Patterns of change. Linguistic innovations in the development of classical mathematics
type: Item
instance of: scholarly article
title: Patterns of change. Linguistic innovations in the development of classical mathematics
zbMATH Open document ID: 1153.01002
author: Ladislav Kvasz
published in: Science Networks. Historical Studies
publication date: 2008-05-26T00:00:00Z
review text: In this book the author expounds his ``linguistic approach'' for reconstructing the development of mathematical disciplines from a philosophical and historical point of view. Kvasz considers three patterns of change which he names ``re-codings'', ``relativizations'', and ``re-formulations''. Regarding the first pattern, ``re-codings'', the author generalizes an idea of \textit{G. Frege} [Begriffsschrift (L. Nebert, Halle a. S.) (1879; JFM 11.0048.02)]. ``Re-codings'' are changes which consist in including fundamentally new symbolic or iconic elements into the language. Kvasz describes the transition from elementary arithmetics (symbolic language) to geometry of the ancient Greeks (iconic), to algebra (symbolic), and so on. One of the main theses of the book is that each transition from one symbolic language to a subsequent one (as from elementary arithmetic to algebra) is via an intermediate iconic stage (as geometry in the case just mentioned). With ``relativization'' the author means changes regarding ``the ontological commitments of language, and the way'' in which ``the reference of the linguistic expressions is understood'' (p. 201). Taking up and developing further an idea of \textit{L. Wittgenstein} [Tractatus (1921; Zbl 0117.25106)], Kvasz describes ``relativization'' as change of the ``pictorial form'' (p. 108) of language, and he gives a detailed analysis of the historical changes within geometry (up to Tarski's axiomatic system of 1948 [\textit{A. Tarski}, A Decision Method for Elementary Algebra and Geometry (US Air Force Project Rand, R-109, The Rand Corporation, Santa Monica, California) (1948; Zbl 0035.00602 )]), and algebra (up to modern group and ring theories). He also shows similar conceptual stages in both lines of development. ``Re-formulations'' are changes which consist in the expression of mathematical ``definitions, theorems, proofs, or axioms'' (p. 225) in an equivalent, more specialized, or more general mode. Kvasz discusses this pattern o…
Mathematics Subject Classification ID: 01-02
Mathematics Subject Classification ID: 01A05
Mathematics Subject Classification ID: 00A30
Mathematics Subject Classification ID: 00A35
Mathematics Subject Classification ID: 97C30
Mathematics Subject Classification ID: 97D20
zbMATH DE Number: 5278964
zbMATH Keywords: linguistics
zbMATH Keywords: development of mathematical disciplines
zbMATH Keywords: Frege
zbMATH Keywords: Wittgenstein
zbMATH Keywords: Kuhn
zbMATH Keywords: Lakatos
zbMATH Keywords: Piaget
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5278964