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Propositions as Intentions

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Abstract

I argue against the interpretation of propositions as intentions and proof-objects as fulfillments proposed by Heyting and defended by Tieszen and van Atten. The idea is already a frequent target of criticisms regarding the incompatibility of Brouwer’s and Husserl’s positions, mainly by Rosado Haddock and Hill. I raise a stronger objection in this paper. My claim is that even if we grant that the incompatibility can be properly dealt with, as van Atten believes it can, two fundamental issues indicate that the interpretation is unsustainable regardless: (1) it is hard to determine, without appealing to propositional intentions on pain of circularity, what intention a proof-object should be understood as a fulfillment of; (2) due to a difficult fulfillment dilemma, it is unclear, at best, what the object of an intention corresponding to a proposition is.

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Notes

  1. Brouwer’s philosophy itself lacks a foundation to its own satisfaction [(van Atten, 2006, ch. 5.5).

  2. See especially Tieszen (1989) and van Atten (2004, 2006, 2017). The phenomenological interpretation is also appealed to some extent in Sundholm (1983) and Martin-Löf (1985), figures prominently in Franchella (2007), and is assumed in the background of Martin-Löf (1993) and van der Schaar (2011).

  3. Roughly, a construction is a proof-object of a proposition if it satisfies certain conditions associated with it.For example, a proof-object of the conjunction \(A \wedge B\) usually takes the form of a pair containing a proof-object of A and a proof-object of B (Troelstra and van Dalen, 1988).

  4. This and the following English translations are my own.

  5. Except that instead of sentences I refer directly to propositions. Negation and disjunction are explained in (Heyting 1931, pp. 113–114) and implication in (Heyting, 1934, p. 14). In his explanation of implication, Heyting actually uses the term ‘proof’ (Beweis).

  6. See e.g. (Sundholm & van Atten, 2008, §6).

  7. In an unpublished manuscript on set theory dated from 1920 in the Husserl Archives in Cologne, Husserl seriously considered constructivism as a way to avoid the threat of paradoxes, possibly under the influence of Weyl (see Rosado Haddock, 2010, pp. 28–30). There are no signs of constructivism left in FTL, his conclusive treatise on logic and mathematics.

  8. This theory of sets is, however, abandoned by Husserl shortly after the publication of Philosophy of Arithmetic (Husserl, 1981). Regardless, I want to examine the issues in connection with this particular view because it is closer to the theory of sets advocated by (Tieszen 1989, p. 143).

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Acknowledgements

I would like to thank Mark van Atten, Philipp Berghofer, Miriam Franchella, Claire Ortiz Hill, James Kinkaid, Ivo Pezlar, and an anonymous reviewer for comments on an early version of this paper. I am also indebted to Jairo da Silva and Göran Sundholm for helpful discussions on Heyting and the phenomenological approach to intuitionism.

Funding

This research was supported in part by the Lumina quaeruntur fellowship number LQ300092101 from the Czech Academy of Sciences.

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Bentzen, B. Propositions as Intentions. Husserl Stud 39, 143–160 (2023). https://doi.org/10.1007/s10743-022-09323-3

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