Structure-similarity as a cornerstone of the philosophy of mathematics. (English) Zbl 0848.00005

Grattan-Guinness, Ivor

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. 91-111 (1992).

The concept of “structure similarity” is used to reconsider several parts of the philosophy of mathematics. The author distinguishes several types of such similarities: “The chief concern is with the content of a mathematical theory, especially the way in which its structure relates to that of other mathematical theories (which I call ‘intramathematical similarity’, to that of the scientific theory to which it is of hire (‘scientific similarity’), and to empirical interpretations of that scientific theory in reality (‘ontological similarity’)” (p. 91). The paper discusses the way how a mathematical theory can have meaning, not what it may mean (p. 93). The power of the concept is shown in case studies from the history of mathematics concerning intramathematical similarities between algebra and geometry, presentations and icons, modelling continua, linearity and nonlinearity, and finally “mathematical psychology and the algebra of thought”. The new terminology is presented as a tool for creating a general philosophy of mathematics, which should especially serve as an alternative to those philosophies of mathematics inspired by the axiomatic method. Axiomatic systems miss an intuitive character, their structure-similarities are rarely evident (p. 103).
The general philosophy of mathematics should be able to accomodate the two traditions of the “philosophy of mathematics” “which starts out from logic, set theories and the axiomatisation of theories, but rarely gets much further” (p. 104), and the “opposite absurdity practised by mathematicians, which respects (much of the) range of their subject but tends to adopt the metaphilosophy of ignoring the logico-philosophy of the subject” (ibid.). In particular the philosophy of forms, reasonings and structures could be helpful in this programme: “[…] in this philosophy mathematics is seen as a group of problems, topics and branches in which forms and reasonings are chosen and developed in a variety of structures exhibiting differing levels of content. This, briefly, is how this philosophy of mathematics means, and also how this philosophy of mathematics means” (p. 108).
For the entire collection see [Zbl 0839.00019].

Reviewer: V.Peckhaus (Erlangen)

MSC:

00A30

Philosophy of mathematics

Keywords:

structure similarity