FOM: Foundationalism
John Mayberry
J.P.Mayberry at bristol.ac.uk
Sat Sep 12 07:25:58 EDT 1998
It may be, as Stephan Ferguson says, that Quine has discredited
(Cartesian) foundationalism in philosophy, but there seems to me to be
a clear sense of "foundationalism" in which to discredit
foundationalism in mathematics is to discredit mathematics itself.
I don't want to trigger off yet another debate about the
meanings of words, but it seems to me obvious that mathematics not only
has foundations, but cannot do without them. This is because
mathematics, unlike any of the other sciences, deals in rigorous proof
and exact definition.
If you are going to prove something, you have to start
with propositions that you and your interlocutors accept as true
without proof. And if you are going to define something, you have to
start with concepts that you and your interlocutors understand without
prior definition. The propositions that mathematicians accept as true
without proof, and the concepts that mathematicians accept as
meaningful without prior definition, then constitute the foundations
upon which mathematical proof and mathematical definition rest.
Of course in practice mathematicians start their proofs from
previously established theorems and their definitions from previously
defined concepts. But you can always demand that proofs or definitions
be supplied, and in this way you can trace a given proof or definition
back to its ultimate presuppositions. When you have arrived at
propositions that don't require, or admit of, proof, or at concepts
that don't require, or admit of, definition, then you have reached
bedrock, and there the genuine foundations of mathematics are to be
found.
Naturally you may be disappointed, or scandalised, by what you
find. Historically, mathematicians' ideas of what requires proof or
definition have changed. But by and large, the change has always been
in the direction of greater depth, clarity, and precision, since once
you have seen a genuine difficulty, you cannot subsequently simply
ignore it.
So I say that the idea of "mathematics without
foundations", if the term "foundations" is understood in the sense I
have just given (and that seems to me to be its most basic sense), is
simply a contradiction in terms.
-----------------------------
John Mayberry
J.P.Mayberry at bristol.ac.uk
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