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
505 670-9918
Santa Fe, NM
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