danny.frederick at btinternet.com
Thu Jan 7 07:37:38 EST 2010
'Proof' is sometimes distinguished from 'derivation.' A derivation of q from
p shows how q follows from p. But since p may be false, the derivation of q
from p does not amount to a proof of q. For a proof of q we also need a
demonstration that p is true. Obviously, if 'demonstration' is taken to mean
derivation, we get an infinite regress. So if we are to have a proof, then
demonstration must cover also the mere showing or exhibiting of the truth of
a proposition. For this to be possible there must be propositions that are
in some way self-evident to us, that is, axioms that can be known to be true
without being derived. But, since we lack the power to know any proposition
self-evidently, there is no such thing as proof - at least, not for fallible
creatures like us. When people talk of 'proof' they normally mean
'derivation;' if they don't, they are just mistaken.
When the complete steps of a formal proof are made explicit, what has
happened is that a valid argument (one such that it is necessary that if its
premises are true then so is its conclusion) has been turned into a formally
valid argument, that is, one that can be mechanised. When mathematicians
provide 'proofs' they offer (if they are successful) valid arguments from
accepted premises; but these will rarely be formally valid arguments (to
produce such would in most circumstances be a pointless waste of time - what
matters is that it can be done).
Incidentally, 'Proofs and Refutations' was written by Lakatos, not Popper,
though it did apply Popper's theory of scientific knowledge to mathematical
I have not forgotten that I owe Roger a reply on a related topic. I will get
around to it eventually!
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