HDTenEightyP/NetNews huggingface dataset:
From: ndanner@silver.ucs.indiana.edu (norman danner)
Newsgroups: sci.math
Subject: Re: Most important event in the history of mathematics
Date: 29 Sep 92 00:28:08 GMT
Organization: Indiana University

[Lot's o' contenders deleted for space conservation, rather than out
of disrespect...]

Hand's down, the first and second incompleteness theorems.

I think the most important result really depends on the area of mathematics.
In particular, most analysists will proabaly disagree with me.  But I
think there's a strong argument to be made that the incompleteness theorems
were the most important results when it comes to philosophy of mathematics.
Just my $.02

-- 
Norman Danner                    |"There are three kinds of mathematicians:
"The guy with the hair."         | Those who can count, and those who can't."
ndanner@indiana.edu              | --???
-----------------------------------------------------------------------------
Newsgroups: sci.math
From: jane@mathematik.uni-Bremen.de (wayne tvedt)
Subject: Re: Hyperbolic space?
Organization: University of Oldenburg, Germany
Date: Tue, 29 Sep 1992 12:17:34 GMT
X-Newsreader: TIN [version 1.1 PL6]

Allan Adler (ara@zurich.ai.mit.edu) wrote:
>Maybe it is time someone wrote for hyperbolic space the book analogous to
>Flatland.

A French guy, Jean-Pierre Petit, has written some nice cartoon books about
curved spaces, relativity, and like topics.  I have some German editions
from Physik-Verlag.

Begins _The Geometricon_:

  "This book is neither a textbook nor a coursebook.  It is simply the 
  story of Anselm Lanturlu (German:  "Wuesstegern") and one of his 
  excursions into the land of geometry." 

Includes little experiments in differential geometry you can do with
scissors and tape.  Who needs 24-bit color workstations?  Unfortunately, 
Petit's _The Black Hole_ will never be used in American primary schools, 
because it starts off with Anselm and his friend Herr Albert toking on 
a hookah. 

Newsgroups: sci.math
From: kyee@bnlux1.bnl.gov (kenton yee)
Subject: Re: Mattel says "Math is tough."
Organization: Brookhaven National Laboratory, Upton, NY 11973
Date: Mon, 5 Oct 1992 21:21:15 GMT

In article <BvLyKC.LFH@rice.edu> iltiss@owlnet.rice.edu (Steven Patrick Iltis) writes:
>        Mattel has just released a new product. A Barbie doll that says
>"Math is tough".
>        This thoughtless attack on mathematics education in general and
>women in particular should not be allowed to happen without a significant
>protest from the mathematics and scientific community.

  Why do you assume women will be turned off by math simply
 because it is tough?   I know lots of people who like math
  BECAUSE it is tough.  



Newsgroups: sci.math
From: mkrogers@unix.amherst.edu (MICHAEL K ROGERS)
Subject: Re: various number systems
Organization: Amherst College
X-Newsreader: TIN [version 1.1 PL6]
Date: Tue, 29 Sep 1992 14:48:10 GMT

lloyd mitchell (lmitchel@nyx.cs.du.edu) wrote:
: I am currently teaching a pre-calculus course at a local community college.
: We have just covered the quadratic formula, which has served as an intro-
: duction to complex numbers.  Now, I would like to expose my students to
: other numbering systems.  My goal is not to teach them the ins and outs of
: the systems, but to expose them to other ways of thinking about numbers.
: 
: I'm aware of quaternions and perplex numbers.  Are there other number
: systems out there about which I should tell them?

There's a book published by Springer-Verlag: "Numbers",
Ebbinghaus, et. al.  It is sometimes at a high level,
but it might give you some ideas.

I don't know what perplex numbers are, but here are some ideas.
They should probably know about the relationship of the
rationals to the reals.  (They probably know a little already.)

You could add to the real numbers a new invented number called
epsilon with the following properties:
	a) epsilon > 0;
	b) for every real number  x > 0,  epsilon < x .
You think of epsilon as an "infinitessimally small" number.
You now take the closure under addition, multiplication, etc.
You can use this new field of numbers to do calculus without
limits (Nonstandard Analysis).  There are also a book by Berlekamp,
Conway, and Guy called "Winning Ways for Your Mathematical Plays"
(Academic Press) and another by Conway called "On Numbers and Games"
(also Academic Press) which use similar number systems to analyze
games.

There are also the p-adic numbers where p is a prime.  This
would probably take you too far afield.  For those who don't
know about this field:  The p-adic rationals can be described
as the set of formal infinite power series
	bn * p^-n + ... b1 * p^-1 + a0 + a1 * p + a2 * p^2 + ...
where the coefficients are between 0 and p-1 (inclusively).
Addition, multiplication, etc. are done is a way analogous
to decimal addition, multiplication, etc. (with carries
when a coefficient get too big or becomes negative).  There
is a notion of absolute value which has similar properties
to the ordinary absolute value (triangle inequality, product
rule), but it is remarkable different.  For example, the absolute
value of p^n gets smaller as the integer n gets bigger.  The
rational numbers are a subset of them and have a similar
relationship to them as they do to the real numbers.
(They are dense.)  They can be explained is a fairly elementary
way, but I don't think the students would get any appreciation
of them in less than a week or two.

	-mike
Newsgroups: sci.math
From: torkel@sics.se (Torkel Franzen)
Subject: Re: ZFC etc. (was Re: Report on Philosophies of Physicists)
Followup-To: sci.math
Organization: Swedish Institute of Computer Science, Kista
Date: Sat, 26 Sep 1992 23:26:26 GMT

In article <22846@optima.cs.arizona.edu> gudeman@cs.arizona.edu 
(David Gudeman) writes:

   >but the previous researchers certainly did assume the
   >existence of an unrestricted comprehension axiom.  They only began
   >restricting it when the various paradoxes began to arise.

  Which previous researchers, and how does it emerge that they assumed
an unrestricted comprehension axiom?

   >In fact
   >Cantor's proof that the cardinality of the set of subsets of S is
   >larger than the cardinality of S relies pivotally on the creation of a
   >set that degenerates to Russell's paradoxical set if the theorem were
   >not true.

  It's unclear what you mean to say here. Russell did indeed derive his
paradox by applying Cantor's proof to a supposed universal set. What
does this have to do with whether or not anybody assumed an unrestricted
comprehension axiom?
From: RVESTERM@vma.cc.nd.edu
Newsgroups: sci.math
Subject: Re: Mattel says "Math is tough."
Date: 4 Oct 92 23:36:26 GMT

uh, as a math grad student, i must agree with barbie.
Newsgroups: sci.math
From: anshu@cs.UAlberta.CA (Anshuman Tyagi)
Subject: Queueing Theory Problem
Organization: University of Alberta, Edmonton, Canada
Date: Tue, 6 Oct 1992 04:48:54 GMT


Hi there,

Can someone provide me the refrence to the following problem:

Probability of having n2 customers at time t when at time 0 the number
of customers in the queue were n1 and the server started serving the
queue at this time. The server is busy throughout time t i.e. the queue 
is never empty between time 0 and t. The arrival and departure processes
are Poisson.

Thanks,

Anshuman.
anshu@cs.ualberta.ca

Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Mattel says "Math is tough."
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Tue, 6 Oct 92 01:42:44 GMT

In article <5805@tuegate.tue.nl> wsadjw@urc.tue.nl writes:
>In article <92278.183626RVESTERM@vma.cc.nd.edu> RVESTERM@vma.cc.nd.edu writes:
>>uh, as a math grad student, i must agree with barbie.
>
>Would people get upset if barbie said "Running marathons is tough"?
>Or "relativity is tough"?  Or "Homer is tough"?

They never would have made a Barbie doll say any of these things.  

>Or is it that nowadays in North American culture anything more advanced
>than 1+1 is considered (a) math (b) tough ?

Math is generally considered tough, but especially for girls, who should
be practicing combing their hair and trying on new outfits.

In any event, this has apparently made it onto TV here in LA (didn't see
it), and also in the newspaper it apparently gives a location where you
can return your Barbie doll (haven't read it yet - but I told my math
class you can exchange it for one that says "The Hahn-Banach theorem is
very useful!")
From: hbtnn@gruc18.nor.chevron.com (Brian Tanner)
Newsgroups: sci.math
Subject: Re: Square root of i
Date: 29 Sep 92 21:40:12 GMT
Organization: Chevron

In article <1992Sep25.163119.18316@ima.isc.com>, karl@ima.isc.com (Karl Heuer) writes:
|> In article <1992Sep24.160754@gruc18.nor.chevron.com> hbtnn@gruc18.nor.chevron.com (Brian Tanner) writes:
|> >[expand (a+bi)^2 = 0+1i and solve for a and b]
|> >
|> >too much work.
|> >use r * exp(i theta) notation...
|> 
|> I don't think that's simpler.  When I first solved this (4th grade, as I
|> recall), I knew how to handle quadratic equations, but not the trigonometry
|> required by your solution.
|> 
|> Karl Heuer   karl@ima.isc.com   uunet!ima!karl

Depends on your definition of simpler.
I find rotating a vector, then doing a quick conversion, using a 45-45-90
triangle, a lot easier, and less error prone, than trying to solve 
simulaneous equations equations.

(And yes, Charles, there are two roots: -rexp(i*pi/4).  There may even
 be more, but I'm too lazy to prove it either way)

Brian
From: brnstnd@nyu.edu (D. J. Bernstein)
Newsgroups: sci.math
Subject: Re: Topos theory for mere mortals (?)
Date: 27 Sep 92 21:57:27 GMT
Organization: IR

In article <1992Sep27.121129.114@gems.vcu.edu> fritz@gems.vcu.edu (J. Fritz) writes:
> The point is that these statements, if translated into bare ZF, would
> look quite different.

Why would anyone engage in such a translation? We were talking about
what is _convincing_. This bears very little relation to what remains
convincing when stated in ``bare ZF.''

The question of exactly _which_ set of real numbers, or _which_ set of
integers, we happen to be talking about, is about as interesting as the
question of whether the 2-adic integers are sequences of bits or
sequences of integers mod 2^m for various m. Nobody cares. Every theorem
about these objects is implicitly preceded by a statement of the form
``Let Z be an integer set, Q a rational set containing Z, .... By the
way, there exist objects satisfying these assumptions.'' When it's more
convenient one might instead assume ``Let Z be an integer set, Q the set
of equivalence classes of pairs of...'' The most basic fact about these
objects is that the different constructions are all isomorphic. In this
case, everyone knows that any internal localization Q of Z is the same
as the external localization Q of Z, so nobody bothers saying so
explicitly. Nobody cares exactly _which_ rationals you use.

---Dan
Newsgroups: sci.math
From: ara@zurich.ai.mit.edu (Allan Adler)
Subject: Re: Opinions Sought (Why major in math?)
Organization: M.I.T. Artificial Intelligence Lab.
Date: Thu, 1 Oct 1992 08:23:19 GMT


One of the failures of our educational system is that no one in the
so-called real world knows what mathematics is and therefore no one
knows what to do with a mathematician. This tends to be prejudicial to
the employment options of mathematicians.

That seems like a good reason not to major in math. However, I am often
reminded of my orientation when I started college in 1964 and to my
surprise I find that I was actually given some good advice, for example:
"The job you are preparing yourself for may not exist by the time you
graduate."

I think that if one takes this advice to heart, the logical conclusion
is not to organize your education around giving yourself the design
specifications you think that an employer wants to see. Instead, organize
it around giving yourself the best critical faculties and the greatest
versatility in conducting your own education after you graduate. Use your
time in school to enrich yourself culturally, technically and intellectually 
in every way that you can. There is no point in depriving yourself of this
with the illusion that you will be better off economically for having
done so. An employer will happily screw you anyway, no matter what
illusions you brought with you.

So if you like math, then study it. And no matter what your major is,
don't let yourself be circumscribed by it, because whatever it is,
there is much more to intellectual and real life.

In connection with the practice of inconveniencing oneself in order
to do what one thinks an employer might want, I would like to introduce
here a concept which I have found useful in some of my own thinking
on this subject, namely the concept of "metawork". 

What is metawork? Well, work is work you do in the hope of being rewarded
with money. Metawork is work you do in the hope of being rewarded with
work. Metawork is the stuff that exploitation feeds on. There is no limit
to the amount of metawork you can be suckered into doing and always it is
with the same justification: a hint of a rumour that someone will reward
you with remunerated work. The trouble with metawork is that it is so
abundant that an economic niche has been created for those parasites
who thrive on the metawork of others. And these parasites are sheltered
in their niche by the fact that they never actually promised you anything
in exchange for doing metawork and are under no obligation to reward
you for having done it. Why, it can even be argued that you wanted to
do metawork, for reasons that were generated by your own illusions
about how things really work.

When you rearrange your entire education when no one actually offered
you anything in exchange, you are doing metawork and accordingly you
are the instrument of your own exploitation. 

Allan Adler
ara@altdorf.ai.mit.edu
From: dsg@impmh.uucp (Dave Gordon)
Newsgroups: sci.math
Subject: Re: Hyperbolic space?
Date: 1 Oct 92 16:13:24 GMT
Organization: Integrated Micro Products Ltd

In article <1992Sep24.000116.26133@pellns.alleg.edu> frisinv@alleg.edu writes:
>
>This may sound weird, but could someone tell me what it is and what it  
>would look like to move around in hyperbolic space. 
>
>The concept would fit in very well with a story I'm writing.

Check out "Inverted World" by Christopher Priest.

It's set on a hyperboloid analogue of the earth, with the added
complication that the ground moves (away from the pole, towards
the equator - at least in the hemisphere where the story is set).
There are space/time distortion effects, getting more pronounced
as one moves away from the 45 degree area.  The inhabitants
(descendants of colonists from earth) live in a city, which must
constantly move (relative to the ground) in order to stay in the
same place (relative to the world).

There are two versions of this story; the original short story, and
an expanded, novel-length version.  Personally, I prefer the short story,
but the other version is OK, if a little less consistent.

On the other hand, Chris Priest once explained that the inspiration for
actually writing the story comes from a time when he was discussing
a hyperbolic world with (scientist) friends, and he realised that
"there was a hole in the physics so big I could drive a city through it".

-- 

                        Dave
===============================================================================
        You are in a twistly little maze of standards, all different
===============================================================================
From: bennett@math.ksu.edu (Andy Bennett)
Newsgroups: sci.math
Subject: Re: Lebesgue measure in Fractal Dimension
Date: 6 Oct 1992 10:44:51 -0500
Organization: Dept. of Mathematics, Kansas State University
Distribution: usa
NNTP-Posting-Host: hilbert.math.ksu.edu

sassoj@rs6412.ecs.rpi.edu (John J. Sasso Jr.) writes:
>   Now, the Cantor set has Lebesgue measure 0 for the Lebesgue measure defined
>in spaces R^1, R^2, etc...  However, the Cantor does have a fractal dimension
>(or Hausdorff dimension) between 0 and 1 (I cannnot recall the value), and
>I'll call this dimension Df.  Is it possible to define a Lebesgue measure in
>a space of dimension Df (R^Df) such that the measure of the Cantor set is a 
>value other than 0?

It is possible to define a measure certainly, though it isn't Lebesgue 
measure as you noted. It is Hausdorff measure. The only introductory text
I know that covers this topic is Billingsley's "Probability and Measure"
which uses Hausdorff measure to discuss measures on curved 2-dimensional
subsets of R^3 and similar problems (the book predates interest in fractals)
I will steal freely from that text in what follows.

Let A be a set in R^k. For positive m and s let

h_{m,s}(A)=c_m*inf(Sum(diam B_n)^m) 

where the inf extends over countable
coverings of A by sets B_n with diameters < s and c_m is a constant to be
determined later. As s decreases the inf is taken over 
a smaller collection so h_{m,s} increases and we can define

h_m(A)=limit as s goes to 0 of h_{m,s}(A).

This is the m-dimensional outer Hausdorff measure of A. Now we can do the
usual process of restricting the domain of h_m to the sigma-algebra of
sets satisfying the Caratheodory condition and obtain the Hausdorff measure.
This is a nice Borel measure and everything. We do have to pick the 
normalizing constant c_m. The general rule is

c_m=2^{-m}\pi^{m/2}/\Gamma(1+m/2).

Note that with this definition of c_m m-dimensional Hausdorff measure
agrees with m-dimensional Lebesgue measure on sets where both are defined.

It is now easy to define Hausdorff dimension. We can prove that if
h_m(A)=0 then h_l(A)=0 for l>m and if h_m(A)>0 then h_n(A)=infinity
for n<m. So we let the Hausdorff dimension of a set be 

d(A) = inf{l:h_l(A)=0} = sup{n:h_n(A)=infinity}

Note that h_{d(A)}(A) may =0 or =infinity or anything in between.

This should get you going. I haven't seen any books on fractals or the
like that get into this much measure theory but there must be some out
there. Maybe somebody else has a good reference for further reading and/or
the exact Hausdorff dimension and corresponding Hausdorff measure of the
Cantor middle-thirds set.

Andy Bennett 

-- 
Andrew G. Bennett         bennett@math.ksu.edu         If you count too
Dept. of Mathematics      Voice: (913) 532-6750        much you turn
Kansas State University   Fax:   (913) 532-7004        purple.  -  SARAH
Manhattan, KS 66502       STRICTLY MY OWN OPINIONS                    
Newsgroups: sci.math,rec.puzzles,comp.theory
From: cs202144@umbc8.umbc.edu (cs202144)
Subject: Re: more on .999... = 10.0
Organization: University of Maryland, Baltimore County Campus
Date: Mon, 5 Oct 1992 14:36:46 GMT

This probably isn't a theorem, but I believe it can be proved. Any repeating
set of numbers can be placed over as many nines as there are numbers to be
repeated. For example...If you want the number 0.54545454... then place the
54 over 99 (since there are two digits) and you get 54/99 which works and
happens to reduce to 6/11. If you look at just integers you will notice the
pattern:

1/9       = .11111111...
2/9       = .22222222...
3/9 = 1/3 = .33333333...
4/9       = .44444444...
5/9       = .55555555...
6/9 = 2/3 = .66666666...
7/9       = .77777777...
8/9       = .88888888...
9/9 = 1/1 = .99999999...

Wow does that prove it...That 10x method does work since all the books do it
that way, but my way uses less algebra and more visualization.
From: pbond@spitws100.sbil.co.uk (Philip Bond)
Newsgroups: sci.math
Subject: Re: Most important event in the history of mat
Date: 2 Oct 92 12:36:04 GMT
Reply-To: pbond@spitws100.sbil.co.uk
Organization: Salomon Brothers, Ltd.

My vote is split : The first event was the development of the decimal 
system. Previous cultures ( Greek, Roman etc. ) failed to develop beyond
geometry mainly due to a lack of this system. With the invention of 
decimal representation mankind really took a giant leap forward. Of 
course it took time to develop but I think ( my opinion only ) that the
Arab mathematicians should be given credit for bringing the system to 
fruition.
	The second development was the calculus.


Phil.





My humble opinion only. Usual disclaimers apply.
Newsgroups: sci.math
From: sbiederman@mentorg.com (Steve Biederman)
Subject: Re: "Solvability" in Chess
Organization: Mentor Graphics Corporation
Distribution: sci
Date: Tue, 29 Sep 1992 21:14:58 GMT

In article <7396@armltd.uucp> dseal@armltd.co.uk (David Seal) writes:
>
> But it really *is* an assumption that chess is a zero-sum game, not a
> provable fact. In fact, it can often be shown that in a particular game,
> this isn't the case. E.g. consider the case of a player in a chess
> tournament who needs just half a point more to clinch first place in the
> tournament. For him, a draw is just as valuable as a win. If he's playing
> someone from the middle of the rankings, to whom the usual win > draw > loss
> order applies, the values of a draw to the two players have a greater sum
> than the values of a first player win. I.e. it is *not* a zero-sum game.

    These players are not playing the game called Chess.  They
are playing the game called Chess Tournament.  These are obviously not
the same game.  Chess is obviously a zero-sum game.  Other games
layered on top of Chess may or may not be.  (Which makes me wonder about
situations where the opposing players think themselves to be playing
*different* games.  E.g., Black thinks he is playing Chess, but
White has been convinced by a gambler to throw the game, so White is
actually playing the game "Lose in a non-obvious manner so as to keep 
my legs unbroken and make some money at the same time".  Thinking about 
such situations makes my head hurt, so I won't.  Unless any sci.math'ers 
want to comment on them . . .)

                            Steve Biederman        sbiederman@mentorg.com
From: edgar@function.mps.ohio-state.edu (Gerald Edgar)
Newsgroups: sci.math
Subject: Re: set theory
Date: 6 Oct 92 12:29:35 GMT
Distribution: sci
Organization: The Ohio State University, Dept. of Math.
NNTP-Posting-Host: function.mps.ohio-state.edu

In article <1992Oct6.021037.6554@math.utexas.edu> epstein@rene.ma.utexas.edu (Paul Epstein) writes:
>Let's not assume the continuum hypothesis.  Let S be a subset of the real
>numbers with cardinality less than that of the reals.
>Does S have Lebesgue measure zero?
>

Certainly such a set has inner measure zero, so the question is
whether such a set must be Lebesgue measurable at all.

The answer is:  yes or no.

The assertion is independent of ZFC+notCH.  It is a consequence of
Martin's Axiom.  If the continuum is a real-valued measurable cardinal,
then it is false.  There is a model of ZFC where the continuum is aleph_2,
and there is a set of cardinal aleph_1 with positive outer measure.
{Forcing enthusiasts can add aleph_2 random reals to a model of CH.
Any aleph_1 of them can serve as our set.}

I found this material in section 4 of Fremlin's paper in Manuscripta
Math 15 (1975) 219--242.

-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
Newsgroups: sci.math
From: epstein@rene.ma.utexas.edu (Paul Epstein)
Subject: set theory
Organization: University of Texas at Austin Mathematics
Distribution: sci
Date: Tue, 6 Oct 1992 02:10:37 GMT

Let's not assume the continuum hypothesis.  Let S be a subset of the real
numbers with cardinality less than that of the reals.
Does S have Lebesgue measure zero?

(I heard this question through a faculty member at Austin Texas though
it has probably been asked before).

Paul Epstein
Xref: sparky sci.math:12508 sci.engr.control:103 sci.math.num-analysis:2876