Surely You're Joking, Mr. Ullrich !
Han de Bruijn
book "Complex Made Simple" by David C. Ullrich. On page 200 it reads:
(An event is said to occur "almost surely" if it happens with proba-
bility 1. If it sounds to you like this means the event is certain
to occur, we assure you that it does not; the distinction between
"almost surely" and "certainly" will become clear when you study
formal measure-theory-based probability theory.)
?? Another kind of "counter intuitive" mathematics again ??
Han de Bruijn
José Carlos Santos
What's your problem concerning that sentence exactly? I ask this because
it is the kind of thing that I could have written myself.
Best regards,
Jose Carlos Santos
Toni...@yahoo.com
wrote:
== Not at all, my dear HdB: it simply is Mathematicas, capital M,
though for you it seems to be mandarin chinese mixed with totonaca and
aztec.
No worries: not all have the required talent to understand basic
mathematics, and WM still loves you.
Regards
Tonio
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
>Sentences between (parentheses) are often the most interesting in the
>book "Complex Made Simple" by David C. Ullrich. On page 200 it reads:
>
>(An event is said to occur "almost surely" if it happens with proba-
> bility 1. If it sounds to you like this means the event is certain
> to occur, we assure you that it does not; the distinction between
> "almost surely" and "certainly" will become clear when you study
> formal measure-theory-based probability theory.)
No, I'm not joking there.
>?? Another kind of "counter intuitive" mathematics again ??
Hard to answer that, since what's intuitive varies from person
to person. You wouldn't find it the slightest bit surprising
if you _had_ studied formal measure-theory-based
probability.
Define A = [0,1] and B = (0,1). So A is a closed interval
and B is an open interval.
Suppose you choose a point of A "at random". Call that
randomly-chosen point x.
(i) What is the probability that x is in B?
(answer: 1).
(ii) Is x = 1 impossible?
(answer: no. All the points of A are the same here;
if x = 1 were impossible then x = anything else
would also be impossible, but x does equal
_something_.)
So the event "x is in B" has probability 1 but is
not certain.
Anything about that that bothers you?
>Han de Bruijn
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
erlan...@gmail.com
Suppose you drop a stick to the floor. What is the the probability
that the stick will point _exactly_ in the north/south direction?
If you think about it, you realize that the probability must be zero,
since the north/south direction is an infinitely small proportion of
the set of all directions (which we assume are equally probable).
Likewise, the probablity that the stick points in _any_ particular
direction is zero. But the stick must point in _some_ direction after
you drop it to the floor. This means that an event with probability
zero must occur.
So, you see that events with probability zero need not be impossible,
and their complements, events with probability one, need not be
certain!
This is of course an idealization: there are no infinitely thin sticks
and it is hardly meaningful to say that a real life stick points in an
_exact_ direction. But mathematics deals with idealizations, and must
do so; otherwise, it wouldn't be mathematics!
Han de Bruijn
Certainly so.
The difference between the closed set A and the open set B is marginal.
Too marginal, to be precise. Therefore:
(i) What is the probability that x is in B?
Exactly the same as the probability that x is in A.
(ii) Is x = 1 impossible?
No. For some good reasons; see above.
So the event "x is in B" = "x is in A" has probability 1 and is certain.
Han de Bruijn
Angus Rodgers
>But mathematics deals with idealizations, and must
>do so; otherwise, it wouldn't be mathematics!
Most interesting and useful mathematics does so, and has done
so ever since Euclid and his predecessors, but I don't think
it's true that all mathematics /has/ to. For instance (I should
apologise for mentioning this yet again, without exhibiting any
complete development of the idea) it is possible to define what
is meant by a finite sequence (e.g. the sequence of buttons on
a shirt, or the sequence of written letters [strictly, tokens
of letters] in a written word) without any idealisation of the
concrete phenomena, and the result still looks like mathematics.
(Very similar arguments to Dedekind's /The Nature and Meaning of
Numbers/, but with a successor function that is partial rather
than total, and may not even be injective.) Attempts have also
been made (at least since Nicod's /La géométrie dans le monde
sensible/ (1923): see <http://en.wikipedia.org/wiki/Jean_Nicod>)
to develop geometry as a part of physics, without idealisation.
However, it seems that the heart and soul of mathematics do lie
in some sort of idealisation, and it is a somewhat grim business
to try to do without it. (At the moment I'm not even trying, but
I mean to get back to finiteness, pointless geometry, and so on,
later on, when I'm better acquainted with the "heart and soul"!)
It might be more true to say that mathematics always deals with
abstractions (which occasionally, but seldom, remain absolutely
faithful to the phenomena from which they have been abstracted).
--
Angus Rodgers
Han de Bruijn
> On Mar 11, 11:07 am, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL>
> wrote:
>
>>Sentences between (parentheses) are often the most interesting in the
>>book "Complex Made Simple" by David C. Ullrich. On page 200 it reads:
>>
>>(An event is said to occur "almost surely" if it happens with proba-
>> bility 1. If it sounds to you like this means the event is certain
>> to occur, we assure you that it does not; the distinction between
>> "almost surely" and "certainly" will become clear when you study
>> formal measure-theory-based probability theory.)
>>
>>?? Another kind of "counter intuitive" mathematics again ??
>
> == Not at all, my dear HdB: it simply is Mathematicas, capital M,
> though for you it seems to be mandarin chinese mixed with totonaca and
> aztec.
> No worries: not all have the required talent to understand basic
> mathematics, and WM still loves you.
The idea that _you_ "have the required talent to understand basic
mathematics" has, so far, NOT become evident from some _substantial_
contributions from your side. All I've seen from you are just CHEAP
hanger-on comments like "I'm so glad that *I* understand mathematics
so well while you suckers don't".
Han de Bruijn
Jesse F. Hughes
> The difference between the closed set A and the open set B is marginal.
> Too marginal, to be precise. Therefore:
>
> (i) What is the probability that x is in B?
>
> Exactly the same as the probability that x is in A.
>
> (ii) Is x = 1 impossible?
>
> No. For some good reasons; see above.
>
> So the event "x is in B" = "x is in A" has probability 1 and is certain.
>
> Han de Bruijn
I don't understand how the claim following "so" is supposed to follow
from the above claims. (As well, your notion of the event is a bit
strange! The event is simply "x is in B given that x is in A.")
You just agreed that x = 1 is possible and hence, it seems, that "x is
in B given that x is in A" is *not* certain. Curiously, you instead
immediately claim that it *is* certain!
--
Jesse F. Hughes
"I'm better than you, and you know it."
-- James Harris
Dave L. Renfro
Han de Bruijn wrote (in part):
> The difference between the closed set A {the closed
> interval [0,1]} and the open set B (the open interval
> (0,1)} is marginal. Too marginal, to be precise.
Here's a less "marginal" example. Let A be the closed
unit square [0,1] x [0,1] in the plane and let B be
the union of all the points in A that belong to some
line of the form y = mx + b or to x = c, where m, b,
and c are rational numbers. In particular, every point
in A on some horizontal line y = b (b rational) and
every point in A on some vertical line x = c (c rational)
is included, every point in A on any of the infinitely
many diagonal lines y = x + b (b rational), every point
in A on any of the infinitely many diagonal lines y = 2x + b
(b rational), every point in A on any of the infinitely
many diagonal lines y = -(3579/17771)x + b (b rational), etc.
Then the planar area of set B is zero (B is the union of
countably many line segments), so the probability
that a randomly chosen point in A lies in B is zero.
Dave L. Renfro
Han de Bruijn
As long as there is an _exact_ distinguishing between irrational and
rational numbers: YES. But make that distinguishing a tiny bit fuzzy
and your argument is NOPE. How do we make a fuzzy line segment? Start
reading on page 11 of:
http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/VaagZien.pdf @
http://hdebruijn.soo.dto.tudelft.nl/www/programs/delphi.htm#VZ
Han de Bruijn
Dave L. Renfro
> As long as there is an _exact_ distinguishing
> between irrational and rational numbers: YES.
I guess I'm included among those who would make
such an exact distinction, for otherwise there would
be little of interest to me in mathematics. Kind of
like a linguist assuming the existence of words,
a biologist assuming the existence of an external
world, a sci.math poster assuming that other posters
are real people and not MIT machine intelligence
experiments, etc.
Dave L. Renfro
Achava Nakhash, the Loving Snake
> On Wed, 11 Mar 2009 04:57:11 -0700 (PDT), erlandga...@gmail.com wrote:
> I mean to get back to finiteness, pointless geometry, and so on,
>
>> Angus Rodgers
Interesting. When I was in high school I thought I thought all
geometry was pointless, but after looking at "Geometry Revisited" by
Coxeter and Greitzer, I learned to love the subject, while remaining
reasonably ignorant.
Regards,
Achava
Dave L. Renfro
>> I mean to get back to finiteness, pointless geometry, and so on,
Achava Nakhash wrote:
> Interesting. When I was in high school I thought I thought
> all geometry was pointless, but after looking at "Geometry
> Revisited" by Coxeter and Greitzer, I learned to love the
> subject, while remaining reasonably ignorant.
I thought Angus Rodgers was making an allusion to
pointless topology, but maybe I was reading too much
into his words.
http://en.wikipedia.org/wiki/Pointless_topology
Dave L. Renfro
Angus Rodgers
Yes, and no: I wasn't alluding to the theory of locales and frames,
specifically, but rather to the general idea of trying to do a kind
of geometry without taking "point" as a primitive concept. (Or, if
you prefer, redefining "point" in such a way that a "point" is not
to be thought of, even loosely, as something infinitesimally small
- but that seems to me a confusing use of the word in this context,
although of course it is strictly defensible, as a special case of
the extended use of the word "point" in general topology.)
I imagine Achava was joking, but I may be reading too much into his
words! :-)
--
Angus Rodgers
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
Just observing that the probability of getting a rational value if one
could pick a real number truly "at random" must be zero.
Virgil
Possibly in physics, but NOT in mathematics.
In mathematics the distinction between x < 1 and x <= 1 is non-trivial,
however trivial it may be in physics.
Those idiots who regard mathematics as merely a subset of physics only
show their ignorance.
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
The idea that HdB has "the required talent to understand basic
mathematics" has, so far, NOT become evident, and is counter-indicated
by his evident misrepresentations of it.
While in physics, there may be little practical distinction between
"x < 1" and "x <= 1", in mathematics, the distinction can be and usually
is quite important.
That HdB does not know that shows his ignorance of the distinction
between physics and mathematics.
Virgil
Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
> Dave L. Renfro wrote:
>
> > Han de Bruijn wrote (in part):
> >
> >>The difference between the closed set A {the closed
> >>interval [0,1]} and the open set B (the open interval
> >>(0,1)} is marginal. Too marginal, to be precise.
> >
> > Here's a less "marginal" example. Let A be the closed
> > unit square [0,1] x [0,1] in the plane and let B be
> > the union of all the points in A that belong to some
> > line of the form y = mx + b or to x = c, where m, b,
> > and c are rational numbers. In particular, every point
> > in A on some horizontal line y = b (b rational) and
> > every point in A on some vertical line x = c (c rational)
> > is included, every point in A on any of the infinitely
> > many diagonal lines y = x + b (b rational), every point
> > in A on any of the infinitely many diagonal lines y = 2x + b
> > (b rational), every point in A on any of the infinitely
> > many diagonal lines y = -(3579/17771)x + b (b rational), etc.
> >
> > Then the planar area of set B is zero (B is the union of
> > countably many line segments), so the probability
> > that a randomly chosen point in A lies in B is zero.
>
> As long as there is an _exact_ distinguishing between irrational and
> rational numbers: YES.
In mathematics, at least, there is such an exact distinction.
That HdB's physics does not include mathematics is evidenced by the fact
that in his physics there is no such exact distinction.
Kaba
> Just observing that the probability of getting a rational value if one
> could pick a real number truly "at random" must be zero.
So does it hold that a random real number in an interval [a, b[ is
almost surely irrational?
Virgil
Kaba <no...@here.com> wrote:
Yes, at least if the number is to be selected in such a way that gives
each real number in that interval equal probability of being selected as
any other.
Jon Slaughter
"David C. Ullrich" <dull...@sprynet.com> wrote in message
news:rj8fr4hsj0j1dvqet...@4ax.com...
The problem is that you can't choose any x, the probability of "choosing an
x", is 0. I know you realize this but you are mixing the two concepts up
here. The probability of x being in B is 1 because the probability of x = 0
or 1 is 0. The probability of choosing in x in A is 0. The probability of
choosing some small interval x is proportional to it's length assuming a
uniform density.
All these ideas are consistent with finite probabilities except that you are
implying there is some non-zero probability of choosing an element in
[0,1]... after all, you said
"Suppose you choose a point of A "at random""
and this is simply impossible. This is what is confusing HdB IMO. You are
treating the set as if it were finite and using those concepts and same
terminology along with setting up similar implications. Everything you have
said is perfectly true except your hypothesis is wrong(that is "choosing a
point").
There is a huge difference between a finite but extremely small probability
and an infinitesimally small or zero probability. You should be more clear
when dealing with infinite sets since it sounds as if your using the same
logic when we really need to approach them differently.
As we both know, the only way to get a non-zero probability is to look at
sets that contain an infinite number of elements. Any finite set must have a
0 probability.
> (ii) Is x = 1 impossible?
>
> (answer: no. All the points of A are the same here;
> if x = 1 were impossible then x = anything else
> would also be impossible, but x does equal
> _something_.)
You are correct that the probability of x is the same as the probability of
every other element in the set but since we can't even choose those elements
it is an invalid conclusion.
Now, you can say "Sure we can choose them". And in some sense we can. But
how do we choose a real number from the set [0,e] for any e>0? Since every
choice as the same probability(assuming uniformly w.l.o.g.) the probability
of choosing one must be 0. But if every element has probability 0 then no
element can be chosen.
(and zero here techinically is infinitesimally small)
Ask yourself how you can construct a RNG to choose a real number from the
interval [0,1]? (and please don't point to the existing RNG's that simply
work on a finite set(usually 0 to 2^32 and then normalize the results).
As we both know there is a huge difference between infinite and finite(no
matter how large it is). You are making it seem as those difference did not
exist. I'm not saying your wrong but that you are implying that and if this
is what HdB is whining about then he is correct.
again, the problem is with the statement:
"Suppose you choose a point of A "at random". Call that
randomly-chosen point x."
You are implying that such a "process" can be carried out... and this is
simply not true. Since we can't choose such a point your argument is
meaningless. Note that HdB is reading "choose" as if there is a non-zero
probability that it can be done. You mean it entirely different, at least I
imagine. So here the argument is simply about termoniology and semantics.
I agree with HdB(assuming this is his problem... I haven't been following
the thread) in that your wording is a bit ambiguous. It would be better to
reword it without talking about choosing elements from an infinite set or at
the very least make it clear that it is only hypothetically possible.
"The probability of choosing any real number from A is 0. The probability of
choosing any real number from B is 0. Hence the probability of choosing 1
from A is 0 and we can add any finite zero probability elements to a set
without changing the probabilities of any of the elements in that set"
(of course you would definitely want to make that more rigorous by using the
axioms of probability)
Realize I'm not getting on to you but just trying to explain how HdB is most
likely seeing it. It's clear to me but only because I know what your talking
about. If someone doesn't understand it and is learning such things for the
first time then it can be confusing(specially since they are thinking in
terms of finite sets).
Similarly how it would be unwise to say the probability of each element is 0
as I have done. We should technically say it is infinitesimally small. If it
were 0 then the total probability would be 0 which contradicts the axioms of
probability.
quasi
<Jon_Sl...@Hotmail.com> wrote:
>Similarly how it would be unwise to say the probability of each element is 0
>as I have done. We should technically say it is infinitesimally small. If it
>were 0 then the total probability would be 0 which contradicts the axioms of
>probability.
No, it doesn't contradict the axioms of probability.
The axioms only require a probability function to be countably
additive.
quasi
Christopher Henrich
Virgil <Vir...@gmale.com> wrote:
That probability would have to be 0 .
A better way to describe a "random" selection is to say that the
probability that the selected number lies in any segment is proportional
to the length of that segment.
Here's another description of how to pick a real number "at random" that
may make this idea more intuitive. A real number is determined by its
representation as a decimal fraction; this representation is generally
non-terminating. You could pick a real number at random by randomly
choosing each decimal digit. Now, a rational number is represented by a
decimal fraction that either terminates or becomes periodic. That is,
the sequence of digits eventually repeats indefinitely, either 00000...
or 9999.... or some other pattern repeated infinitely often. But if the
digits are chosen independently, then the chances of any such infinite
repetition are nil.
--
Christopher J. Henrich
chen...@monmouth.com
http://www.mathinteract.com
Achava Nakhash, the Loving Snake
>
> - Show quoted text -
I most certainly was not joking about Coxeter and Greitzer. It is a
wonderful little book and I recommend it everyone. Many things that
you wanted to know about plane old Euclidean geometry are demonstrated
here in a very elegant manner. I cannot recommend it too highly. I
must admit that I haven't finished it, but what I said is certainly
true for a great deal of it.
Regards,
Achava
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
"precise" is a very funny word to use here, since we have
no clue what you mean by "marginal" in the first place.
But never mind that:
>Therefore:
>
>(i) What is the probability that x is in B?
>
>Exactly the same as the probability that x is in A.
>
>(ii) Is x = 1 impossible?
>
>No. For some good reasons; see above.
>
>So the event "x is in B" = "x is in A" has probability 1 and is certain.
You seem to be contradicting yourself here - you agree that
x = 1 is not impossible, and then you say (or rather I think
you meant to say) that it's certain that x is in A if and only if
x is in B.
Maybe there was a typo somewhere.
Han de Bruijn
Oh no (for me at last) it's just mighty confusing, that's all ..
Han de Bruijn
Toni...@yahoo.com
> Tonic...@yahoo.com wrote:
> > On Mar 11, 11:07 am, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL>
> > wrote:
>
> >>Sentences between (parentheses) are often the most interesting in the
> >>book "Complex Made Simple" by David C. Ullrich. On page 200 it reads:
>
> >>(An event is said to occur "almost surely" if it happens with proba-
> >> bility 1. If it sounds to you like this means the event is certain
> >> to occur, we assure you that it does not; the distinction between
> >> "almost surely" and "certainly" will become clear when you study
> >> formal measure-theory-based probability theory.)
>
> >>?? Another kind of "counter intuitive" mathematics again ??
>
> > == Not at all, my dear HdB: it simply is Mathematicas, capital M,
> > though for you it seems to be mandarin chinese mixed with totonaca and
> > aztec.
> > No worries: not all have the required talent to understand basic
> > mathematics, and WM still loves you.
>
> The idea that _you_ "have the required talent to understand basic
> mathematics" has, so far, NOT become evident from some _substantial_
> contributions from your side.
== Says you, my good chap. You already asked in the past and were
answered about my thesis. Whether you ask it from my university's
library or not is your problem
All I've seen from you are just CHEAP
> hanger-on comments like "I'm so glad that *I* understand mathematics
> so well while you suckers don't".
>
== Now you blatantly lie, HdB: please do quote any of my posts where I
imply something even close to that.
That you are unable to understand some parts of mathematics is not a
shame in itself. Almost all of us don't know some parts in
mathematics. The shame resides in your pathetic tries to claim that
something you don't know about is wrong. That's the shame, not knowing
stuff.
Well, at least you're not alone in that...
Regards
Tonio
Pd By the way, you know that "hanger-on" is the same as parasite,
right?
Angus Rodgers
>I most certainly was not joking about Coxeter and Greitzer.
Of course not. I simply assumed that everyone had heard of pointless
topology (galathaea has mentioned it several times), and by extension,
pointless geometry (AKA point-free geometry). I therefore assumed that
you were making a joke, by pretending to think that I mean "pointless"
in the sense of uninteresting or useless.
It's odd, the misunderstandings that can happen on the Internet
(especially in view of the title of this thread).
>It is a
>wonderful little book and I recommend it everyone. Many things that
>you wanted to know about plane old Euclidean geometry are demonstrated
>here in a very elegant manner. I cannot recommend it too highly. I
>must admit that I haven't finished it, but what I said is certainly
>true for a great deal of it.
It's going on my list. Thanks for the recommendation.
--
Angus Rodgers
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
If there was no typo then you really _are_ saying
one thing and then saying the exact opposite one
sentence later. You see that?
>(for me at last) it's just mighty confusing, that's all ..
The only reason it's confusing is that you're convinced
that math should be the way you feel it should be.
If you got over that there'd be nothing confusing
about this, or any number of other things that
seem to confuse you.
Really. Thinking that probability 1 implies certainty
is perfectly natural. But it's _wrong_. Many people
have explained why it's wrong, in various related ways.
If you had a logical reason for thinking it's right _that_
would justify some confusion. But you have no such
reason - the only reason you think it's right is that
you've always thought that. So get over it - it's
simply not so, and there's nothing confusing about
it unless you're unable to get over the idea that it
_is_ so.
Han de Bruijn
> On Mar 11, 3:05 pm, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
>
>>Tonic...@yahoo.com wrote:
>>
>>>On Mar 11, 11:07 am, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL>
>>>wrote:
>>
>>>>Sentences between (parentheses) are often the most interesting in the
>>>>book "Complex Made Simple" by David C. Ullrich. On page 200 it reads:
>>
>>>>(An event is said to occur "almost surely" if it happens with proba-
>>>> bility 1. If it sounds to you like this means the event is certain
>>>> to occur, we assure you that it does not; the distinction between
>>>> "almost surely" and "certainly" will become clear when you study
>>>> formal measure-theory-based probability theory.)
>>
>>>>?? Another kind of "counter intuitive" mathematics again ??
>>
>>>== Not at all, my dear HdB: it simply is Mathematicas, capital M,
>>>though for you it seems to be mandarin chinese mixed with totonaca and
>>>aztec.
>>>No worries: not all have the required talent to understand basic
>>>mathematics, and WM still loves you.
>>
>>The idea that _you_ "have the required talent to understand basic
>>mathematics" has, so far, NOT become evident from some _substantial_
>>contributions from your side.
>
> == Says you, my good chap. You already asked in the past and were
> answered about my thesis. Whether you ask it from my university's
> library or not is your problem
Sure. And that's the _only_ thing you've ever accomplished. Right?
> All I've seen from you are just CHEAP
Then take a better look, I'd say.
>>hanger-on comments like "I'm so glad that *I* understand mathematics
>>so well while you suckers don't".
>>
> == Now you blatantly lie, HdB: please do quote any of my posts where I
> imply something even close to that.
>
> That you are unable to understand some parts of mathematics is not a
> shame in itself. Almost all of us don't know some parts in
> mathematics. The shame resides in your pathetic tries to claim that
> something you don't know about is wrong. That's the shame, not knowing
> stuff.
>
> Well, at least you're not alone in that...
>
> Pd By the way, you know that "hanger-on" is the same as parasite,
> right?
Not that I'm a aware of. IMO <EN>hanger-on</EN> = <NL>meeloper</NL> ;
I have not the intention to attach another (i.e. worse) meaning to it.
Han de Bruijn
Dave L. Renfro
> Of course not. I simply assumed that everyone had heard
> of pointless topology (galathaea has mentioned it several
> times), and by extension, pointless geometry (AKA point-free
> geometry). I therefore assumed that you were making a joke,
> by pretending to think that I mean "pointless" in the sense
> of uninteresting or useless.
I was curious how often it's come up in usenet (more accurately,
in a google-groups search), and I got 48 hits for "pointless
topology". That's less than I would have guessed. I note 3 are
by me, the first being in January 2001 (but the earliest mention
goes back several years before this). I first heard of it back
in 1983 when P. T. Johnstone's "The point of pointless topology"
article came out in Bull. Amer. Math. Soc., a paper that's freely
available on the internet by the way.
Dave L. Renfro
Dik T. Winter
> Not that I'm a aware of. IMO <EN>hanger-on</EN> = <NL>meeloper</NL> ;
> I have not the intention to attach another (i.e. worse) meaning to it.
hanger-on: one that hangs around a person, place, or institution especially
for personal gain.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Achava Nakhash, the Loving Snake
>
> - Show quoted text -
Mea culpa time. Of course I was joking about geometry being
pointless. However, even in point of fact, I had never heard of
pointless topology, although it wasn't too hard to figure out what it
had to be. So I learned something. Probably will even look at the
article Dave Renfro recommended in his reply to this same post.
Regards,
Achava
Jon Slaughter
"quasi" <qu...@null.set> wrote in message
news:604hr4hnfb79rcv3d...@4ax.com...
They also require the probability of the certain event to be 1.
http://en.wikipedia.org/wiki/Probability_axioms
Second axiom.
If the probabilities of every "point" in the set were 0 then every set of
those points would be zero and the entire space would have probability 0.
"This is often overlooked in some mistaken probability calculations; if you
cannot precisely define the whole sample space, then the probability of any
subset cannot be defined either." (from wiki)
which is exactly the problem. In this case the points have infinitessimally
small but non-zero probability. If this wasn't the case how could the we
define a probability space on it?
Also they say
"These assumptions can be summarised as: Let (?, F, P) be a measure space
with P(?)=1"
which is impossible if every event in the space has probability 0.
Angus Rodgers
Loving Snake" <ach...@hotmail.com> wrote:
>[...] in point of fact, I had never heard of
>pointless topology, although it wasn't too hard to figure out what it
>had to be. So I learned something. Probably will even look at the
>article Dave Renfro recommended in his reply to this same post.
As well as that category-theoretic approach, there are some older
ideas, mentioned in the list of references to the paper "Geometry
Without Points", listed, under the heading "Point-free Geometry",
near the bottom of this Web page:
<http://www.dmi.unisa.it/people/gerla/www/>
(No endorsement is implied, especially as I am averse to anything
"fuzzy"!)
I could probably dig up some other references, although I've been
trying hard not to think about this stuff for years. My efforts
to work out something in this general direction foundered, more
than a decade ago. (Partly because I was having to do mathematics
while still somehow regarding the entire world of mathematics as
mysterious and inaccessible! Hard to explain, and my psychological
asides tend to be unpopular, so I won't try!) I tried to work out
some kind of mereology, using partial order structures satisfying
various postulates, including what I eventually learned was called
(in forcing theory) "separativeness". According to my notes, this
characterises dense subsets of Boolean algebras. (I think there is
a good account in Jech, /Set Theory/, but I don't have the book to
hand. It doesn't treat pointless geometry, or topology, as such.)
When I'm more comfortable with mathematics, I'll go back to this
aborted project and try to work out if it really was a dead end,
or if I just made a mess of it, and something could be salvaged.
--
Angus Rodgers
Toni...@yahoo.com
>
> >>The idea that _you_ "have the required talent to understand basic
> >>mathematics" has, so far, NOT become evident from some _substantial_
> >>contributions from your side.
>
> > == Says you, my good chap. You already asked in the past and were
> > answered about my thesis. Whether you ask it from my university's
> > library or not is your problem
>
> Sure. And that's the _only_ thing you've ever accomplished. Right?
>
== Not even close, but that is what =you= asked for back then and what
you got. Of course, I have no web page with nonsensical articles
inserted there.
> > All I've seen from you are just CHEAP
>
> Then take a better look, I'd say.
== Now you're addressing yourself: ==you== wrote the first line above!
=:)
> >>hanger-on comments like "I'm so glad that *I* understand mathematics
> >>so well while you suckers don't".
>
> > == Now you blatantly lie, HdB: please do quote any of my posts where I
> > imply something even close to that.
>
> > That you are unable to understand some parts of mathematics is not a
> > shame in itself. Almost all of us don't know some parts in
> > mathematics. The shame resides in your pathetic tries to claim that
> > something you don't know about is wrong. That's the shame, not knowing
> > stuff.
>
> > Well, at least you're not alone in that...
>
> > Pd By the way, you know that "hanger-on" is the same as parasite,
> > right?
>
> Not that I'm a aware of. IMO <EN>hanger-on</EN> = <NL>meeloper</NL> ;
> I have not the intention to attach another (i.e. worse) meaning to it.
>
== Well, you didn't address the accusation of being a liar, which by
itself is interesting.
And about meeloper: it seems to be "a person who places expediency
above principle", according to http://lookwayup.com/lwu.exe/lwu/toEng?s=d&w=meeloper&slang=Nld
So hanger-on is NOT meeloper, and more interesting: you seem to think
that either meeloper or hanger-on is an appropiate name to call me,
and manyh others, when we try to educate you and correct your stupid
ideas about mathematics.
Well...ok, I suppose. As long as you feel fine with that.
Regards
Tonio
Dave Seaman
> "quasi" <qu...@null.set> wrote in message
> news:604hr4hnfb79rcv3d...@4ax.com...
>> On Wed, 11 Mar 2009 22:23:44 -0500, "Jon Slaughter"
>> <Jon_Sl...@Hotmail.com> wrote:
>>>Similarly how it would be unwise to say the probability of each element is
>>>0
>>>as I have done. We should technically say it is infinitesimally small. If
>>>it
>>>were 0 then the total probability would be 0 which contradicts the axioms
>>>of
>>>probability.
>> No, it doesn't contradict the axioms of probability.
>> The axioms only require a probability function to be countably
>> additive.
> They also require the probability of the certain event to be 1.
Correct.
> http://en.wikipedia.org/wiki/Probability_axioms
> Second axiom.
> If the probabilities of every "point" in the set were 0 then every set of
> those points would be zero and the entire space would have probability 0.
No, that is not what the second axiom says. In fact, the second axiom
does not mention additivity at all. The third axiom mentions
sigma-additivity, which means countable additivity. It does not apply in
the case of uncountable sets.
<http://en.wikipedia.org/wiki/%CE%A3-additivity>
(That strange character in the URL is supposed to be a capital Greek
letter Sigma. There is a link to that page from where sigma-additivity
is mentioned in the third axiom of the page you cited.)
> "This is often overlooked in some mistaken probability calculations; if you
> cannot precisely define the whole sample space, then the probability of any
> subset cannot be defined either." (from wiki)
True, but irrelevant. Nobody is trying to define probabilities without
precisely defining the whole sample space.
> which is exactly the problem. In this case the points have infinitessimally
> small but non-zero probability. If this wasn't the case how could the we
> define a probability space on it?
<http://en.wikipedia.org/wiki/Probability_space>
A probability space is first of all a measure space. Measures, by
definition, are real-valued and therefore there are no infinitesimals
involved.
<http://en.wikipedia.org/wiki/Measure_(mathematics)>
Besides, even if you tried the approach of assigning an infinitesimal
"probability" to each point in an uncountable set and then adding up the
infinitesimals, the idea wouldn't get you very far. For example, if your
probability space is the unit interval [0,1], then there are exactly as
many points in [0,1/2] as in the whole space [0,1]. How do you propose
to assign infinitesimal probabilities so that they will add up properly
for every possible subset?
An example of a probability space in which each point has probability
zero is the unit interval [0,1] with Lebesgue measure.
<http://en.wikipedia.org/wiki/Lebesgue_measure>
> Also they say
> "These assumptions can be summarised as: Let (?, F, P) be a measure space
> with P(?)=1"
> which is impossible if every event in the space has probability 0.
Wrong. The page you quoted talks about sigma-additivity, which means
countable additivity.
--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
Jon Slaughter
"Dave Seaman" <dse...@no.such.host> wrote in message
news:gpbet5$7d7$1...@mailhub227.itcs.purdue.edu...
I've got better things than to argue with you about it. I did not use sigma
additivity but said the second axiom which says the certain effect must have
probability of 1.
http://en.wikipedia.org/wiki/Probability_space
If you read the "General case" you see they explicitly address this problem.
In this case there are many "Events" we can define that are non-zero.
Also "The case p(w) = 0 is permitted by the definition, but rarely used,
since such can safely be excluded from the sample space."
My point is that Ullrich is using the terminology from a discrete
probability space to describe an infinite probability space. This is exactly
where the confusion is occuring.
If you goto the "non-atomic" case you'll see the problem if p(w) = 0 for all
events w. It has nothing to do with sigma-additivity.
"Then a limiting procedure allows to ascribe probabilities to sets that are
limits of sequences of elementary sets, or limits of limits, and so on."
Note the idea of a "limit" which is what I summed up as an "infinitessimal".
Again, if you want to argue about it then do it with a probability book.
Dave Seaman
>> Correct.
>>> http://en.wikipedia.org/wiki/Probability_axioms
>>> Second axiom.
>> <http://en.wikipedia.org/wiki/%CE%A3-additivity>
>> <http://en.wikipedia.org/wiki/Probability_space>
>> <http://en.wikipedia.org/wiki/Measure_(mathematics)>
>> <http://en.wikipedia.org/wiki/Lebesgue_measure>
>>> Also they say
I agree that you did not use sigma-additivity. In fact, your failure to
use sigma-additivity is precisely where your mistake lies.
Probability measures are not additive. Period. They are merely
sigma-additive. If you try to pretend that they are additive when
applied to uncountable unions, then you are displaying a complete
misunderstanding of measure theory.
> http://en.wikipedia.org/wiki/Probability_space
> If you read the "General case" you see they explicitly address this problem.
> In this case there are many "Events" we can define that are non-zero.
In the case I cited (Lebesgue measure on [0,1]), there are no such
events. If you apply the "General case" to that space, you find that
each point has probability 0. As stated in the "General case":
Otherwise, if the sum of probabilities of all atoms is less than 1
(maybe 0), then the probability space decomposes into a discrete
(atomic) part (maybe empty) and a non-atomic part.
In other words, Lebesgue measure on [0,1] is completely non-atomic.
> Also "The case p(w) = 0 is permitted by the definition, but rarely used,
> since such can safely be excluded from the sample space."
That applies only to the discrete case. Lebesgue measure on [0,1] is an
example of the Non-atomic case, quoted two sections below that.
> My point is that Ullrich is using the terminology from a discrete
> probability space to describe an infinite probability space. This is exactly
> where the confusion is occuring.
What made you think Ullrich was talking about a discrete probability
space? His example was Lebesgue measure on the line, which is the
non-atomic case.
> If you goto the "non-atomic" case you'll see the problem if p(w) = 0 for all
> events w. It has nothing to do with sigma-additivity.
It's not a problem at all, and sigma-additivity (in the definition of a
measure) is precisely the property of a measure that makes such examples
possible without violating the definition.
> "Then a limiting procedure allows to ascribe probabilities to sets that are
> limits of sequences of elementary sets, or limits of limits, and so on."
> Note the idea of a "limit" which is what I summed up as an "infinitessimal".
Infinitesimals have nothing to do with probability measures.
> Again, if you want to argue about it then do it with a probability book.
I learned my probability theory from Feller's book decades ago. Your own
quotes from the web undermine your case, as I have shown.
Victor Eijkhout
> ?? Another kind of "counter intuitive" mathematics again ??
I'm sure TU Delft has courses in measure theory. Take one. It's fun.
This sort of counter-intuitive fact will come up even in a one semester
course.
Victor.
--
Victor Eijkhout -- eijkhout at tacc utexas edu
Jon Slaughter
"Dave Seaman" <dse...@no.such.host> wrote in message
All it has shown is your ego and that you don't read the posts. I was
precisely stating this is why HdB is confused because Ullrich is using the
terminology as if it were finite.
In any case people like you get a gold star and get put on a very special
list. I call it the asshole list but you can call it what you like.
Dave Seaman
The post of yours that I originally responded to did not mention Ullrich
or HdB at all. You only retreated to that claim after I called you out
on all your nonsense about infinitesimals and about it being impossible
for a measure to be completely non-atomic. When I mentioned that
countable additivity is part of the definition of a probability measure,
you completely missed the point, suggesting that it somehow was not
relevant because you hadn't mentioned it.
> In any case people like you get a gold star and get put on a very special
> list. I call it the asshole list but you can call it what you like.
No doubt, I am an asshole for explaining exactly why your claims are
utter nonsense. People who make such claims keep finding that sci.math
is full of assholes.
My questions were too difficult for you to answer, I see. You ignored
every single one.
Let's review:
What did Ullrich say that made you think he was talking
about finite probability spaces? As I have already mentioned,
his example was of an infinite probability space, using
Lebesgue measure. The terminology was completely standard.
Where in measure theory do you find any mention of
infinitesimals? Exact quote, please.
How do you propose to obtain the measure of [0,1] = 1 by
adding up infinitesimals, in such a way that [0,1/2] has
measure 1/2, though the number of points is the same?
What, exactly, are you claiming I didn't read?
Why do you keep pretending that countable additivity is not
part of the definition of a measure?
You asked me to argue from a "probability book". Here's the definition
of a probability measure, from Feller, _Introduction to Probability
Theory and Its Applications_, Volume II, second edition, p. 115:
Definition 1. A probability measure P on a sigma-algebra
U of sets in S is a function assigning a value P{A} >= 0
to each set A in U such that P{S} = 1 and that for every
countable collection of non-overlapping sets A_n in U
P{ union A_n } = sum P{A_n}.
Notice the explicit mention of countable additivity. Notice that
Lebesgue measure on [0,1] satisfies the definition of a probability
measure, even though each point has measure zero.
David C. Ullrich
<Jon_Sl...@Hotmail.com> wrote:
>
>[...]
>
>There is a huge difference between a finite but extremely small probability
>and an infinitesimally small or zero probability.
Yes. One difference being that (in standard probability theory) there
is no such thing as an "infinitesmal" probability.
>You should be more clear
>when dealing with infinite sets since it sounds as if your using the same
>logic when we really need to approach them differently.
>
>As we both know, the only way to get a non-zero probability is to look at
>sets that contain an infinite number of elements. Any finite set must have a
>0 probability.
That's true in some cases, not in others.
>
>> (ii) Is x = 1 impossible?
>>
>> (answer: no. All the points of A are the same here;
>> if x = 1 were impossible then x = anything else
>> would also be impossible, but x does equal
>> _something_.)
>
>You are correct that the probability of x is the same as the probability of
>every other element in the set but since we can't even choose those elements
>it is an invalid conclusion.
>
>Now, you can say "Sure we can choose them".
I do? I don't recall saying that.
In fact there is no talk of choosing this or that in actual
mathematical probability theory - that's just a figure
of speech when we're talking about things informally.
>And in some sense we can. But
>how do we choose a real number from the set [0,e] for any e>0? Since every
>choice as the same probability(assuming uniformly w.l.o.g.) the probability
>of choosing one must be 0. But if every element has probability 0 then no
>element can be chosen.
No, that simply does not follow.
>(and zero here techinically is infinitesimally small)
>
>Ask yourself how you can construct a RNG to choose a real number from the
>interval [0,1]?
If you mean an actual physical RNG I'm not going to ask myself
that - of course you can't. So what? The topic was probability
theory, not RNG;s.
>(and please don't point to the existing RNG's that simply
>work on a finite set(usually 0 to 2^32 and then normalize the results).
>
>As we both know there is a huge difference between infinite and finite(no
>matter how large it is). You are making it seem as those difference did not
>exist.
I am? Wow.
>I'm not saying your wrong but that you are implying that and if this
>is what HdB is whining about then he is correct.
>
>again, the problem is with the statement:
>
>"Suppose you choose a point of A "at random". Call that
>randomly-chosen point x."
>
>You are implying that such a "process" can be carried out... and this is
>simply not true.
No, I'm attempting to take the statement "let x be a random
variable, uniformly distributed on [0,1]" and rephrase it
in a way that Han might understand.
> Since we can't choose such a point your argument is
>meaningless. Note that HdB is reading "choose" as if there is a non-zero
>probability that it can be done. You mean it entirely different, at least I
>imagine. So here the argument is simply about termoniology and semantics.
>
>I agree with HdB(assuming this is his problem... I haven't been following
>the thread) in that your wording is a bit ambiguous. It would be better to
>reword it without talking about choosing elements from an infinite set or at
>the very least make it clear that it is only hypothetically possible.
>
>"The probability of choosing any real number from A is 0. The probability of
>choosing any real number from B is 0. Hence the probability of choosing 1
>from A is 0 and we can add any finite zero probability elements to a set
>without changing the probabilities of any of the elements in that set"
>
>(of course you would definitely want to make that more rigorous by using the
>axioms of probability)
>
>Realize I'm not getting on to you but just trying to explain how HdB is most
>likely seeing it. It's clear to me but only because I know what your talking
>about. If someone doesn't understand it and is learning such things for the
>first time then it can be confusing(specially since they are thinking in
>terms of finite sets).
>
>Similarly how it would be unwise to say the probability of each element is 0
>as I have done. We should technically say it is infinitesimally small.
Before talking about what we should "technically" say you should
really learn the math. Technically saying that probability is
infinitesmally small is nonsense - technically saying it is 0 is
exactly right.
>If it
>were 0 then the total probability would be 0 which contradicts the axioms of
>probability.
Nonsense. You don't have that axiom quite straight.
David C. Ullrich
<Jon_Sl...@Hotmail.com> wrote:
Curious that you're arguing about it, then.
> I did not use sigma
>additivity but said the second axiom which says the certain effect must have
>probability of 1.
How _did_ you get from "the probability of {x} is 0 for every
x" to "the probability of the entire space is 0" then?
>http://en.wikipedia.org/wiki/Probability_space
>
>If you read the "General case" you see they explicitly address this problem.
>In this case there are many "Events" we can define that are non-zero.
>
>Also "The case p(w) = 0 is permitted by the definition, but rarely used,
>since such can safely be excluded from the sample space."
>
>My point is that Ullrich is using the terminology from a discrete
>probability space to describe an infinite probability space.
Whether or not that's your point, it's not so.
>This is exactly
>where the confusion is occuring.
>
>If you goto the "non-atomic" case you'll see the problem if p(w) = 0 for all
>events w.
Of course you can't have p(w) = 0 for all events w! That's not what
was claimed.
In the case where, to give an informal description, we are choosing
x at random from [0,1], it is true that for any specific number, the
probability that x = that number is 0. But there are more events
than "x = some specific number". For example, "0 < x < 1/2"
is an event with probability 1/2.
> It has nothing to do with sigma-additivity.
>
>"Then a limiting procedure allows to ascribe probabilities to sets that are
>limits of sequences of elementary sets, or limits of limits, and so on."
>
>Note the idea of a "limit" which is what I summed up as an "infinitessimal".
>
>Again, if you want to argue about it then do it with a probability book.
Wow.
I'll give you a hint. The person who needs to study the math
behind all this is _you_. The things you're saying are simply
wrong - it says so in books.
You need a book on actual mathematical probability theory,
of course. That is, a book that assumes measure theory
as a prerequisite.
David C. Ullrich
<Jon_Sl...@Hotmail.com> wrote:
>
>"Dave Seaman" <dse...@no.such.host> wrote in message
>news:gpbrto$f48$1...@mailhub227.itcs.purdue.edu...
>> [...]
>>
>>> Note the idea of a "limit" which is what I summed up as an
>>> "infinitessimal".
>>
>> Infinitesimals have nothing to do with probability measures.
>>
>>> Again, if you want to argue about it then do it with a probability book.
>>
>> I learned my probability theory from Feller's book decades ago. Your own
>> quotes from the web undermine your case, as I have shown.
>>
>
>All it has shown is your ego and that you don't read the posts. I was
>precisely stating this is why HdB is confused because Ullrich is using the
>terminology as if it were finite.
>
>In any case people like you get a gold star and get put on a very special
>list. I call it the asshole list but you can call it what you like.
Are you the guy who made that post a little while ago complaing
bitterly about the fact that he got no respect in some group,
even though the things he said were right?
If I have the name wrong then never mind. If you _are_ that
guy, some hints:
First, more or less everything you've said about probability
has been simply wrong.
Second, saying "read a book" was a bad idea, because it's
precisely _you_ who needs to read a book about all this.
If you can find a serious book on probability theory
(again, that means one that assumes measure theory as
a prerequisite, that being what probability theory is
based on) that says anything about adding infinitesmal
probabilities that will be incredible - you need to tell
us the name of that book.
Finally, and perhaps most important - _you_ are
the person who started calling people assholes here.
_If_ he'd said that he must be right because he read
a book I'd think that that was a reasonable thing to
say, but some people might think it was obnoxious.
But _you_ _told_ him to read a book, and when
he simply replied that he _had_ done so you called
him an asshole. Not a good way to make friends.
Now you tell us what book _you_ read, where it
explains that the probability that x = whatever
should "technically" be described as infinitesmal
instead of zero.
Jesse F. Hughes
> All it has shown is your ego and that you don't read the posts. I was
> precisely stating this is why HdB is confused because Ullrich is using the
> terminology as if it were finite.
>
> In any case people like you get a gold star and get put on a very special
> list. I call it the asshole list but you can call it what you like.
Near as I can figger, when you call someone an asshole, you mean that
he has given clear and respectful correction of your errors.
--
Jesse F. Hughes
"I thought it relevant to inform that I notified the FBI a couple of
months ago about some of the math issues I've brought up here."
-- James S. Harris gives Special Agent Fox a new assignment.
Arturo Magidin
[...]
> In any case people like you get a gold star and get put on a very special
> list. I call it the asshole list but you can call it what you like.
Is there an "'s" missing somewhere in that last sentence?
--
Arturo Magidin, sans .sig
Michael Press
<94d00ad3-b886-4810...@d2g2000pra.googlegroups.com>,
"Achava Nakhash, the Loving Snake" <ach...@hotmail.com> wrote:
> I most certainly was not joking about Coxeter and Greitzer. It is a
> wonderful little book and I recommend it everyone. Many things that
> you wanted to know about plane old Euclidean geometry are demonstrated
> here in a very elegant manner. I cannot recommend it too highly. I
> must admit that I haven't finished it, but what I said is certainly
> true for a great deal of it.
Agree. They knit together all those results in
the geometry tool box. Great book.
--
Michael Press
Han de Bruijn
> And about meeloper: it seems to be "a person who places expediency
> above principle", according to
> http://lookwayup.com/lwu.exe/lwu/toEng?s=d&w=meeloper&slang=Nld
>
> So hanger-on is NOT meeloper, and more interesting: you seem to think
> that either meeloper or hanger-on is an appropiate name to call me,
> and manyh others, when we try to educate you and correct your stupid
> ideas about mathematics.
> Well...ok, I suppose. As long as you feel fine with that.
Ah, "opportunism". Thanks for another suitable name-calling.
Han de Bruijn
Han de Bruijn
> Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
>
>>?? Another kind of "counter intuitive" mathematics again ??
>
> I'm sure TU Delft has courses in measure theory. Take one. It's fun.
> This sort of counter-intuitive fact will come up even in a one semester
> course.
Guess I'm already too old to learn :-(
Han de Bruijn
Toni...@yahoo.com
wrote:
> Victor Eijkhout wrote:
> > Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
>
> >>?? Another kind of "counter intuitive" mathematics again ??
>
> > I'm sure TU Delft has courses in measure theory. Take one. It's fun.
> > This sort of counter-intuitive fact will come up even in a one semester
> > course.
>
> Guess I'm already too old to learn :-(
>
Regards
Tonio
> Han de Bruijn
Jesse F. Hughes
That seems to sum up the pointlessness of these discussions.
--
"People make mistakes. Better to live today and learn the truth, than
to be one of those poor saps who died deluded, thinking they knew
certain things that they just didn't. Thinking they had proofs that
they didn't." --James S. Harris, almost too sad for a .sig
Han de Bruijn
> Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
>
>>Victor Eijkhout wrote:
>>
>>>Han de Bruijn <Han.de...@DTO.TUDelft.NL> wrote:
>>>
>>>>?? Another kind of "counter intuitive" mathematics again ??
>>>
>>>I'm sure TU Delft has courses in measure theory. Take one. It's fun.
>>>This sort of counter-intuitive fact will come up even in a one semester
>>>course.
>>
>>Guess I'm already too old to learn :-(
>
> That seems to sum up the pointlessness of these discussions.
Apart from the smiley .. But _this_ discussion will end soon, as far as
I am concerned. David Ullrich's book has been an eye-opener, sort of ..
Han de Bruijn