[FOM] Permanent Value?
Peter John Apostoli
apostoli at cs.toronto.edu
Fri May 14 12:49:53 EDT 2004
> Mr. Friedman asked genuinely, I think, whether or not Philosophers have
> made contributions to "enduring knowledge" in the sense of things known
> in the last five years comparable to the kinds of knowledge that
> mathematicians (and we'll assume he meant "pure mathematicians" for the
> moment) make regularly.
I too missed this interesting post from Prof. Dr. Friedman. I'd like to
point out the limits of formalized axiomatic foundations of mathematics by
comparing ZF with an alternative approach to the set theoretic foundations
of mathematics based upon pure semantics. The later approach offers a
degree of scientific unification undreamed of in Prof. Friedman's
But first, and as a side remark, we should keep in mind that
mathematical work done within the framework of ZF has only as much
"lasting value" as the framework itself. As no one knows whether that
framework in even deductively consistent, how much "lasting intellectual
value" this work has may still be an open question.
ZF is not a model of set theory. It is a set of axioms which allow
sustantive questions of set theoretic truth to be replaced by linguistic
question regarding formal derivability. As such, it is a theory of
"representations", rather than a theory of set-like objects (sets). Due to
deductive incompleteness, ZF falls far short of specifying the fine
grained structure of the set theoretic universe that an adequate ontology
is required to give.
We speak of "the" set theoretic universe because
philosophers have developed a canonical model for naive set theory. The
model, constructed using at most 4th order arithmetic, is the unique
solution to Russell's paradox. The model is canonical in the sense that it
is the canonical model of an associated (non Kripkean) modal logic. In
particular, it contains every logically possible set as an element, that
is, every logical possibility is represented.
However, this solution to
Russell's paradox requires that the universe of sets be "granulated" under
the set-theoretic indiscernibility relation. In other words, Russell's
paradox is tanamount to a proof that the set theoretic continuum has a
granular structure. Sets come surrounded by a Halo or granule of
infinitesimally close ("modal") counterparts. This granular structure of
sets directly relates the canonical universe to the areas of infinitesimal
analysis, rough set theory, quantum computing and renormalization theory.
The granularity of the set theoretic continuum is a mathematically
demonstrable truth (first established in Cocchiarella's formal systems
T,T* from the denial of Russell's contradiction), which asserts that the
continuum has a fundamental "Planck length". It is upon this discrete
basis that the phenomenal continuum (the characteristic smoothness of the
continuum) emerges as a large-scale approximation. By Goedel's
incompletness theorem, this fine grained detail could never be derived
from an axiomatic set theory such as ZF.
This model does more than provide
a categorical consistent conception of set. It unifies set theory with
modal logic, the foundations of physics, the theory of rough sets and
rough set approaches to informatics and engineering science more
generally. Such theoretical unification is unheard of in mainstream FOM
but is the earmark of successful science these days (when traditional
academic divisions between the sciences are eroding).
Therefore, the canonical universe of sets is a better scientific
explanation of sethood that ZF. And its still an open question whether ZF
provides any explanations at all.
How's that for lasting value in the FOM?
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