More Constructivism

From: "andrew cooke" <andrew@...>

Date: Fri, 20 Jan 2006 12:03:14 -0300 (CLST)

This is related to the thread I had on AskMe -

Since I've given up Mefi in disgust and sorrow, I thought I'd follow up
here with some links to related items on Wikipedia.

Curry-Howard correspondence (lots of interesting stuff here) -
http://en.wikipedia.org/wiki/Curry-Howard_correspondence - "the close
relationship between computer programs and mathematical proofs [...] There
are a number of ways to think about the Curry-Howard correspondence. In
one direction, it operates on the level compile proofs into programs. Here
proof was, originally, limited to proofs in constructive logic  typically
in a system of intuitionistic logic."

And another way to look at the excluded middle - Peirce's law -
http://en.wikipedia.org/wiki/Peirce's_law

Vaguely related - Heyting algebras -
http://en.wikipedia.org/wiki/Heyting_algebra - "... arise as models of
intuitionistic logic, a logic in which the law of excluded middle does not
in general hold"

Andrew

Type Theory and Functional Programming

From: "andrew cooke" <andrew@...>

Date: Fri, 20 Jan 2006 12:07:32 -0300 (CLST)

Related to the above

Free book - http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/

Andrew

Djinn - Types to Functions

From: "andrew cooke" <andrew@...>

Date: Fri, 20 Jan 2006 17:59:48 -0300 (CLST)

Starting to read through the types book I realisd I didn't know what a
proposition was, which seemed kind of worrying.  Searching lambda turned
up this post from Shae - http://lambda-the-ultimate.org/node/view/1178

It doesn't answer my question, but it does blow my mind.

Andrew

Formalizing Constructive Mathematics

From: "andrew cooke" <andrew@...>

Date: Sat, 21 Jan 2006 09:46:33 -0300 (CLST)

This is a section of chapter 3 of TTFP, starting at p 64
(http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/)

[...] [Bee85] http://www.amazon.com/gp/product/0387121730 provides a very
useful survey addressing [the many schools fomalizing constructive
mathematics] and we refer the reader to this for more detailed primary
references.  The text covers theories like Intuitionistic set theory
(IZF), Feferman's theories of operations and classes [Fef79]
http://www.amazon.com/gp/product/0444853782, as well as various formalized
theories of rules, all of which have been proposed as foundations for a
treatment of constructive mathematics.

One area which is overlooked in this study is the link between category
theory and logic, the topic of [LS86]
http://www.amazon.com/gp/product/0521356539.  This link has a number of
threads, including the relationship between the lambda calculus and
cartesian closed categories, and the category-theoretic models of
intuitionist type theory provided by toposes.  The interested reader will
want to follow the primary references in [LS86]