[FOM] Falsify Platonism?

[W.Taylor@math.canterbury.ac.nz]

Quoting Monroe Eskew <meskew at math.uci.edu>:

>> Granted, we tend to suppose that we "know" what the basic properties of
>> the numbers are. But maybe we don't know. Maybe we only think we know.
>> That, indeed, is one of the central worries that people have about
>> Platonism: It looks at least conceivable that the mathematical facts, as
>> they are in mathematical heaven, could come arbitrarily apart from what
>> we think we know.

> I'm sorry, but I find this whole style of argument to be ridiculous.

I tend to agree.

> Look at the axioms of PA.  I need not list them here.  They are such
> basic and bone-headed statements.  They follow from the concept of
> natural numbers.

Yes.  I alluded to this in my earlier reply.  Well said Monroe.

It is a nonsense to say, "Yes, natural numbers are really there but we
could be wrong about their basic properties" !!

The whole point about natural numbers IS their (basic) properties.
They ARE their properties.  One might perhaps call this structuralism,
but I still prefer just to call it realism.

If these properties don't hold, then we're not speaking of natural numbers,
but something else - it might be groups, fields, reals, graphs, whatever,
but NOT naturals.   Whatever they are, the naturals ARE the things (if any)
that satisfy:    functional FOL(=) with 0, S  such that

1.  ~(Sx = 0)
2.  Sx = Sy  ->  x=y
3.  [P0 ^ (Px -> PSx)] ->  Py.

(I have played fast-&-loose with quantifiers for brevity.)

If one prefers relational rather than functional FOL there will be another
axiom or two to ensure S is a function.  That is merely technical.
Whether one regards 3 as a schema or otherwise is mostly merely technical.
How one introduces addition and multiplication, and whether one wants to
similarly introduce exponentiation (for convenience) is mostly merely  
technical.

The above three things ARE natural numbers.  Whether or not there are other
models is merely technical.

If the above three things do not hold, we are simply NOT talking about
natural numbers.   I took all this for granted - sorry if I was too sweeping.

And on this basis I claim again - falsifying numerical Platonism
(as outlined above) would mean finding a contradiction in the conclusions,
as Edward Nelson famously still hopes for!

I did not really follow Tim Chow's demurral about some sentences being true
(or applicable) and not others.  Perhaps he might like to elaborate.
____

And again I repeat, Platonism (realism) regarding set theory is a whole nother
story!!

-- Bill of Basics



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