# Constructive Recursive Functions, Church’s Thesis, and Brouwer’s Theory of the Creating Subject: Afterthoughts on a Parisian Joint Session

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## Abstract

The first half of the paper discusses recursive versus constructive functions and, following Heyting, stresses that from a constructive point the former cannot replace the latter. The second half of the paper treats of the Kreisel-Myhill theory CS for Brouwer’s Creating Subject, and its relation to BHK meaning-explanations and Kripke’s Schema. Kripke’s Schema is reformulated as a principle and shown to be classically valid. Assuming existence of a verification-object for this principle, a modification of a proof of conservativeness of Van Dalen’s, is shown to give a relative BHK meaning explanation for the Kreisel-Myhill connective. The result offers an explanation of why Kripke’s Schema can be used as a replacement of the Theory of Creating Subject when formulating Brouwerian counter-examples. It also shows that the Theory of Creating Subject is classically valid.

## Keywords

Recursive Function Recursion Equation Natural Deduction Constructive Function Existential Quantifier## Notes

### Acknowledgements

I was an invited speaker at the Joint Session in Paris and spoke on the notion of function in constructive mathematics. However, problems of health in 2007–2008 unfortunately prevented me from submitting anything to its Proceedings, whence it was a pleasant surprise, as well as a challenging task gladly undertaken, when Michel Bourdeau and Shahid Rahman requested that I write an introductory essay for the Proceedings volume. Accordingly, in these *Afterthoughts* I return to some of the things I said in my Paris talk in 2006, as well as present later reflections, caused by rethinking some of the issues and rereading many of the original sources. The material on Brouwer’s Creating Subject that is presented here was not part of my Paris lecture. It stems from research carried out during a Visiting Professorship at Lille, February to April 2012, and I am grateful to my host Shahid Rahman and his students for being such a keen audience. The criticism of the Kreisel-Troelstra-Dummett account was first presented in a Lille seminar April 5, 2012, with, appropriately enough, several of the participants of the Paris Joint Session also present, to wit, my Paris hosts Van Atten, Bourdeau, and Fichot.

I am indebted to Yannis Moschovakis for his valuable comments in the Paris discussion, but above all for sharing with me his correspondence with Alonzo Church and granting permission to publish this important document. I am also indebted to Thierry Coquand for discussion of Heyting’s views, as well as for providing me with a copy of the little known paper by Skolem that is referred to in his contribution to the present volume. Andrew Hodges and Tim Krabbé answered questions respectively on Alan Turing and on Langstaff. Michèle Friend gave helpful comments on the first, non-technical part. Furthermore, as always in recent years, I am indebted to Mark van Atten and Per Martin-Löf for stimulating conversations and valuable comments. In particular, Van Atten, by means of incisive questioning, forced me to revise my formulations on Leibniz and *calculemus*. He, as wel as Zoe McConaughey, helped greatly in producing the final draft of the manuscript. The IHPST, Paris, in the persons of Michel Bourdeau and Jean Fichot, offered generous hospitality on two occasions in May and November 2012.

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