The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...) the famous Hilbert-Brouwer controversy in the 1920s. -/- The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. (shrink)

En este ensayo respondo negativamente a la pregunta del título al sostener que el enunciado “La suma de los ángulos internos de un triángulo es igual a 180°” es contingentemente verdadero. Para ello, intento refutar la tesis de Ramsey de que las verdades geométricas necesariamente son verdades necesarias, así como la tesis de Kripke de que no puede haber proposiciones matemáticas contingentemente verdaderas. Además, recurriendo a la concepción fregeana sobre lo a priori y lo a posteriori, sostengo que hay verdades (...) geométricas que pueden ser a priori sin tener que serlo. -/- . (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)

Although much creativity research has suggested that creativity is influenced by cultural and social factors, these have been minimally explored in the context of mathematics and mathematics learning. This problematically limits who is seen as mathematically creative and who can enter the discipline of mathematics. This paper proposes a framework of creativity that is based in what it means to know or do mathematics and accepts that creativity is something that can be nurtured in all students. Prominent mathematical epistemologies held (...) since the beginning of the twentieth century in the Western mathematics tradition have different implications for promoting creativity in the mathematics classroom, with fallibilist and social constructivist perspectives arguably being most conducive for conceiving of creativity as a type of action for all students. Thus, this paper proposes a framework of creative mathematical action that is based in these epistemologies and explains key aspects of the framework by drawing connections between it and research in the field of creativity. (shrink)

The purpose of this dissertation is to present historical evidence in favor of the thesis that many forms of dichotomy appearing in the history of epistemology are related to the duality represented by the mathematical concepts of continuity and discreteness. Parts 1 and 2 give a descriptive and historical account of epistemological dichotomies appearing during the development of mathematics and logic. In part 3, the implications of these dichotomies for general philosophy are explored by means of a collage of analytic, (...) historical, and factual materials, accompanied by brief commentaries, and aimed at establishing the following points: Acceptance of the primordial intuition of continuity distinguishes the constructive model from the structural model in epistemology; The constructive model emphasizes meaningful thought processes of the reflecting and acting subject, while the structural model emphasizes publicly recognizable and communicable thought processes based upon logic, linguistics, or natural law; An activity which translates between the constructive model and the structural model by means of symbols which include iconic components is to be called "aesthetic"; The function of aesthetics is to bring about harmonious and productive relations between individuals and the larger wholes to which they belong. ;The conclusion is reached that, from the point of view of the individual, aesthetics is important for motivation and for the development of cognitive powers and moral sensibilities; from the point of view of the species, aesthetics is important for long-term survival. Therefore, logic and aesthetics are of equal and complementary importance and deserve equal consideration in the study of epistemology. (shrink)