Detlef Spalt

Detlef D. Spalt (* 1952 ) is a German math historian and philosopher of mathematics.

Spalt studied mathematics at the TU Darmstadt from 1970 to 1975 and received his doctorate there in 1981 under Detlef Laugwitz (From the myth of mathematical reason) . and then worked for a long time at the TU Darmstadt. It deals with the fundamentals of analysis , combining historical and philosophical considerations. He wrote two monographs on the basics of analysis. His theses are controversial and unorthodox (for example, in December 1992 his habilitation, Die Vernunft im Cauchy-Mythos, was rejected by the mathematics department of the TH Darmstadt, it was published as a book in 1996). He gave guest lectures at the universities of Salzburg (several times), Marburg and Frankfurt am Main (2016).

Like his academic teacher Laugwitz (who is one of the co-founders of nonstandardanalysis ), in a debate (from the 1980s) on the foundations of analysis with Augustin-Louis Cauchy, he initially took the view that Augustin-Louis Cauchy was a forerunner of nonstandardanalysis (advocated the idea Abraham Robinson in the 1960s ). It was particularly concerned with Cauchy's sum theorem in his Cours d'analysis from 1821 (see Uniform Convergence ), which is incorrect according to the usual interpretation of the prerequisite as point-wise convergence . If one assumes that Cauchy, who explicitly introduces infinite small quantities in his Cours d'Analyse, also considers infinitesimally adjacent nonstandard values ​​instead of the real number range in the sense of nonstandard analysis, this is equivalent to the assumption of uniform convergence in usual analysis.

In his dissertation from 1981, Spalt developed the thesis that Cauchy's sum theorem could be saved with Laugwitz's version of non-standard analysis. Laugwitz followed suit with several modifications. Later, however, Spalt came to the conclusion that Cauchy in his time (and especially his mathematicians-contemporaries) lacked the logical basis for the application of nonstandard analysis in the modern sense (whether according to Robinson or Laugwitz). In an essay from 2002 (Cauchy's Continuum) , he interpreted Cauchy's proof as using what Constantin Carathéodory later called continuous convergence (from which the uniform convergence follows). According to Spalt, his concept of function deviated radically from that of his predecessors in the 18th century, such as Euler, and described an extended object (the functional value) that depended on another extended object (the variable). According to Spalt, Cauchy thus fundamentally differed from the views of his contemporaries and remained misunderstood then and later. After Spalt, the Weierstrass School was a codification of a generally recognized version of analysis that replaced the previously existing diversity and made the interpretation of the history of mathematics difficult.

In a monograph published in 2015, he extends his close-to-source, conceptual-historical investigation of the history of analysis to its early years in the 17th century ( René Descartes , Isaac Newton , Gottfried Wilhelm Leibniz ) and its development in the 18th century (Johann I Bernoulli, Euler, Lagrange ), but also deals with the 19th century in detail (Bolzano, Cauchy, Riemann, Weierstrass, Cantor, etc.).

Fonts

• with Laugwitz: Another View of Cauchy's Theorem on convergent series of functions - An essay on the methodology of historiography in mathematics, TH Darmstadt: 1988
• Editor: Computing with the Infinite. Contributions to the development of a controversial subject, Springer 1990
  • in it by Spalt the introduction and: The infinite set of numbers in the change from Bolzano to set theory - or: Cantor as the father of bourgeois mathematics.
• Bolzano's doctrine of measurable numbers 1830-1989 , Archive for History of Exact Sciences, Volume 42, 1991, pp. 15-70
• The mathematical and philosophical foundations of the Weierstrasse number concept between Bolzano and Cantor , Archive for History of Exact Sciences, Volume 41, 1991, pp. 311-362
• Precision as a constant, terms in the change of mathematical thinking. On analysis in the 19th and 20th centuries, in: Czermak (Ed.), Philosophy of Mathematics. Files from the 15th International Wittgenstein Symposium Part 1, Vienna 1993, pp. 45–59
• Reason in the Cauchy myth. Harri Deutsch, Frankfurt am Main 1996
• Curt Schmieden's Non-standard Analysis - A Method of Dissolving the Standard Paradoxes of Analysis, Centaurus, Volume 43, 2001, pp. 137-175
• Cauchy's continuum. A historiographical approximation via Cauchy's sum theorem , Arch. Hist. Exact Sci., Vol. 56, 2002, pp. 285-338
• What functional terms did Leonhard Euler use? , Historia Mathematica, Volume 38, 2011, pp. 485-505.
• The introduction of the concept of value in analysis or the constitution of differential and integral calculus as a mathematical teaching , Archives Internationales d'Histoire des Sciences, Volume 65, 2015, pp. 117–193
• Analysis in transition and in conflict. A history of formation of their basic concepts, Verlag Karl Alber 2015
• A Brief History of Analysis. For mathematicians and philosophers , Springer, 2019.

Individual evidence

1. ↑ Detlef Spalt in the Mathematics Genealogy Project (English)
2. ↑ Information about his person in his book Die Analysis im Wandel und im Widerstreit
3. ^ Review by Thomas Sonar, Mathematische Semesterberichte, October 2017