[FOM] Falsify Platonism?

[Vaughan Pratt]

On 4/29/2010 4:38 PM, Jon Awbrey wrote:
> So we have the hypothesis of a "numberist", who thinks that it makes sense to
> speak of numbers as real objects, and we have the hypothesis of a "numeralist",
> who says that "numbers are only numerals".

Andrej Bauer made the point that the natural numbers can be presented as 
an initial algebra.  While he had in mind the initial algebra with 
signature 0-1 (one constant and one unary operation), with no nontrivial 
equations, there is also the free monoid on one generator, aka the 
initial pointed monoid with signature 0-0-2 (constants 0 and 1 along 
with an associative multiplication that makes 0 a left and right 
identity, with no nontrivial equation governing the constant 1).  This 
can be identified with the Roman numerals I, II, III, IIII, IIIII, ... 
(forget V, X, L, etc.) together with the empty string, closed under 
concatenation.

A "numeralist" who identified the natural numbers with the finite words 
on a one-letter alphabet (namely the letter "I") then becomes very hard 
to distinguish from a Platonist.

What does Peano arithmetic have to offer mathematics that isn't already 
provided by the naive Roman numerals, suitably understood as forming the 
free monoid on one generator "I"?

How could any nontrivial theory communicable with arbitrarily long 
finite words over an alphabet of one or more letters be more consistent 
than the theory of numbers understood as finite words over an alphabet 
of one letter?

Vaughan Pratt