Did Errett Bishop and L.E.J. Brouwer treat the real numbers differently?

Question

I know how Brouwer treated the real numbers and I know both mathematicians followed constructive mathematics rules. I want to understand if there had been some some disagreement between them regarding the real numbers.

Answer 1

They treated the continuum differently: the differences are more on real functions.

You can see :

• Errett Bishop, Foundations of constructive analysis (1967), Ch.1 A constructivist manifesto, page 1-on,

and the introductive discussion into :

• Anne Troelstra & Dirk van Dalen, Constructivism in mathematics: An Introduction. Volume I (1988), page 28:

[Bishop] cannot, like Brouwer, show that all real functions are continuous.

For an explicit criticism by Bishop of Brouwer's "metaphysical speculation", see:

• Errett Bishop & Douglas Bridges, Constructive Analysis (1985), page 9.

See also Myhill's review of Bishop's book, with a summary comparison of Bishop's and Brouwer's concepts:

An important difference is that the notion of "free choice sequence" is dropped from Bishop's mathematics and the only sequences used are lawlike ones. Another difference is that no results are stated which contradict theorems of classical mathematics; for instance, instead of the intuitionistic result that all real functions are continuous, we have the metatheorem that the only functions that can be proved to exist are provably continuous. Both these differences make the mathematics in the book look far more familiar to the classical mathematician than do Brouwer's.

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Useful comments can be found also into :

• Michael Beeson, Foundations of constructive mathematics (1985), Ch.III Some Different Philosophies of Constructive Mathematics, page 47-on.

And see also Constructive Mathematics.