FOM: categorical non-foundations; three challenges for McLarty

[Stephen G Simpson]

Echoing Harvey's posting of 24 Jan 1998 00:13:02, I pose three very
specific challenges for McLarty.

Note that challenges 1 and 2 below concern specific technical issues.
Challenge 3 is more philosophical, but all three challenges are
directly relevant to McLarty's wild claims about "categorical
foundations".

Challenge 1. McLarty needs to explicitly state fully formal axioms of
topos theory plus natural number object etc.  These axioms are to be
compared side-by-side with the very concise yet fully formal axioms of
set theory which were posted by Harvey in 23 Jan 1998 00:54:20.  (The
purpose of the comparison is to see that the set theory axioms are
much, much simpler than the topos axioms.)

Challenge 2. McLarty needs to forthrightly concede that topos with
natural number object is not sufficient for undergraduate real
analysis, specifically the standard undergraduate material on
calculus, ordinary differential equations, partial differential
equations, power series, and Fourier series.

Challenge 3. McLarty needs to forthrightly concede that, as Harvey put it,
 > there is no coherent conception of the mathematical universe that
 > underlies categorical foundations in your sense.

Why doesn't McLarty come clean on these points?  McLarty's refusal to
come clean on these points is hampering the discussion.  Several FOM
subscribers have shown that they understand the nature and importance
of these points.

-- Steve