Physics Gamma, continued: Chapters 4 - 8

The discussion in these chapters has to do with to apeiron and to peperasmenon - translated in your text as 'the infinite' and 'the finite', respectively. It may be helpful to bear in mind that 'apeiron' also means "unlimited", "non-limited", or "indefinite". 'Peperasmenon' can also mean "limited", "having been limited", or "definite". Both words are adjectival in form, but when used with the definite article 'to' they are taken as nouns: "the unlimited", "the infinite"; "the limited", "that which has been limited", etc. (much as in English 'the rich' means "those who are rich", 'the presidential hopeful' means "the one who hopes to be president", etc.) I mention all of this because some of what Aristotle says may seem not to make sense under the translation used in your text. Having these additional translation options will not necessarily clear up all problems and questions; some philosophical problems are likely to remain, and at least some of these will be discussed below.


A. The central questions of Chapters 4-8:
--Does the apeiron exist (with respect to magnitude, motion, and/or time)? That is, is there unlimitedness, or infinity, with respect to magnitude, motion, and/or time?

--If there is some such apeiron (infinity, unlimitedness, infinite thing, unlimited thing), in what way or sense does it exist? In what way or sense is it apeiron?

--If there is some such apeiron, what is it? For example,

------Is it an aspect or attribute of some thing or kind of thing - as, for example, if there were something that was infinitely (or indefinitely) long?

------Is it an "essential attribute" of some nature (203b35)? (That is, equality is considered an "essential attribute" of quantity: while quantities are not all equal, still when there is equality, quantity must be involved, since quantities are the only sort of entity that can be equal, according to Aristotle. Or, color would be considered an "essential attribute" of the visible: not all visible things are the same color, but if something has color it must be visible; only visible things have color. In this way, then, would apeiron, infinity, unlimitedness, be an "essential attribute" of magnitude, motion, or time?)

------Is it something that exists independently of other things, an independent feature or factor in the universe (a "substance", 203b35)? Something else?


B. Aristotle warns at 203b30 that "many impossibilities result whether the apeiron is posited (laid down) to exist or not to exist". He will try to find a way of looking at the things studied by physics that makes sense (presents things coherently and consistently) and that allows us to avoid as many of these impossibilities as possible.

C. For purposes of the Physics, at very least, various senses of 'apeiron' must be distingushed (204a1-8):
1) That which is "infinite" or "unlimited" in the sense of being by nature unable to be "gone through" or "passed through", such as a voice: one cannot traverse or cut or even limit a voice. One can limit the volume of a voice, or cut off the duration of a voice's sounding, but these are not limitations of the voice itself. The voice itself is not affected in these cases; it's still the same voice no matter how long it is heard or how loud it is.

2) a) That which has an unending or hardly ending(?) path or orbit or way out (I'm not sure what is meant here: possibly that which is circular or spherical, and/or that which has no way of being accomplished due to some practical obstacle - going to a galaxy several light-years away, for example).

b) That which is in one sense the kind of thing that can be gone through or traversed, but which in fact cannot be traversed or has no limit. Examples of this would be objects or periods of time that are infinitely long, or infinitely divisible. There are supposed to be objects of definite size and periods of time of definite duration; but perhaps in some sense some objects and periods could be infinitely long, and/or infinitely divisible. (We will see in what if any sense there are such objects or periods.)

Aristotle will mainly be treating things that fall under #2, especially 2b. He further distinguishes that which is considered to be apeiron with respect to addition and that which is considered to be apeiron with respect to division or removal. (A thing can be apeiron with respect to one or both.) 'Apeiron with respect to addition' refers to that which grows indefinitely due to addition of units of a given size. The set of counting numbers {1,2,3,...} is apeiron with respect to addition; ones are continually being added. Whatever whole number you can think of, there is always a higher one (see 207a1-10, 207b1-5). 'Apeiron with respect to division (or removal)' refers to that which becomes indefinitely or unlimitedly small by successive divisions into (or removals of) parts in a certain proportion. Suppose that a certain length is divided in half, then divided in half again (to get 1/4 of the original), then divided in half again (to get 1/8 of the original), etc. No matter how many successive halvings are made, something will always be left: the length will not be reduced to zero*.


D. His responses to the questions in Part A above are these:
--The apeiron (with respect to magnitude, motion, and time - i.e., Aristotle is not talking about mathematics here) cannot exist independently and apart from things that can be sensed. That is, there cannot be said to be "an infinite/unlimited" that is not an attribute of things that can be sensed. The apeiron cannot be a substance (a thing that exists on its own and not as an attribute or aspect of something else). It will be an attribute of magnitude, motion, and time, that is, magnitudes, motions, and times can be apeiron (204a).

--There cannot be an apeiron sensible body: there couldn't be more than one spatially apeiron body, because either there would be no room for all or they would limit each other or something else would limit them and hence they would not be unlimited. There could bot be only one infinite body composed of a finite element and an infinite element, because then the finite element would be overcome. But there also could not (for purposes of physics) be a "simple" (not composite; uniform) infinite body, either as composed of one element or as that from which all elements (simple basic parts or components of things; for the Greeks, usually air, earth, fire, water; but often also or instead, "the hot", "the cold", "the wet", "the dry"; or perhaps others) are generated. If there is only one kind of entity with nothing outside of it and no internal divisions, there will be no contraries and hence no change. This will not necessarily be a problem, unless of course you want to have the kind of things physics studies: things that grow and change (204b-206a). (Thus the criticism of Heracleitus at 205a4 will only be relevant if he is taken to be talking about the things physics studies, as they are studied by physics.)

--From this, Aristotle says (end of Ch. 5, beginning of Ch. 6), we can see that the apeiron in physics does not exist in "actuality" (energeia or entelecheia, depending on the context), i.e. it does not exist as something that is completed or wholly at work at any one time (or place). Rather, it exists "potentially" (by/in dunamis), that is, as a capacity. It's easy to see how time could be apeiron by addition only potentially; time doesn't all exist at once. But we should also note that a finite interval of time (or of magnitude; or a body of definite size) will be apeiron with respect to division, and it's important to see that although in one sense the interval can be completed, it is apeiron from another perspective, and from that other perspective, it is not completed, and not wholly in effect. Or, it exists "actually" in the sense that a day exists "actually", namely as in the process of occurring (once a day has occurred, or is completed, it no longer exists, and the whole day does not exist at once). It exists "potentially" in the sense of being that which is always able to have more taken: "Thus the apeiron is that outside of (beyond) which, with respect to taking a quantity, there is always some part [yet] to be taken" (207a8-9). This "part" will be a unit of a specific size in the case of the apeiron by addition; in the case of the apeiron by division, it will be a piece that stands in the same ratio to the piece taken before it as that piece stood to its predecessor.

This addresses magnitude and time; but what about motion? The specifics will be dealt with in Book Epsilon (E). However, it can be seen from the discussion here that motion will be able to be apeiron in so far as magnitude and time are able to. Motions occur over time, so if time intervals can be apeiron then motions will not be prevented from being so. Motions often involve distance or expansion or contraction, and hence magnitude, so in so far as a thing can be said to increase or decrease or traverse distance indefinitely, motions will be able to be considered to be apeiron.

Note that in his discussion, Aristotle takes the case of magnitude and the case of time to be parallel. It might be asked whether this is valid in all relevant respects. He speaks of the "actuality" and completion (or lack thereof) of time intervals in the same way as he speaks of distance intervals. Distances don't necessarily invoke time, although the notion of traversing them does; we traverse (or try to traverse) distances "in" time. Time intervals obviously invoke time, and when we traverse them we don't necessarily traverse anything else. One might ask whether passing through a time interval "occurs in time", or whether passing through a time interval "is" time (or a part of it); or both; or neither.


E. From 204b10 through the end of Ch. 5, Aristotle argues that considered from the point of view of physics (i.e. as far as physics is concerned, or from the point of view of physics), it would seem that no apeiron body could be either simple (or one kind alone, and/or without parts) or composite (composed of simple bodies, hence of multiple kinds, and/or having parts) (204b12). His conclusion is that in or for physics, and based on what he has said so far, no apeiron body can exist in actuality - that is, no body can exist, for physics, that is completely and completedly apeiron in all respects (for apeiron and completion don't go together), or that functions as apeiron in all respects. For example, if there was actually infinite air and finite amounts of everything else, the air would have overwhelmed everything else with its qualities. (In other respects, there might be an apeiron aspect to the quantities of each element, but the fact that there would not be such an aspect in this respect, and in the respect of each element having its own definite identity, suggest that in or for physics, no body or element/consituent can be wholly or in all respects or unconditionally apeiron, and that is what Aristotle wants to show.) If motions always occur between contraries, as A. has proposed, no element or constituent had better be actually infinite, or its properties will have overwhelmed their contraries.
But what if they were all infinite? One problem for Aristotle is that it would then seem that each element (or apeiron thing) would take up the whole universe. Those people who are familiar with mathematics might argue that although the odd numbers are infinite (increase infinitely) and the even numbers are infinite, there is "room" for both in the counting numbers. Without investigating here whether mathematical objects and physical bodies are relevantly similar (or comparable in this respect), we can look to Aristotle's other argument: If all bodies or elements were actually infinite in spatial extent, and the universe was infinite in extent, where and how would objects move? That is, if air moves "up" with respect to earth, and earth moves down with respect to air, but the universe extended infinitely in all directions, it would have no specific center and so there would be no definite "up" and "down". Then how would things move; which way would they go? Today, we say that gravity moves bodies (and conceivably energy) toward other bodies; things move toward whatever bodies are close enough to attract them, and vice-versa. Does this mean that the universe can be infinite in extent?

--Not necessarily. If we take it, with many scientists today, that the universe is expanding outward from some center, then we do not have to assume that the universe is "in actuality" infinite at all; it could just be "potentially infinite", as A. says. If we assume that it is expanding, and that a finite amount of matter and energy is moving outward through an infinite void, we run into problems that will be discussed below. If we assume that the universe is "in actuality" infinite and has no center, we still have problems, for we must explain why things are located where they are, why they are scattered over an infinite expanse (we can account for why things travel definite distances, but does that mean we can account for how they could get to be infinite distances apart in only a finite number of years, or all at once?), and so on. Also we might run into the same problem about infinite void that will be discussed below.

But Aristotle also says, at the beginning of Ch. 6, that the apeiron cannot be taken without qualification not to exist. (Another way to translate this would be to say that we cannot take it, if we wish to avoid absurdities, that the apeiron without qualification does not exist.) In other words, we would run into problems if we said that nothing that is apeiron in any respect exists in any way, in or for physics. For one thing, if we took it that there was nothing apeiron at all in the physical world, we'd have to say that magnitudes will not always be divisible into magnitudes. But, you might ask, what's wrong with that? What's wrong would be that aside from any logical or conceptual problems that might arise, physics won't work because we would not be able to provide an account of motion. (We still might have problems doing that given what A. says, but they won't be the same problems, and they will be problems that A. will try to keep out of physics - which does not mean they are not relevant or important, or that they have no effect on what physics does or means.)

Here's the problem, or one of them: For physics, we will need to suppose that motion over a given magnitude (distance) is continuous over that distance (more on this in Book E). The reasons we will need to suppose this include the fact that if the motion is discontinuous, i.e. broken, we will need to account for what makes it stop and what gets it going again. If magnitudes were not always divisible into magnitudes, there would be some smallest magnitude, some smallest unit of measure. Each magnitude would be composed of a bunch of these units, stuck together. But there will always be something in between each pair of these units - for if there were not, the magnitude would in fact be infinitely divisible (because it would in fact be continuous and could be divided anywhere). So what is between the units? (Remember that we are talking about magnitude itself, not about objects that have magnitude.) If we say that in fact the units are in some way separated, we cannot understand motions as being continuous, because we could not account for what happens "between" units: non-existence? non-motion? We would not know, and therefore could not say that we knew what motions involved.

Looked at another way, if we said that there was a smallest magnitude then we could not say that motions started at all points of a given magnitude; motions could not be said to start between units, and/or could not be said to start at different points within a unit (because units are not divisible). That might be OK (if strange) if time was also and in a correlate way not infinitely divisble, i.e. if there was said to be a smallest unit of time. But then what would happen "between" those units of time? In principle, nothing would happen, if happening must be in time. But how could there be said to be intervals between units of time? Would they not have to occur "in time"? And if they did not, what does happen between units of time? We could not tell; our perceptions are in time, or are of things in time. Anything could happen, or nothing might happen - either of which would really destroy any claim we had to be talking about causes in physics, and any claim we had about recognizing how anything works, or recognizing what happens. For if nothing went on between units of time, we could not explain how motions continued over a series of units. If something unrecognizable went on or could in principle go on in between units of time, we could not say that we knew that these unrecognized things had no effect on what we do perceive and recognize**, and hence we could not say that we knew how anything worked, or indeed exactly what happened in a given case.

Thus it can be seen that Aristotle's insistence on the existence of the apeiron as potential (and not also "actual") for physics has to do with making sure that certain incoherences and contradictions can be avoided in physics, and with making it possible (or trying to make it possible) for certain things to be described or accounted for coherently and consistently within physics. Notice that he does not say that the things he rules out are wrong or ridiculous overall, nor that the things he asserts will not engender problems.


F. Why does Aristotle say that if void (the empty) or place is infinite, there must be infinite body too (203b30)?
(Recall that he will conclude that there can't be infinite body anyway.)

This claim at 203b30 may seem odd to us today; we're told that the physical universe is infinite in extent, and that this includes infinite "empty" space (space with no matter "in" it). The amount of matter (which is like Aristotle's "body" in that it's said to be that which occupies space or place) could be limited, we say; it is just distributed through infinite space, or is expanding or moving outward infinitely (depending on what theory you accept). Why would Aristotle disagree with this?

Part of the answer will come in his discussion of void and body in Book Delta, but something can be said now.

If the universe is taken to be infinite in extent, could it be occupied by, or involve, a finite amount of body (stuff that takes up space and can be sensed)? This might be thought to be possible if that body were to be expanding infinitely. That expansion would have to work in one (or both) of these 2 ways:

(1) The quantity of body might be increasing infinitely; i.e. more of it would be getting generated. But then it would not be [a] finite (limited) body at all, but rather a potentially infinite (unlimited) body, and Aristotle would be right after all.

(2) The quantity of body might be staying the same, but the volume might be expanding constantly. If the volume is increasing without the addition of any more body (and air counts as body), then what is being added must be "empty" (not-occupied-by-body) space or place (i.e. there are new "places" coming to be, places with no body "in" them). --Why wouldn't Aristotle accept that?

Once again, I think, the reasons have to do with a consideration of the conditions that would make physics coherent, or on any level understandable***.


Let us put aside for a moment the modern notion that all of the "bodies" we observe contain at least some "void" (empty or matter-less space) within them. We say today that gases are composed of molecules scattered sparsely throughout void, that liquids are somewhat denser concentrations of molecules amid void, and that solids are mostly molecules of matter with very little (but some) void between them. Even within molecules, there are void spaces through which the electrons that make up the atoms of the molecules travel and share orbits; and within atoms, we say, there are void spaces through which electrons and some other subatomic entities move. As I say, though, let's put all of that aside temporarily, for we will return to it.


In Aristotle's time, the only motions that were known were said to be those of bodies through other bodies, or motions of bodies in themselves - motions of rocks through air or water; changes of surface colors; bread "rising" (getting bigger); and so on. Air was not thought by everyone to contain empty space; Aristotle will argue later that motion does not require the existence of void. Aristotle and many others thought of air as a very light, transparent, fluid material; they thought that body did not have to be opaque or heavy (and they had no reason to think that what surrounded the stars was or was not like what surrounds us on Earth). Similarly, the expansion of bread was thought to have to do with the production (from some other substance that takes up less space) and release of air inside it; contraction of things put into fire was thought to have to do either with release of air or water or with the ability of things to bend and fold (as in the flexing of muscles) or with some sort of mixing of things; and so on. Thus no body was known to contain void, and all motions were thought to take place within or among bodies with no void around them (or at least, there was not generally held to be any proof that there was matter-less space,as we will see). But why would void not be a good choice or even a viable possibility for explaining the differences in density between, say, air and rock; or for explaining how things move? That is, why would Aristotle insist on arguing against it?


Simply put, the problem with accepting a complete void that was independent of bodies would be that it would make physics, the study of that which grows and changes, fundamentally incoherent. (This is not to say that the alternative hypotheses have no problems.)

The basic difficulties have to do with determining how bodies would behave in void, how if at all void would affect bodies or their motions (and vice-versa), and in general how causality (or any regularities that could be used in explanations) could be ascribed to a universe that contained or involved void.

For example, consider (a) the view that bodies contain no void within themselves, but that there is or can be void between bodies (e.g. in outer space, or if you think that gases are composed of very tiny solid bodies suspended in void); and then (b) the view that there is void both within bodies and in between bodies.

(a)Suppose that there is no void within bodies, but there is void in between bodies (or at least between some of them). We take it that when bodies collide with each other, each resists the other to some extent. But what does something that lacks body do when it meets with a body, or vice-versa? There is not "contact" in the sense that there is contact between bodies, but what happens? We can say that that which lacks the qualities that body has will not do what body does, but what will it do? Just because void lacks the qualities that body has, does not necessarily imply that void will have no effect on body. Just because we cannot perceive void directly (through the 5 senses) does not necessarily mean that it will have no effect on that which we can perceive through the 5 senses. But we don't know what effect it will have, since we can't perceive it directly: we could not test any hypothesis about what void did, because there would be no way to distinguish actual void from a situation in which bodies were present but unperceived (hidden in some way).

But suppose that we took it that void lacked all qualities that bodies have. One way to determine how bodies will behave when confronted with void, we might think, would be to consider what those same bodies do when they are confronted with smaller and smaller or less and less dense (less and less powerful) other bodies; and then extrapolate what would happen as those smaller bodies decreased to zero.

The problem with that procedure is that the difference between a very small or weak body and no body at all is not just quantitative but qualitative; and not just quantiative and qualitative but a matter of existence versus non-existence. (An analogy might be the difference, in terms of effects on the world, between a person living 75 years, that person living 20 years, and that person's never having been born at all.) We cannot argue that the difference between the effects of the smallest body and the effects of no body is the same as the difference between the effects of two of the smallest body put together and the effects of one smallest body alone. We just don't know, because we cannot perceive the absence of body directly. Even in arithmetic, we say that we can divide any number by any positive number (or negative number, in algebra), but that we cannot divide a number by zero. We approximate the value of all numbers divided by zero to be the same, namely infinity; but first of all this is an approximation (the actual value is simply unmeasurable, "off the scale", a discontinuity), and second of all we should remember that no other "finite" integer divisor produces the same quotient when divided into all integers.

We also must ask whether movements of bodies displace void. If so, how? If not, in what sense is void still present when bodies are in it? These questions would have to be answered in order to explain how motion works or what motion involves. If body displaces void, they must have something in common (which they are not supposed to), and must both be "in" a place; what would that place be, if neither body nor void; and would it be in a place? If body does not displace void, or vice-versa, why should we think that void is present; why must it always be present; and how does it allow movement?

(b)But what about the modern hypothesis that all bodies contain void within them? In terms of interactions involving bodies traveling through void, it would seem that the same problems could arise: we just do not know what void would do alone. We say it would produce no resistance, but one might still ask what if anything it does do. Also, one might ask how it could be (or be known to be, or be known not to be) involved in causal interactions with bodies. Things are made more complicated by the idea that what we call bodies contain some void; for then we must ask which properties or attributes of bodies are due to the non-void "parts", which attributes are due to the void "parts", and which are due to the combination. If both "parts" are required in order for bodies to be what they are and to be perceptible as bodies, or if we don't know that they are not both required (because separating them, if we could do that, would destroy the bodies as bodies), then we do not know what exactly void alone (or non-void alone) does.


But our modern models of the universe involve vast stretches of empty, matterless space. What's wrong with that?

There is no denying that our modern models have had quite a lot of predictive success (also a fair amount of inter-model conflict). But that does not mean that they are internally consistent or coherent, or that they do not have any problems.

The models we use liken physical space to mathematical (geometric) "space", or try to. But there is a potentially important disanalogy. Mathematical space is composed of points. Points are conceived as dimensionless, although when put together they form (or are) mathematical space, which has dimension. To find the distance between objects (cubes, spheres, triangles, etc.) or points "in" mathematical space, we draw lines, and we have equations for measuring them. Mathematical objects do not "occupy" mathematical space in the sense of being held by it or in the sense of displacing it; they ARE mathematical space, or parts of it. Mathematical space extends infinitely in all directions, and if you are not interested in objects in any part of it, you can treat that part of space as a bunch of points that simply constitute empty distance.

In modern physics, we use mathematical methods to figure out distances, velocities, etc. of bodies too far away or too small for us to measure directly. For these purposes, we assume that physical space is relevantly similar to mathematical space. We also assume that what is between planets, and also what is between subatomic particles, is void, space that lacks what we today call "matter". But this means that in an important way, we take physical space to be DIFFERENT from mathematical space. In mathematics, both the objects and the "space" they are in are taken to be the same thing: points. Objects are parts of space, not separate things that are contained by space. But in physics, objects (bodies: planets, molecules, and so on) are not supposed to be the same as the space that they are "in"; that space is supposed to be the absence of matter. Thus it is not clear that we are justified in using a mathematically-based model of the universe.


This does not mean that void does not exist, only that we have no proof that it does exist, and that our hypotheses about it involve some inconsistencies and unsupported (possibly unsupportable) assumptions.



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*For those who are thinking about the modern mathematical conception of infinite series: Stop it! Aside from the question of whether Greek mathematics could have supported such a notion, we must consider Aristotle's conclusion about the sense in which the infinite can coherently (or most coherently) be said to exist as regards the things studied in physics. He says that the infinite must be taken to exist not in "actuality" (energeia, entelecheia, depending on the context) but in "potentiality" (dunamis). Roughly, that means that in physics we are not to look at the infinite as completed, or as all there (or all functioning) at once, but rather as that of which there is always/will always be more (or less, in a way, in the apeiron by division). Thus as concerns a series which is said to converge to some number as some other number n goes from 1 to infinity, Aristotle might say that in studying the things that physics covers, we don't take n as ever getting to infinity; infinity (as a number) is exactly that which is never reached.

**I realize that Descartes takes it that nothing we are not aware of (or nothing that we could not be aware of) can have an effect on that which we are aware of (see e.g. Discourse Parts 4 and 5), but does he ever prove, support, or defend this? I haven't found anything of the kind in Descartes; but if you do, please let me know.

***I am not arguing or claiming that Aristotle thinks that the universe is or must be understandable by us. Rather, I would propose, Aristotle is examining what conditions we would have to accept as being in place in the universe in order for us to make sense in talking about growth and change, or in order for us to say anything coherent and non-contradictory about growth and change. He is not saying anything about whether those conditions are actually in place, whether they can be known to be in place, whether what we call 'growth' and 'change' are really what we think they are, and so on.

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