The Atlas of Reality

by Robert C. Koons,Timothy Pickavance

  That this is absurd is clear if we imagine a scenario in which there are two such entities. Suppose, for example, that there were two infinitely long rods, each with a circular face behind which the matter extends infinitely far into the distance. These two rods could be located along a common axis, and they could be moving toward one another at a constant velocity. At some point, the two circular faces will collide, and each will carry an infinite momentum behind it. What will happen at the collision? Each possible answer seems absurd. We might suppose that symmetry requires us to suppose that the two rods will bounce off each other, leaving the collision with an equal and opposite velocity. However, it is easy to imagine cases in which symmetry is no help. Suppose, for example, that one rod has a cross-section that has twice the area of a cross-section of the other. Then we would seem to be forced to say that the one infinite momentum is twice as great as the other infinite momentum, which seems to be utter nonsense. At least, when choosing between competing theories and models, physicists treat real infinities as something to be avoided.

  Perhaps, however, physicists are guilty of a mere prejudice against the infinite. In addition, even if they are right to avoid infinities in describing the actual world, it doesn't follow that physical infinities are absolutely impossible. There is a consistent mathematical theory, non-standard analysis, which takes seriously the possibility of infinitely large and infinitely small (or infinitesimal) real numbers. Since such non-standard real numbers satisfy all of the usual laws of real analysis, we could presumably use them to build a theory of infinite forces and masses that would obey the laws of physics. Alternatively, if we were to suppose that the idea of infinite real numbers is a mere mathematical fancy, we might still suppose infinite magnitudes to be possible by supposing that they obey different, less quantitative laws of physics.

  Perhaps the objection to infinite force is rooted in an objection to infinite acceleration and infinite velocity. Imagine an infinitely long pool stick striking a finite ball. The ball would experience an infinite force at the moment of collision. An object under an infinite force would, according to Newton's laws of motion, undergo infinite acceleration. Infinite acceleration would seem to bring an object instantaneously to an infinite velocity, but what sense can we make of an infinite velocity? Where is the object while it is moving infinitely fast? All over the place in a single instant? That seems to be impossible, assuming that objects cannot be wholly located in more than one place at a time. So, we seem to have good objections to infinite velocities and infinite acceleration.

  However, given Einstein's special theory of relativity, ruling out infinite velocity will not enable us to rule out infinite forces, since Einstein's theory tells us that an infinite force would accelerate a massive body instantaneously only to the speed of light. Nothing can go faster than that, no matter how much force is applied. If the universe contained an infinitely great force of gravity, the result would be that everything would move toward the source at the speed of light, disappearing into a massive black hole— an undesirable result, no doubt, but not metaphysically impossible.

  Still, we might conclude that the idea of infinite mass is dubious at best. If so, we have good grounds for denying the very possibility of an entity infinite in volume. If such an entity is impossible, there is also some reason to think that an entity finite in volume but infinite in mass is also impossible if the latter is composed of infinitely many point-masses. Suppose, for example, that we had an entity with finite volume but with varying density. If this entity is composed of an infinity of point-masses, there is no reason that the density can vary within it in any way that can be described mathematically. Let's suppose that the entity is a cube 1 meter long on each of its edges. The first half of the cube has a uniform density of 2 grams per cubic meter, the next fourth has a density of 4 grams per cubic meter, the next eighth a density of 8, the next sixteenth a density of 16, and so on ad infinitum. Each of these infinitely many parts of the cube has a mass of 1 gram. Hence, the cube as a whole has an infinite mass. If we want to rule out the very possibility of infinite mass, it seems that we need to rule out point-masses altogether. To do that, it seems we would have to rule out the possibility of an infinite number of actual places.

  Thus, we have some reason to suppose that any extended entity has only finitely many actual parts. Hence, it is impossible for an entity to have an infinite volume or an infinitely varying density. Consequently, every possible entity has finite mass. If, as seems reasonable, we suppose that infinite velocity is also impossible, then every entity will have some finite kinetic energy and momentum, avoiding the dynamical absurdity of colliding infinities.

  There is an obvious objection to this line of argument against infinitary composition. We have some reason to suppose that infinitely massive bodies are impossible. Why not stop there? Why go further and ban infinite composition? Why suppose that all entities with infinitely many actual parts are impossible? The answer to this objection involves an appeal to a familiar methodological maxim: avoid brute necessities (PMeth 1.2). Further, where possible, we should explain all impossibilities in terms of hypotheses about the essential structure of space and time. This is Structuralism (PMeth 3). The point of Structuralism can be brought out by considering patchwork principles (discussed in Chapter 16). Here again is Infinitary Patchwork:

  PMeta 5.2 Infinite Patchwork. If T is a class of types of events or processes, and for each member of T, it is possible for an event or process of type T to occur, and there is enough room in the history of the world to locate within it instances of each of the types in T without overlap in space and time between the instances, then it is possible for all of the types in T to be realized together.

  If a body can be composed of an infinite number of distinct, non-overlapping material parts, each with its own unique, unshared location, then that body and its parts provide a spatial framework that could be occupied (in other worlds) by bodies of steadily increasing density. So long as there is no upper bound to the possible density of a body, we can use Infinite Patchwork to find a counterpart of our infinitely divided body that has infinite mass in some possible world. If such an infinitely massive body is in fact impossible, then the Patchwork principle entails that there cannot be an infinitely subdivided body in any world, and so Finitism must be true.

  Aristotelian Finitism, the view that space and space-occupiers can never have an infinite number of actual parts, together with the assumption that mass-density can only take finite values, provides a structural account of why finite bodies cannot have infinite mass. This is much to be preferred to the view that the impossibility of infinite mass is a brute, inexplicable necessity.

  On this score, Discrete Finitists are in a somewhat stronger position than Aristotelian Finitists, since Discrete Finitists can plausibly claim that it is impossible for any region or any body to be smaller than some fixed lower bound, the minimum unit of space. Aristotelian Finitists want to place no fixed lower bound on the possible size of bodies. Any body, no matter how small, could be subdivided further. Instead, Aristotelian Finitists claim that it is impossible for a body to be actually infinite in its compositional complexity. This seems somewhat unprincipled. If we can subdivide bodies without limit, why suppose that it is metaphysically impossible for a body to be actually subdivided into an infinite number of parts?

  Aristotelian Finitists could respond by arguing that infinite subdivision of a body is impossible because of the structure of time. No process could complete an infinite number of sub-processes in a finite period of time. We will consider this sort of argument in more detail in Chapter 19.

  There is another route that Spatial and Material Pointillists could take. They can deny that extended bodies ever fill all of the points of any extended region. Instead, all extended things of necessity consist of a finite number of point-masses embedded within a continuous spatial region. The apparent impenetrability of the matter could then be explained in terms of forces of repulsion between
the particles, forces that increase exponentially as the particles approach one another in space. This was proposed by Roger Boscovich in 1763 (1966), building on the atomism of Isaac Newton.

  However, this hypothesis will explain the impossibility of infinite mass or momentum only if the existence of these repellent forces are themselves a matter of metaphysical necessity, that is, only if it is absolutely impossible for there to exist point-masses that do not repel each other in this way. This seems implausible. Even in the actual world, it is likely that some particles, such as photons, neutrons, and neutrinos, exist that do not repel one another in this way. If so, there would be nothing absurd in a situation in which an infinity of neutrinos exist, occupying every point within a region. It seems that we must instead posit some necessity inherent in the very nature of space that prohibits such a situation from happening. It appears that particles like protons are not in fact point-masses, since the concentration of their finite matter and charge into a dimensionless point would result in infinite density, with consequent problems concerning self-interaction. (For example, a charged particle will interact with its own electrical field. If the charge is concentrated in a dimenionless point, this results in a field of unbounded intensity in the neighborhood of the point. Interaction with this unbounded intensity results in paradoxical infinities of force and motion.)

  Even if it were the case that every particle repels every other particle with a force inversely proportional to distance, this would at most explain why it is impossible to compress a finite body into a condition of infinite density. It wouldn't explain why it would be impossible for an infinity of such particles to exist right from the beginning, or for all eternity, in such a space-filling, infinitely dense condition. Such a condition might seem absurd, since it would involve infinite mass and infinite potential energy, but we want to avoid a merely ad hoc prohibition of such infinite quantities. The repellent particles model fails to do so. (A similar problem faces Aristotelian Finitists: if bodies are potentially divisible without limit, why couldn't a body be actually divided into infinitely many parts for all eternity?)

  ARGUMENT AGAINST FINITISM The main argument against Finitism relies on the fact that our best scientific theories, from the time of Newton to the present, represent space as an infinitely divisible continuum through which bodies take differentiable paths. This argument is based on Scientific Realism:

  PMeth 2 Scientific Realism. Other things being equal, adopt the theory that implies that our best scientific theories are straightforwardly true, as standardly represented.

  Since our best physical theories make reference to infinitely many points, curves, and surfaces in space, Scientific Realism supports a metaphysical theory that entails that there really exists such an infinity of spatial entities. This scientific argument against Finitism presupposes that Finitists also embrace Actualism (12.1T), the claim that everything exists. If the Finitists are Possibilists (12.1A.1T) or Meinongians (12.1A.1A), they can argue that Finitism does imply that our best scientific theories are true, since they quantify over all potential spatial entities, of which there are indeed an infinite number. To refute Possibilist-Finitism, Infinitists will have to argue that our best scientific theories entail that infinitely many points, surfaces, and regions are actually in existence.

  Infinitists could argue that field theories do have such an implication, since they attribute causal influence to each point of the field, with the field itself being merely the aggregate of an infinite number of point-intensities. However, quantum field theory suggests that at very small scales, in the vicinity of the Planck length, physical fields must take on a discrete, chunky character, in order to avoid certain dynamic infinities resulting from self-interaction (the problem of re-normalization) (see Georgi 1989).

  18.3.2 Mathematical paradoxes, from Zeno to Tarski

  Second, there are arguments against Pointillism from mathematical paradoxes. The ancient Greek philosopher Zeno (as reconstructed by Skyrms 1983) poses the following paradoxical argument:

  A line segment consists of an infinite number of mutually disjoint parts.

  The concept of magnitude (length, area, or volume) applies to each of the parts.

  The parts of the line segment all have the same magnitude, either zero or positive.

  There are no infinitesimal magnitudes (all positive magnitudes are finite). (Archimedes's Principle)

  The magnitude of a whole is the sum of the magnitude of the parts, taken in sequence.

  Any infinite collection can be ordered into a sequence. (Zorn's lemma, which follows from the Axiom of Choice in set theory)

  The sum of an infinite series of zeros is zero, and the sum of an infinite series of equal finite magnitudes is infinite.

  Therefore, any line segment is infinitely long or has zero length.

  Modern measure theory rejects premise (5). It insists that the measure of a whole is equal to the sum of the measure of its parts taken in sequence only when the sequence is either finite or countably infinite (equinumerous with the counting or natural numbers, 0, 1, 2, etc.). Modern mathematical theory, following Georg Cantor, entails that there are uncountably many points in a line, a very “large” infinity to which the additivity assumed in premise (5) does not apply. However, it is not the case that we cannot make sense of the idea of the summation of an uncountable collection. Given the Axiom of Choice, the process of summing such a collection is straightforward, as Skyrms observes, and the resulting sums will in fact satisfy premise (7), which follows from (5) and (6). If we repeatedly add zero to itself, the sum will remain zero, no matter how long the sequence. The sum of uncountably many zeros is well defined and is, as Zeno argued, equal to zero. Consequently, there is some significant metaphysical cost to be paid for adopting modern measure theory, namely, the rejection of the very plausible premise (5).

  What about giving up premise (4), Archimedes's Principle? In order to do so fruitfully, we would need to have a coherent account of infinite and infinitesimal quantities. The only real candidate for such an account is Abraham Robinson's theory of non-standard analysis, developed in the 1960s. All previous theories of infinitesimals were beset with antinomies and contradictions. However, if we adopt Robinson's theory, we can still derive an unacceptable conclusion, since we can add premise (9), deriving a further absurd conclusion:

  The sum of an infinite series of Robinson infinitesimals is infinitesimal.

  Therefore, any line segment is infinitely long or has zero or infinitesimal length.

  THE BANACH-TARSKI PARADOX According to modern mathematics, building on the work on transfinite set theory by Georg Cantor, the set of points making up a finite segment of a line can be put into a one-to-one correspondence with all of the points in the whole of space! In this sense, there are just as many points in the segment as in any region of space, whether finite or infinite. Consequently, we cannot suppose that the mass or volume of a region corresponds in any way with the number of points in that region.

  However, we might suppose that mass and volume are a function of the number of points together with their arrangement in space. That is, it seems natural to suppose that any two congruent arrangements of points have the same volume. Similarly, two congruent arrangements of qualitatively identical material points should have the same mass and density.

  In addition, it seems plausible that if a region of space can be decomposed into a finite number of subregions, then the volume of the whole should be the sum of the volume of the parts. Likewise, the mass of the whole should be the sum of the masses of the parts.

  Surprisingly, if we put these two natural assumptions together, we get a conclusion that is demonstrably false. It is possible to disassemble a finite region into a finite number of parts (as few as five), move the parts through space without disturbing the mutual relations among the points involved (in other words, preserving congruency), and yet end up with a region twice as large as the original. Similarly, we could disassemble a material body into
five parts, rearrange those parts through rigid motions, and produce a new body with twice the volume and twice the mass of the original. The result is called the Banach-Tarski theorem. There are three spheres, A, B, and C, each with a radius of one unit. Sphere A is composed of five regions E1,…, E5, sphere B is composed of F1 and F2, and sphere C is composed of F3, F4, and F5. Banach and Tarsksi proved that if the decompositions are defined in a particular way, we can prove that E1 is congruent with F1, E2 with F2, and so on. Thus, sphere A can be decomposed into five parts, and those five parts can be re-arranged in space (the first two parts in one group, and the other three in a second group), in such a way that two spheres are produced, each equal in volume to the original. The first sphere has “magically” doubled in volume!

  The Banach-Tarski theorem gives rise to the following argument against Pointillism:

  Any set of points in a specific geometrical arrangement (any “region”) has a definite magnitude (volume).

  If two regions are congruent, then they have the same measure. (Congruence-invariance)

  Measure is finitely additive: the measure of the fusion of a finite collection of disjoint regions is the sum of the measures of the members.

  There are regions E1,…, E5 and F1,…, F5 such that E1 and F1 are congruent, E2 and F2, and so on, E1,…, E5 compose a unit sphere, F1 and F2 compose a unit sphere, and F3,…, F5 compose a unit sphere. (Tarski-Banach theorem)

  Therefore, one unit sphere has the same volume as the sum of the volumes of two unit spheres.

  The standard mathematical solution of the paradox is to stipulate that the sets produced by the construction are unmeasurable. This constitutes a denial of premise (1). The paradox demonstrates that it is impossible to assign any value to any of the five parts. Mathematicians have developed an entire branch of mathematics, Lebesque measure theory, designed to cope with this phenomenon. Lebesque measure theory puts precise mathematical conditions on the class of sets that are measurable, requiring that all other sets of points be treated as lacking volume, and thus also lacking mass or charge. Common sense tells us that whenever an extended region of space can be divided into a finite number of subregions, the volume of the whole must be the sum of the volume of its parts. Measure theory denies this. The additivity of volume only makes sense when the regions correspond to measurable sets of points.