Can Events of Zero Probability Happen?
Shubee
probability theory. My view is that mathematical models of toy
universes do exist that are logically consistent where measure zero
events can happen.
His expressed position was that "Zero probability means that an event
cannot happen." He also stated that "If you change that, you are no
longer talking about probability but about something else, and
whatever you say becomes incomprehensible."
I responded by saying:
"In finite probability spaces, you are correct. In general, all that
is needed for a 'Probability space' is a measure on a sigma-algebra of
events that assigns to each event a number between 0 and 1.
http://en.wikipedia.org/wiki/Measure_%28mathematics%29
http://en.wikipedia.org/wiki/Probability_space "
I believe that my learned opponent is wrong for three reasons, two of
which are purely mathematical. First, it seems that he is unable to
detach himself from his own emotional feelings about the universe to
even consider the meaning of zero probability from a measure-theoretic
point of view. Second, he thinks that the occurrence of a measure-zero
probability event is incomprehensible.
http://groups.google.com/group/sci.physics.foundations/browse_frm/thread/c4de9ae9a364fc79
Who is right?
Shubee
Randy Poe
> Recently, I had a debate with a physicist on the physical meaning of
> probability theory. My view is that mathematical models of toy
> universes do exist that are logically consistent where measure zero
> events can happen.
Absolutely.
> His expressed position was that "Zero probability means that an event
> cannot happen." He also stated that "If you change that, you are no
> longer talking about probability but about something else, and
> whatever you say becomes incomprehensible."
He's incorrect. In the measure-theoretic development
of probability theory (and it's not surprising that
a physics professor wouldn't have seen this theory --
why would he need it?), events can occur which have
measure 0.
Consider any outcome from a uniform or normal distribution,
say x = 3. The probability that x = 3 exactly from any
continuous distribution is 0.
>
> I responded by saying:
>
> "In finite probability spaces, you are correct. In general, all that
> is needed for a 'Probability space' is a measure on a sigma-algebra of
> events that assigns to each event a number between 0 and 1.
>
> http://en.wikipedia.org/wiki/Measure_%28mathematics%29http://en.wikipedia.org/wiki/Probability_space"
>
> I believe that my learned opponent is wrong for three reasons, two of
> which are purely mathematical. First, it seems that he is unable to
> detach himself from his own emotional feelings about the universe to
> even consider the meaning of zero probability from a measure-theoretic
> point of view. Second, he thinks that the occurrence of a measure-zero
> probability event is incomprehensible.
http://groups.google.com/group/sci.physics.foundations/browse_frm/thr...
>
> Who is right?
You're wrong to ascribe dark motives to him.
This would be one of those areas of mathematics where
the physicist can correctly dismiss it as arcane and
of no practical importance to most questions in physics,
like the rigorous handling of delta "functions".
- Randy
David C. Ullrich
mathematical question.
Ask him this: Suppose that X is a real number between 0 and 1,
chosen at random. What is the probability that X = 1/2?
Where to go from there depends on how he replies. It _could_
be, for example, that the way he's using the term,
"choose a real number between 0 and 1 at random" is simply
not something that can "happen"...
On Fri, 11 Jan 2008 06:20:15 -0800 (PST), Shubee <e.Sh...@gmail.com>
wrote:
************************
David C. Ullrich
Richard Tobin
David C. Ullrich <ull...@math.okstate.edu> wrote:
>Whether or not something can "really" "happen" is not a
>mathematical question.
>
>Ask him this: Suppose that X is a real number between 0 and 1,
>chosen at random. What is the probability that X = 1/2?
This seems a little like cheating, because when you instead ask
"suppose X is an integer chosen at random, what is the probability
that it's 2?", the mathematician is just as stuck as the physicist.
-- Richard
--
:wq
Shubee
> On Jan 11, 9:20 am, Shubee <e.Shu...@gmail.com> wrote:
>
> > I believe that my learned opponent is wrong for three reasons, two of
> > which are purely mathematical. First, it seems that he is unable to
> > detach himself from his own emotional feelings about the universe to
> > even consider the meaning of zero probability from a measure-theoretic
> > point of view. Second, he thinks that the occurrence of a measure-zero
> > probability event is incomprehensible.
>
>
> > Who is right?
>
> You're wrong to ascribe dark motives to him.
I have not ascribed dark motives to any physicist that can't reason
like a mathematician because he or she is emotionally wed to their own
pet theories about the universe.
> This would be one of those areas of mathematics where
> the physicist can correctly dismiss it as arcane and
> of no practical importance to most questions in physics,
The dispute was in the context of a specific category of toy universes
where physicists do pretend to speak authoritatively.
http://groups.google.com/group/sci.physics.foundations/browse_frm/thread/c4de9ae9a364fc79/
The other mathematical misconception by the physicist in question,
which is a shame to mention, is that he believes that events described
by fantastically small probabilities can't happen. His argument:
"... the mere calculation of a non-zero probability does not enable
you to say that that event can happen unless you have an absolutely
perfect mathematical model of physics and have taken absolutely
everything into account. That is never the case."
http://groups.google.com/group/sci.physics.foundations/msg/ca3eb017d60ca734
Shubee
Shubee
> Whether or not something can "really" "happen" is not a
> mathematical question.
Ah, but that's not true in the context of my assertions. All physics
is mathematics in David Hilbert's philosophy of physics. [1][2].
1. The Axiomatization of Physics - Step 1 (section 2)
http://www.everythingimportant.org/relativity/special.pdf
2. The Relativity of Discovery: Hilbert's First Note on the
Foundations of Physics (sections 1.1 and 1.2)
http://arxiv.org/abs/physics/9811050
Shubee
Edmund
PMJI,
What exactly is the answer? It's been a long time since I've done
any probability theory so I'm very rusty in it and given that the
domain is Integers, doesn't this also include Set theory (another
topic that I'm not good at)?
P(X=2) = 2/inf. = 0 (from a simplified view), but from a truly
mathematical perspective, wouldn't it be 0.00000....(infinite #
of zeros)1? Or does this make any sense at all?
Ed
Mariano Suárez-Alvarez
> In article <961fo35uieus57unl9fg2n80663l2m0...@4ax.com>,
> David C. Ullrich <ullr...@math.okstate.edu> wrote:
>
> >Whether or not something can "really" "happen" is not a
> >mathematical question.
>
> >Ask him this: Suppose that X is a real number between 0 and 1,
> >chosen at random. What is the probability that X = 1/2?
>
> This seems a little like cheating, because when you instead ask
> "suppose X is an integer chosen at random, what is the probability
> that it's 2?", the mathematician is just as stuck as the physicist.
Well, I do not think responding "depends on your distribution"
qualifies as being stuck, and anyone with a two day course on
probabilities should ask that.
-- m
Virgil
If one has a set which is countably infinite, like the set of integers,
or naturals, or rationals, then it turns out to be impossible to
"chose one member at random", since that would require each of countably
many members to have the same probability of being chosen, but there
cannot be any real number of which countably many "add up" to anything
but zero or infinity.
Shubee
> If one has a set which is countably infinite, like the set of integers,
> or naturals, or rationals, then it turns out to be impossible to
> "chose one member at random", since that would require each of countably
> many members to have the same probability of being chosen, but there
> cannot be any real number of which countably many "add up" to anything
> but zero or infinity.
True but really that's not a problem because, in this instance, we
should be using a finitely-additive probability space, which only
requires an algebra of sets, rather than a sigma-algebra.
http://en.wikipedia.org/wiki/Probability_axioms
Shubee
David C. Ullrich
wrote:
Puhleeze. In the first case the words "with a uniform distribution"
are understood (or added explicitly if it turns out they're not
understood). In the second case the mathematician is not "stuck",
he simply points out that the distribution needs to be specified
since there is no natural default.
>
>-- Richard
************************
David C. Ullrich
David C. Ullrich
wrote:
>On Jan 11, 9:07 am, David C. Ullrich <ullr...@math.okstate.edu> wrote:
>> Whether or not something can "really" "happen" is not a
>> mathematical question.
>
>Ah, but that's not true in the context of my assertions. All physics
>is mathematics in David Hilbert's philosophy of physics. [1][2].
Uh, right. Supposing for the sake of argument that what you
say follows from Hilbert's philosophy does indeed follow:
(i) Are you claiming that Hilbert's "philosophy" is in
fact _correct_ in some absolute sense, that any other
approach to physics is simply _wrong_?
I hope not, that would be silly. Assuming not:
(ii) How do you know that the physicist you're talking
about agrees with this "philosophy"?
I'm not going to take the time to download those pdf's
on a dialup connection. There are certainly useful
points of view according to which all physics is mathematics.
But saying that all physics is mathematics in the sense
relevant to the current discussion is simply silly.
You're claiming that mathematics itself can be used
to determine anything about the nature of the physical
world? That's obvious nonsense. Mathematics can be
used to determine what's possible _given_ certain
axioms - it can't say anything about whether the axioms
are actually true in the real world.
In particular, there's nothing implausible about saying
that "choose a real at random" is not something that
can actually "happen" in the real world - for example
there's no reason anyone has to be certain that there
is anything in the real physical universe that
actually corresponds precisely to the mathematical
notion of "real number" in the first place - it could
well be that everything is discrete.
_If_ you're interested in resolving the difference between
you and the physicist you'd ask what I suggested you
ask and see what he says. Because it's possible that
he'd have no problem with "choose a real between 0 and
1 at random", and in that case you could explain why
he's simply _wrong_ about the impossibility of events
of probability zero happening. That actually seems like
the most likely outcome. It's also possible that he'd
have some objection to the "reality" of the notion
of a random real number - if so then you'd understand
the source of your disagreements: What you're saying
is correct in a certain mathematical model, and he's
simply denying the relevance of that model to the real
world.
But suggesting that Hilbert's axiomatization contains
everything there is to know about actual physics is
just sllly - assuming that the physicist you're talking
to agrees with this, or thinking that he's simple
_wrong_ if he doesn't is even sillier. Measure theory
is not the universe.
>1. The Axiomatization of Physics - Step 1 (section 2)
>http://www.everythingimportant.org/relativity/special.pdf
>2. The Relativity of Discovery: Hilbert's First Note on the
>Foundations of Physics (sections 1.1 and 1.2)
>http://arxiv.org/abs/physics/9811050
>
>Shubee
************************
David C. Ullrich
Richard Tobin
David C. Ullrich <ull...@math.okstate.edu> wrote:
>>>Ask him this: Suppose that X is a real number between 0 and 1,
>>>chosen at random. What is the probability that X = 1/2?
>>This seems a little like cheating, because when you instead ask
>>"suppose X is an integer chosen at random, what is the probability
>>that it's 2?", the mathematician is just as stuck as the physicist.
>Puhleeze. In the first case the words "with a uniform distribution"
>are understood (or added explicitly if it turns out they're not
>understood). In the second case the mathematician is not "stuck",
>he simply points out that the distribution needs to be specified
>since there is no natural default.
I was assuming a uniform distribution in both cases. If you can
choose the distribution there's no problem in either case.
-- Richard
--
:wq
Denis Feldmann
> In article <k6oho397510t8144l...@4ax.com>,
> David C. Ullrich <ull...@math.okstate.edu> wrote:
>
>>>> Ask him this: Suppose that X is a real number between 0 and 1,
>>>> chosen at random. What is the probability that X = 1/2?
>
>>> This seems a little like cheating, because when you instead ask
>>> "suppose X is an integer chosen at random, what is the probability
>>> that it's 2?", the mathematician is just as stuck as the physicist.
>
>> Puhleeze. In the first case the words "with a uniform distribution"
>> are understood (or added explicitly if it turns out they're not
>> understood). In the second case the mathematician is not "stuck",
>> he simply points out that the distribution needs to be specified
>> since there is no natural default.
>
> I was assuming a uniform distribution in both cases.
Try to read. There is *no* such thing aas "uniform distribution" on integers
Virgil
ric...@cogsci.ed.ac.uk (Richard Tobin) wrote:
It is quite easy to show that for a countably infinite set of objects,
such as the integers, a uniform distributions cannot exist.
At least with all probabilities belonging to the set of the standard
reals.
Richard Tobin
Denis Feldmann <feldmann.den...@neuf.fr> wrote:
>>>>> Ask him this: Suppose that X is a real number between 0 and 1,
>>>>> chosen at random. What is the probability that X = 1/2?
>>
>>>> This seems a little like cheating, because when you instead ask
>>>> "suppose X is an integer chosen at random, what is the probability
>>>> that it's 2?", the mathematician is just as stuck as the physicist.
>>
>>> Puhleeze. In the first case the words "with a uniform distribution"
>>> are understood (or added explicitly if it turns out they're not
>>> understood). In the second case the mathematician is not "stuck",
>>> he simply points out that the distribution needs to be specified
>>> since there is no natural default.
>>
>> I was assuming a uniform distribution in both cases.
>
>Try to read. There is *no* such thing aas "uniform distribution" on integers
That was exactly the point: once we switch from the unit interval to
the integers, a uniform distribution becomes as impossible for
mathematicians as for physicists.
I suggest you try reading articles with the assumption that the author
is trying to convey something. Presumably you're not so rude in real
life.
-- Richard
--
:wq
Richard Tobin
Virgil <Vir...@com.com> wrote:
>> >>>Ask him this: Suppose that X is a real number between 0 and 1,
>> >>>chosen at random. What is the probability that X = 1/2?
>>
>> >>This seems a little like cheating, because when you instead ask
>> >>"suppose X is an integer chosen at random, what is the probability
>> >>that it's 2?", the mathematician is just as stuck as the physicist.
>>
>> >Puhleeze. In the first case the words "with a uniform distribution"
>> >are understood (or added explicitly if it turns out they're not
>> >understood). In the second case the mathematician is not "stuck",
>> >he simply points out that the distribution needs to be specified
>> >since there is no natural default.
>>
>> I was assuming a uniform distribution in both cases. If you can
>> choose the distribution there's no problem in either case.
>It is quite easy to show that for a countably infinite set of objects,
>such as the integers, a uniform distributions cannot exist.
Right. The original post was about a physicist not being happy with
zero-probability events that can happen. David Ullrich gave the
example of a uniform distribution on the unit interval as a way to
show the physicist that it must be possible. But a (uniformly
distributed) random integer is not intuitively more impossible that a
random real, so a theory that works for one but not the other is not
much use for addressing the physicist's intuition.
I'll try again. Suppose you said to the physicist: choose a random
integer, what's the probability that it's 2. That would be just as
convincing as the [0,1] case, but it would be wrong. That's why I
said it was cheating: you're using a form of argument that only works
because of the particular example you chose.
-- Richard
--
:wq
tommy1729
> Whether or not something can "really" "happen" is not
> a
> mathematical question.
agreed.
but what should we call such a question then ?
>
> Ask him this: Suppose that X is a real number between
> 0 and 1,
> chosen at random. What is the probability that X =
> 1/2?
good example.
if he answers : "0 , it cant happen."
well than he's an idiot :)
of course he might never bump into such questions perhaps.
on the other hand physics prof are seldom ashamed to "steal math" even if they dont fully understand the concept.
>
> Where to go from there depends on how he replies. It
> _could_
> be, for example, that the way he's using the term,
> "choose a real number between 0 and 1 at random" is
> simply
> not something that can "happen"...
considering that quantum mechanics is considered as a discrete consideration ...
in discrete space probability 0 is "really 0".
***
as for P [ pick [0,1] = 1/2 ] the mathematical discussion somewhat continues though...
( your example in the beginning rewritten )
some would say P = 0 , others would say P = h , with h an infinitesimal small number.
some others might argue h with h^2 = 0 (nilpotent h )
more rare ideas , but certainly sensible would also be
P = t1 [ in venkat's post and mine , if you recall , t1 is the smallest possible number > 0 ( not h ! ) ]
and i can continue : P = 1 / aleph_1 for example.
whereas finitists might even say : P does not exist.
i dont know if there is a real answer , maybe its just about what definitions you chose and what axioms you accept...
Shubee , you are right ;-)
>
>
> ************************
>
> David C. Ullrich
tommy1729
Shubee
wrote:
>
> _If_ you're interested in resolving the difference between
> you and the physicist you'd ask what I suggested you
> ask and see what he says. Because it's possible that
> he'd have no problem with "choose a real between 0 and
> 1 at random", and in that case you could explain why
> he's simply _wrong_ about the impossibility of events
> of probability zero happening.
Sure, on the face of it, it seems possible to reason with a physicist
that believes that conceptualizing events that occur with zero
probability is unfathomable. The problem is, he explicitly said that
even an event of incredibly small probability can't happen.
http://groups.google.com/group/sci.physics.foundations/msg/3d5630f2762edce6
Shubee
Denis Feldmann
You were trying very badly, as your argument were starting by " I was
assuming A", *after* it hasd been said A was impossible.
Anyway, as (from other threads) it becomes clear your argument had no
substance (The general shape is "How can we convince X that there exists
possible events of probability 0" "Answer : give him the example of
uniform distribution on [0,1]" and you conter "But the example of
random integer would not be convincing, so your example is bad...")
Presumably you're not so rude in real
> life.
>
I am with fools like you
> -- Richard
Huang
> In article <fmb03c$1hq...@pc-news.cogsci.ed.ac.uk>,
> rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
>
>
>
>
>
> > In article <k6oho397510t8144lojvekufminbrlg...@4ax.com>,
> > David C. Ullrich <ullr...@math.okstate.edu> wrote:
>
> > >>>Ask him this: Suppose that X is a real number between 0 and 1,
> > >>>chosen at random. What is the probability that X = 1/2?
>
> > >>This seems a little like cheating, because when you instead ask
> > >>"suppose X is an integer chosen at random, what is the probability
> > >>that it's 2?", the mathematician is just as stuck as the physicist.
>
> > >Puhleeze. In the first case the words "with a uniform distribution"
> > >are understood (or added explicitly if it turns out they're not
> > >understood). In the second case the mathematician is not "stuck",
> > >he simply points out that the distribution needs to be specified
> > >since there is no natural default.
>
> > I was assuming a uniform distribution in both cases. If you can
> > choose the distribution there's no problem in either case.
>
> > -- Richard
>
> It is quite easy to show that for a countably infinite set of objects,
> such as the integers, a uniform distributions cannot exist.
>
> At least with all probabilities belonging to the set of the standard
>
> - Show quoted text -
I agree completely with the above.
But please explain -
I will choose a number at random from the unit interval.
My selection will be 0 < X < 1 with probability 1.
Consider 2 subintervals, and my selection process gurantees that I
will choose from one or the other subinterval with probability 1/2.
Divide again, and I will select a number from one of 4 subintervals
each with probability 1/4.
Continue this process indefinitely.
I have countably infinitely many subintervals, each with equal
probability of containing my random number.
Even though the number of subintervals increases exponentially it is
still a coutable set.
Am I wrong here ?
Gus Gassmann
You are wrong. [0, 1] is an uncountable set, and yet it can be
represented by (countable) sequences of zeroes and ones. All this
proves is that 2^oo = c, the cardinality of the continuum. (And to be
perfectly clear about how the sequences correspond to your intervals:
The first digit i1 of .i1i2i3i4... is 0 if you choose the left
interval and is 1 if you choose the right interval, etc.)
Richard Tobin
Denis Feldmann <feldmann.den...@neuf.fr> wrote:
> Presumably you're not so rude in real life.
>I am with fools like you
You really are a most unpleasant person.
-- Richard
--
:wq
Richard Tobin
Huang <huangx...@yahoo.com> wrote:
>I will choose a number at random from the unit interval.
>My selection will be 0 < X < 1 with probability 1.
>Consider 2 subintervals, and my selection process gurantees that I
>will choose from one or the other subinterval with probability 1/2.
>Divide again, and I will select a number from one of 4 subintervals
>each with probability 1/4.
>Continue this process indefinitely.
>I have countably infinitely many subintervals, each with equal
>probability of containing my random number.
>
>Even though the number of subintervals increases exponentially it is
>still a coutable set.
There are contably many subintervals obtained by successively dividing
[0,1] in half, since each of the corresponds to a finite sequence of
choices. But to identify your real uniqely, you need an infinite
sequence of such choices.
-- Richard
--
:wq
David C. Ullrich
wrote:
>On Jan 12, 10:05 am, David C. Ullrich <ullr...@math.okstate.edu>
>wrote:
>>
>> _If_ you're interested in resolving the difference between
>> you and the physicist you'd ask what I suggested you
>> ask and see what he says. Because it's possible that
>> he'd have no problem with "choose a real between 0 and
>> 1 at random", and in that case you could explain why
>> he's simply _wrong_ about the impossibility of events
>> of probability zero happening.
>
>Sure, on the face of it, it seems possible to reason with a physicist
>that believes that conceptualizing events that occur with zero
>probability is unfathomable. The problem is, he explicitly said that
>even an event of incredibly small probability can't happen.
First, if he said that why didn't you say so? There's a big difference
between that and saying that events of zero probability can't happen.
Second, again you should simply ask him a question. First ask him for
an epsilon > 0 such that an event of probability < epsilon can't
happen. Second, calculate an N such that 2^(-N) < epsilon.
Third, ask him to flip a coin N times and tell you what sequences
of heads and tails resulted. Then point out that the probability
of that sequence of heads and tails is < epsilon.
Third, no he _didn't_ say that! He said "Probabilities this low
are generally taken to mean the event could not have happened."
That's _true_.
Yes it is. Suppose I tell you that I was watching a glass
of water the other day, and with no outside energy applied
it just happened that half of it froze solid while the other
half boiled away. Would you believe me?
>http://groups.google.com/group/sci.physics.foundations/msg/3d5630f2762edce6
Huang
>
> - Show quoted text -
Thanks -
Shubee
> On Sun, 13 Jan 2008 21:37:23 -0800 (PST), Shubee <e.Shu...@gmail.com>
> wrote:
>
> >Sure, on the face of it, it seems possible to reason with a physicist
> >that believes that conceptualizing events that occur with zero
> >probability is unfathomable. The problem is, he explicitly said that
> >even an event of incredibly small probability can't happen.
>
> First, if he said that why didn't you say so?
If events of zero probability can happen, then events of fantastically
small probability can happen. Didn't Oh No assert that "Zero
probability means that an event cannot happen"?
http://groups.google.com/group/sci.physics.foundations/msg/ca3eb017d60ca734
> There's a big difference between that and saying
> that events of zero probability can't happen.
That is correct. If I titled this thread, "Can Events of Fantastically
Small Probability Happen?" I probably wouldn't get a serious
mathematical answer and I might even be told that asking such a
fantastically stupid question doesn't belong at sci.math.
> Second, again you should simply ask him a question. First ask him for
> an epsilon > 0 such that an event of probability < epsilon can't
> happen. Second, calculate an N such that 2^(-N) < epsilon.
> Third, ask him to flip a coin N times and tell you what sequences
> of heads and tails resulted. Then point out that the probability
> of that sequence of heads and tails is < epsilon.
David, going that route presupposes that the physicist respects
precise and elegantly stated mathematical reasoning. The way I look at
it, I have already arrived at an apparently insolvable impasse. Didn't
Oh No make it clear that his philosophical perspective supercedes all
established mathematical understanding?
http://groups.google.com/group/sci.physics.foundations/browse_frm/thread/c4de9ae9a364fc79/
> Third, no he _didn't_ say that! He said "Probabilities this low
> are generally taken to mean the event could not have happened."
> That's _true_.
More precisely, those are weasel words in the context of the
discussion. Definition: "A weasel word is used to avoid making a
straightforward statement. Weasel words are also used to deceive,
distract, or manipulate an audience." Weasel wording "conceals the
full picture. In this way, one may evade responsibility for what may
be inferred." http://en.wikipedia.org/wiki/Weasel_words
Please note the meaning of the physicist's whole paragraph in response
to my question:
>Quantum mechanically, is there a nonzero probability for the Red Sea
>to split (Exodus 14:21) and for a man to be fully formed out of the
>inanimate material of the earth in a single day? (Genesis 2:7).
"Although, as in qm, when events are governed by probability, it may
be technically possible to find a non-zero probability for extremely
unlikely events, there must be some doubt about the meaning of the
mathematics. Probabilities this low are generally taken to mean the
event could not have happened."
I interpret that answer as "Yes, but." The key line is "there must be
some doubt about the meaning of the mathematics." Do you really
believe that an expert physicist can rationally justify having doubt
about the meaning of the mathematics?
Oh No's answer "Yes, but," when you mod out all the weasel words in
the whole paragraph, clearly affirms my claim that Oh No "explicitly
said that even an event of incredibly small probability can't happen."
> Yes it is. Suppose I tell you that I was watching a glass
> of water the other day, and with no outside energy applied
> it just happened that half of it froze solid while the other
> half boiled away. Would you believe me?
>
> > http://groups.google.com/group/sci.physics.foundations/msg/3d5630f2762edce6
David, thank you for bringing up this very familiar illustration in
quantum physics. You have proven my point. The accepted and widely
acknowledged answer by the experts in quantum physics is that the
event that you described can happen, although with fantastically
small, non-zero probability.
Now, please consider the meaning of this amusing curiosity. When
mainstream physicists interpret quantum physics and assert that
miraculous events can happen in a glass of water, the meaning of
fantastically small probability is not disputed. When I ask about the
quantum mechanical chances for the Red Sea to part (Exodus 14:21) and
for a man to be fully formed out of the inanimate material of the
earth in a single day (Genesis 2:7), then suddenly those events call
into question the meaning of fantastically small probabilities.
VK
> Recently, I had a debate with a physicist on the physical meaning of
> probability theory. My view is that mathematical models of toy
> universes do exist that are logically consistent where measure zero
> events can happen.
>
> cannot happen." He also stated that "If you change that, you are no
> longer talking about probability but about something else, and
> whatever you say becomes incomprehensible."
>
> I responded by saying:
>
> "In finite probability spaces, you are correct. In general, all that
> is needed for a 'Probability space' is a measure on a sigma-algebra of
> events that assigns to each event a number between 0 and 1.
If we are talking physics (QM) then it is not correct because there
are negative probability (<0), zero probability (0), and positive
probability (>0)
Gonçalo Rodrigues
fed this fish to the penguins:
Now, I'm curious. Since this goes against the frequency interpretation
of probability, what exactly is an event with negative probability?
And where does in QM, events of negative probability appear?
Best regards,
G. Rodrigues
Robert Israel
> On Sun, 13 Jan 2008 21:37:23 -0800 (PST), Shubee <e.Sh...@gmail.com>
> wrote:
>
> >On Jan 12, 10:05 am, David C. Ullrich <ullr...@math.okstate.edu>
> >wrote:
> >>
> >> _If_ you're interested in resolving the difference between
> >> you and the physicist you'd ask what I suggested you
> >> ask and see what he says. Because it's possible that
> >> he'd have no problem with "choose a real between 0 and
> >> 1 at random", and in that case you could explain why
> >> he's simply _wrong_ about the impossibility of events
> >> of probability zero happening.
> >
> >Sure, on the face of it, it seems possible to reason with a physicist
> >that believes that conceptualizing events that occur with zero
> >probability is unfathomable. The problem is, he explicitly said that
> >even an event of incredibly small probability can't happen.
>
> First, if he said that why didn't you say so? There's a big difference
> between that and saying that events of zero probability can't happen.
>
> Second, again you should simply ask him a question. First ask him for
> an epsilon > 0 such that an event of probability < epsilon can't
> happen. Second, calculate an N such that 2^(-N) < epsilon.
> Third, ask him to flip a coin N times and tell you what sequences
> of heads and tails resulted. Then point out that the probability
> of that sequence of heads and tails is < epsilon.
Of course, if he's clever he'll take epsilon so small that he won't be
able to flip a coin N times.
> Third, no he _didn't_ say that! He said "Probabilities this low
> are generally taken to mean the event could not have happened."
> That's _true_.
>
> Yes it is. Suppose I tell you that I was watching a glass
> of water the other day, and with no outside energy applied
> it just happened that half of it froze solid while the other
> half boiled away. Would you believe me?
>
You'd need a heck of a lot of coin flips to get a probability that small.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
galathaea
> On Mon, 14 Jan 2008 12:40:38 -0800 (PST), VK <schools_r...@yahoo.com>
they come about in wigner distributions
quasiprobabilistic theories
and more generalised probabilities
arise from a class of nonclassical logical consequences
they arise naturally in the noncommutative deformation theories
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
galathaea
> Recently, I had a debate with a physicist on the physical meaning of
> probability theory. My view is that mathematical models of toy
> universes do exist that are logically consistent where measure zero
> events can happen.
>
> His expressed position was that "Zero probability means that an event
> cannot happen." He also stated that "If you change that, you are no
> longer talking about probability but about something else, and
> whatever you say becomes incomprehensible."
>
> I responded by saying:
>
> "In finite probability spaces, you are correct. In general, all that
> is needed for a 'Probability space' is a measure on a sigma-algebra of
> events that assigns to each event a number between 0 and 1.
>
> http://en.wikipedia.org/wiki/Probability_space"
>
> I believe that my learned opponent is wrong for three reasons, two of
> which are purely mathematical. First, it seems that he is unable to
> detach himself from his own emotional feelings about the universe to
> even consider the meaning of zero probability from a measure-theoretic
> point of view. Second, he thinks that the occurrence of a measure-zero
> probability event is incomprehensible.
>
> Who is right?
it depends on how you define probability
inside models
existents can have measure zero
but your operational knowledge of a probability can never have zero
measure
as a physicist trapped in experience
only a finitely specified measurement can any ever be accomplished
even if your experience
_really_was_ one of those models with existents
of measure zero in the possibilities of ontological arrangement
you will only learn the result of finite tests of measurement
and the possibilities you could ever measure are finite
so if you look to meaningfulness in operationalism
everything that happens has nonzero probability
VK
> Now, I'm curious. Since this goes against the frequency interpretation
> of probability, what exactly is an event with negative probability?
And what exactly infinite countable set? or 1001-dimensional space?
or... :-)
> And where does in QM, events of negative probability appear?
I guess you don't expect from me a short explanation of QM, because my
own knowledge is lim(0) on this subject. More or less popular texts
are:
http://www.drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm
http://en.wikipedia.org/wiki/Negative_probability
Herman Rubin
galathaea <gala...@gmail.com> wrote:
>On Jan 14, 2:21=A0pm, Gon=E7alo Rodrigues <nos...@invalid.mail> wrote:
>> On Mon, 14 Jan 2008 12:40:38 -0800 (PST), VK <schools_r...@yahoo.com>
>> fed this fish to the penguins:
>> >On Jan 11, 5:20 pm, Shubee <e.Shu...@gmail.com> wrote:
.................
>> >If we are talking physics (QM) then it is not correct because there
>> >are negative probability (<0), zero probability (0), and positive
>> >probability (>0)
>> Now, I'm curious. Since this goes against the frequency interpretation
>> of probability, what exactly is an event with negative probability?
>> And where does in QM, events of negative probability appear?
>they come about in wigner distributions
>quasiprobabilistic theories
>and more generalised probabilities
> arise from a class of nonclassical logical consequences
>they arise naturally in the noncommutative deformation theories
I am unfamiliar with the terminology.
However, if one assumes that there is a joint distribution
of position and momentum, it is computable from the wave
function, and this computation can generate negative
probabilities. But this does not mean that if one takes
simultaneous observations, there is a negative probability
that such events will be observed, but rather that the
pair cannot be simultaneously observed, and the product
of an observation of one together with an immediately
following observation of the other has that uncertainty.
One can define "signed measures" which can have negative
values for some of the sets.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
David C. Ullrich
On Mon, 14 Jan 2008 11:50:16 -0800 (PST), Shubee <e.Sh...@gmail.com>
wrote:
>On Jan 14, 6:12 am, David C. Ullrich <ullr...@math.okstate.edu> wrote:
************************
David C. Ullrich
David C. Ullrich
That doesn't answer the question. Would you believe me if I said
that happened?
************************
David C. Ullrich
David Bernier
> On Jan 14, 6:12 am, David C. Ullrich <ullr...@math.okstate.edu> wrote:
>> On Sun, 13 Jan 2008 21:37:23 -0800 (PST), Shubee <e.Shu...@gmail.com>
>> wrote:
>>
>>> Sure, on the face of it, it seems possible to reason with a physicist
>>> that believes that conceptualizing events that occur with zero
>>> probability is unfathomable. The problem is, he explicitly said that
>>> even an event of incredibly small probability can't happen.
[...]
> Oh No's answer "Yes, but," when you mod out all the weasel words in
> the whole paragraph, clearly affirms my claim that Oh No "explicitly
> said that even an event of incredibly small probability can't happen."
>
>> Yes it is. Suppose I tell you that I was watching a glass
>> of water the other day, and with no outside energy applied
>> it just happened that half of it froze solid while the other
>> half boiled away. Would you believe me?
>>
>>> http://groups.google.com/group/sci.physics.foundations/msg/3d5630f2762edce6
>
> David, thank you for bringing up this very familiar illustration in
> quantum physics. You have proven my point. The accepted and widely
> acknowledged answer by the experts in quantum physics is that the
> event that you described can happen, although with fantastically
> small, non-zero probability.
>
> Now, please consider the meaning of this amusing curiosity. When
> mainstream physicists interpret quantum physics and assert that
> miraculous events can happen in a glass of water, the meaning of
> fantastically small probability is not disputed. When I ask about the
> quantum mechanical chances for the Red Sea to part (Exodus 14:21) and
> for a man to be fully formed out of the inanimate material of the
> earth in a single day (Genesis 2:7), then suddenly those events call
> into question the meaning of fantastically small probabilities.
>
> Shubee
> http://www.everythingimportant.org/creationism
It's not clear how much of Genesis is part of the miracle of creation,
but Chapter 1 v. 16:
(16) "And God made two great lights; the greater light to rule the day,
and the lesser light to rule the night: he made the stars also."
(The earth was created before). In any case, what are the chances
in QM of a standard miraculous event?
What I'm not sure about is what happens to the wave function of the
world when there is an event...
I think you wrote you don't believe in hidden variable theories.
AFAIK, there are at least three "respectable" ideas:
- The collapse of the wave function. ("It just happens")
- Decoherence ( state superposition -> non-superposed through
interaction with outside world)
- many worlds ( the wave function doesn't collapse, dead
Schroedinger cats exist just like their
live counterparts, but don't interact).
The Wikipedia article has quotes from Weinberg, Hawking and Penrose:
< http://en.wikipedia.org/wiki/Schr%C3%B6dinger's_cat >
If one believes the universe only has one go, ever, then the
glass of water curiosity could happen, or not happen, and
man would be around. If the coming of man is thought
of as exceedingly improbable, then it does seem strange that
man is around. But one unanswered question is the number of
bits needed to describe the wave function "at one time".
But with respect to parting of the Red Sea, I put it in the
same category as the glass of water event. Both possible, under
the laws of chance.
Also, I don't believe today's physics is the end of the story.
Around 1900, there was the question of where the sun got its
energy from. Geological evidence of earth being hundreds of millions
of years old didn't square with the thermodynamics of a hot sun cooling
down. The hot sun would be cold in too few millions of years.
Henri Poincaré wrote about the conflicting evidence. He wrote
that physics used math., so made reliable predictions (including
the time for the sun to become cold) and pointed out
that geology then proceeded in many cases by analogy. But he
didn't come to any conclusion (e.g.: the geologists are wrong).
So I'd say current understanding allows tons of molecules to all at once
go one way , or lose kinetic energy to other molecules gain, so
the Red Sea parting and the glass of water anomaly are both
possible. What the understanding will be in hundreds of years is
anybody's guess.
David Bernier
Gonçalo Rodrigues
fed this fish to the penguins:
>On Jan 15, 1:21 am, Gonçalo Rodrigues <nos...@invalid.mail> wrote:
>> Now, I'm curious. Since this goes against the frequency interpretation
>> of probability, what exactly is an event with negative probability?
>
>And what exactly infinite countable set? or 1001-dimensional space?
>or... :-)
>
I am the wrong target for that dart.
Roughly (and even naively), when I said that it goes "against the
frequency interpretation" is because there is a very clear recipe for
going to a statement like "There is a probability p that the system X
is in state x" to an experiment in the lab verifying it.
>> And where does in QM, events of negative probability appear?
>
>I guess you don't expect from me a short explanation of QM, because my
>own knowledge is lim(0) on this subject. More or less popular texts
>are:
>http://www.drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm
>http://en.wikipedia.org/wiki/Negative_probability
Well my knowledge of QM is not 0 and according to the rules of QM
there are no negative probabilities - or so I though, it seems. Will
read the page.
Regards,
G. Rodrigues
Randy Poe
> On Tue, 15 Jan 2008 07:45:42 -0800 (PST), VK <schools_r...@yahoo.com>
According to your second article, the idea was introduced
by Dirac, who seems to be saying that you can have
intermediate things which are treated as probabilities
but which are negative. However, no details are given.
As I read that paraphrase, Dirac is not claiming
that an observable event can have a negative probability
of occurrence, but that negative terms might occur
in the calculation of that probability.
- Randy
galathaea
this arose from the early field theories
the most direct way of quantising a spin 0 field
was the klein-gordon approach
solutions to this equation
though
are unstable
and possess an infinite cascade of negative energy solutions
through the standard interpretation at the time it was first detailed
it was the probability densities themselves that were not positive
definite
and so there were questions raised about its physicality
through a nice concordance of units
the klein-gordon solutions could be reinterpreted as negative charge
densities
though
and really the dirac equation was much more interesting
since the charged particles known had nonzero spin
research into negative probabilities
really didn't take off as a mathematical discipline
until wigner proposed his phase space approach to quantum mechanics
the wigner distribution has a natural probability density
interpretation
and the phase space transform methodology provided a technique
that ensured one could reason about negative probability densities
consistently
even though the densities here are not positive definite
to get meaningful probabilities
one must integrate over physically identifiable measures of state
due the uncertainty principle
there are restriction on minimal phase space volumes observable
and so actual probabilities calculated become again nonnegative
these can be shown to be related to the contextuality of the theory
and fundamentally a result of its nonclassical logic
of course
trajectory theories like bohmian mechanics
already show that the detection probabilities must be classical
David Bernier
[...]
Earlier I wrote that the water glass anomaly and the parting of
the Red Sea might have QM-probability > 0. The two probabilities
could be separated by trillions of trillions of orders of magnitude,
bu that's a detail.
There's a common saying:
"Extraordinary claims demand extraordinary proof".
I subscribe to that while acknowledging that people will disagree
on the balancing of the evidence for and the evidence against some
claim.
For example, this web page about ESP doesn't reflect my
assessment of the odds that ESP exists:
< http://www.quackwatch.com/01QuackeryRelatedTopics/extraproof.html > .
In part, I give a high weight to the negative results in
James Randi's test-takers.
This website of an organization in Philadelphia seems interesting:
< http://www.phact.org/ >
David Bernier
--
Posted via a free Usenet account from http://www.teranews.com
bill
> Recently, I had a debate with a physicist on the physical meaning of
> probability theory. My view is that mathematical models of toy
> universes do exist that are logically consistent where measure zero
> events can happen.
>
> His expressed position was that "Zero probability means that an event
> cannot happen." He also stated that "If you change that, you are no
> longer talking about probability but about something else, and
> whatever you say becomes incomprehensible."
>
> I responded by saying:
>
> "In finite probability spaces, you are correct. In general, all that
> is needed for a 'Probability space' is a measure on a sigma-algebra of
> events that assigns to each event a number between 0 and 1.
>
>
> I believe that my learned opponent is wrong for three reasons, two of
> which are purely mathematical. First, it seems that he is unable to
> detach himself from his own emotional feelings about the universe to
> even consider the meaning of zero probability from a measure-theoretic
> point of view. Second, he thinks that the occurrence of a measure-zero
> probability event is incomprehensible.http://groups.google.com/group/sci.physics.foundations/browse_frm/thr...
>
> Who is right?
>
What is the third reason?
Which of the first two reasons is purely mathematical and not part
philosophical?
Can you suggest a concrete example of a "zero probability event"?
Thank you
Bill J.
Robert Israel
> On Tue, 15 Jan 2008 02:36:45 -0600, Robert Israel
>
>
>
> <isr...@math.MyUniversitysInitials.ca> wrote:
>
> >> On Sun, 13 Jan 2008 21:37:23 -0800 (PST), Shubee <e.Shu...@gmail.com>
Of course not. But that's because of my assessment of the
probabilities of
1) the event actually happening
2) you either lying or hallucinating.
I also might not believe you in situations where you assert
something that is clearly not impossible (e.g. if you were
trying to sell me something).
bill
> Recently, I had a debate with a physicist on the physical meaning of
> probability theory. My view is that mathematical models of toy
> universes do exist that are logically consistent where measure zero
> events can happen.
>
> His expressed position was that "Zero probability means that an event
> cannot happen." He also stated that "If you change that, you are no
> longer talking about probability but about something else, and
> whatever you say becomes incomprehensible."
>
> I responded by saying:
>
> "In finite probability spaces, you are correct. In general, all that
> is needed for a 'Probability space' is a measure on a sigma-algebra of
> events that assigns to each event a number between 0 and 1.
>
> http://en.wikipedia.org/wiki/Measure_%28mathematics%29http://en.wikipedia.org/wiki/Probability_space"
>
> I believe that my learned opponent is wrong for three reasons, two of
> which are purely mathematical. First, it seems that he is unable to
> detach himself from his own emotional feelings about the universe to
> even consider the meaning of zero probability from a measure-theoretic
> point of view. Second, he thinks that the occurrence of a measure-zero
> probability event is incomprehensible.http://groups.google.com/group/sci.physics.foundations/browse_frm/thr...
>
> Who is right?
>
> Shubee
What is "the meaning of zero probability from a measure-theoretic
point of view"?
Bill J.
Math1723
Sometimes it's possible, sometimes it's not.
Example: Let x be a random variable of U[0,1] (uniform distribution
from the closed interval [0,1]).
What is the probability that x = 0? 0, but it can happen.
What is the probability that x = 2? 0, but it cannot happen.
Dave Seaman
A probability space is a measure space in which the measure is positive
and the measure of the entire space is 1.
An "event" in a probability space is merely a set in the corresponding
measure space. An event has probability zero if the measure of the set
is zero.
The only impossible event is the empty set, and the empty set necessarily
has probability zero. However, thre are lots of nonempty sets that also
have probability zero. For example, in the space [0,1] with Lebesgue
measure, any countable set has measure zero.
--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
Virgil
<0f79853c-8192-47bd...@e10g2000prf.googlegroups.com>,
bill <b92...@yahoo.com> wrote:
The probability of choosing a particular real number "at random" from a
real interval (of positive length) is a standard example of just that
sort of probability.
David C. Ullrich
Ok, let's change the question a bit... no, that version is
vulnerable to a similar response.
Ok. I may have to abandon proof and fall back on vehement
assertion. What is and is not "commonly taken to be"
anything is of course not a mathematical question.
Even when we agree that it's _possible_ for a glass of
water to spontaneously half freeze and half boil, the
even is so unlikely that it _is_ in fact "commonly taken
to be impossible". At least in physics, as opposed to
phllosophy. For example the second law of thermodynamics
_is_ taken to be a _law_, and a very fundamental one,
even though the events it says cannot happen are not
literally impossible, just extremely unlikely.
>I also might not believe you in situations where you assert
>something that is clearly not impossible (e.g. if you were
>trying to sell me something).
>
>Robert Israel isr...@math.MyUniversitysInitials.ca
>Department of Mathematics http://www.math.ubc.ca/~israel
>University of British Columbia Vancouver, BC, Canada
>
************************
David C. Ullrich
David Bernier
Are the densities at least real-valued?
I had a look at:
http://www.drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm
and in the line or "equation" :
[2] + [7] = (X + Y - Z) / 2 = -.1036 (per the QM formula),
it seems to me that X, Y and Z can be understood as
non-negative real-valued random variables that are
the results of an experiment.
I guess I follow sections d., e., f. and so on.
I wonder if there's anything more to negative probabilities
than linear combinations (over the reals) of ordinary probabilities ...
Maybe in Quantum Field Theory?
> due the uncertainty principle
> there are restriction on minimal phase space volumes observable
> and so actual probabilities calculated become again nonnegative
>
> these can be shown to be related to the contextuality of the theory
> and fundamentally a result of its nonclassical logic
>
> of course
> trajectory theories like bohmian mechanics
> already show that the detection probabilities must be classical
>
> -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
> galathaea: prankster, fablist, magician, liar
--
tommy1729
> On Jan 11, 10:06 pm, Virgil <Vir...@com.com> wrote:
>
> > If one has a set which is countably infinite, like
> the set of integers,
> > or naturals, or rationals, then it turns out to be
> impossible to
> > "chose one member at random", since that would
> require each of countably
> > many members to have the same probability of being
> chosen, but there
> > cannot be any real number of which countably many
> "add up" to anything
> > but zero or infinity.
>
> True but really that's not a problem because, in this
> instance, we
> should be using a finitely-additive probability
> space, which only
> requires an algebra of sets, rather than a
> sigma-algebra.
>
> http://en.wikipedia.org/wiki/Probability_axioms
>
> Shubee
>
>
>
yep.
tommy1729
> In article
> <961fo35uieus57unl...@4ax.com>,
> David C. Ullrich <ull...@math.okstate.edu> wrote:
>
> >Whether or not something can "really" "happen" is
> not a
> >mathematical question.
> >
> >Ask him this: Suppose that X is a real number
> between 0 and 1,
> >chosen at random. What is the probability that X =
> 1/2?
once again i agree with david :)
>
> This seems a little like cheating, because when you
> instead ask
> "suppose X is an integer chosen at random, what is
> the probability
> that it's 2?", the mathematician is just as stuck as
> the physicist.
no richard, you are as stuck as the physicist :D
no offense.
the probability of X real between 0 and 1 being 1/2
is 1 / aleph_1
the probability of X positive (finite) integer being 2
is 1 / aleph_0
>
> -- Richard
> --
> :wq
regards
tommy1729
" i dont believe it" victor meldrum 666
huya...@gmail.com
> Whether or not something can "really" "happen" is not a
> mathematical question.
>
> Ask him this: Suppose that X is a real number between 0 and 1,
> chosen at random. What is the probability that X = 1/2?
>
> be, for example, that the way he's using the term,
> "choose a real number between 0 and 1 at random" is simply
> not something that can "happen"...
>
> David C. Ullrich
If we are talking about pure math., that is, independent of time and
space of the universe as currently belived/assumed by the mainstream,
then, you maybe right. But, if you take the earth age in seconds (4.5
billion) times the molecular oscillation (10**13), it has not been
enough time to get anywhere yet. An order of 20 zeros at least. Zero
chance is correct.
Hu
Huang
...and now for something completely different...
Physical length may be regarded as being purely existent, or
indeterminately existent.
http://hubpages.com/hub/Continuity-of-Spacetime
Seems like this has some bearing on events of zero measure.
Suppose that the physical length which we observe in nature is really
an "expected length". Then, the measure zero event is .... known
length + random length where random length = 0. Just because it is 0,
you cannot really say that it did not "happen".
tommy1729
LOL !!!
i would say aleph_3 coin flips and a mistake in the current laws of physics are needed.
> --
> Robert Israel
> isr...@math.MyUniversitysInitials.ca
> Department of Mathematics
> http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver,
> BC, Canada
regards
tommy1729