Mark Colyvan, An Introduction to the Philosophy of Mathematics
| Autor | Cristian Soto |
| Cargo | School of Historical and Philosophical Studies, University of Melbourne, Departamento de Filosofía, Universidad de Chile |
Page 93
CRÍTICA, Revista Hispanoamericana de Filosofía. Vol. 46, No. 138 (diciembre 2014): 93–102
Mark Colyvan, An Introduction to the Philosophy of Mathematics, Cambridge University Press, Cambridge, 2012, 188 pp.
Colyvan’s Introduction presents an up-to-date, clearly written journey through the recent debates on the philosophy of mathematics. It offers a valuable examination of some of the current main problems, fulfilling the primary goals of an introductory reading for those working in the discipline. In my judgment, however, it goes beyond, addressing some of the most interesting issues in the philosophy of mathematics and their relevance for the philosophy of science.
Some of the themes explored throughout its pages are these: a chart of the debates on the philosophy of mathematics (Ch. 1); the limits of mathematics (Ch. 2); realist philosophies of mathematics (Ch. 3); anti-realist philosophies of mathematics (Ch. 4); mathematical explanation (Ch. 5); the applicability of mathematics (Ch. 6); inconsistent mathematics (Ch. 7); mathematical notation (Ch. 8); and the examination of twenty philosophically interesting mathematical theorems (Ch. 9). In order to offer a fairly comprehensive review of the volume, in what follows I firstly examine the debate between realist and anti-realist philosophies of mathematics and secondly I briefly focus on the nature of mathematical explanation, the applicability of mathematics, and the viability of the mapping account.
I
According to Colyvan, Benacerraf’s (1983a; 1983b) landmark papers and the indispensability argument, independently elaborated by Quine (1976; 1981) and Putnam (1979), chart the course through the problems in the philosophy of mathematics over the last decades. On a first approach, these problems can be classified into three, namely: first, the elaboration of a uniform semantics for both mathematical and non-mathematical discourse; second, an epistemology of mathematics that satisfactorily addresses the question of how we come to know mathematical entities in the first place; and third, ontological problems related to the nature of mathematical entities. Colyvan makes some interesting comparisons between realist and anti-realist philosophies of mathematics and their attempts to answer these questions.
Following Putnam (1979, p. 70), the author distinguishes realism as a thesis about the objectivity of mathematical knowledge, on one hand, and as a thesis about the existence of mathematical entities, on
Page 94
94 NOTAS BIBLIOGRÁFICAS
the other (pp. 36–38). From a realist perspective, mathematical statements can be regarded as objectively true if there are some mathematical posits that make them true. Once realists have agreed on this, viz., that mathematical existence is a precondition of mathematical knowledge, a new source of possible disagreement emerges from the consideration of the nature of the mathematical entities that supposedly exist. At this point, realism is put forward in different shapes and flavours. For instance, a first form of realism is full-blooded Platonism, which endorses the view that mathematical entities are abstract in nature, having neither spatio-temporal location nor causal powers. A second form is physicalist mathematical realism, which entertains the idea that mathematical entities are physical in nature, belonging to the spatio-temporal causal link (Bigelow 1988; Maddy 1990). And a third form is the structuralist view, which advocates that the subject matter of mathematics is the structural relation, instantiated or not, among mathematical entities (Resnik 1997, pp. 36–41).
Let us look in some further detail at the full-blooded Platonistic version of mathematical realism. Its central tenet can be formulated as follows: every consistent mathematical theory truly describes some part of the mathematical universe (p. 38). Full-blooded Platonism meets the semantic and ontological requirements by postulating a very rich ontology, since it is claimed that statements such as “5 + 7 = 12” and “the atomic number of gold is 72” are both true if we accept that there actually are things such as numbers and chemical properties. Nonetheless, it counts against this view that it does not straightforwardly offer a response to the epistemic problem, viz., how we come to know non-causal, non-spatio-temporally located, abstract mathematical entities in the first place.
Arguments for realism in the philosophy of mathematics usually appeal to some form of indispensability argument (pp. 41–54). Colyvan’s previous contribution (2001a, pp. 6–17) may well be considered the best treatment of its kind. As outlined in the present volume, the general form of the argument is as follows:
(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our current best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.
(C) We ought to have ontological commitment to mathematical entities (p. 43).
Crítica , vol. 46, no. 138 (diciembre 2014)
Page 95
NOTAS BIBLIOGRÁFICAS 95
Prima facie, (P1) relies on the assumption that scientific theories are our best guide to the furniture of reality, whilst (P2) puts forward the idea that in fact scientific theorising quantifies over both mathematical and physical entities. Ergo, (C) concludes that we should accept the existence of mathematical entities just as we typically accept, from a standard scientific realist perspective, the existence of theoretical entities. Furthermore, the argument advocates confirmational holism, which broadly speaking is the view that “theories are confirmed or disconfirmed as wholes” (p. 45). Thus, granted confirmational holism, it can be claimed that the mathematical component of...
Para continuar leyendo
Solicita tu pruebaCOPYRIGHT GALE, Cengage Learning. All rights reserved.
