Koray Akçagüner

This manuscript-based thesis is composed of three individual papers on the philosophy of mathematics. The first two papers, which are Chapters 2 and 3 of the dissertation, focus on the history and philosophy of constructive mathematics. Chapter 2 presents a unified reading of constructivist philosophy from Kant to 1960s while also highlighting the philosophical disagreements between the founders of different constructive systems. It serves as an introduction to the dissertation as a whole. Chapter 3 compares and analyzes constructive systems further, and builds an analogy between them and traditional straightedge-compass constructions. It advances the claim that the limits of constructibility are shaped by the construction tools and forms of representation that we choose to adopt. The third paper, which is Chapter 4 of the dissertation and also the longest chapter, draws on the results of the previous two but has a broader focus; it advances a view of mathematical proofs that is influenced by Kuhn’s account of scientific theory choice. The main claim is that the standards by which mathematical proofs are accepted are influenced by extra-mathematical considerations and value judgments.

This paper explores the concept of mathematical construction by building an analogy between modern computational interpretations and the traditional geometric approach. The focus is on how the tools used for construction, whether computational, such as Turing machines, or geometric, like straightedge and compass, define the boundaries of what can be considered constructible. The paper presents a comparative analysis of geometric construction and computational construction, and argues that mathematical existence is closely linked to the chosen forms of representation and construction tools. The importance of these tools in shaping the concept of constructibility is highlighted with examples, showing how the introduction of new tools enables the construction of new mathematical objects. This perspective contributes to contemporary debates in constructivist philosophy and offers a potential way to deepen our understanding of mathematical construction.

This paper aims to shed light on Henri Poincaré’s intuitionism. Today, intuitionistic philosophy of mathematics is usually associated with L. E. J. Brouwer and the idea of expelling from mathematics the proofs which rest on the law of excluded middle. There is a widespread assumption that intuitionists believe there is a fundamental flaw in classical mathematics and that mathematics therefore requires a radical reform. It is interesting to note, however, that while Kantian philosophy strongly influenced modern intuitionism, Kant neither called for a radical reform of mathematics nor understood intuition in the way the term is commonly used today. This is also true of Poincaré, who made a significant revision to Kant’s philosophy of mathematics and who is usually regarded as a pre-intuitionist or semi-intuitionist. Philosophers such as Warren Goldfarb rightly argued that Poincaré’s concern in invoking intuition was to explain the psychological aspect of mathematical thinking. It is argued in this paper that this psychological aspect was not the whole of Poincaré’s intuitionism as there is a notion of a pure, a priori intuition in his philosophy which he borrowed from the Kantian tradition.