Abstract
The article explores the relationship between the philosopher and historian of mathematics Jacob Klein’s account of the transformation of the concept of number coincident with the invention of algebra and Husserl’s early investigations of the origin of the concept of number and his late account of the Galilean impulse to mathematize nature. Klein’s research is shown to present the historical context for Husserl’s twin failures in the Philosophy of Arithmetic, (1) to provide a psychological foundation for the proper concept of number and (2) to show how this concept of number functions as the mathematical foundation of universal (symbolic) arithmetic. The argument is advanced that one significant result of bringing together Klein’s and Husserl’s thought on these issues is the need to fine-tune Husserl’s Crisis project of desedimenting the mathematization of nature. Specifically, Klein’s research shows that “a ‘sedimented’ understanding of numbers” “is superposed upon the first stratum of ‘sedimented’ geometrical ‘evidences’” uncovered by Husserl’s fragmentary analyses of geometry in the Crisis. In addition, then, to the task of “the intentional-historical reactivation of the origin of geometry” recognized by Husserl as intrinsic to the reactivation of the origin of mathematical physics, Klein discloses a second task, that of “the reactivation” of the “complicated network of sedimented significances” that “underlies the ‘arithmetical’ understanding of geometry.”
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Notes
- 1.
Jacob Klein was born (1899) in Russia (in Libau, which was then in Courland and a part of the Russian Empire and which is now a part of Latvia), educated there, Belgium, and Germany (Ph.D. 1922, Marburg University). He attended Heidegger’s lectures in Marburg (1924–28) and studied (1928–29) with Max Planck and Erwin Schrödinger at the Institute for Theoretical Physics in Berlin before immigrating to the United States in 1938 to escape the Nazis. He was a personal friend of Edmund Husserl’s family. He taught at St. John’s College Annapolis from 1938 until his death in 1978.
- 2.
A letter from Husserl’s wife Malvine to her daughter Elisabeth (March 26, 1937) mentions a “Klein” whom the editor of Husserl’s letters, Karl Schuhmann, identifies as “[d]er Altphilologe Jacob Klein (geb. 1899).” Husserl 1994a, 487. (The reference concerns Klein’s written communication to Malvine expressing his positive assessment of a publication by Jakob Rosenberg, husband of Elisabeth.)
According to Klein’s wife, Klein “visited old Husserl in 1919 in Freiburg—he wanted to study with Husserl. He went to Freiburg and visited Husserl…. But he couldn’t study with Husserl because he couldn’t get a room there, because it was 1919. All the boys came back from the war, and they had preference, so he went to Marburg. Old Husserl said, ‘Well, you study with my old friend Natorp’.” E. Klein, 14.
- 3.
“Phenomenology and the History of Science,” Klein 1940/1985. Hereafter cited as PHS. All citations from this text reflect reprinted pagination. Klein’s contribution was a late addition to the volume. In a letter to Klein dated November 10, 1939, Marvin Farber, the editor of Philosophical Essays in Memory of Edmund Husserl, invited him to submit a paper to the volume. He wrote that Husserl’s son Gerhart “has written to me about your ability to have a paper ready for the E. H. memorial volume within a week, or very soon thereafter,” and that “unusual circumstances … make it possible at this late date to consider another paper.” In a letter to Farber dated November 12, 1939, Klein wrote: “Although the time is very short I can get the article written before the deadline. I shall be grateful to you, if you can extend the time limit to the end of November.” Farber eventually extended the deadline to December 5, in response to Klein’s November 27, 1939 telegram to Farber requesting an extension to that date. In his letter to Farber of November 12, Klein described his proposed paper as follows:
The subject of my paper would be something like Phenomenology and History with special reference to the History of science. I have in mind the Philosophica essay which you mention in your letter and, in addition, Husserl’s article “Die Frage nach dem Ursprung der Geometrie als intentional-historisches Problem” published in the Revue internationale de philosophie (Janvier 1939a). (It goes without saying that I should have to refer to other publications of Husserl as well.)
I should like to add that my intention is not to give simply a commentary on those texts but also to examine the notion of History of science as such.
All of the correspondence referred to and cited above may be found among Klein’s papers, which are housed in the St. John’s College Library in Annapolis, Maryland. I wish to express my thanks to Mr. Elliot Zuckerman, the literary executor of Klein’s estate, for permission to cite from Klein’s correspondence.
- 4.
- 5.
Husserl 1936/1954. Cited hereafter as Crisis.
- 6.
PHS 79.
- 7.
Klein’s article makes repeated references to “Husserl’s notion of ‘intentional history’” (PHS 70; cf. 72–74, 76, 78, 82). However, Klein’s consistent use of quotation marks when referring to the expression “intentional history” is misleading, since he and not Husserl is its originator.
- 8.
PHS 84.
- 9.
PHS 83.
- 10.
PHS 84.
- 11.
PHS 84. Klein refers to Crisis 44, where Husserl discusses the “arithmetization of geometry” and the consequent automatic “emptying of its meaning” as “the geometric signification recedes into the background as a matter of course, indeed drops out altogether.”
- 12.
See Jacob Klein 1934/1936/ 1969. Hereafter cited as GMT.
- 13.
Caton 1971 and Miller 1982. In his review of the English translation of Klein’s articles, Caton (1971, 225) remarked upon Klein’s “failure to cite Husserl as the source of his Husserlian terminology”, that is, the terminology of the “theory of symbolic thinking” and the “concept of intentionality.” It is Caton’s contention that precedence for both of these should go to Husserl. In the case of the former, he appeals to Husserl’s “remarkably similar theory in the Logische Untersuchungen (Vol. II/1, par. 20).” In the case of the latter, he points to how, “by citing the scholastic Eustachias as illustrating the sources of the thinking of Vieta and Descartes,” Klein “ingeniously capitalizes on … [the] genealogy” of intentionality, which Husserl took “from Brentano, who in turn took it from medieval logic.” Miller (1982, 132) writes: “Although Husserl’s own analyses [i.e., in Philosophy of Arithmetic] move on the level of a priori possibility, Klein’s work shows how fruitful these analyses can be when the categories they generate are used in studying the actual history of mathematical thought.” As we shall see below, however, the relationship between Klein’s analyses of natural and symbolic numbers and Husserl’s is more complex than either Caton or Miller is aware. One consequence of this is that the common assumption behind Caton’s and Miller’s remarks here—that Husserl and Klein understand exactly the same thing when it comes to these kinds of numbers and their relationship—cannot withstand critical scrutiny.
- 14.
Carr 1970, xix n.7. This publication date of Koyré’s book is incorrect; it was published in Paris in 1939.
- 15.
Koyré 1945.
- 16.
Schuhmann 1997, 391.
- 17.
She mentions the dates as “‘31, or ‘32″ (E. Klein 14).
- 18.
Loc. cit.
Edmund Husserl’s daughter Elisabeth (Ellie) Rosenberg, one of Klein’s students in a 1933 Plato seminar he taught, invited him to visit her brother Gerhart in Kiel. Klein accepted the invitation, and soon became friends with the extended Husserl family and Gerhart’s wife, Else (Dodo) (E. Klein 17). (Gerhart Husserl divorced Else in 1948; she and Klein were married in 1950 [E. Klein 9].)
- 19.
Klein’s wife’s memory that the ideas concerned “something from one of the [Plato’s] dialogues” is clearly confused, since Koyré’s Plato book, based on lectures he gave in Cairo in 1940, was published in 1945. However, two articles containing parts of Etudes Galiléenes had already appeared in 1937: Koyré 1937a, b, which makes it much more than likely that it is they that contain the unacknowledged ideas borrowed from Klein reported by his wife.
- 20.
Wilson 1976 ii. Wilson, whose source for this information was most likely Klein himself, reports that Klein was engaged in this study from 1935 to 1937 while a fellow of the Moses Mendelssohn Stiftung zur Förderung der Geisteswissenschaften. Klein’s status as a Jew led to his exile from Germany in 1937 and the impossibility of continuing his Galileo studies during those turbulent times.
- 21.
Recently Moran 2012 and Parker (forthcoming) following him have suggested, contrary to Carr’s suggestion that Husserl’s Galileo section may be the result of a visit by Koyré in 1934, that the evidence points rather to Husserl being the source of Koyré’s interest in Galileo. On the one hand, Moran points out, “Reinhold Smid has shown (Hua XXIX il n.2) that Koyré’s last visit with Husserl was in July 1932, prior to the appearance of Koyré’s studies on Galileo that began to appear in artilce form from 1935 on” (Moran 2012, 72). Smid, moreover, also quotes Ludwig Landgrebe, who reported that he met Koyré in Paris in 1937 and Koyré told him he was “very much in agreement with the Galileo interpretation in the Crisis” (Smid 1993, n.2). On the other hand, Moran speculates that “Husserl’s interest in Galileo’s use of geometry was most probably influenced by Jacob Klein, who had published a number of works on the origins of Greek geometry between 1934 and 1936” (Moran 2012, 72–73). Parker, in addition, cites Aaron Gurwitsch, who “recalls that Koyré once remarked that, ‘even though Husserl was not a historian by training, by temperament, or by direction of interest, his analysis provides the key for a profound and radical understanding of Galileo’s work. He submits [Galilean] physics to a critique, not (once again be it said) a criticism’” (Parker forthcoming, 3). Parker also relates that “Gurwitsch points out that some of the preparatory studies for the Crisis date from the late 1920s, perhaps referring to texts dealing with the ‘Mathematisierung der Natur’ written in 1926, and notes also that ‘some of the relevant ideas can be found, at least in germinal form, as early as 1913’” (Parker forthcoming, 24).
This evidence, however, is not only inconclusive but also in one instance flawed. Regarding the chronology, we’ve already seen that Klein’s wife reports 1931 or 1932 as the dates in Paris that Koyré absorbed Klein’s ideas. These dates, then, are consistent with the date Smid (following Karl Schuhmann) reports Koyré last visited Husserl, July 1932. Moran’s suggestion that Klein’s articles on the origins of Greek geometry (which are dated 1934 and 1936 but actually were published together in 1936 in a single volume) most probably influenced Husserl’s understanding of Galileo is very problematic. This is the case because the focus of the articles in question is not geometry but the transformation of the ancient Greek concept of number that occurred with the invention of modern algebra. Neither Greek geometry nor Galileo are thematically treated in Klein’s articles. (See note 10 above, for bibliographic information on the German originals of the articles and their English translation by Eva Brann.) In addition, Gurwitsch’s claim that the preparatory studies for the Crisis date from the 1920s and before, and Parker’s singling out in particular Husserl’s texts on the mathematization of nature in 1926, do not establish that Husserl’s appreciation of Galileo’s role in the establishment of modern mathematical physics and the mathematization of nature in these texts is sufficient to account for his account of Galileo’s role in the mathematization of the life-world in the Crisis, together with Husserl’s presentation of the Greek mathematical context of Galileo’s mathematization in this account. In fact, close study of these texts discloses the basis for the opposite conclusion, namely, the Crisis’s account of the reinterpretation of Euclidean geometry that is sedimented in Galileo’s mathematization of nature is unprecedented in Husserl’s pre-Crisis discussions of Galileo and the mathematization of nature. Finally, neither Koyré’s expression of appreciation for (to Gurwitsch) or agreement with (to Landgrebe) Husserl’s critique of Galileo in 1937 rules out the possibility that Koyré’s appropriation of Klein’s ideas about the relation of Galileo’s physics to ancient Greek mathematics influenced Husserl in their 1932 meeting. It’s clear that Koyré’s appreciation and agreement relate to the aspect of Husserl’s analysis that goes beyond their historical presentation of Galileo, that is, to their phenomenological dimension, regarding which he of course could not have influenced Husserl.
- 22.
- 23.
Husserl 1970/2003. Hereafter cited as PA. German page numbers, which are reproduced in the English translation, will be cited.
- 24.
Husserl 1939b/1975, 127. German page number, which is reproduced in the English translation.
- 25.
PA 203.
- 26.
PA 209.
- 27.
PA 213.
- 28.
Majer 1997, 41–44.
- 29.
Insofar as for Husserl proper numbers begin with the least multiplicity (‘two’) and cardinal numbers begin with ‘1’, this identification is not without its problems. See Majer 1997, 42.
- 30.
Husserl 1994b, c.
- 31.
Husserl 1890, 158/13.
- 32.
Ibid.
- 33.
Ibid.
- 34.
Ibid.
- 35.
Ibid.
- 36.
Op. cit., 159/15.
- 37.
Op. cit., 160/15.
- 38.
Op. cit., 159/14.
- 39.
Loc. cit.
- 40.
Op. cit., 160/15.
- 41.
Op. cit., 160/16.
- 42.
Op. cit., 161/17.
- 43.
PHS 70.
- 44.
Vietae 1591/ 1969. Hereafter cited as Analtyic Art.
- 45.
See GMT and B. Hopkins 2011. Hereafter cited as Origin.
- 46.
Analytic Art 320.
- 47.
Loc. cit.
- 48.
Op. cit., 340.
- 49.
Op. cit., 346.
- 50.
Op. cit., 353 (bold letters in original).
- 51.
Husserl 1974/1969, 48. Page references refer to the German original, which are included in the English translation.
- 52.
On Klein’s view, the prevalent attempt to capture the difference between the ancient and modern concepts of number in terms of the latter’s greater abstractness falls short of the mark of the difference in question, which, as we have seen, cannot be measured in terms of degrees of abstraction but only captured in terms of the transformation of the basic unit of arithmetic from a determinate multitude to the concept of such a multitude.
- 53.
Vieta’s conceptualization of numbers grasped as Anzahlen, that is, determinate amounts of units, at the same time from the conceptual level of their symbolic formulation, is the historical precedent behind Husserl’s conviction that in the case of ordinary arithmetic the system of signs and operation with signs runs “rigorously parallel” to the “system of concepts and operation with judgments” (Stumpf Letter, 159/14). As we have seen, the symbolic level of conceptualization initiated by Vieta treats the concepts of determinate multitude of units (e.g., two units, three units, etc.) as numerically equivalent with their non-conceptual multitudes. Thus, the number two is conceptualized as the general concept of two, which is to say, twoness, while at the same time the numeral 2 is identified with the (non-conceptual) number itself, viz., the determinate multitude of two units. This formulation of Anzahlen from the conceptual level of their symbolic formulation is what, according to Klein, is responsible for what is now the matter of fact identification of ordinary (cardinal) numbers with their signs (numerals). Thus the systematic parallelism between symbolically and conceptually conceived numbers appealed to by Husserl presupposes rather than accounts for the symbolic expression of Anzahlen; this is the case, because what falls under the concepts that are expressed by the system of symbolically employed signs on Husserl’s view are not determinate amounts of units (Anzahlen) but the self-identical and therefore manifestly non-multitudinous general concepts (the individuated species) of the cardinal numbers or the general concept of being a cardinal number as such.
- 54.
Hopkins 2006.
- 55.
Patočka 1996, 35.
- 56.
See Origin ch. 32.
- 57.
GMT 208.
- 58.
Indeed, it is for this reason that Descartes, on Klein’s view, stresses the “power” of imagination, and not the imagination’s images, to assist the pure intellect in grasping the completely indeterminate concepts that it has separated from the ideas that the imagination offers it, because these ideas are precisely “determinate images”—and therefore, intrinsically unsuitable for representing to the intellect its indeterminate concepts. The imagination’s power, however, being indeterminate insofar as it is not limited to any particular one of its images, is able to use is own indeterminateness to enter into the “service” of the pure intellect and make visible a “symbolic representation” of what is otherwise invisible to it, by facilitating, as it were, the identification of the objects of first and second intentions in the symbol’s peculiar mode of being. The imagination’s facilitation involving, as it were, its according its “power” of visibility to the concept’s invisibility.
- 59.
Nagel 1941.
- 60.
Op. cit., 303.
- 61.
Ibid.
- 62.
Ibid.
- 63.
Ibid.
- 64.
Ibid.
- 65.
PHS 84.
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Hopkins, B.C. (2017). Husserl and Jacob Klein. 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_17
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