3. Angel’s Dichotomy
Angel (2001; cf. Priest’s 1999 version of Benardete’s “paradox of the gods”, 1964: 259-60) considered a three-dimensional Euclidean space containing À0 rigid and impenetrable spheres, which interact solely through elastic collisions (which might involve unoccupied accumulation points, as above) governed by Newton’s laws of motion. One particle, M, had unit mass and unit radius, and up until a certain time, t0, it was stationary, with its centre at (0, –1) in a certain plane. In that plane, M was approached, from the positive-y-direction, by À0 smaller unit masses arranged in two endless sequences (for nonnegative integers n), NPn on the negative side of, and PPn on the positive side of (and each limiting towards) the y-axis.
y-axis
NP1 NP2 NP3 … … PP3 PP2 PP1
… 1 …
x-axis
M
At t0, the centres of the smaller masses all lay on the line y = 1, spread out within a unit of (and with their radii tending to zero towards) the y-axis. For each n, NPn and PPn were of equal size. And for each n, NPn and PPn had the same velocity, (0, –2n), so that the smaller masses were all moving (faster nearer the y-axis) towards M.
Because of that unbounded set of speeds, at any time later than t0 there would be a paradoxical situation. No pair of smaller masses could collide with M, since for any such pair, say NPm and PPm, the faster pair (nearer to the y-axis) NPm+1 and PPm+1, would have arrived at M’s position earlier, knocking M away in the negative-y-direction (with M then going too fast for NPm and PPm to catch up with it). So despite the spatial separation of M and the smaller masses at t0, there must be some change in some particle(s) at that time (a before-effect of the collision set-ups, between each pair of smaller masses and M).
Angel mentioned the interesting possibility that his w*-sequence of collision set-ups would cause the before-effect of an action-at-a-distance at t0, which would mean (since only contact interactions were allowed, and before-effects have to be possible if they are to occur) that his scenario was inconsistent; but Perez Laraudogoitia (2003) asserted that a contact resolution was possible, due to the possibility of self-excitations (see below).
For the consequent debate, see later issues of The British Journal for the Philosophy of Science—I’m not especially interested in the details of such (doubly unrealistic) Newtonian mechanics with À0 particles; but note that the more physically implausible Angel’s scenario comes to seem (see below) the more likely it is that, had such scenarios been considered at the end of the nineteenth century (i.e. after Cantor’s transfinites had begun to dominate the calculus, but before the discovery of quantum mechanics), set-theoretical foundations would never have arisen (see my unpublished for a simple reason why they would not have arisen within modern physics).
To begin with, while the purely mathematical theory of Newtonian mechanics with À0 particles may be defined almost arbitrarily, the most natural response to Angel’s paradox would, were such a scenario being taken seriously as physics, be Angel’s because prima facie, although the action at t0 is (instantaneously) at a distance, it is not unreasonable to think of the motion of an object in terms of its worldline (i.e. all the bits of space-time that it ever occupies). Even normal motion requires that the space about to be moved into is (soon enough going to be) vacant, so some degree of “space-time weave” seems to be intrinsic to the concept of motion; and we might expect extremes of such a “weave” to occur when we have À0 particles. And note that the limit of the set of the worldlines of the smaller masses is at zero distance from the worldline of M at t0, whence this response may be no worse than the legitimate before-effect collisions mentioned at the end of §2.
By contrast, Perez Laraudogoitia’s induced self-excitations are hardly the kind of behaviour that we would expect of À0 objects, even were such numbers of objects possible. To begin with, self-excitations (and in particular the kind that I call self-creations) are basically time-reversed momentum-loss (and particle-loss) scenarios; and Perez Laraudogoitia (1996) gave the following (rather elegant) example of momentum-loss.
Consider a unit mass approaching (from the positive-x-direction) an w-sequence of stationary unit masses, at x = 1, x = ½, x = ¼ and so forth (this being the scenario mentioned at the end of §2). The momentum of the approaching mass gets transferred, via an w-sequence of collisions, towards x = 0, where it vanishes. And temporal inversion gives us the spontaneous generation of momentum from x = 0—a self-excitation.
With two such sequences facing each other, forming an (w* + w)-sequence, increasing rates of such self-excitations could lead to those particles approaching unbounded speeds in a finite time (and within the original finite volume bounded by the two open ends). Were the limit of such an unbounded increase in speed the attaining of non-existence (e.g. because such a particle could not be at any point without its worldline being discontinuous), then another temporal inversion would give us À0 particles appearing from nowhere, within any finite volume of space—a self-creation.
And if such self-creating “target particles” (cf. Perez Laraudogoitia 2003: 322-3) appeared in front of the smaller masses of Angel’s scenario (at appropriate times after t0), they could absorb the momenta of the latter via contact interactions (since contact with the accumulation points of the target particles is allowed). Furthermore, since Angel’s scenario would be paradoxical (i.e. impossible) without such self-creations, we arrive at Perez Laraudogoitia’s (2003: 324) resolution—the situation at t0 would induce (i.e. cause) such self-creations.
Nonetheless, there are a couple of reasons why such induced self-creations are physically implausible. Firstly, why should a particle attaining an unbounded speed vanish, rather than have its worldline become perpendicular to time (see below)? But even were self-creations possible, it would still be implausible that Angel’s set-up could cause them to occur, as the following analogy indicates.
Suppose that Benardete’s À0 boards were (more realistically) made of stuff that decays, usually over very long periods of time. Were the decay mechanism indeterministic, there would be some nonzero probability of any particular board decaying completely during any nonzero time period, whence each board might have decayed before the man reached where it was. Therefore a before-effect (stopping the man from reaching any of the boards) need only occur in those cases where such an unlikely sequence of decays did not happen—and conversely (by analogy with Perez Laraudogoitia’s induced self-creations), given that such before-effects are impossible for some reason (e.g. were contact with a board required, for the man’s halting), the decay rates would have to accelerate in such a way (given various other assumptions, as is usual in such thought-experiments).
But in order for that conclusion to mean anything (other than that we were simply ignoring the more interesting cases when such an unlikely sequence of decays did not happen), we would have to assume that the given situation could cause such an acceleration of the decay rates; and so if we assume—rather realistically—that the layout of the boards, together with our approach towards them, cannot cause the decay rates to accelerate, in such a way that they all decay before we reach them (as a mysterious after-effect of our reaching the open end of them), then a before-effect must occur (assuming that it can) in those cases when that after-effect does not occur—which would be almost all of them (given various other assumptions, as usual). The after-effect, for all that it may be a logical consequence of a scenario’s precise description, therefore seems unnatural, whereas the before-effect, for all that it might fail to fit the given definitions (e.g. of contact), seems relatively natural.
Of course, the decay of the boards is an everyday phenomenon that goes beyond the stuff of classical physics, and so our intuitions about w-sequences of such things are not to be trusted (and the whole issue of causation is hardly a model of clarity), but clearly such inductions ought to be viewed with suspicion, at least when we are considering the classical physics rather than the classical mathematics. Were we taking such scenarios seriously as hypothetical physics (e.g. within the counterfactual history mentioned above), we may therefore have rejected Angel’s scenario as badly defined, and redefined ‘contact’.
Even so, we might eventually have rejected À0 because of the physical implausibility of some such scenario; and even for Angel’s scenario, at t0 M was a finite distance from each of the smaller masses, so we can hardly think of them knocking it away in the usual way of massive things. Of course, we ought not to expect the ordinary (since the smaller masses form an endless sequence, whose open end is approaching M), but furthermore M seems to be knocked out of existence at t0, which is certainly counter-intuitive.
Now, if the smaller particles were to collide with M simultaneously (e.g. were M close to y = 1 at t0, and shaped slightly differently), would M simply vanish, or would it vanish at spatial infinity? That it should just vanish seems physically implausible, prima facie, but on the other hand such vanishing may only be the temporal equivalent of a spatial halting (due to the absence of any available points, whose occupation would not make its worldline discontinuous). Still, a principle of the conservation of mass might require that M vanish with an open end to its worldline, and M cannot be non-existent at t0 because it is defined to exist then.
But perhaps M could move away at infinite speed, and vanish at spatial infinity—more precisely, at t0 its worldline would become perpendicular to time, and infinitely long (with an open end at spatial infinity). Note that that would not be similarly ruled out by the definition of the set-up (e.g. were it stipulated that at t0 M was a finite sphere, and hence was not infinitely extended) were our resolution to redefine ‘contact’, so that there was a collision at t0, because of course one cannot assign to masses arbitrary positions and speeds both before and after a collision (since the later speeds may follow from the earlier ones). Also note that if unbounded speeds did correspond to worldlines being perpendicular to time (rather than vanishings) then self-creating sets of particles (as considered above) would not be possible, so that there would then seem to be no contact (with the original definition of ‘contact’) resolution, which would mean that Angel’s set-up was badly defined.
Nonetheless, it may be that a particle simultaneously occupying a continuum of positions would (in our counterfactual history) have been regarded as too unlike motion (at least within classical physics) for such a resolution to be plausible.
So perhaps, whilst allowing the usual definitions and À0 particles, we would have outlawed such arrangements of particles (such as Angel’s) as give rise to contradictory scenarios (see Fletcher 2007: 575). But again, it is highly questionable whether that could be justified within the classical physical world-view.
Alternatively, since the man (of Benardete’s scenario) would plausibly crash through the first À0 boards (which are all thinner than some very small amount), we might redefine the scenario so that the particles were penetrable. And so on—ultimately the issue becomes how well such possibilities model physical reality, and so such hypothetical details (of infinitary Newtonian mechanics) may not now be very interesting, modern physics being quantum-mechanical (see my unpublished).
Still (regarding the likely rejection of À0 within classical physics), it may be worth emphasising the counter-intuitiveness of even such simple scenarios as the man getting crushed beneath À0 unit masses, or a unit mass stopping at (or bouncing off) the open end of an w*-sequence of unit masses.
E.g., we would surely expect the net mass (or charge) of À0 unit masses (or charges) to be À0, whereas unfortunately their net mass (or charge) is undefined—in the first place, À0 is hardly the right kind of number, for a mass (or charge). Of course it is fine to have, in our mathematical models, gaps where things are undefined (and to fill such gaps in any way that seems convenient), but there are surely, in reality, no such gaps (and presumably only one way in which there aren’t).
Surely the halting (or rebounding) mass would, were that scenario modelling something physical, have stopped because it was faced by something massive—something with all the determinateness that we would expect of a transfinite collection of classical masses, and of which we might sensibly say that it was very massive, that its net mass was greater than any finite bound. Furthermore, while the halting mass would, intuitively, be interacting with a very massive set, can an abstract object such as a set be massive? And anyway, could the particle have interacted with that set (massive or otherwise), or would its singleton have had to? The latter seems tidier, but if so then a collision between two particles ought to be regarded as an interaction between singletons, whereas intuitively it would be the physical particles that would be interacting.
And what of particles whose velocities are tending towards an unbounded limit? Do they vanish (with open or closed ends to their worldlines?) or instantaneously occupy a continuum of positions, or do they teleport (with open or closed ends to their worldlines?), or do something even stranger? We may easily visualise an unbounded speed as a vertical line, on a graph of position against time (where a horizontal line is a stationary particle), even though that would not give us a mathematical function, but how plausible is that picture physically? (Incidentally, note that whether or not they are physically plausible, although such undefined velocities, and the undefined masses considered previously, are both simply unbounded limits in modern mathematics, they seem to be structurally different.)