What is this ambiguous computational existence in the positive integers?

Question

I'm reading From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics.

The authors mention Bishop’s Foundations of Constructive Analysis:

The successful formalization of mathematics helped keep mathematics on a wrong course. The fact that space has been arithmetized loses much of its significance if space, number, and everything else are fitted into a matrix of idealism where even the positive integers have an ambiguous computational existence.

What is this ambiguous computational existence in the positive integers?

Answer 1

In order to explain this it is necessary to understand well the view point of Bishop on mathematics (and philosophy of mathematics) in the context of construcivism and intuitionism. This is crucial becasue when he is saying "even the positive integers have an ambiguous computational existence", this is meant from that point of view. See for instance here >>>.

One deeper example of intuitionism in number theory refers for instance to Heyting Arithmetic see here>>>.

But more directed to your question, I think a nice reference I have is the article by Kleene, which is about the interpretation of the intuitionistic number theory >>> here

You may have also a look through L.E.J. Brouwer's work our lectures, and I think Bishop's words very much lean together with Brouwer for instance if you see here>>> and:

for the elementary theory of natural numbers, the principle of complete induction and more or less considerable parts of arithmetic and of algebra, exact existence, absolute reliability and non-contradictority were universally acknowledged, independently of language and without proof. As for the continuum, the question of its languageless existence was neglected, its establishment as a set of real numbers with positive measure was attempted by logical means and no proof of its non-contradictory existence appeared. For the whole of mathematics the four principles of classical logic were accepted as means of deducing exact truths.

This should bring you nearer to the intuitive mystics behind the sentence of Bishop. Very roughly said, by "ambiguous computational existence", Bishop is claiming that the intuitive and constructive, so called exact thought processes are being missed.