[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

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