’s Little Theorem. By looking at arithmetic in this general setting, one receives a deeper understanding of the underlying structures. (shrink)

We approach the philosophy of mathematics via an analysis of mathematics as it is practised. This leads us to a classification in terms of four concepts, which we define and illustrate with a variety of examples. We call these concepts background conventions, context, content, and intuition.

Marcus Giaquinto tells the compelling story of one of the great intellectual adventures of the modern era: the attempt to find firm foundations for mathematics. From the late nineteenth century to the present day, this project has stimulated some of the most original and influential work in logic and philosophy.

Scientific realists often appeal to some version of the conjunction objection to argue that scientific instrumentalism fails to do justice to the full empirical import of scientific theories. Whereas the conjunction objection provides a powerful critique of scientific instrumentalism, I will show that mathematical instrumentalism escapes the conjunction objection unscathed.

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...) about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

Classifies accounts of arithmetic into four sorts according to the resources they appeal to in constructing its subject matter.

If logical truth is necessitated by sheer syntax, mathematics is categorially unlike logic even if all mathematics derives from definitions and logical principles. This contrast gets obscured by the plausibility of the Synonym Substitution Principle implicit in conceptions of analyticity: synonym substitution cannot alter sentence sense. The Principle obviously fails with intercepting: nonuniform term substitution in logical sentences. 'Televisions are televisions' and 'TVs are televisions' neither sound alike nor are used interchangeably. Interception synonymy gets assumed because logical sentences and their (...) synomic interceptions have identical factual content, which seems to exhaust semantic content. However, intercepting alters syntax by eliminating term recurrence, the sole strictly syntactic means of ensuring necessary term coextension, and thereby syntactically securing necessary truth. Interceptional necessity is lexical, a notational artifact. The denial of interception nonsynonymy and the disregard of term recurrence in logic link with many misconceptions about propositions, logical form, conventions, and metalanguages. Mathematics is distinct from logic: its truth is not syntactic; it is transmitted by synonym substitution; term recurrence has no essential role. The '=' of mathematics is an objectual relation between numbers; the '=' of logic marks a syntactic relation of coreferring terms. (shrink)