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Husserl and the Reform of Logic

  • Mark Kac
  • Gian-Carlo Rota
  • Jacob T. Schwartz
Part of the Modern Birkhäuser Classics book series

Abstract

An unbridled and passionate interest in foundations has often been singled out as a characteristic trait of both philosophy and science in this century. Nowhere has this trend been more rampant than in mathematics. Yet, foundational studies, in spite of an auspicious beginning at the turn of the century, followed by unrelenting efforts, far from achieving their purported goal, found themselves attracted into the whirl of mathematical activity, and are now enjoying full voting rights in the mathematical senate. As mathematical logic becomes ever more central within mathematics, its contributions to the philosophical understanding of foundations wane to the point of irrelevance. Worse yet, the feverish technical advances in logic in the last ten years have dashed all hope of founding mathematics upon the notion of set, which had become the primary mathematical concept since Cantor. Equally substantial progress in the fields of algebra and algebraic geometry1 has further contributed to cast a shadow on this notion. At the other end of the mathematical spectrum, the inadequacy of naive set theory had been realized by von Neumann2 since the beginning of quantum theory, and to this day the physicist’s most important method of research remains devoid of adequate foundation, be it mathematical, logical, or philosophical.

Keywords

Ideal Object Inaccessible Cardinal Phenomenological Reduction Genetic Phenomenology Rigorous Foundation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    The fascinating story of the evolution of the notion of set in modern algebra remains to be told, perhaps because it is far from concluded. It began in the thirties, when the discoveries of the nineteenth-century geometers were subjected to rigorous foundation with the help of the newly developed algebraic methods. By way of example, the notion of point, which once seemed so obvious, has now ramified into several different concepts (geometric point, algebraic point, etc.) The theory of categories offers at present the most serious challenge to set theory.Google Scholar
  2. 2.
    Most of von Neumann’s work in pure mathematics (rings of operators, continuous geometries, matrices of high finite order) is concerned with the problem of finding a suitable alternative to Boolean algebra, compatible with the uncertainty principle, upon which to found quantum theory. Nevertheless, the mystery remains, and von Neumann could not conceal in his later years a feeling of failure over this aspect of his scientific work (personal communication from S.M. Ulam).Google Scholar
  3. 3.
    Perhaps the clearest exposition and criticism of this attitude is to be found in Nicolai Hartmann’s Zur Grundlegung der Ontologie,Fourth Part, Berlin, 1934.Google Scholar
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    Logicians will be quick to point out that this list is incomplete: one should at least add a (not a) and the empty set. These concepts have become even more problematic.Google Scholar
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    Following Orteg’ s thesis that phenomenological reduction is a response to the problematization of a belief, expounded in Obras Completas,Madrid, 1955, V, 379–410, and esp. 544–547.Google Scholar
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    This passage is partially adapted from E. Lévinas, “Reflexions sur la ‘technique’ phénomenologique,” in Husserl, Cahiers de Royaumont, Philosophie No. III, Paris, 1959, pp. 95–107.Google Scholar
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    In Les Sciences de l’homme et la phénomenologie,Paris, 1961.Google Scholar
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    The only instance of such a formalization I know of is Alonzo Church’s “A Formulation of the Logic of Sense and Denotation,” in Structure, Method and Meaning, Essays in Honor of Henry M. Sheffer,New York, 1951, pp. 3–24. Unfortunately, Church’s lead seems not to have been followed up, partly because the reading of his paper is a veritable obstacle course. We hazard the hypothesis that Husserl’s Third Investigation could be subjected to similar formalization without excessive retouching.Google Scholar
  9. 9.
    In the well-known note “Une Idée fondamentale de Husserl: l’intentionnalité,” Situations I,Paris, 1947, pp. 31–35.Google Scholar
  10. 10.
    This felicitous adjective is used in this sense by Jean Wahl in L’ouvrage posthume de Husserl: La Krisis,Paris, 1965.Google Scholar
  11. 11.
    Lord Rutherford used to remark acidly that all science is divided into two parts: physics and stamp-collecting.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Mark Kac
  • Gian-Carlo Rota
    • 1
  • Jacob T. Schwartz
    • 2
  1. 1.Dept. of Mathematics and PhilosophyMITCambridgeUSA
  2. 2.Courant InstituteNew YorkUSA

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