Abstract
After briefly showing that, contrary to the received view in analytic circles, Frege’s influence on the evolution of Husserl’s views on logic and mathematics, as well as on the distinction between sense and referent are either insignificant or, as in the last case, totally inexistent, and that Husserl’s account of Leibniz and Bolzano’s influence in Logische Untersuchungen is correct, we turn to Riemann, whose influence is certainly non-negligible. Husserl’s conception of mathematics as a theory of manifolds (or structures) is a generalization of Riemann’s notion of manifold – in fact, a sort of bridge between Riemann and the Bourbaki group. Moreover, Husserl’s conception – since 1892 – of physical geometry as empirical is also strongly influenced by Riemann
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Notes
- 1.
Philosophie der Arithmetik 1891, Hua XII, M. Nijhoff, Den Haag 1970. See Centrone’s paper ‘Husserl on the ‘totality of all conceivable operations’, History and Philosophy of Logic 27 (3), 2006, pp. 211–228.
- 2.
See my ‘Husserl’s Philosophy of Mathematics: its Origin and Relevance’ 2006, reprinted in Against the Current, Ontos Verlag, Frankfurt 2012, pp. 145–181.
- 3.
Logische Untersuchungen (2 vols.) 1900–1901, Hua XVIII & XIX, M. Nijhoff, Den Haag 1975 & 1984.
- 4.
Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie I 1913, Hua III, M. Nijhoff, Den Haag 1950, revised edition 1976.
- 5.
Formale und transzendentale Logik 1929, Hua XVII, M. Nijhoff, Den Haag 1974.
- 6.
Erfahrung und Urteil 1939, sixth edition, F. Meiner, Hamburg 1985.
- 7.
Logische Untersuchungen I, Chapters I–VIII.
- 8.
Ibid. Chapter XI. Compare with Carnap’s Die logische Syntax der Sprache 1934, revised English edition 1937, where the distinction between the two sorts of rules is introduced in p. 4 and then used throughout the whole book.
- 9.
Logische Untersuchungen I, Chapter XI. For Carnap, see preceding footnote.
- 10.
Ibid.
- 11.
Ibid.
- 12.
Logische Untersuchungen II, U. I.
- 13.
Ibid., U. II.
- 14.
Ibid., U. III.
- 15.
Ibid., U. IV.
- 16.
Ibid., U. V.
- 17.
Ibid., U. VI.
- 18.
‘Rezension von E. G. Husserl: Philosophie der Arithmetik I, 1894, reprinted in G. Frege, Kleine Schriften 1967, revised edition, Georg Olms 1990, pp. 179–192.
- 19.
See Evert W. Beth, The Foundations of Mathematics, North Holland, Amsterdam 1965, p. 353.
- 20.
See, e.g. Dagfinn Føllesdal’s Master thesis Husserl und Frege, ein Beitrag zur Beleuchtung der phänomenologischen Philosophie 1964, translation in L. Haaparanta (ed.), Mind, Meaning and Mathematics, Kluwer, Dordrecht 1994, pp. 3–47, as well as his ‘Husserl’s Concept of Noema’, Journal of Philosophy 66 (20), 1969, pp. 680–697.
- 21.
Studien zur Arithmetik und Geometrie, Hua XXII, M. Nijhoff, Den Haag 1983
- 22.
‘Funktion und Begriff’ 1891, reprinted in Gottlob Frege, Kleine Schriften, pp. 125–142. The distinction is the central theme of his famous ‘Über Sinn und Bedeutung’ 1892, reprinted also in Kleine Schriften, pp. 143–162. See also Frege’s paper written in the early 1890s, though only posthumously published, ‘Ausführungen über Sinn und Bedeutung’ in Nachgelassene Schriften, F. Meiner, Hamburg 1969, revised edition 1983, pp. 128–136.
- 23.
‘Besprechung von E. Schröders Vorlesungen über die Algebra der Logik I’ 1891, reprinted in Aufsätze und Rezensionen (1890–1910), Hua XXII, pp. 3–43.
- 24.
‘Zur Logik der Zeichen’, written in 1890, but published for the first time as Appendix B.I to Philosophie der Arithmetik, Hua XII, pp. 340–373.
- 25.
See footnote 24.
- 26.
Wissenschatlicher Briefwechsel, F. Meiner, Hamburg 1976, pp. 94–98.
- 27.
‘Ausführungen über Sinn und Bedeutung’, p. 135. See also p. 134.
- 28.
On Husserl’s philosophy of mathematics see Logische Untersuchungen I, Chapter XI, §§69–70, Formale und transzendentale Logik, 1929, Husserliana XVII, 1974, especially Chapters 2 and 3, and Einleitung in die Logik und Erkenntnistheorie, Husserliana XXIV, 1984, especially Chapter 2.
- 29.
See Logische Untersuchungen I, Chapter X, §§59–61 and Appendix.
- 30.
Introduction to the Logical Investigations, M. Nijhoff, Den Haag 1975, pp. 36–38
- 31.
‘Über die Hypothesen, welche der Geometrie zugrunde liegen’ 1867, third edition, Berlin 1923, reprint, Chelsea, New York 1973.
- 32.
See Logische Untersuchungen I, p. 252, and Formale und transzendentale Logik, p. 97 for references to Riemann in this context.
- 33.
Studien zur Arithmetik und Geometrie, Husserliana XXI, M. Nijhoff, Den Haag 1983.
- 34.
Ibid., pp. 312–347.
- 35.
Die Grundlagen der Arithmetik, §14.
- 36.
‘Über Euklidische Geometrie’, in Nachgelassene Schriften, pp. 182–184.
- 37.
Briefwechsel I, 1994, p. 10.
- 38.
Briefwechsel V, pp. 62, 63 and 83–84, respectively.
- 39.
In more modern parlance, one would say that the probability of space been Euclidean is less than 1, since the probability of any hypothesis or theory ranges from 0 to 1. See also the acknowledgement of the Riemannian restriction of local Euclidicity.
- 40.
The above remark does not include Føllesdal, who certainly has tried to make Husserl known in analytic circles. In fact, it should be pointed out that when he wrote his MA thesis and the paper referred to above, neither Husserl’s ‘Zur Logik der Zeichen’ nor Frege’s Briefwechsel had been published.
- 41.
In this more general and sounder sense, Hume exerted an important influence on Kant, and Quine has exerted an influence on many recent rigorous philosophers, even those like the present author, who disagree with Quine in almost any concrete philosophical issue. But that is precisely the way in which philosophy is made.
- 42.
‘Der Gedanke’ 1918, reprinted in Kleine Schriften, pp. 342–362.
- 43.
Logische Untersuchungen II, U. I, Chapter 3, §§24–26.
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Rosado Haddock, G.E. (2017). Husserl and Riemann. In: Centrone, S. (eds) Essays on Husserl's Logic and Philosophy of Mathematics. Synthese Library, vol 384. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1132-4_10
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