FOM: Foundationalism

[John Mayberry]

	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
-----------------------------