Re: intuitionism

Keith Ramsay wrote:
On Oct 28, 3:04 am, Aatu Koskensilta <aatu.koskensi...@xxxxxx> wrote:
|(I'm sure you know all this
|stuff better than I do, but perhaps these random remarks amuse or
|benefit some random reader.)

I think you've made a good observation here about the way
Brouwerian intuitionism is distinctive. In 1970 Bishop
seems to have felt that Brouwer went overboard with his
concern about philosophical minutia about the continuum.
[...]

I'm going off on a limb here, so it could be wrong.
I've been thinking about the usual proof of the
Hilbert basis Theorem, that R noetherian implies
R[X] is noetherian. As I recall, it's a proof
by contradiction, and a pure existence proof.

I've been thinking that one motivation (at least
for me) for trying to avoid using proofs by contradiction,
and avoiding using ( p .or. ~p ) in a proof, is that
it seems intuitively like there would be a better chance
of obtaining effective existence theorems.

I'm not sure how the proof of the existence of
a maximal ideal goes as regards L.E.M. usually goes,
but in any case, for a particular ring or
effective description of rings, an effective proof
of the existence of a maximal ideal for a given
ring has a chance of yielding something that
can be worked-out on paper (the maximal ideal,
the ring operations in the quotient ring R/M,
M a maximal ideal of R), or implemented
as algorithms, and eventually implemented
in software.

I say "better chance" because I'm not sure what's
permissible in existence proofs in order to go
from existence proof to an "effective" existence
proof, i.e. the kind that will automatically yield
a procedure (recipe) to find, for example,
the rank of an elliptic curve over Q (which
I think is always finite?) , etc.

David Bernier
.

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