[FOM] Falsify Platonism?
Vaughan Pratt
pratt at cs.stanford.edu
Fri Apr 30 00:06:36 EDT 2010
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
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