[FOM] The boundary of objective mathematics

Henrik Nordmark henriknordmark at mac.com
Mon Mar 9 02:03:30 EDT 2009

> I have long felt that objective mathematics is limited to statements
> determined by a recursively enumerable sequence of events, because  
> such
> statements can, at least in theory, be determined by events that occur
> physically.

> For those that question the objectivity of the Continuum Hypothesis,
> what do you think of this proposal for objective mathematics. If the
> answer is not much, where would you draw the boundary and why?

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.

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.

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.

Henrik Nordmark
Institute for Logic, Language and Computation
Universiteit van Amsterdam

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