[FOM] The Field with One Element?

[Harvey Friedman]

On Sat, Jun 18, 2016 at 7:33 PM, Colin McLarty <colin.mclarty at case.edu> wrote:
>
> Two different senses of "foundational" are being used here.
>
> The one-element field is in no way an issue for ZFC foundations versus
> categorical foundations or Homotopy foundations or anything like that.  It
> is even easier from that point of view than Mario Carniero suggests.
>
> The one element field is a matter of finite combinatorics and so it has more
> to do with Subsystems of Second Order Arithmetic.
>
> But the one-element field is a big challenge to the philosophically
> foundational idea that mathematics deals with structured sets.
>
Colin, thanks for the clarification, of course much of it present
already in http://www.cs.nyu.edu/pipermail/fom/2016-June/019914.html

I knew that there was a deeper underlying issue being pointed to, but
I didn't see the obvious thing I wrote down mentioned yet
http://www.cs.nyu.edu/pipermail/fom/2016-June/019916.html

The question at this point: what is the nature of this issue?

The kind of thing I normally think of as a foundational issue is an
issue which is of general intellectual issue in that it makes sense
generally in any systematic field of knowledge, even if it has nothing
particular to do with mathematics.

Prima facie, and prima facie only, this issue about 1 element fields
looks to not be of that character, and looks to do a specialized
mathematical issue. Like, e.g., how do we want to set up the basic
definitions for mathematics, reflecting on a number of celebrated
results. E.g., groups came later out of what amounts to groups of
transformations. Group actions, similarly. Lie groups, fields,
algebraic closures, C* algebras, analytic manifold,,...a virtually
endless list of principal setups that practically everybody agrees are
greatly clarifying and are "right".These definitions and hundreds more
are just the right ones to set up new mathematical theories.

I do not think of the matters in the preceding paragraph as f.o.m.
Rather it is what I call systemization, a perfectly noble enterprise.
HOWEVER, there is a subject called "foundations of systemization",
which seeks to understand what the proper criteria are for a proper
systematization, and how to set up theorems that show what certain
systematizations are proper - or how to classify systematizations
according to their effectiveness. Actually developing particular
systemizations is in the category of "normal professional conceptual
mathematical activity".

Similarly here there is a feeling of how to set up a missing theory,
or largely missing theory, which would meet the usual strong criteria
that mathematicians properly demand of a new mathematical theory.

I know something about this kind of thing, not this actual thing -
some of you watch me at the moment developing Continuation Theory. And
soon I will probably be absorbed with the following issues: right now,
I am "continuing" sets of tuples from a linearly ordered set. I will
want to be "continuing" perhaps arbitrary mathematical structures, but
I need to be doing this in the right settings and in the right senses.

In the case of "continuation theory", the theory is just a garden
variety interesting new mathematical theory, EXCEPT that as you know I
am setting it up because ZFC is not enough rather soon into the
theory. THEN it takes on a deeply f.o.m. aspect. But "continuation
theory" itself? This is meant very much to be NOT f.o.m.

Back to the present one element field situation.

I wonder if the "foundational issue" being suggested can be rephrased
or reinterpreted in a way that is of general intellectual interest, or
at least can be looked at as seriously analogous to other situations
which have been (somewhat) resolved? I understand the rough criteria
that counting formulas should hold universally.

Let me mention a situation that everybody is familiar with, and still
there is a kind of interesting conundrum. Operations on finite sets
and count formulas.

1. Disjoint union. Addition.
2. Cartesian product. Multiplication.
3. Functions from A to B. Exponentiation.

There is the obvious question of "what happened to the sets of
fractional cardinality?" "what happened to the sets of negative
cardinality?"

Conundrum: what is 0^0? According to the function interpretation, it
is definitely 1. But I gather that there is still a constituency for
0^0 = 0.

One can clearly see a theory that extends the notion of finite set to
weighted finite set. That might work out very nicely for 1, nicely for
2, but what about for 3? How do we get a reasonable counting or
combinatorial interpretation of real or even complex exponentiation?

Back to the present 1 element field situation. It would be helpful if
the issue can be rephrased in the most general intellectual terms
possible. More specifically, you indicate that counting the number of
substructures is important here?

Harvey Friedman