prev |
## CHAPTER I |
next |

People have been dividing things and sharing them for millennia. When it comes to dividing something among two or three persons, the "right amount" can often be cut or broken off at once. But when one is dividing something among four persons, it is often easier to divide it in half and then divide each half in half again. Division into smaller portions is most often achieved by a process of repeated or successive division. At some time long past it must have occurred to someone to wonder how long such a process could be continued.

Practical experience sometimes suggests that there is a limit to the process. In dividing something, and dividing the results again, sooner or later one reaches a limit where the remaining parts cannot practically be divided again. For some things this limit is much more obvious than others. Dividing a bag of marbles among children provides an example of an obvious limit. But practical experience also suggests that sometimes there are non-obvious cases. Dividing a pitcher of liquid refreshment among imbibers provides an example of this. One might resort to counting drops, but drops come in different sizes, and there is the matter of the residual film of liquid. We can easily conjecture that the liquid could continue to be divided beyond our ability to distinguish the divisions.

We can use a magnifying glass and a razor blade to divide a droplet of water that we could not perceive as large enough to divide when we looked at it with only the naked eye. We can also use a microscope and appropriately sized tools to divide the droplet that seemed too small to divide when we used only the magnifying glass. The use of higher and higher powers of magnification shows that at each stage an apparently indivisible droplet proved to be divisible when it was looked at with greater magnification. While very high powered devices have been able to distinguish the individual atoms of heavy metals, no such results have been obtained with water. It is our atomic theory of matter that allows us to conclude that there is also an indivisible minimum size for water.

Apart from that modern atomic theory, we can easily generalize that the process of successive division need have no end -- that the process can continue indefinitely. But we are quite aware that our perception is limited. There are smallest amounts -- less than which we cannot perceive. Humans, being the divisive people we are, take sides and argue about such questions. We may reasonably infer that ancient peoples were divided in their opinions even before recorded history.

Atomism is the view that successive division must terminate in some indivisible minimum. The opposing view is the belief that successive division can be continued infinitely. Since division is a process that is applied to something, an immediate dichotomy concerning the question is possible. The question may be asked with emphasis on the object of the process, or it may be asked with emphasis on the process itself. It is from the former that the name 'infinite divisibility' derives and is given to the view opposing atomism. I shall sometimes refer to that view as '*divisionism*'. Divisionism is most often expressed as the view or belief that matter or extension is infinitely divisible. What may be the proper object of the process has varied with the major philosophical positions.

I have already hinted that recognition or perception might influence knowledge of divisibility. I will touch on the epistemological concerns relating to the arguments, but I will primarily be focusing on the metaphysical aspect of the question. The Epicureans argued from the perceptible to the imperceptible by analogy. More recently the question was applied to the conceivable.

One way to organize the perspectives taken by concerns for metaphysical questions, epistemological questions, and questions regarding conceivability is along a subjective-objective dimension. Philosophical perspectives fall along that dimension with realism toward the objective end, idealism toward the subjective end, and phenomenalism somewhere between these two.

In the context of realism, one asks whether matter, extension (space), and duration (time) are infinitely divisible. In the context of phenomenalism, one asks whether perceptions are infinitely divisible. In the context of idealism, one asks whether concepts are infinitely divisible. If one is to focus on the process itself, questions concerning the meaning of 'infinite' arise.

All these questions can be asked with a decidedly metaphysical flavor as well as with a decidedly epistemological flavor, but realism lends itself much more easily to metaphysical questions while phenomenalism lends itself much more easily to epistemological questions. And idealism lends itself more easily to questions regarding conceivability. The analogical relationships I see suggest the visual representation illustrated in figure 1.

In this exposition I shall be concerned mostly with the validity of various arguments for and against each position. I am particularly concerned with mathematical arguments that have been presented and the light they shed on premisses which have been used to support one or another position. To "cut to the chase", my research suggests that there is no valid argument with true premisses which establishes one position over the other. It seems that there are consistent models for both positions. And these models differ by one "axiom" -- the presumption of atomism on the one hand or of infinite divisibility on the other hand.

I also presume a "developmental" perspective consistent with genetic epistemology.^{(1)} I assume that, for the most part, earlier writers assimilated or understood by means of fewer distinctions, and that some problems with earlier views may be resolved by more recent distinctions. However, there are instances when the mere addition of a distinction is insufficient to resolve the issues. The mappings of concepts may have to be significantly reorganized in order to accommodate a newer development.^{(2)} This developmental perspective is also suggested by Furley when he traces the evolution of Atomism as presented by Epicurus:

[This] essay will show how Epicurus' doctrine evolved; it is a modification, adopted for the purpose of meeting Aristotle's criticisms, of a doctrine which the earlier atomists put together to meet and thwart the Eleatic attack on pluralism.

^{(3)}

Some of the earliest writings on this subject are attributed to Zeno of Elea. By Zeno's time the controversy was fairly well developed; the positions were characterized as beliefs in "atomism" and "infinite divisibility". These contrasting beliefs are closely related to the earlier question, whether "things are one" or "things are many". If things are infinitely divisible, any division into parts yields parts which are themselves divisible into parts -- "All things are many" (all the way down). If things are not so divisible then there are things that are not many -- "things are one" (and indivisible).

Even earlier than Zeno, Heraclitus had things to say about the controversy. Interestingly enough, Heraclitus seems to have had the most mature views on the topic, although records of his thoughts are the most scanty. (See page 17.)

Mathematical induction figures prominently in my analysis of infinite divisibility, and I would be remiss not to briefly present it here. Mathematical induction is applied to a statement which is expressed in terms of some arbitrary natural number, usually represented by 'N'. Induction, as one might expect, is a way of reasoning to conclusions about more particulars than may reasonably be examined. Mathematical induction has two premisses and one conclusion. One premiss that must be satisfied is that the statement in question be true for some small value of N. This need not be the smallest, but it usually is and most often is the number 1. The second premiss that must be satisfied is a material conditional going from an arbitrary number N (equal to or larger than the value used in the first premiss) to the next larger number N+1. If these two premisses are satisfied, then one may draw the conclusion that the statement is true for all values of N (greater than or equal to the small value of N). Suppose we refer to the statement as S(X). Premiss 1 would be expressed:

S(A) is true.

Premiss 2 would be expressed:

IF S(N) is true THEN S(N+1) is true (whenever N >= A).

The conclusion that may be drawn is:

For ALL X(>=A), S(X) is true.

This conclusion is justified by the following reasoning. Suppose A is 1. S(2) may be inferred from S(1) and premiss 2 by modus ponens. S(3) may be inferred from S(2) and premiss 2 by modus ponens. This process may be continued until X is as large as you like. Mathematical induction is the shortcut method for deducing the truth of S(X) for all X. Simply by showing that premisses 1 and 2 are satisfied for some statement, S(X), we may use mathematical induction to directly demonstrate the truth of S(K) without going through K-1 applications of modus ponens. Because mathematical induction is only a shorthand for deductive applications of modus ponens, it is strictly truth preserving, as are all valid deductive arguments.

In the literature three primary senses are given for the term "infinite" and its derivatives. These are *arbitrarily large*, *unending*, and *aleph null* (_{0}). Distinguishing carefully among these senses takes one a long way toward resolving Zeno's paradoxes. In many instances substituting the appropriate phraseology in a premiss statement using the term "infinite" renders the truth value of the premiss much more apparent.

The first sense, *arbitrarily large*, is illustrated by James Thomson in "Tasks and Super-Tasks".

[T]o say that a lump is infinitely divisible is just to say that it can be cut into any number of parts.

^{(4)}

*Infinity* is paired with *any number* with the implicit understanding that this number may be as large as you like.

The second sense, *unending*, is illustrated by Russell.

Etymologically, 'infinite' should mean 'having no end'.

^{(5)}

The third sense, *aleph null*, is most precisely captured by the axiom of infinity in the Zermelo-Frankel set theoretic representation for numbers.^{(6)} That axiom is an existence axiom in that it postulates the existence of a number with certain properties. The axiom of infinity can be paraphrased in terms of ordinary natural numbers as follows: There is a number X (infinity) such that 1<X and whenever N<X then N+1<X, where N is any natural number generated from 1 by repeated applications of the successor axiom (+1). This axiomatic definition for infinity (_{0})
explicitly utilizes the structure of mathematical induction.

The notion of "motion" or "velocity" figures into atomism versus divisionism arguments in a number of ways. It is implicit in two of Zeno's arguments and explicit in a third. It is appropriate to present a brief view of the current space-time perspective on motion to provide an explicit background for understanding the arguments presented as they reflect on it.

In mathematical physics *velocity* ("motion") is defined as the rate of change of position with respect to time. When time is taken as a fourth dimension, and one looks at events as having both spatial and temporal coordinates, no "motion" can be seen. When one adopts such a "four-dimensional space-time perspective", one attends to a three-dimensional "object in motion" as a four-dimensional space-time "worm" with its "starting position" at one place-time and its "ending position" at another place-time. The starting position is identified by its having a "lower" time-coordinate. From the four-dimensional space-time perspective a "velocity" is seen as just the slope of a line drawn with both space and time coordinates. It is no different from the rate of change of one spatial dimension with respect to another, such as the slope of a road. For example, on a road with a 7% grade, the road rises 7 feet for every 100
feet of length. Saying the road "rises" is only valid in respect to one's position along the road (and whether one is going up the road or down the road). In order to develop the analogy with motion, references to time must be removed from the notion of physical slope. As one stands in different positions along the road, one's elevation varies depending on one's position.

The physical slope of the road corresponds directly to the ratio of position with respect to time. When an object in three-space is moving in time its spatial coordinates are changing, but only as the time coordinates are changing. Its position coordinate varies with its time coordinate just as one's elevation coordinate varies with one's position coordinate along the road.

When one looks at an object from the four-dimensional space-time perspective one sees all the space-time coordinates of the object. Both the starting point and the ending point coordinates are immediately available. The view is one that could be called "omniscient" in that all space-time positions can be seen "at once".^{(7)} The analogous perspective for viewing the road is from the side. By standing to one side of the road with a 7% grade, far enough back, one can see both the bottom and the top of the hill "at once". In a like manner, "standing to one side of time" allows one to adopt the four-dimensional space-time perspective and see both the beginning and end of the motion of an object "at once".

From the four-dimensional space-time perspective no "motion" is seen at all -- thus exonerating the ancient argument that motion is impossible. However I will discuss the questions mostly from the more conventional, three-dimensional perspective. Keeping in mind the way motion is defined in mathematical physics will provide a consistent view of the problems of infinite divisibility and atomism.

The concept of the atom did not emerge on the scene full-blown. It evolved from a number of earlier views through a gradual process involving a number of stages. Atoms, as we know them, and as most clearly presented by Lucretius, are conceived of as hard, indivisible bits of solid matter that come in various shapes and kinds. They whiz around in empty space colliding with each other, sometimes bouncing off and sometimes sticking to each other. All the "stuff" of the universe is made up of them. Atoms could not exist as even a concept were it not for the co-existence of empty space into which to put them. The universe can be seen as distinguished into bits of matter and space. But without the notion of empty space, the notion of atoms cannot exist.

A "solid" concept of atomism also requires some stability concerning the questions what the stuff of existence may be made of and in how many kinds it can come. I will touch only briefly on this question as it is peripheral to my main interest. But developments in this argument do affect the development of atomism proper, so I present a brief summary.

Atoms (of matter) also represent a synthesis of the notions of dividing and not dividing. The stuff of the universe is divided into bits (atoms), but the atoms themselves can not be divided. Required also is some notion of "size" for matter. Arriving at a birth for the concept of atomism requires that all these questions have undergone some development and some resolution. And all these developments depend upon some notion of existence.

I will not be touching deeply on these early developments. But I will outline them briefly to establish a context for the main discussions. My main focus is on the mathematical arguments that arise as a result of these early arguments.

The question of "being" permeated the early pre-Socratic philosophy. The convoluted arguments centered around what appeared then to be worse than an oxymoron -- the apparently contradictory act of asserting the existence of something in order to deny it. The act of speaking or even thinking something was viewed at the time to have had existential import.

when the goddess points out to her listener that he could neither know nor point out what-is-not (2.7-8), she is precluding reference in thought or speech to the non-existent.

^{(8)}

This made talk of "nothing" or non-existence very problematic. It was the denial of this "void" that lead to monism. In the denial of nothing the early Ionians concluded that everything was one and that motion was impossible. Atomism has its roots in this concept of "the one" or unity -- which later became associated with the idea of indivisibility.

Thales of Miletos is credited with having explained that everything is made of water; that air, ether, and even earth are just different forms of the one substance. As a result, Milesian thought was dominated by corporeal monism,^{(9)} that all things reduce to the one (body) which appears in different forms.^{(10)}

All the Ionians had taken for granted that the primary substance could assume different forms, such as earth, water, and fire, a view suggested by the observed phenomena of freezing, evaporation, and the like. Anaximenes had further explained these transformations as due to rarefaction and condensation (§ 9).

^{(11)}

Thales is credited with opening the question that leads to the atomic theory.^{(12)} It might be reasonable to attribute the contrasting view to Anaximander, of the next generation of Milesians, who was a follower of Thales.

Thales, Anaximander seems to have argued, made the wet too important at the expense of the dry.

^{(13)}

Burnet credits Anaximander with giving some equal standing to the different "elements".

[It] is more natural to speak of the opposites as being 'separated out' from a mass which is as yet undifferentiated . . . .

^{(14)}

But he also begs the question by suggesting that this somehow entails it being made of "particles".

That, of course, really implies that the structure of the primary substance is corpuscular, and that there are interstices of some kind between its particles. It is improbable that Anaximenes realised this consequence of his doctrine.

^{(15)}

No such conclusion is warranted without some presumption of the incompressibility of matter, a later atomic development. The mixing (and separation) of colors shows a non-particulate counter-example. Burnet seems to have "projected" a more modern view into his analysis.

Already the Ionians have a general question regarding whether there is some primary stuff of existence that is divisible into other substances, or there is one of these that cannot be so divided. Anaximander affirmed divisibility while his predecessor Thales affirmed the one.

The Pythagoreans taught that all things were number. And number is an expression of unity or oneness -- the Milesian monism in a less corporeal form. Pythagoras, who was in Kroton from about 532 B.C. to the end of the sixth century, was probably a disciple of Anaximander.^{(16)} He is credited with discovering the problem of doubling the square.^{(17)}

Pythagoras discovered that the square of the hypotenuse was equal to the squares on the other two sides; but we know that he did not prove this in the same way as Euclid did later (I.47). It is probable that his proof was arithmetical rather than geometrical; and, as he was acquainted with the 3 : 4 : 5 triangle, which is always a right-angled triangle, he may have started from the fact that 3

^{2}+ 4^{2}= 5^{2}. He must, however, have discovered also that this proof broke down in the case of the most perfect triangle of all, the isosceles right-angled triangle, seeing that the relation between its hypotenuse and its sides cannot be expressed by any numerical ratio. The side of the square is incommensurable with the diagonal.^{(18)}

In the atmosphere of Milesian monism, it must have been quite disconcerting to be unable to find whole numbers giving a ratio for doubling the square. With monism firmly established in the culture, the faith that such a number existed and would be found probably prevented the discovery of what is now known to be . Had it been discovered then, atomism might have been dealt a disabling blow. If the square root of two is to be a number, then number can no longer be strictly a unity.

Heraclitus of Ephesus (fifth century) was known for his theory of flux and his doctrine of the unity of opposites. Most of Heraclitus's works are lost, but a few fragments have been gleaned from various sources; he was quoted by ancient philosophers from Plato on. Enough substance is contained in those fragments to provide a reasonable assessment of his view concerning the present question. The ancient question concerned whether all things were "one" or "many". We may understand "many" to mean "composed of parts" where the term 'parts' is used circularly or recursively. Zeno's *Paradox of Plurality* (page 41) explicates this issue more fully. By "one" we may understand "an indivisible whole". While this pairing may not be exact, I think it naturally evolved into the atomism versus infinite divisibility distinction. Atomism may be an early attempt to resolve the Paradox of Plurality. If so,
it would provide for a true recursive definition for the term 'part'. A part is either an atom or it is something composed of smaller parts. [See the discussion of the Paradox of Plurality on page 73 below.]

For Heraclitus, the question whether all things are one or many is answered by fragment 112:

From out of all the many particulars comes oneness, and out of oneness comes all the many particulars.

^{(19)}

While this hints at Hegel's synthesis of thesis and antithesis, it also suggests that anything that is one is also divisible. One could interpret this as an affirmation of infinite divisibility. But his doctrine of the unity of opposites actually mandates that he affirm both views. His theory of flux, in which things are continually changing into their opposites, is moderated by his principle of balance. That principle is best stated in fragment 33 and can be understood as a conservation law.

[The] resultant amount is the same as there had been before.

^{(20)}

On this account, it would seem, Heraclitus would not have subscribed to the naive view that if something were infinitely divisible then it would be divisible into nothing at all. Such a premiss would violate his principle of balance. Yet that premiss is exactly the one which has gone unchallenged for millennia. I shall return to this premiss when it is more explicitly stated. [See the discussion under the Paradox of Plurality on page 73 below.]

Parmenides taught, in opposition to Heraclitus, that being was "a solid, homogeneous, extended body"^{(21)}, and that it was spherical and unchanging. In many ways this reaffirms the earlier Milesian view of the one, but the similarity to the later conception of an atom is readily apparent.

Empedocles seems to have attempted to synthesize the views of his predecessors. Thales made everything out of water; Heraclitus made everything out of fire; Anaximenes made everything out of air;^{(22)} Anaximander gave none of these primacy. Empedocles made everything out of all of these, including earth, and, as such, provided the first theoretical forerunner of modern atomic chemistry.

Empedocles called his elements 'roots', and Anaxagoras called his 'seeds', but they both meant something eternal and irreducible to anything else, and they both held the things we perceive with the senses to be temporary combinations of these.

^{(23)}

But Anaxagoras apparently thought Empedocles's system flawed. He did not think that four elements could produce all the substances we see. According to Burnet, Anaxagoras was misunderstood by both Aristotle and the Epicureans. Burnet explains that Anaxagoras's "seeds" were infinitely divisible but differed in their proportions.

He therefore substituted for the primary 'air' a state of the world in which 'all things were together, infinite both in quantity and in smallness' (Fr. I). This is explained to mean that the original mass was infinitely divisible, but that, however far division was carried, every part of it would contain all 'things' , and would in that respect be just like the whole.

^{(24)}

The "flavor" of the disputes of the time were perhaps eloquently expressed by Zeno in his paradoxes, to which we now turn.