Abstract
What is the role and the significance of the continuity in the phenomenology of spatiality? And what is the function of spatial continuity in a phenomenological epistemology? Under the title of continuity falls one of the greatest apories of phenomenology and one of the greatest achievements of the phenomenological epistemology (metric continuum formulated by Hermann Weyl) . Moving from these premises, the essay will face up to the following issues:
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1.
The determination of geometrical subjectivity, that is the definition of the 0-point of consciousness and the corresponding modal (dimensions) and relational (distances) field;
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2.
The definition of metric space as:
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2.1.
the elementary operation of the proportioning between 0 (consciousness) and counter-0 (mode and spatial relations);
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2.2.
the middle-level between the composition of sensible fields and the formalization of geometrical-physical space;
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2.3.
the result of the nullification of egoity and the constant of correlation between the electromagnetic field and gravitational field ;
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2.1.
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3.
The definition of contingent a priori as:
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3.1.
the character of incomplete formalization of Euclidian geometry;
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3.2.
the occasional nature of the elements (e.g.: extension and color) between which there is material-apriorical relationship (e.g.: non-independence);
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3.3.
the projection, through continuum or sequence of integers, of the actually given upon the background of a priori possible;
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3.1.
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4.
The definition of limes as:
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4.1.
the empirical-ideal determination of magnitude, interval or correlation;
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4.2.
the feature, first topological and then metrical, of the sensible spatial fields;
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4.2.1.
then the essential element for the overlay and the coincidence of the sensible fields;
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4.2.1.
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4.3.
the reflexive presupposition of measurement.
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4.1.
Once it has been so proven that the same metrics is the contingent a priori of spatiality, and of its possible continuization, the final objective of the essay is to prepare the ground for the analysis of the central and most problematic point of natural phenomenology, the theme that has sanctioned its failure and irrelevance, that is: the relationship between space and matter.
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Notes
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Husserl defines the transcendence in immanence in this manner also in Husserl (1973a: 170).
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Cfr. Becker (1923: 544).
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The second side of the pair evidently refers to the solution formulated by Weierstrass, regarding the exclusion of infinitesimals in determining the continuum, which constitutes for Husserl not so much a decisive acquisition of his formation, as one of aspects of the mathematical heritage that is most present in his philosophical reflection. In formal terms, in the first case one obtains the limit of a sequence, such that n1, n2, n3… tends infinitarily to (→) nn, in the second the limit of a function, f(x), which can be formulated as follows: “if, given an ε, there is an η0, such that, for 0 < η < η0, the difference f(x0 ± η) − L is less in absolute value than η, then L is the limit of f(x), for x = x0” (Heine 1872: 184); or in an extremely simplified manner: “the number 1/3 is not the limit of the series 3/10 + 3/100 + 3/1000··· +3/+ 10n +···; but it is the sequence associated with this series” (Boyer 1968: 606).
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That the understanding of the limit is phenomenologically exposed to this dual possibility is amply demonstrated by the divarication that separates the treatments conducted in that regard by two of the students of Husserl who most contributed to the maturing of Phänomenologie der Wahrnehmung, that is, Schapp (1910: 149–157), on the ideality of the limit, and Hofman (1912: esp. 60), on the concept-limit of the “visual perceivable thing”, (1912: 62–67), on the measure of a visual magnitude, and (1912: 101–103), on spatial limits.
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It does not seem rash to grasp even in this use of apeiron—in which culminates Koyré’s regression to the Zenonian paradoxes more than ten years after Husserl’s rejection of his project for the doctoral dissertation (Koyré 1999)—a testimony of his by now mature separation from his Gottingen master; in this regard, see Zambelli (1999). Husserl’s interest in resolving the link between Mengenlehre and apeiron—demonstrated by the manuscripts Ms. A I 35 (Menge. Die Paradoxien. Die Insolubilia. Insbesondere auch die Paradoxien der Mengenlehre, 1912), pp. 4 et seq., and Ms. A III 12 (Morphologische Realität gegenüber der physikalischen. Das organische Individuum. (1) Die formale Logik des Apeiron, der Apophantik. (2) Formal Logik der Kontinuität, 1926)—is widely reflected in the pages of the Jahrbuch; see esp. volume VI, in which appear London (1923) and Lipps (1923). Cfr. Peckhaus (1990).
- 14.
On the return of apeiron in the history of phenomenology, see Heidegger (1946), Heidegger (1991: esp. 109–113), Fink (1957, 1960), Levinas (1961: esp. 163 f.), Derrida (1962: 48 and 151), Derrida (1967: esp. 142), which, betraying as much the archaic Anaximandrean meaning as the later Platonic one, makes of the non-limit the unity that saves from the limit—Plato, Philebus, 24 a3: “tò àpeiron pollà esti”.
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Husserl (1974a : 32–34). The contingent a priori renders the essential orientation of occasional expressions already, first in the deictic and then in the adverbial determination of the place and then in scientific praxis, cfr. Husserl (1922: 86, 87). On the link between the contingent a priori of the science of space and the facticity of nature, cfr. Husserl (1973c: 287–290). On the phenomenological notion of contingency, see Pöggeler (1986: esp. 17).
- 20.
On the relationship between metric and contingent a priori, see, for the geometric side, Volkert (1994: 271), and for the linguistic-ontological side, see Kripke (1980: esp. 55 ff.) For an acute analysis of the measure-measurement plexus see, beyond the classic (Helmholtz 1887; Nagel 1931; Kuhn 1961; Sokolowski 1987).
- 21.
It is precisely Becker’s taking up (1927: 317), of the apeiron and its omnitemporal character as aei, that leads us from the analysis of the symbolic-transcendent transfinity to a metaphysics of contingency, to a metaphysics of that transcendent “(relatively) close to us and therefore accessible to a ‘divining interpretation’ (“Deutung”)”, that is, nature (Becker 1927: 328). Cfr. Peckhaus (2005).
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- 23.
On Cantor’s role in Husserl’s formation, and particularly in the decision of harking back in ideal-realistic fashion to Platonism as capable of throwing light precisely on the relationship between “sense and type of mathematical knowledge,” cfr. Schuhmann (1977: 19 and 26). A much broader question, to which we shall only make passing reference, is that of Cantor’s influence on Husserl’s philosophical vocabulary, which is documented, besides in the rigorous distinction between stetig and kontinuerlich (Cantor 1883: 903, 919) and the use of immanent-transient (Cantor 1883: 895–896; Husserl 1966: 291, 300), above all in the reversal of the coupling reel and real in Husserl (1922: 411, 632, 633), compared to Cantor (1883: 882).
Abbreviations
- Hua::
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Husserliana Gesammelte Werke, Den Haag/Dordrecht/Boston/Lancaster, M. Nijhoff, and, from vol. XXVII, Dordrecht/Boston/London, Kluwer.
References
Albertazzi, L. (Ed.). (1992). Unfolding perceptual continua. Amsterdam-Philadelphia, John Benjamins.
Becker, O. (1923). Bemerkungen zur phänomenologischen Begründung der Geometrie und ihre physikalischen Anwendungen. Jahrbuch für Philosophie und phänomenologische Forschung, VI, 385–560.
Becker, O. (1927). Mathematische Existenz. Jahrbuch für Philosophie und phänomenologische Forschung, VIII, 441–809.
Bergson, H. (1889). Time and free will. An essay on the immediate data of consciousness (Engl. ed. by F. L. Pogson). New York: Dover Publications (2001).
Bergson, H. (1922). Duration and simultaneity (Engl. ed. by L. Jacobson). The Bobbs-Merrill Company (1965).
Boyer, C. B. (1968). A history of mathematics. New York: Wiley.
Brentano, F. (1976). Raum, Zeit, Kontinuum. hrsg. von S. Kröner- R. M. Chisholm. Hamburg: Felix Meiner.
Cantor, G. (1883). Foundations of a general theory of manifolds: A mathematico-philosophical investigation into the theory of infinite (Engl. ed. by W. Ewald), in Id., From Kant to Hilbert, II. Oxford: Clarendon Press (pp. 878–920) (1996).
Conrad-Martius, H. (1923). Realontologie. Jahrbuch für Philosophie und phänomenologische Forschung, VI, 159–333.
Conrad-Martius, H. (1958). Der Raum. München: Kösel-Verlag.
Casati, R., & Varzi, A. (1999). Parts and places. The structures of spatial representation. Cambridge: MIT Press.
Cohen, H. (1883). Das Princip der Infinitesimal-Methode und seine Geschichte. Berlin: Dümmler.
Costa, V. (1999). Estetica trascendentale fenomenologica. Sensibilità e razionalità nella filosofia di Edmund Husserl. Milan: Vita e Pensiero.
Derrida, J. (1962). Edmund Husserl’s origin of geometry: An introduction (Engl. ed. by J. P. Leavey). Lincoln and London: University of Nebraska Press (1989).
Derrida, J. (1967). Violence and metaphysics: An essay on the thought of E. Levinas. In Id., Writing and difference (Engl. ed. by. A. Bass). Routledge, London-New York (pp. 97–192) (2001).
Duval, R. (1990). Durée ou discontinuité de l’apparaître: le choix phénoménologique. In S. Ijsselin1g (Ed.), Husserl-Ausgabe und Husserl-Forschung (pp. 81–106). Dordrecht: Kluwer.
Fink, E. (1957). Nachdenkliches zur ontologischen Frühgeschichte von Raum – Zeit – Bewegung. Den Haag, M. Nijhoff.
Fink, E. (1960): Nietzsche’s philosophy (Engl. ed. by G. Richter). London-New York: Continuum.
Friedman, M. (1985). Kant’s theory of geometry. The Philosophical Review, 44(4), 455–506.
Heidegger, M. (1946). Der Spruch des Anaximander. In Id., Holzwege (1950), GA, Bd. 5, hrsg. von F.-W. von Herrmann, Frankfurt a. M: Klostermann, pp. 321–373 (1977).
Heidegger, M. (1991). Grundbegriffe (1941), GA, Bd. 51, hrsg. von P. Jaeger. Frankfurt a. M.: Klostermann.
Heine, E. (1872). Die Elemente der Functionenlehre. Journal für die reine und angewandte Mathematik, 74, 172–188.
Helmholtz, H. (1887). Numbering and measuring from epistemological viewpoint (Engl. ed. by W. Ewald). In Id., From Kant to Hilbert, II (pp. 727–752). Oxford: Clarendon Press (1996).
Herbart, J. F. (1829). Allgemeine Metaphysik nebst den Anfängen der philosophischen Naturlehre. In Id., Sämtliche Werke (Ed.), vol. VIII, hrsg. von K. Kehrbach-O. Flügel, Neudruck der Ausgabe, Legensalza 1893, Scientia, Aalen 1964.
Hirsch, E. (1982). The concept of identity. Oxford: Oxford University Press.
Hofmann, H. (1912). Untersuchungen über Empfindungsbegriff. Archiv für Psychologie, XXVI, 1–135.
Husserl, E. (1911). Philosophie als strenge Wissenschaft. In Id., Aufsätze und Vorträge (1911–1921), Hua XXV (pp. 3–62).
Husserl, E. (1913): Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie, Hua III/1.
Husserl, E. (1922). Logische Untersuchungen. Zweiter Band, Hua XIX/1.
Husserl, E. (1966). Zur Auflösung des Schemas Auffasungsinhalt-Auffasung. In Id., Zur Phänomenologie des inneren Zeitbewusstseins, Hua X (pp. 269–334).
Husserl, E. (1973a). Zur Phänomenologie der Intersubjektivität. Zweiter Teil (1921–1928). Hua XIV.
Husserl, E. (1973b). Der Widersinn des transzendentalen Realismus. Gegen den Einwand des Solipsismus (1924), Text nr. 17, in Hua XIV (pp. 341–356).
Husserl, E. (1973c). Ding und Raum. Vorlesungen 1907, Hua XVI.
Husserl, E. (1974a). Formale und transzendentale Logik. Versuch einer Kritik der logischen Vernunft (1929), Hua XVII.
Husserl, E. (1974b). Das ideale Erkenntnissubjekt der formalen Logik und die formale Apriorität rein rationaler Gegenstände. Kontingent-Materiales und formales Apriori (1920/21), Ergänzender Text V, in Hua XVII, pp. 379–393.
Husserl, E. (1983). Versuch über der Raum, in Hua XXI.
Husserl, E. (1984a). Einleitung in die Logik und Erkenntnistheorie. Vorlesungen 1906/07, Hua XXIV.
Husserl, E. (1984b). Fours letters from Edmund Husserl to Hermann Weyl (ed. by D. van Dalen). Husserl Studies, 1, 1–12.
Husserl, E. (1986). Aufsätze und Vorträge (1911–1921), Hua XXV.
Husserl, E. (2001a). Natur und Geist. Vorlesungen Sommersemester 1927, Hua XXXI.
Husserl, E. (2001b). Die Bernauer Manuskripte über das Zeitbewusstsein (1917/18), Hua XXXII.
Husserl, E. (2003). Der formal allgemeine Beweis des transzendentalen Idealismus und die Undenkbarkeit von realem Sein ohne leibliche Subjektivität. Wesentliche Beziehung der Weltkonstitution zu einer offenen Vielheit von Subjekten. Geburt und Tod jedes Subjekts als apriorische Notwendigkeiten (1914–15), Text nr. 7, in Hua XXXVI (pp. 132–145).
Husserl, E. (2005a). Einführung in die Phänomenologie der Erkenntnis (1909), Hua, Materialen, VII.
Husserl, E. (2005b). Logische Untersuchungen. Ergänzungsband. Zweiter Teil. Texte für Neufassung der VI. Untersuchung. Zur Phänomenologie des Ausdrucks und der Erkenntnis (1893/94–1921), Hua XX/2.
Husserl, E. (2009). Untersuchungen über Urteilstheorie (1893–1918), Hua XL.
Husserl, E. (2009a). Uneigentliche Affirmationen (1893/94), in Hua XL.
Husserl, E. (2009b). Urteile verschiedener Art aufgrund der blosse Vorstellung (1908), Text nr. 17, in Hua XL.
Husserl, E. (2009c). Empirische und apriorische Aussagen über das erscheinende Ding (1908), Text nr. 19, in Hua XL.
Husserl, E. (2012). Zur Lehre vom Wesen und zur Methode der eidetischen Variation (1891–1935). Hua XLI.
Ingarden, R. (1922). Intuition und Intellekt bei Henri Bergson. Jahrbuch für Philosophie und phänomenologische Forschung, V, 286–461.
Kant, I. (1787). Kritik der reinen Vernunft, hrsg. von J. Timmermann-H. Klemme. Hamburg: Meiner (1956).
Kerszberg, P. (1986). Le principe de Weyl et l’invention d’una cosmologie non-statique. Archive for History of Exact Sciences, 35(1), pp. 1–89.
Kerzsberg, P. (2007). Perseverance and adjustment: On Weyl’s phenomenological philosophy of nature. In L. Boi, P. Kerszberg, & F. Patras (Eds.), Rediscovering Phenomenology. Phenomenological essays on mathematical beings, physical reality, perception and consciousness (pp. 173–194). Dordrecht: Spinger.
Koyré, A. (1922). Bemerkungen zu den zenonischen Paradoxen. Jahrbuch für Philosophie und phänomenologische Forschung, V, 603–628.
Koyré, A. (1999). Insolubilia. Eine logische Studie über die Grundlagen der Mengenlehre, hrsg. von P. Zambelli. Giornale critico della filosofia italiana, VI/18, 303–350.
Kripke, S. (1980). Naming and necessity. Oxford: Basil Blackwell.
Kuhn, T. (1961). The function of measurement in modern physical science. Isis, 52(2), 61–93.
Levinas, E. (1961). Totality and infinity. An essay on exteriority (Engl. ed. by A. Lingis). Pittsburgh : Duquesne University Press (1969, 200720).
Lipps, H. (1923). Die Paradoxien der Mengenlehre. Jahrbuch für Phänomenologie und phänomenologische Forschung, VI, 561–571.
Lohmar, D. (1998). Erfahrung und kategoriales Denken. Hume, Kant und Husserl über vorprädikative Erfahrung und prädikative Erkenntnis. Dordrecht: Springer.
London, F. (1923). Über die Bedingungen der Möglichkeit einer deduktiven Theorie. Jahrbuch für Phänomenologie und phänomenologische Forschung, VI, 335–384.
Mancosu, P. (2010). The adventure of reason. Interplay between philosophy of mathematics and mathematical logic, 1900–1940. Oxford: Oxford University Press.
Marty, A. (1916). Raum und Zeit, hrsg. von J. Eisenmeier, A. Kastil, O. Kraus, Halle, Niemeyer.
Meinong, A. (1896). Über die Bedeutung des Weber’schen Gesetzes. Beiträge zur Psychologie der Vergleichens und Messens. Leopold Voss: Hamburg-Leipzig.
Meinong, A. (1903). Bemerkungen über den Farbenkörper und das Mischungsgesetz. Zeitschrift für Psychologie und Physiologie der Sinnesorgane, 33, 1–80.
Mulligan, K. (1991). Colors, corners and complexity; Mejnong & Wittgenstein on some internal relations. In B. C. van Fraassen et alii (Eds.), Existence and explanation: Essays in Honor of Karel Lambert (pp. 77–101). Dordrecht: Kluwer.
Nagel, E. (1931). Measurement. Erkenntnis, 2, 313–335.
Ortiz Hill, C. (1997). Did Georg Cantor influence Edmund Husserl? Synthese, 113(1), 145–170.
Peckhaus, V. (1990). Hilbertprogramm und Kritische Philosophie. Das Göttinger Modell interdisziplinär Zusammenarbeit zwischen Mathematik und Philosophie. Göttingen: Vandenhoeck und Ruprecht.
Peckhaus V. (Ed.). (2005). Oskar Becker und die Philosophie der Mathematik. München: Wilhelm Fink Verlag.
Peirce, C. S. (1898). The logic of continuity. In Id., K. L. Ketner (Ed.), Reasoning and the logic of things. The Cambridge Conferences Lectures of 1898 (pp. 242–268). Harvard University Press, Cambridge-London (1992).
Pettoello, R. (1986). Idealismo e Realismo. La formazione filosofica di J. F. Herbart. Florence: La Nuova Italia.
Pöggeler, O. (1986). Kontingenz und Rationalität in der phänomenologischen Wissenschaftstheorie. In Aa. Vv., Vernunft und Kontingenz. Rationalität und Ethos in der Phänomenologie (Phänomenologische Forschungen, 19) (pp. 10–37).
Putnam, H. (1995). Peirce’s Continuum. In K. L. Ketner (Ed.), Peirce and contemporary thought: Philosophical inquiries (pp. 356–365). New York: Fordham University Press.
Rang, B. (1990). Phänomenologie der materiellen Natur. Klostermann: Frankfurt.
Reinach, A. (1921). Über das Wesen der Bewegung (1914). In Id., Gesammelte Schriften, hrsg. von E. Stein, Halle, Niemeyer.
Riemann, B. (1868). On the hypotheses which lie at the foundation of geometry (Engl. ed. by W. Ewald). In Id., From Kant to Hilbert, II (pp. 652–661). Oxford : Clarendon Press (1996).
Scaravelli, E. (1973). Scritti kantiani. La Nuova Italia: Florence.
Schapp W. (1910). Beiträge zur Phänomenologie der Wahrnehmung, hrsg. von C. F. Graumann, Wiesbaden, B. Heymann 1976.
Schuhmann, K. (1977). Husserl-Chronik. Denk- und Lebensweg Edmund Husserls, Den Haag, M. Nijhoff.
Sider, T. (1997). Four-dimensionalism. The Philosophical Review, 106(2), 197–231.
Sokolowski R. (1987). Measurement. American Philosophical Quarterly, 24(1), 71–79.
Volkert K. (1994), Anschauliche Unmöglichkeit versus logische Unmöglichkeit zur erkenntnistheoretischen Diskussion über die nichteuklidische Geometrie. In G. Meggle, V. Wessels (Ed.), Analyomen, 1 (Proceedings of 1st Conference Perspectives in Analytical Philosophy). De Gruyter: Berlin.
Weyl, H. (1918). The continuum: A critical examination of the foundation of analysis (Engl. ed. by S. Pollard, T. Bole). New York : Dover Publications (1987).
Weyl, H. (1919). Raum, Zeit, Materie. Vorlesungen über allgemeine Relativitätstheorie (3rd ed.). Berlin: Springer.
Wittgenstein, L. (1975). Philosophical remarks (R. Rhees (Ed.), Engl. trans. by R. Hargreaves, R. White).
Wittgenstein, L. (1991). Remarks on color (1950–51) (G. E. M. Anscombe (Ed.), Engl. trans. by L.L. McAlister-M. Schättle). New Jersey: Wiley-Blackwell.
Zalamea, F. (2012). Peirce’s logic of continuity: A conceptual and mathematical approach. Chesnut Hill: Docent Press.
Zambelli, P. (1999). Alexandre Koyré im “Mekka der Mathematik”. Koyrés Göttinger Dissertationsentwurf. N. T. M., n. s., 7, 208–230.
Zanfi, C. (2013). Bergson e la filosofia tedesca 1907–1932. Quodlibet: Macerata.
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Masi, F. (2017). Antinomies of Measure. Phenomenology and Spatial Continuity (Husserl, Becker, Weyl). In: Catena, M., Masi, F. (eds) The Changing Faces of Space. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-66911-3_9
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