Martin H. Krieger. Doing Mathematics: Convention, Subject, Calculation, Analogy. Singapore: World Scientific Publishing, 2003. Pp. xviii + 454. ISBN 981-238-2003 (cloth); 981-238-2062 (paperback) (original) (raw)

Why Is There Philosophy of Mathematics at All–A Book Review

Notices of the American Mathematical Society, 2014

Book Review desk work again and again proves useful in dealing with the world outside the mathematician's office. How by just sitting and thinking we (or some of us) can arrive at results applicable to the world around us has puzzled thinkers from Kant to Wigner [4]. The two features are separate. That results should prove applicable to the physical universe even though they were obtained by pure desk work, without controlled experimentation on or systematic observation of the material world, can be surprising even if what the desk work produces is not compelling deductive proofs but "only" suggestive heuristic arguments. And with the two factors being separate, the material in the book is divided into two more or less separate parts, though with a lot of back and forth between them: one devoted to proof, the other to applications. Neither the part about proof nor the part about applications is concerned only with their role in perennially drawing the attention of philosophers to mathematics. And, beyond the general division into these two broad topics, the book is rather loosely organized and digressive, not to say rambling, in a way that makes it quite impossible for the reviewer to summarize its contents in an even halfway adequate fashion. The analytical table of contents goes on for six pages, and there is nothing I would leave out, but this means that even to list the topics addressed would take up more space than is reasonable for a review. One thing just leads to another: If a philosophical view is stated, some mathematical example will be wanted to illustrate it, but then at least an informal explanation of the key concepts in the example will be wanted also, and perhaps a capsule bio of the author or authors of the relevant result or results, and even perhaps in cases where they December 2014 Notices of the Ams 1345

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: 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.

MATHEMATICAL DOING AND THE PHILOSOPHIES OF MATHEMATICS

Contemporary Perspectives in Philosophy and Methodology of Science. Edtied W. J. González-Jesús Alcolea. Ed. Netbiblo, pp. 209-231, 2006

split between Mathemagtical Doing-Philosophical Thought: the Mathematician's Work, Mistakes, Philosophy;Non-accumaltive doing: The mathematician's error; Some exemplary schemes: Lagrange, The "italian school" of projective or Algebraic Geometry; French school or School of París, Current situation (Petefr Lax, Reflexive judgement; Summing Up. Escisión Hacer matemático-Pensar filosófico. El matemático en su trabajo con sus errores, su filosofía. Algunos esquemas ejemplares. A modo de conclusión

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.