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Founding Mathematics on Semantic Conventions

Part of the Synthese Library Book Series (volume 446)

DOI: 10.1007/978-3-030-88534-2

ISBN: 978-3-030-88533-5

Publisher: Springer International Publishing - Springer Journals

Nov 5, 2021

1 - 10 of 10 Chapters

Available

Introduction

Hansen, Casper Storm

Pages: 1 –8

[This short introduction presents the author’s motivation for developing a nominalistic mathematics, an explanation for what he understands by “nominalism”, and an overview of the book. His semantic conventionalism is briefly contrasted against platonism, formalism, and constructivism. In...
Available

Classical Mathematics and Plenitudinous Combinatorialism

Hansen, Casper Storm

Pages: 9 –27

[This and the following two chapters discuss and criticize some existing systems of mathematics. This one is about classical mathematics as inspired by Georg Cantor, and especially its use of undefinable sets. The author refutes some arguments for actual infinity, and suggests that the picture...
Available

Intuitionism and Choice Sequences

Hansen, Casper Storm

Pages: 29 –48

[This chapter deals with intuitionism. According to L.E.J. Brouwer, it is possible to have a mathematics with an austere ontological basis of mental acts, and nevertheless accept undefinable real numbers. That is allegedly made possible by freely-proceeding choice sequences, i.e., sequences...
Available

From Logicism to Predicativism

Hansen, Casper Storm

Pages: 49 –65

[The main alternative to the classical mathematicians’ combinatorial sets are logical sets. Later in the book, a version of such sets is adopted, but this chapter traces the history of the notion through Gottlob Frege’s logicism, Bertrand Russell’s type theory, and Hermann Weyl’s predicativism....
Available

Conventional Truth

Hansen, Casper Storm

Pages: 67 –88

[This chapter contains a solution to the liar paradox. Unlike most other proposed solutions, it does not consist in a claim about how truth values are distributed over some language that allows for self-reference. Instead, it takes one step back: to what truth conditions are in general, namely, a...
Available

Semantic Conventionalism for Mathematics

Hansen, Casper Storm

Pages: 89 –111

[This chapter discusses the indispensability argument for the existence of abstract mathematical objects, Willard Van Orman Quine’s argument for the non-conventional status of logic, and the rule-following problem. It also contains a sandbox version of a semantic-conventionalist system, and the...
Available

A Convention for a Type-free Language

Hansen, Casper Storm

Pages: 113 –143

[This chapter and the two that follow it lay out an extensive semantic-conventionalist system, and are therefore technical in nature. Here, the author develops the specific gappy logic that he thinks is most useful for a mathematical system that is to include real numbers. Though it is strongly...
Available

Basic Mathematics

Hansen, Casper Storm

Pages: 145 –167

[Here, the framework provided in the previous chapter begins to be filled up with mathematical content, starting from arithmetic and other relatively basic areas of mathematics. The chapter demonstrates in detail how mathematics can be constituted by an open-ended language, where the classical...
Available

Real Analysis

Hansen, Casper Storm

Pages: 169 –223

[The process of filling in the author’s proposed framework with mathematical content continues in this chapter. Here, the general subject is classically uncountable areas of mathematics, the challenges of which require creative solutions. The specific topics covered are functions and their...
Available

Possibility

Hansen, Casper Storm

Pages: 225 –237

[The nominalism of this book assumes that there are facts about the merely possible. This final chapter discusses the modal metaphysics that is presupposed, as well as the status of the totality of all possible real numbers and its relationship to the open-ended mathematical systems that can...

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