Subject: Re: limit ordinals and ordinal arith question
From: angilong@u.washington.edu ('mathochist' Angela Long)
Date: 12 Nov 1996 04:22:51 GMT
Daniele Degiorgi wrote:
>In article <327A8400.6DCF@cs.umbc.edu>, peterson@cs.umbc.edu writes:
>>I'm trying to learn about limit ordinals. Does anyone
>>have any recommendations on particularly readable texts
>>on the subject?
I recommend Keith Devlin's "The Joy Of Sets." It's a good general
introduction to set theory and includes ordinal and cardinal math.
>>A specific problem I'm having is that, form the text
>>I'm working from, I'm having trouble understanding
>>the difference between
>> w + 1 {were w is ord assoc with set of nat numbers}
>>and
>> 1 + w
>I think this is an improper usage of +.
No, it's the perfectly standard ordinal +. A + B means the
ordinal that is isomorphic to the ordered set that is formed
by "lining up" the elements of A followed by the elements of
B. So 1 + w is the ordinal isomorphic to {A,0,1,2,...},
which is w. But w + 1 is the ordinal isomorphic to
{0,1,2,...,A}, which is clearly not w, since it has a last
element. In fact, since w is an initial segment of w + 1,
w + 1 > w. Further, w + 2 is the ordinal isomorphic to
{0,1,2,...,A,B}, which clearly has w + 1 as an initial
segment, and also clearly is not isomorphic to w + 1, since
w + 2 has both a last and a next-to-last element. So
w + 1 < w + 2.
--
-- Angi
Subject: Re: Cantor and the reals
From: davis_d@spcunb.spc.edu (David K. Davis)
Date: Tue, 12 Nov 1996 03:38:31 GMT
WAPPLER FRANK (fw7984@csc.albany.edu) wrote:
:
: NovexZ wrote:
: > Can someone give me the demonstrations that Cantor use to prove that the
: > set of the reals is an uncountable set?
: > Thanks
:
: As far as I understand it:
:
: Suppose you initially stipulate a recursive method to enumerate real numbers
: and (start to) produce a list of them (let's say represented as digits)
: accordingly.
:
: But now stipulate a procedure which can change individual digits of those
: number representations in the list. Change the first digit of the first
: number, the second digit of the second number, etc. (that's another
: recursive method).
:
: Doing that you have generated a new digital representation of one real number
: which was not (could never have been) in the list initially proposed.
:
: It is concluded that therefore the (any) initial recursive method to enumerate
: real numbers must be incomplete. (Proof via Cantor's Diagonal Method)
:
: Thanks for asking, though. Frank W ~@) R
I think this is basically right but there's no need to speak of recursive
procedures here. Consider ANY enumeration of the reals (between say 0 and
1 - to avoid annoying complications). Then every such number can be
represented as an infinite string of digits following a '.', (all digits
are zero after a certain point to make all strings infinite). We've made
no significant assumptions here except that such an enumeration can be
done. But now construct another such real: take as the first digit any
digit not equal to the first digit of the first enumerated number, take as
the second digit any digit not equal to the the second digit of the second
enumerated, etc. This number differs from each enumerated number in at
least one digit. Hence, our enumeration is incomplete - we didn't list all
the reals between 0 and 1. But since we've made NO commitment to the
details of this enumeration whatsoever, then ANY enumeration whatsoever is
self-contradictory.
Now it happens that it's also impossible to devise a procedure or a
computer program that will list just the true theorems of arithmetic.
(It's very easy to write a program that will eventually produce all the
true theorems if you don't mind false ones mixed in - just list all
possible strings of symbols - some will be gibberish, some false, some
true). But the list of such theorems IS enumerable if we don't confine
ourselves to constructive enumerability. Just go down the all strings
list, put a one by the first true theorem, a two by the second true
theorem, and so on. Conceptually such an enumeration certainly exists, but
we can't do such an enumeration mechanically because no such mechanical
procedure exists. That's Goedel's famous result. And his proof does use
Cantor's fabulous diagonal method, but it's a little more involved that
Cantor's proof - but attainable, at least in outline.
My only point is that Cantor wasn't the least bit concerned about
effective (procedurally doable) enumerability, rather only abstract
enumerability (i.e. can such an enumeration be self-consistently posited
to exist?)
There was (is?) a school of mathematics that doesn't like the way Cantor
(and practically everyone after him) does mathematics. They object to
non-constructive mathematics, and other may not object but are interested
in seeing how far one can without using the freedom Cantor gave us.
I'm blathering. Others can tell you much more than I.
-Dave D.
Subject: Re: Sets, classes, contradictions, etc. (Practical application!)
From: angilong@u.washington.edu ('mathochist' Angela Long)
Date: 12 Nov 1996 04:47:19 GMT
Dave Rusin wrote:
>David K. Davis wrote:
>>Kralor (ms-drake@students.uiuc.edu) wrote:
>>: Please try not to bruise me...I'm just a naive college student who's
>>: curious. I recently started reading about set theory and all the
>>: contrivances that are used to eliminate paradoxes such as Russell's. I
>>: was just wondering if the entire situation could be resolved by an axiom
>>: which doesn't allow sets to be members of themselves, or does this lead
>>: to other problems? Thanks for any help--
>>I'll just say that this is not enough. You also get in trouble if a set is
>>a member of a member of itself and so on. So you need a stronger axiom
>>to prevent that. But let someone else say what axiom - it's been too long.
Foundation. Letting e represent "is an element of," the axiom of
foundation (or well-foundedness, or regularity) is probably most
simply stated as: There does not exist an infinite chain of sets
such that ...e X_2 e X_1 e X_0. So if we had, for example, X e X,
there would be the chain ...X e X e X, which is not allowed. And
if we had X e Y and Y e X, we would have ...X e Y e X, no good.
>mathematically perverse kind of person I am, I instructed the machine
>"put sack in sack". It was clever enough to say "You can't do that."
>Unsatisfied, I tried
> put sack in bottle
> OK
> put bottle in sack
> OK
>All future attempts to retrieve either item met with "You can't get at it".
ROFL! So it's not so much that these sets can't be *formed*,
it's more that once they're formed, they self-annihilate.
The axiom of foundation could be stated: 1. you can only re-
trieve elements from a set if the elements are not trapped in
an infinite containment loop, and 2. all elements of sets are
retrievable.
Note that there has also been work done more recently (Aczel)
on non-well-founded set theory, in which sets can be allowed
to be members of themselves. Non-well-founded sets are some-
times called "hypersets." It turns out to be a consistent
theory, but you still have to have some "boundaries" to your
universe; for example, you can't have a power set being a
subset of its base set. (Try forming such a set and see
what happens! First all power sets must have the empty set,
and the empty set "sucks in" the power set of the empty set,
which sucks in its own power set... which sucks in the en-
tire well-founded universe, and that's just the beginning!)
--
-- Angi
Subject: Re: GOD
From: Le Compte de Beaudrap
Date: Mon, 11 Nov 1996 22:40:36 -0700
On Thu, 7 Nov 1996, Alan Silver wrote:
> Bob Massey wrote ...
> >Acording to M. Luther,"reason is evil". Any 'idea' that does not embody
> >the love of Jesus or use Biblica foundations will send you to Hell.
> >The 3R's: Religion Rots Reason! Paradise needs no scientists or
> >preachers!
>
> Why is it that you assume that the word "religion" automatically means
> some kind of Christianity?
Amen! (sorry, couldn't resist! ;) I find that (just about) any religion
can be as unreasoningly fanatic as christianity can be sometimes.
But back on the original topic, if christianity preaches "Thou shalt
not think for thyself", how does one account for people like Netwon, Descartes,
Euler? Each of them was very religious, very christian, and each one was a
mathematician, philospher, and physicist (even by today's definition, as
opposed to the definition in their time, when one group almost inevitably
implied the other two). None of them was condemned by any church at any time
(to my knowledge). So, ~(For all cases): (Religion Rots Reason).
Unproved, and with three counterexamples anyhow.
Niel de Beaudrap
----------------------
jd@cpsc.ucalgary.ca
Subject: Re: Determining irreducible polynomial
From: Bill Dubuque
Date: 11 Nov 1996 22:40:50 -0800
pecampbe@mtu.edu (Paul E. Campbell) wrote:
>I've been studying codes recently, which mostly tend to rely on using
>extensions over the finite field GF(2).
>
>So, other than exhaustive checking or simply calling the libraries built into
>Maple or Mathematica, how does one determine whether a polynomial is
>irreducible? Is there a way to find an irreducible polynomial without
>resorting to some sort of random or systematic search given a degree and
>a field?
The polynomial Q(X) in GF(p)[X] of degree n is irreducible iff
n
p
X = X (mod Q(X))
and for all primes q dividing n
n/q
p
gcd( X - X, Q(X) ) = 1.
The proof is an easy exercise using only basic properties of finite fields.
Using repeated squaring to computer powers, this gives an O(n^3*ln(p))
algorithm, assuming that the degree n is quickly factorizable -- which is
always the case in current practice.
Note that this practical polynomial irreducibility test is an analog of the
impractical Pocklington-Lehmer integer primality test (e.g. see Section
3.4.3 of Cohen's text A Course in Computational Algebraic Number Theory).
Special parameterized classes of irreducible polynomials are known in
various cases, e.g. for classical results see Chapter V of A. Albert's
text Fundamental Concepts of Higher Algebra (a most useful reference
for classical results on Finite Fields).
No doubt there are many new results given the recent intense applications
of finite fields to cryptography, etc., e.g. see the Math Review below.
-Bill Dubuque
------------------------------------------------------------------------------
95m:11136 11T06
Niederreiter, Harald (A-OAW-I)
An enumeration formula for certain irreducible polynomials with an
application to the construction of irreducible polynomials over the binary
field. (English. English summary)
Appl. Algebra Engrg. Comm. Comput. 1 (1990), no. 2, 119--124.
------------------------------------------------------------------------------
The paper answers some questions posed by Meyn. The author obtains an explicit
formula for the number of polynomials $f(x) = x\sp n + a\sb {n-1}x\sp {n-1}
+ \cdots + a\sb 1x + 1$ irreducible over $ {\bold F}\sb 2[X]$ with $a\sb
{n-1} = a\sb 1 = 1$. Such polynomials are called $A$-polynomials. In
particular, it follows from that formula that such polynomials exist for all
$n \ge 2$ with $n \ne 3$. The importance of such polynomials is provided by
the following result. For a polynomial $F(X) \in {\bold F}\sb 2[X]$ of
degree $d = \deg F$, define the $Q$-transform as follows: $F\sp Q(X) = x\sp d
F(X + 1/X).$ If $f(X)$ is an $A$-polynomial then every term of the
recursively defined sequence $f\sb 0(X) = f(X)$, $f\sb i(X) = f\sb {i-1}\sp
Q(X)$ is irreducible over ${\bold F}\sb 2$.
Reviewed by Igor E. Shparlinski
Subject: Re: Vietmath War: Wiles FLT lecture at Cambridge
From: Le Compte de Beaudrap
Date: Mon, 11 Nov 1996 23:05:34 -0700
On 7 Nov 1996, Archimedes Plutonium wrote:
> In article <327FD551.4A31@postoffice.worldnet.att.net>
> kenneth paul collins writes:
>
> > Please, what are "p-adics"?
>
> Each of them are Infinite Integers. Around 1901 Kurt Hensel in Germany
> extended the integers through a series operation.
>
> The Finite Integer such as 1 is supposedly finite, nothing to the
> right or left of it.
>
> Infinite Integers all of them have an endless string of digits to the
> leftward, thus 1 is .....000000001 or 231 is .....00000231 but not
> every Infinite Integer repeats in zeros, for instance the Infinite
> Integer
> ....9999999999998 is equivalent to -2
>
> and who knows if these two Infinite Integers have any remarkable
> qualities
> ....951413.
>
> ....172.
>
> But you can quickly see that if you accept the Infinite Integers as
> the real live and true integers and look at the finite integers as a
> sham, a cutsy but crude setup that is all foggy and imprecise, a
> Newtonian first approximation of what numbers are, then all of
> mathematics is changed. No longer do you have Cantor diagonal baloney.
> No longer do you have Number Theory stockpiled with ancient unsolved
> and easy to state problems. No longer do you have hundreds and
> thousands of pages of proofs for easy problems such as FLT or Goldbach
> using every piece of incoherent field of mathematics to tackle it with.
>
> But all of the above is useless to tell any mathematician. It is far
> easier to convince the Pope that Jesus was just an ordinary human being
> than it is to convince a 1993 professor of mathematics that his
> understanding of "finite integer" is all wet.
>
> I wait for the physicist to show that Infinite Integers-- the p-adics
> are essential in physics. I think it is the Quantized Hall Effect. Once
> the physicists report this, then the house-of-cards of mathematics all
> comes a fallin down.
>
> A mathematical proof is nothing more than a physics experiment that
> uses just a pen and a piece of paper. And just like in physics, where
> it takes but one experiment to ruin a theory, the same for mathematics,
> that when a physics report comes in that finite integers are not
> adequate in describing the Quantized Hall Effect but that the p-adics
> are necessary and sufficient thereof. That will be the day that physics
> will have destroyed mathematics and will build her back anew and
> better.
>
> Mathematics from Cantor until 1993 has become more philosophical than
> it has become scientific and it will pay the price for its vagrancy,
> its truancy, and its vandalism meanderings
>
> Noone but me can see that mathematics is nothing but physics and is a
> subdepartment of physics, but how could anyone see that unless they had
> a Atom Totality theory where mathematics is but a mirror reflection of
> how many atoms and atom characteristics.
>
>
I'm not disputing p-adics, so please don't bite. I actually
consider the prospect very interesting (I like it when I find that there
is an idea in math, physics that I haven't contemplated!).
I would merely like to point out that, while math was constructed
to model things in the real world, it was actually far more likely to be
modelling a transaction (my 7 sharp rocks for your arrowhead, say), and
not, for instance, the speed of a running gazelle. The way physics was
built from Gallileo to Descartes to Newton, physics is rather a branch of
mathematics, the mathematics of physical behaviour. And, if finite
integers are seen to be insufficient in physics, it implies either that
our understanding of the physical world is lacking, and/or mathematics as
it stands is not complete or properly defined. The argument would then
come down to why the former is not a sufficient reason for physics'
insufficiency.
Niel de Beaudrap
----------------------
jd@cpsc.ucalgary.ca
Subject: Re: Where's the symmetry?
From: Dirk Laurie
Date: Mon, 11 Nov 1996 09:30:12 +0200 (SAT)
>
> Can anyone recall why (in Abstract Algebra) the compositions of mappings
> of sets into their permutations are called "symmetric" groups? Since each
> element of the group is a permutation, why not call it a permutation group?
> (since composition is the "natural" operation for permutations)
>
> Symmetry implies to me something like commutative and associative properties
> etc. These groups are not necessarily commutative and all groups must (by
> definition) be associative, so what is this special symmetry?
>
Take a square of paper. You can rotate it by 90, 180, 270 degrees, flip
it around either diagonal, or around either axis (i.e. the line halfway
between to sides). Each of these operations leaves the square looking
the way it did before, and we call such a transformation a "symmetry"
of the square. Obviously doing two such operations in a row also leaves
the square invariant, so the set of all symmetries forms a group: the
"symmetry group" of the square.
Now take a set of n objects, and look at the transformations that keeps
this set invariant. It's not illogical to call them the "symmetries" of
the set. Since by definition a set does not depend on the order in which
you specify the elements, there is one symmetry for each possible such
order: one symmetry per permutation of the numbers 1 to n.
> While we're at it, aren't most of the terms used in Abstract Algebra (group,
> ring, ideal, integral domain, vector space) somewhat vague and inconsistent?
> ('Coset' is a great name, however). Hmm, can't think of anything better
> though. (How about 'algebraic sets'?)
>
"When I use a word, it means precisely what I choose it to mean: neither
more nor less." Mathematicians like to quote Lewis Carroll because almost
alone among mathematicians the Reverend Charles Lutwidge Dodgson was able
to disguise the standard habits of mathematicians as delightful nonsense,
entertaining even non-mathematicians in the process.
There comes a time when one needs a short, catchy word to describe an
object - perhaps because you will need to refer to it often - and then
you either coin a new one (not so easy) or redefine a familiar one.
Dirk Laurie
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Subject: Re: Solve this Please
From: Le Compte de Beaudrap
Date: Mon, 11 Nov 1996 23:29:02 -0700
On Fri, 8 Nov 1996, Lyle VonSpreckelsen wrote:
> Solve this
>
> Three Pipes supply an oil storage tank. The tank can be filled by
> pipes A and B running for 10 hours, by pipes B and C running for 15
> hours, or by pipes A and C running for 20 hours. How long does it
> take to fill the tank if all three pipes run?
>
> Got my andvanced math teacher (MR. V.) stumped
>
> J.D
What I am about to write may look long and inelegant, but it
really isn't. Read on.
______________________________________________________________
Let V = volume of tank,
a, b, c = the "flow rates" of pipes A, B, C
Therefore, 10(a + b + 0) = V {eqn1}
15(0 + b + c) = V {eqn2}
20(a + 0 + c) = V {eqn3}
{eqn2} - {eqn1} gives: -10a + 5b + 15c = 0
or (b + 3c)/2 = a
Substitute this into {eqn3} to get: 10(b + 3c) - 20c = V
or 10b + 10c = V.
But from {eqn1}: 10a + 10b = V
We have found: 10c + 10b = V
Therefore: 10a = 10c
or a = c.
Now, {eqn3} states that 20a + 20c = V
Therefore 40a = V
a = V/40
c = V/40
{eqn1} states that 10a + 10b = V
Therefore 10b = 3V/4
b = 3V/40.
Now that a, b, and c are known, what is the time "t" such that (a+b+c)t = V?
The question is now very straightforward. (5V/40)t = V,
t = 40/5 = 8.
Combined, the pipes fill the tank in 8h.
This is the most straightforward (ie, direct) way of solving the problem.
No more direct way can be found (please hit me if I'm wrong).
Niel de Beaudrap
----------------------
jd@cpsc.ucalgary.ca
Subject: Re: What's 0 divided by 0??
From: Le Compte de Beaudrap
Date: Mon, 11 Nov 1996 23:42:46 -0700
On Thu, 7 Nov 1996, Jeremy J. Olson wrote:
> x
> - = 1
> x
>
> although:
>
> x
> - = INF
> 0
>
>
> And if x = zero!?!?! Then which is it!?!?!??!?!
>
> Jeremy J. Olson
> olson@ici.net
> http://www.ici.net/cust_pages/olson/olson.html
>
>
It's not quite as simple as "one or the other". Division by zero
is always a subject for debate, I find.
For every value other than x=0, x/x = 1.
For every value LOWER THAN x=0, x/0 = -oo (oo = inifinity)
For every value GREATER THAN x=0, x/0 = +oo
For the value x=0, x/x and x/0 are both undefined.
That is to say, as of yet, no specific
value is attributed to either.
There is a concept called a limit (forgive me if I seem condescending: I
don't know where you are in your maths), with a notation that reads like
this:
lim f(x) = the limit, as x approches a, of a function "f" of x.
x -> a
The function of x is a mathematical expression like "x squared", "x
cubed", "x times 17", or something like that. The limit says that we are
looking at how th function behaves when x is almost, but not quite, equal
to a specific value a.
So, lim 15x = 15, because when x is nearly 1, "15x" is nearly 15.
x -> 1
2
Similarly, lim x = 4, because when x is nearly 2, x squared is nearly 4
x -> 2
The expression lim x/x is equal to 1.
x -> 0
This does NOT mean that x/x = 1 when x = 0. This means only that,
when x is VERY NEAR to 0, but NOT 0, x/x = 1. But at x = 0, x/x is
undefined/indeterminate.
This is my long winded answer to your simple question. Hope this
helps. Like I said, on some aspects of this question, there is no one
answer as far as some people are concerned.
Niel de Beaudrap
----------------------
jd@cpsc.ucalgary.ca
Subject: Re: Implicits again
From: mlerma@pythagoras.ma.utexas.edu (Miguel Lerma)
Date: 12 Nov 1996 07:30:54 GMT
Murat Erdem (erdemu@boun.edu.tr) wrote:
> Does anybody know what should be the constraint to force a bivariate
> implicit polynomial to be closed (bounded) ?
> (e.g. x^2+y^2-25=0 is a closed one since it represents a circle).
I think all you need is to remove all its terms except those of
greatest degree and check that the (hogeneous) polynomial obtained
has no non trivial zeros (those non trivial zeros are points at "infinity"
in the sense of the projective plane, and also informs us about the
direction of asymptotes). In your example that means that x^2+y^2=0
only for x=0 and y=0. A negative example is xy-1=0 (which is "unbounded"
in your sense), and that can be proven by checking that xy has non
trivial zeros, for instance x=1, y=0 (which at the same time informs
us that xy=1 has an asymptote in the direction of the vector (1,0),
the other asymptote corresponds to x=0, y=1). A more complicated
example could be:
p(x,y) = x^5 + 3 x^4 y - 2 x^2 y^3 + 7 x^2 y - 5 x y^2 + 3 x + 9
Is p(x,y)=0 a bounded set? Remove all terms except those of greatest
degree, which is 5 in this case:
q(x,y) = x^5 + 3 x^4 y - 2 x^2 y^3
and check if it has non trivial zeros. To do that, put y=1 and
x=1, obtaining respectively
f(x) = x^5 + 3 x^4 - 2 x^2
g(y) = 1 + 3 y - 2 y^3
If any of those polynomials has a zero then q(x,y) has non
trivial zeros, and p(x,y)=0 would be unbounded. Otherwise
(if none of f(x) and g(y) has zeros) then q(x,y)=0 only for
(x,y)=(0,0), and p(x,y)=0 will be bounded. In the above example
the set turns out to be unbounded, for instance g(-1)=0, so
q(1,-1)=0, which says that p(x,y)=0 has an asymptote in
the direction of (1,-1). Also f(0)=0, so q(0,1)=0, and
(0,1) is the direccion of another asymptote.
I hope this helps.
Miguel A. Lerma
Subject: Re: insights into the quantum Hall effect; SCIENCE 25OCT96; p-adics
From: Le Compte de Beaudrap
Date: Tue, 12 Nov 1996 00:26:33 -0700
On 8 Nov 1996, Archimedes Plutonium wrote:
> --- quoting SCIENCE, 25OCT96 ---
>
> UPON REFLECTION
>
> In theoretical physics, it is sometimes the case that the solution
> to one problem can be used to solve another by the proper
> transformation of the system, such as switching the role of electrical
> fields and charges with their magnetic analogs in electromagnetism (see
> the Perspective by Girvin, p. 524). Shahar et al. (p. 589) measured the
> current-voltage characteristics of a fractional quantum Hall effect
> fluid and its nearby insulating state and found that the results are
> essentially identical for the two states when current and voltage are
> interchanged. The existence of this duality symmetry for charge and
> magnetic flux may lead to new theoretical insights into the quantum
> Hall effect.
>
> --- end quoting SCIENCE, 25OCT96 ---
>
> Since my discovery that the finite integers, the counting numbers, or
> called Naturals in mathematics are a fake and that the p-adics or
> Infinite Integers are the real true integers, I have looked to physics
> to straighten-out the mess. I have looked to the Quantum Hall Effect
> with its strange numbers to clear the mess that is mathematics.
> Once the world sees a part of physics where the p-adics are essential
> and where the finite integers just do not work, that day my friends is
> a spectacular day here on Earth, for on that day physics subsumes
> mathematics, just as physics subsumed chemistry in the Schroedinger
> equation.
As soon as chemistry stopped being alchemy, it was a branch of
physics. It happened in the late 19th, not the early 20th, century. To
quote Rutherford: "All science is either physics or stamp collecting."
That was from long before the full glory of Quantum Physics.
As well, how can physics ever subsume math!? That's impossible!
While findings in physics may force math to change, it is the laws of
physics that are expressed in math, NOT the laws of math expressed in
physics. I defy you to, using Maxwell's equations, prove that 1+1=2,
without depending on the proof asked for to accomplish it.
If physics predicts a mathematical property, THEN has physics
subsumed math. You state that physics CONTRADICTS math, or shows that
math is insufficient; that means that the laws of mathematics do not form
a proper base as defined.
>
> Mathematicians have for centuries acted like high priests, acted
> superior to the sciences. Almost laughable here at the close of the
> 20th century that mathematicians pander off as true a monsterous 100
> page proof of Fermat's Last Theorem, when just around the corner the
> physicists will show the birdbrain mathematicians that the Quantized
> Hall Effect is written in p-adics. Essentially required p-adics and the
> finite integers just do not work.
Excuse me, but are you not also being a mathematician in
contributing to mathematics? (Are you not also acting like a High Priest
impersonator by writing this article? "The end of the finite integers is
a'comin, and all of the unbeleivin' mathematicians of the world will be
thrown into the fires of hell!")
As well, where does the impression of mathematicians feeling
superior to scientists arise? I never heard of this, and many
mathematicians were also physicists. Were you scared by a
mathematician in your childhood?
As an aside, I take opposition to your calling me a birdbrain,
despite the fact that I haven't breathed a word against p-adics
themselves yet. And until I have sufficient reason, I won't.
>
> What does all of this mean? It means simply that mathematics since
> Cantor in the late 1800s has been mostly gibberish, goon squad
> gibberish.
Maybe. As Quantum Physics has "classical" and "renormalised"
versions of theories, so may mathematics under p-adics. (What does
"p-adic" stand for, anyway? Just curious.)
>
> There are many people in this world who still believe that a
> mathematics proof such as say the 4 Color Mapping Proof or Wiles FLT
> has more bases in reality than any physics experiment, whether you take
> a shoddy one or a highly refined physics experiment. But it is this
> general feeling , this general notion that mathematics proofs are
> higher in trustworthiness of truth than physics experiments which has
> come to a shattering end and a shattering close by 1993. All it takes
> for mathematics to come rolling down from the top of the mountain is
> for physics to show one area of physics where p-adics are essential.
> When that happens then physics will forever more be King of the
> Mountain and mathematics will grub, grub along the base of the
> mountain, and whipped into shape by the physicists.
>
Firstly, I agree that physics EXPERIMENTS have more basis in
reality than math: math is a formal system which seems to work, and
physics experiments are measurements of reality itself. However,
THEORETICAL physics depends on math intimately. Physics theory without
math boils down to: "light is very very very very fast."
Secondly, how did you "derive" (for lack of a better word)
p-adics? Are p-adics a consequence of observation, as the existance of
the neucleus of an atom is? Or are p-adics the only way math and physics
able to coexist? If the latter, I submit that you have found a math that
is better for a basis of physics. P-adics are a part of physics (as
opposed to math) IF AND ONLY IF p-adics exist by the observations of
physics. Have you observed a 2 today? Not two objects, not ink in the
symbolic representation of "2", but an actual 2? No such thing exists,
one cannot "observe" a number, nor can one observe a class of numbers.
Thusly, p-adics are a part of MATH, NOT PHYSICS.
> I will look at all reports of the Quantized Hall Effect, for it is in
> these strange quantum numbers, these strange fractional quantum numbers
> that I believe they are strange looking only because they are not based
> on the Naturals = Finite Integers but instead,
>
> the Quantized Hall Effect is based on Naturals = P-adics and that the
> strange looking numbers are really p-adics or n-adics and their
> strangeness evaporates instantly when these numbers are put into p-adics
>
Ah, so then you did not observe a p-adic, you merely concluded
that to use a p-adic instead of a finite integer solved your problems.
Your problems of reconciliation of THEORETICAL PHYSICS with EXPERIMENTAL
PHYSICS, not PHYSICS with MATH. By improving math, you makephysics
consistent. If p-adics are indeed an improvement, I applaud your efforts.
However, in trying to convince (convert?) others to see things your way,
you have begun to sound more like a fanatical Nazi than a rational
philosopher of any type.
P-adics are math. New math, math brought about due to problems in
physics, but math nonetheless. By insulting math, you insult yourself.
Whether you are aware of it or not, you are a mathematician, and are
trying to bring about a mathematical, and not a physical, revolution.
Physics will not envelop math, as you envision: math will not be whipped,
kiss physics' feet, or be put into concentration camps. Theoretical
physics will still be the middle man between experimental physics and
math, trying to predict the former by use of the latter. It will merely
be the first time that an inadequacy in physics will necessitate a change
in math, is all, just as inadequacies have necessitated better
experimental procedures all these centuries.
I put it to you that you are either a physicist who has been
either abused, teased, or put down by mathematical peers, and that you
are trying to insult them by saying that physics is infinitely superior
to math. In that, you are gravely mistaken. All quests for truth are
equally valid, and while some may be based on others (ie, just as PHYSICS
is based on MATH and observation), all searches for truth are noble, and
light up our world with their insights.
This has been my humble opinion, amplified by way of reaction to
extreme comments about math. (Newton's 3rd law: For every action, there
is an equal but opposite reaction...a qualitative law of physics, which
needs math to be of any concrete use.) I neither oppose nor promote
p-adics, but I do oppose the way that the promoter(s) of p-adics seem to
go to great lengths put math down, especially with shock tactics like
"Physics Envelops Math" and "Math forced to grub, grub, grub". Let
us discuss things, and think things out, like reasoning beings: That's Why
God Gave Us Brains.
Math a branch of Physics? IMHO, impossible. P-adics valid? No
comment. After all, I haven't enough of a basis to have an opinion. If
only all people were like that, all the time...
Niel de Beaudrap
----------------------
jd@cpsc.ucalgary.ca
Subject: Re: Concepts of Time
From: Le Compte de Beaudrap
Date: Tue, 12 Nov 1996 00:55:43 -0700
On 8 Nov 1996, paul thomas wrote:
> In article <32830D66.7078@webworldinc.com>, mike@webworldinc.com says...
> >
> ** snip **
> >Tom Maciukenas wrote:
> ** snip **
> >> The same can be said of our knowledge of reality. Sure, there are some
> >> things we can never know (whether God created the world two seconds ago
> >> or not is one good example). But that shouldn't cause us to doubt the
> >> things we DO know, or the things we CAN know. What we DO know is that the
> >> world APPEARS to be XX billion years old. And if God went to all that
> >> trouble to make it seem that way, isn't it polite to oblige Him by
> >> believing it? :^)
> >
> >I think I like this argument. :)
>
> I too think it is an interesting argument. Such craftsmanship and attention
> to detail deserves to be rewarded.
> >
> > It's a lot softer than an observation that a longtime and very dear friend
> > made when we were rooming together during college. He speculated that
> > religion was invented to satisfy people who can't accept "I don't know" as
> > an answer. I thought this was very cool.
> **snip**
> > Agnostically yours,
> > Mike.
>
> There are other reasons religions were invented (such as a thirst for
> the promise of eventual vengence/justice in an unfair world), but is certainly
> part of the mix. I agree with Tom's comment about "What we DO know is that
> the world APPEARS...", but I am not sure that everyone shares that
> perspective. It implies an existentialist philosophy with which many people
> are uncomfortable. I (and some other existentialists) live my life as if it
> matters and is understandable even though I know deep down inside that life is
> fraught with contradictions and absurdities.
Hear, hear! It all comes down to what I like to think of as the
conundrum of fate (which is actually easily solved). It goes like this:
"If free will exists, then it would behoove me to believe in it. However,
if fate exists, then I can't possibly help the fact that I don't believe
in it; after all, I'm fated to believe in free will."
Similarly, if my life is important, I had better try. If it is
inconsequential, though, then it can do no harm that I tried my best to be
of consequence, futile as it may have been.
Besides, I'm sure we could never stand to not understand why things
are. We're fated to be this way ;)
Niel de Beaudrap
----------------------
jd@cpsc.ucalgary.ca
Subject: Re: Monotony
From: Maurizio Paolini
Date: Tue, 12 Nov 1996 09:31:22 +0100
ghidrah wrote:
>
> Arild Kvalbein wrote in article
> <459.6888T1227T2938@online.no>...
> > Anybody feel like discussing a function's monotony?
> >
> > For example, is x^3 raising in x=0 or is it at a stand-still? Is it
> raising
> > for all x-values?
> > Example 2: Is x^2 falling in x<0 or is it actually _x<=0_ ? If the
> latter, is
> > it both raising and falling in x=0?
> Look at the derivative in each case. If you differentiate x^3 with respect
> to x you get 3x^2. 3x^2 > 0 for every x not equal to 0. So x^3 is
> strictly increasing for every x not equal to 0. However, at x = 0, 3x^2 =
> 0, so the slope of the tangent line is zero at x = 0. Therefore, x^3 is
> neither increasing nor decreasing at x = 0.
> ...
Attention: to my knowledge, it depends on the definition of
monotonicity.
One widely used definition says that a function is strictly
increasing (in the whole domain of definition) if for any two
distinct points x, y it holds [f(x) - f(y)](x - y) > 0.
With this definition (which is very natural in many fields of
mathematics) x^3 IS strictly increasing!
If we restrict x^2 to [0,\infty), it also becomes strictly
increasing.
Constant functions are examples of nondecreasing (which could be
misleading) functions, satisfying [f(x)-f(y)](x,y) >= 0 for
any x, y.
--
Maurizio Paolini paolini@dimi.uniud.it
---------
http://www.mat.unimi.it/~paolini/
"Quello che non so e' quasi tutto. Quello che so e' qualcosa che, per
quanto limitato, e' pero' importante." (E. de Giorgi)
Subject: Re: Autodynamics
From: dean@psy.uq.oz.au (Dean Povey)
Date: 12 Nov 1996 08:57:53 GMT
dean@psy.uq.oz.au (Dean Povey) writes:
[Stuff deleted]
>>> From the Web page:
>>> "[Autodynamics] explains the perihelion advance of Mercury, Venus, Earth
>>> and Mars, and all Binary Star precessions for which we have data."
>>Where is your data posted?
>Well, I didn't come up with this theory so I must confess I don't know. But
>this is a good point. I'll email someone in the SAA and see if they can
>put the data up on their web page.
Oops sorry, put this down to a late night and too much reading/coding
:(. *Smacks self in forehead* The figures are there for all and
sundry, they can be found at: http://www.autodynamics.org/Cosmos/Gravity.html
Here is an extract:
=====>
SR and AD Comparison
The general relativity equation for advance of the Mercury perihelion is:
6 pi GM
T = ---------- [Pardon my ascii, DGP]
c^2 r (1 - e^2)
Where e = eccentricity, c = light speed and G, M, and r have the usual
meaning in this paper.
This equation yields, in a century:
42.4" for Mercury
8" for Venus
4" for Earth
1" for Mars
In AD gravitation, the perihelion advance for each planet is
proportional to the square root of the division of the solar mass by
the orbital radius power 3.
Tp = sqrt(M / r^3) [ditto: DGP]
If the Mercury value is taken as 43", the values for the other planets are:
Venus = 16.8"
Earth = 10.4"
Mars = 5.5"
[These] values are equal to Hall's empirical values and close to the
expected values calculated by Newcomb.
The big difference between SR and AD occurs when the distance to the
Sun is smaller. If the distance is 1/3 of the Mercury radius the values
are:
GR = 129"
AD = 223"
If the radius is 1/5 of Mercury the values are:
GR = 214"
AD = 481"
We see that the proportional difference increases with decreasing
radius. Putting a probe close to the Sun can easily test, in a short
time, if its perihelion advance is a natural phenomenon beyond the
planetary perturbation and if the values are given by GR or AD
gravitational equation.
<==========
The rest is on the web page. There are also figures for the binary star
DI Herculis.
Sorry about that.
Dean.