Patterns of change. Linguistic innovations in the development of classical mathematics
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 of changes ``which happen often and form the content of the everyday work of mathematicians'' (p. 225) with regard to ``concept-formation'', ``problem-solving'', and ``theory-building''. The final chapter of the book gives a comparison of the author's approach and the theories of \textit{T. S. Kuhn} [The Structure of Scientific Revolutions (University of Chicago Press) (1962)], Lakatos [Proofs and Refutations (1963-64; Zbl 0334.00022)] whose contribution is exclusively embedded into the context of re-formulations, and \textit{J. Piaget} [Psychogenesis and the History of Science (Columbia University Press, New York) (1983)]. A major difficulty of the book is that the author's perception of ``language'' (which seems to be rather general and comprehensive) is not explained in sufficient detail. At many places the limiting lines between language and the realm of cognition can not be seen. Explicit linguistic analyses of mathematical texts are quite rare. Nevertheless, Kvasz' approach provides an interesting tool for philosophers and historians which should be further discussed and applied.
scientific article; zbMATH DE number 5278964