[FOM] Profound Flexibility

[Harvey Friedman]

As an illustration of the profound flexibility of f.o.m., I call attention
to the recent Avron/Slater/Holmes (and others) thread. The issue there is,
roughly, how do we want to treat natural numbers in the foundations of
mathematics?

Avron and Holmes and others like a treatment in which natural numbers are
identified (?) with certain sets. This has a long history involving such
great mathematicians as von Neumann.

Slater objects that natural numbers are prior to set theory, and hence such
identifications cannot be literally true (or otherwise defective, say, in
grammar). Slater then suggests that the foundations of mathematics is in
serious intellectual trouble, as it is either incoherent or false.

It is well known that for various well defined purposes in the foundations
of mathematics, this issue makes no difference whatsoever.

However, one can of course dwell on other purposes in the foundations of
mathematics, for which the issue makes a difference. If one goes so far as
to make such other purposes paramount, then the issue is absolutely
critical. 

It is standard practice in f.o.m. to treat notions that are regarded as
particularly fundamental - not to be adequately explained in terms of any
simpler notions - as PRIMITIVES in formalizations.

It may be the case that for some completely unknown reason (to me) that
various participants have apparently forgotten this "old" posting of mine:

http://www.cs.nyu.edu/pipermail/fom/2003-May/006614.html

Actual mathematicians doing actual mathematics teaching (particularly at the
undergraduate level) also make decisions about the status of the natural
numbers, as least implicitly. As far as I can tell, they generally opt for
the natural numbers as primitives, within a set theoretic framework, where
sets are also primitives. The natural numbers are not sets, but rather
urelements. 

Some like the added convenience of having only one (two) primitive notions -
set and membership. Or maybe three - sets, membership, equality.

I don't support the idea of f.o.m. picking one solution to this situation
and forcing it on the mathematicians once and for all.

What I do support is to use this situation as a perfect illustration of the
profound flexibility of f.o.m. Namely

1. We know the appropriate formal systems that correspond properly to any
attitude towards this issue that is at least coherent.

2. We also know the exact relationship between these appropriate formal
systems. 

3. We also have illuminating general notions of comparison between different
formalizations of mathematics that illustrate what we mean when we say that
"it doesn't make any difference in practice".

4. We can also fully accommodate competing viewpoints SIMULTANEOUSLY. E.g.,
we can simultaneously support

i) naïve natural numbers (primitive);
ii) von Neumann natural numbers;
iii) Dedekind natural number systems;
iv) Russell natural numbers;
v) etcetera.

E.g., we will be able to prove in such formalizations that, e.g., no naïve
natural number is a von Neumann natural number, etcetera.

So we have "big tent" f.o.m. to borrow a phrase from USA politics.

All of this has been known for a very long time. See just what good hands
f.o.m. has been in (long before I entered the picture)?

Since all of this is so well known, the serious issues in comtemporary
f.o.m. lie elsewhere.

Harvey Friedman