danny.frederick at btinternet.com
Thu Jan 7 16:19:46 EST 2010
Yes, in times of old, mathematicians and others used to think that things
could be proved and not merely derived (in fact, some still do). And Euclid
was the model (though Euclidean proofs were informal and Russell thought
Russell himself was originally in the Euclidean mould: the idea behind his
logicism was to establish maths on firm foundations. But the discovery of
the paradox put paid to that idea. He then claimed that mathematical axioms
were hypotheses supported ('inductively') by the theorems that could be
derived from them.
Then came Hilbert's formalism and the revival of Euclideanism. Then Godel's
theorems put paid to that.
With the advent of logical positivism, the idea of self-evident axioms was
ostensibly dropped in favour of conventions (though this probably goes back
to Poincare if not before). But this was hopeless. For one thing, it still
relies on self-evidence since it assumes that we can simply see that the
conventions do not involve a latent contradiction that someone like Russell
might turn up. For another thing, the theory is inconsistent, as Quine
showed in 'Truth by Convention' and 'Carnap on Logical Truth' (both in his
'Ways of Paradox').
It is not quite right to say that, according to Popper, it is only when "p"
has been falsified that we achieve growth in our objective knowledge of
things. It is the ATTEMPT to falsify that yields knowledge, whatever the
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