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Constructive mathematics.

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Frank Wimberly-2

Constructive mathematics.

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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Frank Wimberly-2

Re: Constructive mathematics.

Also

the Axiom of Choice can be shown to entail the cardinal rule of classical logic, the law of excluded middle—the assertion that A∨¬AA∨¬A for any proposition AA. To be precise, using the rules of intuitionistic logic within our augmented language LL, we shall derive[15] the law of excluded middle from AC1L conjoined with the following additional principles:

Predicative Comprehension:
∃X∀x[X(x)↔ϕ(x)]∃X∀x[X(x)↔ϕ(x)], where ϕϕ contains no bound function or predicate variables.

Extensionality of Functions:
∀X∀Y∀F[X≈Y→FX=FY]∀X∀Y∀F[X≈Y→FX=FY], where X≈YX≈Y is an abbreviation for ∀x[X(x)↔Y(x)]∀x[X(x)↔Y(x)], that is, XX and YY are extensionally equivalent.

Two Distinct Individuals:
0–≠1–0_≠1_, where 0–0_ and 1–1_ are individual constants.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Fri, Jul 31, 2020, 1:03 PM Frank Wimberly <[hidden email]> 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

Joe Spinden

Re: Constructive mathematics.

In reply to this post by Frank Wimberly-2

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
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
Zoom Fridays 9:30a-12p Mtn GMT-6  bit.ly/virtualfriam
un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
archives: http://friam.471366.n2.nabble.com/
FRIAM-COMIC http://friam-comic.blogspot.com/ 

-- 
Joe

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