FOM: Arithmetic/Geometry

[Harvey Friedman]

I have casually followed the arithmetic/geometry thread of Shipman,
Tennant, Vorobey, Sazonov, Silver, Mayberry, Kanovei, and others. But I am
at a loss to pinpoint an exact statement of the disagreement. I think
making some distinctions would help get to the bottom of the disagreement,
if there is any.

Consider the following four:

1. Arithmetic as motivated by casual considerations of physical reality.
1'. Arithmetic taken as statements about physical reality.
2. Geometry as motivated by casual considerations of physical reality.
2'. Geometry taken as statements about physical reality.

Mathematicians are generally concerned only with items 1 and 2. They focus
successfully on 1 and 2 because of the startling fact that everything they
want to know is based on only such casual considerations, and the great
power of deductive reasoning. The main point is that for 1 and 2, physical
experimentation seems completely useless and irrelevant. Although, even
this point is subtly misleading or incorrect. The running of computer
programs can be viewed as a kind of physical experimentation, which is now
a common way of establishing new results in 1 and 2. However, it is also
"clear" that such results are always deductive consequences of the basic 1
and 2, but the deductions are themselves too large to be obtained by humans
without resorting to the "physical experimentation of actual computing." I
could follow this line of thought further, to good effect, but I will stop
here. E.g., how do I know that this physical experimentation can be
"replaced" by deductions? That involves a whole host of issues such as "why
should I believe a computer?" In any case, it seems inconceivable that
physical experimentation - in the broadest possible sense - is ever going
to falsify results in 1 and 2. One can follow this train of thought, back
and forth, with a determined sceptic - like Wittgenstein(?)(!) - and
generate state of the art f.o.m.

As far as 2' is concerned, it has become part of the folklore of physics
that this kind of geometry is affected by physical experimentation and
observation, and their is an underlying objective reality to this. I think
that the situation in contemporary physics is so conceptually murky - with
such things as quantum mechanics, string theory, etc. - that the content
and status of 2' is rather unclear. For example, in light of modern
thinking in physics, what exact meaning, operationally and otherwise, can
really be assigned to such questions as "is the parallel postulate true?"
"is space indivisible?" I have my doubts about the meaningfullness and/or
objectivity of this sort of thing.

And for 1', I don't see people even considering it. However, one could
interpret 1' to involve something like this: Is the axiom of successor
true? Another way one may want to put this is: are there infinitely many
physical objects? Of course, I don't know what a physical object is, and
that seems also to be very murky right now. Spinoff for f.o.m.: study
arithmetic without the axiom of successor. I don't want to say more about
this right now, except that this also leads to some specific state of the
art projects in f.o.m.

Now there are profoundly interesting similarities and difference and
relationships between 1 and 2. A lot of this quickly leads to unexplored
state of the art f.o.m. if systematically pursued. But before going further
into this on the fom, I'll stop here and see whether people in this thread
find it useful to cast the issues in this general framework. In particular,
in the light of this framework, what precisely is the disagreement about?