Are mathematical theories reducible to non-analytic foundations? (English) Zbl 1302.00054
Summary: In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process.
MSC:
| 00A30 | Philosophy of mathematics |
| 03E25 | Axiom of choice and related propositions |
| 03F40 | Gödel numberings and issues of incompleteness |
Keywords:
axiom of choice; finitistic; first-level idealization; Gödel’s incompleteness theorems; individual-substrate; infinite; intentionality; second-level idealization; Skolem-Löwenheim theoremReferences:
| [1] | Adams R (1979) Primitive thisness and primitive identity. J Philos 76(1):5–26 · doi:10.2307/2025812 |
| [2] | Bell LJ, Slomson BA (2006) Models and ultraproducts. Dover Publications, New York |
| [3] | Cohen P (1966) Set theory and the continuum hypothesis. W.A. Benjamin, New York, MA · Zbl 0182.01301 |
| [4] | French S (1989) Identity and individuality in classical and quantum physics. Australas J Philos 67(4):432–446 · doi:10.1080/00048408912343951 |
| [5] | Føllesdal D (1969) Husserl’s notion of noema. J Philos 66:680–687 · doi:10.2307/2024451 |
| [6] | Goldblatt R (1985) On the role of the Baire category theorem and dependent choice in the foundation of logic. J Symbol Logic 50(2):412–422 · Zbl 0567.03023 · doi:10.2307/2274230 |
| [7] | Halpern DJ, Levy A (1971) The Boolean prime ideal theorem does not imply the axiom of choice. In: Axiomatic set theory. AMS Proceedings, pp 83–134 · Zbl 0233.02024 |
| [8] | Hill CO (2010) Husserl on axiomatization and arithmetic. In: Hartimo M (ed) Phenomenology and mathematics. Springer, Berlin |
| [9] | Hill CO, Rosado Haddock GE (2000) Husserl or Frege? Meaning, objectivity, and mathematics. Open Court, La Salle · Zbl 0979.03001 |
| [10] | Husserl E (1962) Die Krisis der Europäischen Wissenschaften und die Transzendentale Phänomenologie. Hua, Band VI, hgb. W. Biemel. M. Nijhoff, Den Haag |
| [11] | Husserl E (1966) Zur Phänomenologie des Inneren Zeibewusstseins. Hua, Band X, hgb. R. Boehm. M. Nijhoff, Den Haag |
| [12] | Husserl E (1974) Formale und Transzendentale Logik. Hua, Band XVII, hgb. P. Janssen. M. Nijhoff, Den Haag |
| [13] | Husserl E (1975) Logische Untersuchungen. (Prolegomena zur Reinen Logik), Hua, Band XVIII, hgb. E. Holenstein. M. Nijhoff, Den Haag · JFM 23.0058.01 |
| [14] | Husserl E (1976) Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie, Erstes Buch, Hua, Band III/I, hgb. K. Schuhmann. M. Nijhoff, Den Haag |
| [15] | Husserl E (1984) Logische Untersuchungen. Hua, Band XIX1, (zweiter Band, erster Teil), hgb. U. Panzer. M. Nijhoff, Den Haag · JFM 23.0058.01 |
| [16] | Husserl E (2002) Logische Untersuchungen, Ergänzungsband, Erster Teil, Hua, Band XX/I, hgb. U. Melle. Kluwer, Dordrect |
| [17] | Kanamori A (2008) Cohen and set theory. Bull Symbolic Logic 14(3):351–378 · Zbl 1174.03001 · doi:10.2178/bsl/1231081371 |
| [18] | Kleene SC (1980) Introduction to metamathematics. North-Holland Publication Co, New York |
| [19] | Krause D, Coelho AMN (2005) Identity, indiscernibility, and philosophical claims. Axiomathes 15:191–210 · doi:10.1007/s10516-004-6678-5 |
| [20] | Kunen K (1982) Set theory. An introduction to independence proofs. Elsevier, Amsterdam · Zbl 0443.03021 |
| [21] | Lavine S (1994) Understanding the infinite. Harvard University Press, Cambridge · Zbl 0961.03533 |
| [22] | Livadas S (2010) Impredicativity of continuum in phenomenology and in non-cantorian theories. In: Carsetti A (ed) Causality, meaningful complexity and embodied cognition. Springer, Berlin, pp 185–199 |
| [23] | Livadas S (2011) The expressional limits of formal language in the notion of quantum observation. Axiomathes, 1–25. doi: 10.1007/s10516-011-9168-6 . (Online ISSN 1122-1151) |
| [24] | Lohmar D (2002) Elements of a phenomenological justification of logical principles, including an appendix [...] on the transfiniteness of the set of real numbers. Philos Math 10(3):227–250 · Zbl 1025.03006 · doi:10.1093/philmat/10.2.227 |
| [25] | Popper K (1934) Logik der Forschung. Springer, Wien · Zbl 0010.24202 |
| [26] | Rosado Haddock GE (1987) Husserl’s epistemology of mathematics and the foundation of platonism in mathematics. Husserl Stud 4(2):81–102 · doi:10.1007/BF00365248 |
| [27] | Schoenfield J (1967) Mathematical logic. Addison Wesley Publication, Reading |
| [28] | Simmons FG (1963) Introduction to topology and modern analysis. McGraw-Hill Kogakusha Ltd, Tokyo · Zbl 0105.30603 |
| [29] | Sokolowski R (1974) Husserlian meditations. Northwestern University Press, Evanston |
| [30] | Tieszen R (2005) Phenomenology, logic, and the philosophy of mathematics. Cambridge University Press, Cambridge · Zbl 1113.00004 |
| [31] | van Atten M (2006) Brouwer meets Husserl: on the phenomenology of choice sequences, synthese library, vol 335, Springer, Dordrecht · Zbl 1119.03001 |
| [32] | van Atten M, Van Dalen D, Tieszen R (2002) Brouwer and Weyl: the phenomenology and mathematics of the intuitive continuum. Philos Math 10(3):203–226 · Zbl 1032.01025 · doi:10.1093/philmat/10.2.203 |
| [33] | van Dalen D (2004) Logic and structure. Springer, Berlin · Zbl 1048.03001 |
| [34] | van Fraasen B (1991) Quantum mechanics: an empiricist view. Clarendon Press, Oxford |
| [35] | Woodin HW (2001) The continuum hypothesis I & II. N Am Math Soc 48–6 (resp. 567-576 & 681-690) |
| [36] | Woodin HW (2011) The realm of the infinite. In: Heller M, Woodin WH (eds) Infinity: new research frontiers. Cambridge University Press, Cambridge |
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