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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.

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    related posts : [are-all-functions-in-bishops-constructive-mathematics-continuous](http://mathoverflow.net/questions/164694/are-all-functions-in-bishops-constructive-mathematics-continuous) and [did-bishop-heyting-or-brouwer-take-partial-functions-seriously](http://mathoverflow.net/questions/224405/did-bishop-heyting-or-brouwer-take-partial-functions-seriously). – 2017-01-07

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

You can see :

and the introductive discussion into :

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

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

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.

Useful comments can be found also into :

And see also Constructive Mathematics.

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    I find it quite hard to understand, do you have a simple example that shows exactly the point that thier observation to the real functions is changing ? I am getting that Bishopp stand as neutral between classical and intuitistic math. I don't really understand if he, as brouwer, thinks that every real function is continuous. – 2017-01-07
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    Bishop's system is consistent with all real functions being continuous, but is also consistent with classical mathematics. Brouwer claimed a prove that all real functions are continuous, so his system would not be consistent with classical mathematics. – 2017-01-07
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    Does bishopp's real number defenition follows the law of trichometry ? – 2017-01-07
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    @NotSure: trichotomy is not provable in Bishop's system, nor in any of the usual systems of constructive mathematics. In the end, trichotomy should be viewed as a property of the reals that is closely tied to classical logic. – 2017-01-09