# Sets

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### Gene W. Smith

Apr 16, 1993, 8:01:57 AM4/16/93
to
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.

>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

Apr 16, 1993, 10:24:28 AM4/16/93
to
In article <1993Apr16.1...@sun0.urz.uni-heidelberg.de>

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

Apr 17, 1993, 1:23:18 AM4/17/93
to

>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.

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

Apr 17, 1993, 8:38:18 AM4/17/93
to
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

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.

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

>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

Apr 18, 1993, 12:28:12 PM4/18/93
to
In article <1993Apr17....@sun0.urz.uni-heidelberg.de>

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

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

>>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.

>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

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.

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

Apr 19, 1993, 1:02:43 AM4/19/93
to
Gene W. Smith writes in response to Zeleny:

[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

Apr 28, 1993, 4:11:44 AM4/28/93
to
In article <1993Apr17....@sun0.urz.uni-heidelberg.de> 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

... 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
>
>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

Apr 29, 1993, 12:19:38 PM4/29/93
to
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.

>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

Apr 29, 1993, 1:12:09 PM4/29/93
to
In article <1993Apr29.1...@sun0.urz.uni-heidelberg.de> gsm...@lauren.iwr.uni-heidelberg.de (Gene W. Smith) writes:
>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

Apr 30, 1993, 11:47:04 AM4/30/93
to
In article <C6994...@rice.edu> r...@owlnet.rice.edu (Robert Camp
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

Apr 30, 1993, 9:20:19 AM4/30/93
to
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:

>>>>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
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

Apr 30, 1993, 4:44:58 PM4/30/93
to

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

Apr 30, 1993, 7:27:11 PM4/30/93
to
In article <1993Apr30....@ulysses.att.com>
m...@ulysses.att.com (Michael L. Siemon) writes:

>In article <C6994...@rice.edu>
>r...@owlnet.rice.edu (Robert Camp Miner) 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
>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?

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

May 1, 1993, 1:49:12 PM5/1/93
to
zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
>MLS:
>>... Gene's usage is completely common in mathematics. It is

>>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.

### Torkel Franzen

May 1, 1993, 3:06:02 PM5/1/93
to
In article <1993May1.1...@samba.oit.unc.edu>

>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

May 1, 1993, 4:35:05 PM5/1/93
to

>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:

MLS:
>>>... Gene's usage is completely common in mathematics. It is
>>>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

May 1, 1993, 6:21:22 PM5/1/93
to
In article <TORKEL.93...@anhur.sics.se>
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

May 2, 1993, 11:13:56 AM5/2/93
to
In <C6BDM...@rice.edu> r...@owlnet.rice.edu (Robert Camp Miner) writes:

>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 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

May 2, 1993, 7:53:12 PM5/2/93
to
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". 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

May 2, 1993, 9:19:07 PM5/2/93
to
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).
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

May 2, 1993, 10:14:59 PM5/2/93
to
(Robert Vienneau) writes:

>zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
>>MLS:
>>>... Gene's usage is completely common in mathematics. It is
>>>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

>>> 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

May 3, 1993, 12:16:55 AM5/3/93
to
In article <C6FBo...@cantua.canterbury.ac.nz>, w...@math.canterbury.ac.nz (Bill Taylor) writes:

>
. . .

> 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

May 3, 1993, 1:02:12 AM5/3/93
to
m...@panix.com (Michael Siemon) writes:

> ...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

May 3, 1993, 1:20:39 AM5/3/93
to
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? >-- > >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 unread, May 3, 1993, 1:48:53 AM5/3/93 to In article <C6FBo...@cantua.canterbury.ac.nz> 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 unread, May 3, 1993, 4:05:14 AM5/3/93 to yIn article <C6FBo...@cantua.canterbury.ac.nz> w...@math.canterbury.ac.nz (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 unread, May 3, 1993, 9:41:04 AM5/3/93 to >Of course, most mathematicians agree (often without great thought) >that [the axiom of choice] 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. 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 unread, May 3, 1993, 9:40:30 AM5/3/93 to 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.

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

May 3, 1993, 4:04:46 PM5/3/93
to
In article <TORKEL.93...@isis.sics.se> tor...@sics.se (Torkel Franzen) writes:
>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

May 3, 1993, 5:27:14 PM5/3/93
to
In article <1993May3.0...@husc3.harvard.edu>, zel...@husc10.harvard.edu (Mikhail Zeleny) writes:
>
. . .

> 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

May 3, 1993, 9:03:34 PM5/3/93
to
Since I was the one to bring up the Axiom of Choice, I feel obligated to
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

May 4, 1993, 7:31:41 AM5/4/93
to
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

May 4, 1993, 10:05:37 AM5/4/93
to
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.
>
> 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

May 4, 1993, 10:35:57 AM5/4/93
to
>
>

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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

May 4, 1993, 11:18:21 AM5/4/93
to
In article <1993May4.1...@mp.cs.niu.edu> ric...@mp.cs.niu.edu (Neil Rickert) writes:
>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.
>>
>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.
>
I am not prepared to have a well-founded opinion about the *validity*
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

--
Dan Heyman d...@bellcore.com

### Michael Weiss

May 4, 1993, 9:14:27 AM5/4/93
to
In article <1993May4.1...@mp.cs.niu.edu> ric...@mp.cs.niu.edu
(Neil Rickert) writes:
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.

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

May 4, 1993, 1:10:22 PM5/4/93
to
In article <1993May4.1...@walter.bellcore.com> d...@wind.bellcore.com
(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

May 4, 1993, 2:48:09 PM5/4/93
to
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.

>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

May 4, 1993, 1:02:05 PM5/4/93
to
In article <TORKEL.93...@anhur.sics.se> tor...@sics.se (Torkel Franzen) writes:

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

May 4, 1993, 4:30:34 PM5/4/93
to
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.
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

May 4, 1993, 6:17:22 PM5/4/93
to
In article <1993May4.1...@husc3.harvard.edu> ku...@kovalevskaia.harvard.edu (Tal Kubo) writes:
>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

May 4, 1993, 6:21:13 PM5/4/93
to
In a lighter vein I remember a cartoon showing two robed monks in
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

May 5, 1993, 3:14:56 AM5/5/93
to
In article <1993May4.2...@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.

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

>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

May 5, 1993, 5:28:43 AM5/5/93
to
In article <COLUMBUS.9...@strident.think.com> columbus@strident.
think.com (Michael Weiss) writes:

>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
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

May 5, 1993, 6:15:32 AM5/5/93
to
I just reread Goedel's article this morning, but I find your last post so
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

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

May 5, 1993, 11:53:14 AM5/5/93
to

> 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.

not even speak for every mathematician in the world, but I do believe
that I'm basically correct.
further if I've done so.)

Simen Gaure
University of Oslo

### Simen Gaure

May 5, 1993, 12:20:19 PM5/5/93
to
> 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.

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

May 5, 1993, 2:40:30 PM5/5/93
to
>> 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.

> [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

May 5, 1993, 12:13:59 PM5/5/93
to
In article <SGAURE.93...@sylow.uio.no>, sgaure@sylow (Simen Gaure) writes:
>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

May 5, 1993, 2:32:16 PM5/5/93
to
In article <SGAURE.93...@sylow.uio.no>
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.

>not even speak for every mathematician in the world, but I do believe
>that I'm basically correct.
> further if I've done so.)

>Simen Gaure
>University of Oslo

cordially,

### Herman Rubin

May 5, 1993, 3:51:17 PM5/5/93
to
In article <1993May5.0...@husc3.harvard.edu> ku...@zariski.harvard.edu (Tal Kubo) writes:
>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

>>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.