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Philosophers of science giving advice about inconsistent theories (IN
CAPITALS):

https://netfiles.uiuc.edu/brhoads/www/book/chapt1.pdf
"Internal inconsistency is a defect in a theory that gives grounds for
seeking a better theory. Proving that a theory is inconsistent has the
same import as showing that it has false empirical consequences:
inconsistency guarantees that some part of the theory is false.
Unfortunately, this does not require that we throw out the entire
theory. Inconsistency only requires some modification in the theory,
and logic provides little guidance as to what modification is
appropriate. A classic example will illustrate the point. Special
relativity has two key postulates: the laws of physics are the same
for all frames of reference moving with constant relative velocities,
and the velocity of light is the same in all reference frames. Other
physicists besides Einstein contemplated the reconstruction of physics
on the basis of these two postulates, but abandoned the idea because
it seems easy to show that these postulates are mutually inconsistent.
Einstein notes the problem early in his first relativity paper, 'On
the electrodynamics of moving bodies', and announces that he will show
that these are only apparently inconsistent. He then argues that the
inconsistency actually requires a third proposition: whether two
events (at a distance from each other) are simultaneous is an
objective fact. Before Einstein's analysis, physicists had not
considered this claim to be a hypothesis subject to reconsideration,
and this is the hypothesis that Einstein rejects in order to eliminate
the inconsistency. There is an additional twist that we should note.
An inconsistent theory may be extremely useful if it is deployed with
sufficient care. The classic example here is Bohr's theory of the atom
which postulates that certain electron orbits are stable, but does so
in the context of classical electrodynamics which implies that no
orbits are stable. The theory's inconsistency was recognized as a
defect and this defect was eliminated by the new quantum theory of
Schrödinger and Heisenberg. But inconsistency is not a defect that
immediately blocks legitimate application of the theory. In general,
inconsistency is a formal defect in a theory, but THE DISCOVERY THAT A
THEORY IS INCONSISTENT DOES NOT REQUIRE ANY SPECIFIC RESPONSE FROM
THEORISTS."

Theorists do follow the advice of course. Yet there are embarrassing
views:

http://labnews.co.uk/laboratory_article.php/4514/2/2/attack-on-great-pillar-of-physics
"Precisely because Einstein's theory is inconsistent, its supporters
have drawn on contradictory principles in a way that greatly expanded
their apparent ability to explain the universe. Most crazes die out
when it becomes obvious that they were overblown. The amazing thing
about Einstein's theory of relativity is that it has kept going. It is
built on contradictions, but these very contradictions means that
almost anything 'proves' that it is right. It is a bit like a theory
where you say 1 + 1 = 2, but also that 1+ 1 = 3."

Pentcho Valev
pvalev@yahoo.com