# [FOM] The natural numbers as marks on paper

Harvey Friedman friedman at math.ohio-state.edu
Thu Feb 23 01:31:12 EST 2006

```On 2/22/06 11:22 PM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> No, I am not talking about some abstract platonic ideal of marks
> on paper.  I'm talking about marks on paper.

ULTRAFINITIST: What is a mark? What is paper? What is a mark on paper? How
big does a mark have to be in order to be a mark, and not a defect in the
paper?

Friedman:

>> It's not actual marks on actual paper. If it were, then only you
>> have natural numbers, because you can't relate your marks on your
>> paper with my marks on my paper.
>
> But certainly I can relate my marks on paper with your marks on
> paper.  I think this is actually done in real life arithmetic
> fairly often.

ULTRAFINITSIT: How can you do this? There are all sorts of problems as we
are not even in the same city. The relation between my marks and your marks
cannot really be done by running actual cords from where you are to where I
am. So are you thinking of "virtual cords" or "abstract cords"?

> It doesn't seem like a very strong argument.

Standard in f.o.m., in the sense that it is easy to complain about
practically any f.o.m. argument, and just as easy to countercomplain.

>Granted that I can't
> personally make 2^1000 marks on paper because of physical and
> biological limitations, I still know what it means to talk about
> this many marks on paper

ULTRAFINITST: Why? Do I have to accept this on faith? What does "this many"
mean? Aren't you begging the question? Sounds like you are a Platonist.

> in a way that I don't know what it means
> to talk about 2^1000 as an abstract entity, a von Neumann ordinal.

ULTRAFINIST: You seem to want to set up some notion of one-one
correspondence between some marks on one paper and some marks on another
paper in another city. What is a one-one correspondence? Is that an abstract
object? It sounds like you are using a notion of finite relation between
marks on one paper and marks on another, in another city. That can't be a
physical entity. So it must be an abstract object.

> The latter is an abstract object in a completely different sense.
> It is supposed to be a unique canonical entity whereas my marks on
> paper are not.

ULTRAFINIST: I don't see the advantage of your invocation of a theory of
abstract relations between marks-on-paper and marks-on-paper, where the
papers may be different. In fact, you are invoking this abstract notion even
if there are very few marks on each of the two papers. The situation gets
even more problematic if there are "lots of marks" on the paper. There are
also problems with overlapping marks, marks that were in the original paper
to begin with, etcetera.

SUMMARY: It is easy to complain about all levels, and it is easy to