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**Constructivism and objects of mathematical theory.**
*(English)*
Zbl 0848.00007

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. 296-313 (1992).

The author opposes the opinion of most philosophers of mathematics that “both human history in general and the history of mathematics in particular” are epistemologically irrelevant (p. 296). But doing mathematics implies to place it into one’s own context, to connect it with other fields of experience “and therefrom results an interest in history” (ibid.). The author holds that our ideas about what human history is, influence our conceptions concerning mathematical epistemology. Following G. Markus [Language and production (Dordrecht, Reidel) (1986)] the author sees the main productive tension in mathematics not in the analytic-synthetic distinction, but in the existence of two comprehension schemata: the paradigm of language and the paradigm of production.

Kant attempted to understand mathematics as a goal driven activity by reconciling empiricism and apriorism but concentrating “on activity or construction as the one fundamental element of epistemology” (p. 297). Kant’s conception of the synthetical a priori character of arithmetical propositions is faced to the criticism of B. Bolzano who advocated recursive justifications, thus showing “the transition from the arithmetic functions as mere processes and activities to their transformation into particular objects of consideration” (p. 298). Also discussed is the attack of Hegel who accused Kant’s constructivism to be subjectivism, an argument which the author finds in similar form in Ph. Kitcher [The nature of mathematical knowledge (Oxford, Oxford UP) (1984; Zbl 0519.00022), p. 55].

The author gives H. G. Grassmann prominence for being responsible for forcing a new interest in the relation between meta-theory and theory in mathematics: “In Grassmann’s hands \({[\dots]}\) axiomatics turns into a system of meta-theoretical statements in the service of mathematics conceived as a science of forms” (p. 300). The new meta-level discourse is illustrated with the concept of function, standing for a conceptual approach in mathematics, and the problem of formal identities, being related to the constructivist approach. The essential rôle of the principle of continuity for the concept of function is discussed at length (pp. 307-311).

For the entire collection see [Zbl 0839.00019].

Kant attempted to understand mathematics as a goal driven activity by reconciling empiricism and apriorism but concentrating “on activity or construction as the one fundamental element of epistemology” (p. 297). Kant’s conception of the synthetical a priori character of arithmetical propositions is faced to the criticism of B. Bolzano who advocated recursive justifications, thus showing “the transition from the arithmetic functions as mere processes and activities to their transformation into particular objects of consideration” (p. 298). Also discussed is the attack of Hegel who accused Kant’s constructivism to be subjectivism, an argument which the author finds in similar form in Ph. Kitcher [The nature of mathematical knowledge (Oxford, Oxford UP) (1984; Zbl 0519.00022), p. 55].

The author gives H. G. Grassmann prominence for being responsible for forcing a new interest in the relation between meta-theory and theory in mathematics: “In Grassmann’s hands \({[\dots]}\) axiomatics turns into a system of meta-theoretical statements in the service of mathematics conceived as a science of forms” (p. 300). The new meta-level discourse is illustrated with the concept of function, standing for a conceptual approach in mathematics, and the problem of formal identities, being related to the constructivist approach. The essential rôle of the principle of continuity for the concept of function is discussed at length (pp. 307-311).

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:

constructivism; empiricism; apriorism; meta-theory; continuity principle; concept of identity; concept of function### Citations:

Zbl 0519.00022
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\textit{M. Otte}, 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. 296--313 (1992; Zbl 0848.00007)