Conference rusure::math

    Answer: An "event" with probability, P(event)=0
    
    Have fun !
    
    George

    but if P(e)=0, this means that miracles never happen, but they do,
    for example , i did graduate from school.., so may be it should be 
    P(e)= 0.000001 or something like that.
    
    humm.. i think there should be more to it than just probability.
    
    i'll go into deep thought and try to figure something out.

    Re: .2
    
    P(e)=0 does not strictly mean the event never happens.

    Miracle = event conditional on the universe
    
    	Pr(Miracle|Secular Universe) = 0
    
    	Pr(Miracle|Deistic Universe) > 0
    
    :-)
    
    Dick

    First of all, keep in mind that it is not universally accepted that
    miracles must have zero liklihood, or even be "without materialist
    explanation".  Check out any introduction to modern religious thought
    on the subject.

    And, as previously pointed out, a probability of zero does not mean the
    same thing as impossible.  Zero probability events occur all the time.
    *Any* specific value from a continuous or piecewise continuous
    probability distribution has zero probability, yet one of them *will*
    occur.

    Furthermore, neither major theory of mathematical probability
    (frequentist and subjectivist; which serve respectively as the basis of
    classical and Bayesian statistical theories) would accept the
    description of an impossible event occuring as meaningful.  It would
    just mean that your evaluation of probability was in error.

    You need to go to a causitive (physical) system of laws which makes
    specific predictions of the probability of events (or at least can
    exclude absolutely certain events from occuring).  You can then
    say that if an event excluded by those laws occurs then it is a
    miracle.  The problem then is to distinguish "laws" from theories
    about what those laws are.  This is an essential problem in making
    the idea of the literally "supernatural" (literal not being the only
    available model of the concept) coherent.

				    Topher

              <<< Note 1521.5 by CADSYS::COOPER "Topher Cooper" >>>
                                 -< Comments. >-

>    Zero probability events occur all the time.

>    *Any* specific value from a continuous or piecewise continuous
>    probability distribution has zero probability, yet one of them *will*
>    occur.

    In a mathematical idealization perhaps, but in the "real world"
    as opposed to the real line there are no impossible events that
    will occur anyway and this is "probably" even true taking into
    account quantum electrodynamics.

    For example, there is no way to write down an "arbitrary" real
    number with a finite number of digits or even a finite algorithm.

    How then can one choose an arbitrary real number when it cannot be
    written down?

    - Jim

RE: .6 (Jim)

    "If a tree fell in a forrest, and nobody wrote about it, would it make
    a sound?"

    A process makes a "choice" without writing it down -- and that choice
    may be taken from a continuous distribution.  We model that choice by
    making and recording a measurement which by necessity has finite
    precision and is therefore "actually" discrete.  We may -- frequently
    will -- ignore that fact and idealize/approximate the model of the
    system back to a continuous one.

    The distribution of the quantity is continuous.  The distribution of
    our recorded measurements is discrete.  The former is the "real world",
    though the latter is what we have to deal with to be practical.

    QM (not specifically QED) complicates this picture somewhat, but does
    not fundamentally change it.  It might be said that the radial
    component of the position of an orbiting electron has a strictly
    discrete distribution, but this presumes that we know the location of
    the nucleus precisely, which is an idealization.  In QM quantization
    makes some (not all) classically continuous distributions discrete but
    uncertainty makes them continuous again.

    The probability of a continuous quantity such as "position" suitably
    defined (e.g., a mean of the square-root of the wave-function over
    space) taking any particular value is zero -- but it does take some
    value.  The probability of us making/recording any particular
    measurement is, as you say, non-zero.

				    Topher

    Max Born made the distinction between an "indeterminate" and an
    "unknown" quantity during his discussions of quantum phenomena.  A
    quantity is made determinate when the measuring procedure is invoked. 
    It becomes known when some human observer notices its result.
    
    Dick

On a completely different tack, a miracle is a command which violates
Dijkstra's `law of the excluded miracle' (in A Discipline of Programming, p 18).
In this religion, a command C is defined by a `predicate transformer', which
takes a predicate GOAL to the weakest predicate [C]GOAL such that if C is activated
in a state satisfying [C]GOAL, then it is guaranteed to terminate in a state
satisfying GOAL. A command is said to be miraculous if the weakest precondition
for obtaining False (something impossible) is different from False (i.e. not
impossible).

  [ Maybe this isn't what was wanted. But it is the only definition of `miracle'
    I've seen in a mathematical context. ]

One nonmathematical definition of "miracle" that I have read
is something like it is any event the a priori probability of
which is judged to be no more than 10%.

Dan

As an example of Topher's mention in .5 of how events of probability 0 can
occur:

	Time has been going on forever before you were here.  It will go on
	forever after you die.  Hence compared to how much time
	there is, the probability of the "present time" ever falling within
	your pitifully small lifespan is 0.

	Yet here we are !


/Eric

    ref .11
    
     but time has not been going on forever! time started with the creation
     of the universe long time ago, and time will no longer exist when the
     universe falls back again on itself.
    
    /nasser
    

RE: .12

    This is a matter of current debate.  It is widely, almost universally
    (if you'll pardon the expression) believed by physicists that the
    universe as we know it was created between 10 and 20 billion years
    ago in "the big bang".  It is, for most purposes, convenient to label
    that as t=0.  Many cosmological models include the assumption that
    there is no such thing as "negative" time.  Other models do not
    make that assumption.  Still other models say that the "singularity"
    at t=0 (i.e., the beginning of time) is a so called "pseudo-
    singularity" which means that it is a singularity in the equations
    which appear only when you choose the wrong set of co�rdinates.  In
    the correct reference frame, what we call t=0 is mapped into t=-oo.

    What this boils down to is that it is unknown whether Eric's
    assumptions are physically realistic, so that his example cannot be
    said to be invalid.

				    Topher

Time didn't start when the universe did.  There was a point "five minutes
before the big bang".  So I still claim we waited "quite a while" to be
born, and we'll wait "quite a while" after we die.

Does this disturb you ?  It does me.

/Eric

> Time didn't start when the universe did.  There was a point "five minutes
> before the big bang".  
    
    This sounds analagous to the paradox "What's at the edge of the universe, 
    and what's on the other side of it?"
    
      John

    Re .14:
    
    > There was a point "five minutes before the big bang".
    
    Not necessarily.  Time, like space, is curved, and might not extend
    infinitely into the past.  Consider our coordinates on the surface of
    the Earth in latitude and longitude.  If you were to travel south, your
    latitude would continually decrease (measured from -90 at the South
    Pole to +90 at the North Pole).  Ultimately, you would reach the
    "singularity" where latitude "stopped".  Yet physically you would see
    nothing strange -- if you kept going in the direction you were heading,
    you would have just as smooth a ride at -90 degrees latitude as you did
    just before and just after it.  The "space" that is Earth's surface is
    smooth at every point even though it is impossible to place a
    coordinate system on it that is smooth at every point.
    
    Similarly, if you were to travel backward in time, you might find that
    there is only a certain distance back you can go, and that attempts to
    go further back just yield travel in a direction you were not
    expecting.  There would not be any more time prior to the Big Bang
    than there would be Earth's surface more southern than the South Pole.
    
    
    				-- edp

    Time could not have existed before the creation of the Universe, i.e.,
    space-time.  It may have pre-existed the Big Bang (which created the
    Universe-as-we-know-it), however, depending on whether or not the
    Universe-in-some-form pre-existed the Universe-as-we-know-it, and there
    may not have been a meaningful creation of the Universe-in-some-form.
    There may be a moment for which there is no moment in any meaningful
    sense which is less than it -- or there may not.

				    Topher

re .16,

>			The "space" that is Earth's surface is
>    smooth at every point even though it is impossible to place a
>    coordinate system on it that is smooth at every point.

Define "coordinate system". :-)  Apparently it must be ruling
out the obvious smooth covers of a sphere by

	(p,q) -> (R cos p sin q , R sin p sin q , R cos q)

(non unique) or by

	identity map on {(p,q,r) in R^3 | p^2 + q^2 + r^2 = R^2}

(individual coordinates not independent) or by the first example
restricted to

	{(p,q) in R^2 | 0 <= p < 2 pi and 0 < q < pi} union {(0,0),(0,pi)}

(domain not open).

Dan

If I travel South and keep going, I eventually recognize that I've gotten
back to where I started from.

If your analogy holds, then looking back in time, we'd expect some sort of
loop that would connect back to the present.  There's no evidence of this.
Similarly, if the universe is "curved", if we just take off from Earth and
try to keep traveling away, we'll eventually come back home, right ?

/Eric

    re .19
    
    Actually if you travel South you eventually come to a point where you
    can't travel South anymore. You have reached the 'edge of the world' -
    a singular point where there is nothing any further South. Having got
    there you have no choice but to travel North.
    
    Of course if you say you travel East or West then you do get back where
    you started.
    
    Andrew

    Re .19:
    
    > If your analogy holds, then looking back in time, we'd expect some
    > sort of loop that would connect back to the present.  There's no
    > evidence of this.
    
    What evidence of it have you tried to collect?  Did you go on such a
    journey and fail to return to your starting point?
    
    There is in fact evidence that space-time is curved.  The evidence
    includes our own experiments and our observations of the rest of the
    universe.  It is pretty weird for space-time to be so curved that
    travel in time for one observer appears to be travel in space for
    another, but it is not known to be impossible, and it is not so weird
    once all the evidence is considered.
    
    
    				-- edp

>    universe.  It is pretty weird for space-time to be so curved that
>    travel in time for one observer appears to be travel in space for
>    another, but it is not known to be impossible, and it is not so weird
>    once all the evidence is considered.

    Travel in time for one observer appearing to be travel in space for
    another is a basic consequence of special relativity.  GR says that
    *any* curvature (i.e., gravitational "field") has the same effect,
    albeit generally to small to be observed directly.

				Topher

    Did time exist before the universe was created? According to Doctor of
    Philosophy, Leonard Peikoff, no. "Time is a measurement of motion; as
    such, it is a type of relationship. Time applies only within the
    universe, when you define a standard -- such as the motion of the earth
    around the sun. If you take that as a unit, you can say: ``This person
    has a certain relationship to that motion; he has existed for three
    revolutions; he is three years old.'' But when you get to the universe
    as a whole, obviously no standard is applicable. You cannot get outside
    the universe. The universe is eternal in the literal sense:
    non-temporal, out of time."
    
    Dwayne
    

    Re .18:
    
    > Define "coordinate system". :-)
    
    Actually, two of your three examples don't have problems with
    "coordinate system"; they are coordinate systems but aren't "smooth".
    
    The first and the third aren't smooth.  Points with coordinates close
    to each other should be close to each other in the surface, and
    vice-versa.  Yet (0,q) and (pi,q) approach each other in the surface as
    q approaches zero, but the coordinates never get closer to each other
    than pi.
    
    I guess the second case would depend upon a definition of "coordinate
    system" to restrict the possibilities -- one coordinate per dimension
    in the space.
    
    
    				-- edp
            

>    The first and the third aren't smooth.  Points with coordinates close
>    to each other should be close to each other in the surface, and
>    vice-versa.

Having just the first is sufficient to make it smooth.  The
"vice versa" doesn't add anything to smoothness.  If points
have multiple coordinates then obviously the converse won't
hold.

Dan