FOM: Objectivity of logical/mathematical truth?
rhersh at math.unm.edu
Sat Dec 20 14:46:46 EST 1997
On Wed, 17 Dec 1997, Solomon Feferman wrote:
> In his reply yesterday to Joe Shipman on conclusiveness, Moshe Machover
> goes to the heart of the philosophical question: if one is not a platonic
> realist, Kantian or empiricist, and if one believes (as I do) that
> mathematics is "socially constructed", how can logical/mathematical truth
> be objective in character? What we have to accept is that more or less
> objective communication is possible on a whole spectrum of socially
> constructed concepts from nationality, marriage, money, English
> grammar, the calendar, position in the university, to chess and
> mathematics. Perhaps this pushes the question back to a wider and more
> puzzling question, but if one takes the possibility of such more or less
> objective communication as a given, then the question rather becomes: what
> is it about the conceptual and inferential structure of mathematics that
> makes it such a distinctive and supremely objective part of human
> objective communication?
> Sol Feferman
> You ask, "What is it about the conceptual and inferential structure
of mathematics that makes it such a distinctive and supremely objective
part of human objective communication?"
I suggest holding this puzzle by the other end. We observe,
experience, recognize, that a certain part of human objective
communication is "distinctive and
supremely objective." We should give a name to this interesting
and important part of human communication. In fact, such a name has
already been given, and retained over the centuries. It's called
"mathematics." "Mathematics" is what we call the supremely
objective part of human communication.
That still leaves Kant's question, "How is
mathematics possible?" If that question is
ever answered, it will be answered by neurophysiologists and
sociologists, not by logicians or philosophers. However, even if
we don't know how mathematics is possible, we do know it is
possible--which is the principal thing to know.
Compare with Heidgger's question, "Why does anything exist?"
Physicists don't try to figure out why anything exists. The main
thing, the starting point for everything else, is this: there is a
world, there is existence.
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