Re: Cantorian pseudomathematics


Jesse F. Hughes wrote:
|"Keith Ramsay" <kramsay@xxxxxxx> writes:
|> You seem not to believe me when I say it, but there's a
|> simple, straightforward way of applying the "mathematical
|> theories must make predictions" rule that tells us to dump
|> the same two axioms, and nothing else. Take "computational
|> predictions" to mean only the most obvious kind of thing,
|> namely, statements of the form (n)P(n) where n ranges over
|> the integers and P is a primitive recursive predicate.
|> Then say that an axiom system should be no more complicated
|> than necessary to entail all conclusions of that form that
|> it does.
|
|Sorry, Keith, but I don't understand what you mean by that last
|sentence. Can you say that again, maybe a bit slower and with smaller
|words?

I'm not overly fond of requests for clarification that imply
the best you can possibly do is to throw up your hands, and
that you have trouble with such long words as "conclusions"
and "complicated".

In the philosophy of science one sees frequent references to
simplicity and strength as criteria for the quality of a
theory. This is often what is actually meant when people use
slogans like "theories should make predictions". Obviously
a theory that makes scarcely any predictions isn't what they
mean to suggest. They want it to be "strong": they want it
to make predictions readily, to make predictions about as
broad a range of phenomena as one can manage. They also
usually don't have in mind that it's okay if superfluous
ingredients are included in a theory that play no role in
enabling it to make predictions.

Since David Petry is proposing applying some form of this
"a theory should make predictions" slogan to mathematics,
I'm simply suggesting that an appropriate way to do that
would be to adopt analogous properties of a mathematical
theory as criteria for its being a theory of good quality.
It should make as many predictions as possible, and it should
not be more complicated than is needed to make them.

For the sake of argument I'm suggesting considering first
the consequences of a theory that are most obviously
computational. Namely, say that a fact is thoroughly testable
if it says that a certain computer program which is run to
test it will not find a counterexample. Or, equivalently,
that the fact is a matter of a certain program not terminating
(if it continues to be supplied to additional memory as it
needs it). Lots of mathematical results are essentially of
this form. We could write a simple program to test Fermat's
Last Theorem, by checking for counterexamples in some order,
and FLT would be equivalent to saying that the program would
never find one. It's less obvious, but we could do the same
for the Riemann hypothesis.

My point is that by this criterion the set theory on which
he heaps so much abuse makes predictions that do not follow
without using it. My secondary point is that I don't arrive
at this conclusion by mindlessly defending the status quo; I
agree that insisting upon the two criteria above is plausible
and would have consequences for how we do mathematics if we
were to agree to it. In particular, it would entail dropping
the axioms that, whether coincidentally or not, also happen
to be the nonconstructive ones (in the more common usage of
the term). But he's not happy with just that, for reasons
that as far as I'm concerned have nothing to do with wanting
mathematics to make computational predictions.

Keith Ramsay

.

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