Mathematics in philosophy. (English) Zbl 0847.00007

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. 192-201 (1992).
The author discusses “the way or ways in which mathematics has been applied to, or used in, philosophy” (p. 192). He focusses on the influence of mathematics on the “two great opposing theories in epistemology”: rationalism and empiricism (ibid.). He sees the origins of the former in Plato’s philosophy which was closely related to the axiomatising, anti-empiristic ancient Greek mathematics. The author discovers Plato’s influence still in the work of the great 17th century rationalists, Descartes, Leibniz, and Spinoza, who were working on a “Universal Mathematics” while stressing the a priori character of the principles of knowledge. The author, however, argues from an empiristic view that has, as he claims, “to moderns […] all the plausibility” (p. 192). He sees a transition from rationalism to empiricism, “despite Kant’s desperate reargued action” (p. 197). He even claims that “mathematics is really empirical”, although indirectly, due to “the lack of immunity from what is after all empirically-based criticism” (ibid.).
The paper ends with some observations concerning the influence of probability theory on decision theory and inductive reasoning.
[Reviewer’s comment: Claiming the empirical character of mathematics the author seems to confuse theory and application. He overemphasises the dichotomy between rationalism and empiricism in epistemology and undervalues the role of deduction in philosophy which is central in both directions. In maintaining that empiricism “definitely superseded” rationalism (p. 193) the author does not reflect the fact that there are still influential rationalistic directions in epistemology like critical rationalism, constructivism, and new ontology, all of them closely related to mathematics].
For the entire collection see [Zbl 0839.00019].


00A30 Philosophy of mathematics
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)