Jack asked:

Mathematicians have tried to find certainty in the world through proofs and logic. Along the way Kurt
Gödel proved that mathematics would never be completely certain.

I thought one could not prove a negative. How did he do that?

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'Mathematicians have tried to find certainty in the world through proofs and logic.'

Mathematicians not only have tried to 'find certainty', they have found it. Mathematicians are free to
use 'proofs' and 'logic,' i.e., formal reasoning, in deriving one mathematical proposition from another
and in deriving propositions from theorems. There is absolutely nothing wrong with the 'proof' that 2 +
2 = 4; 137 x 59 = 8083; that the empty set is a subset of every set; that if a load bearing surface will
support a weight of 5 kg. per square cm., it will support a weight of 1kg. per square cm. And so on.
Gödel's two theorems do not impinge on the possibility of such proofs and demonstrations;
indeed, Gödel himself makes use of logic in arguing for his first incompleteness theorem, of
which the second is a corollary. How could he not use logic here?

'Along the way Kurt Gödel proved that mathematics would never be completely certain.'

Gödel really did no such thing. Try convincing a shopkeeper that you ítdon really owe her 25
pence if you've given her 75 pence for an item that costs a Pound, or try convincing a physicist that
the speed of light is not the limiting velocity in the universe because of anything Gödel 'proved.'
All that Gödel proved is that for formal systems there will always be sentences whose truth is
not decidable within that system. There will always be some sentence, P, such that if a system, S, is
complete, the truth or falsity of P will be as a matter of logicundecidable within S. 'You can't prove me
either true or false,' P might be imagined as saying. 'And if you do try, you will generate another
equally undecidable sentence, P-prime and so on.' Gödel's results were damaging for Hilbert's
dream of completely formalizingmathematics.

'I thought one could not prove a negative. How did he do that?'

I think that what you have in mind here is something like proving that the Abominable Snowman
doesn't exist, which has nothing to do with mathematics or with Gödel. It is often said that such
a 'negative' cannot be proved because there is always the chance that someday we might find such a
creature. Certainly our failure to have found the Yeti so far is not a logical demonstration that Yetis
don't exist, but this is a failure not of mathematics or of logic, but of exploration and discovery, if it is a
failure at all.

When Leverrier 'discovered' the planet Vulcan and used it to explain the perturbations in the orbit of
Neptune, he warned that it would always be unobservable because it was, from an earthly viewpoint,
always behind the sun. So, how can we prove that (as we now know) Vulcan does not exist? Simply
because Vulcan, the sun, and the earth, would have constituted a straight-line solution to the
three-body problem, a solution that is demonstrablyunstable.

Do not look for ramifications of Gödel's results where there are none.

Paul Trevor