Re: Choice sequences, intuition, etc


On Nov 6, 9:17 pm, Bill Taylor <w.tay...@xxxxxxxxxxxxxxxxxxxxx> wrote:
|OK, let's see if I've cottoned on a bit more. You [Herman] imply,
and
|Bridges has told me outright, that what Intuitionism is about,
|is not so much Platonic-style "truth", as *assertability*.

I wish to warn again against thinking this way. Bridges is
of course an authority on these things, but I think I have
valid qualms here.

My main problem here is with putting it in terms of "what
Intuitionism is about", as if we were considering both the
whole thing, and speaking of it in terms of what its purpose
is.

I'll use again an analogy I used before, which might not
seem entirely fair, but which I think may make my point
clear. Suppose you have a sick person being tended to by
a doctor who is a materialist and a religious person who
believes in the supernatural (in particular, in the
existence of a soul that is what the person ultimately is,
the essential self of the person).

From the doctor's point of view, the person essentially is
their body. Now the doctor may nuance this view in certain
ways. He might say for example that physical health is not
his only concern because he also cares about leaving his
patients in a good state of mental health, but regarding
mental health as merely being a more subtle aspect of their
physical state. He might note that what makes a person well-
off is not strictly confined to their own body; how the
person is connected to their social network is vital, and
hence things like explaining the patients' treatment to his
friends while they are visiting could be a useful part of
the therapy. But in the end, the doctor essentially regards
the person as being a body, and that is what he treats.

From the priest's point of view, all this fails to take
properly into account the patient's soul. One could say
perhaps correctly that what the medical treatment is about,
is not so much religious-style well-being of the person's
true self, as about the person's physical health. Putting
it that way though tends to quietly prejudice against the
doctor and in favor of the priest. The priest may suppose
that the medical treatment is vital, but needs to be
supplemented with prayer, for the sake of the patient's
soul, which though intimately connected with his body is
nevertheless a distinct item having separate needs. The
doctor probably has heard of the studies showing some
medical value in practicing rituals that calm a person down
(lowering their blood pressure for example) and see prayer
as possibly having such a value, but not because it
addresses something metaphysical.

If anyone were to claim that the doctor's practice is less
motivated by a desire to help the patient be well (as
opposed to merely tending to his body) I think the doctor
is entitled to object. The near identity between a person's
"self" and body, in his view, is no reason to attribute to
him less concern for the one and more for the other. The
reason for all his concern for the body is precisely
because that is how he sees himself best helping the person
as such.

When I first saw intuitionism and other kinds of
constructivism it seemed fairly strange. I read some
disparaging remarks about the meaningfulness (in the sense
of "having meaningful content") of the law of excluded
middle, and it seemed very beside the point. One wants to
know whether it is actually the case that P and whether it
is actually the case that Q, and if you can't get that,
then finding out that at least one of them is true (even if
you have no way to tell which it is) is again something one
wants to know. I had this sort of impression of being able
in some limited and indirect way circumvent the limitations
of "what we can prove" to get some partial knowledge of
"what is true anyway, even if we can't prove it". It seemed
like a case of my getting more of these "indirect truths"
than the constructivist, on account of my not being
pointlessly skeptical about what we rationally recognize as
being true about "truth in itself".

I now see this experience as having a strange kind of
cognitive illusion built into it. I thought of myself as
"getting" something (a piece of the truth) that these
other people were "not getting", but on closer examination
the extra bit I was "getting" turned out not to be very
much like I thought it was. One useful reflection was on
the fact that in the cases I was considering, we were both
"getting" the fact that assuming P and Q were both false
led to a contradiction: ~(~P & ~Q). That is what I would
now describe as the "brute fact" being considered. Or one
could say equivalently, ~~(P or Q). Now the insistence that
"well, that P or Q is so is *really true* in *things as
they are independent of what we know about them*" starts to
seem much more hollow.

So here I was, serving as a kind of priest-in-training for
the Platonist cabal, thinking that through double-negation
elimination, I was reaching beyond the mere concern for
constructible objects into concern for pure existence,
while the good constructivist doctors note that the benefits
of such rituals, such as they may be, all are derived from
the impact that they have on your proof theory.

What the non-constructivist "gets" out of accepting double
negation elimination (or equivalently the law of excluded
middle) is at most the opportunity to simplify what they're
saying by dropping out double negations that would otherwise
possibly be needed, and whatever side-benefits that making
such simplifications everywhere might bring. In the process
they succeed simply in obliterating the difference in
meaning between proving "P or Q" and proving "~~(P or Q)",
and so on.

It's not so easy to see whether the benefits outweigh the
loss of meaning. One might think that after all these years
I would have a more definite opinion on that, but I haven't
ever had this sort of evaluation as part of my job, not
really when I taught, and now that I'm not paid to do
mathematics, not at all.

I think there's a temptation to relegate the difference
between constructivism and other philosophies to a matter
of personal taste. I can say that I have enough of an
intellectual curiosity about what's true without having to
assume the law of excluded middle to pursue that off and on.
(I wish I had a better grasp on what assuming the axiom of
choice does to mainstream mathematics.) That's also true of
various logicians and theoretical computer scientists who
have less of a favorable view of constructivism as a
philosophy. Here I'm admitting that an element of personal
taste enters into it. Perhaps those who like to describe it
in terms of personal taste are tacitly wishing that the
difference could be laid to rest that way. I think that
this element of personal taste is a red herring, though.

It's as if you went into an ice cream parlor and discussed
your preferences for chocolate or vanilla ice cream,
assuming that it was all just personal taste, only to find
that the real issue for some of you was the possibility that
one of them was more HEALTHY than the other. The classical
mathematician is familiar with cases where a person likes
a field in mathematics because of their personal tastes. A
person might like constructive mathematics because of a
personal taste for it. But I think that that is much less
of an issue than people think that it is, as opposed to the
belief that one or the other way of doing mathematics is
HEALTHIER for mathematics than the other. That is, is one
superior based on the SAME criteria as people consider as
standard for judging the vitality of a mathematical field,
including in particular how well it helps us get at truth.

Keith Ramsay
.

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