Accepted
Version Before Final Changes (22 Aug 2002)
This article
is in five parts:
(i) A
simultaneous collision that produces paradoxical indeterminism (involving À0 hypothetical
particles in a classical three-space) is described.
(ii) By
showing that a similar paradox occurs with long-range forces, the underlying
cause is seen to be that collections of such objects are assumed to have no
intrinsic ordering. The resolution of allowing only
finite numbers of particles is defended (as being the least ad hoc) by
looking at:
(iii) w-sequences, in the context of a
very basic supertask, and
(iv) w*-sequences (reversed w-sequences), in the form of paradoxical results from the recent
literature.
(v) The possibility of supertasks generating À0 particles is
considered.
(i) When we consider the behaviour of À0 hypothetical physical objects (within classical three-dimensional
space) we use results from analysis such as that the combined effect of À0 vectors is their mathematical sum, that is the limit of the sequence of
partial sums of the vectors. Infinite sums are dependent upon the ordering of
their terms: for example, the limit of the partial sums of (–1)nn(–1),
for natural numbers n, can be any real number, which one it is depending
upon the order of summation. However, we
usually assume that objects (in a hypothetical objective space) are not
intrinsically ordered, but simply coexist. When each (–1)nn(–1)
is the value of something physical, so that there could be an objective
summation by physical conjunction, the existence of the other mathematical
possibilities (in addition to the natural
sequence implicit in the (–1)nn(–1) notation, chosen for simplicity) could conflict with
this assumption of simple coexistence.
The
following collision in Euclidean three-space involves
hypothetical (but not necessarily rigid) spheres (of density 3/(4p), for mathematical simplicity) whose only interaction is that they
stick together upon collision, so that kinetic energy is not necessarily
conserved. They will, however, obey a principle of the conservation of momentum
in the following form: the sum of the momentums of the interacting particles
before the collision must be equal to the sum of their momentums after the
collision. Moving in a plane, À0 particles head towards a single sphere
of mass 13/14 and centre c, and
simultaneously collide with it so that they merge into a single unit mass. For
each natural number n, a particle of radius rn = 1/2(n+1) approaches c at an
angle of qn = p(2(n+1)–3)/2(n+2) relative to some zero direction from c,
in the collision plane. This particle has a speed of vn = 1/(4n.sinqn.rn3).
Particles also approach c at angles of (p–qn), with
the same rn and vn. Each pair with the same
index n thereby contributes momentum of 1/(2n) in the direction (3/2)p. Also approaching c, at angles of (p+qn) and
(2p–qn), are particles with the same rn
but with speeds un = 1/((4n–2).sinqn.rn3), and these pairs
contribute momentum of 1/(2n–1) in the direction (1/2)p.
After the collision the total unit mass
will move with some (possibly zero) component of velocity in the (3/2)p direction of the collision plane (a disappearing mass could be assigned
an infinite velocity). Because momentum is conserved, this is also the value of
the sum of the components of the momentum vectors in the (3/2)p direction before the collision, these being (–1)nn(–1), for natural numbers n
(a divergent sequence in the case of infinite velocity). Because the masses
have this total momentum before the collision, the ordering corresponding to
this value of the sum cannot be imposed upon them by the collision itself. It
might not even depend upon there being a collision, in the sense that in a
similar set-up (wherein additional masses interacted to prevent the above
collision, say) there would be the same intrinsic ordering, the same total
momentum.
So
we might expect the summation of the momentum vectors before the collision to
be the cause of the resulting velocity. Perhaps À0 particles have some ‘hidden ordinal’ behind their transfinite
cardinality? The obvious one would be given by the magnitudes of some physical
property, such as mass or momentum (these giving the same ordering as the
natural one that is implicit in the notation of this example), but there seems
to be no objective reason why any such choice would be the correct one. So,
whether all similar masses would move with a similar velocity after any similar
collision is also unclear. But it is clear that any finite subset of particles
would also have an intrinsic ordering, if the infinite collection did, even if
it did not in itself need one. Since this might conflict with our basic notion
of a finite cardinality, we might question whether we should allow the
possibility of having À0 of these particles. There are
then two issues: whether allowing À0 objects is otherwise
unproblematic (which is the subject of part (iii) below), and just how flexible
our notion of cardinality, of a finite set of objects, is (which is too wide a
subject for this article to consider).
(ii) Instead of constant density and unbounded
speeds, we could consider constant speeds and unbounded densities for the
incoming particles and obtain a similar paradox. The simultaneous collision
would have to be an elastic one between rigid particles. If the particles
arrived at the central sphere on two circles, at the top and bottom of it, with
the denser particles towards one vertical plane through the origin, then they
would not inhibit the indeterminate nature of its motion in that plane’s
perpendicular direction. And instead of unbounded densities we could have
needle-shaped particles (pointing towards the central mass) of unbounded
length, giving both bounded speed and bounded density. And, of course, an
elastic collision between rigid particles with unbounded speeds and densities
could be described in which the total momentum was similarly indeterminate.
Similar
vector conjunctions can occur in other physical models, just so long as those
models allow actual infinites of directed quantities, so the cause of this
paradoxical behaviour is not the idealized physics of classical collision
mechanics. Consider a Euclidean
three-space wherein the interactions include the classically gravitational:
each mass m generates a gravitational force, at a distance r from
its centre of mass c, which is a vector of magnitude Gm/r2
directed towards c, where G is a constant. Pairs of small
spheres, of unit mass, can be positioned symmetrically about an x-axis
in order to obtain a value for the x-component of the gravitational
force at the origin of G times the sum of all the (–1)nn(–1),
and finite y- and z-components.
For
example, (using Cartesian coordinates) masses with their centres at (–1, +1, 0)/21/4 and (–1, –1, 0)/21/4 give a net x-component of
force at the origin of –G. Similarly, masses at 31/2(–1, 0, +1)/21/4 and 31/2(–1, 0, –1)/21/4 give a net x-component of force
at the origin of –G/3. Other particles can be positioned so as to give
the desired x-components, without them having to get smaller to fit into
the three-space, because the number needed to generate forces between –G/r
2 and –G/(r+2)2 (both of the form –G/(2n–1))
increases as 4r, whereas the volume of a hemispherical shell between
radii of r and (r+2) increases as 12r2 (and
similarly for forces of the form –G/2n).
Although other force fields
could be superimposed without losing the paradoxical indeterminism, for
simplicity I shall assume that there are no other forces. Now, any small unit
mass that was at the origin would either accelerate away or not, and the
specific value (possibly zero) of this acceleration is also one possible value
of the gravitational force at the origin, corresponding to one particular order
of summation of the force vectors. If we had also stipulated that any
acceleration a of a mass m must be due to a force of magnitude ma,
then such indeterminism would become contradictory, unless there were ordinals
hidden behind cardinalities of sets of objects. A similarly indeterministic
electrostatic force at the origin would be caused by equal charges occupying
these same positions, of course. These hypothetical force fields need not propagate
instantaneously (that is, the particles do not have to occupy the appropriate
positions simultaneously) since a finite speed of propagation would only impose
an ordering if the sources of the fields were collinear.
Yet another model for this
paradox could involve fluid flows into and out of a reservoir, which fills or
empties at an indeterminate rate. Each such model requires the type of ‘hidden
ordinals’ to be specified somehow, in order to complete its mathematical
description, so the resolution of assuming their existence might begin to seem ad
hoc. And after all, there is a vast range of hypothetical matter-types,
beyond those considered here, with which À0 objects in a
three-space would actually have a well-defined finite total momentum (for
example, and whether this interacted in a paradoxical way or not) only if there
was a hidden ordinal associated with their mathematical sum. We could instead
explain this, along with all of the other well-known counter-intuitive
behaviours resulting from infinite numbers of objects, in terms of that very
feature: infinite numbers of objects.
(iii) Even something as elementary as repeatedly
putting non-interacting balls into a box can be counter-intuitive if we allow
ourselves À0 balls.
Repeated endlessly (starting with an empty box and a plentiful supply of balls)
we would have one ball in the box, then two, and so on. Each successive
ball-addition being done in half the time of the previous one, then after twice
the time it took to put the first ball in the box there would be À0 balls in the
box. We would have an (w+1)-sequence, for the number of balls
in the box: 1, 2, 3… À0. This is a consequence of single additions; at no stage
are À0 balls put
into the box in one operation. The kinematical possibility (in the sense of
finite accelerations and a maximum speed) of putting À0 balls into a
box was shown in Allis and Koetsier [1995], in the context of the
Littlewood-Ross paradox.
But
now consider the task of removing a ball from this box of À0 balls.
Removing one would leave À0, and so on. Upon completion of this supertask there
could be an empty box, since the subset of the plentiful supply of balls that
initially went into the box would already have behaved in such a manner. The
number of balls would stay the same until we were no longer removing them, when
it would suddenly be zero. Counter-intuitiveness arises because if it had been
a large finite number suddenly becoming zero, then that would have been because
they were all removed at that time. Also, for this (w+1)-sequence, the intermediate states of the box (between there
being 0 and À0 balls in the
box) are completely different to those for the (w+1)-sequence
given by the additions. The appearance of two separated collections of
intermediate states, with catastrophic leaps between them of À0 balls, might
be caused by an unrealistic stipulation, of the completion of an endless
sequence of actual processes, but might also arise from considering the
behaviour of a finite number of balls.
There
are À0 natural numbers
by definition, but that physical objects are sufficiently like natural numbers
to allow us À0 of them is
less clear, however. How the sequences given by ball-additions and
ball-removals have different intermediate states is reminiscent of how every
single natural number is in the first (no more than) 0% of the totality of them
(in their natural ordering), and whilst that property is a reasonable one for a
potential infinity (of an endless sequence) it can seem less plausible as one
possessed by an actual collection of objects. Clearly À0 is rather
insensitive to the removals of the balls that it numbers, whereas countable
ordinals are too sensitive to rearrangements. A shepherd with an infinite
number of sheep would want them to have a ‘number’ that was sensitive to the
loss of some of them, but not to them milling about. Those two desirable
properties would be possessed by an infinite hypernatural number of
things, but unfortunately we put a transfinite cardinal number of balls
into the box. In this context, perhaps the hidden ordinals that À0 objects seem
to require are just one more symptom of the inappropriateness of the number À0 for a number
of objects.
(iv) There is, of course, a (1+w*)-sequence from À0 to 0 that does go through all the natural numbers, which
is most easily described as the time-reverse of the sequence for
ball-additions. The number of balls in the box for this is: À0, …2, 1, 0.
But there is also a (1+w*)-sequence corresponding to the
time-reverse of the ball-removals. That would be a way of putting À0 balls into a
box, one-by-one, so that, at any time after starting to do it, there
would already be À0 in the box,
and one would merely be about to put the final finite number in: 0, …À0, À0. This is only counter-intuitive, of course, but a more
implausible situation is easily constructed around it.
If
the balls each had a unit mass, for example, and there was a person underneath
the box, then at all times after starting this sequence of ball-additions they
would have been crushed by an infinite weight. But at the instant of death the
box would have been empty. For each single ball, before its addition the person
would already have been crushed by an infinite weight, so we need only
stipulate that death must occur at some such instant of addition for this to
become contradictory. If the box had been a person, and the balls been
poisonous, there would have been a (physically less plausible) situation where
the person could not be killed by any ingestion, since they would already be
full of poison, and yet they would have to be poisoned for the same reason.
This situation resembles the paradoxes of Hawthorne [2000], counter-intuitive
discoveries about the nature of change, which may not be entirely palatable in
themselves.
This
sort of (1+w*)-sequence also lies at the heart of
paradoxical supertasks like the one described in Angel [2001]. The universe of
Angel [2001] was a three-dimensional Euclidean space, containing À0 perfectly
rigid spheres interacting solely through elastic collisions governed by
Newton’s laws of motion. One particle, M, had unit mass and unit radius, with
its centre at (0, –1) in terms of the Cartesian
coordinates (x, y) of a frame of reference which had M stationary
before a certain time t0. M was approached from the positive y
direction by À0 smaller
unit masses (which I shall call here the small particles) arranged in
two endless sequences, NPn and PPn (for
nonnegative integers n), on the negative and positive sides of the y-axis
respectively. These sequences of trajectories had as their limit the y-axis,
the radii of the small particles tending to zero in such a way as to allow them
each to collide with M. The velocities of particles NPn and
PPn were both (0, –2n), and they were of
equal size. At t0 the centres of the small particles all lay on the line y = +1.
Because
of the unbounded speeds, at any time later than t0 there
would be a paradoxical situation: if M collided with a pair of small particles then it must have collided with a
pair earlier instead (since each such collision knocks M out of the way of
later-arriving small particles), and yet there is no earliest pair for it to
have collided with. So, despite the spatial separation of M and the small particles at t0, these
potential collisions must be avoided by some change in the momentum of some
particle(s) at that time. The w*-sequence of normal, yet
contradictory, collision set-ups is avoided by an emergent
action-at-a-distance, at t0. One solution would have been to
limit the possible numbers of objects to the finite, of course, but the more
interesting solution was to discover this emergent possibility, evidence of a
“space-time weave even in Newtonian collision mechanics” (Angel [2001], p.
357), which some may regard as a paradoxical notion, of course.
(v) One reason for us to allow À0 objects would
be if a supertask could split up a single continuous object into À0 bits. A
simple basis for an object-splitting supertask might be a ball bouncing with
geometrically reducing height, so that it bounced an infinite number of times
in a finite time. The ball could cause the splitting of some thin object upon
which it was bouncing, each time it bounced, for example. That possibility
presupposes assumptions (such as that matter could be perfectly continuous)
which might be unrealistic, but even with hypothetical universes, a desire to
limit the numbers of things to the finite could be based upon w-sequences not being actually completable (that is, if they were
innately only potentially infinite), in which case we could not make the usual
leap from being able to always perform a next task, to being able to perform a
supertask.
The
bouncing ball supertask is said to give “as compact an illustration as we can
expect of the logical consistency of completing an infinity of acts in a finite
time,” (Earman and Norton [1996], p. 235). The time-reverse of the ball coming
to rest would involve the ball beginning, not to rise above the floor, but to
have already bounced an infinite number of times, so this sequence is
counter-intuitive. Although “the issue is not whether the idealized ball could
be realized in our world” (ibid., p. 236), real quasi-rigid balls do stop
bouncing in a conceptually simple way, after a finite number of bounces, but
perhaps after the bounces have become imperceptible, and it is perhaps this
which makes the supertask so compelling. But consistency depends upon the type
of universe in which the ball is bouncing, and may also require such things as
hidden ordinals, or a Newtonian space-time weave, if this supertask could be
given an object-splitting function.
Laraudogoitia
[1998] gives an example of a higher-order supertask, which may begin in an
empty (or partially empty) three-space, not unlike those of section (ii) above,
and which results in it becoming spontaneously occupied by À0 objects. So
it would not be sufficient to be merely considering a finite number of
particles in such a hypothetical universe, the transfinite numbers of particles
would have to be impossible there. That would be the case if À0 was a
potential infinity, applicable to numbers but not to objects, and there would
also then be no possible object-splitting supertask, even if matter was
continuous. The assumption that such supertasks are impossible may be
more reasonable than some of the consequences of allowing À0 objects. If
the unacceptable properties of those universes follow clearly from assuming À0 objects to
exist in an otherwise acceptable universe, then it seems ad hoc to say
that those are problematic universes, rather than problems with having À0 objects.
References:
Angel, L. [2001]: ‘A Physical Model of Zeno’s Dichotomy’,
The British Journal for the Philosophy of Science, 52, pp.
347-58.
Allis, V. and Koetsier, T. [1995]: ‘On Some Paradoxes of the
Infinite II’, The British Journal for the Philosophy of Science, 46,
pp. 235-47.
Earman, J. and Norton, J. D. [1996]: ‘Infinite Pains: The
Trouble with Supertasks’, in ‘Benacer-raf and his Critics’, edited by Morton,
A. and Stich, S. P., Oxford: Blackwell.
Hawthorne,
J. [2000]: ‘Before-Effect and Zeno Causality’, Noûs, 34, pp. 622-33.
Laraudogoitia,
J. P. [1998]: ‘Infinity Machines and Creation Ex Nihilo’, Synthese, 115,
pp. 259-65.