Re: Choice sequences, intuition, etc


On Oct 7, 10:54 pm, Bill Taylor <w.tay...@xxxxxxxxxxxxxxxxxxxxx>
wrote:
|Keith Ramsay wrote (in two articles):
|> I don't see anything worse, more ambiguous or less clear,
|> about free choice sequences than classical [stuff]
|
|Well, as I understand it from your comments, this is quite
|true provided we agree that what is really being defined
|here is a *property* of (Free Choice) Sequences. And that
|sequences per se are really little different from orthodox.
|As I replied before, the issue (to me) is one of type-ing.

I think that an intuitionist and you would each regard the
other as putting the cart before the horse here. It doesn't
seem to me like you're really accepting the idea that by
specifying a different method of constructing the sequence,
one is specifying a different type of entity. Nor would the
intuitionist really accept the idea that he is picking an
entity from the stock of entities known to you and then
winnowing out some subset of their properties to be given
special consideration.

[...]
|> But if one has any interest at all in
|> what views were held by Brouwer, Bishop and so on, and how
|> that motivated their way of doing mathematics, you might be
|> better off just forgetting these axioms.
|
|Regarding Brouwer I'm sure you're right. Regarding Bishop,
|I doubt it. As to his motivations, someone else observed
|that his claimed main motivation (as written in his preface)
|was to keep mathematical statements "as meaningful as possible".
|I have no doubt this is true. And, as I implied earlier,
|he (Bishop) seems to have found that this could be done best
|by eschewing ALL doubtful concepts, and that this included
|LEM *and* FCS's.

Classical mathematicians seem to be strongly drawn to the
impression that their philosophical critics are motivated
by some form of skepticism, a fearful desire to avoid
notions that carry some risk of being faulty (which the
classical mathematician considers far-fetched). I think
that in many cases this impression forms the basis for
fairly misleading overall depictions of these philosophies.

Bishop describes in his analysis textbook some doubts one
could have about the validity of arguments in it. One such
doubt is about the rule of ex falso quodlibet, false->X for
any X. (Intuitionistic logic with that rule removed is known as
"minimal logic".) On your theory, one might expect him to
have thought that such a rule should be avoided for
dubiousness. I don't see him as having that attitude at all.
He points out that he could, if need be, reformulate his
arguments to avoid the use of such a rule. But for his
purposes this is both necessary and unproductive. On the
other hand, he writes elsewhere that reformulating arguments
to avoid negative statements-- by rewriting them positively--
led to an improvement in the content.

Concern for content goes to show why Bishop didn't take
"minimalism" still further. One doesn't have to assume as
much as he did. Logicians spend a lot of time studying
axiom systems for mathematics that are weaker. If you want
to avoid dubiousness, how about refraining from induction
on undecidable predicates? But if there's a strengthening
of content by reducing how much you assume in this way, it's
not something that he could see, in the way that requiring
integer existence to mean the constructive thing does.

|It is significant that neither in his
|big book or elsewhere, did he attempt to use FCSs.

Well, as Bridges seems to have pointed out to you, one
doesn't really get much computational "mileage" out of the
concept.

|I think we may provisionally conclude that Bishop did NOT
|think that a Free Choice Sequence was a meaningful idea.

No, I don't agree. In his textbook, when he mentions
Brouwer's proof that real-valued functions on [0,1] are
uniformly continuous, if you were right, one would expect
that he would say that this was just invalid, and based on
some invalid idea. Instead he says that accepting it as a
proof would destroy the character of mathematics. Then later
he gives it a kind of partial defense, as a possibly useful
metamathematical result: without making certain kinds of
extraneous assumptions, any real-valued function on [0,1]
that you'll be able to construct will be uniformly
continuous. (In the text, he refers to real-valued functions
on R and continuity, but given his definition of continuous
as "uniformly continuous on bounded intervals" this is
equivalent.)

|Incidentally, Douglas Bridges related two Bishop anecdotes,
|which he regards as being his core original motivations.
|
|The first, was when he was doing multivariate complex variable,
|and looking at the spaces in multi-dimensional complex co-ordinates,
|and found to his amazement that after he'd proved certain
|things existed that he could NOT visualise them, or their spaces,
|because of the non-constructive nature of them. As visualisation
|was always a core component of his math, he was dismayed.

It'd be interesting to know the specific "things" that he
found unvisualizable. It's been pointed out that one of
his relatively famous results, the Bishop-Phelps theorem,
relies upon the axiom of choice. He did plenty of functional
analysis like that... which would make it interesting to
know what kind of space would seem peculiar in this way.

|The second came when he was teaching a course in
|"mathematics for gracious living", :) , i.e. math for
|humanities students. He found that many of them were
|very smart cookies, and they came up after his initial
|logic lectures complaining that - "Hey this is rubbish
|what you said about the implication connective,
|(the classical one), it *obviously* doesn't mean that at all!"
|After discussing and thinking about it, he realised that
|they had a point, and began to explore constructivist
|ideas more seriously.

I also heard a version of this one (in mine it was
"math for poets") described as the start of his journey.
Somehow the distinction in meaning between an implication
and its contrapositive entered into it.

|So, that is a summary of Bishop's informal motivations,
|and AFAICS it all adds up to a MINIMALIST approach
|to math and its basic assumptions, and little else,
|and that this agrees with his "meaningful" statements,
|contrary to what some posters have claimed about this.

Minimalism doesn't constitute an approach. It wouldn't
explain why he didn't become still more "minimalist". The
puzzlement expressed by Beeson in his book also is well
explained by the fact that Bishop's aim wasn't to avoid
making doubtful assumptions per se (of which more are
needed if you "believe in" the Goedel Dialectia
interpretation as an exact translation), but rather to "get
under the skin" of the implications being interpreted using
it, to turn a proposition merely stating that A "somehow"
implies B into one that lays bare some computational
content in the manner in which A implies B.

One might also expect a minimalist to spend more time on the
axiomatic foundations of what he is doing. Bishop never has
seemed to have been very much interested in inventorying
what axioms he needed.

So as I wrote before, if you aren't interested in his actual
philosophy, you can learn something about the mathematics
by just considering mathematics done without the axiom of
choice or law of excluded middle, but if you want to know
the "practical" philosophy of it, pondering the axioms with
which other people formalized it, and how they relate to
other sets of axioms, is just barking up the wrong tree.

There are lots of cases to be sure where the goals of
getting at computational meaning and of reducing the
assumptions needed to a minimum are lined up. But where
they don't line up, it's the computational meaning that
Bishop lines up with. Logicians spend a lot of time
studying reductions of assumptions that weren't of much
interest to Bishop as far as I can see. His failure to
assume either that all functions are computable or that
all real functions were continuous was apparently due to
their not actually giving one much in the way of
computational meaning. He thought they were fine as
ways of recognizing what not to bother to try to prove
constructively (such as the existence of a nonrecursive
function N->N, or a non-uniformly-continuous function
on [0,1]). Were Friedman to succeed well enough in
extracting "nice" computational meaning from large cardinal
axioms, I trust Bishop would've taken that as his cue
to start using them in some way.

[...]
|> The philosopher Dummett has argued that knowing how to use
|> a language should be nearly sufficient to allow one to know
|> what the sentences in it mean, and even what the individual
|> terms mean. (Frege had a similar idea that the meaning of
|> a term is given by how it contributes to the meaning of a
|> sentence in which it's used.)
|
|I tend to agree with all this, but disagree that Brouwer
|is exemplifying it in any way.

I wouldn't attribute these ideas to Brouwer, but I think
they serve as a useful point of comparison.

Ontology, the branch of philosophy dealing with the nature
and meaning of existence, always seems very slippery. My
suspicion is that overall one of the biggest sources of
difficulty is a gut feeling that for a thing to "really"
exist should have more significance than it actually does.
This manifests in the philosophy of mathematics as a worry
that either mathematical objects don't really exist, or that
we should have to do something more to justify the claim
that they do really exist.

People often disagree with Dummett's suggestion in some way,
but we can treat it as a kind of razor, a rule of thumb for
what kind of thing it is that we should have to know about
the sentences we use, in order to know what they really
mean. Frege's idea implies that the same sort of razor can
be applied to component parts of sentences, provided we
respect the wholistic nature of the sentence. Even if we
don't accept them as completely accurate, alleged
departures from them deserve hightened scrutiny.

In our case, I would defend free-choice sequences on the
basis that one can without any particular trouble learn how
to reason with them, and what meaning a mathematical
proposition referring to them has.

We can then apply Frege's idea to the question of what the
term "free-choice sequence" means. What does it contribute
to the meaning of a sentence containing it? Above, you write
that the free-choice sequence itself is little different
from the corresponding notion of sequence in classical
mathematics, but that I've supplied you with a different
notion of "property" for them. Well, an elementary sentence
referring to a free-choice sequence will be composed of the
reference itself to the sequence, and a reference to the
property being attributed to it. So you are in effect
relocating the contribution of the reference to the meaning
of the sentence to the other part of the sentence.

I think this is somewhat analogous to the way in which we
might say that an element of the integers mod 17 is
essentially an integer, but that we define a distinct
notion of equality for elements of the integers mod 17,
namely, equivalence mod 17. (One tends to represent them
as sets, but this is not crucial.) A property of an integer
only counts as a property of an integer mod 17 if it
respects equivalence mod 17. If this is how you want to
think of the integers mod 17, and free choice sequences,
I don't see a problem with it, just as long as you realize
that people also are liable to say that an integer mod 17
is "really not the same thing" as an integer, and that a
free choice sequence is "really not the same thing" as a
generic sequence, and that both points of view are valid
in a sense.

Keith Ramsay
.