FOM: Does it matter?
Joseph Shoenfield
jrs at math.duke.edu
Thu Apr 23 11:27:28 EDT 1998
When I was a youth, I was somewhat interested in the mind-matter
problem; but it was finally settled for me by the immortal answer:
never mind, it doesn't matter. (Does anyone know the author of
this remark?) I think some of the longest, dullest, and most
abrasive discussions on fom are about things that don't matter, at
least to people whose primary interest is the nature and structure
of mathematics. Let me give a couple of examlples (without, I
hopre, insulting anyone).
(1) Was Lagrange's theorem true of pebbles on a Jurassic beach?
Mathematicians understand this theorem, know that it is true, and
know how to apply it (eg, to undecidablity of certain theories of
the integers). Even if we could agree whether or not it was true
in Jurassic times, it would not affect the activity of mathemati-
cians or their understanding of the theorem.
(2) Are Boolean algebras the same as Boolean rings? The
exact send in which this two notions are equivalent is well known
and explained in many texts. Again mathematicians know how to use
the equivalence, eg, by using theorems on rings to prove facts about
Boolean algebras. Even if we could decide whether they are "really"
the same, it wouldn't matter.
What does matter? There has been much discussion of the state-
ment: ZFC is a foundation of mathematics. To most of us, this means:
all of accepted mathematics can be translated into ZFC in a fairly
straightforward way. I think most of us agree that this statement
is true. Why does it matter? There has been some discussion of
this, sometimes not well-considered. For example, it has been
stated that it gives a precise definiton of acceptable mathematics.
Perhaps it does, but not in a way that matters. The mathematicians
who verified the correctness of Wiles theorem did not concern them-
selves with ZFC. I think the statement matters because it has prac-
tical applications of the following kind. Cohen proved, by an
ingenious study of certain models, that CH is not provable in ZFC;
he then used the statement to conclude the much more interseting
result that CH is not provable by accepted mathematics. This
application alone wouls seem yo me to justify the hard work which
has been put into verifying the statement.
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