[hist-analytic] Clarity Is Not Enough
danny.frederick at tiscali.co.uk
Sun Feb 22 11:43:59 EST 2009
Like you I will be brief, partly because I am too out of touch with this
stuff to say too much (I have not done this sort of thing since I was an
undergrad), and partly because I am busy with something else. But, for what
it is worth, here goes.
It seems to me arbitrary to consign disagreements about methods of proof to
the philosophy of mathematics. First, the people who disagree over such
things are often leading mathematicians, such as Brouwer, Hilbert, Gentzen
and so on. Second, even if they were all to say that they are doing
philosophy when they disagree, we would have, I think, to regard that as a
joke. It matters to mathematics what counts as an acceptable proof because
if something has not been (acceptably) proved, then it may be false.
As I mentioned in an earlier mail, Lakatos' 'Proofs and Refutations' gives
examples from the history of mathematics of propositions once accepted by
mathematicians as 'indubitably proven' which were later rejected as false.
It seems plain that this can only be a recurring phenomenon where there is
doubt about the reliability of the method used to prove a theorem.
The fact that the consistency of arithmetic can be proved in a stronger
theory only raises the question of the consistency of the stronger theory.
The fact that no one questions the consistency (if it is a fact) is not a
proof of consistency. If a new paradox turns up, some propositions currently
accepted by some as indubitable may be rejected as false.
Russell's discovery of the paradoxes was a criticism of established results,
leading Frege to remark that 'arithmetic totters' (or something similar) and
leading to progress in the form of different attempts to avoid the
antinomies, some more elegant than others.
On the question of an empirical refutation of a mathematical proposition, I
indicated earlier that the dividing line between maths and physics has not
always been clear (is it now?), so some propositions may have been regarded
as belonging to mathematics until they were refuted empirically.
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