Structural features in Ernst Schröder's work. Part I
DOI:
https://doi.org/10.12775/LLP.2011.022Keywords
structure, universal algebra, relationAbstract
In this paper articulated in two parts we propose a structural interpretation of Schröder’s work, pointing out his insistence on the priority of a whole in comparison with its parts. The examples are taken from the diverse areas in which Schröder was active, with a particular interest in his project of an absolute algebra. I am regretting for the bad quality of my English, hoping that notwithstanding the reader can grasp at least the fundamental tracts of my reasoning.References
Daniel S. Alexander, A History of Complex Dynamics, from Schröder to Fatou and Julia, Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig–Wiesbaden, 1994.
John T. Baldwin, “The birth of model theory: Löwenheim’s theory in the frame of the theory of relatives, by Calixto Badesa, Princeton University Press, NJ, 2004”, Bulletin of the American Mathematical Society 47 (2010), 1: 177–185, review.
Stefan Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Ph.D. thesis, University of Lvov, 1920, published in the Fundamenta Mathematicae, 3 (1922), pp. 133–181.
V.G. Boltjanskij and V.A. Efremovič, Anschauliche kombinatorische Topologie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1986.
Alan F. Beardon, Algebra and Geometry, Cambridge University Press, Cambridge, 2005.
Joseph Bak and Donald J. Newman, Complex Analysis, Springer Verlag, Berlin–Heidelberg–New York, 2010, third edition.
Otto Boedicker, Erweiterung der Gauss’schen Theorie der Verschlin-gungen mit Anwendungen in Elektrodynamic, Union Deutsche Verlagsgesellschaft, Stuttgart, 1876.
Davide Bondoni, “On mereology”, unpublished Thesis, in Italian, 2001.
Davide Bondoni, The Problem of the Observer in Physics, arXiv:1006.0187v1 (2010).
Davide Bondoni, Variations on a Theme by Schröder, LED Edizioni, Milan, 2010, in Italian.
Davide Bondoni, “Ernst Schröder, between mathematics and logic” (in Italian), Lettera Pristem Matematica 78 (2011): 48–53.
Davide Bondoni, “Math and poetry as exemplified in Ernst Schröder’s work”, Notices of the American Mathematical Society 58 (2011), 2: 232.
Davide Bondoni, “Quality versus quantity”, Notices of the American Mathematical Society 58 (2011), 6: 766.
Davide Bondoni, “Zbl 00168.00201, Schröder, Ernst, Der Operationskreis des Logikkalküls (German). Unveränderter Nachdruck der 1. Aufl. 1877. Stuttgart: B.-G. Teubner. VIII, 37 S. (1966)”, Zentralblatt Math (2011), review.
Marco Bramanti, Carlo D. Pagani, and Sandro Salsa, Mathematical Analysis 1, Zanichelli editore, Bologna, 2008, in Italian.
John P. D’Angelo, An Introduction to Complex Analysis and Geometry, American Mathematical Society, Providence, Rhode Island, 2010.
Richard Dedekind, Stetigkeiten und irrationale Zahlen, Richard Dedekind, Gesammelte mathematische Werke (Braunschweig) (Robert Fricke, Emmy Noether and Öystein Ore, eds.), Druck und Verlag von Friedr. Vieweg & Sohn Akt.-Ges., 1932, Dritter Band, pp. 315–334.
Richard Dedekind, Was sind und was sollen die Zahlen?, Richard Dedekind, Gesammelte mathematische Werke (Braunschweig) (Robert Fricke, Emmy Noether and Öystein Ore, eds.), Druck und Verlag von Friedr. Vieweg & Sohn Akt.-Ges., 1932, Dritter Band, pp. 335–391.
Richard Dedekind, Continuity and Irrational Numbers, From Kant to Hilbert, a Source Book in the Foundations of Mathematics (Oxford) (William Ewald, ed.), Oxford University Press, 1996, Volume II. English translation by W.-W. Beman, revised by William Ewald pp. 765–779.
Richard Dedekind, Was sind und was sollen die Zahlen?, From Kant to Hilbert, a Source Book in the Foundations of Mathematics (Oxford) (William Ewald, ed.), Oxford University Press, 1996, Volume II. English translation by W.-W. Beman, revised by William Ewald, pp. 790–833.
Jean-Pierre Dedieu, Points Fixes, Zéros et la Méthode de Newton, Springer Verlag, Berlin–Heidelberg–New York, 2006.
Paul A.-M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, Oxford–New York, 1958, fourth edition (revised). Dirac inserted a note in 1967 about his revision.
Duden, Deutsches Universalwörterbuch, Duden Verlag, Mannheim–Leipzig–Wien–Zürich, 2007, sixth revised and ampliated edition.
Roman Duda, “Die Lemberger Mathematikerschule”, Jahresbericht der Deutschen Mathematiker–Vereinigung 112 (2010), 1: 3–24.
Roman Duda, “The Lvov School of Mathematics”, Newsletter of the European Mathematical Society 78 (2010): 40–50.
Adolf Eckstein, Geistiges Deutschland, deutsche Zeitgenossen, Adolf Ecksteins Verlag, Berlin–Charlottenburg, 1901.
Avner Friedman, Foundations of Modern Analysis, Dover Publications Inc., New York, 1982.
Gianni Gilardi, Analysis, McGraw-Hill, Milan, 2011, in Italian. Second edition.
Georg Wilhelm Friedrich Hegel, System der Wissenschaft, Erster Teil: die Phänomenologie des Geistes, Joseph Anton Goebhardt, Bamberg und Würzburg, 1807.
Jean-Baptiste Hiriat-Urrut and Claude Lemaréchal, Fundamentals of Convex Analysis, Springer Verlag, Berlin–Heidelberg–New York, 2001.
Edmund Husserl, Logical Investigations, Routledge & Kegan Paul Ltd, London–New York, 1970, volume II. Translated by J.-N. Findlay from the second German edition of Logische Untersuchungen (D. Moran, ed.).
Edmund Husserl, Logische Untersuchungen, Felix Meiner Verlag, Hamburg, 2009.
A.-N. Kolmogorov and A.-P. Yushkevich (eds.), Mathematics of the 19th Century, Mathematical Logic, Algebra, Number Theory, Probability Theory, Birkhäuser Verlag, Basel–Boston–Berlin, 2001, second revised edition. Translated from Russian by A. Shenitzer, H. grant and O.-B. Sheinin.
Stanislav Leśniewski, Collected Works, vol. 1, Kluwer Academic Publishers, Dordrecht–Boston–London, 1992, edited by Stanisław J. Surma, Jan T. Srzednicki, D.-I. Barnett.
Leopold Löwenheim, “Über Möglichkeiten im Relativkalkül”, Mathematische Annalen 76 (1915): 228–251.
Leopold Löwenheim, “On possibilities in the calculus of relatives”, From Frege to Gödel, a Source Book in Mathematical Logic (Cambridge–Massachusetts) (Jean van Heijenoort, ed.), Harvard University Press, 1967, pp. 232–251.
Roger D. Maddux, “Relevance logic and the calculus of relations”, The Review of Symbolic Logic 3 (2010), 1: 41–70.
Pons, Großwörterbuch Englisch, Ernst Klett Sprachen GmbH, Stuttgart, 2008.
Rainer Wüst, Mathematics for Physicians and Mathematicians, Wiley-VHC Verlag GmbH & Co. KGaA, Weinheim, 2009, Volume 1: Real Analysis and Linear Algebra. Third edition. In German.
Rainer Wüst, Mathematics for Physicians and Mathematicians, Wiley-VHC Verlag GmbH & Co. KGaA, Weinheim, 2009, Volume 2: Analysis in Multidimensional Spaces and Fundamentals in Domains. Third edition. In German.
Simeon Reich and David Shoikhet, Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces, Imperial College Press, London, 2005.
Simeon Reich, David Shoikhet, Mark Elin, and Victor Goryainov, “Fractional iteration and functional equations for functions analytic in the unit disk”, Computational Methods and Function Theory 2 (2002), 2: 353–366.
Simeon Reich, David Shoikhet, and Victor Khatskevich, “Schröder’s functional equation and the Koenigs embedding property”, Nonlinear Analysis 47 (2001): 3977–3988.
Simeon Reich, David Shoikhet, and Victor Khatskevich, “Abel–Schröder equations for linear fractional mappings and the Koenigs embedding problem”, Acta Universitatis Szegediensis – Acta Scientiarum Mathematicarum 69 (2003): 67–98.
Ernst Schröder, ‘Ueber iterirte Functionen”, Mathematische Annalen 3 (1870), 2: 296–322.
Ernst Schröder, “Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen”, Mathematische Annalen 2 (1870), 2: 317–365.
Ernst Schröder, Lehrbuch der Arithmetik und Algebra für Lehrer und Studierende, Teubner, Leipzig, 1873.
Ernst Schröder, Abriss der Arithmetik und Algebra für Schüler an Gymnasien und Realschulen, Teubner, Leipzig, 1874.
Ernst Schröder, Über die formalen Elemente der absoluten Algebra, Schweizerbart, Stuttgart, 1874.
Ernst Schröder, “Ueber v. Staudt’s Rechnung mit Wurfeln und verwandte Processe”, Mathematische Annalen 10 (1876): 289–317.
Ernst Schröder, “Note über den Operationskreis des Logikcalculs”, Mathematische Annalen 12 (1877): 481–484.
Ernst Schröder, Ueber v. Staudt’s Rechnung mit Würfen und verwandte Processe, Repertorium der literarischen Arbeiten aus dem Gebiete der reinen und angewandten Mathematik, II. Band (Leipzig) (Gustav Zeuner and Leo Koenisberg, eds.), Teubner, 1879, pp. 81–85.
Ernst Schröder, Exposition of a Logical Principle, as disclosed by the Algebra of Logic, but overlooked by the Ancient Logicians, Report of the Fifty-Third Meeting of the British Association for the Advancement of Science, held at Southport in September 1883 (London) (John Murray, ed.), Spottiswoode and Co., 1884, p. 412.
Ernst Schröder, “Eine Berechtigung zum ersten Bande meiner Algebra der Logik”, Mathematische Annalen 36 (1890): 602.
Ernst Schröder, Über das Zeichen, Braun, Karlsruhe, 1890.
Ernst Schröder, Der Operationskreis des Logikkalkuls, Wissenschaftliche Buchgesellschaft, Darmstadt, 1966.
Ernst Schröder, Vorlesungen über die Algebra der Logik, vol. 3, Chelsea Publishing Company, Bronx–New York, 1966.
Yu. A. Shashkin, Fixed Points, American Mathematical Society, Providence, 1991, translated from the Russian by Viktor Minachin.
Georgi E. Shilov, Elementary Functional Analysis, Dover Publications Inc., New York, 1996, revised English edition. Translated and edited by Richard A. Silverman.
Adolf Trendelenburg, Historische Beiträge zur Philosophie, vol. 3, Verlag von G. Bethge, Berlin, 1867.
N. Ya. Vilenkin, Successive Approximation, Pergamon Press, New York, 1964, translated and adapted from the Russian by Micheal B.-P. Slater and Joan W. Teller.
John von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer Verlag, Berlin–Heidelberg–New York, 1996, second edition with a preface by Rudolf Haag.
Georg Karl Christian v. Staudt, Geometrie der Lage, Verlag von Bauer und Raspe, Nürnberg, 1847.
Georg Karl Christian v. Staudt, Beiträge zur Geometrie der Lage, Verlag der Fr. Korn’schen Buchhandlung, Nürnberg, 1856, Erstes Heft.
Georg Karl Christian v. Staudt, Beiträge zur Geometrie der Lage, Verlag der Fr. Korn’schen Buchhandlung, Nürnberg, 1856, Zweites Heft.
John A. Wheeler and Wojciech H. Zurek (eds.), Quantum Theory and Measurement, Princeton University Press, Princeton, New Jersey, 1983.
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