[FOM] Set theory and "folk" theory

[Dean Buckner]

Dean Buckner wrote:

> >My view is that set theory is wrong, in the sense it does not capture our
>>ordinary intuitions about sets and things.

Martin Davis replied:

> This makes about as much sense as saying that physics is wrong because it
doesn't capture our ordinary intuitions about work and energy. A scientific
discipline is not intended to capture our ordinary (fallible) intuitions,
but rather to improve on them. Particularly on FOM, set theory is of special
interest in
providing a foundation for mathematics where "our ordinary intuitions" have
proved utterly misleading.
>

Of course, where the natural sciences are concerned.  If you read
Aristotle's Meteorologica, you get some quite whacky explanations of why
things are the way they are, e.g. his theory in 1.8 of why the Milky Way is
"milky".  People now study these things, but for quite different reasons
than people in the 12C studied them.  Their interest now is because we think
they are false (in an interesting way), their interest then was because they
were thought to be true (in an important way).

But isn't it obviously different for the unnatural sciences?  You can
profitably read anything Aristotle wrote on philosophy or logic - not
because interestingly false, but because there still remains the question of
whether it is importantly true (as well as interesting).

Also, there are two senses in which our intuitions may turn out to be false
(i) there is something wrong with the fundamental assumptions underlying our
intuition, though our reasoning from those assumptions OK.  This is
the case with Aristotle, whose powers of reasoning are unparalleled, except
he starts form something clearly silly (e.g. that everthing composed of 4
elements).  Or (ii) it's the reasoning itself which is faulty - you find
this a lot in pseudo-sciences, where tendency to reason from a very small
number of (perfectly sound) cases to a general conclusion.

So in which sense are our orindary intuitions about sets fallible or wrong?
Is our reasoning suspect?  Or are our fundamental assumptions flawed?

I'm convinced it's the latter.  Our ordinary folk schema, I believe, merely
embodies different assumptions from set theory, and who is to question
fundamental assumptions?  (If we could, they woudn't be fundamental).

The difficulty is to say, in the case of the folk scheme, what those
assumptions are.  They are laid out clearly in set theory (that is its
principal strength).  Not so in the case of the folk scheme, though I
believe that a comprehensive analysis of the grammar of numerical statements
(a la Wittgenstein) will lead us to the right answer.

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.

>From which, as I said, we're some way off.


Dean Buckner
London
ENGLAND

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