Subject: Re: Question: Jobs in Pure Math
From: brundage@ipac.caltech.edu (Michael Brundage)
Date: Fri, 20 Dec 1996 16:21:35 -0800
In article <599uko$iap@news-central.tiac.net>, numtheor@tiac.net (Bob
Silverman) wrote:
> jms4@po.CWRU.Edu (James M. Sohr) wrote:
> > I know I would love the work I would do as a mathematician, so
> >that is not what is debated. Generally, it is such matters such
> >as pay scale, work hours, and job availability. My engineering
> >pals contend that mathematicians aren't much of wage earners, and
> >that the job market is glutted. I contend otherwise.
>
> As a professional mathematician who has worked in industry for
> over 15 years I have to say that your friends are right.
>
> Math professors are paid less than their comp. sci. or engineering
> counterparts. Mathematicians don't get much respect in the workplace
> and they get paid less than their counterparts.
>
> The job market is WORSE than glutted. It sucks. In fact, it is worse than
> that. Math depts. are telling incoming grad students that there are no jobs
> available. Academic jobs are impossible to get unless you are a Fields
> Medalist. Industry has the attitude:"what do we need mathematicians for?"
>
> >Can anyone who is currently a mathematician
> >or math professor describe the current job market?
[...]
> >And finally, would you recommend your job to someone
> >intereseted in mathematics, or would you suggest they do something
> >else (ie. study applied math or computer science, etc.)?
>
> Salary sucks, the job market is worse than that and I would recommend
> to anyone to stay away, unless you just want a career in math for the beauty
> of it. Can you say "starving artist"?
I would like to echo all of the sentiments of Bob Silverman (and that's why
I included his remarks almost verbatim). If you are worried about being
successful financially, or having a secure job any time soon after you
finish your degree, don't go into mathematics. I have a B.S. in
mathematics from Caltech, and an M.S. in math from the University of
Washington. I stopped there in large part because it was clear the job
market was getting worse and worse for mathematicians, with no signs of
significant improvement anytime soon.
One consequence of this dire situation is that (in my opinion) math
departments are not particularly interested in the students they have, and
many of the best students are leaving (or avoiding it in the first place)
in droves. You would do well to follow suit.
On the other hand, the unemployment rate in computer programming is almost
negligible (and can be entirely accounted for by people moving between
jobs). I'm currently working at Caltech as a computer programmer for an
astrophysicist here. The work is very challenging mathematically --
involving difficult integral equations, statistics, and efficient
algorithms, among other things -- and in fact this post was just
interrupted by my boss, who had a straightforward (but advanced) question
about vectory geometry relating to a problem with gamma ray burst
detection. Also, the job
pays about four times what math graduate school did (and I would say that's
low for the industry as a whole, where salaries of $50K to $70K per year
appear to be the norm). Professionally, I could not be happier, and I feel
reasonably secure that when I leave my position here, I will not have
significant difficulties finding a job elsewhere.
In addition, I have plenty of free time to continue mathematical pursuits;
I recently solved a problem in number theory, am in correspondence with my
former advisor about my thesis topic, and remain active in graph theory
research. Of course, it helps tremendously that an excellent mathematics
library is close at hand, so that I can stay current with the literature.
The only thing I miss is teaching, and I could easily volunteer to do some
of that either in local schools or at the university.
This is all in stark contrast to the situation for young mathematicians.
You would spend many years flitting from one two-year appointment to
another -- if you're lucky enough to find a job in the first place. The
salary would be terrible (in some cases worse than graduate school), you
would have very little opportunity to do any research (because unless you
find one of the few "cushy" positions available, you would spend an
inordinate amount of time with your overwhelming teaching responsibilities
and the application process for your next position, which would come all
too soon), and you would watch with envy as your friends who bailed earlier
are having the time of their lives. [Note: I haven't been through this
process myself; as I indicated, I am one of those who bailed early.
However, this is based on numerous conversations I've had with others who
stuck it through, and the many grim stories told in the CoYMN, so I think
it is probably an accurate description of the life of the average young
mathematician these days. Of course, your mileage could vary...]
My advice is that if you enjoy mathematics (and maybe think you even want
to do math research) but are not absolutely fanatical about it ("starving
artist" sums it up nicely), then pursue math on the side in addition to
some other field (such as computer science or perhaps economics or
molecular biology) that has a hope of providing you with a satisfying
career. As an added bonus, your mathematics will likely benefit from the
cross-polination of ideas from the other field. If you pursue pure math
only, then when you try to cross over into another discipline you will meet
with resistance -- questions like, "Why did you pursue math??" (instead of
whatever discipline the job you currently seek belongs to).
In any case, this is all just my opinion, and is not the opinion of my
employer. Any relation to persons or institutions real is accidental.
.
Michael Brundage
Infrared Processing Analysis Center, Caltech
brundage@ipac.caltech.edu
Subject: Re: Question: Jobs in Pure Math
From: voloch@max.ma.utexas.edu (Felipe Voloch)
Date: 21 Dec 1996 01:58:54 GMT
On an interview in the Math. Intelligencer, Serre was asked how to
encourage people to become mathematicians. He said that one should
discourage them. If they still want to become mathematicians then
they should be encouraged. I don't recall if he said how.
Unemployment for fresh PhD's is (according to the AMS Notices in Dec)
at 9%, down from 14% in the last two years. A lot of fresh PhD's are
subemployed, too. However I would paint the bleak picture the posters
below do. There are jobs out there. Graduates from U. Texas have been
successful in getting reasonable jobs in the last few years. Also, the
salary is not as good as, say computer science or engineering but is
enough to live decently.
Felipe
Michael Brundage (brundage@ipac.caltech.edu) wrote:
: In article <599uko$iap@news-central.tiac.net>, numtheor@tiac.net (Bob
: Silverman) wrote:
: > jms4@po.CWRU.Edu (James M. Sohr) wrote:
: > > I know I would love the work I would do as a mathematician, so
: > >that is not what is debated. Generally, it is such matters such
: > >as pay scale, work hours, and job availability. My engineering
: > >pals contend that mathematicians aren't much of wage earners, and
: > >that the job market is glutted. I contend otherwise.
: >
: > As a professional mathematician who has worked in industry for
: > over 15 years I have to say that your friends are right.
: >
: > Math professors are paid less than their comp. sci. or engineering
: > counterparts. Mathematicians don't get much respect in the workplace
: > and they get paid less than their counterparts.
: >
: > The job market is WORSE than glutted. It sucks. In fact, it is worse than
: > that. Math depts. are telling incoming grad students that there are no jobs
: > available. Academic jobs are impossible to get unless you are a Fields
: > Medalist. Industry has the attitude:"what do we need mathematicians for?"
: >
: > >Can anyone who is currently a mathematician
: > >or math professor describe the current job market?
: [...]
: > >And finally, would you recommend your job to someone
: > >intereseted in mathematics, or would you suggest they do something
: > >else (ie. study applied math or computer science, etc.)?
: >
: > Salary sucks, the job market is worse than that and I would recommend
: > to anyone to stay away, unless you just want a career in math for the beauty
: > of it. Can you say "starving artist"?
: I would like to echo all of the sentiments of Bob Silverman (and that's why
: I included his remarks almost verbatim). If you are worried about being
: successful financially, or having a secure job any time soon after you
: finish your degree, don't go into mathematics. I have a B.S. in
: mathematics from Caltech, and an M.S. in math from the University of
: Washington. I stopped there in large part because it was clear the job
: market was getting worse and worse for mathematicians, with no signs of
: significant improvement anytime soon.
: One consequence of this dire situation is that (in my opinion) math
: departments are not particularly interested in the students they have, and
: many of the best students are leaving (or avoiding it in the first place)
: in droves. You would do well to follow suit.
: On the other hand, the unemployment rate in computer programming is almost
: negligible (and can be entirely accounted for by people moving between
: jobs). I'm currently working at Caltech as a computer programmer for an
: astrophysicist here. The work is very challenging mathematically --
: involving difficult integral equations, statistics, and efficient
: algorithms, among other things -- and in fact this post was just
: interrupted by my boss, who had a straightforward (but advanced) question
: about vectory geometry relating to a problem with gamma ray burst
: detection. Also, the job
: pays about four times what math graduate school did (and I would say that's
: low for the industry as a whole, where salaries of $50K to $70K per year
: appear to be the norm). Professionally, I could not be happier, and I feel
: reasonably secure that when I leave my position here, I will not have
: significant difficulties finding a job elsewhere.
: In addition, I have plenty of free time to continue mathematical pursuits;
: I recently solved a problem in number theory, am in correspondence with my
: former advisor about my thesis topic, and remain active in graph theory
: research. Of course, it helps tremendously that an excellent mathematics
: library is close at hand, so that I can stay current with the literature.
: The only thing I miss is teaching, and I could easily volunteer to do some
: of that either in local schools or at the university.
: This is all in stark contrast to the situation for young mathematicians.
: You would spend many years flitting from one two-year appointment to
: another -- if you're lucky enough to find a job in the first place. The
: salary would be terrible (in some cases worse than graduate school), you
: would have very little opportunity to do any research (because unless you
: find one of the few "cushy" positions available, you would spend an
: inordinate amount of time with your overwhelming teaching responsibilities
: and the application process for your next position, which would come all
: too soon), and you would watch with envy as your friends who bailed earlier
: are having the time of their lives. [Note: I haven't been through this
: process myself; as I indicated, I am one of those who bailed early.
: However, this is based on numerous conversations I've had with others who
: stuck it through, and the many grim stories told in the CoYMN, so I think
: it is probably an accurate description of the life of the average young
: mathematician these days. Of course, your mileage could vary...]
: My advice is that if you enjoy mathematics (and maybe think you even want
: to do math research) but are not absolutely fanatical about it ("starving
: artist" sums it up nicely), then pursue math on the side in addition to
: some other field (such as computer science or perhaps economics or
: molecular biology) that has a hope of providing you with a satisfying
: career. As an added bonus, your mathematics will likely benefit from the
: cross-polination of ideas from the other field. If you pursue pure math
: only, then when you try to cross over into another discipline you will meet
: with resistance -- questions like, "Why did you pursue math??" (instead of
: whatever discipline the job you currently seek belongs to).
: In any case, this is all just my opinion, and is not the opinion of my
: employer. Any relation to persons or institutions real is accidental.
: .
: Michael Brundage
: Infrared Processing Analysis Center, Caltech
: brundage@ipac.caltech.edu
Subject: Graph layout for Interaction Nets?
From: Vladimir Alexiev
Date: 20 Dec 1996 21:21:01 -0700
For my thesis I'll have to draw a bunch of Interaction Nets (IN). These are
undirected graphs (sometimes with self-loops and multi-edges) with the
following features:
- edges leave the nodes at "node ports".
- some node ports have to be marked. Eg a "principal port" has an OUTGOING
arrow, a "multiport" has a dot, sometimes text has to be attached.
- there may be free edges, so invisible nodes should be supported.
- often two or more INs are drawn in succession, separated by a bold
"rewrites to" arrow ==>; thus "subgraphs" or "clusters" should be
supported.
An inept ASCII example would be
| |
__|____ __^_
(append ) (cons)
-v---|- / \
| | e _\____
__^_ y ==> (append)
(cons) -v--|-
/ \ | |
e x x y
(here v and ^ mark principal ports)
I drew a couple of INs using xfig and I quickly realised that I need an
automatic layout program (you give it some kind of textual description, it
lays out the nodes and the edges). Is there any TeX package I could use?
What have the authors on IN used?
I looked at three graph layout programs:
- dot by S North at AT&T;
- VCG by G Sander at Univ Saarbrucken
- Graphlet by M Himsolt at Univ Passau
but none of them does quite what I need.
Dot and VCG have "node ports" but these are intended to represent records
(structures), and one can't mark the exit point of the edge. So these are
just subdivisions of the node in cells, and not the node ports I need.
Some alternatives could be:
- place the port mark at the edge itself. All the programs only seem to
allow edges like --- <--> <-- --> and not >-- >--< --<, and no other edge
marks except arrows. (The following notation is also acceptable for INs:
->- -><- -<-.)
- dot allows Postscript-programmable edge "styles", but do I have enough
info to draw an edge mark with the correct position and orientation?
- use a textual mark on the edge. All the programs allow labels in the
middle of an edge, but not at the ends.
- use special node shapes whose geometry indicates the place of the ports.
Perhaps only dot allows programmable node shapes, but one can't specify
the exit points of the edges. Actually I'd prefer not to fix the exit
points so as not to over-constrain the layout.
Before I go ahead and look into other layout programs (and I have a list of
about 8 more), can anyone make some suggestions? I suppose I'll need to do
a bit of programming, can someone make suggestions on this? Maybe one of
the authors will be willing to help me a bit, if not with programming then
with some advice on where to start?
If you respond, please CC any replies to me since I don't read these
newsgroups regularly. If someone expresses interest, I'll post a summary.
TIA, Vlad
Subject: Re: Why can't 1/0 be defined???
From: tkidd@hubcap.clemson.edu (Travis Kidd)
Date: 21 Dec 96 03:26:24 GMT
marnix@worldonline.nl (Marnix Klooster) writes:
>> > Two natural requirements are imposed on the quotient: it must exist
>> >and it must be unique.
>> Why? What does this get us? Can't we merely use the above definition
>> and then *prove* that Q exists and is unique if B is not zero? And if
>> B=0, then those requirements are not met. What's the harm?
> Nothing, as long as you don't try to define `the' quotient of
>something and zero.
OK. "The" implies uniqueness, which isn't true if B=0.
Fair enough.
>If A and B are zero, then any Q satisfies
>the above equation. So what is 0/0 then: an expression with
>multiple values?
Yes.
>This gets you into trouble real quick.
Does ln 1 get you into trouble real quick? It has multiple values.
>If you
>define division so that
> a/b = q == a = b*q (1)
>(where == is logical equivalence) holds for all `numbers' a, b,
>and q, you're in for trouble. (I assume there are distinct
>`numbers' 0, 1, and 2, with 0*1=0*2=0.) Taking a=0 and b=0 you
>find that
> 0/0 = q
>for every `number' q with 0*q=0.
No, for *any* number q, not every number. One at a time.
>Thus 0/0 is equal to both 1 and
>2, and by transitivity of equality 1 and 2 are equal to each
>other: a contradiction.
-1 = 1^(1/2) = 1 : a contradiction.
>In mathematical parlance, definition (1)
>above is not sound, since it violates the uniqueness requirement.
What uniqueness requirement?
> As another example, elsewhere in this thread you argue that
>0*oo (zero times infinity) is equal to any number. Naively,
>transitivity then gives
> 0*oo = 1 and 0*oo = 2, therefore 1 = 2
>and you have again introduced a contradiction.
A similar contradiction arises when 1^(1/2)=1 and 1^(1/2) = -1.
> Conclusion: current mathematics does not really like
>expressions that can have more than one value.
So much for roots and logs!
>This is
>understandable, really, since it requires major rethinking of the
>formal underpinnings of mathematics: either every expression has
>at most one value, or you need to modify your notion of equality.
Equality is still equality. If anything needs to be modified,
it is the notion that one cannot divide by 0.
>I believe this can be done, but not in the context of current
>mainstream mathematics.
It's easy. Just acknowledge the special cases and move on.
>Marnix
-Travis
Subject: Re: Number Theory Puzzle (corrected with solution!)
From: rhoads@crayola.rutgers.edu (Glenn Rhoads)
Date: 20 Dec 1996 22:17:57 -0500
>I have a puzzle to which I don't know the solution, and unfortunately am not
>even sure I correctly remember the problem statement.
>There are 2 mathematicians, call them A and B.
>X and Y are positive integers > 1.
>A knows the sum of X and Y, i.e., A knows the value of S=X+Y.
>B knows the product of X and Y, i.e., B knows the value of P=X*Y.
>A and B have the following conversation:
>A : You CAN'T know the sum.
>B : I still don't know the sum.
>A : I don't know the product yet.
>B : Now I know the sum.
>Find X and Y.
The following puzzle from the rec.puzzles archive is probably the one
you have in mind.
Mr. S. and Mr. P. are both perfect logicians, being able to correctly deduce
any truth from any set of axioms. Two integers (not necessarily unique) are
somehow chosen such that each is within some specified range. Mr. S. is
given the sum of these two integers; Mr. P. is given the product of these
two integers. After receiving these numbers, the two logicians do not have
any communication at all except the following dialogue:
* Mr. P.: I do not know the two numbers.
* Mr. S.: I knew that you didn't know the two numbers.
* Mr. P.: Now I know the two numbers.
* Mr. S.: Now I know the two numbers.
Given that the above statements are absolutely truthful, what are the two
numbers?
----------------------------------------------------------------------------
The answer depends upon the ranges from which the numbers are chosen.
The unique solution for the ranges [2,62] through [2,500+] is:
SUM PRODUCT X Y
17 52 4 13
The unique solution for the ranges [3,94] through [3,500+] is:
SUM PRODUCT X Y
29 208 13 16
There are no unique solutions for the ranges starting with 1, and there are
no solutions for ranges starting with numbers above 3.
A program to compute the possible pairs is included below.
----------------------------------------------------------------------------
#include
#define SMALLEST_MIN 1
#define LARGEST_MIN 10
#define SMALLEST_MAX 50
#define LARGEST_MAX 500
long P[(LARGEST_MAX + 1) * (LARGEST_MAX + 1)]; /* products */
long S[(LARGEST_MAX + 1) + (LARGEST_MAX + 1)]; /* sums */
find(long min, long max)
{
long i, j;
/*
* count factorizations in P[]
* all P[n] > 1 satisfy <<1>>.
*/
for(i = 0; i <= max * max; ++i)
P[i] = 0;
for(i = min; i <= max; ++i)
for(j = i; j <= max; ++j)
++P[i * j];
/*
* decompose possible SUMs and check factorizations
* all S[n] == min - 1 satisfy <<2>>.
*/
for(i = min + min; i <= max + max; ++i) {
for(j = i / 2; j >= min; --j)
if(P[j * (i - j)] < 2)
break;
S[i] = j;
}
/*
* decompose SUMs which satisfy <<2>> and see which products
* they produce. All (P[n] / 1000 == 1) satisfy <<3>>.
*/
for(i = min + min; i <= max + max; ++i)
if(S[i] == min - 1)
for(j = i / 2; j >= min; --j)
if(P[j * (i - j)] > 1)
P[j * (i - j)] += 1000;
/*
* decompose SUMs which satisfy <<2>> again and see which products
* satisfy <<3>>. Any (S[n] == 999 + min) satisfies <<4>>
*/
for(i = min + min; i <= max + max; ++i)
if(S[i] == min - 1)
for(j = i / 2; j >= min; --j)
if(P[j * (i - j)] / 1000 == 1)
S[i] += 1000;
/*
* find the answer(s) and print them
*/
printf("[%d,%d]\n",min,max);
for(i = min + min; i <= max + max; ++i)
if(S[i] == 999 + min)
for(j = i / 2; j >= min; --j)
if(P[j * (i - j)] / 1000 == 1)
printf("{ %d %d }: S = %d, P = %d\n",
i - j, j, i, (i - j) * j);
}
main()
{
long min, max;
for (min = SMALLEST_MIN; min <= LARGEST_MIN; min ++)
for (max = SMALLEST_MAX; max <= LARGEST_MAX; max++)
find(min,max);
}
-- Glenn Rhoads
Subject: n twin-prime clusters
From: pmontgom@cwi.nl (Peter L. Montgomery)
Date: Sat, 21 Dec 1996 06:04:54 GMT
Define an n twin-prime cluster as an ordered set of n pairs of twin
primes in actual order as they occur from the lowest value
to the highest value such that no other isolated prime
occurs in their range.
For example, the set 5,7 11,13 & 17,19 is a 3 twin-prime cluster
since there are no other primes between the range 5-19 and all
primes that occur are twin prime pairs. Note that 11,13 17,19 is
also a prime decade. (n,n+2,n+6,n+8) constitute a prime decade
if all four are primes.
We discover, by computer search from 1 to 2^31 the following
items of interest:
The first 4 cluster is 9,419 (10,4,22) decade: 9,431 where
(10,4,22) denotes the difference between the prime pairs and
decade denotes a prime decade starting with 9,431.
The first 4 cluster that consists only of two prime decades is
1,006,301 (4,22,4) decades: 1,006,301 1,006,331
The first 5 cluster occurs at 909,287 (10,16,10,10) with no decades
The first 5 cluster that contains two prime decades occurs much
later, at 1,432,379,951 (4,22,4,22) decades: 1,432,379,951 and
1,432,379,981
The first 6 cluster appears at 325,267,931 (4,10,10,16,10) with the
decade 325,267,931. The second 6 cluster does not appear until
412,984,667 (58,22,46,76,28) with no decades.
Unexpectedly, the first 7 cluster appeared at 678,771,479 with
spacings (10,58,4,58,28,10) so it contains the decade: 678,771,551
All these clusters were found around 3pm PST Thurs Dec.19,96.
The following questions come to mind:
1) Are the numbers of 4,5,6 and 7 clusters infinite?
2) Does any 6 cluster contain only 3 prime decades?
3) When does the second 7 cluster appear?
4) Can n-clusters contain floor(n/2) prime decades?
This has been verified for 3,4, & 5 clusters.
5) The first few sets of 4 clusters consisting of 2
prime decades has spacing (4,22,4). Why 22?
6) As the size of the primes grow larger, the spacing
between the n cluster pairs grows larger. What is
the relationship?
7) Can anything be proved about n-clusters?
8) Is there any literature or references about twin-prime
n-clusters? Prime decades have been studied, how about
these interesting clusters, with no isolated primes?
Can the reader find more 7 clusters? Does an 8 cluster exist?
Please respond to randall_rathbun@rc.trw.com, not to the poster.
--
Peter L. Montgomery pmontgom@cwi.nl San Rafael, California
Mother: Your room is a mess. Me: I don't see a mess.
Mother: You need glasses.
Subject: Re: Golden Ratio
From: jgamble@ripco.com (John M. Gamble)
Date: 21 Dec 1996 05:44:10 GMT
In article <595rqa$j4g@freenet-news.carleton.ca>,
Angel Garcia wrote:
>
[snipped text about the number vs. it's recipricol]
>>>The literature about 'fi' is literally monstruous in current
>>>century (due mostly to botanists, biologists who have studied it
>>>'ad nauseam' in connection with Fibonacci and sunflowers).
>>> But the very very very bad boy about such confusion came, almost
>>>certainly, with that mediocre scientist (but excellent artist and
>>>writer) Theodore Cook with his Feidias who used 1/fi instead of fi.
>>
>> When did Theodore Cook do this?
> In 1914: a most famous book "The curves of life" which has been
>quoted constantly in Art and Architecture circles. Th. Cook is responsible
>for the "Phi" (sic !) capital greel letter for Feidias or Phidias or
>Fidias (the famous Artist-in-chief for the Parthenon, hired for such
>momentous job by the great statesman Pericles).
> Cook knows only elementary math., but writes very well. After him
>several researchers in Art have been pushing the Golden Section as
>CANON of human proportions. Before him NOBODY had ever mentioned
>explicitly such CANON: in fact the Vitruvius-Leonardo Canon of
>around 1490 uses something shlightly 'slender' than the Golden Ratio
>and consequently MORE BEAUTIFUL than Cook's canon who, incidentally,
>it seems to have instaured the malpractice of PHI= 1.618 against
>traditional reliqua-sectio=0.618..
Martin Gardner finds an even earlier reference, in German,
Der Goldene Schnitt, by Adolf Zeising, published 1884. This is
in Gardner's Second Scientific American Book of Mathematical
Puzzles and Diversions.
"Zeisong argues that the golden ratio is the most artistically
pleasing of all proportions and the key to the understanding of
all morphology (including human anatomy), art architecture and
even music. Less crankish but comparable are Samuel Colman's
Nature's Harmonic Unity (1913) and Sir Thomas Cook's The Curves
Of Life (1914)."
Incidently, Gardner states that Mark Barr, a US mathemetician,
gave the ratio its name phi around 1910, which is prior to Cook.
>--
>Angel, secretary of Universitas Americae (UNIAM). His proof of ETI at
>Cydonia and complete Index of new "TETET-96: Faces on Mars.." by Prof.
>Dr. D.G. Lahoz (leader on ETI and Cosmogony) can be studied at URL:
> http://www.ncf.carleton.ca/~bp887 ***************************
--
-john
net address: 72330.501@compuserve.com
jgamble@ripco.com
Subject: Re: Vietmath War: war victims; blinded victims
From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium)
Date: 21 Dec 1996 05:32:10 GMT
In article <32B98C91.2839@mindspring.com>
Richard Mentock writes:
> I'm so stupid, I studied math instead of business.
> I'm so stupid, I believed my senator and governor.
> I'm so stupid, I'm responding to this. I'm so stupid, but it's fun.
>
> --
> D.
>
> mentock@mindspring.com
> http://www.mindspring.com/~mentock/index.htm
Oh I can believe it Mr. Mentock. So let us see you in action.
Tell me if this proof of Euclid's Infinitude of Primes, indirect proof
is a valid proof or an invalid proof. I kind of think you would have
been better off majoring in business.
BEYOND NUMERACY, John Allen Paulos, Alfred A. Knopf, New York 1991, a
BORZOI BOOK
--- quoting Page 95 of Paulos book ---
The mathematical logician Kurt Goedel was one of the preeminent
intellectual giants of the twentieth century, and, assuming the
survival of the species, will probably be one of the few contemporary
figures remembered in 1,000 years. A number of recent books about him
notwithstanding, this judgement is not a result of hype or an incipient
fad (although it is made infinitesimally more acceptable by the
similarity among the words "God," "Godel," and "Godot"). Neither is it,
despite a tendency for all disciplines to foster professional myopia, a
case of mathematicians' self-congratulation. It's simply true.
--- end quoting page 95 of Paulos book ---
--- start quoting Page 185 of Paulos book ---
" Euclid showed, however, that there is no largest prime and that
therefore there are an infinite number of primes.
Euclid's demonstration of this is such a beautiful example of what is
often called an indirect proof that I will risk arousing your
mathematical anxiety and reproduce it. We assume at the outset that
there are only finitely many primes and try to derive a contradiction
from this assumption. Thus, we list the prime numbers 2, 3, 5, . . .,
151, . . . P; P we will take to be the largest prime number. Now form a
new number N by multiplying all the primes in the above list together.
Thus N = 2 x 3 x 5 x . . . x 151 x . . . x P.
Consider the number (N + 1) and whether 2 divides it evenly (with no
remainder) or not. We see that 2 divides N evenly since it is a factor
of N. Therefore 2 does not divide (N +1) evenly, but leaves a remainder
of 1. We see also that 3 divides N evenly since it too is a factor of
N. Therefore 3 does not divide (N + 1) evenly, but also leaves a
remainder of 1. Similarly for 5, 7, and all the prime numbers up to P.
They each divide N evenly, and therefore each leave a remainder of 1
when divided into (N +1).
What does this mean? Since none of the prime numbers 2, 3, 5, . . .,
P divides (N +1) evenly, the number (N + 1) is either itself a prime
number larger than P, or it is divisible by some prime number that is
larger than P. Since we assumed that P was the largest prime number, we
have a contradiction: We have established the existence of a prime
number larger than the largest prime number. Therefore our original
assumption that there are only finitely many prime numbers must be
false. End of proof. QED.
--- end quoting Page 185 of Paulos book ---
Tell me Mr. Mentock, whether you think Paulos has given a valid proof
of Euclid's Infinitude of Primes? My god PU, I love being alive!
Subject: Re: Vietmath War: war victims; blinded victims
From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium)
Date: 21 Dec 1996 05:39:40 GMT
In article <59dtrd$cu8@svin12.win.tue.nl>
Andre Engels writes:
> "I am poor at mathematical reasoning, but poorer still are you at
> understanding what I write."
I know this. But the trouble is that you don't know this. And I care
little in what you write.
So here, let me show you something. Tell me, yes, tell me if you think
this proof by G.H. Hardy is a valid proof or not.
--- quoting A MATHEMATICIAN'S APOLOGY by G.H. Hardy pages 92-94 ---
I can hardly do better than go back to the Greeks. I will state
and prove two of the famous theorems of Greek mathematics. They are
'simple' theorems, simple both in idea and in execution, but there is
no doubt at all about their being theorems of the highest class. Each
is as fresh and significant as when it was discovered-- two thousand
years have not written a wrinkle on either of them. Finally, both the
statements and the proof can be mastered in an hour by any intelligent
reader, however slender his mathematical equipment.
1. The first is Euclid's (Elements IX 20. The real origin of many
theorems in the Elements is obscure, but there seems to be no
particular reason for supposing that this one is not Euclid's own)
proof of the existence of an infinity of prime numbers.
The prime numbers or primes are the numbers (A)
2,3,5,7,11,13,17,19,23,29,... which cannot be resolved into smaller
factors. (There are technical reasons for not counting 1 as a prime.)
Thus 37 and 317 are prime. The primes are the material out of which all
numbers are built up by multiplication: thus 666 = 2x3x3x37. Every
number which is not prime itself is divisible by at least one prime
(usually, of course, by several). We have to prove that there are
infinitely many primes, i.e. that the series (A) never comes to an end.
Let us suppose that it does, and that 2,3,5,..., P is the complete
series (so that P is the largest prime); and let us, on this
hypothesis, consider the number Q defined by the formula Q =
(2x3x5x..xP) + 1. It is plain that Q is not divisible by any of
2,3,5,...,P; for it leaves the remainder 1 when divided by any one of
these numbers. But, if not itself prime, it is divisible by some prime,
and therefore there is a prime (which may be Q itself) greater than any
of them. This contradicts our hypothesis, that there is no prime
greater than P; and therefore this hypothesis is false.
The proof is by reductio ad absurdum, and reductio ad absurdum,
which Euclid loved so much, is one of a mathematician's finest weapons.
It is a far finer gambit than any chess gambit: a chess player may
offer the sacrifice of a pawn or even a piece, but a mathematician
offers the game.
---end quoting A MATHEMATICIAN'S APOLOGY by G.H. Hardy pages 92-94 ---
Simple question there for you Andre Engels, is that a valid proof or
not?
Tell us, li'l math birdbrain.
Subject: Re: Average
From: "Robert E Sawyer"
Date: 21 Dec 1996 08:05:44 GMT
As I said in the earlier posting,
"Average", at least as it's used in the US, is a term for any of a wide
variety of measures of location, including mode, median, and any of the
various means (arithmetic, harmonic, geometric, etc).
(This is both a "dictionary definition" and accepted statistical usage.)
Actually, in my professional & academic experience (textbooks included),
statistical usage doesn't differ from the dictionary definition(s) (e.g.
The Random House Dictionary, or Webster's Ninth Collegiate), viz., as any
one of various quantities typical or representative of a set of quantities
("as a mean, mode, or median", etc).
Of course, depending on their audience, some writers may use the general term
"average" in only one of its specific meanings (often as the arithmetic mean).
But it's precisely to make the important point that the arithmetic mean is only
one of a class of "averages" that this latter term is given the broader meaning.
soen@pacbell.net
____________________
Jim Smith UT wrote ...
| I have never seen or heard the word "average" used in a scientific,
| technical, educational or professional journal, book, presentation, or
| acticle where it's usage indicated anything other than a "mean".
Subject: Re: SNOWFLAKE symmetry - who can explain it?
From: erg@panix.com (Edward Green)
Date: 21 Dec 1996 02:31:18 -0500
I find it hard to believe that with all those extremely appropriate
groups in the header (well, maybe not alt.physics.new-theories) you
have not gotten an acceptable answer, whoever you are! (sprout?)
But here goes...
Dendritic crystal growth: Under some growth conditions crystals grow
in the form of dendrites, or, to the English speaking world, little
trees. Why do they do this? Not crucial to the answer, but basically
because the rate limiting process is extraction of, or dumping to,
the environment. Snowflakes are extracting water vapor and dumping
heat into the surrounding air. This heat removal capacity and water
vapor is depleted from the near air, so the crystal growth front is
unstable.
But not, evidently, unpredictable. It is predictably unstable, and
equivalent conditions of temperature, pressure and water vapor
concentration give identical dendrites. Ok?
Hexagonal symmetry: ice crystals possess hexagonal symmetry. I guess
you accept this.
Varying micro environment: If we grew an army of snowflakes under
identical and carefully controlled conditions, then we should get an
army of identical snowflakes by the above hypothesis. But snowflakes
are not all identical. Further building our hypothesis, we surmise
each flake falls through a succession of micro-environments that is
essentially unique.
Ok... here comes the key hypothesis.... drum roll please....
*The micro-environment is changing as the flake falls, but the change
is effectively identical across the dimensions of the flake*
And that's it. Each of six identical crystal faces is seeing an
identical succession of micro-environments. We therefore have
identical dendritic growth at the six faces, creating the familiar
six symmetrical arms of the snowflake. Different snowflakes however
fall through a different succession of micro-environments, and the
result is a profusion of snowflake types.
Ok? That is my hypothesis. It fits the known laws of physics and
properties of crystal growth. It may be refined or modified, and
there may even be room for a bit of chaos for chaos fans,
(maybe neighboring snowflakes all fall through *almost* identical
micro-environments, but the arm pattern is sensitively determined by
small variations at seeding... or maybe the variations are determined
by the shape of the dust grain that does the seeding --notice we can
only allow the sensitive dependence at seeding, otherwise we would
not get identical arms).
And if that's not good enough, I'd like to hear different from a
meteorologist. Maybe the arms emit crystalline vibrations and wind
up in tune with each other. ;-)
E. Green
Professional Amateur
