Abstract
We are hearing a great deal at this conference about phenomenology and its connection with ontology, logic and mathematics. I want to put some of this into historical perspective by talking about Kant and Brouwer, two thinkers who are known for their analyses of consciousness, and for taking these analyses from phenomenology to the ontology, and ultimately to the logic of ordinary empirical discourse and of mathematics. Specifically I want to use the case studies of Kant and Brouwer in order to precisely formulate just how one can connect phenomenology with ontology, and, once that connection is made, how it in turn effects matters of logic. For it seems to me that these connections are not always well understood in contemporary discussions of Kant and Brouwer, and of these issues in general.
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References
See for instance Brouwer’s”Discours Final” in Les Methodes Formelles en Axiomatique”, Colloque CNRS, Paris, 1950, p. 75, (reprinted in Brouwer’s Collected Works, v. 1, A. Heyting (ed.) North Holland, 1975, p. 503).
See Chapter II of Brouwer’s Ph. D. thesis, Over de grondslagen der wiskunde, Amsterdam, 1907, (translated as “On the foundations of mathematics”, in Collected Works, I, pp. 11–101); Brouwer’s innaugural lecture”Hei wezen der meetkunde”, Amsterdam, 1909, (translated as “The nature of geometry” in Collected Works, I, pp. 112–120); and “Intuitionism and Formalism”, Bull. Amer. Math. Soc., v. 20, 1913, pp. 81–96, (reprinted in Collected Works, I, pp. 123–138).
A. Heyting,”Die formalen Regeln der intuitionistischen Logik”, Sitzungsber. preuss. Akad. Wiss., 1930, pp. 42–56.
See for instance G. Brittan Kant’s Theory of Science, Princeton University Press, 1978; and C. Posy, “Dancing to the Antinomy”, Amer. Phil. Quart., v. 20, 1983, pp. 81–94.
See Critique of Pure Reason, A105.
See Husserl’s Cartesian Meditations, 53–54.
This is discernable in Brouwer’s emphasis on the notion of “repeatable phenomena” in”De onbetrouwbaarheid der logische principes”, Tijdschrift voor wijsbegeerte, 2, 1908, pp. 152—158, (translated as “The unreliability of the logical principles in” in Collected Works, I, pp. 107–112); and in the notions of “iterative complexes” and “causal sequences” in his “Consciousness, Philosophy and Mathematics”, in Proc. of 10th. Internat Cong, of Phil”, Amsterdam, 1948, pp. 1235–1249, (reprinted in Collected Works, I, pp. 480–494).
See for instance Brouwer’s summaries in “Historical Background, Principles and Methods of Intuitionism”, in S. African J. Sci., v. 49, 1952, pp. 139–146 (reprinted in Collected Works, I pp. 508—515); and “Points and Spaces”, Canad. J. Math., v. 6, 1954, pp. 1—17, (reprinted in Collected Works, I pp. 522–538).
See A 524/B 552 ff.
See C. Posy, “Brittanic and Kantian Objects”, in New Essays on Kant, B. den Ouden (ed.), Peter Lang Publishing Co., 1987.
See A 792/B 820.
I. e., — A is true if and only if A is false, (A&B) is true if and only if A and B are, (AvB) is true if and only if A is or B is, etc.
See S. Kleene,”Introduction to Metamathematics”, van Nostrand, 1952.
See, e. g., “The Philosophical Basis of Intuitionistic Logic”, in Truth and Other Enigmas, Harvard University Press, 1978, pp. 215–247.
In my review of Dummett’s Elements of Intuitionism in History and Philosophy of Logic, v. 2, 1981, I pointed out that this assumption of Heyting’s and Dummett’s is not universally shared and has mixed textual evidence.
See for instance Brouwer’s”De non-aequivalentie van de constructieve en de negatieve orderelatie in het continuum”, Indag. math., v. 11, 1949, pp. 37–39, (translated as “The non-equivalence of the constructive and the negative order relation on the continuum” in Collected works, I pp. 495–496); and chapter 1 of Brouwer’s [1946] Cambridge Lectures on Intuitionism, D. van Dalen (ed.), Cambridge University Press, 1981. In “On Brouwer’s Definition of Unextendable Order” (History and Philosophy of Logic, v. 1, 1980) I argued that this tensed notion of truth is helpful in interpreting Brouwer’s earlier work as well.
See, for instance,”De onbetrouwbaarheid der logische principes” (details in note 7) and”Intuitionistische Betrachtungen über den Formalismus”, Sitzber. preuss. Akad. Wiss., 1928, pp. 48—52, (reprinted in Collected Works, I pp. 409–414).
See”De onbetrouwbaarheid der logische principes” (details in note 7).
See A 480–81/B 508–9.
See for instance Dummett’s “Realism” in Truth and Other Enigmas, pp. 145–165.
Actually it is fair to say that for both Kant and Brouwer there is a context in which the empirical project is itself subsidiary to more primary “practical” concerns. This is a well known thesis in Kant interpretation. See R. Kroner, Kant’s Weltanschauung, (Tübingen, 1914; translated by J. E. Smith, Chicago, 1956) for a classic statement of this thesis. In Brouwer’s case note the primacy of “cunning”, the “move from means to end” and the general pragmatic attitude in the works cited in notes 1,2 and 7.
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Posy, C.J. (1991). Mathematics as a Transcendental Science. In: Seebohm, T.M., Føllesdal, D., Mohanty, J.N. (eds) Phenomenology and the Formal Sciences. Contributions to Phenomenology, vol 8. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2580-2_8
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