Cantor on Infinitesimals. Historical and Modern Perspective

Piotr Błaszczyk; Marlena Fila

Bulletin of the Section of Logic (2020)

  • Volume: 49, Issue: 2
  • ISSN: 0138-0680

Abstract

top
In his 1887's Mitteilungen zur Lehre von Transfiniten, Cantor seeks to prove inconsistency of infinitesimals. We provide a detailed analysis of his argument from both historical and mathematical perspective. We show that while his historical analysis are questionable, the mathematical part of the argument is false.

How to cite

top

Piotr Błaszczyk, and Marlena Fila. "Cantor on Infinitesimals. Historical and Modern Perspective." Bulletin of the Section of Logic 49.2 (2020): null. <http://eudml.org/doc/295595>.

@article{PiotrBłaszczyk2020,
abstract = {In his 1887's Mitteilungen zur Lehre von Transfiniten, Cantor seeks to prove inconsistency of infinitesimals. We provide a detailed analysis of his argument from both historical and mathematical perspective. We show that while his historical analysis are questionable, the mathematical part of the argument is false.},
author = {Piotr Błaszczyk, Marlena Fila},
journal = {Bulletin of the Section of Logic},
keywords = {infinitesimals; infinite numbers; real numbers; hyperreals; ordinal numbers; Conway numbers},
language = {eng},
number = {2},
pages = {null},
title = {Cantor on Infinitesimals. Historical and Modern Perspective},
url = {http://eudml.org/doc/295595},
volume = {49},
year = {2020},
}

TY - JOUR
AU - Piotr Błaszczyk
AU - Marlena Fila
TI - Cantor on Infinitesimals. Historical and Modern Perspective
JO - Bulletin of the Section of Logic
PY - 2020
VL - 49
IS - 2
SP - null
AB - In his 1887's Mitteilungen zur Lehre von Transfiniten, Cantor seeks to prove inconsistency of infinitesimals. We provide a detailed analysis of his argument from both historical and mathematical perspective. We show that while his historical analysis are questionable, the mathematical part of the argument is false.
LA - eng
KW - infinitesimals; infinite numbers; real numbers; hyperreals; ordinal numbers; Conway numbers
UR - http://eudml.org/doc/295595
ER -

References

top
  1. [1] P. Błaszczyk, Nota o rozprawie Otto Höldera "Die Axiome der Quantität und die Lehre vom Mass", Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia, vol. 5 (2013), pp. 129–144. 
  2. [2] P. Błaszczyk, A purely algebraic proof of the fundamental theorem of algebra. Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia, vol. 8 (2016), pp. 5–21. 
  3. [3] P. Błaszczyk, J. Major, Calculus without the concept of limit. Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia, vol. 4, (2014) pp. 19–38. 
  4. [4] P. Błaszczyk, K. Mrówka, Euklides i Arystoteles o ciągłości. Część I. Euklides. Filozofia Nauki, vol. 21 (2013), pp. 91–115. 
  5. [5] B. Bolzano, Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwei Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung Liege, Gottlieb Hasse, Prague (1817). 
  6. [6] B. Bolzano, Paradoxien des Undendlichen, Reclam, Leipzig (1851). 
  7. [7] B. Bolzano, Theorie der reellen Zahlen in Bolzanos handschriftlichen Nachlasse, Hrsg. K. Rychlik, Tschechoslovakien Akademie der Wissenschaften, Prague (1962). 
  8. [8] B. Bolzano, Reine Zahlenlehre, [in:] J. Berg (Hrsg.), Bernard Bolzano Gesamtausgabe, Bd. 2A8, Frommann-Holzboog, Stuttgart (1976). 
  9. [9] K. Borsuk, W. Szmielew, Foundations of Geometry, NHPC, Amsterdam (1960). 
  10. [10] N. Bourbaki, Théorie de la mesure et de l'intégration. Introduction, Université Henri Poincaré, Nancy (1947). 
  11. [11] G. Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Springer, Berlin (1932). 
  12. [12] G. Cantor, Über die einfachen Zahlensysteme, Zeitschrift für Mathematik und Physik, Neunzehnter Jahrgang, vol. 14 (1869), pp. 121–128, [in:] [11, pp. 35–42]. 
  13. [13] G. Cantor, Zwei Sätze über eine gewisse Zerlegung der Zahlen in unendliche Produkte, Zeitschrift für Mathematik und Physik, Neunzehnter Jahrgang, vol. 14 (1869), pp. 152–158, [in:] [11, pp. 43–50]. 
  14. [14] G. Cantor, Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen (1872), [in:] [11, pp. 92–102]. 
  15. [15] G. Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (1883), [in:] [11, pp. 165–208]. 
  16. [16] G. Cantor, Mitteilungen zur Lehre von Transfiniten (1887), [in:] [11, pp. 378–439]. 
  17. [17] G. Cantor, Beiträge zur Begründung der transfiniten Mengenlehre (1895), [in:] [11, pp. 285–311]. 
  18. [18] G. Cantor, Beiträge zur Begründung der transfiniten Mengenlehre (1897), [in:] [11, pp. 312–351]. 
  19. [19] L. C. Cohen, G. Ehrlich, The Structure of the Real Number System, Van Nostrand, Toronto (1963). 
  20. [20] J. H. Conway, On numbers and games, Academic Press, London (1976). 
  21. [21] J. H. Conway, On Numbers and Games, AK Peters, Natick (2001). 
  22. [22] J. W. Dauben, Georg Cantor. His Mathematics and Philosophy of Infinite, Princeton University Press, Princeton (1990). 
  23. [23] R. Dedekind, Stetigkeit und irrationale Zahlen, Friedrich Vieweg & Sohn, Braunschweig (1872). 
  24. [24] R. Descartes, La Géométrie, Leiden (1637). 
  25. [25] P. Du Bois-Reymond, Die allgemeine Functiontheorie, Lauppschen, Tübingen (1882). 
  26. [26] P. Du Bois-Reymond, Théorie Générale des Fonctions, Nice (1887). 
  27. [27] P. Ehrlich, The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes, Archive for the History of Exact Science, vol. 60 (2006), pp. 1–121. 
  28. [28] P. Ehrlich, The Absolute Arithmetic Continuum and the Unification of all Numbers Great and Small, Bulletin of Symbolic Logic, vol. 18(1) (2012), pp. 1–45. 
  29. [29] L. Euler, Introductio in Analysin Infinitorum, Lausanae (1748). 
  30. [30] L. Euler, Institutiones Calculi Differentialis, Saint Petersburg (1755). 
  31. [31] W. Ewald, From Kant to Hilbert, vol. II, Clarendon Press, Oxford (1996). 
  32. [32] H. Gonshor, An Introduction to the Theory of Surreal Numbers, Cambridge University Press, Cambridge (1986). 
  33. [33] H. Grassmann, Lehrbuch der Arithmetik, Enslin, Berlin (1861). 
  34. [34] R. Harshorne, Geometry: Euclid and Beyond, Spinger, New York (2000). 
  35. [35] H. Hankel, Zur Geschichte der Mathematik in Altertum und Mittelalter, Teubner, Leipzig (1874). 
  36. [36] T. L. Heath, The works of Archimedes, Dover, New York (1912). 
  37. [37] J. L. Heiberg, Archimedis Opera Omnia cum Commentariis Eutocii, vol. I-II, Teubner, Leipzig (1880–1881). 
  38. [38] J. L. Heiberg, Euclidis Elementa, Teubneri, Lipsiae (1883–1888). 
  39. [39] G. Hessenberg, Grunbegriffe der Mengenlehre. Vandenhoeck und Ruprecht, Göttingen (1906). 
  40. [40] D. Hilbert, Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals in GÜttingen, Teubner, Leipzig (1899). 
  41. [41] D. Hilbert, Über den Zahlbegriff, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 8 (1900), pp. 180–184. 
  42. [42] D. Hilbert, S. Cohn-Vossen, Anschauliche Geometrie, Springer, Berlin (1932). 
  43. [43] O. Hölder, Die Axiome der Quantität und die Lehre vom Mass, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Classe, vol. 53 (1901), pp. 1–63. 
  44. [44] O. Hölder, The Axioms of Quantity and the Theory of Measurement, Translated by J. Mitchell, Journal of Mathematical Psychology, vol. 40 (1996), pp. 235–252. 
  45. [45] H. Meschkowski, W. Nilson Hrsg., Georg Cantor – Briefe, Springer, Berlin (1991). 
  46. [46] K. Shamseddine, M. Berz, Analysis on the Levi-Civita field, a brief overview, [in:] Contemporary Mathematics 508, American Mathematical Society (2010), pp. 215–237. 
  47. [47] O. Stolz, Vorlesungen über Allgemeine Arithmetik, Teubner, Leipzig (1885). 
  48. [48] O. Stolz, J. A. Gmeiner, Theoretische Arithmetik, Teubner, Leipzig (1902). 
  49. [49] H. Weber, Lehrbuch der Algebra, Vieweg, Braunschweig (1895). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.