HDTenEightyP/NetNews huggingface dataset:
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: various number systems
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Wed, 30 Sep 92 00:13:24 GMT

In article <1992Sep29.180205.26078@galois.mit.edu> jbaez@riesz.mit.edu
(John C. Baez) writes, in a typo-ridden post...

>one can easily demonstrate the relationship between the quaternions
>and relations.  Let i, j, and k be rotations of the face by 180 degrees in 
      rotations!
Newsgroups: sci.math,rec.puzzles,comp.theory
From: karr@cs.cornell.edu (David Karr)
Subject: Re: more on .999... = 10.0
Organization: Cornell Univ. CS Dept, Ithaca NY 14853
Distribution: inet
Date: Tue, 29 Sep 1992 03:50:48 GMT

In article <1a6tc7INNo0u@uwm.edu> markh@csd4.csd.uwm.edu (Mark) writes:

>So ACTUALLY, x differs from 10 by an infinitesimal power of 10 as was pointed
>out above.  Which decimal position it differs in depends on which 9.999...
>you chose, since there's a huge number of distinct 9.999...'s that differ at
>some infinite decimal position but are the same at all finite decimal positions.

I am tempted to write, "Now we have heard it all."

Personally, I'm happy to allow you as many infinite decimal positions as
you want, even uncountably many, and manipulate them according to whatever
rules you choose, as long as you don't refer to them when evaluating the
number.  I need only the 9's in finite decimal positions (though I need
*all* of them) to get up to 1.


In article <1a6tjqINNo19@uwm.edu>, markh@csd4.csd.uwm.edu (Mark) goes on:

>All models of the real number axioms have infinitesimals in them, except one.
>So choose any garden variety interpretation of the real number system, it has
>infinitesimals in it with probability 1.

If you take a random selection of models of real numbers out of the
many analysis books on my shelf, I believe you will find
infinitesimals in it with probability 0.

Of course you can bias the search differently.  And if you chose a
random point in time uniformly over recorded history, and examine a
text from that time about the shape of the Earth, you will find (with
very high probability) that the Earth is flat.

I know infinitesimals were utilized to prove many important theorems
in the eighteenth century and thereabouts, but they also led people to
"prove" many falsehoods.  I thought they were abandoned for that
reason.  What twentieth-century models of the real numbers have them?
From: tao@ginger.princeton.edu (Terry Tao)
Newsgroups: sci.math
Subject: Re: Roman Numerals
Date: 5 Oct 92 15:00:29 GMT
Organization: Princeton University
Originator: news@nimaster

Some people hypothesize that brackets were put over numbers to symbolize
multiplication by 1000, hence (V) is 5000.  In fact, since (I) has a faint
resemblance to M, this could explain the symbol M (rather than the traditional
explanation that M is the first letter of mille, the latin for a thousand; this
doesn't hold water, as all the other letters are not initials of latin numbers,
except possibly C.)

also some neat spinoffs: D, 500, is half of (I), i.e. I).

The romans would probably have found it hard to use numbers higher than 3999
anyway, at least for the man in the street.  Only imperial tax collectors or
people like Erasthones would need to use them.
From: angelo@blade.stack.urc.tue.nl (Angelo Wentzler)
Newsgroups: sci.math
Subject: Last one this century!
Date: 29 Sep 92 15:23:01 GMT
Organization: MCGV Stack, Eindhoven University of Technology, the Netherlands.
X-Newsreader: TIN [version 1.1 PL6]

Early this year there was a short discussion about palindrome dates here.
Somebody said that the next one would be somewhere in 2011 (1-10-2011) but
I just realized this is totally wrong!!
TODAY is a palindrome date! (29-9-1992)

So, now that I got this off my chest, I will quietly remove myself and get a
life.

Angelo Wentzler
-- 
angelo@stack.urc.tue.nl (internet)       Eindhoven University of Technology (nl)
Kerkstraat 30, 4581 BW, Vogelwaarde             Den Dolech 2, 5612 AZ, Eindhoven
What happens when an unstoppable, undeflectable force hits an unmovable,
indestructable object? I don't know. Gives me a headache, I guess.
Newsgroups: sci.math
Subject: Logarithms - how did they start/
From: sl870@uranus.ee.usu.edu (Ullas Gargi)
Date: 1 Oct 92 01:10:34 MDT
Distribution: world
Organization: Utah State University

A sitter from a netter:

How did the first guys (Napier et al) calculate the logs of numbers?
If I wanted to sit down and manually calculate the log of a non-trivial no.
(like 10), how would I do it? A Taylor expansion formula?
From: amnell@klaava.Helsinki.FI (Marko Amnell)
Newsgroups: sci.math
Subject: Re: Most important event in the history of mathematics
Date: 30 Sep 92 11:01:20 GMT
Organization: University of Helsinki

In article <hdev.717797409@dutiag> hdev@dutiag.tudelft.nl 
(Hans de Vreught) writes:

>wft@math.canterbury.ac.nz (Bill Taylor) writes:
>
>>Got to be the discovery of irrationals by a Pythagorean (whose name escapes me)
>>Its supreme significance is affirmed by its unique (apocryphal?) status as 
>>the only time a mathematician has been killed for making a shattering
>>mathematical discovery ! 
>
>His name was Hippasos of Metapontum. The legend goes that his disclosure of
>the discovery of irrational numbers resulted in the death penalty by drowning.
>It certainly was his most important event in the history of mathematics...

The whole story is pure legend.  I challenge you to cite any reputable
scholarly evidence that Hippasus discovered the irrational numbers.  As
someone pointed out earlier in another thread in sci.physics, the
discoverer was not executed, but perhaps did perish in a shipwreck.
It is true that the Pythagoreans kept their discovery secret for some
time because of unpleasant consequences in geometry...

The alleged mathematical achievements of Hippasus are addressed in Iamblichus'
_On the Pythagorean Way of Life_.  I will cite two relevant passages

  About Hippasus they say that he was one of the Pythagoreans but that
  because he was the first to publish and construct the sphere of the
  twelve pentagons he died at sea as an impious man.  He acquired the
  reputation of discovering it, although everything belongs to That
  Man (that is how they refer to Pythagoras, never calling him by his
  name).  Iamblichus 88.

  Some say that the divinity punished those who made Pythagoreans' views
  public.  For the man who revealed the construction of the vigintangle
  perished at sea an an impious man.  (The vigintangle is the
  dodecahedron, one of the so-called five solid figures, when it extends
  into a shpere.)  Others said that it was the man who spoke about
  irrationality and incommensurability who suffered this fate.
  Iamblichus 247.

Hippasus was an average Presocratic cosmologist, he was certainly not
a great mathematician.  The name of the discoverer of irrational numbers
is lost to us.  Let us simply call him That Man.  :-)

-- 
Marko Amnell
amnell@klaava.helsinki.fi
Graduate Student in Philosophy
From: cxm7@po.CWRU.Edu (Colin Mclarty)
Newsgroups: sci.math
Subject: Re: Topos theory for mere mortals (?)
Date: 29 Sep 1992 15:58:34 GMT
Organization: Case Western Reserve University, Cleveland, OH (USA)
Reply-To: cxm7@po.CWRU.Edu (Colin Mclarty)
NNTP-Posting-Host: slc12.ins.cwru.edu


In a previous article, arodgers@dcs.qmw.ac.uk (Angus H Rodgers) says:

>In <19o64aINNb9f@usenet.INS.CWRU.Edu> cxm7@po.CWRU.Edu (Colin Mclarty) writes:
>
>>	What an elementary topos IS, is a category something like
>>SET.  Any two "sets" have a cartesian product, for any equation
>>there is a "set" of solutions which may of course be empty, for any
>>two "sets" thre is a "set of all functions" from the first to the
>>second.  And there is a "set" of truth values. [...]
>
>The following point, in essence, has just come up in our e-mail
>correspondence (and I'm sorry that I haven't given you time to reply,
>but in my original message this point may have been entangled with
>other confusions of my own making); but it can be put so simply and
>briefly that I think it may deserve some space on the Net.
>
>The adjunction defining Cartesian closedness determines a (natural)
>bijection between hom-sets: [(A x B) -> C] --> [A -> (C^B)].
>^^^^^^^^^             ^^^^
>If topos theory is a foundation for maths, independent of any (other)
>form of set theory, does this mean that it is no longer Categorically
>Correct to use such old-fashioned language as I have just underlined?
>More to the point, perhaps, is this sort of language consistently
>avoidable when doing elementary topos theory?
>
	Yes it is, and it is avoided by being more explicit.  The
elementary description of the adjunction is:  For any objects B and
C there is an object C^B which you should think of as the "object
of arrows from B to C",  (it is customarily called the function
space, or hom-object, or exponential), and an "evaluation arrow"
ev:C^BxB--->C with this property:  Given any arrow  f:AxB--->C
there is a unique arrow u:A--->C^B such that ev(u(a),b)=f(a,b).

	In short, u takes each value a in A to the function
u(a) which in turn assings to any b in B the value f(a,b).

	Notice that the elementary definition is quite complete,
using no idea more complicated than products.  The homset
definition requires you to define homset functors and natural
isomorphisms.  OF COURSE the higher powered approach has its
uses, but not in presenting categorical foundations.  

	The higher powered approach is naturally "metatheoretic".
If you want to look at some category from outside you can use this,
and you can do it using comma categories in the category of all 
categories.  There is a more subtle approach that is quite useful 
when it applies, using "enriched categories" which now that I 
think of it I am not going to explain here.  

>--
>Angus H. Rodgers  (PhD student), | arodgers@dcs.qmw.ac.uk  
>Dept. of Computer Science, Queen | [ +44 | 0 ] 71 975 5241   
>Mary & Westfield College,  Mile  | This space has not been left blank,
>End Rd., London E1 4NS, England  | intentionally or otherwise.
>

Colin McLarty
Newsgroups: sci.math
From: rz@mips.complang.tuwien.ac.at (Richard Zach)
Subject: Re: Mattel says "Math is tough."
Reply-To: zach@csdec1.tuwien.ac.at
Organization: Technische Universit"at Wien
Date: Tue, 6 Oct 1992 09:27:30 GMT

Has anyone brought this up with the AMS or MAA?
Are they doing anything, eg. petitioning?

-- 
Richard Zach                         Technische Universitaet Wien
[zach@csdec1.tuwien.ac.at]     Abteilung fuer Formale Logik 185.2
Xref: sparky sci.math:12828 sci.math.stat:2082
Newsgroups: sci.math,sci.math.stat
Subject: Re: Stat problem...
From: <ROCHC@CUNYVM.BITNET>
Date: Tuesday, 6 Oct 1992 15:07:26 EDT
Organization: City University of New York/ University Computer Center

     Thank you to everyone who sent me their brilliant explanations...
From: dak@kaa.informatik.rwth-aachen.de (David Kastrup)
Newsgroups: sci.math
Subject: Re: Fourier bignum multiplication
Date: 26 Sep 92 09:53:10 GMT
Organization: Rechnerbetrieb Informatik  /  RWTH Aachen

is@groucho.dev.uga.edu (Bob Stearns) writes:

>I have seen it asserted, and am willing to believe, that x*x can be more
>efficiently implemented by FFT^-1(G(FFT(x)). I need a pointer to either
>the specific code or the algorithm. In particular, are the FFT's real or
>pseudo-complex, can I do the whole thing in integer arithmetic (for
>speed), just what is G, and are there any special cases to watch out
>for?

>Your assistance in this search would be greatly appreciated.

>P.S. How large must x be before the simple multiplication algorithm,
>     which is O(log^2(x)) is actually beaten by the FFT algorithm?
>     That is, just what are, approximately, the leading coefficients
>     of the big-Os?
Have you tried the GNU multiple precision math package? (mp-something
on gnu-archives)
They use a conglomerate of plain multiplication, and Nussbaumer's method
seriously beating the usual methods, and I believe some other algorithms
depending on size. No need to reinvent the wheel.
From: bennett@math.ksu.edu (Andy Bennett)
Newsgroups: sci.math
Subject: Re: Easy (?) probability question
Date: 2 Oct 1992 15:56:22 -0500
Organization: Dept. of Mathematics, Kansas State University
NNTP-Posting-Host: hilbert.math.ksu.edu

tao@ginger.princeton.edu (Terry Tao) writes:
]The variance of X is the minimum value of
]E((X-a)^2)
]for a variable a over the reals.  It is minimized, of course, when a is the
]expectation of X.
]Now it is obvious that if Z = min{X, k}, then the optimal value of a for Z
]is less than k, as E(Z)<k.  But for a < k,
]E((X-a)^2) \geq E((Z-a)^2)
]as you can no doubt easily verify.
]That's it!

No it isn't. Just because E[(X-a)^2]>=E[(Z-a)^2] for a<k doesn't imply
the variance of X is greater than the variance of Z in any way I can
see. You need something more to complete the proof.

]alternatively:
]var(Z) = E((Z-E(Z))^2) \leq E((X-E(Z))^2) \leq E((X-E(X))^2) = var(X)
]the first inequality comes from the fact that X > Z if Z > E(Z), and X=Z
]otherwise, and the second you can figure out yourself. 

The second inequality is false. As you note above, E[(X-a)^2] is minimized
for a=E[X] so E[(X-E[X])^2] certainly doesn't dominate E[(X-E[Z])^2].

-- 
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                    
Xref: sparky sci.math:12645 rec.puzzles:6589
From: kubo@birkhoff.harvard.edu (Tal Kubo)
Newsgroups: sci.math,rec.puzzles
Subject: Re: Conway's Recursive Sequence
Date: 3 Oct 92 01:22:10 GMT
Article-I.D.: husc3.1992Oct2.212210.16039
Organization: Dept. of Math, Harvard Univ.

In article <1992Oct2.185624.15808@cs.yale.edu> ferenbaugh-charles@CS.YALE.EDU (Charles Ferenbaugh) writes:
>  [on representing the successive differences of Conway's
    sequence by a Pascal's triangle ]
>
>I checked with Conway by email, and he suspects that this is probably
>equivalent to one of the combinatorial presentations already known.
>But at any rate, it's prettier than the presentation in the Mallows
>article.

Thanks for responding. A lot of unpublished work has been done on this
fascinating sequence, by Conway (over 10 years ago), Hofstadter (as early
as 1975) and others.  The purpose of my earlier posting was to try to
contact others who have worked on this problem.

The Pascal's triangle pattern is correct, and very pretty.
It is indeed equivalent to one of the known presentations.  See
Mallows' correction note in the June-July 1992 issue of American
Mathematical Monthly.

Ravi Vakil and I are writing an article on the Conway sequence,
in which we derive the Pascal triangle and some other representations.
Other goodies include a mysterious appearance of Catalan numbers,
some interesting recursive graphs which describe the Conway sequence
and its generalizations, and a proof of a conjecture of Mallows about
the limiting distribution of the sequence.  A draft of the article
should be available in a day or two (in AMSLaTeX format).  Interested
parties should contact me by e-mail.

-Tal Kubo    kubo@math.harvard.edu  or  kubo@zariski.harvard.edu
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: sci.math
Subject: Re: consequences of the Axiom of Choice
Date: 1 Oct 92 18:29:41 GMT
Reply-To: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Organization: The Wistar Institute of Anatomy and Biology
In-reply-to: brock@NeXTwork.Rose-Hulman.Edu (Bradley W. Brock)

In article <1992Sep30.183133.15484@cs.rose-hulman.edu>, brock@NeXTwork (Bradley W. Brock) writes:
>3) Flasque sheaves have no cohomology.

Reminds me of a Robert Gunning lecture many years ago.  Gunning is an
amazing lecturer, always bubbling over physically and verbally (eg, you
heard his voice trailing down the hall as he told all what he'd do next
time, and his ability to invent and quickly say mathematical tongue
twisters with no warning was jawdropping, and so on.)  Gunning was
proving the above, or some other basic fact about sheaves, and although
most of the class was familiar with the beginning stuff, he was going
slowly so that the lecture notes could become a book.  One of the
members of the class was infamously egotistical (he later gave this
side of his personality up when he got tenure, but at the time, jokes
and stories about him were legendary, eg, the Somebody-Somebody Index
Theorem, named after himself twice) and Somebody's choice of questions
and comments were also annoying, since every last one was to show us
how wonderful Somebody was, yet Gunning always handled him patiently.
So comes this proof, using AC, and unsurprisingly, Somebody interrupted
this proof to inform us that the same result follows without AC, but
instead--(film at 11! as we'd say nowadays)--by assuming that the base
space was paracompact.  And Gunning got all excited and cheerful, and
said (in about two seconds, from three different positions in the room,
waving his hands and jumping on his toes) "you can assume the axiom of
choice, you can assume paracompactness, you can assume whatever you
want!!!!!"  Everyone else laughed, and Somebody was much more subdued
for the rest of the semester.  And I for one, never cared again much
about AC assumptions except when it actually matters. 
-- 
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
From: jbaez@riesz.mit.edu (John C. Baez)
Newsgroups: sci.math
Subject: Re: Topos theory for mere mortals (?)
Date: 29 Sep 92 18:12:06 GMT
Organization: MIT Department of Mathematics, Cambridge, MA

In article <1992Sep29.135105.9450@dcs.qmw.ac.uk> arodgers@dcs.qmw.ac.uk (Angus H Rodgers) writes:
>In <19o64aINNb9f@usenet.INS.CWRU.Edu> cxm7@po.CWRU.Edu (Colin Mclarty) writes:
>
>>	What an elementary topos IS, is a category something like
>>SET.  Any two "sets" have a cartesian product, for any equation
>>there is a "set" of solutions which may of course be empty, for any
>>two "sets" thre is a "set of all functions" from the first to the
>>second.  And there is a "set" of truth values. [...]
>
>The following point, in essence, has just come up in our e-mail
>correspondence (and I'm sorry that I haven't given you time to reply,
>but in my original message this point may have been entangled with
>other confusions of my own making); but it can be put so simply and
>briefly that I think it may deserve some space on the Net.
>
>The adjunction defining Cartesian closedness determines a (natural)
>bijection between hom-sets: [(A x B) -> C] --> [A -> (C^B)].
>^^^^^^^^^             ^^^^
>If topos theory is a foundation for maths, independent of any (other)
>form of set theory, does this mean that it is no longer Categorically
>Correct to use such old-fashioned language as I have just underlined?
>More to the point, perhaps, is this sort of language consistently
>avoidable when doing elementary topos theory?

Yup.  These homs you are working with are, in a suitable kind of
category, not SETS but OBJECTS in the category itself.  I would
call this a category with "internal homs" but I believe there is
a more precise term.  A topos has these, in any event.  Then your
adjunction becomes not a bijection but a natural isomorphism.

It's a good point: one can indeed completely get rid of sets
and do topos theory from scratch in first-order logic.  I really
recommend MacLane and whatshisname's new book, which treats
this.  (Sorry, I always forget that name.)
From: chrisman@wheatena.berkeley.edu (chrisman)
Newsgroups: sci.math
Subject: Irrational?
Date: 6 Oct 92 00:59:51 GMT
Organization: U.C. Berkeley Math. Department.
NNTP-Posting-Host: wheatena.berkeley.edu

In article <COLUMBUS.92Oct2172559@strident.think.com> columbus@strident.think.com (Michael Weiss) writes:
>
>I believe the term "irrational" actually derives from "not a ratio", i.e.,
>of two integers.
>

Does anybody know if the common-use word "irrational" (meaning
nonsensical) is actually derived from the mathematical term?
From: hrubin@mentor.cc.purdue.edu (Herman Rubin)
Newsgroups: sci.math
Subject: Re: Most important event in the history of mathematics
Date: 29 Sep 92 00:51:02 GMT
Organization: Purdue University Statistics Department

In article <1992Sep28.223806.27669@u.washington.edu> petry@pythagoras.math.washington.edu (David Petry) writes:
>In article <1992Sep28.174237.26784@Cookie.secapl.com> frank@Cookie.secapl.com (Frank Adams) writes:

>>All right; what was the most important event in the history of mathematics?
>>My vote is in; what do the rest of you think?

I would place Diophantus' discovery of the idea of using symbols for quantities
as the most important.  Second, I would put the discovery that an axiomatic
approach could have lots of consequences, which leads to the idea that a
simple structure can be very powerful.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@snap.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
From: wft@math.canterbury.ac.nz (Bill Taylor)
Newsgroups: sci.math
Subject: No definable choice function.
Date: 2 Oct 92 02:02:02 GMT
Organization: Department of Mathematics, University of Canterbury

Gerald Edgar writes

>Explicitly find a countably infinite set S of (unordered) pairs,
>for which a choice function is not obvious.  [More technical details.
>The set S should be explicitly and uniquely specified in the language of ZF.
>It should be provable in ZF that it is a countable set of pairs.]

>An UNCOUNTABLE set of pairs with no obvious choice function is
>easy: take the set of ALL pairs of sets of real numbers.

A very nice wording for a fascinating question !

Gerald Edgar's uncountable example is probably the simplest such.  But here's
another that I like, that seems more "visualizable".

Consider the set 2^w, of 0/1 sequences.
Now consider the set of equivalence classes of these by tail-equality: 
i.e. two sequences are called equivalent if they agree in all but finitely many
terms. Now consider the pairs given by any sequence-class and its co-class,
that is obtained by exchanging 0s & 1s.

This class of pairs (apparently) has no definable choice function.

I call this "visualizable" because one can imagine an 
equivalence-class-of-sequences to be just a single sequence with its front end
flickering and changing at random between 0 and 1, down to varying
finite lengths, (rather like the tenths and hundredths on a digital clock),
while the great infinite tail remains fixed.

-----

The basic question remains, though.

Is there are clearly definable COUNTABLE collection of pairs, for which no
(obvious) definable chioce function exists.

Give yourself more leeway, and try this one:
is there ANY definable countable collection of nonempty sets,
for which no (obvious) definable choice function exists?  

I.e. a countable set for which (countable) choice is needed for a choice function.

I would dearly love to see an example of this. I believe I know of one, but it
is very hairy and quite possibly screwed-up; I would prefer to see other
examples before posting it.

------------------------------------------------------------------------------
            Bill Taylor              wft@math.canterbury.ac.nz
------------------------------------------------------------------------------
      This sentence does not have the property it claims to be lacking.
------------------------------------------------------------------------------
From: petry@pythagoras.math.washington.edu (David Petry)
Newsgroups: sci.math
Subject: Re: finding the units digit of a big number
Date: 29 Sep 92 01:16:06 GMT
Organization: University of Washington, Mathematics, Seattle

In article <1992Sep29.004246.14564@news2.cis.umn.edu> hoffman@n1.math.umn.edu (John R. Hoffman) writes:
>
>
>I was going through an old math contest that I took in high school and came  
>across a similiar question (which I couldn't solve)
>
>What is the units digit in the decimal expansion of 
>
>(15 + \sqrt{220})^19 + (15 + \sqrt{220})^82.
>
>
>Any simple solution would be appreciated.  ( I think calculators were not  
>allowed during the exam)


The key is to note that if f(n) = (15+sqrt(220))^n + (15-sqrt(220))^n,
then f(n+2) = 30*f(n+1) - 5*f(n),  and that 15-sqrt(220) is very small.


David Petry
Newsgroups: sci.math