Re: The Law of the Excluded Middle again (long)


On Dec 4, 5:43 pm, Randy Poe <poespam-t...@xxxxxxxxx> wrote:
|There are no absolutes here. These are two different
|axiomatic places to start mathematics, and not the
|only possible two. As I've said at great length to
|some of our more vocal posters in other threads,
|mathematics is all about starting with some axioms
|and seeing where they lead. There's no one set of
|Axioms You Must Use.

True, but different axiom systems play different roles, some
much more prominent than others. Some have an intended model
and some don't. Some are used in a more foundational manner
than others.

The way I like to define "constructivism" is as the belief
that it would be better if constructive mathematics were the
mainstream. Some systems, like ZFC, which are nonconstructive,
are currently given a certain pride of place, but they may
not be the best for the job.

People talking in an offhand way about the law of excluded
middle seem often to have the feeling that it's so very
natural that they find it hard to imagine not assuming it.
It's not a case of feeling that one "must" use it, but
it sort of verges on it.

|And actually I'm still a little confused about what
|is "constructivism" and what is "intuitionism". According
|to the Wiki article (take that for what it's worth),
|it is intuitionists who do not take the LEM as an axiom.

Intuitionism is a specific kind of constructivism. L.E.J.
Brouwer used the word to describe his own philosophy of
mathematics. Intuitionists, like constructivists generally,
don't assume LEM.

Keith Ramsay
.

Relevant Pages

  • Re: Choice sequences, intuition, etc
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  • Re: The Law of the Excluded Middle again (long)
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