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

[Andrej Bauer]

On Sat, May 16, 2009 at 7:42 AM, Monroe Eskew <meskew at math.uci.edu> wrote:
> I disagree with this characterization.  I know a prominent set
> theorist who made the following analogy:  Most mathematics is "the
> software" and set theory is "the machine language."  Set theorists
> don't ask that people change their methods.  What they do is study a
> system that unifies almost all ordinary mathematics.  Every classical
> result can be translated into and proven within set theory, but we
> also want to justify the ordinary methods in terms of set theory.
> This kind of thing shows that (a) in the eyes of set theorists,
> ordinary methods are not in need of revision, and (b) models of set
> theory are models of classical mathematics, properly interpreted.  (b)
> can be useful.

Like most things in life, this is not a black and white issue. Of
course it is hugely important to have a unifying framework for
mathematics, which is the axiomatic method. Of course set theory has
contributed importantly to unifying modern mathematics and explaining
things about the foundations. Nevertheless, too much reliance on
classical set theory has its negative effects, such as:

- it makes topologists blind to the topological notion of overtness,
which is just as important as compactness (Paul Taylor has said more
about this in a parallel post)

- it makes analysts blind to the possibility of axiomatizing their
subject directly in terms of nilpotent infinitesimals. Luckily,
physicsts never much liked the epsilons and deltas and have kept the
tradition alive (and built radio, TV, transistors, lasers and GPS
using the infinitesimals that do not exist according to orthodox
mathematics)

- it makes set theorists somewhat of a community of outcasts because
they shake the foundation all the time by considering silly things
such as large cardinals. Luckily Harvey Friedman is making excellent
efforts in persuading the mathematicians that ZFC is not the end of
the story.

- it makes the lives of computer scientists difficult by shattering
the connection between computaton and logic to pieces, so that they
have to bolt on computability as an afterthought (as a friend of mine
once said).

Andrej