[FRIAM] Constructive mathematics.

[Joe Spinden]

I'll bite.  Why should a real numbers not be defined as an equivalence 
class of Cauchy sequences of rational numbers?

Joe

On 7/31/20 1:03 PM, Frank Wimberly wrote:
>
> Glen et al,.
>
> This morning Glen raised some interesting questions about the 
> limitations of constructive mathematics if I understood correctly.  My 
> undergraduate advisor, Errett Bishop wrote a book called Foundations 
> of Constructive Analysis.  Fortunately I inherited a copy from 
> Reuben.  I will read a few sentences aloud and use my cellphone to 
> transcribe them.  Pardon the lack of punctuation; I hope the result is 
> readable.  These come from Chapter 1,  A Constructivist Manifesto.
>
> when a classical mathematician claims he is a constructivist he 
> probably means he avoids the axiom of choice this axiom is unique and 
> its ability to trouble the conscience of the classical mathematician 
> but in fact it is not a real source of unconstructive etiz of 
> classical mathematics....the axiom of choice is used to extract 
> elements from equivalence classes where they should never have been 
> put in the first place for instance the real number should not be 
> defined as an equivalence class of cauchy sequences of rational 
> numbers there is no need to drag in the equivalence classes...almost 
> every conceivable type of resistance has been offered to a 
> straightforward realistic treatment of mathematics even by non 
> constructivists brower who has done more for constructive mathematics 
> than anyone else thought it necessary to introduce a revolutionary 
> semi mystical theory of the Continuum Weyl a great mathematician who 
> in practice suppress is constructivist convictions Express the opinion 
> that idealistic mathematics finds its justification and its 
> application to physics Hilbert who insisted on constructive eating and 
> metamathematics but believe the price of it constructed mathematics 
> was too great was willing to settle for consistency brouwer's 
> disciples join forces with the logicians in attempts to formalize 
> constructed mathematics other seat constructive truth in the framework 
> of recursive function theory still others look for a shortcut to 
> reality a point of Vantage which will suddenly revealed classical 
> mathematics in a constructive light none of these substitutes for a 
> straightforward realistic approach has worked it is no exaggeration to 
> say that a straightforward realistic approach to mathematics has yet 
> to be tried it's time to make the attempt...
>
> The book follows
>
>
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz,
> Santa Fe, NM 87505
>
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>
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-- 
Joe

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