A Modest Proposal
Keith Ramsay
The title of the book is "Foundations of Constructive
Analysis". Perhaps it should've been written like this. (We'll get to
what the "r" means in a moment.)
Chapter 8, Section 1
R-DECOMPOSITIONS OF R-MEASURES
R-measures mu and nu on a r-locally-compact space X are said to be
r-mutually singular, if, roughly speaking, their masses are
distributed over disjoint sets. This concept is easily expressed in a
precise way in terms of the r-values of the r-measures on
test r-functions.
Definition 1. Two r-positive r-measures mu and nu on a
r-locally-compact space X are *r-mutually-singular* if for each
r-nonnegative test r-function and each epsilon r-> 0 there exists a
test r-function g with 0 r-<= g r-<= f such that
(1.1) r-integral g d mu + r-integral (f-g) d nu r<= epsilon.
It is natural to ask whether an arbitrary r-measure mu on X is the
difference of two r-positive r-measures. Even more natural is the
question whether it is difference of two r-mutually-singular
r-positive r-measures. Before giving r-necessary and r-sufficient
conditions (Theorem 1 below) we prove a simple lemma.
Lemma 1. Let mu be an r-measure on a r-locally-compact space X, and
let f and g be test r-functions with 0 r-<= g r-<= f. Then
(1.2) r-integral phi d mu r-<= r-integral g d mu + epsilon
(phi r-member of C(X), 0 r-<= phi r-<= f)
r-if and r-only-if
(1.3) r-integral (phi1 - phi2) d mu r-<= epsilon
(phi1, phi2 r-member C(X), 0 r-<= phi1 r-<= f-g, 0 r-<= phi2 r-<=
g).
Keith Ramsay
P.S. The r stands for "remember, we're using the constructive version
of this concept here, we wouldn't want you to forget that".
David C. Ullrich
I think this is very unclear, bound to cause confusion.
Instead of 'r' you should use 'rwutcvotchwwwytft'.
Hmm, that doesn't work either, people will forget what
'rwutcvotchwwwytft' means. Ok, then use
'rrwutcvotchwwwytftmrwutcvotchwwwytft'
(for "remember, rwutcvotchwwwytft means ...").
Good start, though - should do a lot to make
constructive analysis more clear once you get
the notation right.
DU.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Bill Taylor
where they seem more at home; and to avoid further polluting
the main thread with this worthless squabble.
|> There's no need for such a locution; "there exists an explicit..."
|> would be quite helpful to both sides.
> This would be a handy phrase in some cases except that
> it's already acquired a different and informal connotation
> in classical mathematics,
It hardly seems all that different, but if you say so. OK then,
make it "there exists a specific..." . What does it matter?
As long as it's different, short and clear.
> Given a continuous function f, where f(0)<0<f(1), the
> smallest zero of f is an explicit example of a zero of f
Hmmmm. I would have not altogether said so. It's a *particular*
zero, but not really an *explicit* one, all that amount.
Explicit tends to mean "given in some sort of (semi-)closed
algebrao-calculus form. But this is now splitting hairs.
The point is, natural language is *full* of words that have as yet
no technically restricted meaning: let's make use of the facilities!
- and avoid avoidable confusion or misrepresentation.
|>I'm sure such brief and
|>simple expressions could be found in most cases.
|>I already mentioned "inhabited set".
> What substitute for "real number" would you recommend?
What's wrong with "constructive number"?
However, getting back to something meatier than these word problems,
you have raised the point that it would be needlessly cumbersome for
const/intuitionists to keep using longer-winded terms, writing
for one another. This is a fair point, though as I've observed,
Brouwer does already sometimes, and Species, Spread and Bar are
good examples. (The whole matter is clearly not nearly as gross
and cumbersome as you are making out, a little mischievously!)
And in technical papers, which are only going to be read by others
who already know the subject, abbreviations to orthodox terms are
maybe quite acceptable. BUT:- in papers/articles/books/lectures
to orthodox mathies who are hoped to become more sympathetic to
the topic, it is quite needlessly confusing, if not actually
polemic in intent.
-- Wordly William
Bill Taylor
> Perhaps the main thing is that it makes it appear as though
> the constructive concept were somehow derived from
> the classical concept.
You persist in misinterpreting my remarks this way! It's not
a matter of what is a subset of what, but GETTING THROUGH to people
raised in a different mode. There, one can be polemic, or practical.
Insisting on confusing terms is not practical, and is often polemic.
> Good constructive mathematics is elegant and general.
No doubt. Should it also be understandable to as many as possible?
> Your proposal would munge the elegance of it and make it look like
I disagree.
> There is a natural, straightforward way to derive classical
> concepts from corresponding constructive concepts.
Yes yes. And it IS straightforward, but hardly natural!
> I understand that classical mathematicians prefer to disguise the
> complexity of their constructions, by ma[s]king their uses of [LEM]
And YOU accuse ME of making shabby suggestions, later!
You make this sound like a vast, dishonest conspiracy!
> The classical mathematician doesn't feel like they need to warn us
> that they're doing classical mathematics;
QUITE! Because as whats-is-name said, it is *the accepted standard*.
It is for others to make warnings.
> but for you, it seems that
> merely warning the reader that constructive mathematics is being
> done is not enough?
No, it is not. Not when/if honest proselytising is to be done.
> (And I doubt very much that you actually intend to do
> very much reading of constructive mathematics, do you?)
You're very free with the ad hominems these days!
> In this country lots of gay people want to be allowed to marry,
A stunningly worthless comparison of no value in this context.
> When one is doing something simple, one is entitled to use simple
> terminology.
Yes; but not at the price of confusion or possibly dishonest polemics.
|> And quite reasonably, I still aver. The confusion
|> otherwise engenderable is both unnecessary and
|> still (I suspect) a little bit dishonest.
> This kind of suspicion strikes me as shabby.
See above. Shabby is as shabby is seen.
> I don't know what you
> think I think I could be accomplishing by pretenses here.
Oh not you! Sorry if I gave that impression. But Brouwer was
famously and outrageously polemic, and was NOT above commandeering
standard terms to try to ease their re-interpreted acceptance.
This IS shabby. And having set a trend, I get the feeling from
some later authors that they are (perhaps unconsciously) following
the same procedure.
> Constructive
> mathematics should be pursued without needlessly muddling it up
> in order to satisfy the demands of people who can't be bothered to
> remember whether the mathematics they are reading is constructive
> or not, and that's really what I think.
And that is a VERY shabby paragraph too! If const/intuitionists
don't WANT the orthodox to understand them, then your remarks
are entirely appropriate, if a little elitist.
> I still don't know what you think of the analogous case
> of nonstandard analysis.
That is a far more technical matter; see my remarks earlier about
papers for equals vs articles for newbies. Even so, they are
typically very careful with things like e.g. "the standard part
of a number", which is somewhat (socially) comparable with
"a constructive number", which consies/inties don't bother to use.
> to keep from confusing
> people who've somehow forgotten that what they're reading is
> nonstandard analysis?
Shabby sarcasm consistent with the thread title.
-- Withering William
Bill Taylor
|> |>But however it may have been, it is surely an unwarranted
|> |>confusion (if not worse!) to use the same terms, knowing
|> |>that by many they will be misinterpreted.
|
|>> You're the only person I know who has this problem.
|
|> Seemingly you have a limited mathematical acquaintance.
> And how would you know?
From the above. Obviously!
> When you write "mathies",it always suggests to me a lack of interest
> in distinguishing between people who actually *work* at their math
My word, what leaps of imagination to indulge in!
> I know people who have diligently worked to understand constructive
> mathematics,
Likewise, and not one has managed to cotton on to Choice sequences,
including Bridges himself. It is not constructive math that is
the chief problem here, but intuitionist math. (As I keep saying!)
> I don't know how it is you can feel so confident that there
> is some "royal road" here.
Where is the Halmos of Intuitionism?
> I don't know why you think intuitionism, per se, is something to be
> popularized.
You regard Halmos as a "populariser"!? I guess that tells a story.
> It has historical interest,
If you mean to imply it has historical interest *only*, then fine.
> For constructivism, read Bishop, or Bishop and Chang. I don't know
> what problem you have with that,
Good grief, how often have I got to repeat that I have no problem
with the Bishop/Bridges approach! Only and occasionally with
some terminology. And THAT could be improved, as many here have
already agreed *including* the main proponent of the thread;
your infantile sarcasm of the thread title article notwithstanding.
-- Backchatting Bill
Keith Ramsay
wrote:
> > There is a natural, straightforward way to derive classical
> > concepts from corresponding constructive concepts.
>
> Yes yes. And it IS straightforward, but hardly natural!
So what's "unnatural" about assuming the law of excluded middle?
Keith Ramsay
Bill Taylor
> > > concepts from corresponding constructive concepts.
>
> > Yes yes. And it IS straightforward, but hardly natural!
>
> So what's "unnatural" about assuming the law of excluded middle?
You were alluding to the double-negation elimination.
That is rather unnatural and contrived.
b
Daryl McCullough
>On Jan 7, Keith Ramsay wrote:
>
>> Perhaps the main thing is that it makes it appear as though
>> the constructive concept were somehow derived from
>> the classical concept.
>
>You persist in misinterpreting my remarks this way! It's not
>a matter of what is a subset of what, but GETTING THROUGH to people
>raised in a different mode. There, one can be polemic, or practical.
>Insisting on confusing terms is not practical, and is often polemic.
I don't see how it is confusing. The best way (in my opinion) to
understand intuitionistic logic is to go through all the ways that
one may form a statement, that is, go through each logical
connective, and say, for each one, what is required to prove
statements formed using that connective. I cannot imagine why
you would consider it confusing that those connectives are
the same as the classical connectives. As a matter of fact,
every rule for intuitionistic logic is *valid* for the corresponding
classical connective. The difference is that some proof rules that
are classically valid are not intuitionistically valid.
I don't understand why that is considered confusing. It's like
learning abstract algebra. It may use operations that are
familiar from arithmetic, like +, *, but some facts that hold
for arithmetic, such as x*y = y*x, aren't valid for
abstract algebra.
So what exactly is confusing about the intuitionistic connectives?
They are pretty straight-forward. The fact that they don't mean
exactly the same thing as the classical connectives shouldn't
be *confusing*.
I think what you might be disturbing and/or mysterious about
intuitionistic logic is that there isn't a straight-forward
translation of intuitionistic statements into classical
statements. There isn't a straight-forward classical model
theory for intuitionistic logic. But as a "logic", that is,
a set of rules for deriving theorems from assumptions, there
is nothing confusing about it.
--
Daryl McCullough
Ithaca, NY
Keith Ramsay
wrote:
|I think what you might be disturbing and/or mysterious about
|intuitionistic logic is that there isn't a straight-forward
|translation of intuitionistic statements into classical
|statements. There isn't a straight-forward classical model
|theory for intuitionistic logic. But as a "logic", that is,
|a set of rules for deriving theorems from assumptions, there
|is nothing confusing about it.
Indeed. It seems to me that often learning how to work with
it would be easier if one could just set aside one's prior
acquaintance with classical logic and deal with intuitionistic
logic on its own terms for at least until it's familiar. This is
one reason why making comparisons between classical and
constructive concepts is a slippery business.
Keith Ramsay
Keith Ramsay
|> complexity of their constructions, by ma[s]king their uses of [LEM]
|
|And YOU accuse ME of making shabby suggestions, later!
|You make this sound like a vast, dishonest conspiracy!
I don't think I do, at least not to everybody. (And please don't make
suggested insertions within words. The word was "making", not
"masking".) If I had it to do over, I'd use some more neutral term
like hiding than disguising, and I'm sorry if you found it sounded
accusatory.
I got to thinking about the hidden implicit double negations by
considering the cases when we hide information that might be
clarifying (for some), but consider it acceptable. I don't think
anybody seriously supposes that all information that some would like
to see highlighted *ought* to be included all the time.
The other cases where you seem to think I'm bashing classical
mathematicians seem to be much like this one.
On Jan 17, 4:48 am, Bill Taylor <w.tay...@math.canterbury.ac.nz>
wrote:
|> Constructive
|> mathematics should be pursued without needlessly muddling it up
|> in order to satisfy the demands of people who can't be bothered to
|> remember whether the mathematics they are reading is constructive
|> or not, and that's really what I think.
|
|And that is a VERY shabby paragraph too! If const/intuitionists
|don't WANT the orthodox to understand them, then your remarks
|are entirely appropriate, if a little elitist.
You're not allowed to redefine my terms here. I said "muddling up";
you're not substitute "be understood" merely because you think that
what *I* consider a muddling, you happen to think would help us to be
understood.
I think the people who are able to cope, here, are the ones who
most deserve to be catered to.
On Jan 17, 4:48 am, Bill Taylor <w.tay...@math.canterbury.ac.nz>
wrote:
|> (And I doubt very much that you actually intend to do
|> very much reading of constructive mathematics, do you?)
|
|You're very free with the ad hominems these days!
You've based your case for your idea, for how best to make
constructive mathematics understandable, on your own personal
experience, that of one other poster who said "I agree" to you (but
seems not to want to get involved in discussing why), plus some
unstated other mathematicians of your acquaintance. That essentially
makes it into an argument from personal experience, which makes the
basis for that experience relevant.
I don't consider it disparaging to say that a person hasn't read much
constructive mathematics, or doesn't intend to. It also seems pretty
unlikely that after all this time of treating it as a bit of light,
fluffy entertainment, you're about to turn to dealing with it
seriously. I'm not asking you to. It's not as though I'm very high on
the seriousness scale with it either.
Those who have spent at least a moderate amount of time studying it
are far, far less likely to be suffering from confusions over things
like the concept of "sequence", "proof", "integer", etc. Those who
plan to spend at least a significant amount of time studying it in the
future are much more likely to be bored with and/or worn out by the
kind of re-working of terminology that would be required to comply
with the demand you made in your "rant".
On Jan 19, 4:10 am, Bill Taylor <w.tay...@math.canterbury.ac.nz>
wrote:
Well, I'd say it's pretty natural, but this is beside the point. One
can get there even more simply (as I described in my previous posting)
by interpreting a classical X as meaning "if LEM then X". This has
less of an appearance even of being contrived. The double-negation
interpretation is valued for more subtle reasons which aren't so
important here.
You described the original posting in this thread as infantile in its
sarcasm. I have to wonder whether you imagine that there's some more
practical way to comply with your demand.
If we want to mix classical and constructive concepts together without
mixing them up (unlike when we simply say, here's a bunch of
mathematics done one way or the other), I argue that it's best to be
systematic about the terminology one uses for it. It's like the way
"locally" is used. We have the concepts of "locally connected",
"locally simply connected", "locally compact", and so on. If we were
to invent terms for each independently, like "unstringy", "slick", and
"tidy", then we'd have to remember what each of these is the "local"
version of. That's a waste of mental effort, a muddling, not a useful
thing.
And then, if one is unwilling for whatever reason to apply a
conversion to your classical concepts (so that classical mathematics
is labelled as consequences of the law of excluded middle and the
axiom of choice), then you might as well label your constructive
concepts in a very simple way, like prepending "r-". I thought perhaps
seeing a sample of what Bishop winds up looking like would be at least
a little enlightening to you, but perhaps I was mistaken.
On Jan 17, 4:48 am, Bill Taylor <w.tay...@math.canterbury.ac.nz>
wrote:
|> I still don't know what you think of the analogous case
|> of nonstandard analysis.
|
|That is a far more technical matter; see my remarks earlier about
|papers for equals vs articles for newbies. Even so, they are
|typically very careful with things like e.g. "the standard part
|of a number", which is somewhat (socially) comparable with
|"a constructive number", which consies/inties don't bother to use.
I got a bit of a chuckle at your choice of term "socially" here. And
no, it's not a far more technical issue.
In at least some systems of nonstandard analysis, the objects termed
"standard" correspond to what people most often mean by the term: the
"standard reals" are in a natural 1-1 correspondence with the reals,
the "standard integers" are essentially integers, and so on. The
original terms, then, are systematically used to refer to things that
are quite different. There are "integers" that are greater than all
the standard integers 0,1,2,3,...
Systematically relabelling not the constructive terms but the
classical ones is also typical for constructivism. It's just that in
the case of nonstandard analysis, your judgement of them is not being
driven by an effort to rationalize a pet peeve of yours.
On Jan 17, 4:48 am, Bill Taylor <w.tay...@math.canterbury.ac.nz>
wrote:
|A stunningly worthless comparison of no value in this context.
I'm pleased to be able to say that you find the analogy stunning, even
if you don't think it's applicable. This again was a matter of
recalling cases in which language is used in an unethical way, and I
picked it because it's an especially breathtaking case. It seems
common that it's not the minority with the unusual practices that is
doing the abuse, but a certain subset of the majority that makes an
argument a bit like this:
"We in the majority, because we have the weight of tradition on our
side, deserve to maintain the traditional linguistic usage around the
distinction between us and the minority. To change the usage now would
be confusing. Change here is an underhanded way of trying to promote
the minority's interests at the expense of others'."
In both cases, the argument is wrong on almost every possible
level. The people making the argument are pretending to speak for the
majority but are in fact just an especially aggrieved special
interest. The usage has the weight of tradition behind it only if one
is willing to do a little cherry-picking in the history of the
terminology. The majority (let alone those trying to speak for it) has
no special right to insist that the terms continue to be used as
before. The alleged confusingness is overstated. The minority is in
fact being completely open about their motives for wanting to change
the terminology. People in the majority are given a misleading sense
of being "the normal ones" by the usage; it tends to leave one *not
thinking* about the underpinnings of one's language. The aggrieved
members of the majority are mistaken in describing the interests of
the minority as being in conflict with the majority.
It seems usually no real harm is intended. It's not as though one sets
out to engineer the language to be abusive. But there's a fundamental
disparity of sympathy which makes putting stumbling blocks in the way
of the minority seem like a small matter, and little bits of comfort
to people uneasy about change seem like a big matter.
On Jan 19, 4:10 am, Bill Taylor <w.tay...@math.canterbury.ac.nz>
wrote:
|> I know people who have diligently worked to understand constructive
|> mathematics,
|
|Likewise, and not one has managed to cotton on to Choice sequences,
|including Bridges himself. It is not constructive math that is
|the chief problem here, but intuitionist math. (As I keep saying!)
Poor Bill! You just wanted to have a little entertaining rant and then
have us all move on, didn't you? A little peculiar that you take
giving some others a good hard slam a form of light entertainment,
however.
At sports matches, it seems pretty common to see people who do feel
like that. Yell at the opposite team. Yell at the referee. Yell at
your own team. But I wish you wouldn't treat mathematics like a
match. People disagree, but it's not essentially a fight. The main
purpose of it is not the entertainment of people who don't
participate, but stand to the side, watch, and make loose speculations
about what might be done better by the players on the field.
We've had a chance to see the kind of criticism you give when you're
imagining that you're just having fun, slamming the "opposition", and
also the kind you give when you're annoyed. Neither is especially
constructive.
|> I don't know why you think intuitionism, per se, is something to be
|> popularized.
|
|You regard Halmos as a "populariser"!? I guess that tells a story.
You overvalue your ability to read between the lines.
Popularization is a relative term. Anything that moves esoteric,
inaccessible, specialized material further toward being easier to
understand by a larger audience is popularization. Halmos was for sure
a popularizer, in writing a cozy set theory book for the
(mathematical) masses. It's not meant disparagingly.
Learning intuitionism but not for the sake of understanding Brouwer
better? I don't know what the attraction would be, but that's okay
too.
|> It has historical interest,
|
|If you mean to imply it has historical interest *only*, then fine.
No, not exclusively.
|> For constructivism, read Bishop, or Bishop and Chang. I don't know
|> what problem you have with that,
|
|Good grief, how often have I got to repeat that I have no problem
|with the Bishop/Bridges approach! Only and occasionally with
|some terminology. And THAT could be improved, as many here have
|already agreed *including* the main proponent of the thread;
|your infantile sarcasm of the thread title article notwithstanding.
Have you realized how much of their terminology is "EXTREMELY
confusing" according to your rant?
People can feel free to explain, if they like, how else they think
they could improve the terminology. Marking it systematically as in
the original posting in this thread is not so attractive. Changing all
the same terms non-systematically strikes me as worse. I have to
wonder whether some of the people who you think agree with you would
indeed favor either one of these.
You didn't distinguish in your rant between terms like "sequence",
which differ merely because of this systematic difference between
assuming or not assuming the law of excluded middle, and terms like
"continuous" where Bishop, for example, defines it as uniform
continuity on compact sets, where compact is in turn defined as
complete and totally bounded (and these are equivalent to the usual
classical terms only for some broad class of metric spaces).
I can understand why a person might not like this latter alteration of
definition, and I consider that more debatable. My opinion is that
Bishop had the right idea (and that people do not in fact treat old
definitions as sacrosanct when subtly different ones work better for
the purpose at hand) but it's less of a big deal.
Keith Ramsay
Bill Taylor
Well, the thread seems to have degenerated, and the substantive
points are just being repeated.
>>You're very free with the ad hominems these days!
> You've based your case for your idea, for how best to make
> constructive mathematics understandable, on your own personal
> experience,
Oh beg pardon! Saying "I find such and such confusing", is
actually a more modest way of saying "such and such is confusing",
which I dare say you'd have found something to complain about too.
Poor Keith! You just wanted to "win" a debate by maintaining
the appearance of a reasonable, steady, writer; and got caught
out in an ad hominem. (One of a string.) And no amount of
wriggling and whitewash is going to conceal that.
> one other poster who said "I agree" to you
Actually two, same as for you. But intellectual debates are
not decided by vote, are they, as you seem to want? They are
decided by making and defending various points. And if it DID
come to a vote in the math community as to which was more sensible,
orthodox math or intutuitionism, it's pretty clear which would win,
I think. So let's both of us leave out this absurd question
of who has the most "supporters".
> That essentially makes it into an argument from personal
> experience, which makes the basis for that experience relevant.
No, as I said, any first-person was merely a form of deference.
We ALL give (only) our personal views, almost all the time,
whether or not it is couched in 1st-person grammatical form.
You have been ad homineming all along, and introducing a very
rancorous tone. Quite unnecessarily.
> Those who've spent at least a moderate amount of time studying it
...will write in specialist journals, as I said some time ago,
and none of us will complain about that one way or the other.
> by interpreting a classical X as meaning "if LEM then X".
I would still call this somewhat unnatural.
But as you say, this is beside the point.
> You described the original posting in this thread as infantile
> in its sarcasm.
As indeed it was. You also raised the point that there were
a limited number of symbols, and much math uses the limited
supply in various ways, depending on context. Considering that
I'd already made this very point myself, in my opening rant,
(which concentrated solely on words and phrases),
it's unclear what you meant to achieve by this.
.....
So in sumary, we are still faced with a stand-off:
I find this double-use of terminology confusing (and possibly
with a tincture of mendacity, though I will not pursue that
point, even though at least Brouwer deserved it, I fancy.)
So; I find it confusing - you do not. Both of us have support;
both of us have forcefully put our views, but there is little
evidence to back up either of us. Certainly the *potential* for
confusion is there. Your main evidence is your ludicrous sarcasm
of r-terms, which was very far from the point, that being the use
of ordinary everyday words.
The point of "1st-come-1st-served", as a general principle,
has been met with a silly political analogy whose cultural
baggage puts it right out of court.
I might claim a win on points, here.
-- Bill
David C. Ullrich
<w.ta...@math.canterbury.ac.nz> wrote:
>Replying to: Jan 22, 7:30 pm, Keith Ramsay <kram...@aol.com>
>
>Well, the thread seems to have degenerated, and the substantive
>points are just being repeated.
>
>>>You're very free with the ad hominems these days!
>
>> You've based your case for your idea, for how best to make
>> constructive mathematics understandable, on your own personal
>> experience,
>
>Oh beg pardon! Saying "I find such and such confusing", is
>actually a more modest way of saying "such and such is confusing",
The two don't seem equivalent to me - if you meant the second
when you stated the first it's possible you should try to be more
clear.
If such-and-such is perfectly standard terminology and notation
(as in the present case) then "oh? so what - who are you?" is
a perfectly reasonable reply to the first. While a reasonable
reply to the second might be "oh yeah? How many people
agree?"
The first is not something that's subject to debate, but
whether the first _matters_ does have a lot to do with
who's speaking and what his qualifications are. The
second on the other hand would certainly be a good
point if true, but it's not the sort of self-verifying
assertion that the first is.
See the difference now?
>which I dare say you'd have found something to complain about too.
>
>Poor Keith! You just wanted to "win" a debate by maintaining
>the appearance of a reasonable, steady, writer; and got caught
>out in an ad hominem.
Giggle. You trapped him. Poor Keith, dangling on the hook.
Guffaw.
>(One of a string.) And no amount of
>wriggling and whitewash is going to conceal that.
>
>> one other poster who said "I agree" to you
>
>Actually two, same as for you. But intellectual debates are
>not decided by vote, are they, as you seem to want?
Erm, "debates" about what's the best notation and terminology
certainly are settled by vote. (Well, of course they're often not
settled at all, but there would be nothing inappropriate about
using a vote to settle _that_ sort of question, if it were practical
to do so.)
David C. Ullrich
Keith Ramsay
[...]
|[Bill Taylor:]
|>Oh beg pardon! Saying "I find such and such confusing", is
|>actually a more modest way of saying "such and such is confusing",
Saying "This is EXTREMELY confusing" is a more
arrogant way of saying it.
It seemed like it would be better to discuss either specific
things that were confusing, or the confusion experienced
by others, but you kept going with your own experience.
|The two don't seem equivalent to me - if you meant the second
|when you stated the first it's possible you should try to be more
|clear.
|
|If such-and-such is perfectly standard terminology and notation
|(as in the present case) then "oh? so what - who are you?" is
|a perfectly reasonable reply to the first. While a reasonable
|reply to the second might be "oh yeah? How many people
|agree?"
|
|The first is not something that's subject to debate, but
|whether the first _matters_ does have a lot to do with
|who's speaking and what his qualifications are. The
|second on the other hand would certainly be a good
|point if true, but it's not the sort of self-verifying
|assertion that the first is.
That one is confused is also not yet a complaint until one
establishes somehow that part of the responsibility lies
with the maker of the text about which one is confused,
in particular by way of the text itself having some propensity
to be confusing.
|>which I dare say you'd have found something to complain about too.
You may dare to predict such a thing, but you will find that
"I have troubles with such-and-such thing and perhaps you
could help" nearly always is more pleasantly received, by
_anybody_, than to say some of us have created your woes
(perhaps on purpose to trick people) and _owe_ it to the
community at large (because of a historical entitlement of
some kind) to change our ways. I'm not so very different
that way. People posting to sci.* to say that they are
confused about this or that is very commonplace, and
people can see for themselves how I respond to that.
People also vary greatly in their attitude toward "speaking ill of
the dead", but I think now that Brouwer is gone he still deserves
to have people stand up for him. You implied that he was
deceptive in some cases, but I don't know of any. I think he was
relatively much a man of his honor, and while sometimes
cranky and abrasive seldom if ever insincere. I'd be happy to
be corrected if I'm mistaken about that.
Keith Ramsay
Bill Taylor
>|>Oh beg pardon! Saying "I find such and such confusing", is
>|>actually a more modest way of saying "such & such is confusing",
> Saying "This is EXTREMELY confusing" is a more
> arrogant way of saying it.
Quite so! So you see the bind you have fallacistically
wrapped me in! If I say "This is such and such..", I will be
declared arrogant, and if I say "I think this is such and such...",
it leaves me open to allegedly justified ad hominems.
How is one to debate in such conditions!?
It reminds me of one time when my elder brother charged me with...
## "You're a very *argumentative* sort of person, aren't you!"
So then I could neither agree nor disagree without supporting
his charge!
> I think now that Brouwer is gone he still deserves
> to have people stand up for him. You implied that he was
> deceptive in some cases, but I don't know of any.
As I explained earlier, his whole approach was (at least mildly)
deceptive in its choice of pre-booked terminology.
But clearly you attach no significance to such evidence.
> while sometimes cranky and abrasive seldom if ever insincere.
You seem to think that one cannot be both sincere AND deceptive.
Indeed one can, especially if one has made a habit of the
deceptiveness, growing little by little from small beginnings
to become habituated to the point where one no longer notices
one's habitual deceptions. To follow your example (in adducing
wildly emotive and irrelevant analogies), one could look at Hamas.
They are undoubtedly sincere in their attitudes, but have been
casting nets of ("justified") deception for so long that
they are now constantly deceptive without realizing it.
But getting back to the point of the thread, the final section
of 3.1 paragraphs, from my last post, still stands unchanged
by any further comment.
-- Bill as before.
Daryl McCullough
I think that it is difficult to know what would have been a
*less* confusing way to present intutionistic/constructive logic.
You suggest using different terms, such as "identifiable real"
or something, rather than "real". But I don't think that the
difference between intuitionistic mathematics and classical
mathematics is merely one of terminology. It isn't that the
meaning of "real" has changed; you can't simply translate
between the two logics by saying "When an intuitionist says
X, he means the same thing as when a classical mathematician
says Y". The problem is that there *is* no faithful translation
of intuitionistic statements into classical concepts. The two
systems have different notions of what constitutes a proof
of something. It's not that the *objects* (reals, naturals, etc.)
are different, it's that there is a different standard for proving
things about those objects.
On the other hand, the double-negation translation of classical
logic into intuitionistic logic seems faithful, in the sense that
any classical proof of a formula corresponds in a very direct way
to an intuitionistic proof of the translated formula. The proofs
are almost the *same*. But this translation is a way to explain
*classical* mathematics to someone brought up on constructive
logic, it's not a way to explain constructive mathematics to someone
brought up on classical mathematics.
Bill Taylor
helpfully and politely observed:
> I think that it is difficult to know what would have been a
> *less* confusing way to present intutionistic/constructive logic.
As I have said many times, the simplest way to avoid avoidable
confusion, is to avoid using terms used in one context, for
similar but essentially different things in another context.
> You suggest using different terms, such as "identifiable real"
> or something, rather than "real".
Exactly so. We can still use "real" when it doesn't make any
difference, but a new term when it does. As to "identifiable real",
that's a very good term, but one word is usually better than two,
so maybe "ident" would be a useful coinage? "Placement"?
> But I don't think that the difference between intuitionistic
> math and classical math is merely one of terminology.
Oh indeed not! I have never thought that, it has been clear
right from the start that that was not the case, and I doubt
that anything I've written can be legitimately construed that way.
> It isn't that the meaning of "real" has changed;
Hmmmmm.
> you can't simply translate between the two logics by saying
> "When an intuitionist says X, he means the same thing as when
> a classical mathematician says Y".
Indeed not. But this is a matter of "mere constructivism",
rather than Intuitionism per se, and I have no difficulty
with that distinction from "classical" math.
As I have said many times.
> The problem is that there *is* no faithful translation
> of intuitionistic statements into classical concepts.
Indeed not! Which should make it all the more important,
I would have (naively?) thought, to maintain a distinction
in terminology when appropriate.
> The two systems have different notions of what constitutes
> a proof of something.
Indeed so. I think we can all agree on that.
> It's not that the *objects* (reals, naturals, etc.)
> are different,
It's not *merely* that, no. But it is a reasonable view,
in math at least, that to a large extent "things" are in
some sense identifiable with what one may say about them,
so that changes in proof methods automatically imply changes
in ontology. Would you not agree, at least somewhat?
> But this translation is a way to explain
> *classical* mathematics to someone brought up on constructive
> logic, it's not a way to explain constructive mathematics
> to someone brought up on classical mathematics.
That is indeed a very interesting distinction! And though
it's not relevant, and we don't go by voting, I wonder:
*are* there in fact any mathies "brought up on" constructive math?
I dare say there may be one or two in Holland, but even there
not so many, I gather. What is the general opinion here?
Are there many?... any?
-- Wondering William
Daryl McCullough
>
>On Feb 3, 3:42 am, stevendaryl3...@yahoo.com (Daryl McCullough)
>helpfully and politely observed:
>
>> I think that it is difficult to know what would have been a
>> *less* confusing way to present intutionistic/constructive logic.
>
>As I have said many times, the simplest way to avoid avoidable
>confusion, is to avoid using terms used in one context, for
>similar but essentially different things in another context.
I don't think they *are* different things. In going from
classical mathematics to constructive mathematics, it isn't
that reals or naturals change, but that the tools for
*reasoning* about them change.
For any kind of objects, you can apply constructive
or classical *reasoning* to those objects.
>> The problem is that there *is* no faithful translation
>> of intuitionistic statements into classical concepts.
>
>Indeed not! Which should make it all the more important,
>I would have (naively?) thought, to maintain a distinction
>in terminology when appropriate.
Well, the difference is precisely in what counts as
a "proof" in the two systems. So, it might make sense
to use different terminology for logical connectives,
implies_constructive,
and_constructive,
or_constructive,
exists_constructive,
all_constructive,
but I don't see how it is at all helpful to use
different terminology for reals.
>> It's not that the *objects* (reals, naturals, etc.)
>> are different,
>
>It's not *merely* that, no.
It's not that at all. The double-negation translation
shows that you can use constructive logic to prove
all the *classical* facts about the reals. So considering
the *objects* to be different in the two systems
doesn't make sense.
The difference between the classical "there exists a
real such that ..." and the constructive "there exists
a real such that ..." is completely in the changed
meaning of "there exists".
>But it is a reasonable view,
>in math at least, that to a large extent "things" are in
>some sense identifiable with what one may say about them,
>so that changes in proof methods automatically imply changes
>in ontology. Would you not agree, at least somewhat?
If I use constructive logic to reason about Bill Taylor,
I'm not talking about a different Bill Taylor, am I?
Constructive logic, just like classical logic, can be
applied to anything, reals, Kiwis, etc. ("Kiwi" isn't
offensive, is it?)
>> But this translation is a way to explain
>> *classical* mathematics to someone brought up on constructive
>> logic, it's not a way to explain constructive mathematics
>> to someone brought up on classical mathematics.
>
>That is indeed a very interesting distinction! And though
>it's not relevant, and we don't go by voting, I wonder:
>*are* there in fact any mathies "brought up on" constructive math?
I consider myself bilingual. I learned set theory after I learned
constructive logic, although I learned classical analysis
(differential equations, calculus) before I ever learned the
constructive versions.
I don't "speak" choice sequences very well; I'm not
sure I understand what are the rules for reasoning
about them. But that's a case in which Brouwer specifically
*changed* the terminology. He calls them "choice sequences"
rather than just plain "infinite sequence".
David C. Ullrich
<w.ta...@math.canterbury.ac.nz> wrote:
>On Feb 3, 3:42 am, stevendaryl3...@yahoo.com (Daryl McCullough)
>helpfully and politely observed:
>
>> I think that it is difficult to know what would have been a
>> *less* confusing way to present intutionistic/constructive logic.
>
>As I have said many times, the simplest way to avoid avoidable
>confusion, is to avoid using terms used in one context, for
>similar but essentially different things in another context.
I find that statement very confusing. Because the word "term"
means various different things in different contexts.
Seriously. Find an English dictionary. Note that the number
of definitions far exceeds the number of words. Revising
things to give a one-to-one correspondence between the
two is going to require a massive expansion of the language;
more than half the words in the new English will be
brand new. That should lead to much less confusion.
David C. Ullrich
Keith Ramsay
On Feb 1, 9:43 pm, Bill Taylor <w.tay...@math.canterbury.ac.nz> wrote:
|On Jan 30, 6:35 pm, Keith Ramsay <kram...@aol.com> wrote:
|
|>|>Oh beg pardon! Saying "I find such and such confusing", is
|> Saying "This is EXTREMELY confusing" is a more
|> arrogant way of saying it.
|
|Quite so! So you see the bind you have fallacistically
|wrapped me in! If I say "This is such and such..", I will be
|declared arrogant, and if I say "I think this is such and such...",
|it leaves me open to allegedly justified ad hominems.
People's experience of confusion is only a very weak basis for making
a *complaint* against an author unless we have reason to believe that
the readers have put forth some reasonable effort at understanding.
I'm not talking about what it takes to become an expert in the field,
or what it takes to be a serious student, just some indication that
you've taken something like the time that would be required to justify
a declaration of "EXTREMELY confusing". After a few minutes I'm
willing sometimes to say that something was somewhat confusing. After
an hour I might call it fairly thoroughly confusing. To call it
confusing in the extreme without having spent hours and hours just
seems hyperbolic.
If someone finds the constructive usage of terms such as "sequence"
confusing, and it isn't the kind of transient confusion that arises
for someone only very casually acquainted with the topic, I'd like for
some elaboration on how that can occur.
People who learn nonconstructive mathematics first often find it
confusing to learn, but not because the royal road to it has been
blocked somehow, but because changing your whole point of view on
mathematics is difficult.
|How is one to debate in such conditions!?
|It reminds me of one time when my elder brother charged me with...
|## "You're a very *argumentative* sort of person, aren't you!"
|So then I could neither agree nor disagree without supporting
|his charge!
There are plenty of things you could have said, but for some reason
didn't say.
For one, it's baffling to me how you can imagine that an "argument" of
this kind can be put forward without more specific examples of
confusing text. We hear that the constructive usage of "sequence" was
confusing to you at some point, but not when or where or how. How much
more content there would've been in a discussion in which specific
examples were at issue. I supplied one example which you dismissed as
"infantile".
We also don't hear you describe how long it might have taken you or
any of these other baffled people to sort out such a confusion in your
own mind. Hours? Part of an hour? A few minutes? 30 seconds of
frustration? Teachers hear ad nauseum from students who dislike
calculus and give in after a few minutes of frustration, and declare
that calculus is all SO hopelessly confusing. We all want to avoid
that happening, but we also don't simply take their word for how bad
it is.
You find it objectionable that I should hazard a guess that much of
this "SEVERE" confusion is the result of approaching the subject like
a tourist just interested in seeing a few "cool" sights and then going
home to all the things that *actually* matter to you. You don't say
much to counter that, though. You don't know the local language? The
hand gestures are unfamiliar? Not surprising, Mr. Tourist.
Arguing that expository articles should be done with more attention to
the issue of differences in meaning is far more plausible a claim. But
even there, I don't have any idea what expository articles on
intuitionism or constructivism you might have found unhelpful this
way.
I have no problem with people adopting a "tourist" approach so long as
they realize the limitations of it.
|> I think now that Brouwer is gone he still deserves
|> to have people stand up for him. You implied that he was
|> deceptive in some cases, but I don't know of any.
|
|As I explained earlier, his whole approach was (at least mildly)
|deceptive in its choice of pre-booked terminology.
|But clearly you attach no significance to such evidence.
In 1908 Brouwer wrote a thesis on the unreliability of rules of logic
such as the law of excluded middle. He was advised to set all that
aside and establish his career on less controversial stuff, and took
that advice. But in 1916 he wrote a paper on free choice sequences.
I've read the recognition that classical mathematicians were assuming
the axiom of choice dated to 1910. The fact that one asserts "A or not
A" *in spite of* sometimes being unable even in principle to prove
either one, although suspected by various authors previously, was only
accepted by Hilbert in 1930 or so.
In what sense was the terminology "pre-booked"? Everything was squared
away already... when?
At the time it wasn't clear that there was any way to make the two
systems mutually compatible, so it couldn't be regarded as merely a
difference in terminology.
|> while sometimes cranky and abrasive seldom if ever insincere.
|
|You seem to think that one cannot be both sincere AND deceptive.
|
|Indeed one can, especially if one has made a habit of the
|deceptiveness, growing little by little from small beginnings
|to become habituated to the point where one no longer notices
|one's habitual deceptions. To follow your example (in adducing
|wildly emotive and irrelevant analogies), one could look at Hamas.
|They are undoubtedly sincere in their attitudes, but have been
|casting nets of ("justified") deception for so long that
|they are now constantly deceptive without realizing it.
I don't count deceptive talk let alone habitually deceptive talk as
sincere, even if the person has managed to talk themselves into
believing it.
But let me just say, I'm unaware of Brouwer having pulled the wool
over his own eyes in such a way, certainly not any more than Hilbert
et al. or you or me. And you aren't providing any concrete examples of
such.
Keith Ramsay
Bill Taylor
>> to avoid using terms used in one context, for
>> similar but essentially different things in another context.
>
> I don't think they *are* different things. In going from
> classical mathematics to constructive mathematics, it isn't
> that reals or naturals change, but that the tools for
> *reasoning* about them change.
Well, this is opposed to my more general remark below;
which I therefore assume you disagree with.
> Well, the difference is precisely in what counts as
> a "proof" in the two systems.
For mere constructivism, that is so. But for Intuitionism,
there seems to be somewhat more to it.
> but I don't see how it is at all helpful to use
> different terminology for reals.
To keep differences as clear as possible, constantly.
>> But it is a reasonable view,
>> in math at least, that to a large extent "things" are in
>> some sense identifiable with what one may say about them,
>> so that changes in proof methods automatically imply changes
>> in ontology. Would you not agree, at least somewhat?
You did not agree with this, and it is the remark I referred to
just above. It is a point in the philosophy of math,
and de philosophicus non est disputandum.
I assume you disagree.
It seems to be chiefly a mere grammatical point,
(as is often the case with those opposed to Platonism/realism).
Realists tend to think of math concepts in terms of nouns;
and others of different bents, in terms of verbs. This difference
in grammatical taste should not, perhaps, be allowed to cloud
our underlying views of what is what.
> Constructive logic, just like classical logic, can be
> applied to anything, reals, Kiwis, etc.
But the uses of logic in math are hardly cognate
with its uses in the physical world! I have always
insisted that these two arenas are scarcely related,
philosophically, as you will recall. I guess that is
another irreconcilable difference between us.
> ("Kiwi" isn't offensive, is it?)
Au contraire! It is a term of pride!
(Except when applied to our national airline, OC!)
> I consider myself bilingual. I learned set theory after I learned
> constructive logic,
Good heavens! That is amazing.
> although I learned classical analysis
> (differential equations, calculus) before I ever learned the
> constructive versions.
That is certainly more like par for the course.
> But that's a case in which Brouwer specifically
> *changed* the terminology. He calls them "choice sequences"
> rather than just plain "infinite sequence".
Yes, that is to be noted, and applauded. But perhaps he might
have just gone a few extra yards? After all, "sequence" has
a pretty fixed and definite meaning in classical math.
It's not nearly as bad as what many "mere constructivists"
subsume under the heading of "sets", mind you; those things
are a very crook-backed, limping version of classical sets.
And standard terms like "the axiom of choice" when applied
to their sets is VASTLY different from the classical meaning.
Again, it is to Brouwer's credit (as I noted earlier) that he gives up
the term "set" completely, in favour of spreads and species.
A pity about those extra yards.
-- Bedevilled Bill