Sets
Gene W. Smith
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
>How convenient. Too bad you were not on hand as a publicist for the
>US armed forces, when they decided to go on their vacation, away from
>Vietnam.
Sure. I'm leaving for a one-week workshop on 'dessins d'enfant' Sunday.
Write your victory proclamation now.
>In the first place, if set theory is not your specialty (most high
>school kids these days know that numbers are sets), you would be well
>advised to abstain from making bombastic pronouncements on a subject,
>whose basic notions you obviously do not understand.
"Bombastic pronouncements on basic notions you obviously do not
understand" sums up your "ideas" pretty well. Real mathematicians
have no difficulty understanding that defining a cardinal number by
way of an ordinal number is a convention, and that other equally valid
definitions are possible. We need something to count in order to
define cardinals, and ordinals are handy for the purpose, but by no
means necessary.
Real mathematicians also understand that defining a number by a
particular set is also a conventional construction, and that many such
constructions are possible. If high school students are being taught
confused mush like "numbers are sets", I can't help them, but I see no
reason to lower my understanding to the level you share with them.
I suggest you tell us precisely which set the following are:
2
-5/3
sqrt(2)
sqrt(-1)
pi
and *why* they must be precisely these sets, and no other.
Here is another question for you: is a real number
(1) An infinite decimal expansion with sign, modulo equivalence,
(1) A Dedekind cut,
(2) The ring of Cauchy sequences of rational numbers modulo
the maximal ideal of null sequences,
(4) Some nonstandard model Q* of Q modulo the maximal ideal
of infinitesimal elements,
(5) Continued fractions with sign modulo equivalence,
(6) Something else?
How is it possible that these, defined as sets, differ?
Still another question: "2" is a natural number, an integer (element
of Z), a rational number, a real number, an element of \bar Q, an
element of Q[x], and so forth. Defined as sets, however you do it,
these are all different. If a number is a set, how is this possible?
--
Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University
gsm...@kalliope.iwr.uni-heidelberg.de
Gene W. Smith
gsm...@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:
>(4) Some nonstandard model Q* of Q modulo the maximal ideal
> of infinitesimal elements,
This should of course be the ring of finite elements of Q* modulo
the maximal ideal of infinitesimals.
Mikhail Zeleny
gsm...@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:
>In article <1993Apr16.0...@husc3.harvard.edu>
>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
>>How convenient. Too bad you were not on hand as a publicist for the
>>US armed forces, when they decided to go on their vacation, away from
>>Vietnam.
>Sure. I'm leaving for a one-week workshop on 'dessins d'enfant' Sunday.
>Write your victory proclamation now.
No need: I'll just send you a copy of this article, as I've done
before. It is against my principles to withdraw from a debate without
either reaching an agreement, or conceding defeat; consequently I must
assume that my interlocutor follows the same rule. War is hell, --
pull out, and you lose.
>>In the first place, if set theory is not your specialty (most high
>>school kids these days know that numbers are sets), you would be well
>>advised to abstain from making bombastic pronouncements on a subject,
>>whose basic notions you obviously do not understand.
>"Bombastic pronouncements on basic notions you obviously do not
>understand" sums up your "ideas" pretty well. Real mathematicians
>have no difficulty understanding that defining a cardinal number by
>way of an ordinal number is a convention, and that other equally valid
>definitions are possible. We need something to count in order to
>define cardinals, and ordinals are handy for the purpose, but by no
>means necessary.
Real mathematics aspires to nomological force that transcends the
certainty of a convention; to do otherwise, is to practice sociology
of mathematics. The reason cardinals are defined through ordinals, is
that a particular conception of sets has prevailed over all others,
thanks to the efforts of Zermelo. To take the common agreement on
this point as purely conventional, would undermine all metaphysical
conceits of mainstream mathematical practice. Is your work concerned
with numbers, or just talk about numbers?
>Real mathematicians also understand that defining a number by a
>particular set is also a conventional construction, and that many such
>constructions are possible. If high school students are being taught
>confused mush like "numbers are sets", I can't help them, but I see no
>reason to lower my understanding to the level you share with them.
I see nothing in this commonplace to vitiate the point I made earlier.
>I suggest you tell us precisely which set the following are:
>
>2
>-5/3
>sqrt(2)
>sqrt(-1)
>pi
>
>and *why* they must be precisely these sets, and no other.
No need. Surely you will have read Benacerraf's classic paper on just
that subject. Again, the conventional nature of, say, von Neumann's
definition of the integers, or the Wiener-Kuratowski definition of the
ordered pairs, does not demonstrate that *all* reductive mathematical
definitions, such as the identification of the cardinality of X with
the least ordinal \alpha equinumerous to it (assuming AC), are purely
conventional in nature. Some of them just happen to achieve the
status of real definitions, reflecting what things are. If the
iterative conception of sets is true, then the aforementioned
reduction represents a real definition of cardinal numbers.
>Here is another question for you: is a real number
>
>(1) An infinite decimal expansion with sign, modulo equivalence,
>(1) A Dedekind cut,
>(2) The ring of Cauchy sequences of rational numbers modulo
> the maximal ideal of null sequences,
>(4) Some nonstandard model Q* of Q modulo the maximal ideal
> of infinitesimal elements,
>(5) Continued fractions with sign modulo equivalence,
>(6) Something else?
All or none of the above, depending.
>How is it possible that these, defined as sets, differ?
Either because:
(a) the precise manner of effecting a set-theoretic reduction of reals
is immaterial to the theories of reals,
or because:
(b) the natural definition of the reals remains to be discovered.
Take your pick. At the moment, I vote for the latter.
>Still another question: "2" is a natural number, an integer (element
>of Z), a rational number, a real number, an element of \bar Q, an
>element of Q[x], and so forth. Defined as sets, however you do it,
>these are all different. If a number is a set, how is this possible?
Objection: "2" is none of the above, but a *name* capable of denoting
either of the above, and more, depending on the context of its use.
>--
> Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University
> gsm...@kalliope.iwr.uni-heidelberg.de
cordially,
mikhail zel...@husc.harvard.edu
"Le cul des femmes est monotone comme l'esprit des hommes."
Gene W. Smith
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
>Real mathematics aspires to nomological force that transcends the
>certainty of a convention; to do otherwise, is to practice sociology
>of mathematics. The reason cardinals are defined through ordinals, is
>that a particular conception of sets has prevailed over all others,
>thanks to the efforts of Zermelo.
First the Z-man says that we are seeking a "nomological force" which
"transcends the certainty of a convention". One can't get more
certain than a convention, so I suppose this nomological force
transcends it in some other, unconventional manner.
He then asserts we are to avoid mere "sociology of mathematics".
Following on this he turns around and asserts that a particular
concept of set has prevailed for a sociological reason, to wit the
advocacy of Zermelo.
I'll give you a couple of free clues, Z-man: real mathematicians tend
to dislike gibberish like "aspires to nomological force that
transcends the certainty of convention". This is Rule #1 in the
sociology of mathematics. Rule #2 is like unto it, and says that an
argument is not just a bunch of words strung together, with names
dropped in like raisins into a bowl of cereal. Arguments involve
thought, and are not just rhetorical florishes and nonsensical appeals
to authority.
>To take the common agreement on this point as purely conventional,
>would undermine all metaphysical conceits of mainstream mathematical
>practice.
The "metaphysical conceits" of practicing mathematicians are various.
You might even be able to find one who thinks that the axioms of ZFC
were handed to us by God on tablets of stone, though I have never met
such a person. People of the Platonist/realist school who think that
there is such a thing as actual sets will of course think that ZFC
does not completely describe such sets. Other "metaphysical
conceits" will lead to other points of view on the status of ZFC.
None of this is actually mathematics, so it can't undermine mainstream
mathematical practice in the slightest.
In short, you are babbling.
>Is your work concerned with numbers, or just talk about numbers?
Why don't you explain to us what you see as the difference? I would
say it is about numbers, and that someone working with talk about
numbers is in some other field.
>>Real mathematicians also understand that defining a number by a
>>particular set is also a conventional construction, and that many such
>>constructions are possible. If high school students are being taught
>>confused mush like "numbers are sets", I can't help them, but I see no
>>reason to lower my understanding to the level you share with them.
>I see nothing in this commonplace to vitiate the point I made earlier.
You said "numbers are sets". Now you agree that to call this
"confused mush" is "commonplace". This is a contradiction.
>No need. Surely you will have read Benacerraf's classic paper on just
>that subject.
If you have an argument, give it. If you don't, shut up. Name
dropping gets you 0 (as in {}) respect from your typical
mathematician, by the way.
>Again, the conventional nature of, say, von Neumann's definition of
>the integers, or the Wiener-Kuratowski definition of the ordered
>pairs, does not demonstrate that *all* reductive mathematical
>definitions, such as the identification of the cardinality of X with
>the least ordinal \alpha equinumerous to it (assuming AC), are purely
>conventional in nature. Some of them just happen to achieve the
>status of real definitions, reflecting what things are.
Do you know what "conventional" means? Since identifing the
cardinality of X with the least equinumerous ordinal is by no means
necessary to define the cardinality of X, it is a convention. It is a
convention in precisely the same way as the von Neumann or
Wiener-Kuratowski definitions, and for precisely the same reason.
Maybe you would like to spell out the actual difference, instead of
babbling drivel about "what things are".
>If the iterative conception of sets is true, then the aforementioned
>reduction represents a real definition of cardinal numbers.
What the hell? Are you saying V=L implies that cardinals=ordinals, or
what are you on about?
>All or none of the above, depending.
Thank you for that clear and helpful answer. I strongly suggest you
do *not* try to make a career in mathematics or philosophy.
>>How is it possible that these, defined as sets, differ?
>Either because:
>(a) the precise manner of effecting a set-theoretic reduction of reals
>is immaterial to the theories of reals,
Oh no! Could this be??? Could it be that this is how things work in
mathematics?
>or because:
>(b) the natural definition of the reals remains to be discovered.
Right. What does this mean, if anything?
>Take your pick. At the moment, I vote for the latter.
Of course. It doesn't actually mean anything, so you vote for it.
Mikhail Zeleny
gsm...@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:
>In article <1993Apr17.0...@husc3.harvard.edu>
>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
>>Real mathematics aspires to nomological force that transcends the
>>certainty of a convention; to do otherwise, is to practice sociology
>>of mathematics. The reason cardinals are defined through ordinals, is
>>that a particular conception of sets has prevailed over all others,
>>thanks to the efforts of Zermelo.
>First the Z-man says that we are seeking a "nomological force" which
>"transcends the certainty of a convention". One can't get more
>certain than a convention, so I suppose this nomological force
>transcends it in some other, unconventional manner.
>
>He then asserts we are to avoid mere "sociology of mathematics".
>Following on this he turns around and asserts that a particular
>concept of set has prevailed for a sociological reason, to wit the
>advocacy of Zermelo.
All science is conducted in more or less conventional manner; however it
does not follow that the laws of science are limited to the force of a
convention. The manner in which ZFC and NBG prevailed over alternative,
non-Cantorian theories, indicates that the reasons for this prevalence
are more than merely sociological. Of course, to appreciate this point,
you would have to think about the issues involved, rather than content
yourself with spewing forth your customary disdainful invective.
>I'll give you a couple of free clues, Z-man: real mathematicians tend
>to dislike gibberish like "aspires to nomological force that
>transcends the certainty of convention". This is Rule #1 in the
>sociology of mathematics. Rule #2 is like unto it, and says that an
>argument is not just a bunch of words strung together, with names
>dropped in like raisins into a bowl of cereal. Arguments involve
>thought, and are not just rhetorical florishes and nonsensical appeals
>to authority.
Reply #1 is that the fact that real physicists likewise tend to dislike
metaphysical arguments, does not in any way vitiate the merits of the
latter; similarly, your disdainful ignorance of metamathematical issues
reflects solely on your parochialism, rather than on any fact of the
matter. Likewise, any purported sociology of mathematics that fails to
recognize Georg Cantor, Alonzo Church, Jean Dieudonn\'e, or Saunders
MacLane as real mathematicians, is utterly full of shit. Reply #2 is
that failure to appreciate the merits of any given argument, is just as
likely to lie with the addressee, as with the addresser.
>>To take the common agreement on this point as purely conventional,
>>would undermine all metaphysical conceits of mainstream mathematical
>>practice.
>The "metaphysical conceits" of practicing mathematicians are various.
>You might even be able to find one who thinks that the axioms of ZFC
>were handed to us by God on tablets of stone, though I have never met
>such a person. People of the Platonist/realist school who think that
>there is such a thing as actual sets will of course think that
...the axioms of...
> ZFC
>does not completely describe such sets. Other "metaphysical
>conceits" will lead to other points of view on the status of ZFC.
>None of this is actually mathematics, so it can't undermine mainstream
>mathematical practice in the slightest.
Again, concerns about actual mathematical practice fall in the purview
of psychology and sociology. Since in this debate, the key issue is
concerned with the interpretation of mathematical results, and since
mathematics does not concern itself with questions of meaning, it
immediately follows that none of this is actually mathematics; however,
your earlier conclusion that such considerations are bereft of any
merit, will only obtain on the additional assumption of parochialism.
>In short, you are babbling.
As is shown above, you are jumping to conclusions.
>>Is your work concerned with numbers, or just talk about numbers?
>Why don't you explain to us what you see as the difference? I would
>say it is about numbers, and that someone working with talk about
>numbers is in some other field.
Precisely so; this is the best reason why the subject of mathematician's
consensus *cannot* be ruled to be decided *solely* in virtue, and by the
force of, the said consensus.
>>>Real mathematicians also understand that defining a number by a
>>>particular set is also a conventional construction, and that many such
>>>constructions are possible. If high school students are being taught
>>>confused mush like "numbers are sets", I can't help them, but I see no
>>>reason to lower my understanding to the level you share with them.
>>I see nothing in this commonplace to vitiate the point I made earlier.
>You said "numbers are sets". Now you agree that to call this
>"confused mush" is "commonplace". This is a contradiction.
Forgive me for concentrating on your first sentence, in order to
downplay the amount of content-free verbiage you tend to produce.
Once again, pointing out the multiplicity of possible conventions,
does nothing to demonstrate equal validity thereof.
>>No need. Surely you will have read Benacerraf's classic paper on just
>>that subject.
>If you have an argument, give it. If you don't, shut up. Name
>dropping gets you 0 (as in {}) respect from your typical
>mathematician, by the way.
That was an allusion; if you stand in need of more hints, the reference
is to a publication in the _Philosophical Review_ 74 (1965): 47-73. In
analytic philosophy, as in mathematics, our conventional practice is to
take earlier arguments as a given. As says Gilbert Gottfried, go and
figure.
>>Again, the conventional nature of, say, von Neumann's definition of
>>the integers, or the Wiener-Kuratowski definition of the ordered
>>pairs, does not demonstrate that *all* reductive mathematical
>>definitions, such as the identification of the cardinality of X with
>>the least ordinal \alpha equinumerous to it (assuming AC), are purely
>>conventional in nature. Some of them just happen to achieve the
>>status of real definitions, reflecting what things are.
>Do you know what "conventional" means? Since identifing the
>cardinality of X with the least equinumerous ordinal is by no means
>necessary to define the cardinality of X, it is a convention.
Agreed.
> It is a
>convention in precisely the same way as the von Neumann or
>Wiener-Kuratowski definitions, and for precisely the same reason.
This is demonstrably false.
>Maybe you would like to spell out the actual difference, instead of
>babbling drivel about "what things are".
The difference is, that only in the latter case there is no compelling
reason to prefer one alternative over another.
>>If the iterative conception of sets is true, then the aforementioned
>>reduction represents a real definition of cardinal numbers.
>What the hell? Are you saying V=L implies that cardinals=ordinals, or
>what are you on about?
On the iterative conception of the cumulative hierarchy V, which in no
way presupposes the truth of V=L, see the introductory textbooks by
Drake or Shoenfield.
>>All or none of the above, depending.
>Thank you for that clear and helpful answer. I strongly suggest you
>do *not* try to make a career in mathematics or philosophy.
The answer has been elaborated below. Concerning your kind suggestion,
expert opinions differ.
>>>How is it possible that these, defined as sets, differ?
>>Either because:
>>(a) the precise manner of effecting a set-theoretic reduction of reals
>>is immaterial to the theories of reals,
>Oh no! Could this be??? Could it be that this is how things work in
>mathematics?
It could, if by "how things work" you mean strictly the social aspects
of your discipline; it could not, if you are referring to the way things
really are.
>>or because:
>>(b) the natural definition of the reals remains to be discovered.
>Right. What does this mean, if anything?
It means that there will come a time when the relationship between the
reals and the integers is sufficiently well understood to suggest a
clearly superior structural definition of the former in terms of the
latter.
>>Take your pick. At the moment, I vote for the latter.
>Of course. It doesn't actually mean anything, so you vote for it.
You are living proof of the adage that it takes a great mind to achieve
understanding in spite of personal resentment.
>--
> Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University
> gsm...@kalliope.iwr.uni-heidelberg.de
cordially,
Ozan S. Yigit
[mostly elided]
In short, you are babbling.
Why do you keep baiting this platonist twit? You already knew
what to expect...
oz
Robert Camp Miner
>In article <1993Apr17.0...@husc3.harvard.edu>
>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
>
>>Real mathematics aspires to nomological force that transcends the
>>certainty of a convention; to do otherwise, is to practice sociology
>>of mathematics. The reason cardinals are defined through ordinals, is
>>that a particular conception of sets has prevailed over all others,
>>thanks to the efforts of Zermelo.
>
>First the Z-man says that we are seeking a "nomological force" which
>"transcends the certainty of a convention". One can't get more
>certain than a convention
... he says apodictically. (P) One can be certain that one can't get more
certain than a convention. If one can be certain of (P), then you're
merely spouting nonsense. If not, then why should we take your statement,
expressed as if it were apodictic, seriously? Why shouldn't we simply
assume that you think in cliches?
>, so I suppose this nomological force
>transcends it in some other, unconventional manner.
>
>He then asserts we are to avoid mere "sociology of mathematics".
>Following on this he turns around and asserts that a particular
>concept of set has prevailed for a sociological reason, to wit the
>advocacy of Zermelo.
>
>I'll give you a couple of free clues, Z-man: real mathematicians tend
>to dislike gibberish like "aspires to nomological force that
>transcends the certainty of convention".
I haven't read the response of the Z-man (what an original and altogether
witty sobriquet!), but your metaphysics doesn't
entitle you to the notion of "real mathematician." And exactly how do
the ten words above (the ones enclosed by inverted commas) constitute
"gibberish"?
--
------------------------------------------------
Die Welt ist alles, was der Fall ist.
r...@owlnet.rice.edu
------------------------------------------------
Gene W. Smith
Miner) writes:
>>First the Z-man says that we are seeking a "nomological force" which
>>"transcends the certainty of a convention". One can't get more
>>certain than a convention
>... he says apodictically. (P) One can be certain that one can't get more
>certain than a convention. If one can be certain of (P), then you're
>merely spouting nonsense.
Get real, Schlemiel. I was talking about the certainty of
mathematical statements, in case you hadn't noticed. (P) is not a
mathematical statement. Moreover, I did not say that (P) was certain,
or more or less certain, than anything.
You are the one spouting nonsense.
>If not, then why should we take your statement, expressed as if it
>were apodictic, seriously? Why shouldn't we simply assume that you
>think in cliches?
It *is* a cliche in mathematical circles. Why should we assume that
you can distinguish stating a cliche, and "thinking in terms" of
cliches? Why should we assume that you actually have an argument, and
are not simply practicing a few random rhetorical devices? Why should
we assume that even you know what the hell your point is?
>>I'll give you a couple of free clues, Z-man: real mathematicians
>>tend to dislike gibberish like "aspires to nomological force that
>>transcends the certainty of convention".
>I haven't read the response of the Z-man (what an original and
>altogether witty sobriquet!), but your metaphysics doesn't entitle you
>to the notion of "real mathematician."
No, but the fact that I *am* a real mathematician does. Do you
actually have a point, by the way?
>And exactly how do the ten words above (the ones enclosed by inverted
>commas) constitute "gibberish"?
If you like them, why don't you explain them?
Robert Camp Miner
>In article <C66pF...@rice.edu> r...@owlnet.rice.edu (Robert Camp
>Miner) writes:
>
>>>First the Z-man says that we are seeking a "nomological force" which
>>>"transcends the certainty of a convention". One can't get more
>>>certain than a convention
>
>>... he says apodictically. (P) One can be certain that one can't get more
>>certain than a convention. If one can be certain of (P), then you're
>>merely spouting nonsense.
>
>Get real, Schlemiel. I was talking about the certainty of
>mathematical statements, in case you hadn't noticed. (P) is not a
>mathematical statement. Moreover, I did not say that (P) was certain,
>or more or less certain, than anything.
>
>You are the one spouting nonsense.
You said that one can't get more certain than a convention. You still
haven't explained how you can be so certain, if I may use the word, of
the truth of this statement. Do you believe that "one can't get more
certain than a convention" is true only of mathematical statements, or
that it is true of mathematical statements plus some other kinds of
statements, or that it is true of all statements?
>
>>If not, then why should we take your statement, expressed as if it
>>were apodictic, seriously? Why shouldn't we simply assume that you
>>think in cliches?
>
>It *is* a cliche in mathematical circles. Why should we assume that
>you can distinguish stating a cliche, and "thinking in terms" of
>cliches? Why should we assume that you actually have an argument, and
>are not simply practicing a few random rhetorical devices? Why should
>we assume that even you know what the hell your point is?
"It *is* a cliche in mathematical circles." Of course; if it's a cliche
it must be right; it wouldn't be a cliche if it weren't right. Forgive me
for being so impertinent as to question a cliche. If you weren't so
obsessed with showing the world what an aggressively puerile blowhard you are,
you might understand the problematic nature of statements such as "One can't
get more certain than a convention." You mistake a platitude for a
profundity.
>>>I'll give you a couple of free clues, Z-man: real mathematicians
>>>tend to dislike gibberish like "aspires to nomological force that
>>>transcends the certainty of convention".
>
>>I haven't read the response of the Z-man (what an original and
>>altogether witty sobriquet!), but your metaphysics doesn't entitle you
>>to the notion of "real mathematician."
>
>No, but the fact that I *am* a real mathematician does. Do you
>actually have a point, by the way?
Yes.
>>And exactly how do the ten words above (the ones enclosed by inverted
>>commas) constitute "gibberish"?
>
>If you like them, why don't you explain them?
>
Another masterly evasion of the question. Have you considered a career
in politics?
>
>
>
>--
> Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University
> gsm...@kalliope.iwr.uni-heidelberg.de
--
auf wiederhoeren
- rcm
Gene W. Smith
Miner) writes:
>>Get real, Schlemiel. I was talking about the certainty of
>>mathematical statements, in case you hadn't noticed. (P) is not a
>>mathematical statement. Moreover, I did not say that (P) was certain,
>>or more or less certain, than anything.
>You said that one can't get more certain than a convention. You still
>haven't explained how you can be so certain, if I may use the word, of
>the truth of this statement.
The proof of a mathematical statment which is true by definition is
"true by definition", or in other words is the null proof. If I am
doing group theory, and want to prove that xx^{-1} = 1, then I just
notice that this is an axiom, i.e. a convention. This type of
proof is the least capable of being wrong, and hence the most
certain.
>Do you believe that "one can't get more certain than a convention" is
>true only of mathematical statements, or that it is true of
>mathematical statements plus some other kinds of statements, or that
>it is true of all statements?
I'm not interested in this question. If you are, pursue it
yourself. It has nothing to do with what I was saying.
>"It *is* a cliche in mathematical circles." Of course; if it's a cliche
>it must be right; it wouldn't be a cliche if it weren't right.
You accused me of thinking in cliches. Are you advocating thinking in
the denial of cliches, or what?
>Forgive me for being so impertinent as to question a cliche. If you
>weren't so obsessed with showing the world what an aggressively
>puerile blowhard you are, you might understand the problematic nature
>of statements such as "One can't get more certain than a convention."
>You mistake a platitude for a profundity.
I didn't say it was a profundity. I just said it. By the way, why
the hysterical and insulting tone? Do you normally react this way to
statements which appear to be obvious?
>>Do you actually have a point, by the way?
>Yes.
What?
>>>And exactly how do the ten words above (the ones enclosed by inverted
>>>commas) constitute "gibberish"?
>>If you like them, why don't you explain them?
>Another masterly evasion of the question. Have you considered a career
>in politics?
Speaking of masterly evasions by aggressively puerile blowhards, I
note the following: showing something is *not* gibberish is easy, so
long as it is not gibberish. Showing something *is* gibberish is
another matter. So you, Boyo, are doing the evading here.
Michael L. Siemon
>In article <1993Apr29.1...@sun0.urz.uni-heidelberg.de> gsm...@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:
>>>>First the Z-man says that we are seeking a "nomological force" which
>>>>"transcends the certainty of a convention". One can't get more
>>>>certain than a convention
>>
>>>... he says apodictically. (P) One can be certain that one can't get more
>>>certain than a convention. If one can be certain of (P), then you're
>>>merely spouting nonsense.
>>
>>Get real, Schlemiel. I was talking about the certainty of
>>mathematical statements, in case you hadn't noticed. (P) is not a
^^^^^^^^^^^^^^^^^^^^^^^^^^
>>mathematical statement. Moreover, I did not say that (P) was certain,
>>or more or less certain, than anything.
>You said that one can't get more certain than a convention. You still
>haven't explained how you can be so certain, if I may use the word, of
>the truth of this statement. Do you believe that "one can't get more
>certain than a convention" is true only of mathematical statements, or
You really *don't* get it. You can't even notice it when it is pointed
out to you!!!! Gene's usage is completely common in mathematics. It is
"convention" when we say 1! = 1. We have TOTAL certainty about this, as
we have arbitrarily MADE it so. It is not a matter of empirical discovery
or of extended (and hence possibly fallacious) proof. It is merely so,
because we SAY it is so. This sense of "convention" is common in math,
but not limited to it: conventions in contract bridge bidding have much
the same character. But Gene's context was, obviously then, explicitly
now, that of mathematical usage of number -- which Zeleny was going on
about in his usual nonsensical rant, posturing about nothing at all in
the most stupendous words he could dredge from his dictionaries (or the
murky recesses of his soi disant mind). Gene called him on it, in a way
that he more than deserved, doing so with a completely unobjectionable
statement about numbers and conventional constructions in mathematics.
What's *your* problem? Unlike Zeleny's attempts to browbeat everybody
with irrelevant technicalities out of theoretical studies of logic, Gene
was not making grandiose metaphysical claims to which you are required
to subscribe. Does it bother you to see Zeleny shown for the ignorant
fool he is?
--
Michael L. Siemon "Stand, stand at the window
m...@panix.com As the tears scald and start.
m...@ulysses.att.com You shall love your crooked neighbor
-standard disclaimer- With your crooked heart."
Robert Camp Miner
Okay, so you and Gene believe that "one can't get more certain than a
convention" is true *at least* of mathematical statements. I requested
an explication of the epistemological basis for the aforementioned
proposition. Zeleny, if I recall correctly (my apologies to MZ if I
recall incorrectly), wanted the same thing. Neither you nor Gene have
given us anything but cliche and bluster.
>
>What's *your* problem? Unlike Zeleny's attempts to browbeat everybody
>with irrelevant technicalities out of theoretical studies of logic, Gene
>was not making grandiose metaphysical claims to which you are required
>to subscribe. Does it bother you to see Zeleny shown for the ignorant
>fool he is?
>--
>Michael L. Siemon "Stand, stand at the window
>m...@panix.com As the tears scald and start.
>m...@ulysses.att.com You shall love your crooked neighbor
>-standard disclaimer- With your crooked heart."
You don't argue with Zeleny's claims; you just dismiss them as "irrelevant
technicalities." Would you please supply criteria for distinguishing
between (a) technicalities and non-technicalities; (b) irrelvant and
relevant technicalities?
As for showing that Zeleny is an ignorant fool, it wouldn't bother me in
the least. But you haven't done this (and I suspect you *can't* do this).
I ask why a particular statement of Zeleny's is "gibberish," and Gene
refuses to tell me. I would expect more from a "real mathematician," as
he likes to call himself.
Mikhail Zeleny
>In article <C6994...@rice.edu>
>r...@owlnet.rice.edu (Robert Camp Miner) writes:
>>In article <1993Apr29.1...@sun0.urz.uni-heidelberg.de>
>>gsm...@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:
GWS:
>>>>>First the Z-man says that we are seeking a "nomological force" which
>>>>>"transcends the certainty of a convention". One can't get more
>>>>>certain than a convention
RCM:
>>>>... he says apodictically. (P) One can be certain that one can't get more
>>>>certain than a convention. If one can be certain of (P), then you're
>>>>merely spouting nonsense.
GWS:
>>>Get real, Schlemiel. I was talking about the certainty of
>>>mathematical statements, in case you hadn't noticed. (P) is not a
> ^^^^^^^^^^^^^^^^^^^^^^^^^^
>>>mathematical statement. Moreover, I did not say that (P) was certain,
>>>or more or less certain, than anything.
Mr Smith, you are continuing to give a tedious textbook example of a
mendacious exercise in plausible deniability. Having made an
obviously untenable, because self-refuting, categorical claim, you are
attempting a retroactive revision, that in any event will not suffice
to get you out of trouble.
RCM:
>>You said that one can't get more certain than a convention. You still
>>haven't explained how you can be so certain, if I may use the word, of
>>the truth of this statement. Do you believe that "one can't get more
>>certain than a convention" is true only of mathematical statements, or
MLS:
>You really *don't* get it. You can't even notice it when it is pointed
>out to you!!!! Gene's usage is completely common in mathematics. It is
>"convention" when we say 1! = 1. We have TOTAL certainty about this, as
>we have arbitrarily MADE it so.
This is rich. So far, I have only seen Leon Trotsky and Alfred
Korzybski advance the thesis that it is not the case that 1 = 1,
inasmuch as one 1 is on the left side, while the other is on the
right. It is news for me that the public opinion has embraced
Dialectical Materialism and/or General Semantics, to the extent of
arbitrarily promoting this claim to the status of a "convention".
MLS:
> It is not a matter of empirical discovery
>or of extended (and hence possibly fallacious) proof. It is merely so,
>because we SAY it is so. This sense of "convention" is common in math,
>but not limited to it: conventions in contract bridge bidding have much
>the same character.
The cheesy doctrine adumbrated above is known as formalism. Since it
would be impossible for me to add anything to Gottlob Frege's famous
dissection of Thomae in _The Foundations of Arithmetic_, or Georg
Kreisel's more recent lampooning of Abraham Robinson and Paul Cohen in
the AMS Set Theory symposium, I shall limit myself to this mention of
their definitive efforts.
MLS:
> But Gene's context was, obviously then, explicitly
>now, that of mathematical usage of number -- which Zeleny was going on
>about in his usual nonsensical rant, posturing about nothing at all in
>the most stupendous words he could dredge from his dictionaries (or the
>murky recesses of his soi disant mind). Gene called him on it, in a way
>that he more than deserved, doing so with a completely unobjectionable
>statement about numbers and conventional constructions in mathematics.
To say that the questions of usage exhaust the issue of meaning
constitutes a highly partisan move in the philosophy of mathematics.
I have reasons to believe that this move is unwarranted, and moreover,
that it leads to counterintuitive consequences. Having described
these reasons time and again, I shall be happy to repeat myself, once
you and Smith stop frothing at the mouth for reasons that have nothing
to do with the subject of this discussion.
MLS:
>What's *your* problem? Unlike Zeleny's attempts to browbeat everybody
>with irrelevant technicalities out of theoretical studies of logic, Gene
>was not making grandiose metaphysical claims to which you are required
>to subscribe. Does it bother you to see Zeleny shown for the ignorant
>fool he is?
Doesn't it bother you that your venomous bad faith belies your
self-professed religious creed?
>--
>Michael L. Siemon "Stand, stand at the window
>m...@panix.com As the tears scald and start.
>m...@ulysses.att.com You shall love your crooked neighbor
>-standard disclaimer- With your crooked heart."
cordially,
Robert Vienneau
>MLS:
>>... Gene's usage is completely common in mathematics. It is
>>"convention" when we say 1! = 1. We have TOTAL certainty about this, as
>>we have arbitrarily MADE it so.
>
>This is rich. So far, I have only seen Leon Trotsky and Alfred
>Korzybski advance the thesis that it is not the case that 1 = 1,
>inasmuch as one 1 is on the left side, while the other is on the
>right. It is news for me that the public opinion has embraced
>Dialectical Materialism and/or General Semantics, to the extent of
>arbitrarily promoting this claim to the status of a "convention".
I can never tell when Mr. Z is being ironical, but I assumed 1! = 1 was
part of the series:
0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, ...
Can you go on now?
>...MLS:
>> It is not a matter of empirical discovery
>>or of extended (and hence possibly fallacious) proof. It is merely so,
>>because we SAY it is so. This sense of "convention" is common in math,
>>but not limited to it: conventions in contract bridge bidding have much
>>the same character.
>
>The cheesy doctrine adumbrated above is known as formalism. Since it
>would be impossible for me to add anything to Gottlob Frege's famous
>dissection of Thomae in _The Foundations of Arithmetic_...
>
>MLS:
>> But Gene's context was, obviously then, explicitly
>>now, that of mathematical usage of number -- which Zeleny was going on
>>about in his usual nonsensical rant...Gene called him on it, in a way
>>murky recesses of his soi disant mind).
>>that he more than deserved, doing so with a completely unobjectionable
>>statement about numbers and conventional constructions in mathematics.
>
>To say that the questions of usage exhaust the issue of meaning
>constitutes a highly partisan move in the philosophy of mathematics...
>
>MLS:
>>What's *your* problem? Unlike Zeleny's attempts to browbeat everybody
>>with irrelevant technicalities out of theoretical studies of logic, Gene
>>was not making grandiose metaphysical claims to which you are required
>>to subscribe. Does it bother you to see Zeleny shown for the ignorant
>>fool he is?
MLS, inasmuch as he is advocating a philosophical position at all,
sounds more like Wittgenstein than Hilbert to me. He draws an anology
to a game, refuses to admit that statements about mathematics are to be
analyzed as if they were in a mathematical formalism themselves, talks
about language use in a way that others read as asserting that meaning
is use, opposes metaphysics...
I wonder if MLS would agree with the proposition that metamathematics in
the style of Hilbert and Godel is just more mathematics. Working
mathematicians need not treat such mathematics as of any more importance
than any other branch. It's not beneath or more basic than the rest of
mathematics.
So far, I think Mr. Z is "losing" this argument, but then I was against
his position from the start. It seems to me that to support his
position, Mr. Z needs to demonstrate why taking one stand on such
matters as the truth or falsity of the Axiom of Choice is demonstratably
about the basic furniture of the world. The way I read Godel, he thought
there was a correct decision about the AC, even after both its truth and
falsity were shown to be consistent with the other usual axioms of set
theory. I never understood Godel's position here.
Mr. Z could also begin by giving some reason to choose among Gene
Smith's less specialized examples. For instance, I cannot see any
meaning to the assertion that defining reals as Cuts yields a more
natural definition than defining them as equivalence classes of Cauchy
sequences of rationals, or vice versa. Apparently, Mr. Z thinks some day
we'll know that one definition or another is right.
So, Mr. Zeleny, instead of attacking your opponents, why not try to
answer something like the original questions posed to you on this
thread (at least the part I've seen)? So far, I do not think you have
been able to come up with any answers, much less ones that are
convincing.
Robert Vienneau
--
The opinions expressed are not necessarily those of the University of
North Carolina at Chapel Hill, the Campus Office for Information
Technology, or the Experimental Bulletin Board Service.
internet: laUNChpad.unc.edu or 152.2.22.80
Torkel Franzen
Robert....@launchpad.unc.edu (Robert Vienneau) writes:
>The way I read Godel, he thought
>there was a correct decision about the AC, even after both its truth and
>falsity were shown to be consistent with the other usual axioms of set
>theory. I never understood Godel's position here.
Don't you rather have the Continuum Hypothesis in mind? There is no
need to speak about there being "a correct decision" in the case of the
Axiom of Choice: it is simply true on the intended interpretation. If
you find this difficult to understand, you have yet to explain on
what grounds. The axiom of extensionality is also independent of the
remaining axioms: are you saying that it is difficult to understand
why anybody should say that the axiom is true of sets?
Mikhail Zeleny
>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
MLS:
>>>... Gene's usage is completely common in mathematics. It is
>>>"convention" when we say 1! = 1. We have TOTAL certainty about this, as
>>>we have arbitrarily MADE it so.
MZ:
>>This is rich. So far, I have only seen Leon Trotsky and Alfred
>>Korzybski advance the thesis that it is not the case that 1 = 1,
>>inasmuch as one 1 is on the left side, while the other is on the
>>right. It is news for me that the public opinion has embraced
>>Dialectical Materialism and/or General Semantics, to the extent of
>>arbitrarily promoting this claim to the status of a "convention".
RV:
>I can never tell when Mr. Z is being ironical, but I assumed 1! = 1 was
>part of the series:
> 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, ...
>Can you go on now?
Only until the next ellipsis, and the next equivocation.
MLS:
>>> It is not a matter of empirical discovery
>>>or of extended (and hence possibly fallacious) proof. It is merely so,
>>>because we SAY it is so. This sense of "convention" is common in math,
>>>but not limited to it: conventions in contract bridge bidding have much
>>>the same character.
MZ:
>>The cheesy doctrine adumbrated above is known as formalism. Since it
>>would be impossible for me to add anything to Gottlob Frege's famous
>>dissection of Thomae in _The Foundations of Arithmetic_...
MLS:
>>> But Gene's context was, obviously then, explicitly
>>>now, that of mathematical usage of number -- which Zeleny was going on
>>>about in his usual nonsensical rant...Gene called him on it, in a way
>>>murky recesses of his soi disant mind).
>>>that he more than deserved, doing so with a completely unobjectionable
>>>statement about numbers and conventional constructions in mathematics.
MZ:
>>To say that the questions of usage exhaust the issue of meaning
>>constitutes a highly partisan move in the philosophy of mathematics...
MLS:
>>>What's *your* problem? Unlike Zeleny's attempts to browbeat everybody
>>>with irrelevant technicalities out of theoretical studies of logic, Gene
>>>was not making grandiose metaphysical claims to which you are required
>>>to subscribe. Does it bother you to see Zeleny shown for the ignorant
>>>fool he is?
RV:
>MLS, inasmuch as he is advocating a philosophical position at all,
>sounds more like Wittgenstein than Hilbert to me. He draws an anology
>to a game, refuses to admit that statements about mathematics are to be
>analyzed as if they were in a mathematical formalism themselves, talks
>about language use in a way that others read as asserting that meaning
>is use, opposes metaphysics...
Hilbert's position, as recently analyzed by Hallett in a forthcoming
Amherst symposium publication, and especially as reflected in the
intuitive geometry course recorded by Cohn-Vossen, is miles away from
the naive Thomae-style formalism, suggested in the analogy with a card
game (recall that Thomae famously referred to chess in this context
some 40 years before Wittgenstein). In any event, my comment about
reducing meaning to use was meant to apply to Wittgenstein and his
epigoni like Dummett and certain other local figures. Evidently, the
differences between their views and classical, pre-Hilbertian
formalism, are far less significant than you make them out to be.
RV:
>I wonder if MLS would agree with the proposition that metamathematics in
>the style of Hilbert and Godel is just more mathematics. Working
>mathematicians need not treat such mathematics as of any more importance
>than any other branch. It's not beneath or more basic than the rest of
>mathematics.
The reference to the practice of working mathematicians can only carry
so much weight in a foundational discussion. While one conservative
desideratum on a viable philosophy of mathematics would rule out
radical revisionism, yet another would systematically circumscribe the
authority of a "working mathematician" in the task of interpreting his
own results. On this, see Kreisel on Cohen and Robinson, and recall
Socrates disclaiming the onerous title of the best authority on his
own words.
RV:
>So far, I think Mr. Z is "losing" this argument, but then I was against
>his position from the start. It seems to me that to support his
>position, Mr. Z needs to demonstrate why taking one stand on such
>matters as the truth or falsity of the Axiom of Choice is demonstratably
>about the basic furniture of the world. The way I read Godel, he thought
>there was a correct decision about the AC, even after both its truth and
>falsity were shown to be consistent with the other usual axioms of set
>theory. I never understood Godel's position here.
G\"odel evidently thought that extensive study of the consequences of
any given proposition, would result in accretion of its concequences,
and their bearing on the practicioners' intuitive faculties, in such a
manner as to throw light on the plausibility of the original
hypothesis. I see absolutely nothing objectionable, or even
controversial in this view.
RV:
>Mr. Z could also begin by giving some reason to choose among Gene
>Smith's less specialized examples. For instance, I cannot see any
>meaning to the assertion that defining reals as Cuts yields a more
>natural definition than defining them as equivalence classes of Cauchy
>sequences of rationals, or vice versa. Apparently, Mr. Z thinks some day
>we'll know that one definition or another is right.
One major motivation of set-theoretic resesarch can be described as
the search for a natural relation between the integers and the reals.
As witnessed by our failure to settle the CH, this question is far
from being solved. As witnessed by our awareness that ZF^2 *does*
settle the CH, albeit in a fashion currently unknown to ourselves,
there is every reason to suppose that such a relation exists, and that
continued study of higher-order logic will in the long run throw light
on its structure. As a rank amateur in this subject, I feel utterly
unwarranted in pursuing this point any further.
RV:
>So, Mr. Zeleny, instead of attacking your opponents, why not try to
>answer something like the original questions posed to you on this
>thread (at least the part I've seen)? So far, I do not think you have
>been able to come up with any answers, much less ones that are
>convincing.
> Robert Vienneau
It is a long leap indeed to conclude from anyone's personal inability
to answer any given question, the essential imponderability of its
subject matter. As a sound methodological principle underlying all
scientific research, I would suggest assigning the burden of proof to
the timorous skeptic. As for the issue of attacking one's opponents,
I will leave you with the task of tallying and evaluating the noxious
epithets on either side of this exchange, paying special attention to
the matter of casting the first stone.
cordially, | Personne n'est exempt de dire des fadaises.
mikhail zel...@husc.harvard.edu | Le malheur est de les dire curieusement.
Mikhail Zeleny
tor...@sics.se (Torkel Franzen) writes:
Some plausible grounds for rejecting (uncountable) AC are constituted
by the considerations of descriptive set theory, supporting the Axiom
of Determinacy. See Martin's article in the Handbook of Mathematical
Logic, and Martin and Kechris in _Analytic Sets_.
Set theory without the Axiom of Extensionality has been discussed by
Sol Feferman and Nicholas Goodman. Naturally, the salient question is
whether such developments are analytic of the intuitive notion of set.
Michael Siemon
>Okay, so you and Gene believe that "one can't get more certain than a
>convention" is true *at least* of mathematical statements. I requested
No, fergawssake. One can't get more certain than one is of a statement
that embodies mathematical convention. Why must you insist on misreading
this? There are plenty of mathematical statements that are NOT conven-
tions -- and may not even be true :-). My note explicitly mentioned,
in contrast to conventional statements, theorems with long (and maybe
fallacious) proofs, which we hold to be true, and matters (e.g. the
FLT) which have what appears to be some kind of empirical support (if
that notion makes any sense in math, which it may or may not). Proof
is a dicey notion, with considerable cultural influence on what is and
what is not regarded as acceptable. Mathematicians and others will argue
at length about the status of mathematical "truth" and whether or not it
*is* anything more than our explicit conventions working out obscurely
in proofs.
Neither Gene nor I committed to any position on such "epistemological"
matters -- partly because the epistemology is quite irrelevant to (most)
mathematical practice.
>an explication of the epistemological basis for the aforementioned
>proposition.
The "epistemological basis" is that conventions are rules, in Wittgenstein's
sense. That *should* be too obvious to need pointing out, but it is MZ's
attempt to bluster and obscure the obvious which should require something
more in the way of justification than pseudo-authoritative citing of logic
in ways that simply don't apply to the case.
>Zeleny, if I recall correctly (my apologies to MZ if I
>recall incorrectly), wanted the same thing. Neither you nor Gene have
>given us anything but cliche and bluster.
Go back to Gene's article. He gave several explicit examples of the kind
of usage in mathematics, particularly with regard to mathematical under-
standing of (and rules about) numbers, which operate as sufficient counter-
example to Zeloony's attempts to lay down laws about things he knows not.
If you do not understand the examples, nor understand WHY counterexamples
are determinative in this matter, you could ask about THEM. Instead you
are going on about irrelevancies, as if there were a Zeloony-like attempt
to hint obscurely at grandiose epistemological schemes. Not so; Gene was
simply pointing out that the Green Machine grinds out nonsense when seen
against actual mathematics.
--
Michael L. Siemon I say "You are gods, sons of the
m...@panix.com Most High, all of you; nevertheless
- or - you shall die like men, and fall
m...@ulysses.att..com like any prince." Psalm 82:6-7
Bill Taylor
|> There is no
|> need to speak about there being "a correct decision" in the case of the
|> Axiom of Choice: it is simply true on the intended interpretation.
This is a bit high-handed ! Presumably the "intended interpretation" here is
that of the cumulative hierarchy; the sets built up from phi (or N) by powerset
iterated suitably often. Almost everyone would agree that extensionality,
foundation, powerset, replacement apply to this; and that these are thus true
"analytically". But the same can hardly be said for choice. There is not much
in the concept of these iterations that suggests choice should hold automatically,
as the other axioms do.
Of course, most mathematicians agree (often without great thought) that it IS
true in this model, but as a fact, rather than "by definition", as suggested
above. And a substantial minority would have grave doubts, even think otherwise.
Indeed, the very fact that AC is still very often mentioned as a hypothesis
to any important theorem depending essentially on it, shows that it is very
far from being regarded as true analytically, even by those who accept its actual
truth. As Bishop said,
"AC is unique in its ability to trouble the conscience of the ordinary
mathematician".
---
elsewhere, Mikhail Zeleny says...
|> As a sound methodological principle underlying all
|>scientific research, I would suggest assigning the burden of proof to
|>the timorous skeptic.
I'm not sure which remark this was a reply to, nor why the disparaging word
"timorous" has been applied, but it doesn't really sound like a very defensible
statement in itself. Surely the whole thrust of scientific research is that
a reasonable skepticism must always be maintained (whether timorous or not); and
that the burden of proof, (or at least substantial evidence), is always on those
who make positive assertions.
Whether the same applies to math as it does to science is more questionable,
but it is probably a reasonable starting point.
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
The intuitionist confuses knowledge with truth;
The constructivist confuses ignorance with impossibility.
-------------------------------------------------------------------------------
Chris Menzel
: Set theory without the Axiom of Extensionality has been discussed by
: Sol Feferman and Nicholas Goodman. Naturally, the salient question is
: whether such developments are analytic of the intuitive notion of set.
And by Gilmore somewhat before Feferman in his seminal 1974 paper "The
consistency of partial set theory w/o extensionality" ({\it Axiomatic
Set Theory, Symposia in Pure Math XIII, Part II, American Math
Society, 147-153). I believe Gilmore's paper was the motivation
behind Feferman's initial forays into this area (if memory serves).
Gilmore reports his efforts directed toward extending set theory in
ways that look somewhat more Fregean (as I read him); the result ended
up being inconsistent with extensionality, suggesting that the
intuitions at work were more in line with *property* theory rather
than set theory. The same thing happened to Maddy with her theory of
proper classes (JSL early 1980's).
--
Christopher Menzel Internet -> cme...@tamu.edu
Philosophy, Texas A&M University Phone ----> (409) 845-8764
College Station, TX 77843-4237 Fax ------> (409) 845-045
Michael Siemon
(Robert Vienneau) writes:
>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
>>MLS:
>>>... Gene's usage is completely common in mathematics. It is
>>>"convention" when we say 1! = 1. We have TOTAL certainty about this, as
>>>we have arbitrarily MADE it so.
>>This is rich. So far, I have only seen Leon Trotsky and Alfred
>>Korzybski advance the thesis that it is not the case that 1 = 1,
et bleeding cetera ...
>I can never tell when Mr. Z is being ironical, but I assumed 1! = 1 was
>part of the series:
> 0! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, ...
>Can you go on now?
I am not sure if Zeleny is capable of irony. I have certainly never
noted any in his posts (but then, I seldom give them much attention.)
It certainly seemed to me clear that I was refering to the factorial
function (I understand that Whitehead was given to naming this "shriek"
as against our current barbarous pronunciation as "bang"). In any
case, Zeleny can hardly parse a line, as the space between '!' and '='
rules out the interpretation he attempts to put on my statement. Some-
how, this does not surprise me -- he is little prone to accuracy in
reading or in thought.
>>> It is not a matter of empirical discovery
>>>or of extended (and hence possibly fallacious) proof. It is merely so,
>>>because we SAY it is so. This sense of "convention" is common in math,
>>>but not limited to it: conventions in contract bridge bidding have much
>>>the same character.
>>The cheesy doctrine adumbrated above is known as formalism. Since it
No. One *might* get to formalism if one were to go beyond the position
I sketched to the (to my mind excessive) claim that *all* mathematical
statement was of this character. As a not-entirely-reconstructed Platonist,
I am unwilling so to assever.
>>would be impossible for me to add anything to Gottlob Frege's famous
>>dissection of Thomae in _The Foundations of Arithmetic_...
May I point out how characteristic of sieur slime (and irrelevant) this
is?
>>To say that the questions of usage exhaust the issue of meaning
>>constitutes a highly partisan move in the philosophy of mathematics...
More irrelevance. When Zeleny's "meaning" is blatantly contradicted,
he takes refuge in idiotic charges. I haven't the faintest notion what
might "exhaust" the issue of meaning in mathematics, save that our dear
net.fool has little trace of understanding any of it.
>MLS, inasmuch as he is advocating a philosophical position at all,
>sounds more like Wittgenstein than Hilbert to me. He draws an anology
I am indeed more sympathetic to W. (qua philosopher) than to H. Z. may
be inclined to attribute both my sympathies and W.'s philosophy to that
sexual inclination which he so mindlessly loathes. Sort of like Hitler's
dislike of "Jewish" art and physics :-)
>to a game, refuses to admit that statements about mathematics are to be
>analyzed as if they were in a mathematical formalism themselves, talks
>about language use in a way that others read as asserting that meaning
>is use, opposes metaphysics...
My goals are far less ambitious than you attribute to me. I am greatly
interested in "foundations" (though not terribly well-informed about the
current state of this rather recherche endeavor -- I rely on sane voices
like Torkel Franzen to keep me at least minimally clued in.) It is not
the case that mathematical practice PROVES anything, metaphysically (the
_Meno_ notwithstanding :-)) or epistemologically. Rather, some aspects
of the practice are sufficent to call a lie when armchair philosophasters
like Z. maunder on in their ignorance.
>I wonder if MLS would agree with the proposition that metamathematics in
>the style of Hilbert and Godel is just more mathematics. Working
Of course it is "more mathematics." I'm not sure what sense is to be
given to "just" in this case -- not all mathematics is "the same thing"
as other math, so far as I can see. The endless divergences of analysts,
algebraists and geometers point to at least *apparent* dissimilarities
(I write as a geometer with some algebra, to whom analysis is mostly a
mystery.)
Number theory is a more interesting realm to test the interpenetration of
logic or metamathematics and "real" (pardon the tendentious phrasing :-))
math. Torkel, for one, has expounded on the manner in which Goedel's
results and those derived from them have curious *arithmetic* consequence
for Diophantine equations -- than which it is hard to get more traditional
as mathematics goes, as there are traces of this stuff even in Babylonian
and Egyptian sources, long before Diophantos himself took up the study.
If Zeleny would limit himself to starting various hares along the race
course of mathematico-philosophical inquiry, he'd have no opposition from
me. Instead, he insists on spouting pseudo-profound nonsense, and then
blustering when called on it.
>mathematicians need not treat such mathematics as of any more importance
>than any other branch. It's not beneath or more basic than the rest of
>mathematics.
The "important" branches of mathematics are, quite simply, the ones *I*
want to know about. :-) A definition that changes over time.
m...@waikato.ac.nz
>
. . .
> elsewhere, Mikhail Zeleny says...
>
> |> As a sound methodological principle underlying all
> |>scientific research, I would suggest assigning the burden of proof to
> |>the timorous skeptic.
>
> I'm not sure which remark this was a reply to, nor why the disparaging word
> "timorous" has been applied, but it doesn't really sound like a very defensible
> statement in itself. Surely the whole thrust of scientific research is that
> a reasonable skepticism must always be maintained (whether timorous or not); and
> that the burden of proof, (or at least substantial evidence), is always on those
> who make positive assertions.
>
> Whether the same applies to math as it does to science is more questionable,
> but it is probably a reasonable starting point.
>
It seems that the first rule of debate is 'saddle your opponent with the
burden of proof' ! I have never warmed to either of the approaches which
go either
"if you don't accept XXX you are beyond the pale"
or
"I'm standing here with my arms folded and you're not
going to convince me of anything".
That's why I was delighted to find the quote in my sig.
> -------------------------------------------------------------------------------
> Bill Taylor w...@math.canterbury.ac.nz
> -------------------------------------------------------------------------------
> The intuitionist confuses knowledge with truth;
> The constructivist confuses ignorance with impossibility.
> -------------------------------------------------------------------------------
>
Speaking of sigs this seems like a pretty lofty dismisal untypical
of Bill. From what unassailable platform do you cast these jibes?
--
Murray A. Jorgensen [ m...@waikato.ac.nz ] University of Waikato
Department of Mathematics and Statistics Hamilton, New Zealand
----------------------------------------------------------------------------
Douter de tout ou tout croire, ce sont deux solutions e'galement commodes,
qui l'une et l'autre nous dispensent de re'fle'chir. Henri Poincare'
Mikhail Zeleny
> ...bluster and obscure the obvious...
> ...pseudo-authoritative citing of logic...
> ...Zeloony's attempts to lay down laws about things he knows not...
> ...characteristic of sieur slime...
> ...net.fool...
> ...armchair philosophasters
>like Z. maunder on in their ignorance.
That last bit is my personal favorite.
So evocative of the First Meditation.
>If Zeleny would limit himself to starting various hares along the race
>course of mathematico-philosophical inquiry, he'd have no opposition from
>me. Instead, he insists on spouting pseudo-profound nonsense, and then
>blustering when called on it.
Methinks Mr Siemon vastly overestimates the cognitive value of his
flatulent invective. On which see below.
>--
>Michael L. Siemon I say "You are gods, sons of the
>m...@panix.com Most High, all of you; nevertheless
> - or - you shall die like men, and fall
>m...@ulysses.att..com like any prince." Psalm 82:6-7
cordially, | Personne n'est exempt de dire des fadaises.
Mikhail Zeleny
cme...@kbssun1.tamu.edu (Chris Menzel) writes:
>Mikhail Zeleny (zel...@husc10.harvard.edu) wrote:
>>Set theory without the Axiom of Extensionality has been discussed by
>>Sol Feferman and Nicholas Goodman. Naturally, the salient question is
>>whether such developments are analytic of the intuitive notion of set.
>And by Gilmore somewhat before Feferman in his seminal 1974 paper "The
>consistency of partial set theory w/o extensionality" ({\it Axiomatic
>Set Theory, Symposia in Pure Math XIII, Part II, American Math
>Society, 147-153). I believe Gilmore's paper was the motivation
>behind Feferman's initial forays into this area (if memory serves).
Shame on me for omitting the most accessible source. I keep thinking
of something Dana Scott was said to have shown in the Sixties, but
fail to come up with an actual reference.
>Gilmore reports his efforts directed toward extending set theory in
>ways that look somewhat more Fregean (as I read him); the result ended
>up being inconsistent with extensionality, suggesting that the
>intuitions at work were more in line with *property* theory rather
>than set theory. The same thing happened to Maddy with her theory of
>proper classes (JSL early 1980's).
I agree on distinguishing properties from sets. In any event, were
Feferman's results actually published, or even developed beyond a
proposal stage?
>--
>
>Christopher Menzel Internet -> cme...@tamu.edu
>Philosophy, Texas A&M University Phone ----> (409) 845-8764
>College Station, TX 77843-4237 Fax ------> (409) 845-045
cordially,
Mikhail Zeleny
w...@math.canterbury.ac.nz (Bill Taylor) writes:
>In article <TORKEL.93...@anhur.sics.se>,
>tor...@sics.se (Torkel Franzen) writes:
>> There is no
>>need to speak about there being "a correct decision" in the case of the
>>Axiom of Choice: it is simply true on the intended interpretation.
>This is a bit high-handed ! Presumably the "intended interpretation"
>here is that of the cumulative hierarchy; the sets built up from phi
>(or N) by powerset iterated suitably often. Almost everyone would
>agree that extensionality, foundation, powerset, replacement apply to
>this; and that these are thus true "analytically".
I believe you are about to incur the wrath of Randall Holmes.
> But the same can
>hardly be said for choice. There is not much in the concept of these
>iterations that suggests choice should hold automatically, as the
>other axioms do.
I heard it said that AC is true in the cumulative hierarchy in virtue
of the arbitrary nature of set membership, whatever that might be.
>Of course, most mathematicians agree (often without great thought)
>that it IS true in this model, but as a fact, rather than "by
>definition", as suggested above. And a substantial minority would have
>grave doubts, even think otherwise. Indeed, the very fact that AC is
>still very often mentioned as a hypothesis to any important theorem
>depending essentially on it, shows that it is very far from being
>regarded as true analytically, even by those who accept its actual
>truth. As Bishop said,
>"AC is unique in its ability to trouble the conscience of the ordinary
>mathematician".
On the other hand, some of the exquisitely guilty pleasures to be
found in egregiously non-constructive reasoning, may be readily
encountered in Comfort's _Theory of Ultrafilters_ (has anyone got a
spare copy?).
>---
>
>elsewhere, Mikhail Zeleny says...
>>As a sound methodological principle underlying all
>>scientific research, I would suggest assigning the burden of proof to
>>the timorous skeptic.
>I'm not sure which remark this was a reply to, nor why the disparaging
>word "timorous" has been applied, but it doesn't really sound like a
>very defensible statement in itself. Surely the whole thrust of
>scientific research is that a reasonable skepticism must always be
>maintained (whether timorous or not); and that the burden of proof,
>(or at least substantial evidence), is always on those who make
>positive assertions.
When taken out of context of the question of whether such issues as
the truth of the CH, or the nature of the real numbers, can ever be
definitively settled, my statement certainly is not meant to be taken
at face value. However I insist that assuming that every issue in
mathematics should be regarded as potentially tractable, until proven
otherwise, is a workable, nay, desirable starting hypothesis.
>Whether the same applies to math as it does to science is more
>questionable, but it is probably a reasonable starting point.
I would be most reluctant to judge mathematics by the standards of
empirical science, and vice versa, Polya's arguments notwithstanding.
>-------------------------------------------------------------------------------
> Bill Taylor w...@math.canterbury.ac.nz
>-------------------------------------------------------------------------------
> The intuitionist confuses knowledge with truth;
> The constructivist confuses ignorance with impossibility.
>-------------------------------------------------------------------------------
>
cordially,
Torkel Franzen
(Bill Taylor) writes:
>This is a bit high-handed !
No highhandedness was intended; I was merely elliptically assuming
an interpretation on which the axiom of choice is true, since the
suggestion was that it is hard to understand how anybody (specifically
Godel) could take the axiom of be true although it is independent of the
remaining axioms. I agree that the cumulative conception does not in
itself make the axiom convincing.
Ian Sutherland
>true in this model, but as a fact, rather than "by definition", as suggested
>above. And a substantial minority would have grave doubts, even think
>otherwise.
This is something one sees asserted every once in a while. My
experience tends to suggest that this minority is not very
substantial. Does anyone have any idea how substantial this minority
actually is? I'm not saying this has anything to do with mathematical
truth by the way, just curious.
>Indeed, the very fact that AC is still very often mentioned as a hypothesis
>to any important theorem depending essentially on it, shows that it is very
>far from being regarded as true analytically, even by those who accept its actual
>truth.
I don't think this shows that AC is not regarded as true any more than
mathematicians trying to find constructive proofs means they doubt the
principle of the excluded middle. It's just parsimony.
--
Ian Sutherland
i...@eecs.nwu.edu
Sans Peur
Herman Rubin
>: Set theory without the Axiom of Extensionality has been discussed by
>: Sol Feferman and Nicholas Goodman. Naturally, the salient question is
>: whether such developments are analytic of the intuitive notion of set.
The poster gives some slightly earlier examples, but the use of these
is quite old. All of the early models of the independence of the Axiom
of Choice, from the 1920s until the 1950s, used individuals which are
not sets. These models are rather easy to construct and use, and it
is not difficult to prove relative consistency.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hru...@snap.stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
john baez
>yIn article <C6FBo...@cantua.canterbury.ac.nz> w...@math.canterbury.ac.nz
>(Bill Taylor) writes:
[Franzen writes that the axiom of choice is obviously true in the
intended interpretation.]
> >This is a bit high-handed !
>
> No highhandedness was intended; I was merely elliptically assuming
>an interpretation on which the axiom of choice is true...
I almost feel like at chiding Franzen for having adopted my
wise-owl mannerisms and attempting to mystify the layfolk with his
original statement... even though I agreed with it. (Come on, the
product of lots of nonempty sets is obviously nonempty, right?)
m...@waikato.ac.nz
>
. . .
> On the other hand, some of the exquisitely guilty pleasures to be
> found in egregiously non-constructive reasoning, may be readily
> encountered in Comfort's _Theory of Ultrafilters_ (has anyone got a
> spare copy?).
>
Was that 'The Joy of Ultrafilters' ? Let me suggest that trying to
prove anything about these elusive wraith-like beings is enough to
make a constructivist of anyone.
. . .
Robert Vienneau
admit that my knowledge of set theory and logic is almost entirely pop.
I have always wondered when applying statistics whether the existence of
nonmeasurable sets should give me pause or affect my practice in the
slightest.
I freely concede Torkel Franzen's point that more than independence from
the remaining ZF(C) axioms is needed to make the truth or falsity of an
axiom troublesome for the Platonist. Maybe somebody more competent than
I can coherently argue that the many ways ZFC can be extended do create
a problem for an interpretation of mathematics as exploring some already
existing world of abstact entities. I was under the impression that AC
and GCH provided just some of the consistent ways ZF could be extended.
And does the L...(?)-Skolem theorem have any bearing on the subject of
mathematical truth?
That said, feel free to continue the discussion. I'll just read for
awhile.
Torkel Franzen
unc.edu (Robert Vienneau) writes:
>Maybe somebody more competent than
>I can coherently argue that the many ways ZFC can be extended do create
>a problem for an interpretation of mathematics as exploring some already
>existing world of abstact entities.
The observation that the axiom of choice is evidently true does not
necessarily have anything to do with Platonism. It is quite sufficient
to say that as we imagine the world of sets, choice sets clearly
exist. Whether or not this world of sets has any existence outside
our imagination is irrelevant.
In the case of a question where no answer is evident, such as the
continuum problem, there is of course no reason, on such a view, why
the question must have any answer. But, as Godel emphasized, for the
continuum problem to be meaningful, it is not necessary to take a
Platonistic view.
Neil Rickert
>necessarily have anything to do with Platonism. It is quite sufficient
>to say that as we imagine the world of sets, choice sets clearly
>exist. Whether or not this world of sets has any existence outside
>our imagination is irrelevant.
>
> In the case of a question where no answer is evident, such as the
>continuum problem, there is of course no reason, on such a view, why
>the question must have any answer. But, as Godel emphasized, for the
>continuum problem to be meaningful, it is not necessary to take a
>Platonistic view.
One can take a somewhat different view on this, albeit one that some
mathematicians might find troubling.
The axiom of choice leads to proofs of many very strong, very useful,
and very important theorems. These theorems have proved their value
when applied to practical applications. On the other hand, most of the
results that have been proved assuming the continuum hypothesis are
rather obscure and with little useful application.
Because of this, one might say that there is strong empirical evidence
in favor of the axiom of choice, but there is not enough evidence to
warrant taking a position on the continuum hypothesis. Or, to put it
more crudely, if we reject the axiom of choice, many mathematicians
might have to find work driving a taxi, but rejection of CH carries no
such penalty.
Angus H Rodgers
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YAY! Zeleny's back!
I haven't seen such a pretty '>' pattern for months.
I'm sure there's a coded message in there somewhere ...
--
Gus Rodgers, Dept. of Computer Science, Queen Mary & Westfield College,
Mile End Road, London, England +44 71 975 5241 arod...@dcs.qmw.ac.uk
Daniel P Heyman
>In article <TORKEL.93...@isis.sics.se> tor...@sics.se (Torkel Franzen) writes:
>>
>> The observation that the axiom of choice is evidently true does not
>>necessarily have anything to do with Platonism. It is quite sufficient
>>to say that as we imagine the world of sets, choice sets clearly
>>exist. Whether or not this world of sets has any existence outside
>>our imagination is irrelevant.
>>
>mathematicians might find troubling.
>
>The axiom of choice leads to proofs of many very strong, very useful,
>and very important theorems. These theorems have proved their value
>when applied to practical applications. On the other hand, most of the
>results that have been proved assuming the continuum hypothesis are
>rather obscure and with little useful application.
>
of AC, but I don't think it can be called "evidently true". When I
was a grad student I roomed with a fellow who was a student of Tarski.
He (my room mate) told me that Tarski told him that the motivation of
the Banach-Tarski theorem was to prove something so physically outlandish that
mathematicians would deny AC. They were quite miffed that their theorem
had just the opposite effect.
--
Dan Heyman d...@bellcore.com
Michael Weiss
mathematicians might find troubling.
The axiom of choice leads to proofs of many very strong, very useful,
and very important theorems. These theorems have proved their value
when applied to practical applications. On the other hand, most of the
results that have been proved assuming the continuum hypothesis are
rather obscure and with little useful application.
Because of this, one might say that there is strong empirical evidence
in favor of the axiom of choice, but there is not enough evidence to
warrant taking a position on the continuum hypothesis. Or, to put it
more crudely, if we reject the axiom of choice, many mathematicians
might have to find work driving a taxi, but rejection of CH carries no
such penalty.
In fact, Goedel made just this argument (I mean the less crude version, not
the taxi-cab) in his philosophical article on the Continuum Hypothesis.
The title is (if memory serves) "What is Cantor's Continuum Hypothesis?",
and it is reprinted in Bennaceraf and Putnam's anthology "The Philosophy of
Mathematics".
Goedel argued for a strong Platonist position. I am rather mystified at
Torkel's position-- he seems to be saying that the axiom of choice is true
in a mathematical universe that exists in our imagination, but perhaps only
there. This raises two questions: first, what is the difference between
saying the universe of sets "exists in our imagination" and "exists 'out
there'"; second, who's to say everyone's imaginary universe is the same?
Torkel Franzen
(Daniel P Heyman) writes:
>He (my room mate) told me that Tarski told him that the motivation of
>the Banach-Tarski theorem was to prove something so physically outlandish
>that mathematicians would deny AC.
To be sure, the idea that general point sets have some "physical"
significance is one that we have every reason to reject. But why should
this tell against the axiom of choice?
Torkel Franzen
think.com (Michael Weiss) writes:
>Goedel argued for a strong Platonist position.
This is putting it rather too strongly, since Godel emphasized in
the very article you refer to that the existence of our our
set-theoretical intuitions suffices to give meaning to the continuum
problem, whether or not there is any Platonist universe of sets.
>This raises two questions: first, what is the difference between
>saying the universe of sets "exists in our imagination" and "exists 'out
>there'";
Just my point. As far as the justification of the axiom of choice is
concerned, it makes no difference whether the universe of sets is a figment
of my imagination or not. Whatever universes of sets may exist, the one I
am talking about is one in which the axiom of choice is true. And whether
or not that universe has any existence outside my imagination, I can give
no other justification for the axiom than its agreeing with my conception
of what sets are.
>second, who's to say everyone's imaginary universe is the same?
Obviously they are not, since some people say they don't think the
axiom of choice is evident. Nevertheless it is a striking fact that
so many people agree in their fantasies.
Michael Weiss
In article <COLUMBUS.9...@strident.think.com> columbus@strident.
think.com (Michael Weiss) writes:
>Goedel argued for a strong Platonist position.
This is putting it rather too strongly, since Godel emphasized in
the very article you refer to that the existence of our our
set-theoretical intuitions suffices to give meaning to the continuum
problem, whether or not there is any Platonist universe of sets.
As I recall the article, Goedel held that we have an imperfectly developed
intuition about the "real" universe of sets, and that we could hope to
strengthen this intuition to the point where we might be able to determine
the true value of 2^aleph_0. But I should reread the article, before
I claim any more.
>second, who's to say everyone's imaginary universe is the same?
Obviously they are not, since some people say they don't think the
axiom of choice is evident. Nevertheless it is a striking fact that
so many people agree in their fantasies.
Alas, this argument is quite vulnerable to the cultural counter-argument.
The small flurry of activity that flared up around AD a number of years ago
demonstrates, I think, that mathematicians will go where the theorems are.
Tying "reality" to fruitfulness is an interesting idea, but I'm not sure
I'm convinced. Are linear operators more real than non-linear operators,
because the theory of linear operators is better developed?
Tal Kubo
>and very important theorems. These theorems have proved their value
>when applied to practical applications. On the other hand, most of the
>results that have been proved assuming the continuum hypothesis are
>rather obscure and with little useful application.
>
Can you give examples of such theorems? Clearly the axiom of choice
has no "useful application" in the sense of assisting computations.
So you must mean that one's theories come out looking nicer because of
AC -- e.g. that certain statements can be made in complete generality,
or the collections of objects considered come out having nicer categorical
properties. I'm skeptical, but await further enlightenment.
Neil Rickert
>In article <1993May4.1...@mp.cs.niu.edu>
>ric...@mp.cs.niu.edu (Neil Rickert) writes:
>>
>>The axiom of choice leads to proofs of many very strong, very useful,
>>and very important theorems. These theorems have proved their value
>>when applied to practical applications. On the other hand, most of the
>>results that have been proved assuming the continuum hypothesis are
>>rather obscure and with little useful application.
>Can you give examples of such theorems? Clearly the axiom of choice
>has no "useful application" in the sense of assisting computations.
Frequently AC is used indirectly, typically in the form of Zorn's lemma.
There are numerous uses in point set topology, functional analysis,
algebra. If I recall correctly the Hahn-Banach theorem is one example.
It is basic to functional analysis, and functional analysis in turn
leads to results useful in physics.
Yes, it is possible to get many of the useful results, or at least
specialized versions of them sufficient for applications to physics,
without first developing functional analysis. But it would be much
harder, and they might not have all been discovered due to the lack of
an effective framework.
m...@waikato.ac.nz
discussion as they emerged from a chapel. One is saying to the
other
'Sects, sects, sects - that's all you ever talk about!'.
I had a political science friend who would change this line to
'Sets, sets, sets - that's all you mathematicians ever talk about!'
Tal Kubo
>>>
>>>The axiom of choice leads to proofs of many very strong, very useful,
>>>and very important theorems. These theorems have proved their value
>>>when applied to practical applications. On the other hand, most of the
>>>results that have been proved assuming the continuum hypothesis are
>>>rather obscure and with little useful application.
>
>>Can you give examples of such theorems? Clearly the axiom of choice
>>has no "useful application" in the sense of assisting computations.
>
>Frequently AC is used indirectly, typically in the form of Zorn's lemma.
>There are numerous uses in point set topology, functional analysis,
>algebra. If I recall correctly the Hahn-Banach theorem is one example.
>It is basic to functional analysis, and functional analysis in turn
>leads to results useful in physics.
Why not just claim that AC is basic to set theory, sets are basic to math,
math is basic to physics, ergo AC is basic to physics? Again, I'd like to
see *any* computation in physics or elsewhere in which AC-dependent
theorems are helpful (even heuristically).
>Yes, it is possible to get many of the useful results, or at least
>specialized versions of them sufficient for applications to physics,
>without first developing functional analysis. But it would be much
>harder, and they might not have all been discovered due to the lack of
>an effective framework.
Are you claiming that AC is a necessary heuristic crutch for development of
functional analysis applicable to physics? As the lady from Missouri said,
"show me". Given the strong claims put forth in your paragraph quoted at
the top of this message, I assume that you will have no trouble finding
better examples than Hahn-Banach.
Tal Kubo
ku...@math.harvard.edu
Torkel Franzen
>But I should reread the article, before I claim any more.
Actually I haven't read it recently either, so let me modify my
comment: Godel's thinking about these things is, as Hao Wang
says, "intricate".
>Alas, this argument is quite vulnerable to the cultural counter-argument.
>The small flurry of activity that flared up around AD a number of years ago
>demonstrates, I think, that mathematicians will go where the theorems are.
I'm not sure what argument you have in mind. I didn't intend any
argument myself, only an observation: it is a striking fact that so
many people agree that e.g. the axiom of choice accords with their
conception of the world of sets. But I see that my comments were not
very clear. My point is only this. In my understanding and my
justification of set theory I don't invoke any objective
reality of sets, and indeed I don't see what use I could possibly make
of the assumption that there is such a reality. That P(R) (the set of
sets of real numbers) or P(P(R)) exists is not a strange metaphysical
assumption, but a pice of mathematical fantasy, if you like, which
appeals to me because I can argue within it in a coherent and rewarding
mathematical way, guided by simple pictures and intuitions. If
somebody claims that these supposed totalities are inconceivable or
meaningless, I don't exhort him to conceive more vigorously or try to
prove to him that what I'm saying makes sense. Rather, I accept that
some people don't feel at home with these concepts, but prefer other
kinds of mathematics. Cantor spoke of pure mathematics as "free
mathematics": mathematics in which we are free to study whatever
concepts and ideas our minds can come up with, without being bound by
any particular set of principles regarding what makes mathematical
sense.
Similarly, if somebody asks me to justify the axiom of choice, what
I do is essentially to invoke ideas such as "in the full powerset,
every possibility of making a choice is realized as a set", "I can
imagine picking an element from each set in the collection". Clearly
there is nothing either profound or compelling about this. If somebody
says that he can imagine this only if the set is finite, or just can't
see what I'm talking about, I'm stumped. From my point of view,
however, the axiom is extraordinarily satisfying: it is the only
principle I have which allows me to exploit explicitly the conception
of sets as extensional totalities independent of all constructions and
definitions, and in the light of this conception it is a principle
highly "pleasing to the mind". And it so happens that many people
agree with me.
However, mathematics is not all pure or free. Mathematics makes
contact with matters where we either have other criteria than those of
pure mathematics for the acceptability of a particular conclusion, or
link the meaning of mathematical statements to matters that are not
mathematical - the use of mathematical models in astronomical
navigation or in building bridges, the mathematics of the termination
or complexity of algorithms, and so on. Hence those very interesting
philosophical questions, such as the question of the nature of the
link between our pure mathematical fantasies (large infinite sets) and
our knowledge of computational matters.
Michael Weiss
much more lucid and appealing than Goedel's arguments that I won't say much
more about it. One of Goedel's proposals became obsolete after Cohen's
results; others are expressed in a needlessly obscure fashion.
We seem to agree that mathematics consists of following the consequences of
mathematical intuitions; that formal axiom systems codify such intuitions;
that extra-mathematical criteria often play a role; and that a belief in the
"objective reality" of mathematical objects is not a prerequisite for this
style of mathematics.
By the "cultural argument" I mean simply the proposal that mathematics is a
cultural activity, and cultural factors play a very large role in
determining what sort of math is considered significant. Applied to AC,
this says that most mathematicians accept AC primarily because it buys you
little or nothing to reject it, and you gain a wealth of nice results if
you admit it.
The cumulative hierarchy does provide a pleasant conceptual foundation for
ZFC, but I dare say that ZF+AD, or NFU, would come to seem equally natural
if one worked with either of them for a while.
Simen Gaure
> Goedel argued for a strong Platonist position. I am rather mystified at
> Torkel's position-- he seems to be saying that the axiom of choice is true
> in a mathematical universe that exists in our imagination, but perhaps only
> there. This raises two questions: first, what is the difference between
> saying the universe of sets "exists in our imagination" and "exists 'out
> there'"; second, who's to say everyone's imaginary universe is the same?
Torkel's position follows closely the 'fundamental' idea of modern
mathematics.
The objects we are studying in mathematics are only required to
exist within our imagination. It's not a very old idea in mathematics,
maybe about a hundred years or so. The ancient greeks ran into
problems because they required numbers to be geometrically
constructible with compass and ruler. Negative numbers
(which in a sense only exist in our imagination) wasn't really
accepted until a few hundred years ago. Complex numbers a bit earlier(!)
Modern mathematics is exclusively concerned with
imaginary objects. All the rest is physics, chemistry, economics,
biology, computer science, psychology, sociology & c. These are
some of the sciences where mathematics is used, and where the
physical interpretation of the mathematical ideas are important.
Mathematics itself doesn't exist outside the mind of the
mathematician. The objects being studied are defined by
exact definitions (at least we like to think so) and the
rules for manipulating the objects are strictly defined.
With no reference to the physical world.
Of course, one is allowed to think in terms of physical
objects, but mathematics as such doesn't refer to it.
(The mathematician, however, exists in the
physical world, and most mathematicians take some 'physical'
facts for granted (e.g. the pigeon hole principle), but these
concepts may also be deduced from simpler axioms.)
The reasons for taking this position are many and often
pragmatic in nature.
One mathematical object may have many different physical
interpretation. I.e. a differential equation may model
a falling body, the growth of a population of flies,
monetary flow in a stock exchange & c. None of these
interpretations have direct relevance to the study of
the equation, even though they may be helpful examples.
Similarly, a group may be interpreted as actions taken
on Rubik's cube, but the study of groups doesn't depend
on any such interpretation.
When one doesn't require a physical interpretation one isn't
limited by the physical universe. This turns out to be useful.
A central notion in several branches of mathematics is the notion of infinity.
This notion can hardly be physically realized at all in this universe.
Still it is very useful, not only for studying 'infinite'
phenomena, but even for analyzing mathematical objects which
do have physical interpretations.
Many mathematicians don't even regard applicability of their
results to the physical world to be very interesting, others do.
Several of these useful examples have lead mathematicians to throw away
the physical world from the world of mathematics.
Mathematicians often think in terms of physical objects, but
the mathematical objects studied are not these.
The (often imaginary) physical objects are only images of the ideas.
Of course, everyone's imaginary universe are not the same. But
one is required to follow the definitions(axioms) agreed upon.
If you for some reason don't accept other people's definitions of
an object, you either have to prove that it is inconsistent or else
you may choose to study some other mathematical object.
It doesn't invalidate the first definition. Everyone may study
whatever mathematical object they like,
but some are more interesting (to some) than others.
The Axiom of Choice has primarily interest in the infinite
mathematical world because it's one of the most useful tools available there.
In the finite world it is by most people regarded as trivial.
You may accept the axiom of choice or you may not. But since
the interest is mainly in infinite sets one is excluded from
any physical interpretation. That is to say, 'exists in our imagination'.
No mathematician would care if you don't accept it.
The truth-value of axioms are not
discussed in mathematics, although one may investigate whether
they are inconsistent, in which case it's no longer an axiom.
I'm sure there is a lot more to be said about this subject, I may
not even speak for every mathematician in the world, but I do believe
that I'm basically correct.
(I may also have mis-interpreted the questions, please read no
further if I've done so.)
Simen Gaure
University of Oslo
Simen Gaure
> functional analysis applicable to physics? As the lady from Missouri said,
> "show me". Given the strong claims put forth in your paragraph quoted at
> the top of this message, I assume that you will have no trouble finding
> better examples than Hahn-Banach.
Hmm, I don't know physics very well, but I do know a little
operator theory. I do believe that modern physics use
mathematics related to my field, although I don't know how,
and I'm not particularly interested in it either.
In development of the theory of e.g. projection theory in von
Neumann algebras, which is closely related to functional
analysis, Zorn's lemma is used heavily, both in proofs
similar to Hahn-Banach and in other theorems.
You may probably develop your specific (countable?) theory without reference
to general theory, but in some instances you'll get a better
picture if you look at it in a general setting.
It wouldn't surprise me if some ideas useful in physics have
originated in this way, as many ideas have originated in physics
and found it's way into mathematics. It isn't easy to answer
your demand "show me", ideas tend to originate in strange places
at odd times. It may be hard to tell whether the axiom of choice
it directly related to your functional analysis.
The awareness of something like the axiom of choice, I'm sure, have
stimulated much thinking about the subject, it might happen that some
of the thinking have had results which are crucial to functional
analysis. You may 'clean' the development afterwards, but the ideas
might simply not have evolved without abstract mathematics.
Simen Gaure
>> Torkel's position-- he seems to be saying that the axiom of choice is true
>> in a mathematical universe that exists in our imagination, but perhaps only
>> there. This raises two questions: first, what is the difference between
>> saying the universe of sets "exists in our imagination" and "exists 'out
>> there'"; second, who's to say everyone's imaginary universe is the same?
> Torkel's position follows closely the 'fundamental' idea of modern
> mathematics.
> [stuff deleted]
> The truth-value of axioms are not
> discussed in mathematics, although one may investigate whether
> they are inconsistent, in which case it's no longer an axiom.
In a proud tradition of commenting my own inconsistencies:
(Well after all, I didn't comment on the following topic)
This means that Torkel's belief that the axiom of choice is
somehow 'more' true than e.g. the continuum hypothesis, is not
subject to mathematical reasoning. Thus, in this respect he's
not following any idea of modern mathematics, but still an idea,
just not a mathematical one. This is rather a discussion of
a mathematicians heuristic reasoning. There is no mathematical
reason to believe in the axiom of choice, just pragmatic ones.
Even very strong pragmatic reasons.
The fact that the axiom of choice seems to be evident, is that we
have physical experience from the finite (and thereby 'physical') case.
But finite mathematics is no more real than it's infinite counterpart.
It's just more comprehensible (and 'physical').
The axiom of choice has to be assumed there too, or proved from
simpler axioms, even if one believes it to be evident.
Most mathematicians don't care about that, I don't either.
The formal structure of finite mathematics
is exactly the same as that of the infinite mathematics,
axioms/definitions, deduction rules, proofs, theorems. Nothing more.
(Although the working practices and intuition of mathematicians may vary;
and thus the appearance of proofs.)
Matthew P Wiener
>In development of the theory of e.g. projection theory in von Neumann
>algebras, which is closely related to functional analysis, Zorn's
>lemma is used heavily, both in proofs similar to Hahn-Banach and in
>other theorems. You may probably develop your specific (countable?)
>theory without reference to general theory, but in some instances
>you'll get a better picture if you look at it in a general setting.
In general, one expects that any physical theory can be finitized as
a calculational method. As such, AC ought to be irrelevant, since
these calculations are absolute.
But the driving force of physics is not the calculations, but to find
frameworks in which to make and understand these calculations. AC is
highly relevant here.
--
-Matthew P Wiener (wee...@sagi.wistar.upenn.edu)
Mikhail Zeleny
sga...@sylow.uio.no (Simen Gaure) writes:
>> Goedel argued for a strong Platonist position. I am rather mystified at
>> Torkel's position-- he seems to be saying that the axiom of choice is true
>> in a mathematical universe that exists in our imagination, but perhaps only
>> there. This raises two questions: first, what is the difference between
>> saying the universe of sets "exists in our imagination" and "exists 'out
>> there'"; second, who's to say everyone's imaginary universe is the same?
>Torkel's position follows closely the 'fundamental' idea of modern
>mathematics.
>
>The objects we are studying in mathematics are only required to
>exist within our imagination. It's not a very old idea in mathematics,
>maybe about a hundred years or so. The ancient greeks ran into
>problems because they required numbers to be geometrically
>constructible with compass and ruler. Negative numbers
>(which in a sense only exist in our imagination) wasn't really
>accepted until a few hundred years ago. Complex numbers a bit earlier(!)
Idealism of the sort adumbrated above is certainly alien to Greek
philosophy. One small question: what is this thing you call "our
imagination"? Am I to assume that your imagination intersects or
otherwise overlaps with mine, to the extent that we both understand
the concept of S_3, or a torus? Are you prepared to stipulate, or at
least suggest identity conditions relating our individual imaginary
realms? Finally, is there really a difference of degree of "real
existence" between the successors of zero and its predecessors?
>Modern mathematics is exclusively concerned with
>imaginary objects. All the rest is physics, chemistry, economics,
>biology, computer science, psychology, sociology & c. These are
>some of the sciences where mathematics is used, and where the
>physical interpretation of the mathematical ideas are important.
This view is doomed to gloss over what Wigner called "the unreasonable
effectiveness" of mathematics in natural sciences. Why should the
study of figments of our imagination have any bearing on the laws
governing the behavior of the objects of our perception?
>Mathematics itself doesn't exist outside the mind of the
>mathematician. The objects being studied are defined by
>exact definitions (at least we like to think so) and the
>rules for manipulating the objects are strictly defined.
>With no reference to the physical world.
>Of course, one is allowed to think in terms of physical
>objects, but mathematics as such doesn't refer to it.
The fact that mathematics does not refer to the physical world, in no
way supports your contention that its objects have no existence
outside of the mind of the mathematician.
>(The mathematician, however, exists in the
>physical world, and most mathematicians take some 'physical'
>facts for granted (e.g. the pigeon hole principle), but these
>concepts may also be deduced from simpler axioms.)
>
>The reasons for taking this position are many and often
>pragmatic in nature.
>
>One mathematical object may have many different physical
>interpretation. I.e. a differential equation may model
>a falling body, the growth of a population of flies,
>monetary flow in a stock exchange & c. None of these
>interpretations have direct relevance to the study of
>the equation, even though they may be helpful examples.
>Similarly, a group may be interpreted as actions taken
>on Rubik's cube, but the study of groups doesn't depend
>on any such interpretation.
>
>When one doesn't require a physical interpretation one isn't
>limited by the physical universe. This turns out to be useful.
Ditto.
>A central notion in several branches of mathematics is the notion of infinity.
>This notion can hardly be physically realized at all in this universe.
As far as I am able to ascertain, this is still an open question.
The idealistic position you appear to suggest is at odds with the
egregiously non-constructive insight afforded by the AC. Most people
have no way of imagining choice sequences, non-principal ultrafilters,
or the Banach-Tarski decomposition of the sphere; some go as far as
suggesting that that's just the point of these objects.
>The truth-value of axioms are not
>discussed in mathematics, although one may investigate whether
>they are inconsistent, in which case it's no longer an axiom.
Surely model theory is a branch of mathematics.
>I'm sure there is a lot more to be said about this subject, I may
>not even speak for every mathematician in the world, but I do believe
>that I'm basically correct.
>(I may also have mis-interpreted the questions, please read no
> further if I've done so.)
>Simen Gaure
>University of Oslo
cordially,
Herman Rubin
>In article <1993May4.2...@mp.cs.niu.edu>
>ric...@mp.cs.niu.edu (Neil Rickert) writes:
........................
>>Frequently AC is used indirectly, typically in the form of Zorn's lemma.
>>There are numerous uses in point set topology, functional analysis,
>>algebra. If I recall correctly the Hahn-Banach theorem is one example.
>>It is basic to functional analysis, and functional analysis in turn
>>leads to results useful in physics.
>Why not just claim that AC is basic to set theory, sets are basic to math,
>math is basic to physics, ergo AC is basic to physics? Again, I'd like to
>see *any* computation in physics or elsewhere in which AC-dependent
>theorems are helpful (even heuristically).
>>Yes, it is possible to get many of the useful results, or at least
>>specialized versions of them sufficient for applications to physics,
>>without first developing functional analysis. But it would be much
>>harder, and they might not have all been discovered due to the lack of
>>an effective framework.
>Are you claiming that AC is a necessary heuristic crutch for development of
>functional analysis applicable to physics? As the lady from Missouri said,
>"show me". Given the strong claims put forth in your paragraph quoted at
>the top of this message, I assume that you will have no trouble finding
>better examples than Hahn-Banach.
Many uses of AC are in cases where an uncountable number of choices are
made. One would have a great deal of difficulty in many situations without
a countable number of choices, but the uncountable cases are not of that
great importance in applications. In measure theory a countable set of
choices is required; in many cases where AC, or lesser versions such as
the Hahn-Banach Theorem are used, it is often the case that constructive
procedures suffice. Analysis would be quite difficult without a little
more than a countable version of AC, and while there are situations where
more is useful, these are not common.
lrud...@vax.clarku.edu
inter alia:
>This view is doomed to gloss over what Wigner called "the unreasonable
>effectiveness" of mathematics in natural sciences. Why should the
>study of figments of our imagination have any bearing on the laws
>governing the behavior of the objects of our perception?
Since it's always nice to hear Mikhail's opinions, I invite him to comment on
(and if possible give the correct citation, and if necessary the corrected
quotation of, for) someone's (I think Kreisel's) question: "Why would it be
more reasonable for humans to have evolved incapable of thinking about the
world they live in?"
Lee Rudolph
Timothy Chow
>functional analysis applicable to physics? As the lady from Missouri said,
>"show me". Given the strong claims put forth in your paragraph quoted at
>the top of this message, I assume that you will have no trouble finding
>better examples than Hahn-Banach.
What about the existence of unbounded operators on an infinite-dimensional
Hilbert space? The only constructions I know rely on AC, but perhaps I am
just revealing my ignorance.
Alaoglu's theorem is usually proved using AC, but I don't know if there
are any direct applications to physics.
Finally, I believe that the uncountability of the reals relies on
countable choice. It seems like it would very difficult to develop a
satisfactory analysis without countable choice at least.
Tal Kubo
>
>In general, one expects that any physical theory can be finitized as
>a calculational method. As such, AC ought to be irrelevant, since
>these calculations are absolute.
In principle you don't even need the law of the excluded middle, as any
constructivist will confirm. The question is not whether the proof can
be sanitized to remove reference to AC, infinity, or LEM, but whether those
concepts help in organizing the calculations. Infinity and LEM seem to be
extremely useful in this respect, but I have yet to see any instance where
AC was of any use at all.
>But the driving force of physics is not the calculations, but to find
>frameworks in which to make and understand these calculations. AC is
>highly relevant here.
If it's so relevant, it should be easy to find some concrete examples of
physics calculations inspired by AC-based reasoning. With all the path
integrals and infinite-dimensional whatnot in quantum theory, one would
expect *some* example of AC being used in an essential way. I'm reminded of
Le Rochefoucauld's line about love: it's like a ghostly apparition which
everyone talks about, but nobody has ever seen. And how come set theory
isn't a graduation requirement for a physics PhD...
Tal Kubo
ku...@math.harvard.edu
Gregory McColm
>ric...@mp.cs.niu.edu (Neil Rickert) writes:
>>>>
>>>>The axiom of choice leads to proofs of many very strong, very useful,
>>>>and very important theorems. These theorems have proved their value
>>>>when applied to practical applications. On the other hand, most of the
>>>>results that have been proved assuming the continuum hypothesis are
>>>>rather obscure and with little useful application.
>>
>>>Can you give examples of such theorems? Clearly the axiom of choice
>>>has no "useful application" in the sense of assisting computations.
>>
>>Frequently AC is used indirectly, typically in the form of Zorn's lemma.
>>There are numerous uses in point set topology, functional analysis,
>>algebra. If I recall correctly the Hahn-Banach theorem is one example.
>>It is basic to functional analysis, and functional analysis in turn
>>leads to results useful in physics.
>
>Why not just claim that AC is basic to set theory, sets are basic to math,
>math is basic to physics, ergo AC is basic to physics? Again, I'd like to
>see *any* computation in physics or elsewhere in which AC-dependent
>theorems are helpful (even heuristically).
>
In view of the fact that
"ZF is consistent" => "ZF + -AC is consistent",
I would be quite surprised if any *computations* used by physicists
required AC in all of its glory (such computations probably could be
carried out in *any* model of ZF). In fact, note that
"ZFC + `there exists an inaccessible cardinal' is consistent" =>
"ZF + DC + `all sets of reals are measurable' is consistent"
(where DC = Dependent Choice is the amount of Choice required in most
proofs using Choice). This suggests that the use of full AC in proofs
are often necessary only to deal with pathological objects (eg, non-
measurable sets) whose occurence in the "real" world could be excluded
by imposing restrictions on the definitions the physicist uses (eg,
we work within a model of ZF + DC that has no nonmeasurable sets).
On the other hand, I suspect that if mathematics was actually done this
way (eg, the standard version of the Hilbert Basis Theorem is false)
life for mathematical physicists would become a lot harder.
The situation for DC is, of course, quite different. DC merely says
that if \forall x \exists y R(x,y), then for any x, there exists a
function f such that x \in domain(f), and for any y \in domain(f),
f(y) \in domain(f) and R(y,f(y)). This assumption is used (implicitly)
all the time in computations, and while strictly weaker than AC, it
is strictly stronger than ZF alone.
-----Greg McColm
Herman Rubin
>In article <SGAURE.93...@sylow.uio.no>
>sga...@sylow.uio.no (Simen Gaure) writes:
......................
>>Modern mathematics is exclusively concerned with
>>imaginary objects. All the rest is physics, chemistry, economics,
>>biology, computer science, psychology, sociology & c. These are
>>some of the sciences where mathematics is used, and where the
>>physical interpretation of the mathematical ideas are important.
>This view is doomed to gloss over what Wigner called "the unreasonable
>effectiveness" of mathematics in natural sciences. Why should the
>study of figments of our imagination have any bearing on the laws
>governing the behavior of the objects of our perception?
>>Mathematics itself doesn't exist outside the mind of the
>>mathematician. The objects being studied are defined by
>>exact definitions (at least we like to think so) and the
>>rules for manipulating the objects are strictly defined.
>>With no reference to the physical world.
>>Of course, one is allowed to think in terms of physical
>>objects, but mathematics as such doesn't refer to it.
This problem is inherent in all forms of symbolism. One might try to
think in terms of physical objects, but I strongly question whether that
is even possible. Whatever we do is in terms of constructs, and this is
even true for all organisms; success is influenced by the accuracy of the
process.
That children, and even teachers, are not aware of this, makes for the
current poor abilities to use mathematics. Looking at things carefully,
the only way that mathematics can be used well is to construct a
mathematical model which supposedly mimics the real world, operate in
that model within the mathematical grammar, and translate the results
back. When the mathematical model is accurate, as it often is in physics
and chemistry, this process is quite successful. When the mathematical
model is inaccurate, as it usually is in the social sciences, and very
often in the biological sciences, the results can vary from good to
catastrophically bad.
Herman Rubin
>In article <1993May5.0...@husc3.harvard.edu> ku...@zariski.harvard.edu (Tal Kubo) writes:
>>Are you claiming that AC is a necessary heuristic crutch for development of
>>functional analysis applicable to physics? As the lady from Missouri said,
>>"show me". Given the strong claims put forth in your paragraph quoted at
>>the top of this message, I assume that you will have no trouble finding
>>better examples than Hahn-Banach.
>What about the existence of unbounded operators on an infinite-dimensional
>Hilbert space? The only constructions I know rely on AC, but perhaps I am
>just revealing my ignorance.
If you mean unbounded operators defined everywhere, it does not require the
full axiom of choice, but it does require something of that nature. But is
this what is needed in physics? Unbounded operators on a dense set can be
explicitly constructed.
>Alaoglu's theorem is usually proved using AC, but I don't know if there
>are any direct applications to physics.
This is weaker than AC, and is equivalent to the prime ideal theorem
(the existence of ultrafilters).
>Finally, I believe that the uncountability of the reals relies on
>countable choice. It seems like it would very difficult to develop a
>satisfactory analysis without countable choice at least.
The uncountability of the reals relies on nothing of the sort; only those
who insist on strict constructibility, which even makes the existence of
discontinuous functions impossible, find anything wrong with the usual
proofs. However, without countable choice, one cannot prove that the
reals are not the countable union of countable sets.
I agree with your last statement; in fact, a little more may be needed,
namely, making a countable number of choices each dependent on the
provious ones. But this is far weaker than AC.
Simen Gaure
> constructivist will confirm. The question is not whether the proof can
> be sanitized to remove reference to AC, infinity, or LEM, but whether those
> concepts help in organizing the calculations. Infinity and LEM seem to be
> extremely useful in this respect, but I have yet to see any instance where
> AC was of any use at all.
Ok, you're probably right. AC in it's general form has probably
no relevance in finite (or even countable) computations.
I thought this discussion included it's use in functional analysis,
that's two quite different things.
One is a general theory, the other is an application.
> If it's so relevant, it should be easy to find some concrete examples of
< physics calculations inspired by AC-based reasoning. With all the path
> integrals and infinite-dimensional whatnot in quantum theory, one would
> expect *some* example of AC being used in an essential way. I'm reminded of
< Le Rochefoucauld's line about love: it's like a ghostly apparition which
> everyone talks about, but nobody has ever seen. And how come set theory
> isn't a graduation requirement for a physics PhD...
I even thought physics included some theories apart from mere calculations,
but I may be wrong, I don't know the subject very well.
Simen Gaure
Univ of Oslo
Matthew P Wiener
>>But the driving force of physics is not the calculations, but to find
>>frameworks in which to make and understand these calculations. AC is
>>highly relevant here.
>If it's so relevant, it should be easy to find some concrete examples of
>physics calculations inspired by AC-based reasoning.
Quantum mechanics.
> With all the path
>integrals and infinite-dimensional whatnot in quantum theory, one would
>expect *some* example of AC being used in an essential way.
Yes. The physics literature usually doesn't bother dotting the exact
mathematics used, so if you're not AC-wise to begin with, you won't
notice just how widespread it is.
> I'm reminded of
>Le Rochefoucauld's line about love: it's like a ghostly apparition which
>everyone talks about, but nobody has ever seen.
Just because you haven't, please don't assume others haven't. Good grief,
I can point to an application of ZFC+CH (or, as the author was savvy to,
Martin's axiom sufficed) in physics if you really like. It's a _local_
hidden variable theory that screws around with measure in a really funky
way--worse than non-measurable sets.
[Perhaps this citation and a brief description belong in the sci.physics
FAQ, since I find myself mentioning it about every two or three months.]
> And how come set theory
>isn't a graduation requirement for a physics PhD...
Why should it be? There's no point in something easy being made extra
difficult for the physicists, now is there?
I recall a Berkeley physicist teaching a course on axiomatic quantum
field theory, and he had an unfriendly attitude regarding set theory.
Upon learning that was my area of specialty, he asked me to show him
an explicit non-measurable set. I asked him to show me an explicit
electron. He laughed.
Simen Gaure
> philosophy. One small question: what is this thing you call "our
> imagination"? Am I to assume that your imagination intersects or
> otherwise overlaps with mine, to the extent that we both understand
> the concept of S_3, or a torus? Are you prepared to stipulate, or at
> least suggest identity conditions relating our individual imaginary
> realms?
not try to prove anything mathematical about it without, either
describe it as a set of equations which we could agree upon,
or by constructing it by topological rules we could agree upon.
Or by some other operational device.
Depending on what I were to prove.
My view of a torus may be completely different from yours,
one may think of it as class of equivalent topological objects
or one may think of it as a donut. The differences in
people's imaginations is due to different abstract intuition.
I see no reason to relate our individual imaginary views
of a torus in any other way. At least not for any purely
mathematical purpose. I would, however, try to do it for teaching
or for explaining steps in a proof.
> Finally, is there really a difference of degree of "real
> existence" between the successors of zero and its predecessors?
No, I don't think so. I've never stated so in this discussion.
But it was believed to be by many brilliant mathematicians in the past,
that's not making their significance any less.
In modern mathematics there isn't. In some people's minds there
is a difference, I don't really care. It's of no significance.
The positive integers are referred to as 'natural numbers',
but the word 'natural' doesn't have its ordinary lexical meaning
in this context.
> [...] Why should the
> study of figments of our imagination have any bearing on the laws
> governing the behavior of the objects of our perception?
I see no particular reason for it.
It's a purely philosophical belief. And from a
pragmatic point of view, it actually works. But I can't prove that it must.
And that's not the objective of mathematics, but of philosophy.
If it one day didn't work, I'd think it was pretty interesting, but
I wouldn't see it as contradictory or anything such.
(Mathematics wouldn't get much economic support though.)
> The fact that mathematics does not refer to the physical world, in no
> way supports your contention that its objects have no existence
> outside of the mind of the mathematician.
It is said that mathematicians are bad philosophers, that may
be the reason why I don't understand you. Or it may be language problems.
>>A central notion in several branches of mathematics is the notion of infinity.
>>This notion can hardly be physically realized at all in this universe.
>
> As far as I am able to ascertain, this is still an open question.
Yes, I think so too, the word 'realized' may be to strong,
'experienced' may be a better word. Although I have some problems
of thinking some infinite object into the Universe. As a matter
of fact, I don't care a pair of dingo's kidneys about it.
> The idealistic position you appear to suggest is at odds with the
> egregiously non-constructive insight afforded by the AC. Most people
> have no way of imagining choice sequences, non-principal ultrafilters,
> or the Banach-Tarski decomposition of the sphere; some go as far as
> suggesting that that's just the point of these objects.
But they're still useful. I can't 'imagine' an ultrafilter,
but they turn out to be useful in my work. So they exist in my mind.
'I think of them, thus they are' ?
I'm just being pragmatic about it.
>>The truth-value of axioms are not
>>discussed in mathematics, although one may investigate whether
>>they are inconsistent, in which case it's no longer an axiom.
>
> Surely model theory is a branch of mathematics.
Can you elaborate a bit on that, I don't quite understand what connection
your comment has to my statement.
Simen Gaure
> think in terms of physical objects, but I strongly question whether that
> is even possible. Whatever we do is in terms of constructs, and this is
> even true for all organisms; success is influenced by the accuracy of the
> process.
>
> That children, and even teachers, are not aware of this, makes for the
> current poor abilities to use mathematics. Looking at things carefully,
> the only way that mathematics can be used well is to construct a
> mathematical model which supposedly mimics the real world, operate in
> that model within the mathematical grammar, and translate the results
> back. When the mathematical model is accurate, as it often is in physics
> and chemistry, this process is quite successful. When the mathematical
> model is inaccurate, as it usually is in the social sciences, and very
> often in the biological sciences, the results can vary from good to
> catastrophically bad.
Agreed.
I wonder if it's time for some clarifying of language in this discussion.
This isn't about the clear exposition above, but a comment
to the whole discussion. This was a handy moment.
When I say, or think of, 'mathematics' I think of a science of
purely symbolic appearance. I.e. pure mathematics.
Sketches and graphs are part of this, they are in a sense extended symbols.
This is a (somewhat simplified) description of my image of mathematics.
When pure mathematics is used in a way which explains the behaviour/appearance
of some physical object/concept, I usually refer to that process as
applied mathematics.
There isn't always a sharp border between these two,
e.g. computer science does often concern itself with 'virtual' machines,
and is in a sense finite pure mathematics, but it also concerns
itself with real machines and specific implementations.
Similar considerations, I believe, goes for some
branches of physics and chemistry.
And when teaching pure mathematics at a lower level
it is common to do this inside the framework of applied mathematics.
E.g. by using familiar concepts like velocity, rotation, counting
of apples etc, and of course, many pure mathematicians think
in terms of similar concepts.
When one refers to the applicability of some mathematical model
to a problem related to the description of some physical object,
I tend to label that as 'applied mathematics' because it involves
the use of knowledge from outside the imaginary world of pure mathematics.
I never use the word mathematics alone when I mean applied mathematics,
although I often use it in this way when I actually mean pure mathematics.
Applied mathematics consists of (at least) two parts, pure mathematics
and model adaptation. Thus, I may use the word 'mathematics' alone
in a discussion about applied mathematics, but only when referring
to the purely mathematical aspects of that particular science.
Thus, when I say 'mathematics', I don't mean mathematical physics,
fluid mechanics or any other natural science.
I'm not quite sure about statistics.
Is this use of language common?
Or is the word 'mathematics' used both for pure and applied mathematics,
this being formally more correct?
(I mean by professionals, not in everyday language.)
This may be a matter of cultural differences.
(Please, this is not a flame bait on pure vs applied mathematics,
and if you strongly object to my classification, please don't
flame me, but explain your reasons.)
john baez
Heck, even general relativity isn't typically a graduation requirement
for a physics PhD.
You guys seem to be talking past each other. Of course no specific calculation
ever uses the axiom of choice in an essential way, because the objects
are concrete and one doesn't have to go rummaging around in large misty
sets to find them. But that doesn't negate the force of what Wiener
says. I'm a mathematical physicist and I use the axiom of choice all
the time. E.g., hand me a Hilbert space and I have no qualms about
assuming it has a basis. The axiom of choice guarantees that it will
ALWAYS have a basis, but in any concrete example it is easy to find a
basis without AC. So, sure, we could skip the axiom of choice and
define a Hilbert space to be a "Hilbert space with a basis". However,
this would simply be a distracting nuisance for the most part. Someday
people may find it worthwhile to keep track of such things more closely,
but not yet.
Look at Reed and Simon's Methods of Modern Mathematical Physics. Delete
all the theorems that depend implicitly upon the Axiom of Choice. Note
how little there is left. Of course, if there was a constructivist coup
one could rewrite the whole book without using the axiom of choice,
using tricks such as the one mentioned above. But the way mathematical
physics is done NOW uses the mathematical universe in which AC is true.
Now, it's true that most physicists are only dimly aware of mathematical
physics, but that's another issue....
john baez
>When I say, or think of, 'mathematics' I think of a science of
>purely symbolic appearance. I.e. pure mathematics.
If this is what you mean by "mathematics," why don't you use a different
term. E.g., pure mathematics.
>I never use the word mathematics alone when I mean applied mathematics,
>although I often use it in this way when I actually mean pure mathematics.
>Thus, when I say 'mathematics', I don't mean mathematical physics,
>fluid mechanics or any other natural science.
>I'm not quite sure about statistics.
>Is this use of language common?
I hope not.
>Or is the word 'mathematics' used both for pure and applied mathematics,
>this being formally more correct?
It's used for both - "pure" and "applied" mathematics are different
aspects of mathematics, which of course interweave in an inextricable
manner. I know you didn't mean your post to be a flame, but as a
mathematical physicist who teaches in a mathematics department, proves
theorems and publishes them in mathematics journals, and otherwise
behaves like a mathematician, it's hard not to a little annoyed.
Tal Kubo
>
> "ZFC + `there exists an inaccessible cardinal' is consistent" =>
> "ZF + DC + `all sets of reals are measurable' is consistent"
>
>(where DC = Dependent Choice is the amount of Choice required in most
>proofs using Choice). This suggests that the use of full AC in proofs
>are often necessary only to deal with pathological objects (eg, non-
>measurable sets) whose occurence in the "real" world could be excluded
>by imposing restrictions on the definitions the physicist uses (eg,
>we work within a model of ZF + DC that has no nonmeasurable sets).
Not only that, but those pathological objects usually require AC for their
construction, i.e. the extra generality afforded by AC is almost always
illusory because the additional objects to which the theorems apply
can arise only through the use of AC. It's like the glazier sending
someone ahead of him to throw rocks through the windows.
>On the other hand, I suspect that if mathematics was actually done this
>way (eg, the standard version of the Hilbert Basis Theorem is false)
>life for mathematical physicists would become a lot harder.
I suspect that physicists are confident that the objects they will actually
encounter are sufficiently well-behaved that all standard theorems apply,
and thus (like the mathematicians) are too lazy to refine their
definitions to cover just the hypotheses actually needed for calculations.
>The situation for DC is, of course, quite different. DC merely says
>that if \forall x \exists y R(x,y), then for any x, there exists a
>function f such that x \in domain(f), and for any y \in domain(f),
>f(y) \in domain(f) and R(y,f(y)). This assumption is used (implicitly)
>all the time in computations, and while strictly weaker than AC, it
>is strictly stronger than ZF alone.
Yes, the question was about AC. I agree that life without DC would
look much different. (I mean set-theory axioms, not electricity...)
Tal
Tal Kubo
>
>>Are you claiming that AC is a necessary heuristic crutch for development of
>
>What about the existence of unbounded operators on an infinite-dimensional
>Hilbert space? The only constructions I know rely on AC, but perhaps I am
>just revealing my ignorance.
How about the derivative operator, on the space of smooth functions on
an interval with the usual inner product?
>
>Alaoglu's theorem is usually proved using AC, but I don't know if there
>are any direct applications to physics.
>
If there are any applications, I'd like to see them.
>Finally, I believe that the uncountability of the reals relies on
>countable choice. It seems like it would very difficult to develop a
>satisfactory analysis without countable choice at least.
Probably. The question was about AC.
Gerald Edgar
>>
>>What about the existence of unbounded operators on an infinite-dimensional
>>Hilbert space? The only constructions I know rely on AC, but perhaps I am
>>just revealing my ignorance.
The operators used in physics are only defined on (non-closed) subsets
of the Hilbert space, so they may be constructed without AC.
>
In article <1993May6.2...@husc3.harvard.edu> ku...@zariski.harvard.edu (Tal Kubo) writes:
>How about the derivative operator, on the space of smooth functions on
>an interval with the usual inner product?
That space is not a Hilbert space. A Cauchy sequence of smooth functions
may converge to a non-differentiable function in the norm induced by the
usual inner product.
The Axiom of Choice is required to prove the existence and/or important
properties of Lebesgue measure. I have argued that Lebesgue integration
is important for physics, but (since physics programs do not require it)
physicists apparently do not agree with me on that.
--
Gerald A. Edgar Internet: ed...@math.ohio-state.edu
Department of Mathematics Bitnet: EDGAR@OHSTPY
The Ohio State University telephone: 614-292-0395 (Office)
Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
Tal Kubo
>>
>>What about the existence of unbounded operators on an infinite-dimensional
>>Hilbert space? The only constructions I know rely on AC, but perhaps I am
>>just revealing my ignorance.
>
>How about the derivative operator, on the space of smooth functions on
>an interval with the usual inner product?
On second thought, this is nonsense, because said space is not a Hilbert
space, and differentiation is not an everywhere defined operator
on its Hilbert space completion.
Chris Menzel
: In article <1s1rub$c...@tamsun.tamu.edu>
: cme...@kbssun1.tamu.edu (Chris Menzel) writes:
: >Mikhail Zeleny (zel...@husc10.harvard.edu) wrote:
: >>Set theory without the Axiom of Extensionality has been discussed by
: >>Sol Feferman and Nicholas Goodman. Naturally, the salient question is
: >>whether such developments are analytic of the intuitive notion of set.
: >And by Gilmore somewhat before Feferman in his seminal 1974 paper "The
: >consistency of partial set theory w/o extensionality" ({\it Axiomatic
: >Set Theory, Symposia in Pure Math XIII, Part II, American Math
: >Society, 147-153). I believe Gilmore's paper was the motivation
: >behind Feferman's initial forays into this area (if memory serves).
: Shame on me for omitting the most accessible source. I keep thinking
: of something Dana Scott was said to have shown in the Sixties, but
: fail to come up with an actual reference.
: >Gilmore reports his efforts directed toward extending set theory in
: >ways that look somewhat more Fregean (as I read him); the result ended
: >up being inconsistent with extensionality, suggesting that the
: >intuitions at work were more in line with *property* theory rather
: >than set theory. The same thing happened to Maddy with her theory of
: >proper classes (JSL early 1980's).
: I agree on distinguishing properties from sets. In any event, were
: Feferman's results actually published, or even developed beyond a
: proposal stage?
Feferman reports the results of his work in this area, both work of
his own and with Peter Aczel, as well as related work, in "Toward
Useful Type-free Theories, I" in Martin (ed.) Recent Essays on Truth
and the Liar Paradox, Oxford, 1984. Last I heard there were no plans
for part II.
--
Christopher Menzel Internet -> cme...@tamu.edu
Philosophy, Texas A&M University Phone ----> (409) 845-8764
College Station, TX 77843-4237 Fax ------> (409) 845-045
Tal Kubo
>
> Of course no specific calculation
>ever uses the axiom of choice in an essential way, because the objects
>are concrete and one doesn't have to go rummaging around in large misty
>sets to find them.
Nobody disputes this. The question is whether there are concrete
calculations which would have been done differently, or not at all,
if physicists had explicitly rejected AC.
> But that doesn't negate the force of what Wiener
>says. I'm a mathematical physicist and I use the axiom of choice all
>the time. E.g., hand me a Hilbert space and I have no qualms about
>assuming it has a basis. The axiom of choice guarantees that it will
>ALWAYS have a basis, but in any concrete example it is easy to find a
>basis without AC. So, sure, we could skip the axiom of choice and
>define a Hilbert space to be a "Hilbert space with a basis". However,
>this would simply be a distracting nuisance for the most part. Someday
>people may find it worthwhile to keep track of such things more closely,
>but not yet.
This jives with what I said earlier: it's not that AC is part of the
baggage that comes with the calculations, just that people are too lazy
to refine their definitions appropriately. It seems to me that the
intuition being used here is not "every space, no matter how large
and wildly nonconstructive, has a basis", but rather something like
"I can find a basis for any space I meet". In other words, it's an
observation about the sorts of spaces which arise in practice, not
anything connected with AC per se. I'm sure that almost all
physicists also behave as though every set were measurable, but does
this really mean that some deep set-theoretic assumption (such as
"not-AC") is lurking behind their calculations?
>Look at Reed and Simon's Methods of Modern Mathematical Physics. Delete
>all the theorems that depend implicitly upon the Axiom of Choice. Note
>how little there is left. Of course, if there was a constructivist coup
>one could rewrite the whole book without using the axiom of choice,
>using tricks such as the one mentioned above.
If this is true then it should be easy to indicate situations where
some calculation would not have been done, or would have been done
differently, or would have been understood differently, had AC been
in disfavor. (I'm talking about full-strength AC or the like, not
the countable versions).
> But the way mathematical
>physics is done NOW uses the mathematical universe in which AC is true.
I think you're a bit too knowledgeable to say this with a straight
face. The way mathematical physics is done now uses a
mathematical universe where theorems can have counterexamples.
James A. Donald
> Ok, you're probably right. AC in it's general form has probably
> no relevance in finite (or even countable) computations.
> I thought this discussion included it's use in functional analysis,
> that's two quite different things.
> One is a general theory, the other is an application.
The calculations in quantum field theory involve infinite and uncountable
quantities, for example sum over all possible feynman diagrams, yet I cannot
see any relevance to set theory in these calculations, but then my pure maths
is perhaps as poor as Gaure's physics.
The topological analysis of spacetime used in proving singularity theorems
in general relativity involves massive use of set theory, and involves all
sorts of infinite uncountable quantities. I cannot see how the
axiom of choice is relevant to that either.
---------------------------------------------------------------------
|
James A. Donald | Joseph Stalin said: "Ideas are more powerful
| than guns. We would not let our enemies have
jame...@infoserv.com | guns, why should we let them have ideas."
Rob Vienneau
read the 1964 version of Godel's "What is Cantor's Continuum Problem?"
This note attempts to explain why I think accepting that undecidable
propositions exist in set theory is incompatible with Godel's
Platonistic view of mathematics. As a point of exegesis, I ask does
anybody know if Godel ever accepted the existence of absolutely
undecidable propositions? That is, did he think that for all purely
mathematic propositions, we will eventually be able to discover some
axioms and (possibly transfinite) method of drawing inferences such
that their truth or falsity will eventually be decided?
As outlined by Godel, the Platonist believes that "mathematical
objects exist independently of our constructions and of our having an
intuition of them individually." This does not mean that the Planonist
thinks mathematical objects exist physically. In fact, Godel brings up
Kant and argues that even our knowledge of physical objects extends
beyond mere sensations. The idea that a physical object has a
permanent existence contains "constituents qualitatively different
from sensations or mere combinations of sensations." Thus, Godel
concludes, something other than sensations must also be immediately
given for us to form our ideas of physical objects. In forming our
ideas of mathematical objects, all we have is this "something else
which is immediately given," but no sensations. We are able to
"recognize [the] soundness and the truth of the axioms" concerning
sets because our "general mathematical concepts" are "sufficiently
clear." (This last phrase reminds me of Descartes' clear and distinct
ideas.)
Now, if mathematical objects exist, any statement about them using
purely mathematical notions must be either true or false. If we can
find a statement that is neither, but can be freely decided either
way, then the Platonic conception of the nature of mathematical
objects must be false.
This does not mean that the demonstration of the consistency of both
the truth and falsity of the Continuum Hypothesis (CH) with the axioms
of ZFC is sufficient to reject sets as Platonic objects. For there may
be additional axioms not in ZFC that we may come to accept and that
imply, say, the falsity of the CH. Evidence for an axiom may include
its fruitfulness in proving new theorems and in deriving shorter
proofs for already known theorems. Godel explictly cites a case whose
mathematics is beyond me, namely "the existence of inaccessible
numbers (which can be proved to be undecidable from the von
Neumann-Bernays axioms of set theory provided that" the N-B axioms are
consistent). The negation of this axiom has a model in the original
system, but does not imply any number-theoretic theorems whose
validity can be checked, while its assertion does imply such theorems.
Godel thinks this is good evidence that the axiom is true in the
original realm of entities which were being modelled. (Note the
negation is not about sets, but something else in a model based on
sets.)
As far as I can see, Godel accepts undecidable axioms only if the
meaing of terms in a system of axioms are left uninterpreted. For
example, the terms "point," "line," "angle," etc. in Euclidean
Geometry can be given several interpretations. Sometimes Euclid's
fifth postulate comes out true and sometimes false. That gives no
problem to the Platonic philosophy of mathematics, since the meaning
of the terms in geometry thereby refer to different physical entities,
not mathematical ones.
But if one accepts that one is free to construct numbers from sets in
any of several acceptable ways and that even in set theory, some
theorems may be forever undecidable, one should reject Godel's version
of Platonism. I do think mathematicians do have this freedom in
creating their calculii and in drawing connections between various
systems. Furthermore, I am unsympathetic to the metaphysics involved
in drawing an analogy between mathematical intuition and some sort of
sense perception. So at least this week I find mathematical Platonism
unconvincing. Maybe if I learn more mathematics I will change my mind,
but I doubt it.
Robert Vienneau
john baez
>In article <30...@galaxy.ucr.edu> ba...@ucrmath.ucr.edu (john baez) writes:
>>Look at Reed and Simon's Methods of Modern Mathematical Physics. Delete
>>all the theorems that depend implicitly upon the Axiom of Choice. Note
>>how little there is left. Of course, if there was a constructivist coup
>>one could rewrite the whole book without using the axiom of choice,
>>using tricks such as the one mentioned above.
>
>If this is true then it should be easy to indicate situations where
>some calculation would not have been done, or would have been done
>differently, or would have been understood differently, had AC been
>in disfavor. (I'm talking about full-strength AC or the like, not
>the countable versions).
I'm not sure why you say that. I don't want to worry my head over the
difference between full-strength AC and the countable versions, since
the way things work is this: Reed and Simon prove a bunch of general
theorems that people feel free to use in calculations. These theorems
use AC through and through. But certainly in any specific calculation I
can think of, one could redo it and get the same answer avoiding AC.
Whether this counts as doing it differently or not is up to you.
>> But the way mathematical
>>physics is done NOW uses the mathematical universe in which AC is true.
>I think you're a bit too knowledgeable to say this with a straight
>face. The way mathematical physics is done now uses a
>mathematical universe where theorems can have counterexamples.
It sounds to me like you're talking about theoretical physics, a wholly
different ball of wax. Mathematical physics is a branch of mathematics
that is just as consistent as the rest. Look at Reed and Simon for a
classic mathematical physics text. Theoretical physics is a branch of
physis, which is not about proving theorems but about getting answers to
physics problems by hook or by crook. Most theoretical physicists
couldn't care less about axiomatic systems.
john baez
(James A. Donald) writes:
>The topological analysis of spacetime used in proving singularity theorems
>in general relativity involves massive use of set theory, and involves all
>sorts of infinite uncountable quantities. I cannot see how the
>axiom of choice is relevant to that either.
Well, if you look at the proofs you will see that the axiom of choice is
being used all over the place. Well, maybe you *won't* see it, because
they sure as hell don't say "and now, by the Axiom of Choice..."! :-)
This is actually a pretty nice example. Whipping out Wald's General
Relativity, going to chapter 9 on the singularity theorems, we see (on a
randomly chosen page, honest!): "If tau were continuous on c(p,q) we
could extend it to a continuous function on all of C(p,q) by setting
tau(mu) = lim tau(lambda_n) where {lambda_n} is a sequence in c(p,q)
which approaches the continuous causal curve mu in C(p,q)." Here what
we are doing is saying that for each n we can find a lambda_n with some
property, and then using AC to form {lambda_n}.
Timothy Y Chow
>calculations which would have been done differently, or not at all,
>if physicists had explicitly rejected AC.
Fools never learn, so let me make another stab on this thread.
Let us call a vector space "automorphic" if it is (a) zero-dimensional,
or (b) one-dimensional, or (c) has a non-trivial automorphism. Since all
vector spaces are automorphic, this is a bit of a pointless definition.
Nobody is likely to develop a serious theory of automorphic vector
spaces. Simiarly, if all sets were measurable, I doubt that measure
theory would have developed in the way that it has. Pathologies often
motivate definitions.
Now measure theory certainly has an impact on the way analysis is done.
It also has had a big impact on probability theory. Both of these are
fundamental to physics.
Alan Smaill
This note harks back to prior discussions in this thread. I've just
read the 1964 version of Godel's "What is Cantor's Continuum Problem?"
This note attempts to explain why I think accepting that undecidable
propositions exist in set theory is incompatible with Godel's
Platonistic view of mathematics. As a point of exegesis, I ask does
anybody know if Godel ever accepted the existence of absolutely
undecidable propositions? That is, did he think that for all purely
mathematic propositions, we will eventually be able to discover some
axioms and (possibly transfinite) method of drawing inferences such
that their truth or falsity will eventually be decided?
There are two distinct uses of "undecidable" that often appear in this
sort of discussion. The notion you have just mentioned we might call
epistemological decidability --
" for all purely mathematic propositions,
we will eventually be able to _discover_ ...".
As outlined by Godel, the Platonist believes that "mathematical
objects exist independently of our constructions and of our having an
intuition of them individually."
Now, here this quote suggests something else -- mathematical
statements being true or false independently of our knowing the fact
of the matter. So this is another way of being "decided" -- that
statement has a truth value. (what can we call it -- metaphysical
decidability?)
This does not mean that the Planonist
thinks mathematical objects exist physically. In fact, Godel brings up
Kant and argues that even our knowledge of physical objects extends
beyond mere sensations. The idea that a physical object has a
permanent existence contains "constituents qualitatively different
from sensations or mere combinations of sensations." Thus, Godel
concludes, something other than sensations must also be immediately
given for us to form our ideas of physical objects. In forming our
ideas of mathematical objects, all we have is this "something else
which is immediately given," but no sensations. We are able to
"recognize [the] soundness and the truth of the axioms" concerning
sets because our "general mathematical concepts" are "sufficiently
clear." (This last phrase reminds me of Descartes' clear and distinct
ideas.)
Now, if mathematical objects exist, any statement about them using
purely mathematical notions must be either true or false. If we can
find a statement that is neither, but can be freely decided either
way, then the Platonic conception of the nature of mathematical
objects must be false.
I'm not sure which notion of "decided" is meant here.
Here the point is not really either of the notions above, I think.
It's that statements with uninterpreted terms are not declarative
statements at all until we fix the interpretation.
But if one accepts that one is free to construct numbers from sets in
any of several acceptable ways and that even in set theory, some
theorems may be forever undecidable, one should reject Godel's version
of Platonism.
Here you seem to be slipping from the second sort of decidability
(which Godel surely believed in) to the first (which I have
no reason to think he believed in).
I do think mathematicians do have this freedom in
creating their calculii and in drawing connections between various
systems. Furthermore, I am unsympathetic to the metaphysics involved
in drawing an analogy between mathematical intuition and some sort of
sense perception. So at least this week I find mathematical Platonism
unconvincing. Maybe if I learn more mathematics I will change my mind,
but I doubt it.
--
Alan Smaill, JANET: A.Sm...@uk.ac.ed
Department of Artificial ARPA: A.Smaill%uk.a...@nsfnet-relay.ac.uk
Intelligence, UUCP: ...!uknet!ed.ac.uk!A.Smaill
Edinburgh University.
Herman Rubin
General measure theory admits most of the problems with no special situations
required. I doubt that much would have been done differently if the
counterexamples to all sets being Lebesgue measurable had not been found.
Mikhail S. Verbitsky
>In article <125...@netnews.upenn.edu>
>wee...@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
>
>>But the driving force of physics is not the calculations, but to find
>>frameworks in which to make and understand these calculations. AC is
>>highly relevant here.
>
>If it's so relevant, it should be easy to find some concrete examples of
>physics calculations inspired by AC-based reasoning. With all the path
>integrals and infinite-dimensional whatnot in quantum theory, one would
>expect *some* example of AC being used in an essential way. I'm reminded of
>Le Rochefoucauld's line about love: it's like a ghostly apparition which
>everyone talks about, but nobody has ever seen. And how come set theory
>isn't a graduation requirement for a physics PhD...
I am not sure about physics, but one can find examples
of using AC even in such a constructive domain of mathematics
as algebraic geometry and number theory. For example, to prove
Hilbert's theorem about zeros one uses the fact that
the algebraic completion of the field of rational functions
over C is isomorphic to C. This is proved using Cauchy-Gamel
transcendency base, and essentially, Zorn lemma.
There is another proof of Hilbert's theorem,
which uses obscure logic machinery (basically,
Tarsky-Seidenberg theorem). Still, no-one knows
yet how to deal without Cauchy bases in the
p-adic Hodge theory.
The simplest question I don't know the answer
on: how to prove that the algebraic completion
of Z_p is isomorphic to C without using
AC? This isomorphism is very important for
p-adic numbers theory. So, at least continual
AC is essential for modern algebraic geometry.
Misha.
Tal Kubo
>or (b) one-dimensional, or (c) has a non-trivial automorphism. Since all
>vector spaces are automorphic, this is a bit of a pointless definition.
>Nobody is likely to develop a serious theory of automorphic vector
>spaces. Simiarly, if all sets were measurable, I doubt that measure
>theory would have developed in the way that it has. Pathologies often
>motivate definitions.
Very good. Has anything in physics been influenced by the
study of pathologies so wild that they require uncountable choice?
>Now measure theory certainly has an impact on the way analysis is done.
>It also has had a big impact on probability theory. Both of these are
>fundamental to physics.
As I suggested before, those who buy this argument ought to simply claim
that AC is basic to set theory, sets are basic to math, math is basic to
physics, ergo AC is basic to physics. They ought to then explain why this
line works for AC but not, say, GCH. (Before answering, consider that in
Halmos' gossip book, he mentions a collaboration with von Neumann wherein
GCH was used to prove theorems in operator theory. This in the days when
von Neumann was helping lay mathematical foundations for QM. And GCH is
older than AC).
Historically, astronomy has had a great effect on the development of
calculus, but we don't see broad claims being made about the necessity for
telescopes in analysis. The historical role of AC in the development of
the relevant mathematics, is quite separate from its logical position in
physics theory. I'm inquiring about the latter.
Tal Kubo
>
>>>Look at Reed and Simon's Methods of Modern Mathematical Physics. Delete
>>>all the theorems that depend implicitly upon the Axiom of Choice. Note
>>
>>If this is true then it should be easy to indicate situations where
>>some calculation would not have been done, or would have been done
>>differently, or would have been understood differently, had AC been
>>in disfavor. (I'm talking about full-strength AC or the like, not
>>the countable versions).
>
>I'm not sure why you say that. I don't want to worry my head over the
>difference between full-strength AC and the countable versions,
I do. I think it's a very interesting question whether abstract
nonconstructive reasoning is relevant to physics. Conflating countable
dependent choice DC, which is very concrete and used all the time, and
full-strength AC, which is egregiously nonconstructive, would be quite
misleading in this context.
>since the way things work is this: Reed and Simon prove a bunch of
>general theorems that people feel free to use in calculations. These
>theorems use AC through and through.
If the theorem really depends on AC, invoking it would stop any
calculation dead in its tracks; AC has the rude habit of rendering
computations ineffective. I think there are various metatheorems
saying how bad a set has to be if its definition uses AC in an
essential way (perhaps weemba can help us out here). There is a
famous result that modulo some large cardinal hypothesis, ZFC is
equiconsistent with ZF + not-AC + "every set of reals is measurable",
i.e. if it takes AC to construct it, chances are it's nonmeasurable.
Hardly the sort of thing that could help out your calculations, or a
theory about such calculations.
> But certainly in any specific calculation I
>can think of, one could redo it and get the same answer avoiding AC.
>Whether this counts as doing it differently or not is up to you.
I'm not asking about sanitizing the computations. The issue is
whether the AC-dependent considerations help, *or even come up*, in
thinking about what or how to calculate. "Every vector space has a basis"
does not seem to help when thinking about physical Hilbert spaces, because
whenever one wants to think in terms of a basis, there is a natural basis
lying around; considerations and intuitions specific to the problem
predominate and the set theory never enters the picture. This means that
one should think twice before proclaiming that AC is doing work in theorems
or calculations which superficially seem to depend on it
Matthew P Wiener
>This in the days when von Neumann was helping lay mathematical
I don't believe this. Cantor implicitly conjectured AC in the all
cardinals are comparable format, and only explicitly conjectured CH,
so far as I remember.
And since ZF+GCH => AC, it's hard to ever call GCH older than AC.
Cameron Laird
.
.
.
>I do. I think it's a very interesting question whether abstract
>nonconstructive reasoning is relevant to physics. Conflating countable
>dependent choice DC, which is very concrete and used all the time, and
>full-strength AC, which is egregiously nonconstructive, would be quite
>misleading in this context.
.
.
I'm hijacking this thread, and turning it into a brief, personal com-
ment on
Gillies, Donald, editor
1992 Revolutions in Mathematics. Oxford
University Press, Oxford
which has been discussed before in some of these newsgroups. At the
end, I explain the connection to AC and egregious non-constructibility.
The book rotates around the "Is not!--Is too!" confrontation Michael
Crowe and Joseph Dauben raise over the question: does Mathematics
have revolutions? Caroline Dumore, in her chapter on "Meta-level
revolutions in mathematics", gives what every right-thinking person
must agree is the right answer: yes and no. No, mathematical
objects (euclidean geometries, the areas of conics) don't become in-
valid, but yes, their metamathematical status (how they're taught,
whether anyone funds them) does change fundamentally through the years.
Today's teaching assistants would red-ink one of Euler's proofs, but we
still agree on the correctness of essentially all his results (results
are mathematical, proofs are metamathematical).
That settled, I turn to the dirty work of a few specific examples.
Dauben discusses non-standard analysis at length in "Appendix (1992):
revolutions revisited", and explains why its adoption constitutes a
revolution. It's a good, cogent explanation, except for its funda-
mental premise: from what I can tell, we're still living in an
epsilon-delta world. Dauben dates Robinson's revolution from 1961;
what sort of a revolution is it in which the population still doesn't
know the slogans of the new ruling class after thirty-two years? I'm
not denying that Robinson has done a great thing, or that we all ought
to learn non-standard analysis, but, from what I've seen of university
classrooms, the revolution hasn't gone far yet.
One that has made more progress pedagogically, though, is that of con-
structivism. I don't pretend at all that Errett Bishop would be
happy with the way mathematics is taught today, but I see more and
more diffusion of the taste for constructivism. "Elegance" is not the
only criterion for judging the expression of a proof; "directness" and
"computability" are now legitimate desiderata for professional work.
More subtly, more and more physicists and mathematicians show an
interest in "weakening" a proof to make it more constructive. To my
eyes, that constitutes a revolution. Like any good revolution, the
participants have a variety of motivations, and many (most?) of them
don't recognize the original prophets. Professor Kubo is right to
wonder whether there *really* is any meritorious mathematical physics
that is non-constructible. What's interesting to me is that the
question would have seemed sophomoric a generation ago.
I've started to re-direct follow-ups.
--
Cameron Laird
cla...@Neosoft.com (claird%Neoso...@uunet.uu.net) +1 713 267 7966
cla...@litwin.com (claird%litwi...@uunet.uu.net) +1 713 996 8546
ani...@draco.unm.edu
john baez
>Is society as a
>whole ready for the technology that is being presented to it?
No, but we'll use it anyway and see what happens.
>What are your feelings about this?
Resignation, and occaisionally glee.
>Do you think that we should even continue to develop new
>technologies?
We will whether or not we should.
>Do you feel that technology should be used for
>destruction?
Only for destruction of evil people who deserve to be destroyed.
Opinions vary on just who those people are.
>Do you feel that capitalism will destroy the world,
>with technology as it's tool?
Time will tell, there's really no way of knowing ahead of time.
Tal Kubo
In article <126...@netnews.upenn.edu>
>
>>This in the days when von Neumann was helping lay mathematical
>>foundations for QM. And GCH is older than AC.
>
>I don't believe this. Cantor implicitly conjectured AC in the all
>cardinals are comparable format, and only explicitly conjectured CH,
>so far as I remember.
The continuum problem was explicitly recognized starting with Cantor.
AC was not isolated until Zermelo's paper circa 1900. In that sense
it's a later development. Euclid implicitly conjectured deep topological
properties of the line and plane, but I wouldn't say topology dates back
quite that far.
>
>And since ZF+GCH => AC, it's hard to ever call GCH older than AC.
The implication wasn't noticed until AC was recognized as such.
ZF + AC ---> Banach-Tarski, but I have no qualms about calling
the latter older than AC.
Tal
Gary Martin
And no way of knowing after the fact, either.
--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
Mar...@cis.umassd.edu
Angus H Rodgers
>[...] we now live in a society that is ruled by
>those who have the most information.
I don't know about you, but over here we're ruled by
a bunch of two-bit know-nothings.
--
Gus Rodgers, Dept. of Computer Science, Queen Mary & Westfield College,
Mile End Road, London, England +44 71 975 5241 arod...@dcs.qmw.ac.uk
lrud...@vax.clarku.edu
>In <1sn248...@lynx.unm.edu> ani...@draco.unm.edu () writes:
>
>>[...] we now live in a society that is ruled by
>>those who have the most information.
>
>I don't know about you, but over here we're ruled by
>a bunch of two-bit know-nothings.
Around here, even at the personal level the standard is rapidly becoming
32-bit know-nothings.
Lee Rudolph
Matthew P Wiener
>nonconstructive reasoning is relevant to physics. Conflating countable
>dependent choice DC, which is very concrete and used all the time,
Note that DC still requires one to believe in infinity in some essential
not really concrete sense. DC does have one march down a choice procedure,
a->b->c->d->.... Any finite amount of this is fully within ZF. It's the
"..."s that are the problem.
In a real calculation, one wouldn't need this much.
> and
>full-strength AC, which is egregiously nonconstructive, would be quite
>misleading in this context.
I'll assume that you'll consider ultrafilters/maximal ideals firmly in
the egregiously non-constructive camp.
>>since the way things work is this: Reed and Simon prove a bunch of
>>general theorems that people feel free to use in calculations. These
>>theorems use AC through and through.
>If the theorem really depends on AC, invoking it would stop any
>calculation dead in its tracks; AC has the rude habit of rendering
>computations ineffective.
But it also has the habit of rendering certain constructions conceivable.
Consider the spectral theorem: every operator, bounded or not, can be
resolved as an integral on its spectrum. It's an extremely powerful
tool, and allows one to extend the holomorphic functional calculus to
more general functions--a great convenience in QM!
At the heart of the proof is the Gelfand-Naimark theorem regarding the
isomorphism between a commutative C*-algebra and its MAXIMAL IDEAL space.
Without some form of egregiously nonconstructive choice, this basic tool
cannot get started. One does not actually _need_ the particulars of the
maximal ideals to use the spectral theorem--rather, one needs the _idea_
of a representation as functions on a compact space.
This is why things like non-commutative geometry are possible. No one
has been able to identify the appropriate spaces yet--just some algebraic
representations.
> I think there are various metatheorems
>saying how bad a set has to be if its definition uses AC in an
>essential way (perhaps weemba can help us out here). There is a
>famous result that modulo some large cardinal hypothesis, ZFC is
>equiconsistent with ZF + not-AC + "every set of reals is measurable",
>i.e. if it takes AC to construct it, chances are it's nonmeasurable.
>Hardly the sort of thing that could help out your calculations, or a
>theory about such calculations.
The basic results are as follows. Starting with the Borel sets, one
forms their continuous images, the analytic sets aka PI^1_1. Their
complements are called the co-analytic sets aka SIGMA^1_1. The first
theorems are that a set is Borel iff it is analytic and co-analytic,
and that this is non-trivial. One now iterates the procedure. The
continuous images of SIGMA^1_1 sets are called PCA sets (P for pro-
jection, as it turns out you can consider just projection maps) aka
PI^1_2. Complements of PCA sets are CPCA or SIGMA^1_2. Continuous
images of SIGMA^1_2 sets are PI^1_3, etc.
This is the so-called projective hierarchy. It is non-trivial, and
the basic questions are just how constructive is this stuff? It turns
out, for example, that one can prove analytic and hence co-analytic
sets are Lebesgue measurable. Beyond that, one can't say too much
without pulling in extra set theory. If you assume the axiom of
constructibility, one has an explicit enough PCA well-ordering of
the reals and can give an explicit PCA non-measurable set. If you
assume the right large cardinals are consistent, one can deduce that
the whole projective hierarchy is measurable.
Roughly speaking, one can say that constructive thinking buys you as
much as the analytic sets, but that constructive thinking modulo the
right set theoretic assumption (much deeper than AC) buys you the rest
of the projective hierarchy. And of course, there are things beyond
to figure out--analogous results hold here too.
(BTW: PI and SIGMA in print are the bold-face capitals.)
Simen Gaure
> Around here, even at the personal level the standard is rapidly becoming
> 32-bit know-nothings.
And over here we've been on 64-bits for some months, but only
at the technical level (DEC Alpha).
Our politicians don't use bits for decisions
they use 'trits' as they always have.
For a discussion of trits, see e.g.
"Seminumerical algortihms", D. Knuth 1981 pp 190--192
Simen Gaure
University of Oslo
Tom Maddox
>In <1sn248...@lynx.unm.edu> ani...@draco.unm.edu () writes:
>
>>[...] we now live in a society that is ruled by
>>those who have the most information.
>
>I don't know about you, but over here we're ruled by
>a bunch of two-bit know-nothings.
Not us. We are ruled by bunches of cunning wealthy who exercise
their control through two-bit know-nothings who make faces and noises
that could mislead you into believing *they* were in charge.
Reaganocracy, it's called.
--
Tom Maddox
tma...@netcom.com
"That's a bird bone chair, Bob. I don't know if I should sit there."
Tom Waits
BARTHOLDI Laurent
Just let the evolution work. If mankind is not fit for survival,
because it breaks down the planet, well I say, let it die! This
will release some free space for a new species, one that hopefully
will be more evolved than us.
Why not have trust in evolution? Let her do her job; and meanwhile,
advance the technology as fast as possible so mankind will disappear
as soon as possible if it isn't fit!
larry (forgive him, he's a mathematician)
Charles Yeomans
>
> In article <C6v8J...@dcs.qmw.ac.uk> arod...@dcs.qmw.ac.uk (Angus H Rodgers) writes:
> >In <1sn248...@lynx.unm.edu> ani...@draco.unm.edu () writes:
> >
> >>[...] we now live in a society that is ruled by
> >>those who have the most information.
> >
> >I don't know about you, but over here we're ruled by
> >a bunch of two-bit know-nothings.
>
> Not us. We are ruled by bunches of cunning wealthy who exercise
> their control through two-bit know-nothings who make faces and noises
> that could mislead you into believing *they* were in charge.
>
> Reaganocracy, it's called.
Actually, Ronald Reagan hasn't been president since 1989 (and some would
argue that he hadn't been president since late 1986). I also note
that most of the rich presidents of the 20th century have been
Democrats. Perhaps FDRocracy would be a more descriptive term.
Furthermore, I would argue, in opposition to Bart Goddard, that society is
becoming increasingly moral, and that we are becoming more so precisely
because we can afford to be moral. While correlation certainly
does not imply causality, it is interesting to note, for instance,
that the decline of slavery coincides rather nicely with the rise
of the industrial revolution.
Charles Yeomans
Matthew P Wiener
>"Elegance" is not the only criterion for judging the expression of a
>proof; "directness" and "computability" are now legitimate desiderata
>for professional work. More subtly, more and more physicists and
>mathematicians show an interest in "weakening" a proof to make it
>more constructive. To my eyes, that constitutes a revolution.
This is just fashion and taste, and not a revolution.
Francis Muir
Cameron Laird writes:
"Elegance" is not the only criterion for judging the
expression of a proof; "directness" and "computability"
are now legitimate desiderata for professional work.
More subtly, more and more physicists and mathematicians
show an interest in "weakening" a proof to make it
more constructive. To my eyes, that constitutes a
revolution.
This is just fashion and taste, and not a revolution.
Replace Laird's "more subtly" with more commonsensically. It is not a
revolution, nor a matter of fashion and taste, but a welcome return to
an older tradition and a time when constructive proofs were what proofs
were. Has the Courant Institute yet returned to doing what they were
set up to do?
Goofy
john baez
I would feel more comfortable with this attitude if I knew that life
would arise infinitely often, or at least very often, in the universe.
Then we could kick back and relax, so to speak. However, this is is not
at all clear. As it is, we cannot count on new intelligent life forms
evolving *on earth* if we kick off. Of course, my bias towards
intelligence is showing here; quite possibly evolution has other ideas
(so to speak).
D. J. Bernstein
colu...@strident.think.com (Michael Weiss) writes:
> Applied to AC,
> this says that most mathematicians accept AC primarily because it buys you
> little or nothing to reject it, and you gain a wealth of nice results if
> you admit it.
It buys you little or nothing to reject the separability of Hilbert
space, and you gain a wealth of nice results if you admit it. Does this
mean we should redefine ``Hilbert space'' as ``separable Hilbert space''?
Obviously the axiom of choice implies lots of results. Of course we want
to use fun axioms which do neat things. There's no reason anyone should
be stopped from using AC.
But I think each use of AC should be _identified_. To _identify_ AC when
it comes up takes very little effort if you do it consistently---and it
serves as a warning to your readers that they have to shift mental gears.
---Dan
D. J. Bernstein
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
> Why should the
> study of figments of our imagination have any bearing on the laws
> governing the behavior of the objects of our perception?
Mathematics is the ultimate intellectual game. It's the game of writing
true statements of the form ``p, q, ... imply r,'' with modus ponens as
the only rule of inference. (This is the direct limit of intellectual
games, one might snobbishly say, as all the intellectual games ever
invented by man can be embedded into it.) When faced with a real-world
problem, why should we avoid analyzing it mathematically?
---Dan
br...@quake.sylmar.ca.us
>Attention all persons of creative thought:
> We stand on a threshold to a new tomorrow.
>With the
>developement of such advanced technologies as virtual reality and
>artificial intelligence, we now live in a society that is ruled by
>those who have the most information.
>As technology becomes a
>dominant governing force in our life,
>I feel that we, as the
>creators of this technology, must ask a question. Is society as a
>whole ready for the technology that is being presented to it?
which have not been invented yet?", then the answer is "Of coursenot.". If what you mean is "Is everyone out there capable of
learning to make fruitful use of every new technology that
comes along." the answer is "probably, for most technologies".
If the REAL meaning of your question is "Will everything
go smoothly as new technologies appear on the market?" the
the answer is "Obviously not.".
My basic reaction to this is to ask "So what?". To imply that
new technologies are bad merely because they are technological
or because they are new is simple-minded and misguided.
Certainly some drugs, some inventions, and some social
institutions are bad and should be avoided, but they have to
be evaluated on a case by case basis on their individual merits,
not given a blanket technophobic condemnation.
>Have
>humans developed the morality and the responsibility required to
>safely use this technology for the betterment of mankind, or will
>this technology be used for the destruction?
you seem to have trouble telling the difference between individual
people. There are people whom I feel great about their having the
nuclear bomb for instance, and others who tend to keep me up at
night.
If you are going to wait until there are no morally imperfect people
in the world, or until all of the implications of new technologies
are discovered, you might as well just outlaw thinking, invention,
and action since they might just result in some consequences you
might not want.
>I pose this question
>to any and all who read this. What are your feelings about this?
you are generalizing in such a way as to condemn everyone and
everything people are or do or think. Your way of thinking
points the way to technophobia and luddism.
>Do you think that we should even continue to develop new
>technologies?
negative consequences of technological change that you forget
that new technologies can create more efficient ways of living,
cures for diseases, longer lives, better food, more affordable
housing, better entertainment, better education, and on and on.
Go ahead and tell a mother whose baby was just saved from
a horrible death that she would be better off without MRI
machines. Go ahead and tell a blind man that he would be
better off if no further laser surgery technology is developed.
Go ahead and tell a poor farmer that he would be better off
without improved grains that can grow on his otherwise useless
land. And tell it too to his hungry neighbor who can't afford to
eat three meals a day.
What kind of cold-hearted monster could deny these people
the benefits of new technologies just because he's a little worried
that he doesn't know what will happen if they are created?
>Do you feel that technology should be used for destruction?
cancer tumors, garbage, and disease organisms is great. Are
you opposed to destroying these things?
>Do you feel that capitalism will destroy the world,
>with technology as it's tool?
call "capitalism" is actually just "mixed economy", but in any
event, capitalism is the political/economic system that allows
people to act in accordance with their values and desires.
In many cases, they use technology to do this. If you don't
like that, I suggest you go live in Cuba, China, or North Korea
where someone will plan your life for you and make sure that
only government-approved ideas, industries, technologies, and
people will be allowed.
>Please, send your thoughts, flames,
>answers, questions, essays, poems, words, letters, pictures, etc..
>to "M_NI...@apsicc.aps.edu". This is an informal survey,
there are on the net?
>and if
>you state your interest, I will send you the results, unless many
>people respond, in which case I will post them.
>
>P.S. - Please respond, as the more people who answer, the more
>accurate the sentiment I can extract.
of your own on this issue? There are certainly a lot of them
implicit in the questions you asked.
--Brian
br...@quake.sylmar.ca.us
>In article <C6v8J...@dcs.qmw.ac.uk> arod...@dcs.qmw.ac.uk (Angus H Rodgers) writes:
>>In <1sn248...@lynx.unm.edu> ani...@draco.unm.edu () writes:
>>>[...] we now live in a society that is ruled by
>>>those who have the most information.
>>I don't know about you, but over here we're ruled by
>>a bunch of two-bit know-nothings.
> Not us. We are ruled by bunches of cunning wealthy who exercise
>their control through two-bit know-nothings who make faces and noises
>that could mislead you into believing *they* were in charge.
> Reaganocracy, it's called.
Perhaps you have not heard, Reagan has been out for some time now. I'm
curious who you think these cunning wealthy conspirators are. Some
sinister jewish cabal no doubt.
(;-)
Do you think that's who's in charge now?
--Brian
br...@quake.sylmar.ca.us
wrote the original request for comments was worse, or this
unthinking anti-judgemental jellyfish who can't evaluate
anything for fear that he might have to disapprove of what
is going on, or might have to approve of it and be called into
question for the faults of that reality.
ucrmath.ucr.edu (john baez) writes:
>In article <1sn248...@lynx.unm.edu> ani...@draco.unm.edu writes:
>>Is society as a
>>whole ready for the technology that is being presented to it?
>No, but we'll use it anyway and see what happens.
whether or not you think technology is good. Don't you
have a better answer than that?
>>What are your feelings about this?
>Resignation, and occaisionally glee.
is just something that can't be opposed...even condemned without
any further action?
>>Do you think that we should even continue to develop new
>>technologies?
>We will whether or not we should.
response. You could at least have an opinion instead of just
hiding behind some kind of deterministic fog.
>>Do you feel that technology should be used for
>>destruction?
>Only for destruction of evil people who deserve to be destroyed.
>Opinions vary on just who those people are.
should be destroyed? Have you ever considered that you might
want to have some well-justified idea of who these evil people
might be? Or are you above all this "good and evil" stuff?
Does that mean that you will simply accept whatever happens
as inevitable no matter how good or bad it is?
Grow a spine.
>>Do you feel that capitalism will destroy the world,
>>with technology as it's tool?
>Time will tell, there's really no way of knowing ahead of time.
comes your way. Do you really think that there's no way to
know whether a social system is going to destroy the world?
Do you claim that there's no way to make any judgements about
technology, people, or institutions? You have really covered the
gamut here of refusing to judge ANYTHING as being better or
worse than anything else. It is the apathetic attitudes of people
like you that allow terrible consequences of things, people, and
institutions develop since you absolve yourself of the need to
evaluate them. You would rather just sit back and let them
happen to you for better or worse.
If ANYTHING is going to "destroy the world" it is not
"capitalism using technology", it is "tyrants using apathetic
resignation in the population".
The worst enemy of the moral man is not the immoral one
who seeks to destroy him, he's at his core impotent. It is
the amoral man who supports the immoral one.
--Brian