[FOM] Great achievements of FOM (or looking for a home)
Irving
ianellis at iupui.edu
Mon May 23 10:47:12 EDT 2011
I think that it would be helpful to examine the historical roots to the
indifference, if not hostility, towards foundations which to Prof.
Friedman refers, in order to fully understand that attitude on the part
of working mathematicians, mathematicians, that is, who prove new
theorems in algebra, geometry, topology, analysis, or whatever their
specialization happens to be.
Like Prof. Friedman, I'll start with a personal experience. I attended
my first AMS meeting in (if I recall correctly) the Summer of 1978, and
sat beside Jean van Heijenoort (with whom I completed my Ph.D. the
previous year) for a talk by Ernest Snapper on "The Three Crises in
Mathematics", in which he argued that the foundational disputes between
logicism, formalism, and intuitionism were, at bottom, metaphysical --
about the status of abstract entities. With that perspective, one can
understand why the mathematician who might otherwise be interested in
foundations might not want to investigate further than van Heijenoort's
"From Frege to Gödel", or wonder if there is anything worth seriously
examining in foundations that is chronologically beyond Gödel.
When the "working" mathematician turns to the problem of foundations
and sees the disarray in which mathematics is left by the set-theoretic
antinomies, the arguments between logicism, formalism, and
intuitionism, and, finally, Gödel's incompleteness theorems, is it any
surprise that those wishing to investigate the properties of, say,
hypergeometric functions, P.D.E.'s, or homotopical algebra, will be
impatient with foundations. (I think that one of the principal points
made by John Dawson in his article on "The Reception of Gödel's
Incompleteness Theorems" is that the theorems surprised few, were
inconsequential to most "working" mathematicians, and exaggerated by
philosophers, who took it to be statement about the limitations of
knowledge in general.)
Now consider Whitehead and Russell's "Principia Mathematica". We have
to note that it took them 362 pages of the first edition to reach the
point of being able to assert that 1 + 1 = 2 is a theorem in their
system. That is, 1 + 1 = 2 IF. It is not until Part III of the second
volume of the Principia, when dealing with cardinal arithmetic, that
arithmetical addition, a + b, is in fact finally defined (Whitehead &
Russell 1912, p. 75, *110.01), and only at *110.643 (Whitehead &
Russell 1912, p. 86) -- that is after another 305 pages of volume I and
yet another 86 pages of vol. II -- that 1 + 1 = 2 is actually finally
stated and proved.
Add to that Russell's famous -- or infamous -- letter to Leon Henkin of
April 1, 1963, asking him whether Gödel's incompleteness theorems mean
that "2 + 2 = 4.001 in
"school-boy arithmetic". Henkin once asked me (while I was working at
the Bertrand Russell Editorial Project) whether Russell had really been
serious when he asked him that question. I replied that I thought that
he was, because I think that, in the back of his mind, Russell was
thinking of one of the first logic textbooks that he had studied, F. H.
Bradley's "Principles of Logic" and the remark (Bradley 1882, vol. II,
p. 399) that 'It is false that "one and one are two". They make two,
but do not make it unless I happen to put them together; and I need not
do so unless I happen to choose. The result is thus hypothetical and
arbitrary.'
Given the claims made by some philosophers that Gödel's theorems are
just one more example not merely of the limitation of logical thought
but of an inherent limitation on knowledge in almost any field, one
might ask what relevance these epistemological issues have for working
mathematicians attempting to examine the properties of rings or fields
or groups or monoids, ..., etc. Under such circumstances, it is not
surprising if we dismiss these as issues for philosophers rather than
for mathematicians.
Finally, let's consider Fermat's Last Theorem. It was, as I recall, an
example that Gödel himself gave of an undecidable problem. It is an
example of an undecidable problem that van Heijenoort gave. Fermat
first wrote down some time between 1637 and 1640 that he had a proof of
FLT, we had to wait until 1995 for Wiles' more than 100-page proof
(supplemented by Wiles and Taylor to fill in the gap concerning Hecke
algebras). One might imagine that if it took Wiles and Taylor some 130
pages to prove FLT using the "shortcuts" of "working" mathematics to
prove FLT while it took Whitehead and Russell close to 700 pages just
to prove 1 + 1 = 2, few would be willing to go through what it would
take using the apparatus of Principia to prove FLT.
I think that the attitude of a good many of the "working"
mathematicians towards foundations is: Now that we know that we CAN do
it in this painstaking foundational way, let's just get on with the
buisness of using our usual techniques and get on with proving
something interesting, knowing that if we really HAD to do it according
to Principia (or ZFC, or whatever preferred theory we choose), we COULD.
I will end with another personal note. The attitudes of the majority of
"working" mathematicians towards foundations which Prof. Friedman
described are not far different from the one that I encounter in
consideration of the history of mathematics, when, after a colloquium
talk on history, I am asked why I care about history. (My usual
response is that, if nothing else, knowing the history of our subject
can help us watch out for, and avoid, the mistakes of our predecessors,
and, if nothing else, protect us from the embarrassment of claiming as
original a theorem that someone else has already proven.) I also have a
different response from philosophers, who question why I worry about
the details of the differences and similarities between, say Schröder
and Russell regarding the logic of relations, and how they got to them,
instead of trying to figure out what they should have done.
I'm not certain how well I have articulated my concerns, or how well I
have described those aspects of the history of foundations that are
behind attitudes towards the subject. But I do think that the history
of foundations has played a role in the ways in which foundations is
viewed by mathematicians and philosophers.
Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info
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