FOM: Time Mag, Einstein, Godel, Foundational Studies

[Harvey Friedman]

Reply to Insall 7:29PM 1/9/00. My posting 3:38PM 1/4/00 "An interesting
poll" is relevant.

> Professor Friedman wrote, on 28 December, 1999:
>
>
>  >There is nothing definitive about Time Magazine's list, of course.
>  >But it is an indication that there is greater general interest in
>  >what Godel and Turing did than what went on in core mathematics,
>  >regardless of how deep and intricate it was. If the core
>  >mathematicians wish to compete with Godel and Turing in the general
>  >intellectual culture of our times, they will want to cast their
>  >subjects in more generally intellectually attractive and generally
>  >understandable terms.
>
>  To an extent, I agree with Professor Friedman here.  But note that
>  Einstein has been declared ``Person of the Century’’ by one of these
>  magazines, and it is his use of non-Euclidean Geometry and other
>  higher-level ``core mathematics’’ that won him that ``title’’.

This is not a reasonable way of stating how Einstein is "man of the
century". He won it for penetrating, spectacular insights into fundamental
issues in physics. The fact that there was some use of classical
mathematics does not distinguish Einstein from any of a number of other
great and not so great scientists in physics and many other disciplines.

>Thus, even
>  if ``core mathematicians’’ do not seem to be competing well with Gödel and
>  Turing (whose popularity I expect turns on the spin of computer scientists
>  mostly, rather than their own abilities in self-promotion), their
>  mathematics is even more popular then that of Gödel and Turing.

What core mathematics do you think is "even more popular than that of Godel
and Turing"? You can't count ancient mathematics. E.g., Godel uses the
Chinese Remainder Theorem. He also uses induction.

If we want to play this game, then all of mathematics can be viewed as
resting on, say, Aristotle, Euclid, etc., and perhaps that's the most
popular mathematics.

>What is
>  most popular from Gödel and Turing anyway?  Is it Gödel’s undecidability
>  proofs?  Not really.

I think so.

>What is popular is a large amount of misquoting of
>  those results, in the form of ``we can never know whether mathematics is
>  consistent’’.

That is not a misquote. It may not be the whole story - in fact, it is
certainly not the whole story - but it is a reasonable general
encapsulation. Of course, there are other somewhat different encapsulations
as well - but so what?

>This neglects the influence of Church’s Thesis in the
>  hypotheses, and applies to human capabilities limitations which are not
> even
>  known for certain to be the limitations on computation devices.

That may be part of the rest of the story.

>In
>  actuality, I would say that core mathematics has fared even better than
>  Foundations and other types of mathematics,

Fared better in what sense? More people do it?

>and this success with the
>  general populace is, to a large extent due to the popularization of modern
>  physics and its mathematical flavour, through popular science fiction.

It is not true that core mathematics has fared better than f.o.m. in terms
of the general population. Of course, it has obviously fared better with
full time professional research mathematicians. Fundamental physics is not
a branch of core mathematics.

>  Results there may be misquoted as well, but the physical results are much
>  more commonly understood, in spite of the existence of many misquotes.

I will agree that fundamental physics has fared better with the general
population than f.o.m. has. Physics is not core mathematics.

> Even
>  where there is a reasonable amount of room for popularization of Gödel’s
> and
>  Turing’s results in science fiction, they are essentially never mentioned.

Is this relevant to the comparison of f.o.m. and core mathematics?

>  I speak of the popularization of theoretical computer science in stories
>  about robots and androids.  Currently, in the computer science and
>  philosophical literature, and in applications to computer engineering,
> there
>  is frequent reference to ``Gödellian self-reference’’, but even this
>  terminology seems not to have been popularized by the futuristic
> literature.

Is this relevant to the comparison of f.o.m. and core mathematics?

>  We must face the fact that people in general do not understand what we do,
>  and therefore they do not care much about it.

Is this relevant to the comparison of f.o.m. and core mathematics?

I have consistent success explaining to a wide variety of people what I do,
and I get them to care about it in a way that is suitable for them. The
level of understanding and interest to be expected varies in a predictable
way given the background and nature of the people involved.

In the realm of intellectual ideas, Godel and Turing stand very tall
compared to other mathematical people in the 20th century.

POSTSCRIPT

I have said that fundamental physics has fared better with the general
population than f.o.m. has. Physics is not the same as core mathematics.

However, this will not always be the case in the following sense. I have no
doubt that there is a largely undeveloped subject called

foundational studies.

And that by far the most well developed part of foundational studies is

foundations of mathematics.

The overall structure and great success of f.o.m. will point the way to the
proper development of foundational studies.

I have written about my conception of foundational studies several times on
the FOM. I would like to cast one principal aspect of foundational studies
in a somewhat different way.

A principal feature of foundational studies is a common language and
framework for casting philosophically coherent presentations of subjects.

The power of new tools for communication that bring together greatly
diverse people onto a common ground, cannot be overestimated. I am thinking
of the incredible power of internet communications.

Foundational studies is destined to become the internet of intellectual
life. And foundations of mathematics will lead the way to the development
of foundational studies.

Foundational studies will cause an even more spectacular revolution in
intellectual life - research, education, exposition - than internet
communications is starting to do now in commercial and social life.

Why more than even the internet? Before the internet, we had mail and
phones. So we did have a lot of effective communication. The internet
provides a new dimension in ease of communication.

But at the present time, we do not have effective communication between
major intellectual cultures. Departments at Universities barely talk to one
another - certainly don't understand each other. Foundational studies will
take us from almost nothing to the equivalent of mail, phone, and internet,
by 2050.