Countable and Uncountable, again.
Bill Taylor
a short list of such topics - they must be simple enough that a layman
has some chance of understanding the question, but complex enough to have
acquired an opus of machinery surrounding them that the learner can hide
behind when he gets stuck and can't or wont do any further work toward
genuine understanding. So they make fine breeding grounds for crackpots.
There are three classic crackpot generators:
1) FLT; (which the recent proof of has surprisingly not diminished one whit);
2) Godel's theorem;
3) Uncountability.
For some reason, the 4 colour theorem never achieved the status of FLT, and
I was never sure why! I'd have thought it was a sitter - even easier to
understand, and prone to pretty pictures and obfuscation. Even "practical".
But it hardly ever attracted much crackpottery, and since its proof, none.
Why? Anyone care to hazard a guess?
Riemann's hypothesis would make superb crackpot manure, but for one thing -
it's too hard to understand, and even the confirmed crackpot feels out of
his depth here! The same remarks *should* apply to uncountability, i.e.
Cantor's theorem. I believe that if the original proof (involving closed
intervals) had taken hold as the "real" one, this would have remained so.
But alas, the diagonal proof is so incredibly easy to understand, AND
so prone to leave vague feelings of discontent, (involving Richard-like
impredicativity, but that's another topic), that it has become THE field
of endeavour for the "upmarket" crackpot. By discussing a matter he knows
to be just beyond the scope of the ordinary layman, he keeps a delicate
frisson of superior feelings, but at the same time there is sufficient room
for obfuscation to maintain a feeling that his betters have "misunderstood"
him, and perhaps wilfully so, even conspiratorially. Even Godel's theorem
cannot *quite* match this heady crackpot-intoxicating mixture, (it is just
a tad too complex), though admittedly it comes close.
I write this article, which is just a rehash of a standard one I write here
often, with the intention of just maybe sorting out a few thoughts in those
learners who may otherwise become crackpots through frustration. I have little
real hope of this, though, and no hope at all of changing the thought patterns
of those confirmed uncountability crackpots we all know and love so well.
One of the troubles, I believe, is the way the meaning of the topic is
always presented. Even the term itself shares in this mini-villainy;
"uncountable" already conjures up images of a vast, unsurveyable vista of
almost incomprehensible magnitude, and learners may find this deeply troubling.
IMHO "unlistable" would be a *far* more preferable term, and much closer to
the heart of the matter. But the terminology is far too entrenched now.
The chief concern though, is the way it's invariably presented in popularizing
books and preliminary courses:- as a measure of the SIZE of sets. This is
not so very heinous, except that it is likely to prepare the ground for
crackpottery, should that tendency arise later. IMHO it is much safer,
and (oxymoron alert!) philosophically more accurate, to regard a set of
any higher cardinality than another, to be NOT bigger, but MORE COMPLEX.
This has many benefits. The two concepts overlap a great deal, but not
entrirely. It leaves open the possibility that two cardinalities may be
non-comparable, as two sets could easily be complex in different ways,
(as can happen in the absence of AC). It still allows the reals to be of
higher cardinality than the naturals, as they are (individually) far more
complex beasts, so the set of all of them inherits this complexity as well.
And complexness is the very feature that lies at the core of Cantor's proof.
And finally, to come back to the current context, it allows the pre-crackpot
to relax in the thought that yes, the reals are greater than the naturals in
a wholly *believable* way - they are not more numerous, merely more complex.
And that could be quite a gain for the math world, by reducing the crackpot
population at birth. Anyway that is my hope. I do it in my own classes.
-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Is aleph_69 a cardinal sin?
-------------------------------------------------------------------------------
Dik T. Winter
> For some reason, the 4 colour theorem never achieved the status of FLT, and
> I was never sure why! I'd have thought it was a sitter - even easier to
> understand, and prone to pretty pictures and obfuscation. Even "practical".
> But it hardly ever attracted much crackpottery, and since its proof, none.
> Why? Anyone care to hazard a guess?
You must have missed Archimedes Plutonium (or was it Ludwig Plutonium at
that time?) proving it. At the same time he proved the 8 colour theorem
in 3d-space. And recently there have also been a few threads about it.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Nico Benschop
>
> And that could be quite a gain for the math world,
> by reducing the crackpot population at birth. Anyway that is my hope.
> I do it in my own classes.
>---------------------------------------------------------------------
> Is aleph_69 a cardinal sin?
>---------------------------------------------------------------------
I could'nt agree more. Bless you, Bill, you made my day ;-)
---------- One is Always Halfway Anyway (AHA;-) -----------
xxxxxxxxxxxxxxxxxxxxxxxxxxxx1.1xxxxxxxxxxxxxxxxxxxxxxxxxxxx
Peano <--------- -----------> Cantor
N == a* 2^N == {a,b}* (2 dim for most;-)
-- NB -- http://www.iae.nl/users/benschop/cantor.htm
http://www.iae.nl/users/benschop/ism.htm (Integer State
Machines)
G. A. Edgar
] There are three classic crackpot generators:
]
] 1) FLT; (which the recent proof of has surprisingly not diminished one whit);
] 2) Godel's theorem;
] 3) Uncountability.
How about angle trisecting?
--
Gerald A. Edgar ed...@math.ohio-state.edu
Randy Poe
Taylor) wrote:
>For some reason, the 4 colour theorem never achieved the status of FLT, and
>I was never sure why! I'd have thought it was a sitter - even easier to
>understand, and prone to pretty pictures and obfuscation. Even "practical".
>But it hardly ever attracted much crackpottery, and since its proof, none.
>Why? Anyone care to hazard a guess?
Doesn't lend itself to a text-based forum. As we speak, there may be
crackpots wandering the halls of academia with pages full of 3-colored
maps.
I like your concept of criteria for crackpot-breeding grounds. Those
familiar with modern physics will recognize that similar things go on
with relativity, quantum theory, and cosmology (especially the Big
Bang and black holes).
One wonders if there would be a way to treat breeding grounds so that
they don't breed, but actually crackpots are probably a very important
part of the ecosystem. Aside from the obvious entertainment value,
they cause us to question and think about our fundamental axioms
(hypotheses in the physics world) and understand them more clearly.
That questioning is always a good thing.
And every once in a great while, an apparent crackpot turns out to be
right. That is of course what lights the fuel of every crackpot.
- Randy
Torkel Franzen
> And every once in a great while, an apparent crackpot turns out to be
> right.
Who do you have in mind here?
David C. Ullrich
wrote:
[...]
>
>And every once in a great while, an apparent crackpot turns out to be
>right. That is of course what lights the fuel of every crackpot.
What's an example in _mathematics_???
> - Randy
>
Christian Bau
I heard this story that Abel got a letter from some young man named Galois
claiming he had found a proof that polynomials of degree 5 or higher have
no closed form solution. He filed that letter under "crackpot" and ten
years later or so he found that it was indeed a correct proof.
Kari Aaltonen
whit);
Fit's me. I am a crackpot. So is Harris. The proof is OK.
2) Godel's theorem;
Fit's me. I am a crackpot. No sufficiently strong formal
systems (containing aritmetic) can be either sound or
complete. Refer also to the Humanity Theory.
3) Uncountability.
I am a crackpot. Do not bother to explain. It is simple.
Can you verify if I am correct, please ?
David C. Ullrich
The proof was included in the letter?
Arturo Magidin
Christian Bau <christ...@isltd.insignia.com> wrote:
>I heard this story that Abel got a letter from some young man named Galois
>claiming he had found a proof that polynomials of degree 5 or higher have
>no closed form solution. He filed that letter under "crackpot" and ten
>years later or so he found that it was indeed a correct proof.
Man. This is real perversion of history...
Abel wrote a paper called "On the solution of the general quintic by
radicals" or words to that effect. It was a standard mathematical
paper, in which he proved that it was impossible to solve the quintic
by radicals.
At the time, he was looking for a position. Hoping to get a
recommendation from him, he sent a copy of his paper to Gauss. The
title, however, misled Gauss, who was (a) too busy anyway; and (b) in
receipt of a large number of alleged proofs of ->how<- to solve the
quintic by radicals. So he didn't get around to reading it until after
Abel had died.
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
mag...@math.berkeley.edu
Randy Poe
wrote:
It seems to me that plate tectonics, the big bang, and the meteoric
extinction of dinosaurs were all theories widely derided by the
establishments at the time, and pushed by their somewhat iconoclastic
proponents until they finally gained experimental verification.
I can't think of any examples from mathematics.
- Randy
Christian Bau
(Arturo Magidin) wrote:
> In article <christian.bau-1...@christian-mac.isltd.insignia.com>,
> Christian Bau <christ...@isltd.insignia.com> wrote:
>
> <words that deserve to be snipped>
>
> Man. This is real perversion of history...
>
> Abel wrote a paper called "On the solution of the general quintic by
> radicals" or words to that effect. It was a standard mathematical
> paper, in which he proved that it was impossible to solve the quintic
> by radicals.
>
> At the time, he was looking for a position. Hoping to get a
> recommendation from him, he sent a copy of his paper to Gauss. The
> title, however, misled Gauss, who was (a) too busy anyway; and (b) in
> receipt of a large number of alleged proofs of ->how<- to solve the
> quintic by radicals. So he didn't get around to reading it until after
> Abel had died.
At least I got one name right :-)
Arturo Magidin
Bill Taylor <mat...@math.canterbury.ac.nz> wrote:
[.snip.]
>For some reason, the 4 colour theorem never achieved the status of FLT, and
>I was never sure why! I'd have thought it was a sitter - even easier to
>understand, and prone to pretty pictures and obfuscation. Even "practical".
>But it hardly ever attracted much crackpottery, and since its proof, none.
>Why? Anyone care to hazard a guess?
It doesn't lend itself to the Net, but it had, in fact attracted very
much amateur interest.
According to Martin Gardner, the most common 'proof' of the 4 color
map theorem send my amateurs consists of proving that K_5 is not
planar (or course, they don't actually phrase it that way: they show
that it is impossible to have 5 mutually adjacent regions on a
plane). Many think this is enough to prove the 4 color map theorem,
not realizing it is merely a necessary condition.
Even so, there is a fair amount of 'crackpottery' associated to it. Of
course, since it has been proven for a while now, the interest has
subsided (although you still get the 'I have an elementary proof, not
like the computer proof they had...').
Torkel Franzen
> It seems to me that plate tectonics, the big bang, and the meteoric
> extinction of dinosaurs were all theories widely derided by the
> establishments at the time, and pushed by their somewhat iconoclastic
> proponents until they finally gained experimental verification.
Plate tectonics was never derided - you're thinking of the theory of
continental drift, for which plate tectonics gives a much-needed
theoretical basis. Continental drift wasn't exactly derided, and
Wegener was not considered a crackpot (indeed he was a respectable
academic). The big bang is still considered absurd by some, but was
the theory invented by a crackpot? Similarly for the meteorite theory.
George Greene
: And finally, to come back to the current context, it allows the pre-crackpot
: to relax in the thought that yes, the reals are greater than the naturals in
: a wholly *believable* way - they are not more numerous, merely more complex.
It would seem that that given that Skolem had proved the existence of
a countable model for the first-order axiomatization of the reals (and
of the first-order axiomatiazation of anything else as well), this might
might be a good place for everybody to pause.
: And that could be quite a gain for the math world, by reducing the crackpot
: population at birth. Anyway that is my hope. I do it in my own classes.
Do you teach LST too?
--
---
"It's difficult ... you need to be united to have any
strength, but internal issues have to be addressed."
--- E. Ray Lewis, on liberalism in America
James Hunter
Torkel Franzen wrote:
The Big Fart is an *unremovable* absurdity though, so it actually
is the cre`me de la cre`me from the shitheads of "science".
Tapio
"Randy Poe" <ran...@visionplace.com> wrote
news:3a894b65...@news.newsguy.com...
As required panem et circenses, it take a freedom to remind:
Consider infinite integers defined as a sum of infinite series, exactly
analogous ways as decimal numbers are defined. Infinite integers (*N) are
still widely denied, i.e. most of people still claim infinite integers do
not exist. (It is worth to mention that infinite integers have also other
names).
The sum of the infinite series was also pointed out to be different than the
limit of the sum of the infinite series that is very radical result. The
basic simplicity is that the sum of the infinite serie must reproduce the
written number, which is different task than the concept that infinite serie
may have a limit-too.
According to "classic" definition two reals were considered to be equal, if
they had the same limit. By making a difference between two concept
(asymptotical limit and tangential limit) it was pointed out that two reals
may have the same asymptotical limit but those two reals are however
different, because they had a difference that is not zero. This was proofed
by eliminating equal members from the required and proposed equivalence
equation, which led to contradiction.
BTW, This among others simplified the requirements concerning the mapping of
positive *N one-to-one with R (0,1) and as a consequence, therefore there is
no one-to-one mapping between positive N and R (0,1). (It is very strange
that integers are claimed to be finite, but it is swallowed that decimal
part can be infinite. And ...then we had problems of mapping of integers
with reals (0,1) ref. Cantor etc.. Cantor considered N not *N)
Actually this is not a very good example as the writer claimed to be a clown
instead of the writer was called to be a troll. ;-)
(FYI: The clown of the circus forces the audience to laugh themselves "Why
the boys in our department did not recognise that?". Nobody laugh for
troll.)
Another point to consider is that the writer never claimed the presented
work was original. Vice versa, the writer pertinently wanted to point out
that the original definitions were interpreted or construed later by
mathematical community in a way that is not consistent with the original
claims, i.e. today's concept is different than what was originally claimed.
Infinite integers were also discovered earlier but definitions were
different (afaik ?).
You have received other examples-too.There are some historical flaws under
your thread concerning Galois.Galois is also a poor example because nobody
claimed that Galois is crackpot (during his days).
Carl Boyer in "A history of Mathematics" writes something like this (this is
my short abstract): His teachers estimated Galois as peculiar
(strange).Cauchy propably lost Galois first article. Fourier examined Galois
second article, but Fourier died and Fourier's paper dissappeared. The 3rd
trial by Galois was examined by Simeon-Denis Poisson, who required some
additional proofs. Finally- Galois wrote his theories during his last night
for his friend Chevalier and Joseph Liouville published Galois papers in
Journal de Mathematique. Gauss was referred by Galois, but I cannot find
anything else concerning Galois_Gauss relationship.
Carl Boyer writes in his book back cover and also in the preface (at least
in my translated edition). My free translation back to English is:
"Mathematics is unique, There is no remarkable corrections in mathematics,
there are only extensions". I unfortunately disagree. I consider:
Mathematics is just like any other science.
Tapio
Ross A. Finlayson
Tapio wrote:
I'm among those who say you can use N, the set of natural numbers, to map to
R(0,1), whether N is only finite or infinite (*N).
The reason I use to justify this argument is that the values of the relevant
properties of my map assignation are the same for "finite" or "infinite" N.
The limits go to zero or the other expected values for either case.
Cantor did not have the luxury of rigorous hyperreals.
It is! Math is not mathematics.
Ross
--
Ross Andrew Finlayson
Finlayson Consulting, Est. 1994
Ross at tiki-lounge.com: http://neurosis.hungry.com/~raf/
"Have a nice day." FARS, DFARS, Berne, USA copyright rules may apply
Steve Leibel
mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
>
> And that could be quite a gain for the math world, by reducing the
> crackpot
> population at birth. Anyway that is my hope. I do it in my own
> classes.
>
I think you're confusing math with sci.math. sci.math is a Usenet
newsgroup where the principle of free speech trumps the principle of
rational discourse. It's not the purpose of working mathematicians to
state their arguments in such a way as to prevent the existence of
irrational discourse on Usenet.
David Petry
David C. Ullrich wrote
Some mathematicians regarded Cantor as a crackpot, and now some
think he is right.
David Petry
>>For some reason, the 4 colour theorem never achieved the status of FLT, and
>>I was never sure why!
For one thing, anyone who attempts to prove the 4 color theorem realizes
he'd better be able to color any map thrown at him, and a little
experimentation might shake his confidence in his ability to do so.
There is no such confidence shaker in FLT.
Torkel Franzen
> Some mathematicians regarded Cantor as a crackpot, and now some
> think he is right.
Which mathematicians regarded Cantor as a crackpot?
Steve Leibel
<tor...@sm.luth.se> wrote:
None whatsoever. A number of posters to sci.math express these
"anti-Cantor" sentiments from time to time but again, this is Usenet.
Torkel Franzen
> None whatsoever. A number of posters to sci.math express these
> "anti-Cantor" sentiments from time to time but again, this is
> Usenet.
There were of course mathematicians who didn't regard set theory
as meaningful (as there are now), but I doubt that any of them
regarded Cantor as anything but a fine mathematician.
David Petry
Torkel Franzen wrote
Kronecker called Cantor "a corrupter of youth".
I don't know much else about it.
Torkel Franzen
>Kronecker called Cantor "a corrupter of youth".
That has nothing to do with being a crackpot.
Daryl McCullough
>IMHO it is much safer, and (oxymoron alert!) philosophically more
>accurate, to regard a set of any higher cardinality than another,
>to be NOT bigger, but MORE COMPLEX.
I don't quite understand this way of looking at it. All countable
sets are not equally complex. The set of all integers is simpler
than the set of all integers that code true statements of arithmetic,
even though they have the same cardinality.
So cardinality and complexity, though maybe they are related, certainly
aren't synonymous, so I'm not sure how complexity helps to explain the
difference between, say, the integers and the reals.
--
Daryl McCullough
CoGenTex, Inc.
Ithaca, NY
Jon and Mary Frances Miller
> There were of course mathematicians who didn't regard set theory as
> meaningful (as there are now), but I doubt that any of them regarded
> Cantor as anything but a fine mathematician.
Kronecker considered Cantor a "bad" mathematician. Not because his math
was wrong, but because it was meaningless, and lured others away from
meaningful mathematics. Of course, other mathematicians (Hilbert a
notable example) disagreed with Kronecker about this.
You might compare with Socrates and the Athenian leaders.
The closest thing to a "crackpot" being later proved right in
Mathematics that I can think of is de Branges' proof of the Bieberbach
conjecture. de Branges had a reputation for not checking his work
carefully enough and was considered by some to be "borderline crackpot",
although that appellation is gone now. Note that even though
"borderline crackpot", he still had an academic position and was
publishing regularly in reviewed journals, not a typical example of
crackpot activity.
As to the other examples cited, Walter and Luis Alvarez' theory of
meteor impact was considered at the time it was first raised not the
theory of crackpots, but at worst a crackpot theory from reputable
scientists. Even then, there were plenty of scientists who gave it
consideration because of the reputation of its proponents who would not
have finished reading a letter from me, tossing it out and muttering
"crackpot".
I think that crackpot/respectability mostly goes the other way (at least
for scientists, if not theories), as James Shockley and Ilya Prigogine
seem to have lost a lot of luster in their later years and many people
feel that Newton went off the deep end when he switched from physics to
theology (and the same for Napier). The latter two are interesting
positions, since both Newton and Napier were writing theology at the
time they were creating their mathematics, and the basic tenets remained
the same throughout their writings.
Jon Miller
Torkel Franzen
> Kronecker considered Cantor a "bad" mathematician. Not because his math
> was wrong, but because it was meaningless, and lured others away from
> meaningful mathematics.
That is not to say that Kronecker considered Cantor a crackpot,
any more than he considered other eminent mathematicians who
embraced set theory crackpots.
> I think that crackpot/respectability mostly goes the other way (at least
> for scientists, if not theories), as James Shockley and Ilya Prigogine
> seem to have lost a lot of luster in their later years and many people
> feel that Newton went off the deep end when he switched from physics to
> theology (and the same for Napier). The latter two are interesting
> positions, since both Newton and Napier were writing theology at the
> time they were creating their mathematics, and the basic tenets remained
> the same throughout their writings.
What writings of Newton's do you have in mind? NB: writings that
he published in his lifetime.
Daryl McCullough
>I think that crackpot/respectability mostly goes the other way (at least
>for scientists, if not theories), as James Shockley and Ilya Prigogine
>seem to have lost a lot of luster in their later years...
What did Prigogine do that is considered crackpot?
James Hunter
Torkel Franzen wrote:
> Jon and Mary Frances Miller <jonatha...@home.com> writes:
>
> > Kronecker considered Cantor a "bad" mathematician. Not because his math
> > was wrong, but because it was meaningless, and lured others away from
> > meaningful mathematics.
>
> That is not to say that Kronecker considered Cantor a crackpot,
> any more than he considered other eminent mathematicians who
> embraced set theory crackpots.
Well Kronecker was a natural number worshipper so he was
really a bigger crackpot than Cantor.
David Petry
Torkel Franzen wrote
>"David Petry" <dpe...@uswest.net> writes:
>
>
> >Kronecker called Cantor "a corrupter of youth".
>
> That has nothing to do with being a crackpot.
I guess I don't know the technical definition of "crackpot".
It's been many many years since I read a biography of
Cantor which, if I recall correctly, surmised that Cantor's
mental illness was at least partly the result of the treatment
he received from the mathematics community after he
proposed his theory. Presumably, many members of the
mathematics community thought of him as less than a
fine mathematician.
Torkel Franzen
> Presumably, many members of the
> mathematics community thought of him as less than a
> fine mathematician.
Name a few of them, and provide some sources for their opinion
of Cantor as a mathematician. Rejecting set theory as meaningless
does not imply describing set theorists as less than fine
mathematicians, let alone regarding them as crackpots.
Mike Oliver
>
> Kronecker called Cantor "a corrupter of youth".
>
<Best Samantha's-little-sister voice from "High Society">
Kronecker was an evil old man, wasn't he?
<voice off>
James Hunter
Torkel Franzen wrote:
Every language that the religious can't understand is
considered to be meaningless. Since that's been known
for over 2000 years there is nothing new there.
Randy Poe
Are you offering us an explanation for your ad nauseam comments that
all of the language of mathematics and physics is meaningless?
- Randy
David Petry
Torkel Franzen wrote
>"David Petry" <dpe...@uswest.net> writes:
>
> > Presumably, many members of the
> > mathematics community thought of him as less than a
> > fine mathematician.
>
> Name a few of them,
Nah, the topic bores me; I'm not going to research it.
Torkel Franzen
> Nah, the topic bores me; I'm not going to research it.
Of course not.
The fact that there isn't any example of any mathematical crackpot
who made good is of course no deterrent to crackpots. "They laughed
at Einstein, too" is good enough for any stout-hearted crackpot.
Arturo Magidin
Except they didn't laugh at Einstein... (-;
Of course, more to the point is that they also laughed at Bozo the
Clown. Much more.
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
mag...@math.berkeley.edu
Torkel Franzen
> Except they didn't laugh at Einstein... (-;
Of course not, but this is one of the thoughts that console the
crackpot.
David Petry
Torkel Franzen wrote
> The fact that there isn't any example of any mathematical crackpot
>who made good is of course no deterrent to crackpots. "They laughed
>at Einstein, too" is good enough for any stout-hearted crackpot.
Yeah, yeah, yeah.
They laughed at Hitler too.
But is that relevant?
Tapio
"David Petry" <dpe...@uswest.net> kirjoitti viestissä
news:hxSi6.864$J4.1...@news.uswest.net...
1) Hippasos (from Kroton or Metapontion) was a Pythagorean.There are at
least three different stories (omitted purposely see web-sites). Anyway - He
was kicked out from mathematics community. It is also told that Pythagoreans
set up for him a gravestone though Hippasos was alive. Exclusion and
taciturnity? Quite often used, but sometimes we do not know as maybe in this
example:
2) Danish Georg Mohr (1640-1697) was maybe thought to be less important,
because his book published 1672 "Eucleides danicus" was no way recognized as
important even in Danmark. The reason can be also a limited edition ?. Mohr
proofed that all such geometric constuctions that are possible with compass
and streightedge are also possible only and only with compass iff two points
are considered to define a line. Thus one instrument is enough. These
constructions are often called Mascheroni constructions. Mascheroni
published the same principle 125 year later than Mohr. Mohr's book was
totally disappeared and it was found by accident in antiqua book store in
Copenhagen in 1928.
Tapio
James Hunter
Randy Poe wrote:
Not really, but since both religions tend to be philosophic
with their ad nauseam Latin gibberish, we just like to remind all
religious shitheads that they are in the process of being digitized.
Bill Taylor
and to the point:
|> All countable sets are not equally complex.
Very true; thanks for reminding us of this.
|> The set of all integers is simpler than the set of all integers that code
|> true statements of arithmetic, even though they have the same cardinality.
Excellent point. The complexity of a set seems to be acquired by two things,
the complexity of its individual elements, and the complexity of the rule
determining which elements are in it. Clearly for the case of A being a
subset of B, A is automatically <= B in the first way, though may be > B
in the other. And as in such cases we also have A <= B in cardinality,
it seems that the first way is more important. But not over-riding.
|> So cardinality and complexity, though maybe they are related, certainly
|> aren't synonymous,
Very true. One could say the same of cardinality and containment.
But whereas containment and complexity are both transitive, complexity
need not be trichotomous, unlike containment. And without AC, cardinality also.
|> so I'm not sure how complexity helps to explain the
|> difference between, say, the integers and the reals.
Well I'd have thought that was obvious from the way Cantor's theorem is proved.
The individual reals are complex enough (simple-mindedly:- "long enough") to
admit of construction in such a suitably cunning way. Whereas the individual
integers aren't. This ties in with my constant articles about the unimpeachable
reality of the integers, compared with the rather more difficult credibility of
some alleged reals.
As I say, your comments are always valuable and welcome, Daryl!
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Bill Taylor W.Ta...@math.canterbury.ac.nz
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And God said:
Let there be numbers
And there *were* numbers.
Odd and even created he them,
He said to them be fruitful and multiply
And he commanded them to keep the laws of induction.
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Bill Taylor
|> They laughed at Hitler too.
They laughed at me when I said I wanted to become a comedian.
But I showed them!
Nobody's laughing now...
-----------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
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Use our home testing kit to determine if you really have a headache!
-----------------------------------------------------------------------------
Steve Leibel
mat...@math.canterbury.ac.nz (Bill Taylor) wrote:
> Well I'd have thought that was obvious from the way Cantor's theorem is
> proved.
> The individual reals are complex enough (simple-mindedly:- "long enough")
> to
> admit of construction in such a suitably cunning way. Whereas the
> individual
> integers aren't. This ties in with my constant articles about the
> unimpeachable
> reality of the integers, compared with the rather more difficult
> credibility of
> some alleged reals.
>
Many people feel this way. And yet if the integers are unimpeachable,
certainly collections of them are unimpeachable as well (just as
collections of honest people are generally honest).
But the collections of integers can be put into 1-1 correspondence with
the reals -- not just the nice reals, but the entire pile of "alleged"
ones, including the ones that can never be described by finite-length
strings or produced by algorithms.
What do you make of that?
Lee Rudolph
>They laughed at me when I said I wanted to become a comedian.
>But I showed them!
>Nobody's laughing now...
One is tempted to say "la commedia e finita", but with Ross and his
like on the rampage, one must resist.
Mike Oliver (if no one else) may possibly appreciate that, having
studied Italian but being then (as now) quite ignorant of opera,
I first heard that phrase uttered by the leader of a gang of Italian
laborers who ran Ken McAloon, me, and another young American
mathematician out of Cahors one rainy evening in the summer of 1969.
Lee Rudolph
Ross A. Finlayson
Lee Rudolph wrote:
That's after the time they had some futurism.
Ross
--
Ross Andrew Finlayson
Finlayson Consulting, Est. 1994
Ross at tiki-lounge.com: http://neurosis.hungry.com/~raf/
"Have a nice day." FARS, DFARS, Berne, USA copyright rules may apply
Dave Seaman
>>They laughed at me when I said I wanted to become a comedian.
>>But I showed them!
>>Nobody's laughing now...
>One is tempted to say "la commedia e finita", but with Ross and his
>like on the rampage, one must resist.
Which are you trying hard to resist? The line itself, or the act that
precedes it in the opera?
--
Dave Seaman dse...@purdue.edu
Amnesty International calls for new trial for Mumia Abu-Jamal
<http://www.amnestyusa.org/abolish/reports/mumia/>
Daryl McCullough
>They laughed at me when I said I wanted to become a comedian.
>
>But I showed them!
>
>Nobody's laughing now...
Do you know where that joke came from? I thought *I* made it up,
years ago, but I must have heard it and forgot the source.
James Hunter
Torkel Franzen wrote:
They really laugh at *Einstein* though, since being intelligent
he discovered Relativity, but also not being a moron he
didn't invent the Big Fart.
John R Ramsden
>
> Torkel Franzen <tor...@sm.luth.se> wrote:
> >
> > ran...@visionplace.com (Randy Poe) writes:
> >
> > > And every once in a great while, an apparent crackpot turns out
> > > to be right.
> >
> > Who do you have in mind here?
>
> It seems to me that plate tectonics, the big bang, and the meteoric
> extinction of dinosaurs were all theories widely derided by the
> establishments at the time, and pushed by their somewhat iconoclastic
> proponents until they finally gained experimental verification.
In case anyone disputes any or all of these, one geological theory that
certainly was universally scorned at the outset, and continued to be in
the dog house until ten years or so ago, was the Snowball Earth theory.
Back in the 1930s a British geologist explained the occurrence of large
lumps of rock in deposits all over the Earth as being "drop stones", i.e.
rocks deposited on the sea floor by melting glaciers, and his conclusion
was that the entire Earth had been covered by ice at the beginning of the
Cambrian (600 million years BC).
This theory is generally accepted now that all the former objections
have been explained. (The main objection was that once a global ice
sheet formed it would be there to stay, due to the high albedo of
ice reflecting radiation away from the Earth. But this overlooks the
steady atmospheric build-up of CO2 from volcanoes, in the absence of
rain to wash it out of the atmosphere, and the consequent green-house
effect.)
The theory also neatly explains the Cambrian "explosion" of multi-celled
species, since life only survived in small pockets under the thin ice
near the equator and in isolated mountain lakes etc, so that when the
ice melted most of the former competition was no longer present to
constrain the survivors.
> I can't think of any examples from mathematics.
Wasn't calculus considered suspect by many in its early days? Bishop
Berkeley described "fluxions" dismissively as the "ghosts of departed
quantities"; but how widespread this skepticism was I'm not sure.
Cheers
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John R Ramsden (j...@redmink.demon.co.uk)
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The new is in the old concealed, the old is in the new revealed.
St Augustine.
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Russell Easterly
> And every once in a great while, an apparent crackpot turns out to be
right.
They didn't have usenet back then.
Nowadays, everyone is a crackpot.
Russell
- Zeno was right. Motion is impossible.