Re: A quote (and question) about intuitionism


On Nov 9, 9:41 pm, David Bernier <david...@xxxxxxxxxxxx> wrote:
|Keith Ramsay wrote:
|> Angus Rodgers wrote:
|> |The following passage was quoted today in another thread:
|> |
|> | It does not make sense to think of truth or falsity of a
|> | mathematical statement independently of our knowledge
|> | concerning the statement. A statement is /true/ if we have
|> | proof of it, and /false/ if we can show that the assumption
|> | that there is proof for the statement leads to a contradiction.
|> | For an arbitrary statement we can therefore not assert that it
|> | is either true or false.
|> |
|> |(Source: A. S. Troelstra and D. van Dalen, "Constructivismin
|> |Mathematics", vol. 1 (1988), p.4.)
|
|> They meant it.
|
|Is that a definition of truth?

Truth is a property of propositions. To know whether a
statement is true, one has to know first what it means, and
then, according to the meaning, the statement is sometimes
true. There's a different idea of what kind of thing "the
meaning of a statement" is in constructivism than one
typically has elsewhere. So I don't think that thinking of
the above merely as a definition of "truth" is liable to
lead to a good understanding of it. The issue is not just
what _property_ of propositions should be called "truth";
it's what _kind_ of thing truth a property _of_.

One attempt as explaining the meaning of a sentence, for
example, is to say that knowing the meaning is the same as
knowing the set of conditions under which it would be true.
(I think many philosophers would dispute such a definition
in some way, but it is some approximation to the way that
"meaning" is understood in lots of philosophies.) In
constructivism, it's closer to say that knowing the meaning
of a sentence is knowing the conditions under which it can
be considered known. (Probably many constructivists would
dispute this definition in some way, but it's similarly
some approximation to the way "meaning" is understood by
them.)

|If it were a property of truth,
|how would they know that it really is a property of truth?
|How could someone know that there are no external truths?

One of the unfortunate features of the sort of claim they
made (above) is that it tends to sound like some kind of
metaphysical claim about reality. That's perhaps the main
reason why I wouldn't put it that way myself, if I were
writing a book about it.

It may help if you know something of Wittgenstein, and his
notion of "bewitchment of language". From the constuctivist
point of view, the non-constructivist is typically suffering
from a kind of bewitchment of language, which leads to a
phony impression of statements having more meaning than
they do. The constructive content has more of a bare,
unvarnished, nonmetaphysical character to it. All the more
reason not to go around saying things that sound like they're
meant in a *more* metaphysical way than usual.

Think about the claim that

(*) "there are statements that are true right now that
are not currently known to be true".

What does it mean? If you look at it from a constructivist
point of view, you understand its meaning as closely tied to
the kind of experience that (in some sense) justifies saying
it. But when you do that, it tends to lose some of its aura
of being an important subtle truth.

One reason for saying (*), for example, which would be
considered appropriate by a lot of constructivists, is the
fact that we have ways of discovering facts about the past.
Since I'm going to discover such-and-such a truth in the
future (where I put an item I lost say), and since it'll be
about the past (which is currently the present), I can say
that this fact, that I will discover in the future, was true
already now. The item is here at a specific spot, even though
I don't know it right now.

All this is fine, but probably seems a little beside the
point. The non-constructive point of view usually wants to
say that there are truths that are not merely not yet
discovered, but not even possible to discover. Truth isn't
just different from "what I know" based on some technicality;
truth somehow completely transcends what I know.

But the constructive point of view, as I said at the start
of this message, takes the "meaning" of statements like (*)
in a different way. The things (like truths that are forever
beyond being exhibited as truths) that one describes as
"existing" from one point of view, in spite of not being
able to exhibit any, are described differently from a
constructive point of view. The constructivist might say,
you haven't exhibited an example of the kind of thing
you claimed exist. To the extent that your claim has any
meaning, however, it must be that you've exhibited
_something else_, which will relate to your actual experience,
and not some experience-transcendent reality. The act of
describing this experience as being of "realizing that there
are, in fact, truths that I don't know yet" is seen as not
adding anything to the content of the claim.

I previously gave, for example, the case of mathematical
statements P and Q, where we know that assuming both are
false leads to a contradiction. Symbolically, ~(~P&~Q).
That's a perfectly sensible constructive way to describe
lots of these cases. (Others may have infinitely many
statements; the essential point is the same.) That we lack
a way of determining that P is true or determining that Q
is true, is equivalent to saying (in the constructive sense)
that we don't know yet that "P or Q" holds. All of this is
within the realm of experience. We experience knowing that
~(~P&~Q) (for various choices of P and Q). We ask whether
"P or Q" holds, and know we don't know yet.

If you think in these terms, you have all the same facts in
front of you, as does a person who thinks in nonconstructive
terms about the same situation. You just don't have the
nonconstructive version of what Dummett called the "metaphysical
picture" to go with it.

Personally, I can't say with confidence that the
nonconstructive "metaphysical picture" should be disposed of
in this way. In that sense I'm not entirely a constructivist.
It's just that I can't see, either, that it serves a meaningful
purpose.

|If it's their definition of truth, it seems to me that others
|are free to give their own definition, which would omit
|the part about "having a proof".
|
|If we let n = 10^(10^100), and
|P be the sentence "the n'th prime is conguent to 9, modulo 10."
|
|I don't really have a problem with the LEM in case the assertion is:
|
|"P or (not P)".
|
|So perhaps some constructivists would say "P or (not P)"
|is not true, and also not false, since there are no constructive
|proofs available.
|
|What would be a response by constructivists to:
|
|" You don't have a proof of "P or (not P)" ,
| but I have one, since it's true from my point
| of view." ?

Before you try to answer any of these questions, first try
to determine what you mean by them.

If understanding the meaning of "P or not P" is
understanding the conditions under which it's true, then
it seems like one is left understanding it as merely
being a tautology, something automatically true.

On the other hand, there are two or three constructive
ways of defining "or" that might make sense here. One
way, that you might call the strict (or "ultrafinitist")
way, would be one by which "A or B" is justified only by
knowing A or by knowing B. A second, weaker, way would be
that we say "A or B" holds if we have a procedure that
would, in principle, demonstrate either that A or that B.
(The nonconstructive sense of "or" would suggest a third
and yet weaker definition of "or".)

You've produced an example of a case where we know that the
second, weaker sense holds but not the stronger one (yet).
Those are the "raw" facts of the case.

The temptation is now to think that after answering these
two or three meaningful questions, that there remains one
more question, not yet answered, as to whether "is P true"
currently has, "in fact", an answer. (Lacking any obvious
way of tackling the question of whether P is actually true,
I'll leave that one aside.) But constructively this looks
a lot like a non-question. The facts are all clear enough.
Yet one (sometimes) has this peculiar temptation to ask
this extra question. What could it actually mean, though?

It would be like if, during the discussion of how to define
"planet", someone had asked, "Yes, we have these possible
definitions, and facts about Pluto. But, definitions and
measurements aside, is Pluto a planet *really*?" Since it's
a matter of definition, this doesn't make much sense.
Likewise, whether "P or not P" is true is a matter of how
you define "or" (as opposed to how you define "true"),
or at least constructivist philosophy seems to treat it so.

As it happens, the strict sense of "or" I describe above
is not as useful as the second sense. There's just not much
to be done with it. The second one is the usual one in
constructive mathematics. Hence one would customarily take
"or" to mean something that makes the claim "P or not P"
true in this case.

One constructivist of my acquaintance would sometimes refer
to it in terms of compulsion. From the nonconstructive point
of view, one is liable to feel compelled to say, yes, one
of them _is_ true. From the constructive point of view, it
feels like a mere matter of semantics.

Keith Ramsay

.

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