Husserl and Jacob Klein

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 , together with 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: to provide a psychological foundation for the proper concept of number (Anzahl), and to show how this concept of number functions as the mathematical foundation of universal (symbolic) arithmetic. This context establishes that Husserl’s failures are ultimately rooted in the historical transformation of number documented in Klein’s research, from its premodern meaning as the unity of a multitude of determinate objects to its modern meaning as a symbolic representation with no immediate relation to a concrete multiplicity. 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 project in The Crisis of European Sciences and Transcendental Phenomenology of de-sedimenting 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.”

Notes

1. Jacob Klein was born in 1899 in Libau, Russia (which was then in Courland and a part of the Russian Empire and which is now a part of Latvia), educated there, and in Belgium and Germany (1922 Ph.D. Marburg University). He attended Heidegger’s lectures in Marburg (1924–28) and studied with Max Planck and Erwin Schrödinger at the Institute for Theoretical Physics in Berlin (1928–29) before emigrating 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, Maryland, from 1938 until his death in 1978.

2. A letter from Husserl’s wife Malvine to her daughter Elisabeth (26 March 1937) mentions a “Klein” whom the editor of Husserl’s letters, Karl Schuhmann, identifies as “Der Altphilologe Jacob Klein (geb. 1899).” Edmund Husserl. Briefwechsel, vol. IX, ed. Karl Schuhmann (in cooperation with Elisabeth Schumann), 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’” (Else [Dodo] Klein, page 14 of a transcript of a tape recording [the original tape recording is apparently lost] among Klein’s papers, which are housed in St. John’s College Library, Annapolis, Maryland; hereafter cited as “Interview.”

3. Jacob Klein, “Phenomenology and the History of Science,” in Philosophical Essays in Memory of Edmund Husserl, ed. Marvin Farber (Cambridge, MA: Harvard University Press, 1940), 143–63; reprinted in Jacob Klein, Lectures and Essays, ed. Robert B. Williamson and Elliott Zuckerman (Annapolis, MD: St. John’s Press, 1985), 65–84; hereafter cited in the text 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 10 November 1939, Marvin Farber, 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 12 November 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 5 December, in response to Klein’s telegram on 27 November 1939 to Farber requesting an extension. In his letter to Farber of 12 November, 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 1939). (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. 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. First published in the Revue internationale de Philosophie 1, ed. Eugen Fink (1939): 203–25. English translation, “The Origin of Geometry,” in The Crisis of European Sciences and Transcendental Phenomenology, trans. David Carr (Evanston, IL: Northwestern University Press, 1970), 370; hereafter cited as “OG” with English page references. Fink’s typescript of Husserl’s original, and significantly different, 1936 text (which is the text translated by Carr) was published as Beilage III in Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie. Eine Einleitung in die phänomenologische Philosophie, ed. Walter Biemel, Husserliana VI (The Hague: Nijhoff, 1954; 1976).

5. First published in Philosophia 1 (1936): 77–176. The text of this article is reprinted as §§1–27 of the text edited by Biemel, cited in the previous note; hereafter cited as Crisis.

6. 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 was its originator.

7. “Sedimentation” is an important concept that Husserl introduced in his last writings to indicate the status of meaning formations that are no longer present to consciousness but that nevertheless can still be made accessible to it. This status pertains both to the temporal modification of the experience of meaning formations and the role that passive understanding plays in the apprehension of the meaning of concepts and words. In either case, it is sometimes possible to render the sedimented formations present to consciousness again in a process called ‘awakening’. In the case of the passive understanding of meaning formations, because it does not reproduce the cognitive activity that originally produced their meaning, Husserl contends that the original meaning becomes diminished and in some sense forgotten. Insofar as the original meaning has not completely disappeared, however, it can still be “awakened” by phenomenological reflection. In the Crisis Husserl attempts to ‘awaken’ the original cognitive activity that gave rise to the meaning formations constitutive of Euclidean geometry, meaning formations that he maintained are “sedimented” in Galileo’s project of mathematizing nature.

8. Klein, PHS, 84. Klein refers to Crisis, 44–45, 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” (44).

9. See Jacob Klein, “Die griechische Logistik und die Entstehung der Algebra,” Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien 3.1 (1934): 18–105 (Part 1), and 2 (1936): 122–35 (Part 2); English translation: Greek Mathematical Thought and the Origin of Algebra, trans. Eva Brann (Cambridge, MA: MIT Press, 1969; reprint, New York: Dover, 1992); hereafter cited in the text as GMT.

10. Hiram Caton, “Review of Jacob Klein’s Greek Mathematical Thought and the Origin of Algebra,” Studi Internationali di Filosofia 3 (1971): 222–26. In his review of the English translation of Klein’s articles, Caton remarked upon Klein’s “failure to cite Husserl as the source of his Husserlian terminology” (225), 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. 2/1, 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.”

J. Philip Miller, Numbers in Presence and Absence: A Study of Husserl’s Philosophy of Mathematics (The Hague: Nijhoff, 1982). He 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” (132).

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.

11. David Carr, “Translator’s Introduction,” Crisis, xix n. 7. This publication date of Koyré’s book is incorrect; it was published in Paris in 1939.

12. Karl Schuhmann, “Alexandre Koyré,” in the Encyclopedia of Phenomenology (Dordrecht: The Netherlands, 1997), 391, referring to Alexandre Koyré, Introduction à la lecture de Platon (Paris: Gallimard, 1945).

13. Klein’s wife mentions the dates as “‘31, or ‘32” (14).

14. Else Klein, “Interview,” 14. 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) (“Interview,” 17). (Gerhart Husserl divorced Else in 1948; she and Klein were married in 1950 [“Interview,” 9].)

15. 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: “Galilée et l’expérience de Pise,” in Annales de l’Université de Paris 12 (1937): 441–53; “Galilée et Descartes,” in Travaux du IX^e Congrés international de Philosophie 2 (1937): 41–47, which makes it much more than likely that it is they that contain the unacknowledged ideas borrowed from Klein reported by his wife.

16. Curtis Wilson, “Preface” to Essays in Honor of Jacob Klein (Annapolis, MD: St John’s College Press, 1976), ii. Wilson, whose source for this information was most likely Klein himself, reports that Klein was engaged in this study from1935 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. See also Klein’s letter to Leo Strauss, 9 November 1934, in Leo Strauss Gesammelte Schriften, Bd. 3, ed. Heinrichand Wiebke Meier (Stuttgart/Weimar: J. B. Metzler, 2001), 521, in which he discusses his plans to publish a study on Galileo, Aristotle’s de coelo and Archimedes; and Klein’s letter to Krüger, 13 February 1930: “Als Habilitationsschrift würde ich in zwei Monaten eine Arbeit über Galileis Dialog im Verhältnis zum de coelo und Timaios fertigsgeotellen.” [“For my qualifying thesis I would complete in two months a work about Galileo’s dialogue in relation to de coelo and Timaeus.”]

17. Recently Dermot Moran (Husserl’s Crisis of the European Sciences and Transcendental Philosophy: An Introduction [Cambridge: Cambridge University Press, 2012]), and Rodney Parker following him (“The History Between Koyré and Husserl,” forthcoming, in Hypotheses and Perspectives within History and Philosophy of Science: Homage to Alexandre Koyré,s 1964–2014, ed. Raffaele Pisano, Joseph Agassi, and Daria Drozdova [Berlin: Springer]) 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, 24.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” (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” (HUA, 24.il 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” (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’” (3, ms). 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’” (24, ms).

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.

18. Oskar Becker, whose article “The Theory of Odd and Even in the Ninth Book of Euclid’s Elements” appeared in the same journal that Klein’s first article appeared in, refers therein to Klein’s article as “a very important work.” Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien 3.1 (1934): 533–53.

19. “Entwurf einer ‘Vorrede’ zu den Logischen Untersuchungen (1913), ed. Eugen Fink, in Tijdschrift voor Philosophie (1939): 106–33; English translation, Introduction to the Logical Investigations, trans. Philip J. Bossert and Curtis H. Peters (The Hague: Martinus Nijhoff, 1975), 127 (German page number, which is reproduced in the English translation.)

20. Husserl holds that, “Two concepts are logically equivalent when each object of the one is also an object of the other, and conversely. That, for the purposes of our interests in forming judgments, symbolic presentations can surrogate, to the furthest extent, for the corresponding authentic presentations rests upon this circumstance” (PA, 194, my emphasis).

21. Ulrich Majer, “Husserl and Hilbert on Completeness: A Neglected Chapter in Early Twentieth Century Foundations of Mathematics,” Synthese 110 (1997): 41–44.

22. 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, “Husserl and Hilbert on Completeness,” 42.

23. “Husserl an Stumpf, ca. Februar 1890,” in Edmund Husserl, Briefwechsel, Band I, ed. Karl Schuhmann (Dordrecht: Kluwer, 1994), 158. English translation, “Letter from Edmund Husserl to Carl Stumpf,” in Edmund Husserl, Early Writings in the Philosophy of Logic and Mathematics, trans. Dallas Willard (Dordrecht: Kluwer, 1994), 13; hereafter cited in the text as “Stumpf Letter,” with page numbers referring to the original and the English edition, respectively.

24. Francisci Vietae, In Artem Analyticem (sic) Isagoge, Seorsim excussa ab opere restituate Mathematicae Analyseo, seu, Algebra Nova (Introduction to the Analytical Art, excerpted as a separate piece from the opus of the restored Mathematical Analysis, or The New Algebra [Tours, 1591]). English translation, f Introduction to the Analytic Art, trans. J. Winfree Smith, appendix to Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, trans. Eva Brann (Cambridge, MA: MIT Press, 1968); hereafter cited in the text as Analtyic Art.

25. See Jacob Klein, GMT, and Burt C. Hopkins, The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein (Bloomington, IN: Indiana University Press, 2011); hereafter cited as Origin.

26. Vietae, Analytic Art, 320.

27. Vietae, Analytic Art, 340.

28. Vietae, Analytic Art, 353 (capitals in original).

29. Edmund Husserl, Formale und transzendentale Logik (The Hague: Nijhoff, 1974); English translation, Formal and Transcendental Logic, trans. Dorion Cairns (The Hague: Nijhoff, 1969), 48; page numbers refer to the original, and are included in the English translation.

30. In 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 in terms of the transformation of the basic unit of arithmetic from a determinate multitude to the concept of such a multitude.

31. 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.

32. Burt C. Hopkins, “Husserl’s Psychologism, and Critique of Psychologism, Revisited,” Husserl Studies 22 (2006): 91–119.

33. Jan Patočka, “The Philosophy of Arithmetic,” in An Introduction to Husserl’s Phenomenology, ed. James Dodd, trans. Erazim Kohák (Chicago, IL: Open Court, 1996), 35.

34. See Klein, Origin, chap. 32.

35. Klein, GMT, 208.

36. 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 its 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.

37. Ernest Nagel, “Review of Philosophical Essays in Memory of Edmund Husserl, ed. Marvin Farber, The Journal of Philosophy 38.11 (22 May 1941): 301–6.