Structure-similarity: Between mathematics and philosophy. (English) Zbl 0849.00017

Grattan-Guinness, Ivor

Czermak, Johannes (ed.), Philosophy of mathematics. Proceedings of the 15th international Wittgenstein-Symposium, August 16-23, 1992, Kirchberg am Wechsel, Austria. Part I. Wien: Hölder-Pichler-Tempsky. Schriftenreihe der Wittgenstein-Gesellschaft. 20/I, 317-333 (1993).

The property of structure-similarity is presented as mediating between the two distinct philosophies of mathematics: the mathematicians’ philosophy of mathematics, dealing with philosophical questions pertaining to a branch of mathematics, and the philosopher’s philosophy of mathematics, centered around some kinds of logics and set-theories, presenting reductionist theories, mathematics itself, however, is missing. The term “structure-similarity” mediates between these two positions in the sense that it is a feature of mathematics of all kinds which can be applied to concerns in which philosophers are interested. The notion chiefly refers to the contents of mathematical theories focussing on the way their structure relates a) to that of other mathematical theories (“intra-mathematical similarity”), b) to that of a scientific theory to which it is applied (“scientific similarity”), c) to the empirical interpretation of that scientific theory in reality (“ontological similarity”). The author clarifies these types by presenting examples from the following fields: the relation between linearity and nonlinearity, the origin of applications of Fourier’s trigonometric series, differential and integral calculus, several types of additions stressing above all Georges Boole’s “mathematical psychology”, various relation types between geometries and algebras, mathematical notations. In concluding the author applies the concept to typical aspects of the philosopher’s philosophy of mathematics: the limitations of axiomatizations, the philosophy of forms, and the philosophy of reasoning.
For the entire collection see [Zbl 0836.00022].

Reviewer: V.Peckhaus (Erlangen)

MSC:

00A30	Philosophy of mathematics
03B05	Classical propositional logic

Keywords:

logicism; incompleteness; structure similarity