[FOM] The boundary of objective mathematics
paul at mtnmath.com
Fri Mar 6 13:28:10 EST 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
Some statements requiring quantification over the reals meet this
definition. For example we can ask if a TM that keeps requesting integer
inputs will eventually halt regardless of what sequence of integer
inputs it receives. The result is determined by what the TM does for
every possible finite sequence of integer inputs. Such questions may be
of practical interest for those existing in an always finite, but
potentially infinite universe. (It is possible that we inhabit such a
universe in spite of what cosmology predicts for our ultimate fate. Of
necessity cosmology is an extremely speculative science.) Thus one might
be interested to know if the human species will evolve an infinite chain
of descendant species. One might even wish to make decisions that make
this more likely.
The Continuum Hypothesis is an obvious example of a statement that does
not meet this condition. Intuitionists are more conservative (at least
in terms of what proofs they accept) but most mathematicians are, I
suspect, far more liberal in the mathematics they believe is objective.
Yet there is increasing scepticism about the objective truth of the
Continuum Hypothesis and similar statements.
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?
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