[FOM] Mathematical Truth
Paul Budnik
paul at mtnmath.com
Thu Dec 31 12:41:32 EST 2009
I want to clarify the approach to mathematical truth I have touched on
in previous posts. This is based in part on comments from reviewers of a
bounced post on this subject.
There needs to be a line between objective mathematical truth and
statements that are true, false or undecidable only relative to a
particular formal system. I think objective mathematical truth requires
at least the theoretical possibility of a connection to physical
reality. By physical reality I mean a system in which a finite space
time region is fully characterized by a finite amount of information as
our universe seems to be. This connection can involve questions about
the unlimited evolution of a potentially infinite universe as long as
that evolution is completely determined by initial conditions. Thus I
think the halting problem for Turing Machines is part of objective
mathematics even though there can be no general way to decide the
question if the TM does not halt.
Long ago I reinvented the U quantifier. Un r(n) is true iff r(n) is true
on an infinite subset of the integers. One can replace an alternating
pair of universal and existential quantifiers on the integers by a
single U quantifier. I realized that the arithmetical and
hyperarithmetical hierarchies could be interpreted as generalizations of
the halting problem. The first step is to ask does a TM have an infinite
number of outputs. Next does it have an infinite number of outputs an
infinite subset of which are the Godel numbers of TMs that have an
infinite number of outputs. Iterating this up to any integer leads to
the arithmetic hierarchy. Iterating it up to any recursive ordinal leads
to the hyperarithmetical hierarchy. One can go further and consider a TM
generating outputs that are labeled either as termination nodes or the
Godel numbers of TMs to be simulated. One can then ask if such a process
is well founded i. e. if one simulates this TM and every output it or
its descendants generates, will every path end with a terminating node.
This is an objective mathematical question that is logically determined
by initial conditions even though it requires quantification over the
reals to state. It is a question that the inhabitants of a potentially
infinite universe might be interested in. They might want to know under
what conditions a biological species would produce in infinite chain of
descendant species.
These ideas and observations led me to the opinion that objective
mathematics is logically determined by a recursively enumerable sequence
of events. Logically determined in this context does not mean
computable. The events are all computable, but the logical relationship
between them is not since it can involve relationships on and between
infinite subsets of the events. This statement is vague, because I do
not think there is any definable limit to objective mathematical truth.
It lies in a strange arena of creativity and objectivity. It is creative
because there are no infinite sequences in physical reality. They are a
human conceptual invention as are the meaningful properties we define on
infinite sequences of events. It is objective because everything that
happens in the sequence is determined by initial conditions.
Existing mathematics has created an algebra of the infinite that is both
fascinating in its own right and extremely productive. I am not against
using whatever works. The concept of cardinals makes sense in my
framework, but it is a relative concept and not an absolute one.
I suspect that computer technology will eventually play a key role in
expanding the foundations of mathematics. It can make it practical to
deal with the combinatorial explosion in obtaining explicit formulations
of the recursive ordinals implicitly defined by ZF. With such explicit
formulations, I think we will eventually come to understand how to go
significantly beyond the recursive ordinals provably definable in ZF. A
result like this could shift the philosophy of mathematics closer to it
computational roots.
Paul Budnik
www.mtnmath.com
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