[FOM] The natural numbers as marks on paper

Nik Weaver nweaver at math.wustl.edu
Wed Feb 22 23:22:17 EST 2006

Martin Davis wrote:

> Alas, one person's marks on papers are another's abstract object
> smelling foully of Platonism. As Weaver perfectly well knows, a
> sheet of paper holds only finitely many marks.

Harvey Friedman wrote:

> It sounds like you are a confirmed Platonist who has some abstract
> concept of marks on paper that is independent of actual paper, actual
> marks, actual possession, actual destruction of paper, etcetera.

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

It doesn't seem like a very strong argument.  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 in a way that I don't know what it means
to talk about 2^1000 as an abstract entity, a von Neumann ordinal.
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.

An analogy.  Friedman talks about how easy it is to explain set
theory to laypeople.  I imagine laypeople would also readily
accept an explanation of how early the average taxpayer filed
his return last year, how much he paid, etc.  Do you believe there
actually is such a person as the average taxpayer?  Not a physical
person, to be sure, but an "abstract" person?

I could say something like "if you earned $12,345.67 last year
then you would have paid such-and-such in taxes" and you could
respond "but nobody actually earned exactly that amount.  Aha!
you must be talking about an abstract person!  So you're as
much a platonist as those of us who believe in the abstract
existence of an average taxpayer!"  Well, no.


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