FOM: On "grasping" Con(ZFC)
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Sun Aug 30 10:23:43 EDT 1998
Robert Tragesser writes "There is an important sense of 'understand' where
it makes much good sense to say that we haven't understood a (mathematical)
proposition until we have proved or disproved it (or have shown that it is
absolutely 'undecidable')."
Even if such a sense of "understand" can be made out, and with interesting
theoretical consequences, it surely cannot be the kind of understanding that
the meaning theorist would want to equate with "grasping" the meaning of a
sentence---i.e. knowing *what the sentence says*, or knowing *what its truth-
conditions are*. One can know *what sentence S says* without knowing *what
the epistemic status of S is*. By epistemic status here is meant 'known to
be true', 'known to be false', or 'known to be unknowable-as-true and to be
unknowable-as-false'. Those three epistemic values correspond to Tragesser's
"proved", "disproved", and "shown [to be] absolutely 'undecidable'", respectively.
Note that none of these three epistemic values has yet been conferred on, say,
Goldbach's Conjecture (that every even number greater than 2 is the sum of two
prime numbers). GC has not been proved; nor has it been disproved; nor has it
been shown to be absolutely undedicable. YET there is no question but that
even a mediocre mathematician understands GC precisely---that is, grasps
*what GC says*.
Indeed, it is only because of such grasp---the kind of grasp that does *not*
depend on knowing the sentence's epistemic status---that one could ever come
to appreciate, eventually, that its epistemic status had been settled by
whatever proof, disproof, of independence proof might one day be offered.
Neil Tennant
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