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It is alleged that the causal inertness of abstract objects and the causal conditions of certain naturalized epistemologies precludes the possibility of mathematical knowledge. This paper rejects this alleged incompatibility, while also maintaining that the objects of mathematical beliefs are abstract objects, by incorporating a naturalistically acceptable account of ‘rational intuition’. On this view, rational intuition consists in a non-inferential belief-forming process where the entertaining of propositions or certain contemplations result in true beliefs. This view is free of any conditions incompatible with abstract objects, for the reason that it is not necessary that S stand in some causal relation to the entities in virtue of which p is true. Mathematical intuition is simply one kind of reliable process type, whose inputs are not abstract numbers, but rather, contemplations of abstract numbers.
Revista Pesquisa Qualitativa
Epistemology and rationality of intuition and imagination in the field of mathematicsThis paper aims to contribute to the clarification of the role of mathematical intuition and imagination in the constitution of mathematical knowledge, evidencing its epistemological and procedural characteristics. For that, an "epistemology of intuition and imagination" in the field of mathematics is outlined (suggested) emphasizing the need to adopt a dynamic conception of mathematics. In this context, intuition and imagination present themselves as forms of mathematical experience that give access, through paths that are not purely logical, to mathematical knowledge. Its epistemological and rationality characteristics, a rational of a non-logical nature, are highlighted by several examples, resources for moving the ideas involved. The epistemological study of intuition and imagination also allows highlighting its ontology, constituted of more relations than objects. From the pedagogical point of view, we discuss the formative character of philosophical studies involving...
The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: What does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics.
Topoi-an International Review of Philosophy
Mathematical Intuition and the Cognitive Roots of Mathematical Concepts2010 •
The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition.
2020 •
Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its nal form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) mathematical truths are not truths about the external world but speci cations (formulations) of mathematical conceptions; (iii) mathematics is rst and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. keywords: mathematical models; mathematical objects; mathematical truths; applicability of mathematics The basic problem of the philosophy of mathematics (not mathematics itself) is to answer the followi...
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