[FRIAM] Constructive mathematics.

[Frank Wimberly]

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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