Re: Choice sequence lessons.

On Dec 15, 5:51 am, Bill Taylor <w.tay...@xxxxxxxxxxxxxxxxxxxxx>
wrote:

RANT BEGINS___________________

I rant about the constructivist's hi-jacking of standard
math-language terms to use in their own interpretation,
rather than coming up with new ones of their own!

I agree!


It is EXTREMELY confusing to have standard words like "truth",
"proof", "there exists", "not" (even!), "sequence" and many
others, have two quite different meanings depending on context.

....

Now it is frequently replied, that no-one has
a right to exclusive use/definition of everyday words,
and that their right is just as good as the orthodox mathie.

BUT THIS IS NOT SO!  The orthodox have "first dibs" on these
disputed terms, from Frege, Cantor, Hilbert, Russell and
others in late C19 to early C20, specifically for the orthodox
("existential") PoV.  The specifically constructive interpretations
came noticeably later, beginning about 1920.  Therefore, I assert,
(uselessly, OC!), that it is up to the those of a constructive
("algorithmic") bent, to find new ones.  

As Keith replies, the history is more complicated. In any case, the
"I was here first" may work on the playground, but I don't think it is
appropriate here. So, while I agree with your main point, I don't
think you should be using history as justification. The reason why
intuitionists should use different terms (or at least, stick
"Intuitionist" in front, such as "there Intuitionistically exists") is
just that there *is* an orthodox meaning, which almost everyone uses.
And, as in many things, numbers count. The word 'revolution,' when
applied to a political event, first meant a *return* to a status quo
(the use came from astronomy). Now it means a *breaking* from a
status quo. Today, if someone came along and started using
'revolution' in its original way, then I think the rest of us would
find that very confusing and inadvisable, even though he would win the
"I was here first" argument. It's confusing because the numbers are
against him; there's an orthodox meaning, and the original meaning is
just not the orthodox any longer.

If understand Keith's reply to you, then he is suggesting that he is
fine with a "paragraph 1", where the Intuitionist explains he is using
his terms in an unorthodox way. IMHO this is not ideal, but it is
satisfactory.


_________________________________RANT ENDS

Well, now that I've had my bleating whinge, and turned off
most of my intended audience, I will continue with the choice
sequence material.

I said I had a new viewpoint on it, and there are two aspects
to this, dealing with things that have hindered my understanding,
(specifically because of that bleat topic!)
_______________

a) The first is, that it now seems to me, that we should NOT be
speaking of "choice sequences" at all! - but of "SEQUENCE SYSTEMS".

That is, a whole system of *generating* a sequence. This highlights
the system itself, rather than an actual sequence, which may never
exist at all in its entirety, as we see with choice sequence
systems.  So an ordinary (fixed) sequence comes with a very
boring "system", which basically amounts to just a definition,
typed and presented in the usual undergrad way.  One might
also be able to speak of *random* sequence systems, for those
who want to do so, as some intuitionists in the past have done.

And mostly, here, we will be speaking of "free choice systems",
which can have all the game-theoretic or other properties
that Herman and/or other inties want!

And it meshes very well with one of Herman's earlier remarks...

You could say that in contrast with Cov, the 'crux' of the type CS
is not lack of knowledge but 'volatility': lack of determinateness

If we understand that we are dealing with sequence SYSTEMS,
rather than mere sequences, then there is no longer much problem.

With this new terminology, I hope everyone would be satisfied.
It certainly makes a lot more sense to me, anyway!!
_____________________________________________________end of (a)

b) The second of our changes is much more minor, and indeed
   hardly a terminology change at all.

It involves a conflation that often causes ruptures in English
(natural language) interpretations of things - the difference
between "ANY" and "ALL".  (!)

For negative statements, they are clearly distinct:-

        NOT ALL of the students will pass.  (some fail)
vs     NOT ANY of the students will pass.  (all fail)

For interogative sentences it is also the case:-

Will ALL  of the students pass?     (extreme optimism)
Will ANY of the students pass?     (extreme pessimism)

But for positive declarative sentences they may co-incide:-

       ANY of the students can pass, if they work hard.
       ALL  of the students can pass, if they work hard.

No doubt all this is old hat to linguistic philosophers and old
grammarians, and there are doubtless exceptional cases
either way.   (Indeed I use one below!)

However, this is, I suggest, at the root of my problems with
Herman's "funny sentence", that we have spent so much time on.

______________________end of (b)

So here it is again:-

## exists x:CS forall k exists n forall i>=n [x(i) = k]     (7)

Let's first see why (7) comes out as true with our semantics.

Yes, now I have no problem at all with your game semantics.
I had no problem before; but I kept getting my wheels stuck,
because of the (IMHO) bad usage of the terms, that co-opting
of orthodox terms as in my rant.   But now all is clear.

With a *choice sequence system*, it is quite clear that the above
is true, on the game-semantics that Herman gave so clearly.

And it clears up the last tincture of unhappiness,
if we use the quantifier language I suggested...

## exists x:CSS for ANY k exists n forall i>=n [x(i) = k]    (7A)

Now it is as plain as day that I can use my free-choice system
to counter ANY k the opponent might put up; as long as
I don't have to be able to counter EVERY k he might put up,
all simultaneously.  The simultaneity seems to make the difference,
here.  As with the following (exceptional) case:-

   I can beat up anyone in the class!
   I can beat up everyone in the class!

The latter clearly doesn't follow from the former,
if simultaneity is required in "all".

So, I think I might be happier now.  I wonder if we might keep
these two linguistic conventions in mind, a & b, as we go on?
............

So, are you still (or again) following me, so far?

I hope so.  I haven't commented on many other details,
as this post is already too long!

Semantics is leading, here. There are no rules of logic (here)
other than 'observations after the fact'.

I think I understand this remark much better now, too.

SO; all in all, if you agree (more or less) with what I've said,
especially points a & b, then I can happily go on and read
the latest three-languages post, and you can prepare
the next lesson for your incredibly thick-headed student.

-- Bone-headed Bill

.

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