Formalization and Intuition in Husserl’s Raumbuch

Caracciolo, Edoardo

doi:10.1007/978-3-319-10434-8_3

Edoardo Caracciolo⁸

Part of the book series: Boston Studies in the Philosophy and History of Science ((BSPS,volume 308))

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Abstract

Husserl planned to deal with a philosophy of Euclidean geometry in the second and unpublished volume of the Philosophie der Arithmetik. Thus, after the publication of the first volume, he outlined an analytical/formal method that should have allowed a pure understanding of any spatial manifold. Nevertheless, dealing with the analytical geometry of Bernhard Riemann, he understood the limits of a pure formal approach. Therefore, he started investigating space representation from a psychological point of view, trying to detect those intuitions grounding geometrical concepts. Notwithstanding, Husserl abandoned this geometrizing space theory too because, in my view, it did not offer a plain explanation of the relations between geometry and space theory, material, and formal concepts. In this paper, I investigate the reasons why in his early studies Husserl wavered from a formalizing to an intuitive approach to the space problem.

This paper deals with the methodological issues that Husserl encountered when he was developing his first space theory. In particular, the present paper tries to throw some light on the impact of intuition and formalization on early Husserl’s geometrical studies. In Il problema dello spazio nel primo Husserl (“Rivista di Filosofia,” vol. CIV, n. 2. agosto 2013), instead, I offer an historical overview on the Raumbuch, mostly focusing on the psychological issues, like Husserl’s critique of Helmholtz’s space theory.

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Notes

1.
Cf. Miller, J.P. 1982. Number in presence and absence. A study of Husserl’s philosophy and mathematichs, 11. The Hague/Boston/London: Nijhoff.
2.
Cf. HUA XII pp. 294–295. The Philosophie der Arithmetik reflects on the foundation of arithmetic that, in the first part of the book, is defined as “science of number,” a subject based on the concept of positive integer. Indeed, this definition is originated from Weierstrass’s studies. Claudio Majolino explains in which way Brentanian psychology answers to an exigency of intuitive researches that Weierstrass left unfulfilled. Cf. Majolino, C. 2004. Declinazioni dello spazio, sul rapporto tra spazialità percettiva e spazialità geometrica nel primo Husserl. Paradigmi XXII(64/65): 223–238. For an overall view on Husserl’s juvenile years cf. Rollinger, R.D. 1999. Husserl’s position in the school of Brentano, Phaenomenologica, 15–21. Dordrecht: Kluwer.
3.
Douglas Willard hypothesize that this second perspective was influenced by Schröder’s algebra of logic: indeed, Husserl was writing a (negative) review on his Vorlesungen über die Algebra der Logik during the composition of the Philosophie der Arithmetik, last chapter. (cf. D. Willard, D. 1984. Logic and the objectivity of knowledge. A study in Husserl’ early philosophy, 109. Athens: Ohio University Press.).
4.
HUA XII p. 132; cf. also pp. 258, 346. For example, numbering is a mechanical operation that “[…] substitutes the names for the concepts, and then by means of the systematic of names and a purely external process, derives names from names, in the course of which there finally issue names whose conceptual interpretation necessarily yelds the result sought” (HUA XII p. 239). On this matter, cf. also Sinigaglia, C. 2000. La seduzione dello spazio, 64–66. Milano: Unicopli.
5.
Cf. HUA XII p. 283.
6.
Cf. HUA XII pp. 7–8.
7.
Cf. HUA XXI pp. 244–249, 396. Cf. also Sinigaglia, C. La seduzione dello spazio, 61, op. cit.
8.
HUA XII p. 193.
9.
In Zur Logik der Zeichen, Husserl explores the wide range of symbolic representations. Among them he numbers the artificial signs (Künstliche Zeichen) of general arithmetic and those conceptual second class signs (symbolichen Vorstellungen der Zweiten Klasse) standing for things that cannot be properly represented; cf. HUA XXI pp. 349–350, 354–356.
10.
Cf. HUA XXI p. 248. On the relation between Husserl and the brentanian logic, cf. De Boer, T. 1978. The development of Husserl’s thought, 91–93. Den Haag: Nijhoff.
11.
Cf. Argentieri, N. 2008. Matematica e fenomenologia dello spazio. In Forma e materia dello spazio, dialogo con Edmund Husserl, ed. P. Natorp, 246, edited by N. Argentieri. Napoli: Bibliopolis, Corrado Sinigaglia proposes a close analysis of the relations between the Philosophie der Arithmetik and the Raumbuch; cf. Sinigaglia, C. 2001. La libera variazione delle forme. Husserl lettore di Riemann. In Logica e politica. Per Marco Mondadori, Fondazione Arnoldo e Alberto Mondadori, edited by M. D’Agostino, G. Giorello, and S. Veca, 377–403. Milano: il Saggiatore.
12.
Cf. HUA XXI pp. 285–286.
13.
Non-Euclidean geometries deny the parallel postulate. This postulate states that if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles. Cf. M. Kline, M. 1972. Mathematical thought from ancient to modern times, vol. I, p. 59, vol. III, p. 865. New York/Oxford: Oxford University Press. On non-Euclidean geometries paternity, cf. Kline, M. Mathematical thought from ancient to modern times, vol. III, 869–870, op. cit.
14.
Cf. Torretti, R. 1984. Philosophy of geometry from Riemann to Poincaré, 285–291. Dordrecht: Reidel. On the fracture between things and representation, cf. Lotze, H. 1899. Microcosmus: An essay concerning man and his relation to the world, 344–353, 573–578. Edinburgh: T. & T. Clark.
15.
Cf. Helmholtz, H. 1867. Handbuch der Physiologischen Optik, 194. Leipzig: Voss; Helmholtz, H. 1876. The origin and the meaning of geometrical axioms. Mind 1(3): 316–318. On the relation between Lotze and Helmholtz, cf. Gehlhaar, S. 1991. Die Frühepositivistsche (Helmholtz) und phänomenologische (Husserl) Revision der Kantischen Erkenntnislehre, 30. Cuxhaven: Junghans-Verlag.
16.
The note can be found in Ms. K I 50/47a. The Grundproblem der Geometrie is published in HUA XXI pp. 312–347.
17.
Cf. HUA XXI pp. 262–311. The Raumbuch structure is presented in a note published in HUA XXI pp. 402–404. For an historical panorama on the Raumbuch birth and on the Tagebuch zum Raumbuch, cf. the Textkritische Anmerkungen published in HUA XXI pp. 469, 485–486; HUA D. I pp. 36–37; Mohanty, J. N. 1999. The development of husserl’s thought. In The Cambridge companion to Husserl, edited by B. Smith and D. W. Smith, 51. Cambridge: Cambridge University Press.
18.
For example, cf. HUA D. III/5 p. 80.
19.
Husserl uses the expression “analytical geometry” in a standard geometrical way; cf. HUA XXI pp. 232, 323.
20.
Cf. HUA XIX p. 259. On the opposition between analytical and synthetical truths, cf. HUA 3-1 pp. 22, 30.
21.
HUA XXI pp. 396–397.
22.
Cf. HUA XXI p. 396. Cf. Hartimo, M.H. 2007. Towards completeness: Husserl on theories of manifolds 1890–1901. Synthese 283; Hartimo, M.H. 2008. From geometry to phenomenology. Synthese 162: 226–227. Selected manuscripts have been published in HUA XXI pp. 234–243, 312–347.
23.
Husserl reads carefully and annotates his 1878 reprint of the 1844 version of the Ausdehnungslehre (Grassman, H. 1878. Die lineale Ausdehnungslehre. Leipzig: Otto Wigand). Cf. Hartimo, From geometry to phenomenology, op. cit., pp. 225–233.
24.
Cf. HUA XXI p. 323. For an historical account on non-Euclidean geometry, cf. HUA XXI pp. 322–347.
25.
On this matter, cf. Sinigaglia, C. La seduzione dello spazio, 24–25, op. cit.
26.
Cf. Riemann, B. 1868. Über die Hypothesen, welche der Geometrie zugrunde liegen. In Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, Vol. XIII, 133. Göttingen.
27.
Ibidem. On this matter, cf. Kaiser-El-Safti, M. Fenomenologia trascendentale versus iletica. Psicologia e fenomenologia in Husserl e Stumpf. In Carl Stumpf e la fenomenologia dell’esperienza immediata, edited by S. Besoli and R. Martinelli, Discipline Filosofiche, Anno XI, numero 2, 247. Macerata: Quodlibet.
28.
HUA XVIII p. 252. Cf. also HUA D. III/1 p. 11. Cf. Parrocchia, D. 1994. La forme générale de la philosophie husserlienne et la théorie des multiplicités. Kairos 5: 137–140.
29.
Cf. HUA XXI p. 345.
30.
Cf. HUA XXI p. 344. For a deeper examination of Husserl’s remarks on Riemann’s arguments, cf. L. Boi, Le problème mathèmatique de l’espace, Springer, Berlin/Heidelberg, 1995, pp. 241–243; Sinigaglia, C. La libera variazione delle forme. Husserl lettore di Riemann, 387–388, op. cit.; Hartimo, M. H. From geometry to phenomenology, op. cit., p. 228.
31.
Cf. Helmholtz, H. 1921. Über die Thatsachen, die der Geometrie zu Grunde liegen (1868). In Schriften zur Erkenntnistheorie, edited by M. Schlick and P. Hertz, 55. New York: Springer. Cf. also Torretti, R. Philosophy of geometry from Riemann to Poincaré, 156–157, op. cit.
32.
Cf. HUA XXI p. 409.
33.
Cf. HUA XXI pp. 348, 407–410. Leaving aside the debate about Husserl’s theory of manifold, it would be useful to refer to the other definitions of manifold proposed in the Prolegomena (cf. HUA XVIII p. 249) and in the Philosophie der Arithmetik (cf. HUA XII p. 81). Husserl himself, in a footnote of the Ideen, provides a brief historical-critical examination of the concept of manifold in his former works: cf. HUA III/1. On the riemannian Zahlenmannigfaltigkeit, cf. Brisart, R. 2003. Le Général et l’abstrait: sur la maturation des Recherces Logiques de Husserl. In Aux origines de la phénoménologie, edited by D. Fisette e S. Lapointe, 39. Paris: Vrin; Majolino, C. Declinazioni dello spazio, sul rapporto tra spazialità percettiva e spazialità geometrica nel primo Husserl, 228–229, op. cit.; Sinigaglia, C. La seduzione dello spazio, 57–58n, op. cit.
34.
Lobačevsky, N. 1898. Neue Anfangsgründe der Geometrie mit einer vollständigen Theorie der Parallellinien. In Zwei geometrische abhandlungen aus dem russischen uebersetzt, mit anmerkungen und mit einer biographie des verfassers, edited by F. Engel and P. Stäckel, 80–82. Leipzig: Teubner.
35.
Cf. HUA XXI pp. 312–314, 322–323.
36.
This last perspective represents the core idea of the Raumbuch and seems to echoes Lobačevsky’s opinion about the importance of synthesis in mathematics, whose “constructive procedure” has to clarify “those representations that are directly connected to the early concepts of our mind” (Lobačevsky, N. Neue Anfangsgründe der Geometrie mit einer vollständigen Theorie der Parallellinien, 80–81, op. cit.).
37.
HUA XXI p. 411.
38.
HUA D. III/1 p. 11.
39.
C. Stumpf, C. 1873. Über den psychologischen Ursprung der Raumvorstellung. Leipzig: Verlag von S. Hirzel.
40.
Cf. HUA XXI p. 281.
41.
This argument may remind a Kantian thesis, but, actually, the Raumbuch displays an anti-Kantian perspective on space. Kant wanted to prove the priority of spatial form over spatial object showing that object cannot be displayed without a surrounding space, whereas space itself can be conceived as object-free. Instead, Husserl speaks in terms of extension: the single fraction of space is an extension as well as the world space. Obviously, the first extension is part of the second one, but – here it is the difference from kantianism – the single place cannot be conceived without conceiving its surrounding places as well as the world space cannot be conceived without its composing parts. On this subject, cf. Kant, I. Kritik der reinen Vernunft, A24, B39. This thesis anticipates, in a different theoretical context, an idea that Husserl will elaborate in the Logischen Untersuchungen. There he notices that every representation has both intuitive and symbolical sides, each one contributing in a different degree to the whole representation. Cf. HUA XIX pp. 610–614.
42.
HUA XXI p. 276.
43.
Cf. Kant, I. Kritik der reinen Vernunft, A 99, 107, 120n; B 201-2n, 218-9, 129–130, 134–135. Victor Popescu highlights the subtle differences between Stumpf’s and Husserl’s mereologies: cf. Popescu, V. 2003. Espace et mouvement chez Stumpf et Husserl, une approche méréologique. Studia Phaenomenologica III(1–2): 115–133.
44.
Cf. HUA XXI pp. 281, 307. Cf. also Stumpf, C. Über den psychologischen Ursprung der Raumvorstellung, 107–109, op. cit. This distinction will be further developed in the Psychologische Studien zur elementaren Logik (cf. HUA XXII pp. 97–98) and in the Logische Untersuchungen (cf. HUA XIX pp. 231–240, 272–274). Cf. Kaiser-El-Safti, M. Fenomenologia trascendentale versus iletica. Psicologia e fenomenologia in Husserl e Stumpf, 236, op. cit.; Majolino, C. Declinazioni dello spazio, sul rapporto tra spazialità percettiva e spazialità geometrica nel primo Husserl, 230–231, op. cit.
45.
According to Stumpf, those judgments describing objective relations are necessary by nature and universally valid. Starting from those judgments, we can develop a set of a priori material laws. Cf. Stumpf, C. 1982. Psychologie und Erkenntinistheorie. In Abhandlung der Königlich Bayerischen Akademie der Wissenschaften, I Classe, 19, 2, München, 494–495. On this subject cf. De Palma, V. 2001. L’a priori del contenuto. Il rovesciamento della rivoluzione copernicana in Stumpf e Husserl. In: Carl Stumpf e la fenomenologia dell’esperienza immediata, edited by S. Besoli and R. Martinelli, Discipline Filosofiche, XI, 2, 316–318. Macerata: Quodlibet.
46.
HUA XXI p. 284. Pursuing this strand of research, in the Dingvorlesung, Husserl will deal with the problem of the tridimensional circularity of the real object.
47.
Cf. HUA III/1 p. 138.
48.
Cf. HUA XXI p. 286.
49.
HUA XXI p. 308.
50.
HUA XXI p. 296. This passage reminds Lobačevsky’s New principles of geometry: “[…] it will be possible to form any body by means of composition, reaching an identity degree beyond which our senses stop perceiving imperfections. […] although we get our first concepts from it [the nature] we owe the rigor of the former to our senses imperfection” (Lobačevsky, N. Neue Anfangsgründe der Geometrie mit einer vollständigen Theorie der Parallellinien, p. 81, op. cit.).
51.
HUA XXI p. 287.
52.
Ibidem.
53.
Cf. HUA XXI pp. 289–290, 294. Not only single objects but the entire intuitive space may be used as a symbolic surrogate of pure geometrical space.
54.
Cf. HUA XXI p. 295.
55.
HUA XXI pp. 271, 295–296.
56.
Cf. HUA XXI p. 296. This passage anticipates the distinction between physical and pure geometry in the Prolegomena; cf. HUA XVIII p. 251.
57.
Cf. HUA D. III/5 pp. 53–54.
58.
Cf. Brisart, R. Le Général et l’abstrait: sur la maturation des Recherces Logiques de Husserl, 39–40, op. cit.
59.
This idea shows up in the Psychologische Studien zur elementaren Logik, cf. HUA XXII p. 104.
60.
HUA XXI p. 262.
61.
XII p. 193. This definition has many similarities with the one that Husserl gives in HUA XXI p. 272. It is worth noticing a minor semantic sliding: the “proper representations” in the Philosophie der Arithmetik are named “intuitions” in the Raumbuch.
62.
Cf. HUA XXI pp. 295–296. Husserl will define the representational status of concepts when he will deal with the categorial intuition in the Logische Untersuchungen. There he also dismantles the intuition/symbolization dichotomy that structures the Raumbuch representational theory.
63.
HUA XXI pp. 295–296.
64.
That reminds an idea from the Philosophie der Arithmetik, where arithmetic is presented as a tool dealing with sets that cannot be intuited because of subjective inability; consequently, since powerful subjectivities, like angels or God, need not to develop arithmetic to handle large sets, then their arithmetical domain (objects and procedures) is almost empty. At the end, “results” are the same: both man and angel represent the same large sets, but the former uses a tool (arithmetic), whereas the latter need it not. Cf. HUA XII pp. 191–192.
65.
Cf. HUA XXI pp. 262, 296, 287.
66.
Cf. HUA XXI pp. 281–283.
67.
Cf. HUA XXI pp. 278–279.
68.
Cf. HUA XXI pp. 278–279, 286. Stumpf explains that many space theories of his time incorrectly mix two strands of research that should be kept separated: epistemology, focusing on immediately evident truths, merges with descriptive psychology, focusing on the genesis of concepts. Thus, the researches on the origin of spatial representation overlap the studies on the nature of geometrical axioms. For this reason, the spatial analysis of the early Husserl displays a geometrical nuance. Cf. Stumpf, C. Psychologie und Erkenntinistheorie, 484, op.cit.
69.
Husserl inherits immanentism from Brentano, and when he is working on the Raumbuch, he still adopts this intentional theory. For example, he stresses the distinction between immanent object (immanente Objekt) and real object; he defines the metaphysical space – i.e., the real space – as transcendent space (transzendent Raum). Cf. HUA XXI pp. 262, 265–266, 270, 305. Paradoxically, he makes the same mistake that he highlights in Helmholtz’s empiricist space theory: according to him, Helmholtz confuses the inner psychological experience with the real external one. Cf. HUA XXI p. 309.
70.
Cf. HUA III/1 pp. 59–60, 108, 115–116.
71.
Cf. HUA III/1 pp. 112–115. It is no accident that several sciences presuppose the axiomatic method called mathesis universalis − whose first model was Euclidean geometry.
72.
HUA III/1 pp. 115–116.
73.
Cf. HUA XVIII p. 252.
74.
Cf. HUA III/1 pp. 133–136; HUA XVII pp. 79–80; HUA XVIII pp. 247–248.
75.
Cf. HUA III/1 pp. 131–132.
76.
Cf. HUA III/1 pp. 133–139.

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Edoardo Caracciolo

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Caracciolo, E. (2015). Formalization and Intuition in Husserl’s Raumbuch . In: Lolli, G., Panza, M., Venturi, G. (eds) From Logic to Practice. Boston Studies in the Philosophy and History of Science, vol 308. Springer, Cham. https://doi.org/10.1007/978-3-319-10434-8_3

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