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On distance and proximity between Dummett and Brouwer

Mark van Atten

Abstract

This paper asks what Brouwer might have replied to Dummett's interpretation of intuitionism. Complementing earlier literature, it treats Dummett's rejection of the ontological approach; the charge of psychologism and solipsism; indefinite extensibility; and predicativity. It is argued that Dummett's direct arguments against Brouwerian intuitionism do not settle the matter, and that, on the latter two themes,Dummett's position comes closer to Brouwer's than his own account suggests. The remaining philosophical distance, however, is substantial.

On distance and proximity between Dummett and Brouwer

Abstract

This paper asks what Brouwer might have replied to Dummett's interpretation of intuitionism. Complementing earlier literature, it treats Dummett's rejection of the ontological approach; the charge of psychologism and solipsism; indefinite extensibility; and predicativity. It is argued that Dummett's direct arguments against Brouwerian intuitionism do not settle the matter, and that, on the latter two themes,Dummett's position comes closer to Brouwer's than his own account suggests. The remaining philosophical distance, however, is substantial.

Paper Structure

This paper contains 15 sections, 2 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Dummett's argument against the ontological route begs the question
  3. Brouwer's intuitionism is not psychologistic, and in some sense not solipsistic
  4. Dummett
  5. Brouwer
  6. Comments
  7. Dummett's indefinitely extensible concept and Brouwer's denumerably unfinished set are similar, but not co-extensive
  8. Dummett
  9. Brouwer
  10. Comments
  11. Brouwer's intuitionism is predicative, for a reason that Dummett sees but does not generalise
  12. Brouwer
  13. Dummett
  14. Concluding remark
  15. Acknowledgement.

Key Result

Theorem 1

This form is decidedly weaker than the full theorem Lawvere1969 in that (a) it is formulated only for sets; (b) it requires representability not for arbitrary mappings from $A$ to $B$, but for one specific $f$; and (c) representability is not a global requirement on the domain of $f$ but only at the Let $h : B \to B$, and define Assume that $g$ is such that diagonal representability holds for $f$

Theorems & Definitions (3)

  • Theorem : Lawvere’s fixed point theorem, diagonal form
  • proof
  • Corollary