[FOM] Set theory and "folk" theory

[Jeremy Clark]

Dean Buckner wrote:
>
> The goal would be a "folk" theory of number that is more than just
folklore
> or psychology, but is importantly true in the sense it is consistent with
> all the fundamental assumptions of the folk scheme, and which is
> sufficiently rich to deal with all the numerical reasoning we require in
> day-to-day life.

I don't think that set theory is a good place to start if you seek a scheme
of the sort
described above. An awful lot of number theory, for example, can be
formulated in
Heyting Arithmetic, which is Peano Arithmetic without the law of excluded
middle. It
depends on what you call 'day-to-day life', which might be adding up
shopping lists for
some, and doing set theory for others. If you mean the former, then Heyting
Arithmetic
will definitely do it, because although we will also reason about sets,
they'll all be finite
(and hereditarily finite), so they can be encoded into numbers easily. I
doubt if real folk
maths would suffer very much if you restricted yourself to integers less
than 10^100,
even.

The idea of a folk theory of number is a tricky concept. In general a 'folk
theory' of something
(like psychology, physics) is, I think, the theory we *actually* use,
inconsistencies and all.
So that our folk theory of number would include *all* of our intuitions
about how numbers
behave. Consistency would be sacrificed in the name of efficacy in such a
theory, because
it has evolved to do a job in real time, and doesn't worry itself with such
ephemera as
G\"odel sentences and subtle paradoxes. (In the same way that folk physics
makes no allowance
for oxygen-starved candles going out.) Thus folk maths is a pretty limited
thing, even more
limited than folk physics because we do more day-to-day physics than maths.
Still I think
that our intuitions about how numbers behave ought to be a starting point
for building up a
theory. It is on these grounds, I think, that ZF set theory can be
criticised as excessive: do we
really need a well-ordering of the reals in order to do analysis? Okay, I've
thrown in choice.
Put it another way: do we even need the *concept* of a well-ordering of the
reals in order
to do analysis?

Jeremy Clark