[FOM] Richard Epstein's view
Timothy Y. Chow
tchow at alum.mit.edu
Fri Mar 30 14:20:49 EDT 2012
On Fri, 30 Mar 2012, Arnon Avron wrote:
> "With normal vision" - so blind people cannot understand claims of the
> form "The formal sentence A is/isn't a theorem of the formal system T"?
They can. To understand an abstract statement, it is not necessary to
understand every single concrete instantiation of it first. One need only
understand *some* concrete instantiations of it, and then possess the
ability to abstract from the concrete instantiations.
> "suitably programmed electronic computer" what is this? "suitable"
> for what? And what is the criterion for being "suitably programmed"?
> Can this be given any sensible (or at least non-circular) meaning
> without understanding first the absolute nature of the truth of
> propositions of the form we are discussing?
You're confusing *meaning* with *truth*. One needs to understand the
*meaning* of an abstract proposition in order to identify which concrete
statements are instantiations of it. But it's not clear to me that one
needs to predicate *truth* of the abstract proposition in order to assign
meaning to it. For example, we all understand the *meaning* of "T is a
theorem of X" regardless of whether T is or is not a theorem of X. The
truth or falsity of the assertion doesn't come into play at all.
Now one can try to develop a theory of the truth of abstract statements,
but it will necessarily rest on a prior notion of the truth of the
concrete instantiations. That is, we would declare an abstract statement
false if and only if there are no true concrete instantiations of it.
Notice that this approach helps explain why we are perplexed about the
"truth" of, say, "strongly inaccessible cardinals exist." Presumably this
statement is true if and only if there is some concrete instantiation of
it that is true, but since we don't seem to have direct access to concrete
instantiations of strongly inaccessible cardinals, we are not sure how to
assess the truth value.
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