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**Applying mathematics and the indispensability argument.**
*(English)*
Zbl 0849.00014

Echeverria, Javier (ed.) et al., The space of mathematics. Philosophical, epistemological, and historical explorations. Revised papers from a symposium on structures in mathematical theories, Donostia/San Sebastian, Basque Country, Spain, September 1990. Berlin: Walter de Gruyter. Grundlagen der Kommunikation und Kognition. 115-131 (1992).

Quine’s and Putnam’s indispensability argument says that “appealing to mathematical objects and mathematical truths figures indispensably in using mathematics in science, and as a consequence, we should consider mathematical objects to be no less real than scientific ones” (pp. 115). Three attempts to undercut the realistic Quine-Putnam argument with anti-realistic approaches are discussed using two examples of applying mathematics, the barnyard use of arithmetic and counting theory and the Hardy-Weinberg law of the distribution of genes in population genetics. These attempts are Hartry Field’s structuralism as presented in his book “Science without number. A defense of nominalism” (Oxford, Blackwell, 1980; Zbl 0454.00015), the statistical approach of Henry Kyborg [Theory and measurement (Cambridge, Cambridge UP, 1984)] and Nancy Cartwright’s treating mathematical models as analogies of physical situations [cf. How the laws of physics lie (Oxford, Oxford UP, 1983)].

Approaches using analogies have to face the problem to handle the tension between logical or mathematical consistency and truth, the last implying a commitment to mathematical objects or truths. “New modalism” as presented, e.g., by H. Field and G. Hellman is discussed as a possible way out. It proposes that “instead of saying that some (finite) set of axioms \(A_1, A_2, \dots, A_k\) is consistent, we say that it is logically possible that \(A_1 \& A_2 \& \dots \& A_k\)” (p. 128). The author concludes that there is no convincing single way of thinking about the possibility of mathematical objects. He himself advocates the position of a Platonic structuralism holding “that mathematics is about structures”, but allowing “for structures to exist uninstantiated by nonmathematical entities” (p. 131).

For the entire collection see [Zbl 0839.00019].

Approaches using analogies have to face the problem to handle the tension between logical or mathematical consistency and truth, the last implying a commitment to mathematical objects or truths. “New modalism” as presented, e.g., by H. Field and G. Hellman is discussed as a possible way out. It proposes that “instead of saying that some (finite) set of axioms \(A_1, A_2, \dots, A_k\) is consistent, we say that it is logically possible that \(A_1 \& A_2 \& \dots \& A_k\)” (p. 128). The author concludes that there is no convincing single way of thinking about the possibility of mathematical objects. He himself advocates the position of a Platonic structuralism holding “that mathematics is about structures”, but allowing “for structures to exist uninstantiated by nonmathematical entities” (p. 131).

For the entire collection see [Zbl 0839.00019].

Reviewer: V.Peckhaus (Erlangen)

### MSC:

00A30 | Philosophy of mathematics |

03A05 | Philosophical and critical aspects of logic and foundations |

### Keywords:

Hardy-Weinberg law of population genetics; mathematical objects; mathematical existence; truth; consistency; structuralism; statistics; analogies; modalism; Platonism### Citations:

Zbl 0454.00015
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\textit{M. D. Resnik}, in: The space of mathematics. Philosophical, epistemological, and historical explorations. Revised papers from a symposium on structures in mathematical theories, Donostia/San Sebastian, Basque Country, Spain, September 1990. Berlin: Walter de Gruyter. 115--131 (1992; Zbl 0849.00014)