Atomism and Infinite Divisibility





Epicurus (341-270 bc)

Epicurus's argument regarding atomism and divisionism is similar to Aristotle's, except that Epicurus arrives at the opposite conclusion. Furley distinguishes among three kinds of indivisibility. They are physical (atom), theoretical, and perceptual. It is Furley's contention that Epicurus's Letter to Herodotus begs the question regarding the atom.

I claim that the passage to be discussed offers no argument at all for the existence of a physical minimum, but assumes it.(1)

But Furley then summarizes Epicurus's premisses and includes a quotation of Epicurus's reasoning.

Nothing comes into being out of nothing or passes away into nothing, and the universe is a closed system -- it has no relations with anything outside it. The irreducible contents of the universe are bodies and space; everything else can be reduced to these. The bodies in question are "physically indivisible and unchangeable, if all things are not to be destroyed into non-being but are to remain durable in the dissolution of compounds -- solid by nature, unable to be dissolved anywhere or anyhow. It follows that the first principles must be physically indivisible bodies".(2)

According to Furley, the argument goes as follows:

Real things cannot be destroyed into 'non-being'; but unless there were a limit to physical divisibility this is what would happen; there therefore must be a limit to physical divisibility.(3)

The argument expands not unlike Aristotle's presentation, but is much simpler because the distinction between place and point is not involved.

  1. No real thing can pass-away into non-being (by division or otherwise).
  2. If something is infinitely divisible then it can be divided into non-being.
  3. If something can not be divided into non-being then it can not be infinitely divisible. (Contrapositive of 2.)
  4. No real thing can be infinitely divisible. (by 1. & 3. using quantitative logic)

An examination of premiss 2 suggests a flaw. When something extended is divided, the parts are extended. When these extended parts are divided, their parts are extended. There is no limit to this process. Since having no limit to the process is what we mean by "infinitely divisible", we always have extended parts. As the number of times an extension is bisected increases, the limit of the size of the remaining extension is indeed zero. But at every stage in the process what is yet to be bisected has non-zero extension. At no stage will dividing a non-zero extension yield zero extension.

There is a special case: removing the end-point of a closed interval removes a "piece" with zero extension, but the remaining part still has the same extension as the original. But this operation can only be performed twice -- once for each end of the line segment. Cantor examined continuing the process by removing individual points in a line. This extended process can continue for countably many removals and still not diminish the extension of the original segment. (Such a set of points as was removed is known as a set of measure zero.)

If it be argued that the removed single points somehow pass away into non-being, the extension of the remaining parts is not diminished at all. Hence even allowing points to pass away into non-being does not cause the object under consideration to pass away into non-being.

In the former case, dividing an extended object into extended objects does not cause the object to pass away into non-being even if the division process is continued to infinity. A trivial proof by mathematical induction on the Nth bisection shows that for all N, the parts are extended.(4) Consequently, it is an error to conclude that that which is infinitely divisible is divisible into non-being. We may conclude that Premiss 2 is false.

Now let us consider the case of a minimum theoretical quantity (idea or conception), as Furley translates Epicurus' "Letter to Herodotus".

(A) Moreover one must not suppose that in the limited body there are infinitely numerous parts, even parts of any size you like.(5)

(B1) Therefore we must not only do away with division into smaller and smaller parts to infinity, so that we may not make everything weak and in our conceptions of the totals be compelled to grind away things that exist and let them go to waste into the non-existent,(6)

(B2) but also we must not suppose that in finite bodies you continue to infinity in passing on from one part to another, even if the parts get smaller and smaller.(7)

Furley interprets Epicurus's argument as meaning that the alternative to theoretical atomism is essentially infinite regress of thought, which is unacceptable.

We must reject infinite divisibility, [Epicurus] says, for otherwise we should make everything weak -- that is to say, when we tried to get a firm mental grasp . . . on the atoms, we should find them crumbling away into nothingness. Every time we thought we had arrived at the irreducible minima, we should have to admit that even these minima are divisible. And so our search for the reality of the atoms would be endlessly frustrated.(8)

The weakness is motivation for rejecting infinite divisibility. The impossibility of completing an infinite sequence of contemplation of parts is grounds for rejecting infinite divisibility. Here's how I see the argument expanded.

  1. We clearly comprehend a whole finite object.
  2. To comprehend a whole object, we must comprehend its parts.
  3. If its parts are infinite in number, then we cannot complete a sequential process of comprehending each part.
  4. Therefore, we cannot comprehend its parts.
  5. Therefore, we cannot comprehend the whole object.

According to Furley, both arguments are theoretical: One deals with what would be left after an infinite number of divisions; the other deals with how such a thing might be comprehended. He suggests these correspond directly to two of Zeno's arguments.

(C1) For when someone once says that there are infinite parts in something, however small they may be, it is impossible to see how this can still be finite in size; for obviously the infinite parts must be of some size, and whatever size they may happen to be, the size <of the body> would be infinite.(9)

It is not obvious that the "some size" that the infinite parts must have does not have zero as a limit. For the argument to hold, "some size" must have a limit greater than zero. The process of bisection reduces the size by half and has a limit of zero. Although every bisection starts with something of "some size" it yields parts which still have "some size". And each bisection yields parts which have "some size", there is no non-zero limiting size. This argument is infected by question begging in assuming that there is a positive limit to "some size" (atomism).

Epicurus suggests that a theoretical or "cognitive" minimum can be conceived of as "next" to something similar, and that this sequential, one at a time, cognition transverses the finite body.

(C2) And if the finite body has an extremity which is distinguishable, even though it cannot be thought of in isolation, it must be that one thinks of the similar part next to this and that thus as one proceeds onward step by step it is possible, according to this opponent, to arrive at infinity in thought.(10)

But by using the term 'next' he introduces the atomic perspective. Furley notices a difficulty but fails explicitly to note the question begging nature of the assumption implicit in the notion of "next".

We are considering someone's suggestion that there are [an] infinite [number of] [parts] in a finite body. Starting from one edge of the body we imagine a minute part of it, 'the extremity', inconceivable in isolation from the body. If we think of the part next to this extremity, we must necessarily think of another distinct part similar to the extremity itself. But according to our imaginary opponent, there are in our finite body an infinite number of such parts. So if we proceed in thought from one such part to another, it must be possible, when we traverse the whole object, to reach infinity in our thinking, which is absurd.(11)

It will be seen that this argument needs support. It is not yet clear why the extremity is a minute part, nor why we can only think of the part next to the extremity as being similar to it. This support is provided in the next sentence, by an analogy with the visual minimum.(12)

According to Furley, Epicurean theory often depends upon analogy with the perceptible for explaining the imperceptible.(13) The existence of a perceptual minimum is taken to support the existence of imperceptible minimum.

(D1) We must observe that the minimum in sensation, too, is neither quite the same as that which allows progression from one part to another, nor wholly unlike it; it has a certain similarity to things which allow progression, but it has no distinction of parts.(14)

(D2) When because of the closeness of the resemblance we think we can make distinctions in it -- one part to this side, one to that -- what confronts us must be equal.(15)

(D3) And we study these parts in succession, beginning from the first, and not all within the same area nor as touching each other part to part, but, in their own proper nature, measuring out the sizes, more of them for a larger one, fewer for a smaller.(16)

(E) This analogy, we must believe, is followed by the minimum in the atom; for in its smallness, clearly, it differs from that which is perceptible, but it follows the same analogy. For we have already stated that the atom has magnitude, in virtue of its analogy with the things of this world, just projecting something small on a large scale.(17)

(F) Further, we must take these minimum partless limits as providing for larger and smaller things the standard of measurement of their lengths, being themselves the primary units, for our use in studying by means of thought these invisible bodies. For the similarity between them and changeable things is sufficient to establish so much;(18)

In the light of modern knowledge of vision systems, there is an element of question begging in the use of the perceptual analogy. The difficulty comes from the structure of the visual receptors in our eyes. The retina of each eye is comprised of an array of thousands of receptor cells, each with a finite size. The fact that there are two kinds of such cells is of no consequence. We know that these cells respond by triggering the discharge of an optical neuron. Aside from the fact that these cells are either discharging or are quiescent is the matter of their physical layout on the retina. Incoming light signals that activate a single receptor cell produce a minimum perceptual experience. It does not matter whether we choose to view light as corpuscular in nature or as a wave. It is simply not possible to have a visual or perceptual experience with less than one whole cell of the retina activated. And since there are a finite number of discrete cells, this amounts to built in atomism of the perceptual apparatus.

The geometry of the eye requires that light from an object stimulating a single receptor cell be within a minimum angle. The most dense concentration of cells occurs within the fovea. (Light from an object strikes the fovea when we look directly at the object). But when only one cell is activated, it is not possible to determine if the object has a relative size smaller than the minimum angle subtended by the cell. Figure 2 shows a diagram of the angle that can subtend one retina cell. Notice that the light from objects smaller than this angle can activate the cell, but because the cell response is simply on or off, there is no information in the perceptual system about how big the image of the object may have been on the cell itself. The perceptual response is simply that it sees the smallest possible activation. (Anything less would be no cells activated at all.)

The visual system has an atomic structure. Its built in bias is to respond in atomic terms. Consequently, using an analogy to argue from the perceptual to the actual brings the atomic structure of the visual system and imposes it via the analogy onto the actual. This is a subtle form of question begging. "There are atoms because I see atoms." But I see atoms because my vision system has an atomic structure, and it shapes it's incoming information into its own atomic structure.

Chrysippus (280-207 bc)

Although Zeno of Citum (334-262 bc) founded the Stoic school, its view is largely known through Chrysippus. The stoics, we are told, rejected atomism. The rejection appears to have centered around problems with infinity. Today it is well known that both the natural numbers and the even numbers are infinite; the even numbers are a part of the natural numbers. There is no difference [in number] between them. It would be inappropriate to interpret 'no difference' in terms of subtraction -- infinity minus infinity is indeterminate. But it appears that Chrysippus may have been onto one of the paradoxes of infinity.

Chrysippus, we are told, held "bodies" to be infinitely divisible, not in the sense that a body could be divided into an infinite number of parts, but in the sense that there was no limit to division. It followed from this, as he observed, that there was no sense in saying that the whole of any extended magnitude contained more parts than any one of its parts. "Man does not consist of more parts than his finger, nor the cosmos of more parts than a man; for division of bodies continues to infinity, and of infinities none is greater or smaller than others." This Stoic doctrine is a more precise and deliberate formulation of a principle first announced by Anaxagoras: "Of the small there is no smallest, but always a smaller, since what exists cannot cease to exist; also there is always a larger than the large . . . ." It is worth noting, too, that Chrysippus appears to have avoided saying that two infinities are equal; he said that no infinity is greater or smaller than another.(19)

In the light of this view, questions would arise concerning what the terms 'more', 'less', and 'same' mean in the context of the infinite. Obviously, lack of recognition that 'same' [size] in the context of infinity could mean different things, allows equivocation to creep into arguments about it. But Chrysippus's care in this matter seems not to have been followed by Lucretius.

Lucretius (99/94-55/51 bc)

Lucretius seems to be aware of what could both be described as a characteristic of and as a problem with infinity. One modern way to show that a set is infinite is to show that it can be placed in one-to-one correspondence with a proper subset of itself. For every integer there is a corresponding even integer. Multiply the integer by 2 to get the even integer; divide the even integer by 2 to get the integer. It is this very thing that causes Lucretius to reject infinite divisibility.

The argument for the existence of minimae partes is worth a little attention, since it seems to be something not found in the Letter to Herodotus. The argument is simply this: if everything is infinitely divisible, then the smallest bodies as well as the largest will be composed of an infinite number of parts, and there will be no difference between them.

This has been said to be directed against the Stoics.(20)

There seems to be something disquieting about a set with a proper subset "as big as" itself. It seems absurd that the very largest of things is the same in number of parts as the very smallest of things. Lucretius rejects this absurdity in favor of atomism.

Arnauld (1612-94) and Infinite Divisibility.

By the time Arnauld published The Port-Royal Logic, many arguments were taken for granted to "prove" the infinite divisibility of extension. Arnauld cites three from geometry himself.

1. Geometry demonstrates that there are certain pairs of lines which do not have a common measure and are called for that reason "incommensurables". An example is the diagonal and the side of a square. If the side of a square and the square's diagonal were each composed of a certain number of indivisible parts, one of these parts would be the common measure of the two lines. Since there is no common measure, it is impossible that the two lines be composed of any number of indivisible parts.

2. Geometry also demonstrates that although there is no square of a number which is twice the square of another number, still it is quite possible that the area of one square be twice the area of another square. If these two squares were composed of a certain number of ultimate parts, then the larger square would contain twice as many parts as the smaller one; and since both figures are squares, there would exist a square number double another square number -- an impossibility.

3. Finally, nothing is clearer than this principle: Two entities of zero extension taken together still do not have any extension; that is to say, an extended whole has parts. Take any two of these parts which we assume to be indivisible. I ask whether the parts have extension. If they do not have extension, they have zero extension and the two taken together cannot have extension; if the indivisible parts have extension, they have parts and are hence divisible.(21)

The first argument depends upon the Pythagorean theorem (A2 + B2 = C2). The length of the diagonal is computed as the positive square root of the sum of the squares of the other two sides (). Suppose the sides of the square are each of length 1. Then the length of the hypotenuse is the square root of 1 squared plus 1 squared, or the square root of 2. It is a simple reductio proof to show that the square root of two is not a rational number.(22) The first argument is valid, but the premisses are not all true. The argument assumes that the Pythagorean theorem applies in the case of discrete metrics, a premiss that is not true. The Pythagorean theorem formula is derived using the premiss that extension is infinitely divisible. (See chapter VII for a detailed demonstration of the dependence.) Consequently, using the Pythagorean theorem in this context begs the question by presuming what the argument purports to prove.

Moreover, a geometric device for dividing a line of unknown length into a fixed number of equal parts using parallel lines and line segments of known fixed length shows commensurability and is illustrated in figure 3. The ratio of the respective segments is not 1:1. But the perspective ratio(23) for different directions in a discrete metric space is not 1:1 either.

The second argument suffers from a similar fate. It assumes that every line has a midpoint. Unfortunately, in discrete metric spaces, not every line has a midpoint. A line with an even number of points does not have a midpoint; only a line with an odd number of points has a midpoint. Moreover, the perspective ratio varies as lines are rotated in the plane; lines at "45" have a 1.414:1 perspective ratio and may have fewer points than a line of the same (continuous) length. To illustrate the difficulty consider a square inscribed inside another square as shown in figure 4.

Consider the same diagram using a discrete metric as illustrated in figure 5. I shall select a size which is odd and has many points. The outer square is 7 points long and has an area of 49 points; the inner, rotated, square has an area of 25 points, although each side has only 4 points. One may conjecture that as the size of the outer square gets large relative to the size of a point, the ratio of the size of the outer square to the inner square approaches 2:1. A square with a side 3 points long and an area of 9 points has an inscribed square with an area of 5 and sides of length 2 points, but a square with only two points on a side has no inscribed square at all.

The third argument, if not outright self-contradictory, merely asserts that to have extension is to be divisible: "if . . . indivisible parts have extension, . . . they . . . are . . . divisible".(24)

Arnauld goes on to add another alleged proof. His demonstration imagines a flat (Euclidean) sea with a ship that is receding in the distance. He constructs a similar triangle argument using the eye of the observer, the light ray coming from the waterline of the ship, one coming from the horizon, and an interceded parallel transparent glass. Figure 6 shows the geometry involved. One is supposed to be convinced by this argument that there is a point in the plane of the glass where the light ray coming from the waterline of the ship passes. The illustration shows that the passage of the "continuous" line goes through actual points in the interceded parallel plane only when the distances are fortuitously correct. Rays that do not intersect points can be called "virtual" lines. Every line requires at least two points, but the remainder of any line may be virtual rather than actual. The ray that intersects the transparent glass does so at a virtual point rather than at an actual one. When extension is measured with a discrete metric, the experiment proposed by Arnauld results in a hypotenuse which gets "thinner" and "thinner". It does so by passing through only as many points as are between the water line and the horizon. The line is so nearly parallel to the horizontal baseline that it extends for long distances without passing through actual points. The perspective ratio is nearly infinity. One cannot "pick out" two points in the transparent glass. Once one line intersects at a point, the other is "too close" to hit an adjacent point and is only a virtual intersection. The computer implementation shows only a single row of dots close to the pane, as the triangle in figure 7 shows. (25),(26)

Hume and Infinite Divisibility

Hume argues against infinite divisibility; he interprets space or extension in terms of the objects which might occupy it. A consequence of the Humean view would be an empirical answer to the question whether extension is infinitely divisible or not.

Hume's argument involves treating space or extension as not distinct from matter. According to Baxter, Hume's view subsumes matter and extension into one structure.

Hume believes that our idea of a region of space is an abstract idea of the following sort: We think of a region by thinking indifferently of various objects that could occupy that region . . . . [The] upshot of this theory is that regions of space have the structure of extended objects.(27)

Under this view, space or extension would have the same structure as matter. Consequently, the findings of modern physics would be doubly relevant. What we learn about matter is immediately generalizable to extension (space). Baxter essentially showed that Hume believes that the smallest things have no extension and that extended things are created by combining these unextended smallest things.(28)

Modern physical theory corroborates this Humean view. According to current theory, all matter is mostly empty space; extended objects are created by combining objects which have (nearly) no extension of their own. The smallest particles are called 'quarks' and are described by physicists as "point-like" entities.

The first evidence for the existence of quarks came about 15 years ago [1969] in experiments that probed nuclei with energetic electrons. They revealed point-like objects (then called partons or quark-partons) inside the neutrons and protons of the nuclei.(29)

The influence of these point-like particles "extends" space and matter and produces extended particles (including protons and neutrons). However, whether matter is infinitely divisible or not is still not settled.

Underneath the standard [physical] model [of matter] is the realm of "compositeness". The standard model holds that everything is built out of six kinds of quarks and six kinds of leptons, and that these quarks and leptons are the most elementary forms of matter. Up to now, whenever physicists have thought they had reached the most elementary constituents of matter, they have been proven wrong. There is a fraction of theorists who think they are still wrong. Believers in compositeness say the quarks and leptons are themselves composite, made of more elementary objects, which may be called preons or technicolor quarks or something else.(30)

Hume's approach would allow the question of infinite divisibility to be settled by empirical physics. But Hume's tack represents a significant departure from Aristotle's thinking. Even if modern physics did provide a definitive answer, Aristotle's difficulties would not be disposed of. Aristotle clearly distinguishes between place and matter in The Physics: "Place is no part of the thing." (211a1). As a result, Hume's approach could be conceived of as predating Aristotle.

(Chapter 6)