[FOM] The Platonistic philosophy of mathematics

[Paul Budnik]

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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