The Essence of Numbers by Frédéric Patras

Author:Frédéric Patras
Language: eng
Format: epub
ISBN: 9783030567002
Publisher: Springer International Publishing

8.4 Frege and Complex Numbers

The judgment of Leibniz , who saw in imaginaries amphibious entities between being and non-being, seems definitively outdated after the discovery of their geometric interpretation. However, the philosophical stakes underlying the existence of generalized number domains are multiple. This is evidenced by the fact that, at the end of the nineteenth century, Frege and others, such as Husserl , will return to these questions and the tacit difficulties they raise.

The text (Frege [54, pp. 68–71]) on which the following remarks are based consists of notes for a review of the Gesammelte Abhandlungen zur Lehre vom Transfiniten [25], a collection of three texts published by Cantor from 1886 to 1888. The purpose of Cantor’s articles is to plead for the acceptance by the mathematical community of the idea of actual infinity. Faced with the objections that are traditionally addressed to the latter, Cantor develops a technical analysis which shows that these objections are based on two misunderstandings: either properties which are valid only for the finite are transposed to infinity, leading to logical paradoxes; or the metaphysical idea of infinity (inaccessible, uncontrollable) is confused with its possible mathematical realizations, so that two ontological levels (metaphysical infinity and mathematical infinity(ies)) are confused and, with them, two orders of reasons. Frege agrees with Cantor , but is more critical with respect to the thought processes at work in the mathematical component of Cantor’s work, with complex numbers playing an emblematic role in his analysis.

Frege’s criticism is based on an analysis of the methods for creating mathematical concepts. Like many posthumously edited Fregean texts, it is expressed in a pictorial and informal way. “Many mathematicians react to philosophical expressions [crediting them with magical properties]. I am thinking in particular here of the following: ‘define’ (Brahmā), ‘reflect’ (Vishnu), ‘abstract’ (Shiva). The names of the Indian gods are meant to indicate the kind of magical effects the expressions are supposed to have.” In Hinduism, life, like the universe, goes through three successive phases: creation, conservation, destruction. Brahmā personifies the absolute, the creation of the world, gods and beings, Vishnu the preservation, Shiva the destruction; Frege sees corresponding analogs of those three attributes in the mathematical world.

Behind Frege’s words, one must read his hostility to uncontrolled processes of conceptual creation. Mathematicians must take control of the theoretical instruments that allow them to work, and this control is most uncertain when new mathematical objects are born whose existence is not a priori guaranteed. Mathematical rigour must not be limited to checking the validity of reasoning about existing theoretical objects and tools, but must also look back to the logical origins of mathematical thought. This is the whole meaning of the Fregean work, at the origins of the modern conception of mathematics whose foundations are based on mathematical logic, the respect of certain rules making it possible, in principle, to guarantee the validity of definitions and to avoid the aporias with which mathematics was hitherto confronted.

Frege justifies his reservations about the Cantorian work on the basis of polemics about the imaginaries.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.