Abstract
Husserl and Cantor were colleagues and close friends during the last 14 years of the nineteenth century, when Cantor was at the height of his creative powers and Husserl in the throes of an intellectual struggle during which he drew apart from people and writings to whom he owed most of his intellectual training and drew closer to the ideas of thinkers whose writings he had not been able to evaluate properly and had consulted too little. I study ways in which Husserl and Cantor might be said to have been alike, while pointing to dissimilarities between them. In particular, I discuss how their ideas overlapped and crisscrossed with regard to mathematics and philosophy, Platonic idealism, abstraction, empiricism, psychologism, actual consciousness and pure logic, Frege’s reviews of their works, metaphysics and mysticism, sets, arithmetization, strange and imaginary numbers and manifolds. I conclude that Cantor was among those of his mentors from whose ideas Husserl drew away and Lotze and Bolzano were among those to whose ideas he drew closer.
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Notes
- 1.
Über den Begriff der Zahl is published in Husserl, Philosophie der Arithmetik. Mit ergänzenden Texten (1890–1901), Husserliana vol. XII, henceforth cited as PdA, 289–338, and in English as “On the Concept of Number: Psychological Analyses” in Philosophy of Arithmetic, Psychological and Logical Investigations with Supplementary Texts from 1887–1901, henceforth cited as PoA, 305–357.
- 2.
Hereto cp. Hill & da Silva 2013, 368.
- 3.
Husserl’s time in Halle is studied in Gerlach and Sepp 1994.
- 4.
PdA 1–283; PoA 1–299.
- 5.
Many of the articles and reviews were published in Aufsätze und Rezensionen (1890–1910), Husserliana vol. XXII. They are translated and published Husserl’s Early Writings in the Philosophy of Logic and Mathematics, vol. V of Husserl’s Collected Works, which also includes translations of short works published in PdA and Einleitung in die Logik und Erkenntnistheorie, Vorlesungen 1906/07, Husserliana vol. XXIV.
- 6.
Husserl, Logische Untersuchungen, Bd I: Prolegomena zur reinen Logik, Max Niemeyer, Halle, 1900 (21913), henceforth cited as PRe. Bd II: Untersuchungen zur Phänomenologie und Theorie der Erkenntnis, Max Niemeyer, Halle a.d.S., 1901 (21913–1921). English translation: Logical Investigations (J. N. Findlay, translator), Routledge and Kegan Paul, New York 1970, henceforth cited as LI.
- 7.
See Cantor letter’s 240, 264, 291 as translated and published in Hill & da Silva 2013, specifically pages 367–369, 374–375, 377.
- 8.
See M. Husserl 1988, §E.
- 9.
Husserl’s training in mathematics is discussed at length in my essay “On Husserl’s Mathematical Apprenticeship and Philosophy of Mathematics” in Hill 2002a.
- 10.
See Hill 1998.
- 11.
See PdA 294; PoA 310.
- 12.
Quotes after Hallett 1984, 7.
- 13.
Dauben 1979, 120.
- 14.
See Dauben 1979, Chapter 6, 120–148, “Cantor’s Philosophy of the Infinite” for an in depth discussion.
- 15.
Cantor 1887/8.
- 16.
See Dauben 1979, 139, 336 n. 29.
- 17.
- 18.
See page 118 of his letter of March 3, 1883 to Gösta Mittag-Leffler as published in Cantor 1991, 118.
- 19.
Op. cit., 244.
- 20.
Cantor 1883, 181, 206 n. 6.
- 21.
Op. cit., 204 n. 1.
- 22.
See Cantor’s letter of September 21, 1895 to Giuseppe Peano as published in Cantor 1991, 365.
- 23.
- 24.
Cantor 1887/8, 406, 420.
- 25.
Op. cit., 418.
- 26.
- 27.
Cantor 1887/8, 379, 387, 411, 418 n. 1.
- 28.
- 29.
- 30.
Cantor 1885, 440.
- 31.
“On the Concept of Number: Psychological Analyses” as published in PdA 303–304; PoA 320–321.
- 32.
Husserl 1906/07 §§11, 13b, 23.
- 33.
Cantor 1887/8, 418 n. 1; PdA 116 n.; PoA 121 n. 3.
- 34.
PdA 294, 337; PoA 310, 354–355.
- 35.
PdA 86; PoA 89–90. Hereto cp. Hill 1999.
- 36.
PdA 116 n. (see above); PoA 121 n. 3
- 37.
- 38.
Ex. Cantor 1883, 168, 170, 201.
- 39.
Cantor 1887/8, 418 n. 1.
- 40.
Cantor 1883, 207 n. 6.
- 41.
- 42.
PdA 295, PoA 311.
- 43.
Husserl 1913, 33.
- 44.
Husserl 1913, 16–17; 34–35.
- 45.
Husserl 1919, 344–345.
- 46.
PRe 185.
- 47.
Husserl 1919, 345.
- 48.
Husserl 1913, 20.
- 49.
Op. cit., 29.
- 50.
Husserl 1905, 37.
- 51.
Husserl 1913, 42.
- 52.
- 53.
- 54.
Husserl 1913, 21–22.
- 55.
See Hallett 1984, 16–18, 34–35, 121, 128–133, 146–158.
- 56.
Cantor 1887/8, 416.
- 57.
Op. cit., 418 n. 1.
- 58.
- 59.
Cantor 1883, 168, 170, 201.
- 60.
Frege 1884, §86.
- 61.
- 62.
Husserl 1913, 35, 222.
- 63.
- 64.
- 65.
Husserl 1902/03, 12–13.
- 66.
Husserl 1906/07, §20.
- 67.
Husserl 1898/99, 233.
- 68.
- 69.
Husserl 1906/07, §21.
- 70.
Husserl 1898/99, 225–255.
- 71.
Husserl 1905, 39.
- 72.
- 73.
See Cantor’s letter to Charles Hermite of January 24, 1894 as published in Cantor 1991, 350.
- 74.
Cited in Hallett 1984, 10.
- 75.
Loc. cit.
- 76.
Dauben 1979, 125.
- 77.
Cited in Hallett 1984, 149.
- 78.
See Cantor’s letters of September 7, 1890 to Giuseppe Veronese, of November 3, 1886 to Axel Harnack, and of March 26, 1887 to Aloys Schmid as published in Cantor 1991, 326, 267, 282 respectively.
- 79.
Cited in Hallett 1984, 36.
- 80.
Cited in Dauben 1979, 298.
- 81.
Dauben 1979, 290–291.
- 82.
PRe 242.
- 83.
- 84.
- 85.
Hill 1994.
- 86.
Frege 1894, 196.
- 87.
Op. cit., 197.
- 88.
Op. cit., 197–198.
- 89.
Op. cit., 204.
- 90.
Op. cit., 205; Hill 1997c.
- 91.
Frege 1894, 205.
- 92.
Frege 1984, 180, 181.
- 93.
Frege 1979, 69.
- 94.
Op. cit., 70.
- 95.
Op. cit., 71.
- 96.
Frege 1894, 201–202.
- 97.
Op. cit., 195.
- 98.
Op. cit., 202.
- 99.
Cantor 1885, 728–729.
- 100.
Frege 1884, §96 note.
- 101.
- 102.
Frege 1894 (‘Review of Dr. E. Husserl’s Philosophy of Arithmetic’), 207; PdA 219; PoA 231.
- 103.
- 104.
Husserl 1929, §27a; also §24 and note.
- 105.
PdA 14–15; PoA 15–16; also PdA 297; PoA 313–314.
- 106.
PdA 11; PoA 12.
- 107.
PdA 14–15; PoA 15–16; PdA 298; PoA 314–315.
- 108.
Husserl 1913, 35.
- 109.
- 110.
LI 41–43.
- 111.
- 112.
Op. cit., 52–91, 421–441.
- 113.
Ex. op. cit., 92–114, 115–120, 121–130, 135–138, 443–451.
- 114.
Op. cit., 109, 123.
- 115.
PdA 218–222; PoA 230–234.
- 116.
Dauben 1979, 240–270.
- 117.
- 118.
Hilbert 1925, 375.
- 119.
Frege 1980, 51.
- 120.
- 121.
- 122.
Dauben 1979, 241.
- 123.
Op. cit., 165–168, 242.
- 124.
Husserl 1906/07, 18d.
- 125.
Husserl Ms. A 1 35.
- 126.
Russell 1903, §489.
- 127.
Hill 1997b.
- 128.
Husserl 1929, §§ 23, 26.
- 129.
Cantor 1883, 165–166.
- 130.
Schuhmann 1977, 7.
- 131.
Loc. cit.
- 132.
- 133.
PdA 294–95; PoA 310–11.
- 134.
- 135.
PdA 294; PoA 310.
- 136.
PdA 12; PoA 13.
- 137.
PdA 7; PoA 7.
- 138.
- 139.
LI 41–42; Husserl 1913, 35.
- 140.
- 141.
PdA 221; PoA 233.
- 142.
Grattan-Guinness 1971, 369.
- 143.
Dauben, 158–59.
- 144.
PRe §70.
- 145.
- 146.
Cantor 1883, 182.
- 147.
- 148.
- 149.
LI 43; Husserl 1913, 16–17.
- 150.
- 151.
PdA 294; PoA 310.
- 152.
Frege 1984, 180; 181.
- 153.
PRe 242.
- 154.
LI 41–42; Husserl 1913, 35.
- 155.
- 156.
Cp. Hill 1998.
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Hill, C.O. (2017). Husserl and Cantor. 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_8
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