Conference rusure::math

    Well, you maybe find the concept more prevailant in religion and
    philosophy in a somewhat more cloudy form.   But that is far from 
    reality although the fundamentalists may not agree with me.
    
    :-) :-)
    
    Eugene

Well, I think the answer is "yes", but in an unexpected way. As to very 
small things, Heisenberg's principle deals with things that are too small
to be both located and dimensioned. Below a certain threshhold chaos theory 
appears to be a better description of the real world than classical 
geometry. At the opposite end of the scale is the issue of whether the 
universe is in fact finite or possibly curved - where the distances and 
times are too large to observe. Timewise is the issue: did anything (even 
time?) exist before the "Big Bang"? Practically, of course, any time 
duration longer than your own lifespan is effectively infinite.

Primitive societies, I hear, counted by "one, two, three, many", admitting 
that the counting numbers are infinite and beyond their comprehension. But
I suppose you were looking for something further outside the math realm. 

Lynn 

    The largest Number that maps 1:1 to elements in the physical universe
    would be the total number of Quarks in the universe, can this number
    be Infinity ?
       
    ( Since The last time I looked, The Quark  was the most elementry
    undivisable "thing" around )
    
     
    /naser
    

    No.  There seems to be (I've heard) a finite mass in the universe
    and a finite number of quarks per subatomic particle and they carry
    mass therefore the number is finite in fact you could probably
    calculate it from estimates of the quantity of matter in the universe.
	Quantum theory does not say that BECAUSE particles are small that
	planck's constant equals the product of the uncertainties of
	momentum and position, it says that h= delta p delta x
	just because they're small relative to
	the particles we'd use to detect them.  Big things have the same
	equation.
	
	The point of fractals is that they appear on all scales.  Witness graphs
	of Digital stock for 1 year or 10 years.  
	Non-linear equations affect droplets and waterfalls, heartbeats and
	three and more moons affecting each other gravitationally.

	If the universe if positive in curvature, it is just as indefinite in
	extent as a circle.  What is a better metaphysical symbol 
    	of the infinite
	that a closed curved universe with no spacial end?  Only in 4 dimensions
	does the universe seem to have an end.


Tom
    

>    would be the total number of Quarks in the universe, can this number
>    be Infinity ?
    
    I believe Eddington's estimate of the number of electrons in the
    universe was on the order of 10^74.  If the estimate is anywhere
    near correct I don't see how the total number of quarks can be
    over about 10^80.  DECWET::PHYSICS is a good place to pursue this
    sort of question.
    
    Infinity is a longgg way past 10^80...
    
      John

In order to deal with *really* large numbers you have to get away from 
thinking about physical measurements and start thinking about, say, the
number of permutations of states the universe could be in, or even more
convoluted things. Like, take that 10^80 in the previous note (actually, I
think Eddington's exponent had three digits - perhaps 273?) and consider
something like 

		10
	    ...
	  10
	10

where there are Eddington's number of exponentiation operators in the 
expression. No, of course you can't write it down. Actually, the mind 
begins to boggle when you have worked out just the first three or four 
exponentionations. But try to imagine it.

When your mind has expanded enough to take THAT in, let your new estimate 
of *the* largest number be the number of exponentiation operators in this 
expression, and stretch again...

The meaning of infinity, as I comprehend it, is that this expansion process
has no limit, no end, no means of full comprehension. Nothing we know about 
in the physical universe can touch it ... but it can exist in our minds,
souls, spirits, whatever else you may believe is a non-physical part of us.

Lynn Yarbrough 

    Re:           <<< Note 1195.4 by MILKWY::JANZEN "Tom 228-5421 FXO/28" >>>

>    No.  There seems to be (I've heard) a finite mass in the universe
    
    This is still an open question.  Observation suggests that the universe
    is infinite in space, and theory suggests (rather weakly) that it is
    finite in both space and time.  
    
>	The point of fractals is that they appear on all scales.  Witness graphs
>	of Digital stock for 1 year or 10 years.  
    
    This may be a proof that there are no true fractals in nature.  DEC
    stock, for example, is not scalable to a scale of minutes or centuries.
    
>	Only in 4 dimensions does the universe seem to have an end.
    
    And that is not certain.  It may have only a beginning.  Or it may have
    an infinite number of singularities on the time line.

    Re:             <<< Note 1195.6 by AKQJ10::YARBROUGH "I prefer Pi" >>>

>In order to deal with *really* large numbers you have to get away from 
>thinking about physical measurements and start thinking about, say, the
>number of permutations of states the universe could be in, or even more
>convoluted things. Like, take that 10^80 in the previous note (actually, I
>think Eddington's exponent had three digits - perhaps 273?)
    
    No, I think that this is correct for Eddington's number.  We should
    include photons in our computation.  I don't know the number of photons
    in a finite universe, but let's take 10^100 as a guess.
    
    Now each of those particles can exist in some state, described by some
    quantum numbers.  I think the largest contribution will come from the
    position and momentum of the particles in the universe as a whole. 
    With some more data we could calculate this, but let's just assume that
    each particle could exist in 10^100 states.
    
    The state of the universe is just a superposition of these states, so
    the number of possible states is just 
    
    	( 10^100 ) ^ ( 10^100 ) = 10 ^ ( 10^102 )
    
    This is getting to be a good sized number, but it is still finite.

    
    
    Here is another way to include a large number in the real world,
    following Feynmann's Quantum Electrodynamics.  Consider the scattering
    of two electrons.  They may exchange a photon, which contributes one
    term to the sum.  They may exchange two photons, which contributes
    another term.  They may exchange three...  I don't know whether the
    exchange is limited to the number of photons which could possibly exist
    in the universe.  If so, then for a finite universe it is bounded.  If
    not, then the scattering of two electrons involves the exchange of an
    infinite number of virtual photons.
    
    
    And here is another way, in a finite and collapsing universe.  Consider
    a photon which sets off around the universe shortly after creation. 
    While the universe is expanding, it will never make the full trip.  But
    when the universe begins to collapse, it will have time to make several
    trips.  As the universe collapses further, it will have time to make
    many trips.  As the diameter of the universe decreases without limit,
    the photon can make an infinite number of trips.  I suspect two things
    will prevent the practical realization of this infinity.  First off,
    the very high density of matter and radiation will make the small
    universe effectively opaque.  Second, quantum gravity may prevent the
    universe from decreasing without limit.

RE: .0 (Dwayne)
    
    > Besides mathematically, do the concepts of "infinite" and
    > "infintesimal" relate to anything in reality?
    
    Absolutely not -- but besides mathematically, the concepts of "one"
    and "two" don't relate to anything in reality.
    
    Numbers finite or transfinite or subfinite are mathematical concepts
    which we model the world with.  Although simple counting seems so
    elementary to us as to be a part of the real world, it is not.  When
    we say "there are two sheep there" we are saying something about a
    way (a useful way, generally) that we perceive and organize the world.
    First we are dividing the essentially undivided world into spatially
    bounded "things", and then we are identifying characteristics of some
    of those "things" as creating a group ("this is a sheep in this general
    space/time region and this is a different sheep in this general space/
    time region, so they are both in the group of "sheep over there"), and
    then we are making a one-to-one mapping from the elements of that group
    to a special set -- the set of "counting numbers".  Twoness is a
    part of our perceptions of the universe not something "real".
    
    That doesn't mean that I think that "two" is a completely arbitrary
    label without *meaning*.  It provides a useful model for certain
    aspects of the universe, and is therefore meaningful.  But it is a
    *meaningful description* of the universe not a part of the universe.
    Similarly, though less elementary, for infinities and infintesimals
    -- they are essential parts of useful and accurate models of the
    universe and are therefore *meaningful physical concepts*.  But they
    are not "real".
    
    Most obviously, our current theories, perceive space-time as a
    continuum without a smallest "distance".  They are therefore modeled
    by real numbers, of which there are an infinite number in any finite
    interval and they are separated (forgive the loose language) by
    infintesimal distances.  These mathematical properties are used for
    doing calculations which produce meaningful and correct predictions
    so they are good models.  If at some point we decide that there is
    a minimum length in space-time, then the models will continue to
    be useful, meaningful models but will no longer be seen as fundamental.
    Instead they will be *reasonable approximations* just as finite lattice
    models are now perceived as reasonable approximations to the underlying
    continuum.
    
    					Topher

RE: .2 (Lynn)
    
    > Below a certain threshhold chaos theory appears to be a better
    > description of the real world than classical geometry.
    
    That's a bit confused.  Chaos theory is about "dynamics" (although
    it frequently uses geometric tools, mathematically) -- a theory
    about how things *change in time*.  Classical geometry is about,
    well, geometry -- how space is structured.
    
    Did you mean that below a certain level "fractal geometries" or
    highly complex topologies (genus approaching infinity), or
    n-dimensional geometries (n>4) seem to be better descriptions of the
    structure of space-time than Euclidean 3+1 space?  These are areas
    of active investigation but it is not known whether or not they
    are accurate models.
    
    Did you mean that quantum uncertainty is a reflection of chaos at
    an even smaller scale?  That has been speculated also, but I don't
    think that it has been put in very concrete terms.
    
    Did you mean that at quantum scales chaotic dynamics dominate?  That is
    flat out wrong according to current knowledge.  There is no known
    quantum process which is chaotic.  The field of "quantum chaotics"
    consists entirely, at this point, of investigation of quantum systems
    whose corresponding classical system are chaotic and the repeated
    discovery that one way or another, the chaotic behavior is suppressed.
    The pattern is very strong, but nobody knows yet whether it is
    universal or even why it should be the common case.  Quantum systems
    are less well behaved than well-behaved classical systems, but they
    are much better behaved than poorly-behaved classical systems.
    
    					Topher

RE: .7 (Neilsen)
    
    >> There seems to be ... a finite mass in the universe
    >
    > This is still an open question.  Observation suggests that the
    > universe is infinite in space, and theory suggests (rather weakly)
    > that it is finite in both space and time.
    
    True, although the "rather weakly" is a matter of opinion.  However,
    a universe which is infinite in space does not need to contain an
    infinite amount of mass.  Just the contrary, the plausible model
    universes which are "open" (infinite in extent) contain less mass than
    those which are "closed" (finite in extent).  The relevant observations
    are those which do not find evidence of enough mass to "close" the
    universe.  The relevant picture is of a small, finite island of mass,
    exanding in an outer shell of photons and neutrinos at the speed of
    light, within an empty (containing only virtual particles of 0 total
    mass) ocean of infinite extent.
    
    					Topher

    Models exist because models reside in the organization of neurons and
    neurons exist.  models do not exist without people and people could not
    make models without neurons or similar.  Therefore, models have
    physical existance (in neurons.). Models do not exist separately from
    physical brains.
    Books hold records of models, not models.
    So the physical universe comprehends itself.
    It thinks, therefore it are.
    Tom

RE: .12 (Tom)
    
    Of course models exist.  But that they exist, as models, does not make
    their content "real".  Not unless you wish to accept the contents of
    dreams or of hallucinations induced by drugs as "real".  The concept
    of "two" and the concept of "infinity" exist as concepts -- that
    doesn't make them parts of the physical universe.
    
    I think that you are making an error in levels of abstraction.  Our
    ideas no more "exist" in our neurons than pressure "exists" in the
    individual molecules of a gas.  Concepts like "motive", "idea" and
    "concept" are not characteristics of a physical brain, but are rather
    approriate only in talking about that abstraction of brain behavior
    which we call "mind".  Mind is also an abstract model of a particular
    part of the physical universe (i.e., a human brain), albeit one which
    seems to be hard-wired into these particular symbol-processors and which
    is recursive in the sense that "model" (in the sense we are using it) is
    only meaningful within that model.  (The concept of "mind" seems to be a
    very useful one for social interaction, and there is a (IMHO) very
    plausible theory that intelligence arose in humans specifically to
    support the complexities of social interaction).
    
    						Topher

    I was trying to be provocative by taking a shortcut, and you proved my
    point. 8-)
    Tom

    Re:             <<< Note 1195.11 by CADSYS::COOPER "Topher Cooper" >>>
    
    This is getting even further afield.  I started a topic on the open
    universe, but I can't remember whether it was in ASTRONOMY or PHYSICS.
    We should probably continue the discussion there.
    
>    a universe which is infinite in space does not need to contain an
>    infinite amount of mass.  Just the contrary, the plausible model
>    universes which are "open" (infinite in extent) contain less mass than
>    those which are "closed" (finite in extent).  The relevant observations
>    are those which do not find evidence of enough mass to "close" the
    
    I believe this is wrong.  It is mass *density* and not mass which
    determines whether the universe is closed or open.  If you assume the
    universe is homogeneous (admittedly without much evidence) then an open
    universe may contain much more mass (perhaps infinitely more) than a
    closed universe.
    
>    universe.  The relevant picture is of a small, finite island of mass,
>    exanding in an outer shell of photons and neutrinos at the speed of
>    light, within an empty (containing only virtual particles of 0 total
>    mass) ocean of infinite extent.
    
    I've never found an explicit discussion of an open universe.  The
    discussions I have found seem to assume that the universe is
    homogeneous, and therefore if it open, it contains an infinite mass,
    distributed over an infinite space, at a density which is constant on a
    large scale.  Space itself is expanding at every point in this
    universe, so it is observationally similar to the universe we see.  And
    since it has existed for a finite time, we cannot now observe parts of
    space which are far away.

    Virtually all contemporary discussion of cosmology has as its base the
    Big Bang or one of its variants.  In general they do *not* assume that
    the Universe=the-space-time-continuum is homogenous.  Rather they
    assume (or derive) that the Universe=all-points-causally-descended-
    from-the-initial-singularity are homogenous.  In closed cosmologies
    these are not necessarily equivalent (generally are not) and in open
    cosmologies they are necessarily different.
    
    The size of the "matter-filled-Universe" is roughly the same in both
    (ignoring issues of defining simultaneity, etc.).  It is a sphere
    of radius in light-years equal to the time since the big-bang.  The
    specific cosmology will determine a bit how to interpret observational
    evidence to get an age, but that's fine tuning.  Roughly speaking,
    given a finite volume, deciding on the density is roughly equivalent
    to deciding on the total mass, subject to uncertainty of the value of
    the finite volume.
    
    This is modified somewhat by some inflationary cosmologies, which, in
    essence, postulate that the Big Bang occured after an initial
    inflationary epic in which space-time expanded much faster than light-
    speed.  The Big Bang, then, started out much larger than a point,
    perhaps much larger than our currently observable universe, but this
    modifies the scale only -- the matter-filled universe is still finite.
    (As I understand this stuff, which is not as well as I would like).
    
    					Topher