865: Structural Mapping Theory/3

[Harvey Friedman]

On Fri, Feb 12, 2021 at 5:10 PM <martdowd at aol.com> wrote:

> Harvey Friedman writes
>
> picking a specific set of counting numbers 0,1,2,... as von Neumann
> ordinals, something horrendously ghastly, artificial, repulsive, and
> wrong headed to most mathematicians
>
DOWD writes

> There is another point of view.  Set theory is a "gift of nature" which allows us to do profound and elegant things like, as a mere exercise, give a formal definition of the natural numbers.

Sure, but the point I am stressing is that set theory is the singular
foundational scheme (modulo equivalent variants) that is based on
RIGIDITY.

RIGIDITY reigns supreme in so many ways in so many contexts. There
must be a lot of RIGIDITY going on in engineering, where engineering
standards are rigidly enforced and allow engineering to raise the
standard of living 100 fold over the years. E.g., rigid standards for
eclectic outlets. Rigid standards for weights and measures.

Or with the game of chess. Extremely rully rigorous standards in the
rules of chess down to the operations of formally sanctioned
tournaments. Look at https://en.wikipedia.org/wiki/Rules_of_chess
There is an incredible amount of RIGID detail here. It is a rather
foundational document.
>
FRIEDMAN WRITES:

> > So what is 1/0? The straightforward answer is that it is undefined. This
> > leads to what is called FREE LOGIC which allows undefined terms.
>
DOWD WRITES:

> Obviously introductory texts on fields will not resort to such drastic measures,  I only looked at one example, Lidl and Niederreiter
> section 1.2.  They use the notation "a^-1", only assuming that "a" is nonzero.  This suggests using two-sorted logic, one sort being the entire field and the other the non-zero elements.  Then "/" would be a function on sort 1 x sort 2.

I think you have this wrong. It is much more drastic to introduce new
sorts for things like this. One would have an enormous catalog of
sorts. Instead one has a few new devices with a few basic unifying
principles.

KATZ WRITES

This is a very intriguing text, and I spent some time perusing it
though I can't claim I read it all.  I have a few problems with some
of the assumptions of your post.

 1. Granted, you seek to present the position that ultimate
foundations are necessary.  But how do you know that?

ME:

Ultimate foundations WHEN IT CAN BE OBTAINED which is rare in certain
areas but common in others, IS ADOPTED AND VIEWED UNIVERSALLY AS
NECESSARY.

It is necessary in the chess world. It is necessary in the sports
world. It is necessary in so much of the military world. It is
necessary in the practical computer world. It is necessary all through
engineering.

KATZ WRITES

It strikes me that in its presentation here, the position is more of
an ardently held opinion than a convincing argument.  Since this is a
"meta-mathematical" (in the sense of largely phiosophical) issue, it
is disappointing that you did not engage with any of the related work
in the recent literature, some of which was extensively debated at
FoM.

ME

It is background material, not even close to the main point. the main
point is what research programs are suggested by my symmetric
semigroup results. Since it is already clear to me that this is going
to be a major new interface between category theory and set theory,
informing people outside FOM what the standard FOM viewpoints are that
I strongly hold, is going to be a useful thing to do communication
wise. But it is background.

KATZ WRITES

 2. I have, of course, encountered this type of opinion before.

ME

It is self evident and totally standard for people who do serious
research in the foundations of mathematics. The RIGIDITY of set theory
is its great strength and it is necessary for ultimate foundations as
far as one can see now..

KATZ

It strikes me as a posture comparable to thinking that Euclidean space
is primary when you study geometry.

ME

Not even remotely analogous. Not even close.

The sense in which Euclidean space is primary is very important and
real but different than the role of set theory. Set theory is the
ultimate arbiter of rigor.

Euclidean geometry is not any ultimate arbiter of rigor. Rather it is
the most fundamental of all geometric ideas as it is so tied up with
fundametnal thought process we all possess.

I wrote a long series of postings on deriving the Euclidean structure
uniquely from fundamental principles. Of course for much more
technical and specialized purposes, one wants something more general
or even something very different to call "geometry".

KATZ WRITES

I had a discussion with a colleague who is a professional
mathematician, who in fact linked both issues (set theory as
foundations of mathematics and Euclidean space as foundations for
geometry).  Other mathematicians may also feel this way but as a
differential geometer I can tell you that my personal opinion is that
such a viewpoint (concerning Euclidean geometry) is naive and moreover
untenable.

ME

Euclidean Geometry is not at all remotely trying to be any ultimate
arbiter of all rigorous geometric thinking. Not even close to that. So
any analogy of that sort between set theory and Euclidean geometry in
that vein is wrong.

A differential geometer may be very interested in various more
specialized matters of less general intellectual interest than
Euclidean geometry which has such high general intellectual interest.
And it may be essential for all kinds of specialized purposes that
Euclidean geometry is not even remotely attempting to do.That has
nothing to do with the deep philosophically fundamental character of
Euclidean geometry.

Look at the huge difference between Euclidean Geometry and set theory.
The first is a way of going about rigorously formalizing extremely
fundamental intuitive ideas. Set theory also does that also but is
trying to do something much more. Namely, to provide an ultimate
foundation for ALL mathematical reasoning. It doesn't tell us how to
do sophisticated mathematical reasoning. But it is the ultimate
.arbiter of rigor.

KATZ WRITES:

 3. Admittedly a dilettante in foundations, I am puzzled by your
postulation of set theory as currently the only clear foundation.  As
you probably know, many in category theory think that they possess
clearer foundations.  It seems to me that stating your opinion more
forcefully than usual is not going to convince many people.

ME

Category theory rates extremely poorly in PHILOSOPHICAL COHERENCE.
This kind of issue has been discussed many times in the literature and
elsewhere. E.g., Solomon Feferman, a very excellent writer on the
interface between math and philosophy (although he had some very wrong
headed dogmatic views sometimes), wrote on this issue.
https://math.stanford.edu/~feferman/papers/Cat_founds.pdf

KATZ WRITES

 4. Some leading mathematicians working on the frontier of math and
physics (and I am thinking of Lou Kauffman in particular) expressed
their professional opinion that some of the entities they are dealing
with are just not sets.  Without professional expertise in such
fields, it seems difficult to make sweeping claims about alleged
ultimate foundations.  I don't recall you engaging Kauffman on the
point he made.  This of course ties back to item 1 above.

ME

Lou Kauffman and I agree completely on the ultimate foundational
viewpoint regarding set theory as the ultimate arbiter of rigor. Of
course we both agree that mathematicians like to think about things,
particularly geometric things, in non set theoretic terms. Like moving
parts, or stretching, or identifying points, and all kinds of things
like that. If you just try to think in terms of sets, you are frozen
and can't do much. BUT THE ULTIMATE ARBITER is the rigid set theoretic
foundations.

Harvey Friedman