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<p>I'll bite. Why should a real numbers not be defined as an
equivalence class of Cauchy sequences of rational numbers?</p>
<p>Joe<br>
</p>
<div class="moz-cite-prefix">On 7/31/20 1:03 PM, Frank Wimberly
wrote:<br>
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Glen et al,.
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<div dir="auto">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.</div>
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<div dir="auto">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...</div>
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<div dir="auto">The book follows</div>
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<div data-smartmail="gmail_signature" dir="auto">--- <br>
Frank C. Wimberly<br>
140 Calle Ojo Feliz, <br>
Santa Fe, NM 87505<br>
<br>
505 670-9918<br>
Santa Fe, NM</div>
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<pre class="moz-signature" cols="72">--
Joe
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