HDTenEightyP/NetNews huggingface dataset:
From: kevin@rodin.wustl.edu (Kevin Ruland)
Subject: Tarski and Hilbert's 10th
Originator: kevin@rodin
Organization: Center for Optimization and Semantic Control, Washington University
Date: Fri, 17 Apr 1992 00:07:16 GMT


Tarski proved that there is an algorith that can translate
any proposition in first order theory of the reals into
a quantifier free formula.  Now, Hilbert's 10th problem
(existance of solutions to diaphantane equations) is unsolvable.
That is there is no algorithm that can determine if an
arbitrary diaphantane equation has a solution.  This means that
there is no result similar to Tarski's for the "first order theory
of the integers".  Okay, everyone know this, but could someone
explain why?  What I mean to ask is what is there about the integers
that makes everything so much harder.  I realize that division might
have something to do with it but I can't lay my finger on the
problems.

Thanks.

Kevin Ruland
kevin@rodin.wustl.edu
From: suchenek@nic.csu.net
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 12 Apr 92 16:56:37 PDT
Organization: Information Resources and Technology

In article <1992Apr11.130906.796@gacvx2.gac.edu>, logic@gacvx2.gac.edu writes:
> In article <1992Apr10.161709.5397@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk (R Loader) writes:
>> In article <1992Apr8.100146.689@gacvx2.gac.edu>, logic@gacvx2.gac.edu writes:
>>> In article <1992Apr7.124948.5325@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk (R Loader) writes:
>>> 
>>>> S4 and S5 (and most other common modal logics), are subsystems of classical
>>>> (non-modal) logic extended with the axiom: A iff \box A.
>>> 
>>> Other stuff deleted....
>>> 
>>>> Ralph Loader,
>>>> Mathematical Institute, Oxford.
>>> 
>>> S4 and S5 are supersystems, not subsystems, of classical logic..  Neither
>>> contain the axiom "A iff \box A", both contain the axiom "if \box A then A".
>>> 
>>> Al Hart
>> 
>> Sorry, My original posting was ambiguous: I meant ". . . are subsystems
>>   of the logic obtained by adding A iff \box A to classical logic"
>> --
>> Ralph Loader, Oxford.
> 
> I see the ambiguity now - missed it during my first post..  however, not having
> extracted your first post, i'm not sure i see the point.  extending classical
> logic with the axiom A iff \box A obviously makes \box completely superfluous.
> 
> confused,
> al hart


He is obviously wrong, as two of us already pointed out (I did in # 1463).
"A iff \box A" does not belong to S4 or S5.

Marek Suchenek
From: suchenek@nic.csu.net
Newsgroups: sci.logic
Subject: Re: Set theory and modal logic
Date: 16 Apr 92 12:30:56 PDT
Organization: Information Resources and Technology

In article <1992Apr13.194859.5426@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk writes:
> In article <1992Apr5.213320.27618@morrow.stanford.edu>,
> BR.MJC@forsythe.stanford.edu (Michael Carroll) writes:
> 
[...]

> In particular, if D is a statement of classical set theory, then ZFC(S5) /- D
> if and only if ZFC(S5) /- D.
[...]
> --
> R Loader, Oxford.

If this is all you wanted to say (modulo a typo in the above statement, I
preassume) then it is an easy excercice, indeed, to figure out that:


(*) For any modal-free theory T and modal-free sentence p,
    T |- p  iff  T + S5 |- p.

This fact (*), however, is irrelevant to whether S5 (or its variants) are
usefull in answering the consistency questions like "Is ZFC + \neg HC con-
sistent?". The real problem is somwhere else. Namely, it has been the
treatment of negation that makes S5 too weak to prove non-trivial (as far
as my taste is concerned) consistency results. In particular, if T is a
modal-free theory and p is a modal-free sentence with neither T |- p nor
T |- \neg p, then neither T + S5 |- Kp nor T + S5 |- \neg Kp. Actually,
what one may need here is some kind of "nonmonotonic" rule of inference
which allows (under certain conditions) to infer \neg Kp from non-provabili-
ty of p.

Currently, there is a lot of research in progress (specially on West Coast)
with people trying to find (a reasonable approximation of) such nonmonotonic
variant of S5. Nonmonotonic S5 seems strong enough to prove e.g. certain 
consistency results (this is not the major concern, though). Please note, 
that nonmonotonic S5 still satisfies (*). Therefore, as I said, (*) is
rather irrelevant in this context.

Greetings,

Marek A. Suchenek
Computer Science
Cal State Dominguez Hills
1000 E Victoria St
Carson CA 90747
From: rickert@mp.cs.niu.edu (Neil Rickert)
Subject: Re: Sets in mathematics
Organization: Northern Illinois University
Date: Fri, 10 Apr 1992 04:31:20 GMT

 I usually disagree with Dave Gudeman when I comment on his postings (on
a very different topic).  But I will defend his comments here.

In article <1992Apr10.024357.15902@midway.uchicago.edu> trivedi@rabi.uchicago.edu (Anil Trivedi) writes:
>In article <15127@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
>>In article  <1992Apr9.090844.12520@midway.uchicago.edu> Anil Trivedi writes:
>>]
>
>AT:
>>]In mathematics, traditional arithmetic, algebra, geometry etc. are much
>>]more stable than any "foundations" anyone has provided for them. Make no
>>]mistake, it is those "foundations" that are going to come and go and 
>>]change with time: 
>>]
>>]THESE ARE CLEVERLY WORDED SUPER-PATTERNS, NOT FOUNDATIONS.
>
>DG:
>>The term "foundational" in this sense does not refer to relatively
>>permanency but to relative primacy.  In other words, mathematics in
>>general is built upon a foundational theory, 
>
AT:
>Where did you get that? You should know that mathematics was built,
>not upon these "foundations", but long before anyone thought of them.

  Did you read what David wrote?  He did not say "mathematics was built
upon these foundations."  He used the present tense, not the past.  You
are talking about the historical development of mathematics, while he
is talking about mathematics as it is currently viewed.

  Historically, the foundations were built upon traditional arithmetic, etc,
just as atoms were built upon matter in the history of physics.  As
currently viewed, mathematics is built upon ZF or similar foundations,
just as matter is built from atoms.

>DG:
>>You may
>>not be personally interested in these questions, satisfied to just
>>"use what works", but that is what makes you a physicist 
>AT:
>Actually, for an outsider, I am very interested; that's how I developed
>these opinions. It is most mathematicians who don't give a hoot about
>your "foundations".  :-)

 Most mathematicians do indeed care about foundations.  That does not mean
they spend a lot of time worrying about them, but they do know of their
importance.

>DG:
>> ...only through
>>foundational studies will we ever (possibly) have any proof that
>>mathematics actually works.

>AT:
>What can I say? It is good to have convictions. Goog luck; and I
>do mean that sincerely. 

 If physicists can have convictions about the importance of quarks and
neutrinos, why shouldn't mathematicians have convictions about
mathematical foundations?

-- 
=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=
  Neil W. Rickert, Computer Science               <rickert@cs.niu.edu>
  Northern Illinois Univ.
  DeKalb, IL 60115                                   +1-815-753-6940
Newsgroups: sci.philosophy.tech
From: ske@pkmab.se (Kristoffer Eriksson)
Newsgroups: sci.philosophy.tech,comp.ai.philosophy
Subject: Re: A rock implements every FSA
Date: 8 Apr 92 22:18:22 GMT
Organization: Peridot Konsult i Mellansverige AB, Oerebro, Sweden

In article <1992Apr6.114955.10762@husc3.harvard.edu> zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
>In article <6741@pkmab.se> ske@pkmab.se (Kristoffer Eriksson) writes:
>>In article <1992Apr1.081958.10553@husc3.harvard.edu> zeleny@zariski.harvard.edu (Mikhail Zeleny) writes: 
>
>MZ:
>>>I note that any description of Chalmers' wacky notion of implementation is
>>>conspicuously absent from the above.
>
>KE:
>>As if it was less wacky to hold the view that a rock counts as an
>>implementation of each and every FSA...
>
>Consider the notion of "reductio ad absurdum".

As far as I can see, the only thing that it reduces to absurdity is Putnam's
own notion of "implementation", which several people (including me, and your
primary target) have criticized over and over again. So what's the point?

(To summarize my own version of the criticism that I gave in another article:
Putnam's mapping of abstract FSA states onto a different (and random) physical
state for each input results in only an incomplete mapping of the FSA's
transition function onto the rock's physical transition function. It leaves
holes in the mapping of the transition function. Therefore, it is logically
false to claim that the rock is accurately described by the abstract FSA,
which I would presume should be a minimum requirement for any reasonable
notion of "implementation". - Your counterattack that there are problems in
reasoning about possible worlds seems totally beside the point; Putnam's
notion of "impementation" is no less flawed because of that. Did I mention
any possible worlds? I might even abstain from mentioning physical laws,
you know, but that might bring up some other bulky issues.)


>KE:
>>Whatever the correct notion of implementation may be, I think it should
>>be one that excludes arbitrary rocks, otherwise it simply is useless, and
>>definitely does not capture the thing we use this notion for.
>
>Agreed, but this doesn't exclude the possibility that the correct notion of
>implementation, like the related notion of interpretation, or the
>fundamental notion of reference, is simply unavailable to a run-of-the-mill
>functionalist.

But in what way does Putnam's argument help in demonstrating that point,
especially if the functionalist in question does not swallow Putnam's
notion of "implementation"?

> Chalmers may yet escape this quagmire,

You haven't even shown him to be IN any quagmire. You have still not
really explained why he should be concerned about your possible worlds
objection. You just keep repeating it with no additional material.

-- 
Kristoffer Eriksson, Peridot Konsult AB, Hagagatan 6, S-703 40 Oerebro, Sweden
Phone: +46 19-13 03 60  !  e-mail: ske@pkmab.se
Fax:   +46 19-11 51 03  !  or ...!{uunet,mcsun}!mail.swip.net!kullmar!pkmab!ske
Newsgroups: sci.philosophy.tech
From: ian@cambridge.oracorp.com (Ian Sutherland)
Subject: Re: Sets in mathematics
Organization: ORA Corp, 675 Mass Ave, Cambridge, MA 02139
Date: Tue, 14 Apr 92 02:26:58 GMT

In article <s85diINNnet@agate.berkeley.edu> epfaith@twinkies.berkeley.edu (Edward Paul Faith) writes:
>In article<15204@optima.cs.arizona.edu>, gudeman@cs.arizona.edu 
>(David Gudeman) writes:
>>I'm
>>inclined to be more interested in the actual _practice_ of mathematics
>>than in the (rather naive) philosophy of its practitioners.
>
>What is the astounding fact here?  The astounding fact seems to be
>that even though mathematicians don't understand what they are doing,
>as you put so diplomatically, nevertheless what they do works.
>Mathematicians make mistakes, but these are eventually recognized as
>such without need for deep reflection as to the nature of mathematics.

It's true, the mistakes are recognized as mistakes without the need
for deep reflection.  They're usually recognized when someone
"proves" a statement whose negation has already been "proven".  What
takes deep reflection is figuring out what's wrong with the existing
approach, and how to fix it so the same kinds of mistake won't happen
again.  Your example of Euclid is misleading.  Euclid is perfectly
usable today, but Newton's calculus is not.  If a student nowadays
submitted proofs of results by the kinds of methods that Newton used,
he would get poor grades.  What's considered a rigorous argument in
analysis now is a direct result of considering the foundations of
analysis.  Your implication that the methods of mathematics don't
change with time, or that those changes have little to do with
foundational considerations, is simply incorrect.
-- 
Ian Sutherland                          ian@cambridge.oracorp.com

Sans peur
From: epfaith@twinkies.berkeley.edu (Edward Paul Faith)
Newsgroups: sci.philosophy.tech
Subject: Re: Sets in mathematics
Date: 10 Apr 1992 19:54:19 GMT
Organization: U.C. Berkeley Math. Department.
NNTP-Posting-Host: twinkies.berkeley.edu

There is an ongoing dispute over the ultimate meaning of the study of
mathematical foundations.

One might think that the study of foundations is a sort of science,
like biology or linguistics.  This would justify saying that for the
usefulness and validity of "foundations" it does not matter whether or
not mathematics was already around long before its foundations were
examined, since living beings were around long before biology was
studied, and languagage was around long before there was a
linguistics.

But the study of foundations is different from biology and
linguistics, in that it is itself a field of mathematics, and
mathematics is what it claims to study.  Biology, in contrast, is not
a living being, at least not in the usual sense.

What keeps "foundations" from being just another field of math, as
opposed to being a commentary on all math?  As mathematicians like to
say, math is defined by being not about anything.  Mathematical
statements are ultimately empty, mere symbol manipulation, until the
user imbues them with some meaning, some extramathematical use.  But
this imbuing is most unmathematical.

Strictly as a field of mathematics, then, the study of foundations is
as vacuous as any field of mathematics.  What then gives it its
extramathematical meaning?  And is this meaning something that a
self-respecting mathematician should trust, or is it just a bunch of
handwaving?

To be more specific, calculus has clear application.  It is loaded
with meaning and utility.  This is not inherent in the math, but
rather is a gift from sciences like physics.  Most importantly, there
is no controversy about this particular way of applying mathematics.
But the extramathematical meaning of "foundations" is philosophically
much more worrisome, not least because it gives primacy to a certain
field of mathematics which on the face of it (just looking over the
theorems from a symbolic point of view) does not seem any more
inherently meaningful than other fields like topology and arithmetic.
Indeed, if I wished to assure myself of the validity of "foundations",
I would probably found it on arithmetic, which I trust more.

"The study of foundations" involves not only math, but also
philosophy.  I have ignored the latter aspect, since the worldly
sciences do not generally seem to need philosophy to back them up, and
I have been trying to show how different "foundations" is from worldly
sciences like biology.  However, the real sticky questions of the
meaning of math and so forth are in the end purely philosophical, and
it is to the philosophers that we must turn (this includes lay
philosophers).  Thus in my view "foundations" has most to offer as a
field of pure philosophy.

But should we really trust philosophy more than we trust arithmetic?
Newsgroups: sci.philosophy.tech
From: ian@cambridge.oracorp.com (Ian Sutherland)
Subject: Re: Sets in mathematics
Organization: ORA Corp, 675 Mass Ave, Cambridge, MA 02139
Date: Sun, 12 Apr 92 20:12:39 GMT

In article <s6elfINNjd1@agate.berkeley.edu> epfaith@twinkies.berkeley.edu (Edward Paul Faith) writes:
>Godel's theorems prove that within a certain calculus one could not construct a
>formal proof of certain facts about that calculus.

Godel's theorems apply to a large class of calculi, not just one.
Perhaps you just misspoke.

>Hilbert wanted to formalize
>all of mathematics into a single calculus (so I recall), and Godel showed that
>this was impossible, since no matter what calculus we develop, if it is
>sufficiently complex, we will be able to prove things about it, without being
>able to carry out the proofs within that calculus.  However, these proofs, while
>not realizable within the calculus, are nevertheless mathematical proofs. This
>is why mathematics cannot be completely formalized.

This is not a correct interpretation of Godel's theorems.  It is true
if we apply Godel's theorem to true theories in the language of
arithmetic.  It is not true, for example, if we apply Godel's theorem
to ZFC.  Godel's second incompleteness theorem applied to ZFC tells
us that if ZFC is consistent, then CON(ZFC) is independent of ZFC.
There is, however, no mathematical proof of the consistency of ZFC.

[Aside: There are those who read this newsgroup who believe that we
can reliably believe CON(ZFC) because of some kind of "inductive"
reasoning (where "inductive" does not mean "mathematical induction",
but a nondeductive form of reasoning).  While this may in some sense
be reasonable, I claim that such "inductive" arguments are not what
are ordinarily referred to as "mathematical proofs", and that such
arguments will never appear in mathematical journals as "proofs" that
CON(ZFC) is a theorem.]

>My careless wording has led you to believe that I agree with Hilbert.  Well, I
>don't, as you can plainly see.  In truth, mathematics uses not a formal system,
>but an enormously sophisticated symbolic system, namely language.

The trouble with natural language is that it is inconsistent.
-- 
Ian Sutherland                          ian@cambridge.oracorp.com

Sans peur
Xref: sparky sci.philosophy.tech:2068 comp.ai.philosophy:4131
From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: sci.philosophy.tech,comp.ai.philosophy
Subject: Re: A rock implements every FSA
Date: 6 Apr 92 15:49:53 GMT
Organization: Dept. of Math, Harvard Univ.

In article <6741@pkmab.se> 
ske@pkmab.se (Kristoffer Eriksson) writes:

>In article <1992Apr1.081958.10553@husc3.harvard.edu>
>zeleny@zariski.harvard.edu (Mikhail Zeleny) writes: 

MZ:
>>I note that any description of Chalmers' wacky notion of implementation is
>>conspicuously absent from the above.

KE:
>As if it was less wacky to hold the view that a rock counts as an
>implementation of each and every FSA...

Consider the notion of "reductio ad absurdum".

KE:
>Whatever the correct notion of implementation may be, I think it should
>be one that excludes arbitrary rocks, otherwise it simply is useless, and
>definitely does not capture the thing we use this notion for.

Agreed, but this doesn't exclude the possibility that the correct notion of
implementation, like the related notion of interpretation, or the
fundamental notion of reference, is simply unavailable to a run-of-the-mill
functionalist.  Chalmers may yet escape this quagmire, if he chooses to
trade his "reluctant dualism" for a full-blown Fregean troilism; however it
seems that every reductive or eliminative materialist will be stuck with a
Davidson-like inscrutability of implementation.

>-- 
>Kristoffer Eriksson, Peridot Konsult AB, Hagagatan 6, S-703 40 Oerebro, Sweden
>Phone: +46 19-13 03 60  !  e-mail: ske@pkmab.se
>Fax:   +46 19-11 51 03  !  or ...!{uunet,mcsun}!mail.swip.net!kullmar!pkmab!ske

N.B.  I haven't answered your article about semantics simply because there
seems to be nothing to say, beyond reiterating that your quasi-Russellian
view admits the possibility of referring to phenomenal data only.


`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'
: Qu'est-ce qui est bien?  Qu'est-ce qui est laid?         Harvard   :
: Qu'est-ce qui est grand, fort, faible...                 doesn't   :
: Connais pas! Connais pas!                                 think    :
:                                                             so     :
: Mikhail Zeleny                                                     :
: 872 Massachusetts Ave., Apt. 707                                   :
: Cambridge, Massachusetts 02139           (617) 661-8151            :
: email zeleny@zariski.harvard.edu or zeleny@HUMA1.BITNET            :
:                                                                    :
'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`'`
From: jeff@aiai.ed.ac.uk (Jeff Dalton)
Newsgroups: sci.philosophy.tech
Subject: Re: Freewill
Date: 9 Apr 92 17:58:05 GMT
Organization: AIAI, University of Edinburgh, Scotland

In article <1992Mar31.204712.22637@cambridge.oracorp.com> ian@cambridge.oracorp.com (Ian Sutherland) writes:

>Don't confuse "philosophers" with the people that read this newsgroup
>(your quotes around the word "philosophers" may indicate that you are
>not a bit confused about this).  If people who posted to "news"
>clearly defined their terms before using them (and if others didn't
>spend a lot of time arguing about whether their definitions were
>"right"), I can easily believe that 99% of all the postings on this
>group (and several others I could name) would be avoided.  If it only
>it could be ...

What _are_ you sugqesting?  That every message include a glossary?
That we all keep track of everyone else's idiosyncratic vocabulary?

Moreover, it is quite wrong to suppose that if everyone defined
their terms and agreed not to argue about what definition was
right all would be well.  If we can't use agreed meanings, it's
difficult to talk at all.  And people who use nonstandard meanings
can make it almost impossible to talk about standard issues
(try talking with Objectivists about possibility some time).

So far as I can tell, most demands for definitions on the net
are nothing more than opportunistic attempts to shift the burden
of proof.  If anyone out there is really convinced that definitions
are important, they should begin by offering definitions, not by
demanding them.

BTW, the philosophical position that most disputes are merely verbal
has been tried and found wanting.

-- jd
From: ske@pkmab.se (Kristoffer Eriksson)
Newsgroups: sci.philosophy.tech,comp.ai.philosophy
Subject: Re: A rock implements every FSA
Date: 4 Apr 92 03:02:29 GMT
Organization: Peridot Konsult i Mellansverige AB, Oerebro, Sweden

In article <1992Apr1.081958.10553@husc3.harvard.edu> zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
>I note that any description of Chalmers' wacky notion of implementation is
>conspicuously absent from the above.

As if it was less wacky to hold the view that a rock counts as an
implementation of each and every FSA...

Whatever the correct notion of implementation may be, I think it should
be one that excludes arbitrary rocks, otherwise it simply is useless, and
definitely does not capture the thing we use this notion for.

-- 
Kristoffer Eriksson, Peridot Konsult AB, Hagagatan 6, S-703 40 Oerebro, Sweden
Phone: +46 19-13 03 60  !  e-mail: ske@pkmab.se
Fax:   +46 19-11 51 03  !  or ...!{uunet,mcsun}!mail.swip.net!kullmar!pkmab!ske
Xref: sparky sci.philosophy.tech:2079 comp.ai.philosophy:4230
From: ske@pkmab.se (Kristoffer Eriksson)
Newsgroups: sci.philosophy.tech,comp.ai.philosophy
Subject: Re: A rock implements every FSA
Date: 8 Apr 92 07:58:38 GMT
Organization: Peridot Konsult i Mellansverige AB, Oerebro, Sweden

In article <1992Apr2.012749.10596@husc3.harvard.edu> zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
>In article <60507@aurs01.UUCP> throop@aurs01.UUCP (Wayne Throop) writes:
>>Exactly so.  And the requirement not to depend upon time is very modest
>>indeed, yet seems to kill Putnam's line of reasoning quite dead.
>
>  Start by individuating a
>material object that persists through time.  Keep in mind the "ship of
>Theseus" problem of loss and replacement of its material constituents.

Is the chair you sit in the same object as yesterday? Is the planet you
walk on the same object as yesterday? Indeed, is your own body the same
object as it was yesterday?

Although these questions may be theoretically challenging, they don't seem
to cause us any trouble in every-day life or in physical sciences. Why
should we make them into any more of a problem in the case of implementing
FSAs?

-- 
Kristoffer Eriksson, Peridot Konsult AB, Hagagatan 6, S-703 40 Oerebro, Sweden
Phone: +46 19-13 03 60  !  e-mail: ske@pkmab.se
Fax:   +46 19-11 51 03  !  or ...!{uunet,mcsun}!mail.swip.net!kullmar!pkmab!ske
Xref: sparky sci.philosophy.tech:2114 comp.ai.philosophy:4326