Modern mathematics doesn’t make complete sense. The unfortunate consequences include difficulty in deciding what to teach and how to teach it, many
papers that are logically flawed, the challenge of recruiting young people to the
subject, and an unfortunate teetering on the brink of irrelevance.
If mathematics made complete sense it would be a lot easier to teach, and
a lot easier to learn. Using flawed and ambiguous concepts, hiding confusions
and circular reasoning, pulling theorems out of thin air to be justified ‘later’
(i.e. never) and relying on appeals to authority don’t help young people, they
make things more difficult for them.
If mathematics made complete sense there would be higher standards of
rigour, with fewer but better books and papers published. That might make
it easier for ordinary researchers to be confident of a small but meaningful
contribution. If mathematics made complete sense then the physicists wouldn’t
have to thrash around quite so wildly for the right mathematical theories for
quantum field theory and string theory. Mathematics that makes complete sense
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tends to parallel the real world and be highly relevant to it, while mathematics
that doesn’t make complete sense rarely ever hits the nail right on the head,
although it can still be very useful.
So where exactly are the logical
