General Intellectual Initiatives

Harvey Friedman hmflogic at
Thu May 19 21:27:36 EDT 2022 Tim Chow

"Let's be honest here.  In the "wider intellectual and scientific
community" today, nobody cares about the foundations of mathematics.  In
the arts and humanities (except for philosophy), nobody cares about
mathematics, period.  In the sciences and engineering (except for
theoretical computer science, which is more or less mathematics), people
care about mathematics only insofar as it is a useful tool.  In
philosophy, only specialists in the philosophy of mathematics care about
the foundations of mathematics.

The last time there was a development in the foundations of mathematics
that carried "general intellectual interest" was when Goedel proved his
incompleteness theorems.  (Well, maybe a bit later, if the foundations of
computer science count as the foundations of mathematics.)  The only
audience for this topic is a small subset of mathematicians and
philosophers.  And within that subset, an even tinier fraction care a lot
about "philosophical coherence" in the sense you mean it."

This has a lot of truth to this but is highly defeatist and therefore
rather misleading. The "nobody" is of cour=e totally refutable. I
would assume Tim means something quantitative.

The world has changed very much and the potential for serious
foundations of mathematics coming into the school curriculum and
touching seriously most of the "gifted" children by the usual
definitions of top 10-20% is very real, through videos and through
teaching the teachers. There is a huge need for a serious kind of math
literacy more than we get now lest China swallow us whole.

I have no doubt that at least if I were younger and worked full time
on it, I could get basic f.o.m. to play a huge role in K-12 at least
for gifted, and cleverly watered down versions for not so gifted. And
it would raise the fundamental level of math literacy in those
important directions that are needed for raising the level of
programming skills. Of course, there are plenty of natural programmers
who don't need to think logically, but just intuitively, but other
people do need a grounding in mathematical rigor to write correct code
and correct specs. I am familiar with this situation in undergrad
because I taught baby logic and discrete math for computer science
majors. And it really shows that they didn't grow up with basic f.o.m.
ideas under their belts in K-12. I even designed stuff for middle
school and took over some middle school classes just to see how the
interaction could go.

On a different but related note, there is another community that I
intend to directly affect personally with f.o.m. and, going against
Tim's grain, intend for them to get excited about the new revelations
in Tangible Incompleteness. The case of 2 dimensions (in my stuff) is
extremely accessible to the extremely gifted. THe extremely gifted
populate several summer math programs one of which is the granddaddy
of them all, the Arnold Ross Program. Plan is to teach a mini course
there this summer and I will see. These kids are extremely gifted, and
maybe about 10% become math professors, and the rest become often
notable physicists, engineers, computer scientists, statisticians,
economists, etcetera. I expect that (at least when I do get my shxx
together properly) f.o.m. will make a lasting impression on these

F.o.m. is really special if done with very modern creative finesse. In
fact of the matter is that it is truly unique. Of course you might say
that I am intoxicated with it because I do it. But there is much more
to the story than that.

First of all there is a captive audience of sorts here. K-12 is around
and math is required to a fair extent. There is no question in my mind
that I basically know how to remake the entire math curricula from K
to 12 heavily informed by the f.o.m. structure reworked and repackaged
in very creative ways. This is not going to be done with category
theory or algebraic geometry.

And yes, a reasonable fraction of the gifted will really feel general
interest revelations in f.o.m. including General and Tangible
Incompleteness. And yes, Tangible Incompleteness is rapidly getting
good enough for this soon.

philosophy, only specialists in the philosophy of mathematics care about
the foundations of mathematics."

This would be very different if f.o.m. were properly and creatively
exposited to philosophers. No question about it. I know from personal
experience. I chose not to write the key books for philosophers
because I thought it was better to develop the subject as I can no
longer take my time for granted.

the arts and humanities (except for philosophy), nobody cares about
mathematics, period."

If they understand a little then they naturally care a litte, and this
can be leveraged to care quite a lot.

Still there is a visible amount of interest in quoting Incompleteness
from time to time. And I have given talks on f.o.m. to hugely general
audiences (not often), and by the way I was Distinguished Humanities
Visiting Professor at Princeton some years back. I could make videos
directed to ANY audience like even arts and humanities, and make them
feel totally satisfied that they really learned something extremely
interesting. It would have to be done differently and even very
differently for every single audience.

I could make a series of 20 videos for 20 different audiences each
carefully crafted for this purpose. No doubt. First consult with
experts in the target areas to get a good feel for how the target
audiences think.

And I don't mean just shoehorning my own personal work. Of course it
has to be broader, and rather systematically so, than that.

Despite what Tim says:

EVERYBODY cares about philosophical coherence (not consciously) when
they are NOT in the subject at hand. I agree that VERY FEW care about
philosophical coherence (at all) when they are IN the subject at hand.

Philosophical coherence is a necessary component for doing anything
like what I am talking about. Period.

Harvey Friedman
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