# [FOM] The Platonistic philosophy of mathematics

Paul Budnik paul at mtnmath.com
Tue Oct 11 12:47:59 EDT 2011

```Feferman said "I believe the Platonistic philosophy of mathematics that
is currently claimed to justify set theory and mathematics more
generally is thoroughly unsatisfactory and that some other philosophy
grounded in inter-subjective /human/ conceptions will have to be sought
to explain the apparent objectivity of mathematics." (see
http://math.stanford.edu/~feferman/papers/newaxioms.pdf ). This posting
assumes the universe is always finite but may be unbounded. It suggests
that most of mathematics is objective in and possibly relevant to such a
universe.

Technology has justified a limited form of Platonism that refers, not to
another universe. but to physical means for idealizing part of finite
mathematics. We cannot construct Plato's perfect circle, but pi has been
calculated to over a trillion decimal places with a high probability
that it is correct. The complexity of finite statements that can be
physically simulated is continually increasing. Pi is the limit of a
convergent recursively enumerable sequence of numbers defined by finite
statements.

Infinity is a human conceptual creation that in some cases can be
objective. A recursively enumerable sequence of finite statements may be
all true or all false. There is no physical reality that corresponds to
these two alternatives, but they are logically determined by events each
of which is plausibly objective. This can be generalized by defining an
objective statement as a logically determined statement about a
recursively enumerable sequence of finite or other objective statements.
This is sufficient to define the hyperarithmetic hierarchy.

Logically determined in this context is a philosophical concept that is
subject to indefinite expansion. For the hyperarithmetic hierarchy it
can be limited to a recursively enumerable sequence of objective
statements that are all true or all false. In general it requires that
all events that directly or indirectly determine the outcome are
recursively enumerable and that their relationship is objective. It is
the latter condition that is indefinitely expandable and correspondingly
subject to error.

Consider an always finite put potentially infinite universe determined
by recursive laws in which a species can in theory have an unbounded
number of direct descendant species. The question will a species have an
infinite chain of descendant species is determined by a recursively
enumerable set of events, the history of the species and all of its
direct and indirect descendant species. Yet this question requires
quantification over the reals to state. Although they are not part of
engineering mathematics today, I speculate that important questions
about divergent creative processes like biological evolution will often
involve quantification over the reals.

In the universe postulated here, Cantor's proof that the reals are not
countable establishes the incompleteness of definability in any formal
system. One can always use diagonalization to define more reals. Large
cardinal axioms can be thought of as proposed extensions of this idea.
They postulate hierarchies of yet to be defined reals, functions on
reals , etc., that /apply to any "correct" formal system/ /that can be
defined/. Statements about uncountable sets do not necessarily have a
definite truth value since the sets they refer to cannot have a definite
meaning in this universe because there is no rigorous way to define the
absolutely uncountable.

Paul Budnik
www.mtnmath.com

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