Högskolan i Skövde

Recently Carrie S. Jenkins formulated an epistemology of mathematics, or rather arithmetic, respecting apriorism, empiricism, and realism. Central is an idea of concept grounding. The adequacy of this idea has been questioned e.g. concerning the grounding of the mathematically central concept of set (or class), and of composite concepts. In this paper we present a view of concept formation in mathematics, based on ideas from Carnap, leading to modifications of Jenkins’s epistemology that may solve some problematic issues with her ideas. But we also present some further problems with her view, concerning the role of proof for mathematical knowledge.

Attempts have been made to analyse features in mathematics within a social constructivist context. In this paper we critically examine some of those attempts recently made with focus on problems of the objectivity, ontology, necessity, and atemporality of mathematics. Our conclusion is that these attempts fare no better than traditional alternatives, and that they, furthermore, create new problems of their own.

One questioned premiss in the indispensability argument of Quine and Putnam is confirmational holism. In this paper I argue for a weakened form of holism, and thus a strengthened version of the indispensability argument. The argument is based on an idea of concept formation in mathematics. Mathematical concepts are arrived at via a sequence of explications, in Carnap's sense, of non-clear, originally empirical, concepts. I identify a deductive and an empirical component in mathematical concepts. In a test situation the use of the empirical component, but not of the deductive one, is corroborated or falsified together with the scientific theory.

This thesis consists of three overlapping parts, where the first one centers around the possibility of defining a measure of the power of arithmetical theories. In this part a partial measure of the power of arithmetical theories is constructed, where “power” is understood as capability to prove theorems. It is also shown that other suggestions in the literature for such a measure do not satisfy natural conditions on a measure. In the second part a theory of concept formation in mathematics is developed. This is inspired by Aristotle’s conception of mathematical objects as abstractions, and it uses Carnap’s method of explication as a means to formulate these abstractions in an ontologically neutral way. Finally, in the third part some problems of philosophy of mathematics are discussed. In the light of this idea of concept formation it is discussed how the relation between formal and informal proof can be understood, how mathematical theories are tested, how to characterize mathematics, and some questions about realism and indispensability.

Mathematical concepts are explications, in Carnap's sense, of vague or otherwise unclear concepts; mathematical theories have an empirical and a deductive component. From this perspective, I argue that the empirical component of a mathematical theory may be tested together with the fruitfulness of its explications. Using these ideas, I furthermore give an argument for mathematical realism, based on the indispensability argument combined with a weakened version of confirmational holism

Using Carnap’s concept explication, we propose a theory of concept formation in mathematics. This theory is then applied to the problem of how to understand the relation between the concepts formal proof (deduction) and informal, mathematical proof.

In this paper I discuss possible ways of measuring the power of arithmetical theories, and the possiblity of making an explication in Carnap’s sense of this concept. Chaitin formulates several suggestions how to construct measures, and these suggestions are reviewed together with some new and old critical arguments. I also briefly review a measure I have designed together with some shortcomings of this measure. The conclusion of the paper is that it is not possible to formulate an explication of the concept.