[FOM] Some Clarifications on Platonism and Formalism
Dmytro Taranovsky
dmytro at mit.edu
Fri Oct 17 17:49:58 EDT 2003
Podnieks: "How is it possible that the most fundamental mathematical
structure allows only an implicit description?"
Formal definitions only allow descriptions of an object in terms of
other objects. The most fundamental concepts are simply understood
without being formally defined. Other concepts are defined in terms of
the basic ones. In fact, almost all (but not all!) mathematical
semantics can be expressed in the first order language of sets and
membership.
Podnieks: "I'm feeling baffled about the well known fact that
'consistent universal arithmetical statements are provable'."
Every true existential statement is provable, so every consistent
universal statement is true. No consistent recursive (or recursively
enumerable) theory proves all true universal statements.
Harvey Friedman: "The imaginary natural number system enjoys some very
fundamental properties that the imaginary set system does not."
More precisely, natural numbers are concrete. If n is a natural number,
then (at least, theoretically) n can be stored and manipulated by a
computer.
Sazonov: "The difference [of the study of works of fiction]
with mathematics is only that in these stories there are no formal
rules of behavior of these personages."
Simple axioms which are obviously true provide an almost complete theory
of integers. Moreover, there is a unique natural way to resolve all
natural incompletenesses in the theory of integers. Most of the
important incompletenesses in PA arithmetic are solved simply by
adding: If for all n, PA proves phi(n), then for all n phi(n). All
known natural arithmetical incompletenesses are solved by invoking the
notion of a set of integers, and adding basic axioms about such sets,
along with projective determinacy.
By contrast any attempt to formalize a work of fiction would lead to a
long list of arbitrary axioms and plenty of natural incompletenesses,
almost none of which are resolved by natural axioms.
Robinson, as quoted by Sazonov: "As far as I know, only a small
minority of mathematicians, even those with Platonist views, accept the
idea that there may be mathematical facts which are true but
unknowable."
Only a small minority of mathematicians work in the areas where
independence from ZFC is dominant. Apparently, the rest do not bother
to determine whether every mathematical incompleteness can be solved by
human civilization.
In his reply to my posting "On Platonism and Formalism"
Sazonov wrote, "Why do we need these subtle considerations whether the
physical reality exists or not?"
Such considerations are needed to dispel the belief that existence means
physical observable existence and that references to mathematical
objects are merely references to appropriate physical systems.
Sazonov: "There is NOTHING in the real world what would
correspond to the real (and even natural) numbers PRECISELY."
If the universe is infinite, then every integer is physically realized
as the number of "cobble-stones" in a particular region of space. Every
real number is probably realized as the distance between two particular
points.
Higher levels of the cumulative hierarchy are probably not physically
realized, but they still exist. They exist because they appear to exist
and there is no evidence to the contrary. They are not vague because
they can be formally defined in the first order language of set theory.
Best Wishes
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm
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