[FOM] Foundations Crucial

[Harvey Friedman]

Chow wrote:

"I'm not sure what you're driving at.  Alternative foundations, like all of
math, must of course be correct.  If there's something incorrect, then of
course there's a problem, but at the same time it should be possible to
identify exactly what is incorrect (or insufficiently justified), and then
engage in the usual mathematical dialogue to straighten it out.  Unless
someone digs in his heels and refuses to acknowledge specific objections
and engage in dialogue (as in the case of Hsiang's purported proof of the
Kepler conjecture), the issue of correctness is something we know how to
deal with.  So I don't understand what you're driving at when you cast
general, vague aspersions like those above.  If you have a specific error
in mind that you're worried is unfixable, let's hear what it is.
Otherwise it just sounds like you're saying that because people have made
mistakes before in this subject area, we should be skeptical about the
subject as a whole.  If that's the objection then we might as well just
give up on all of mathematics.  Let him who has never made a mathematical
error be the first to cast a stone."

We know from experience that "correct mathematical thinking" is an entirely
problematic idea - the more abstract the context the more problematic - and
we have known this much earlier than our experience with naive set theory.

Now that we know how easy it is to be fooled when being "liberated", and
how strong foundations can be of mathematical subjects, the appropriate
standards for mathematical rigor is very high in 2014. Of course, our
ability to supply mathematical rigor is very high in 2014 compared to
earlier periods, and that is a key factor in having the appropriate
standards so high.

I gave an example when I mentioned naive category theory as a form of
unacceptable "liberation of mathematicians". The use of "the category of
all categories" has nothing to do with "making an error". The issue is far
deeper and far more subtle.

I do not advocate putting mathematicians into jail or even firing
mathematicians who use or argue in naive category theory, without some
fundamental accompanying foundational work. However, in 2014,  if they
don't fully acknowledge the full importance of what is missing - especially
if they minimize it - well, that is a different story. In 2014, this would
represent an alarming level of intellectual malpractice.

Doesn't recent work indicate that using the category of all categories in a
category theoretic environment is just as seriously flawed as was using
naive set theory, without its fundamental accompanying foundational work?

Does the way you and Conway refer to "liberation" minimize the great
fundamental importance of fully rigorous foundations? I would concede that
it should not be a requirement that every mathematician do foundations of
mathematics. But every mathematician should recognize its great fundamental
importance in all context, the great general intellectual need for it, the
great insights one gets from it, and what greatly importance things are
missing when one does not have it.

Of course, a critical challenge for foundations is to develop deep
foundations of subjects other than mathematics that in any way compare to
what we have for the usual foundations of mathematics. I predict that we
will see advances along these lines for statistics and physical science
comparable to the usual foundations of mathematics by 2100. The general
intellectual excitement will be comparable to the Goedel era in foundations
of mathematics.

Harvey Friedman
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