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On Why Mathematics Can Not be Ontology

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Abstract

The formalism of mathematics has always inspired ontological theorization based on it. As is evident from his magnum opus Being and Event, Alain Badiou remains one of the most important contemporary contributors to this enterprise. His famous maxim—“mathematics is ontology” has its basis in the ingenuity that he has shown in capitalizing on Gödel’s and Cohen’s work in the field of set theory. Their work jointly establish the independence of the continuum hypothesis from the standard axioms of Zermelo–Fraenkel set theory, with Gödel’s result showing their consistency to the affirmative, while Cohen’s showing it to the negative. These results serve as the cornerstone of Badiou’s mathematical ontology. In it, drawing heavily on Cohen’s technically formidable method of forcing, Badiou makes the latter result the key to his defense of the possibility of a faithful tracing of the consequences in the ‘state’ of an ‘event’ by a ‘subject’. Whereas, Gödel’s result based on the assumption of constructability becomes the pivot for criticism of the general philosophical orientation that Badiou calls ‘constructivism’. Viewed from a position internal to mathematical formalism itself, and taking into account the twentieth century developments in the relevant field, Badiou’s stance seems to be neither appreciative of the actual course of such developments, nor just to the philosophical view point that was actually maintained by Gödel. In the present paper, this concern is intended to be substantiated through an exposition of certain facts pertaining to the said developments as well as to Gödel’s philosophical inclinations.

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Notes

  1. Badiou (2006) (hereafter referred to as BE).

  2. Gödel K (1938) The consistency of the axiom of choice and the generalized continuum hypothesis. In: Proceedings of the national academy of sciences of the U.S.A., vol 24, pp 556–557.

  3. Cohen (1963–1964) The independence of the continuum hypothesis, and ibid., II. Proceedings of the national academy of sciences 50 (1963), 1143–1148 and 51 (1964) 105–110.

  4. It is through the procedure of forcing, that Cohen creates the generic sets of immeasurable cardinality, which negate the continuum hypothesis.

  5. Badiou’s usage of phrases like, ‘sayable’, ‘nameable’ etc. have to be understood against the fact that constructability in mathematics is equivalent in his theory of the ‘events’ to the possibility of linguistic representation and expression in any given situation. And just how the peculiarities of mathematical constructions curtail the scope of the events in the former, such ‘nameability’ controls the potential evental changes in the latter.

  6. See ‘The concept of truth in formalized languages’, in Tarski (1956).

  7. See Wang (1974, p. 9).

  8. See, van Heijenoort (1967, p. 368).

  9. See (BE, p. 284).

  10. Gödel (1995, pp. 254–270).

  11. See Feferman (1988, p. 111).

References

  • Badiou, A. (2006) Being and Event (trans: Feltham O). Continuum, London

  • Feferman S (1988) Kurt Gödel: conviction and caution. In: Shanker S (ed) Godel’s Theorem in focus. Croom Helm, New York, pp 96–114

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  • Gödel K (1995) What is cantor’s continuum problem? In: Feferman S, Dawson JW, Kleene SC, Moore GH, Solvay RM, van Heijenoort J (eds) Kurt Gödel: collected works, vol II. Oxford University Press, New York, pp 1938–1974

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  • van Heijenoort J (ed) (1967) From frege to godel, a source book in mathematical logic, 1879–1931. Harvard University Press, Cambridge

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Correspondence to Shiva Rahman.

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Rahman, S. On Why Mathematics Can Not be Ontology. Axiomathes 29, 289–296 (2019). https://doi.org/10.1007/s10516-018-9406-2

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