FOM: Re: foundationalism (fwd)
rhersh at math.math.unm.edu
Thu Sep 24 16:34:06 EDT 1998
---------- Forwarded message ----------
Date: Wed, 23 Sep 1998 17:06:24 -0600 (MDT)
From: Reuben Hersh <rhersh at math.math.unm.edu>
To: Stephen G Simpson <simpson at math.psu.edu>
Subject: Re: foundationalism
On Fri, 18 Sep 1998, Stephen G Simpson wrote:
> OK, I'm looking forward to your comments.
> I don't really understand the point of your comments about religion,
> Platonism, etc. If the work of Frege, Russel, et al is of scientific
> value, why can't you accept it as such, regardless of the supposed
> religious or Platonistic overtones?
> -- Steve Simpson
This messsage made me sad. It showed how unsuccessful I
have been in communicating my thoughts.
I do, and always have, accepted the work of Frege, Russell,
Hilbert and Brouwer as profound and important contributions to
mathematics and logic.
Permit me to quote myself, from Tymoczko's anthology, "New
Directions in the philosophy of matheamtics", reprinted from Advances
in Mathematics, 1979.
"The work on this program (logicism) played a major role in the
developmenet of logic." (p.15)
"Russell's logic and Hilbert's proof theory became the starting points
for new branches of mathematics. Model theory and other branches of
mathematical logic have become an intrinsic part of the whole
structure of contemporary mathematics." (p. 16)
And from "What is Mathematics, Really?" p 29
`"the creators of foundationist philosophy of mathematics [logicism,
formalism, intuitionism] soughtto turn philosophical problems into
mathematical problems, to make them precise. This bias was fruitful
mathematically. Some of today's mathematical logic descended from the
search for mathematical solutions to philosophical problems. But, even
though mathematcally fruitful, it was philosophically misguided."
Then why do I bother myself about religious or Platonic
overtones? Certainly most mathematicians are not bothered about
The answer is simply that I really care about philosophy. I
can't answer you except by a little autobiography. About twenty
years ago, in the course of my going through all the courses in the
math catalog, I offered to teach a course (in the catalog but never
offered) on Foundations of Math. I searched hard for a good textbook,
but didn't find one. I put the course together from various books
on the subject that I found in the library.
I had expected naively that this would be like teaching
number theory or differential geometry. Follow the text, stay ahead
of the class, and at the end of the semester of teaching a new subject
I would have learned something.
Here in foundations the story was different. There were three
schools, logicism, formalism, and intuitionism. All sought to
repair the foundations of mathematics, after the damage they had suffered
froom the Antinomies. None of them succeeded in their mission. In the
course of an unsuccessful philosophic quest, they all created some
original and important mathematics.
I was bothered not so much by the impasse in which
foundationalist philosophy of mathematics found itself.
I was much more disturbed by the obvious (to me) fact that
all three were incredible. The pictures of mathematics they
offered did not at all resemble the mathematics I knew as student,
teacher, and researcher.
Platonism (including its special case, logicism) invoked
a mathematical world eternal, unchanging, and independent of human
actions. How this ghost world related to the material world wasn't
even recognized as a problem! (Later I learned about Benacerraf
throwing the same problem at his fellow philosophers of math.)
Formalism, removing or denying the content, the meaning, from
mathematics, was equally indigestible, for in my experience the
meaning of mathematical sentences, words, and ideas was exactly what
made them interesting, what made it worth the trouble to analyze them.
Trying to explain math by making it meaningless was to me just as
absurd as trying to understand literature by making it meaningless.
As for intuitionism, I liked its attending to the mind of the
mathematician, but I didn't understand the obsession with LEM,
So I started on a course of reading and talking to friends,
for many years. I was distressed that philosophers of math seemed
uninterested or ignorant of anything but sets, numbers and logic.
It seemed to me that anything deserving to be called the philosophy of
math had to include geometry, analysis, and algebra.
I found two authors who seemed to me to be on the right track.
Imre Lakatos in Proofs and Refutations doesn't bring philosophical
preconceptions into which math must be forced. He presents a living
scene of mathematics being created, from which the reader has to draw
his own conclusions. To me the importance of Lakatos was that he
showed a different way to do philosophy of math, one rooted in what
really goes on in mathematical work.
The second author who impressed me was Leslie White,
an anthropologist whose essay on math is in the last volume of the
4-volume World of Mathematics (Newman.)
He brought in concepts from social science--particularly the
recognition of social institutions as being real, just as real as
physical or mental reality. This seemed to me to be the missing answer
to my puzzle. Mathematics is not a physical reality, not a mental
reality, then what is it? I didn't believe it was transcendental or
typographical. Once the possiblity is mentioned, it seemed obvious
to me that social reality is the kind of reality math is.
Unfortunately, I naively didn't expect that the use of the word
"social" would bring a swarm of gnats and hornets aftaer me. Neither did
I expect that adopting Lakatos' term "foundationalism" for the three
trends would be taken as an attack on a research program led by Harvey
Friedman. These are misunderstandings that I was slow to understand,
and therefore slow to respond to.
The U.S. Army, the constitution of the U.S., Harvard College,
Leavenworth Prison are all social entities. That doesn't make them
vague or socialistic or lacking objectivity. Their real existence
is perfectly objective. Saying math is a social entity doesn't
question it's objectivity at all.
Saying math is social doesn't mean saying it's the same as the
US Army, the constitution, etc etc. Each of these entities is quite
different from the others. What they have in common is that they exist by
the shared understanding or consent or agreement of groups of people.
They can't be explained in a purely individual, isolated way.
I don't know what "postmodernism" is, but I am sure it's
not what I am saying.
One mystery about math is that it's at once outer and
inner. To the student learning it, it's outer. It's out there.
But from a bigger perspective, looking at humanity on our planet,
math is inner, it's in the social thinking of this species. Only
the understanding of math as a social phenomenon can explain this.
If my words against logicism, formalism and intuitionism have
been hurtful, I'm sorry. It seemed to me that all three foundationalist
programs had petered out, had failed. I was arguing that it was time
to set their agendas aside and make a fresh start in the philosophy of
mathematics. No longer identify philosophy of math with foundations
In this letter I have been trying to answer your question, why
I say and write what I do. I have not been trying to defend or justify
my viewpoint, just tell you where it came from as I see it.
Reuben Hersh > >
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