[FOM] The boundary of objective mathematics
paul at mtnmath.com
Mon Mar 9 20:50:58 EDT 2009
Henrik Nordmark wrote:
> What I find interesting is that you seem to suggest that mathematical
> statements can be partitioned into objective statements and non-
> objective statements. Most philosophical camps attempt to have one
> overarching status for all mathematical statements. You seem to be a
> realist for a certain subclass of statements and an anti-realist for
> the rest. Regardless of where exactly where one draws a boundary, I am
> wondering what kinds of problems does drawing a boundary generate.
> Clearly, having one status for all mathematical statements is more
> elegant but that hardly seems like a serious philosophical objection
> against your stance.
The natural instinct is to make a formal system for mathematics as
powerful as possible. This has led to inconsistencies in the past but it
can also lead to a formal system that is consistent but incorrect. For
example it might be consistent but not omega consistent. I suspect that
ZF does not violate any generalization of omega consistency but I still
think it is too powerful and thus in some sense incorrect.
It is incorrect in the sense of treating some relative statements as if
they were absolute because, in some respects, it treats infinite objects
as if they had something like a physical existence. This begins with the
power set axiom. The set of all subsets of the integers definable within
ZF is well defined. One can enumerate all the statements in ZF that
provably define a particular real. But this is not what mathematicians
want the set of all subsets of the integers to mean. However there is no
way to enforce what they want it to mean and when they argue as if what
they mean is objective they are treating a relative statement as if it
I an dividing mathematics into absolute objective statements and
relative ones that are sometimes treated as if they were absolute.
> However, I suspect that your position probably does have some problems
> with it. For example, you seem to accept the existence of real numbers
> and natural numbers, so these would be amongst your objective and true
> statements. But then, one can ask whether there is an infinite subset
> of the reals which is not in a bijection to either the reals nor the
> natural numbers. Since we are dealing with real objects, presumably
> this question must have a definite and objective answer, but this
> seems to contradict your position that CH is a non-objective
> mathematical statement.
Real numbers are problematic because they seem so natural and obvious. I
think we can talk about an arbitrary integer parameter for a recursive
process. And I think we can talk about an arbitrary infinite sequence of
integer parameters. This all makes sense to me in an always finite but
potentially infinite universe. Asking what a TM will do for all possible
infinite sequences of inputs make sense because every event that
determines that can be recursively enumerated. However I think this
question is a human creation. It is not a question about something that
can ever exist. It only asks about the unbounded future of a recursive
process in a potentially infinite universe.
In contrast the CH asks about what infinite sets exist in an absolute
sense. The question is meaningful relative to the sets provably
definable in a formal system but it has no absolute meaning because
infinite sets are human creations and abstractions.
> In all fairness, I do not know your exact position well enough to
> assess whether the example I just gave would actually be problematic
> for the stance you are trying to take. However, it does make me wonder
> if one can develop a philosophical stance with a realist/anti-realist
> dichotomy built into it that is not exceedingly problematic.
The difference I want to make is between an an arbitrary path that is
followed by a recursive process in a potentially infinite universe and a
completed infinite set. This is a distinction that goes back at least
to Aristotle. My position is perhaps close to constructivists, but I do
not demand a constructive proof of a statement. I only demand a
constructive proof that all the events that determine the statement are
themselves determined by finite events.
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