[FOM] Great achievements of FOM (or looking for a home)

[Irving]

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