FOM: RE: Time Mag, Einstein, Godel, Foundational Studies

[Matt Insall]

Professor Friedman's response to me about Einstein's ``Man-of-the-Century''
status brings us back to the question ``What is Mathematics?''  To keep from
the overexpansion that can come from posting a volley of replies back and
forth, I will ``snip'' as carefully as possible.  Please refer to Professor
Friedman's posting of January 13, 2000 20:10 for more details, if necessary.

 Name: Matt Insall
 Position: Associate Professor of Mathematics
 Institution: University of Missouri - Rolla
 Research interest: Foundations of Mathematics
 More information: http://www.umr.edu/~insall


> -----Original Message-----
> From: owner-fom at math.psu.edu [mailto:owner-fom at math.psu.edu]On Behalf Of
> Harvey Friedman
> Sent: Thursday, January 13, 2000 8:10 PM
> To: fom at math.psu.edu
> Subject: FOM: Time Mag, Einstein, Godel, Foundational Studies
>
>
> Reply to Insall 7:29PM 1/9/00. My posting 3:38PM 1/4/00 "An interesting
> poll" is relevant.
>

<snip>

> >
> >  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.


I do not mean to deny the importance of Professor Einstein's work, or the
spectacularity of his insights.  However, what, mathematically, was the
result of his discoveries?  Or, more precisely, what was the effect on
mathematics of his discoveries and their popularity?  I contend that the
entire face of mathematics was affected.  In so-called ``pure'' mathematics,
many departments seem to focus their attention on the parts of ``core
mathematics'' that appear to impinge most immediately on mathematical
physics.  Consider, for example, the core (Mathematical) requirements for an
undergraduate degree in Mathematics:  Calculus, Differential Equations, Some
type of proof-oriented course, Advanced Calculus, Linear Algebra,
Statistics, Real and/or Complex Analysis, Modern Algebra.  (I know that each
university has the freedom to design its own curriculum, but from what I've
seen, these form the core mathematical requirements, usually.  If I am in
error, please let me know.)  The only course in this list that does not
appear to the average non-mathematician to have immediate application in
physics is Modern Algebra.  Is this because the understanding of Mathematics
requires three to five calculus courses, plus real (and/or complex)
analysis?  I doubt it.  Why should there not be a required course in
Mathematical Logic, or Set Theory, for example?  Because students, who
predominantly think of Mathematics as computational in nature, and who seem
to have a hardwired connection between their understanding of Physics and
its applications (as in engineering) and their understanding of Mathematics,
will not continue to enroll in courses at a university that requires a
Foundational approach to the Mathematics they must take.  This is, I think,
a reasonable measure of what the populace thinks Mathematics is, or should
be, because the course offerings at universities, especially at the
undergraduate level, are so sensitive to popular opinion, and in
Mathematics, that popular opinion has no room for the most recent advances
in foundations of mathematics, or for significant change in what one
considers Mathematics to be.  This is quite different from the popular
opinion of what other disciplines are, or should be, like.  For example, it
is *expected* that disciplines such as Computer Science will modify their
upper-level undergraduate courses to accomodate recent advances in the
theoretical aspects of the subject.  Thus, as a measure of the popularity of
a bit of Mathematics, although the declarations of Time or Life magazines
may be questionable, it is their declaration of professor Einstein as ``Man
of the Century'', in my opinion, that most accurately represents popular
opinion about Mathematics, even though his main contribution may not be
Mathematical in nature.  (I say ``may not be'', although I expect to argue
later that it is Mathematical in nature, at least according to one
definition of Mathematics that has been proposed in this forum.)



>
> >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.


Yes, he uses these things.  Perhaps a clarification of what constitutes
``core mathematics'' is needed.  As I indicated above, I call something
``core mathematics'' if a preponderance of universities require Mathematics
majors (undergraduate or graduate students) to study it.  I have not made a
systematic study of this, but I believe if one did such a systematic study,
it would show that what constitutes ``core mathematics'' according to this
definition would be predominantly related to applications to Physics, or
physics-like applications.  Perhaps the Chinese Remainder Theorem falls in
this category, although I doubt it.  Induction may be a part of ``core
mathematics'' according to this definition (induction is commonly taught in
a proof-introduction course, or even in the first linear algebra course).
However, the Mathematics of Gödel and Turing (Mathematical Logic, and its
applications, and the Foundational consequences of their theorems) is
commonly not even available as a core course, except in Computer Science or
Philosophy departments, or sometimes in Mathematics departments that also
offer a Computer Science degree (perhaps because the two disciplines would
not command a large enough enrollment separately).  This is changing (ever
so slowly) as Science and Engineering Departments realize the value of the
study of Mathematical Logic and its applications in their own disciplines,
and begin to send their students to Mathematical Logicians for tutoring in
their research or to take courses in these more foundational subjects.  (I
am currently the beneficiary of such increased interest in the applications
of Mathematical Logic to Engineering, in particular, with some relatively
good graduate students being sent my way.)


BTW, even though many departments require Number Theory for their graduate
students, I do not call this ``core mathematics'', because so many do not
*require* it.  As I have been indicating, ``core mathematics'' seems to be
calculus-oriented, presumably because there seems to be a majority of
people, in and out of the Mathematics community, who think first of calculus
when they think of Mathematics.  The popularity of a bit of Mathematics
seems to be tightly tied to its perceived applicability to the Physical
Sciences and their applications, and this seems to be also predominantly
calculus-oriented.  The following appear to get relatively little attention
in ``core mathematics'':

Discrete Mathematics (except for a small bit of Modern Algebra, usually
restricted to Group Theory and some Ring and Field Theory), Foundational
Studies (even when applied directly to mathematics), Lattice Theory,
Universal Algebra, Mathematical Logic, History of Mathematics, Philosophy of
Mathematics, and Applications of Mathematics outside the physical sciences.

However, the following are commonly required:  Calculus (and its extensions,
including real and complex analysis, and differential equations), Linear
Algebra, Geometry (especially for potential high-school teachers [including
Non-Euclidean Geometry]) and Differential Geometry, Probability and
Statistics (calculus-based).

This seems to me to indicate a preponderance of mathematics required and
popularized to be the type of Mathematics that is needed to understand or
compute in Physics or its applications.


<snip>


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


Perhaps one particular item from the proofs, the popular idea of the ``Liar
Paradox'' (another mistranslation of what Gödel's proof does) is the most
popular thing attributed to these gentlemen.  In fact, Gödel's proof seems
to turn on a ``Truth-teller Paradox'' rather than a ``Liar Paradox''.
(Roughly, in Formal Arithmetic, one can formulate the statement ``I am
consistent.'', but cannot prove this statement from the given axioms, or
even any recursive extension of them.)  [I only write this down because if I
am misunderstanding Gödel's argument, I would like you to let me know.  I am
convinced you already knew this much of the details of Gödel's argument.]
The entire notion of recursive functions, developed in the proofs of Gödel
and Turing, although widely understood among Mathematical Logicians,
Computer Scientists, and Philosophers, seems to be given very little
``airplay'' in any of the popular accounts of Gödel's work.  However, this
is one of the most important aspects of the proof!  Even the popularized
``Liar Paradox'' is not really due to Gödel or Turing, as it was around well
before either of them was alive.  The significance of it in Gödel's work is,
IMHO, in its unique application as an instance of the ``self-aware'' nature
of even first-order parts of the mathematics of Gödel's day.  This
``self-awareness'', which is recognized also in Russel's Paradox and many of
the classical paradoxes, did not have the damning effect in Gödel's work
that it had in the case of the classical paradoxes of set theory, although I
believe many people thought (or even still think) it did have a damning
effect.  In the case of Russel's Paradox, and similar antinomies,
self-reference lead to a contradiction that required a complete
reformulation because an axiom of Frege's system actually was inconsistent
with the others.  But in Gödel's case, it only leads to undecidability of
recursive extensions of formal arithmetic.  This is a far cry, IMHO, from
actual inconsistency, i.e. from the lack of the existence of a model for
such theories.  However, the people outside Mathematical Logic who think
anything at all about Gödel's theorem seem to me to normally not think a bit
about its proof, but only about some vague reformulation of its statement.


<snip>



>
> >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?


Yes.  More people do it.  More people know more details about its results.
More people think that there is nothing more to Mathematics beyond what one
might call ``core mathematics''.  And more support of funding agencies seems
to go to ``core mathematics'' than to areas of Mathematics outside the
``core''.  Moreover, the negative publicity a Mathematics Department gets
with other departments and students is apparently directly related to any
attempts to expand itself beyond the traditional mold:  More f.o.m. is
looked on with disdain from outside.  The more foundational a class is made,
the less students will take it.  The exercise of writing arguments is viewed
as not being Mathematics by students and their parents, because they seem to
think that all there is or should be to Mathematics is computation.  (I have
direct experience with this in classes where I require written exposition at
the undergraduate level.  Fortunately, this view does not prevail among our
graduate students, but they are a much smaller segment of the population.)


>
> >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.


A huge portion of fundamental physics is directly the application of ``core
mathematics''.  Of course, in more advanced physics, certain mathematical
concepts outside the ``core'' are making their mark, such as non-classical
logics, etc.  However, I still find that the physicists I speak with
frequently do not care much about foundational aspects in Mathematics.  They
just want their students to be able to differentiate and integrate
vector-valued functions by the time they leave calculus I.  (Of course, this
is a pipe dream, since vector-valued functions are traditionally only
introduced in calculus III.)  If this requires that all Mathematicians stop
doing any research and spend all their time explaining to students who care
not at all for Mathematics how to use MathCad or Mathematica, and grading
hundreds of homework problems from 70 or more students per class, all the
better, they seem to think.  It seems that administrations and parents of
students agree.  They do not seem to respect fundamental research in
Mathematics as much as they do fundamental research in the Sciences, and in
Engineering, or creative effort in the Arts.  Whether we like it or not, the
uses of Mathematics outside Mathematics have a major effect on what parts of
Mathematics are considered worthwhile by funding agencies, administrations,
and our colleagues in other disciplines.  Culturally, this effect has
additional second-order effects on what parts of Mathematics are considered
worthwhile by our colleagues within our own discipline and subdisciplines.


>
> >  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.
>

No, Physics is not Mathematics, much less ``core mathematics'', but the
Mathematics that is used in Physics, or is developed in response to the
needs of Physicists, is predominantly of the ``core'' variety, IMHO.


> > 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?
>

IMHO, it is relevant to any discussion of what constitutes *popular* parts
of f.o.m. and *popular* parts of ``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?
>

IMHO, it is relevant to any discussion of the comparison of their relative
popularity in our culture.




> >  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.

I realize that you have had such success, and I applaud that.  The first
time I learned anything of your efforts was while I was a graduate student
and I attended a plenary session in which you spoke about your work, at the
AMS bicentennial.  Your presentation was quite interesting and, I thought,
would be quite understandable, at acertain level, to a broad segment of the
population.  However, we are either not all blessed with such abilities, or
we have not done our job in popularizing f.o.m., IMHO, because, as I have
said, f.o.m. is both less popular and less respected, it seems, than
traditional, ``core mathematics''.



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

They certainly do, and I would not intentionally trivialize their
accomplishments.  On the contrary, I would say that, in the light of the way
that popular culture deals with ``core mathematics'' and other mathematics,
their accomplishments have not been accorded the respect that they deserve,
and others who study f.o.m. in their wake do not receive the attention they
deserve, either.



>
> 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.
>

I would agree with this.  I expect it will be a long time, however, before
such a subject develops its own undergraduate curriculum.




> And that by far the most well developed part of foundational studies is
>
> foundations of mathematics.

Again I agree.  I do feel that this is because Mathematicians, who have
frequently also been Philosophers, are very concerned with the universality
and correctness of their results.



>
> 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.

I look forward to the opportunity to read those postings.  I believe I will
find them invigorating.

>
> 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.


You are quite right.  I think some very valuable benefits will come our way
as we increase this facet of our communication with one another.



>
> 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.

I think there are various reasons for this:
1.  Currently, the cost of internet communications is essentially relatively
nil.
2.  Communication by email has an appealing quality I would call
``semi-conversational'', or ``semi-immediacy''.  I can respond to you fairly
quickly, and you can respond to me fairly quickly, as if in a telephone
conversation, but not quite, so that we can give a little more thought to
what we say than when we speak on the telephone.  [Sometimes, though, I do
not seem to reap the benefits of this facet of internet communications,
because I still sometimes hit the *send* button before my brain goes into
action. :-) ]
3.  Archives, such as the one that these postings are saved in, make it easy
to review conversation threads, whether we have been directly involved in
them or not.



>
> 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.


I will agree with this.  The information explosion of the twentieth century
has almost killed collaborative scientific exploration, but not quite.  This
is where some of the funding agencies are, as I would say, ``on the ball''.
Currently, large amounts of funding are available to any and all who will
collaborate across discipline lines.  This is important on an engineering
campus such as ours.  Sometimes, the only way to get students to study a
subject is to show them how it is related to their own discipline outside
Mathematics, and the best way to do this is to learn something about those
other disciplines oneself.  To have time to do this requires that one get a
break from some of the other commitments one is expected to make to the
campus, and one help is the current availability of funding of
multi-disciplinary proposals.