In the first part of this article I investigated the Popperian roots of Lakatos's Proofs and Refutations, which was an attempt to apply, and thereby to test, Popper's theory of knowledge in a field—mathematics—to which it had not primarily been intended to apply. While Popper's theory of knowledge stood up gloriously to this test, the new application gave rise to new insights into the heuristic of mathematical development, which necessitated further clarification and improvement of some Popperian methodological maxims. In the (...) present part I analyze this second phase in the development of Lakatos's Popperian programme in mathematics, and its connection to the methodology of scientific research programmes. (shrink)

This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...) program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals. (shrink)

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without caution, as the use (...) of anachronistic notation tends to impede, rather than enhance, our ability to recognize the emergent nature of mathematical objects. (shrink)

The canonical history of mathematics suggests that the late 19th-century “arithmetization” of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, (...) which reveals a reliance on dynamic conceptual resources. The second is a cognitive-historical case study of an incident in 19th-century mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice. (shrink)

In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathematics has important consequences on our conception (...) of mathematical knowledge, in particular mathematical growth. (shrink)

This article canvasses five senses in which one might introduce an historical element into the philosophy of mathematics: 1. The temporal dimension of logic; 2. Explanatory Appeal to Context rather than to General Principles; 3. Heraclitean Flux; 4. All history is the History of Thought; and 5. History is Non-Judgmental. It concludes by adapting Bernard Williams’ distinction between ‘history of philosophy’ and ‘history of ideas’ to argue that the philosophy of mathematics is unavoidably historical, but need not and must not (...) merge with historiography. (shrink)

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

Lakatos's Proofs and Refutations is usually understood as an attempt to apply Popper's methodology of science to mathematics. This view has been challenged because despite appearances the methodology expounded in it deviates considerably from what would have been a straightforward application of Popperian maxims. I take a closer look at the Popperian roots of Lakatos's philosophy of mathematics, considered not as an application but as an extension of Popper's critical programme, and focus especially on the core ideas of this programme (...) as they appeared where Popper specifically addressed philosophical problems concerning mathematics. (shrink)

In this paper I assess the obstacles to a transfer of Lakatos's methodology of scientific research programmes to mathematics. I argue that, if we are to use something akin to this methodology to discuss modern mathematics with its interweaving theoretical development, we shall require a more intricate construction and we shall have to move still further away from seeing mathematical knowledge as a collection of statements. I also examine the notion of rivalry within mathematics and claim that this appears to (...) be significant only at a high level. In addition, ideas of ‘progress’ in mathematics are outlined. (shrink)