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Philosophy of Mathematics: Making a Fresh Start

Studies in History and Philosophy of Science, vol. 44 (2013), pp. 32-42., 2013

The paper distinguishes between two kinds of mathematics, natural mathematics which is a result of biological evolution and artificial mathematics which is a result of cultural evolution. On this basis, it outlines an approach to the philosophy of mathematics which involves a new treatment of the method of mathematics, the notion of demonstration, the questions of discovery and justification, the nature of mathematical objects, the character of mathematical definition, the role of intuition, the role of diagrams in mathematics, and the effectiveness of mathematics in natural science.

Modernizing the philosophy of mathematics

Synthese, 1991

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematicai theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content.

Philosophy of mathematics Selected readings SECOND EDITION

I am not the autor, it is a collection of essays edited by Hilary Putnam and Paul Benacerraf. The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.

Philosophy of Mathematics in the Twentieth Century: Selected Essays

2014

It surely goes without saying that Charles Parsons is one of the most important philosophers of mathematics in our generation. Through his many publications and his teaching and lecturing, he has spawned and influenced a large body of historically sensitive work in the philosophy of mathematics and an equally large body of philosophically sensitive work in the history of the philosophy of mathematics. This is Parsons' fourth book. It and From Kant to Husserl (Harvard University Press, 2012) collect together most of his essays on other philosophers, with the exception of those essays reprinted in his earlier collection, Mathematics in Philosophy (Ithaca, New York, Cornell University Press, 1983). As indicated by its title, the essays in the present book are on philosophers whose work appeared primarily, or entirely, in the twentieth century. All but one of the essays have been published elsewhere. It is good to have them collected here, for at least two reasons. First, some of the essays appear in collections that are not readily available, including volumes honoring some of the authors discussed, which are sources that are notoriously hard to acquire. Second, and more important, it is interesting and helpful to see similar ideas and themes developed in different contexts. Four of the essays have new postscripts.

The Practical Turn in Philosophy of Mathematics: A Portrait of a Young Discipline

Phenomenology and Mind, 2017

In the present article, the current situation of the so-called philosophy of mathematical practice is discussed. First, its emergence is evaluated in relation to the “practical” turn in philosophy of science and in philosophy of mathematics. Second, the variety of approaches concerned with the practice of mathematics and the new topics being now object of research are introduced. Third, the possible replies to the question about what counts as mathematical practice are taken into account. Finally, some of the problems that are still open in the philosophy of mathematical practice are presented and some possible new directions of research considered.

Philosophy of Mathematics Relevance.docx

Abstract At a time in the history of mankind when man became aware of his environment the changes he observed in the phenomena prompted him to wonder. Then he tried to speculate on what could be at the base of reality. This quest for knowledge came in two distinct forms: First some consider it only in the material aspect of being where as others based it on both material and immaterial aspects. Though it is relatively old for people of the developed nations, the discipline is also relatively new in the sense that the study comes with its discoveries in consonance with the age that studies it. The task of this paper is to examine the philosophy of mathematics and its relevance to national development.