FOM: Understanding Con(ZFC): the corridor test
Stephen G Simpson
simpson at math.psu.edu
Sun Aug 23 19:04:37 EDT 1998
Neil Tennant writes:
> It is common, in the theory of meaning, to appeal to the notion of
> being able, *in principle*, to do something/recognize
> something/carry out some procedure, etc., without regard to its
> feasibility (in terms of space- or time-requirements).
Whose "theory of meaning" is that? Tarski's definition of
satisfaction and truth? Certainly Tarski's definition is a landmark
of foundational research, but it has its limitations. In particular,
it's not very convincing when we come to the psychology of what it
means to understand something. This psychology needs to take account
of the way the human mind works, in particular the mind's need to
conceptualize and simplify by boiling down to essentials.
Let me clarify. When I speak of "understanding", I'm not talking
about some artificial construct of analytical philosophy. Rather I am
talking about conceptual understanding, in the layman's sense.
Conceptual simplicity is the key to this. Harvey once told me that,
in testing the understandability of his independent statements, he
sometimes uses what he calls "the corridor test": can you explain it
to a mathematician while standing in the corridor outside the
mathematician's office. This is the kind of understandability that I
am referring to. Con(ZFC) fails the corridor test, unless the
mathematician in question already knows quite a bit of mathematical
logic and set theory.
> It would be wrong to demand of him/her a knowledge of numeralwise
> representability of recursive functions as a prerequisite for
> *semantic grasp*.
Neil, I don't understand. How am I demanding a knowledge of
numeralwise representability of recursive functions? Such knowledge
seems irrelevant to a conceptual grasp of Con(ZFC).
> Thus Steve would be going against Frege, one of the greatest
> foundationalists, by denying compositionality and insisting on
Perhaps Neil is right and my position is somewhat at odds with Frege.
Didn't Frege rail against what he called "psychologism", i.e. taking
account of the specific nature of the human mind? I was never very
happy with that aspect of Frege.
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