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I'm looking Bishop's Foundations of Constructive Analysis.

I decided to look at the Amazon's page searching for some elucidative review about if it's a worthy read, I've found this review. The first phrase of the second paragraph says:

The point is that, if you want, you can base analysis on the natural numbers rather than on sets.

I don't get the specific meaning of this nor what difference it would make, we build sets of natural numbers on analysis, right? What's the point? A total avoidance of the concept of sets? For what purpose? Bishop mentions a lot about computational existence and computational ambiguity - I'm not really sure but I feel that they are connected with the basing of analysis on natural numbers (instead of sets) somehow.

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I'll try to say something more in a minute, but I wouldn't take this reviewer's comment as whatsoever indicative of the motivation behind Bishop's work. I'm more familiar with the classical intuitionists, but I don't think Bishop's constructivism is primarily about taking the naturals as more directly given than sets, either. – Kevin Carlson Jul 29 '13 at 6:55

This is a very soft question, so my answer may be utterly useless, but I'll make a go at it. The core of the constructivist objection to classical mathematics is that statements asserting the existence of some object are of no use to humans unless they come with some way of describing that object as precisely as desired in a finite amount of time.

Constructivists are frustrated by the fact that we can abstractly define sets and functions which encode information that's not currently available to us. For instance, let $S=\{a\}$, where $a=1$ if Goldbach's conjecture holds and $a=0$ otherwise. For now, we still don't know whether $S=\{0\}$ or $S=\{1\}$. In mainstream mathematics, that's no impediment to claiming $S$ exists, since it's certainly exactly one of the two options. But constructivists think it's useless to make abstract existence claims about objects whose properties we can't describe.

So for Bishop all proofs of existence will come essentially with a computer program that outputs a description of the object proven to exist. That, I think, is at the heart of his approach, moreso than a distaste for sets. I think this is also a somewhat more sympathetic position than the "sets bad, $\Bbb{N}$ good" of the review you quote. But the emphasis on constructions, i.e. on computations, does correlate with the tendency of constructivists to claim with Kronecker that the natural numbers, with their arithmetic structure and the possibility of induction, are intuitively and unshakably given to us. (computations are only interesting if they terminate in a finite time, so we'd better naturally understand what "finite" means!)

I think this is a silly position, one which the history of thought about number pretty strongly denies, but I'm not convinced it can't be gotten around in developing the heart of a constructivist mathematics. If you're still interested, you should read Bishop's first chapter, as my outsider's description of the position pales in comparison to his fervent polemic.

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