You might find relevant Errett Bishop' <https://en.wikipedia.org/wiki/Errett_Bishop>s work on "constructive mathematics". I used to have a copy of his book, "Foundations of Constructive Analysis", but I cannot find it right now. As I recall his constructive definition of a real number is the same as that given here: https://en.wikipedia.org/wiki/Constructivism_(mathematics)#Example_from_real... In his book he has a definition of (in) equality of real numbers and if memory serves in his system the law of trichotomy need not hold (--since he doesn't accept the law of the excluded middle in proofs.) Nevertheless Bishop was able to prove constructive versions of surprisingly many results in analysis. On Mon, Jun 5, 2017 at 10:45 AM, James Propp <jamespropp@gmail.com> wrote:
In recent email conversation with a polite but insistent .999... dissenter, I've been made aware of strands of thought that rejects the mathematical concept of infinity as being unrealistic, incoherent, and repugnant (three different charges requiring three different sorts of defense).
There are books and articles and on-line videos that approach the topic of infinity from different angles, but are there any that address these attacks head-on?
It's possible that what's really being rejected (at least by some dissenters) is not the concept of infinity in isolation but the whole enterprise of making deductions in imaginary worlds (like the world of Euclidean geometry) to draw conclusions that are applicable in our world. A convincing defense of "unrealistic" concepts like a Euclidean point or the set of all counting numbers should in my opinion include an admission that we don't know why the mathematical enterprise has been so successful (the "unreasonable effectiveness" phenomenon).
What's been done along these lines?
One thing that's relevant, sort of, is an essay by Isaac Asimov in which young Isaac gets into an argument with a philosophy professor, challenging him to produce half a piece of chalk and then pointing out that it's probably not exactly HALF. But I think the philosopher was onto something when he got the whole in-class argument started by insisting that mathematicians are idealists because they believe in imaginary numbers. Old Isaac's point in his essaywas that complex numbers, properly understood, are no more fanciful than ordinary fractions; but you can turn this around and say that ordinary fractions, properly understood, are no less fanciful than complex numbers.
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