[FOM] When is it appropriate to treat isomorphism as identity?

[sambin@math.unipd.it]

Quoting Monroe Eskew <meskew at math.uci.edu>:

> I would be interested to see an
> example of an invention or scientific discovery that came about due to
> the person's use of anti-classical mathematics.

Let's leave aside the fight between the parties, classical and "anti-classical"
(although the very choice of this term is very significant), and take up this
suggestion.
>From my "anti-classical" perspective, I have discovered a few facts, about
*classical* mathematics (topology), which escaped mathematicians with a
classical perspective. These are:

- writing formally the definition of open (=every point is in the interior) and
closed (=contains its limit points) in topology, one can see that they are
logically dual to each other (interchange exists-forall and implies-and). This
had not be seen assuming classical logic, because one "simplifies prematurely"
using excluded third and double negation.

- keeping the bases of topologies on the scene, one can see that the essence of
continuity is a commutative square of relations. This had not be seen
classically, since one gets rid of bases "too early" on the assumption they can
be reconstructed from the topology.

Proofs are very elementary, once you "see" them. With classical "eyes", it is
very hard to see these little facts; and in fact they had not be seen for over
80 years. Building on them, one can develop topology in a constructive and very
structural way, showing that logic and topology are much closer to each other
than one would expect before. This is, in my opinion, of great foundational
interest.

I am writing a book on these developments. The two elementary facts above can be
found in my paper
http://www.math.unipd.it/~sambin/txt/SP.pdf

Other examples, of more applicative nature (one noteworthy example is the
raising interest of physicists in constructive logic and mathematics), can be
illustrated by more expert colleagues writing on FOM,  as Bauer, Spitters,
Taylor.

I agree that FOM is of help in softening the "iron curtain" between classical
and "anti-classical" views, at least by showing to the classical minded that
other people exist with a different view; so a constructive (non-classical)
view on the world is possible (ab esse ad posse).

Best regards

Giovanni Sambin

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Dipartimento di Matematica Pura ed Applicata
Universita' degli Studi di Padova
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