Re: A quote (and question) about intuitionism

On Fri, 02 Nov 2007 00:35:29 -0700, Keith Ramsay
<kramsay@xxxxxxx> wrote:

You might find Dummett's writings on verificationism
interesting. He puts forth arguments for (something like)
the position that the meaning of a statement is given by the
conditions under which we're justified in asserting it (i.e.,
when it's been verified). (He's careful to qualify himself,
especially in more recent writings, which is why I wrote
"something like".) A good place to read about this, especially
if you want to read about intuitionism at the same time, is
his book "Elements of Intuitionism". Much of the philosophically
relevant stuff comes early in the book, too, so you needn't
read the whole thing.

(I remember looking at that book once, a long time ago, and
just bouncing off it. But I bounced off a lot of things then
that I might not bounce off now, so it might be worth another
look. In a way, I have always been trying to avoid thinking
about these problems before, whereas now, if a problem is in
my way, I feel bound to think about it. And the question of
the validity of classical logic now seems to be in my way.)

Think a little about what you mean when you consider that P
and "P is known to be true" have potentially distinct truth-
values. You imagine the possibility of P being unknown but
being true anyway. From a constructivist point of view, this
is not much different from imagining that you don't know P,
but some hypothetical external observer does. Or perhaps you
discover it later. If there's any meaning to the notion of P
"just being true anyway although I don't know it" beyond
that, it's a little obscure.

But to me it seems perfectly clear, apart from a crisis of
confidence in my own judgment. That lack of confidence is
partly due to imagining that in someone else's mind there
must be a clear perception that I am making some schoolboy
error, which would be embarrassing to me to have exposed.

I see no way out of this other than simply to say what seems
to me to be obviously true, so that any errors I am making can
be pointed out. I just have to become inured to embarrassment.

So I will pompously state my possibly ridiculous (and certainly
banal and unoriginal) opinions, without further apology (hoping
that this one will suffice).

My first remark is that the negation of P is "P is false",
whereas the negation of "P is known to be true" is "P is
not known to be true", and these seem like clearly different
statements, so the statements being negated must also be
different. But I really do think that this is probably a
silly error (damn, I said no more apologies!), and I don't
want to lay much stress on it. It seems more important to
focus on the really basic point you are asking me to think
about.

Without having thought about it in detail yet, my first impression
is of being faced with an embarrassment of riches. (Embarrassments
do seem to keep cropping up!) Surely there are lots and lots and
lots of things that nobody knows but that are still true or false?
You seem to be giving me a /huge/ target to aim at. It must be a
trap! Let me read a little more first:

If you are thinking of truth as something external, then it
will seem as though the subject whose experience is being
considered needs to be specified, before we can say whether
the truth and what "we" know are really the same. But each
subject can speak from their own experience, and we can also
speak from a collective experience. In each case, the speaker
is describing their own experience. Saying that something is
true is saying that you know it's true, whoever you might be.

Indeed, /saying/ that something is true is saying that you
know (or rather that you believe - to "know" is stronger)
that it's true. No quarrel (or not much) there.

But one can mention without using. And in mathematics one
very often mentions without using.

Obviously it would be absurd for me to say, "P is true, but
I don't know it's true." But it is not at all absurd for me
to say (as I do!), "There exists P such that P is true, but
also such that I don't know P is true." That's sheer modesty
- sanity, even.

For example, you are thinking something right now, about this
argument. What you are thinking (not all of it, but some of
it, and an important part) can be stated in words (and other
symbols). So there is a statement both well-formed and true,
of the form "Keith Ramsay is thinking Q right now". I don't
even know what Q is, but I'm pretty sure there is some such
true sentence.

I don't expect you to disagree (much) with that, because it's
already covered by your phrase, "some hypothetical external
observer does" (the observer in this case probably being you,
observing yourself).

But let me take (more or less at random) a proposition for
which there is no human observer. /Something/ wiped out the
dinosaurs. Perhaps it was a meteor, perhaps a supervolcano,
perhaps a burst of gamma rays, perhaps (probably?) something
else - I don't know! But there is a true proposition of the
form "X wiped out the dinosaurs", although nobody at the
moment can (as far as I know) reasonably claim to be sure
what X is (was).

The weather's getting cold, and I've lost my hot water bottle.
I'm pretty sure it's still in the house somewhere, but bless
me if I can find it, now that I need it. There is (you know
what's coming!) a true proposition of the form "Angus's hot
water bottle is right HERE!", but I don't know (and nor does
anyone else) that that proposition is true (otherwise I would
go and get the bottle and fill it up with hot water right now).

/Someone/ was Jack the Ripper. (OK, that's not absolutely
certain, as there might have been more than one, but that
seems highly improbable.) In the last pages of a detective
story which will (probably) never be written, the identity of
the murderer is revealed. In point of cold fact, some actual
living, breathing human being committed those actual crimes.
Given (quite realistically) a list of all possible suspects,
you could generate a list of damning verdicts, one of which
would be true. But at this late stage we don't seem to have
any way of finding out which.

William Shakespeare (almost certainly, and disregarding all
claims that someone else wrote the plays) had something to
eat on the morning of the day he wrote the words, "To be or
not to be ...". Fat chance that we'll ever find out what he
had for breakfast that day! But it was real, it filled his
stomach, fortified him for the job he had to do that day.

On the side of the Moon hidden from the Earth at the moment
(unless it's a full Moon - I haven't checked), the rays of
the Sun are heating up a certain patch of dust to a certain
temperature. Certain? Well, none of us will ever be able to
draw an accurate temperature map of the entire surface of the
Moon right at this moment; nevertheless the Moon is being
heated up right now.

A tree falls in the forest, and nobody hears; but there is a
sound, which startles a bird off the branch of a nearby tree.
Do we have to invoke that bird as a "hypothetical observer"?

There are various forms of the "Haar condition" on a finite-
dimensional subspace of C[a,b]. Forms (1),(3),(4) are clearly
equivalent, and clearly imply a further condition (2), which,
however, an authoritative textbook on approximation theory
(Powell) states does not imply the others. My professor and
a colleague find an error in Powell's alleged counterexample,
and I verify (as an exercise) the existence of the error.
Moreover, Prof. X claims they proved that (2) in fact does
imply (1),(3) and (4). I get tantalisingly close to proving
this, but can't. Prof. X says he has a decades-old unpublished
manuscript. Perhaps there is an error in that, too? I really
don't know, unless I see it and work at it; but I don't persist
in my questioning about it, and am not /that/ interested. So I
end up in a state of uncertainty (quite tolerable, as I don't
really care!) as to whether (2) implies (1), say. But I don't
doubt there is a fact of the matter. I think it is highly
probable that the matter was settled twenty years ago by my
professor and his colleague; however, as their work was not
fully peer-reviewed, some doubt must remain (and some would
remain even if had been, of course - "certainty" in that
sense is simply not achievable, in a world in which living,
breathing, fallible, flesh-and-blood beings somehow interact
with propositions and proofs). But the point is that it is
quite possible that /nobody/ really knows whether (2) implies
(1) or not (because perhaps is has never been that important
to know for "sure", and I am not the only one who doesn't
care enough to really find out); yet, either (2) implies (1),
or it doesn't.

If I say I know Fermat's last theorem is true, for example,
I might be saying merely that I know this from the work of
others (which happens to be the case). I've seen parts of the
proof but I can't claim to know the whole thing. But I would
be saying that in a different sense than if I said I knew it
from knowing an entire proof in detail.

It's tempting to say in such a case that these are two
degrees of knowledge about the same one truth, as opposed to
two different truths (as I usually would, in fact). But this
distinction between cases where two kinds of knowledge are
considered "knowing different facts" and cases where two
kinds of knowledge are considered "knowing the same fact to
different degrees" seems somewhat conventional and arbitrary.
This perhaps subtle-seeming distinction in points of view is
more significant than it might seem at first.

For example, consider the statement

(*) The Riemann hypothesis is either true or false.

A constructive proof of (*) would require having a method
for determining whether the Riemann hypothesis is true. We
already have a nonconstructive proof of (*), just by the
trivial reasoning that all mathematical statements are either
true or false. From a constructivist point of view, that (*)
is true means one (highly nontrivial) claim, which is not
yet known, and that it follows from the law of excluded
middle, LEM=>(*) is a second claim that we know already
with ease. They are two distinct facts, not one fact known
to one degree of "thoroughness" now and another later.

Two distinct facts, because of two distinct meanings given
to each of the words "true" and "false". Why is this not an
ordinary case of ambiguity? Cannot both senses of "true" and
"false" coexist in the same mental universe?

There's a certain temptation from a nonconstructive point of
view to think that there is just one fact in question here,
(*), and that constructivists are merely interested in
finding a different and less trivial kind of proof of it. Or
maybe there's just perverse skepticism about LEM. I say that
because people say things like that when discussing
constructive mathematics. They'll ask, why the skepticism
about such simple facts?

Why the scepticism about such simple facts? ;-)

A person with a classical realist
point of view who has some philosophical sophistication will
pretty soon of course think of alternative ways of looking at
the situation,

Ouch! :-)

but the initial temptation to think that it's
just one statement being proven in two ways seems fairly
stable.

I think I could be quite happy to regard it as two distinct
statements - if only I understood your sense of "true" and
"false". I /think/ I understand my own sense of the words
(but this whole discussion, this last week or so, has cast
me into a pit of self-doubt).

Just consider the fact that constructivists agree that the
statement (*) cannot be false. Do you find that that leaves
you feeling like you have no choice but to consider it true?

No, because I don't understand what you mean by "false" and
"true"! I might well consider it "true" in my own classical
sense (were it not that seeds of doubt have been sown even as
to that, and now I want to see a /proof/ that (*) is true ...
"true" in my sense, I mean!).

(I have no idea what such a "proof" would look like, but I am
fairly sure I would be much less demanding of it than you.)

What else could it be? Nevertheless, for the constructivist
(*) represents a fact (i.e. a fact about the world that we
experience) that has not yet been experienced as true, hence
we should not claim that it is.

Fair enough (even with my use of the word "true"). I'm not
entirely serious in my use of the word "proof" here. What I
need more closely resembles reassurance than proof. I'd like
to feel the confidence Hilbert felt about the subject, about
"Cantor's paradise" in particular (but not at all necessarily
agreeing with the philosophical views of either man).

(Realism matters to me personally because, for a long time,
at a deep level, and at great personal cost, I entirely "lost
faith" in the existence of mathematics. It's hard to explain
properly, and now is not the time to do so. Now is the time
for me to be learning some maths! But foundational issues do
now obtrude unavoidably, for some obscure reason.)

I am just not happy to go on repeating, stupidly, "Either P
or not P! I mean really, P or not P! Come on, you /must/
believe that!" (etc., etc.) I need a better "argument" (if
only to convince myself).

I consider it a fair litmus test for a philosophy of
mathematics whether other fields of inquiry are treated
consistently with mathematics. Does mathematics have to be
assumed to be an exceptional case, and if so is there a
reason given for doing so? Certainly I think the
constructivist view works the same outside of mathematics as
inside, however.

I feel similarly about the need either not to treat maths
as a special case, or else to explain what is special about
it that makes it one.

There is another aspect of the philosophy that should be
kept in mind when reading philosophical constructivists. Some
ways of putting things are understood by constructivists as
being conventional that one might be tempted to think of as
having a canonical or "God-given" meaning. Take this issue of
the use of time when describing mathematical facts.

There may be a basis for discriminating in this case. The
use of time in mathematical description is to some degree
more slippery. If I say that it rained on Sunday, then you
know the significance of the reference to Sunday. If I say
that the Riemann hypothesis was true on Sunday, it's perhaps
less clear what I mean. From a "Platonist" or realist point
of view it seems more obligatory to say that a mathematical
truth is true at all times. But from a constructivist point
of view, it's much easier to think of this notion as a merely
conventional choice. Using the phrase "it was true on Sunday"
to mean that a proof was available on Sunday sounds much
less eccentric for someone thinking this way. And exactly
what degree of availability of a proof is at issue is also
somewhat arbitrary and up to you to decide. There isn't some
metaphysical significance to the choice.

I prefer to steer clear of that "tensed" mathematical
language if for no other reason, then at least because
it throws people off who aren't expecting it. It doesn't
mean that I believe, however, that the tenseless way of
describing the situation is naturally sound or "objective"
and the other isn't. Putting times on your descriptions
can help to emphasize the operational quality of certain
claims expressed in constructive mathematical language. A
proof of "if P then Q" allows you to make Q true when the
truth of P has been provided to you, for instance.

It just strikes me as very dogmatic to insist (were someone
to insist) on using the word "true" in this tensed fashion,
when the classical/Platonist/realist/whatever mathematician
already has a perfectly clear way of saying exactly the same
things as are said in this way, only using phrases like "had
been proved" rather than "was true". Again it seems to me
like a /loss/ of possibilities of communication, not a gain.

(Perhaps this is another of my long posts that I shouldn't
post, but I feel that would be too depressing, so here goes!
I had intended to write instead about a simple example of
mathematical argument, and ask where it is not clear to a
constructivist. Another time, perhaps.)
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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