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29a Bayle Dictionnaire, “Zeno F & G” Bayle’s article on Zeno carried sceptical arguments to the bastion of dogmatism: intellectual knowledge. The philosophers of Bayle’s day had become increasingly worried about what can be known on the basis of sensory experience, but most of them had been content to take intellectual knowledge to be beyond doubt. Descartes had considered the senses to be only able to tell us whether things are good or bad for us, and often even then only on the basis of past experience. He did not consider them to be able to tell us very much about what objects are really like. And he had had great difficulty proving that our sensory experience can even do so much as give us reason to believe that there is an external world containing bodies. In light of the weaknesses of Descartes’s arguments, later Cartesians had given up on this enterprise of attempting to prove that there is an external world, claiming that this is “too difficult” a task for our inadequate powers of knowledge to take on and that we must accept the existence of an external world as part of the revealed word of God. Locke, for his part, had maintained that our senses cannot lead us to any knowledge of the “real constitution” of individual things responsible for giving them their secondary qualities or powers, that our scientific reasoning is confined to drawing conclusions from “nominal essences” we have ourselves invented, and that we cannot claim knowledge of the continued existence of bodies when not perceived or of the existence of other minds than God. But the major philosophers up to Bayle had maintained that whatever problems we might have obtaining sensory knowledge of matters of fact concerning the existence of particular things, we could at least obtain purely intellectual knowledge of such things as the principles of mathematics. Locke and Descartes had maintained that we can simply “intuit” the existence of certain relations between ideas. All we need to do is remember the ideas, and compare them with one another, and we can intuit that white is not black, that orange is more like red than it is like green, that cubes have six faces, or that two plus three is five. And in a number of cases where we cannot directly intuit the relation between the ideas, we can at least demonstrate it, beyond any possibility of doubt, by means of a chain of intuitions. Locke’s favourite example was the “angle sum theorem,” which is demonstrated by showing the equality of the internal angles of a triangle to the angles making up the straight line produced by extending one of its sides and drawing a line parallel to the opposite side of the triangle from the vertex where the straight line was extended. Bayle’s Zeno article questioned the basis for even that certainty, by resurrecting powerful reasons for concluding that mathematical truths that we consider to be intuitively evident are in fact deeply paradoxical. The arguments begin with a restatement of Zeno’s paradoxes of motion, and hence with sceptical arguments against the coherence of mechanics and all the sciences founded on it, since they presuppose motion. But Bayle went on to offer other arguments that, so he said, Zeno “could have given” to show that extension and space, and so the science of geometry, are incoherent. (As a matter of fact, Zeno authored a composition paradox that is very much like the first of the arguments Bayle offered against the intelligibility of extension. It is unclear why Bayle did not recognize this.) Bayle’s “Pyrrho” article had already laid the basis for scepticism about the truths of demonstration by raising the possibility that I may only have just been created, with a set of memories that represent me as having a past life I never in fact had. Insofar as demonstration involves remembering earlier intuitions while going on to build on them with later ones, this opens the possibility that the memories I rely upon in demonstration may in fact be mistaken because I never actually intuited those relations but was only just a moment ago created with the false memories of having done so. But in “Zeno” Bayle went much further. He attacked the clarity and coherence of the currently occurring intuitions themselves. His success in this enterprise is grounded in the fact that “intuitive” knowledge, as understood by Descartes and Locke, is not coextensive with what we today call “analytic” knowledge. Analytic knowledge is knowledge that is true in virtue of the definition of terms (never mind whether anything exists that is named by those terms). “Triangles are three-angled closed figures” is analytic because that is just the definition of the word, “triangle.” But intuitive knowledge has a broader basis than the meanings of words. It is instead grounded on the content of ideas. When I intuit that orange is more like red than it is like green, I am not simply unpacking what is contained in the definition of the words, “orange,” “red” and “green.” This similarity relation is not in fact contained in the definitions of these terms. The same holds for a number of geometrical truths which do not depend on the definitions of the concepts, but rather on what is found upon comparing ideas of the objects. Even given Descartes’s invention of analytic geometry, which made it possible to reduce geometrical propositions to mathematical ones and demonstrate them by appeal to the truths of arithmetic, does nothing to change the fact that geometrical truths are independently known — and were first discovered — by the technique of using rulers and compasses to diagram the contents of our ideas of the geometric objects and so reveal features of the objects that do not arise from their definitions but from how they are conceived as embedded in a three-dimensional Euclidean space. That the internal angles of a triangle sum to two right angles is not true of a triangle drawn on the surface of a sphere. Think of the triangle made by drawing a line from a point on the Equator of the Earth to the North Pole, then making a ninety-degree angle at the pole, and drawing another straight line down to the Equator. This spherical triangle contains three ninety degree angles, not two. The angle sum theorem is only true of triangles drawn in a “flat” space, not a curved one. Accordingly, the theorem does not follow from the definition of the word, “triangle,” but from inspection of what is contained in the abstract idea of that “complex mode” (as Locke called it). Many geometrical truths are like this. They are based on “intuition” of relations between ideas rather than analysis of terms. In the case of geometrical knowledge, the additional information that goes beyond what is contained in the definition of terms arises from constraints that the nature of space places on objects. It is not the definition of “triangle” that ensures that the internal angles of a triangle will sum to two right angles but the flatness of the space in which the triangle is embedded. That flatness is something we intuit when we inspect our idea of a triangle and not something that follows from its definition. Nor is it something that could simply be added to the definition. “Flatness” is something that is very hard to define. It is, in Locke’s terms, a simple idea. You have to intuit it to know what it means and what it does to triangles. You can’t gather that from a definition. — At least, so it seemed. This is what really gave Bayle’s
arguments their punch. If the paradigm
of intuitive knowledge, intuitions about geometrical objects, depends on
inspecting ideas as they are traced out in space, but there is something
fundamentally incoherent about space, then that makes all our geometrical
knowledge incoherent as well. If, in
addition we follow According to Zeno’s original arguments, an arrow in flight cannot move continuously from one point to the next, a faster runner can never catch a slower one, bodies in motion must move with more than one speed at once, and a runner will not only never be able to finish a race, but will not even be able make the least progress away from the start line. The upshot is that there can be no motion. According to Bayle’s embellishments (and Zeno’s original composition paradox), there is no way that space could actually be built up from spaces, and space and extension must be merely ideal things that could not have any real existence and that can only be conceived because we don’t enter into the details that render their conception incoherent. No real space means no external world — at least no external material world composed of bodies set outside of bodies, whatever we might think of the possibility of the existence of other minds separated from our own in some way that does not involve being differently disposed in space. But though Bayle offered these arguments, he was far from endorsing their conclusions. In a concluding remark to note “G” he declared that he very much doubted that Zeno had ever been seriously of the opinion that there is no motion or extension or body, and he made clear what his own purpose was in repeating and embellishing Zeno’s arguments. The purpose was not to endorse the conclusion but rather to demonstrate just how weak all of our powers of knowledge are and just how far we are dependent on the revelation offered by a more magnificent being if we are to know even the most basic and (supposedly) intuitively obvious things. Those who have once appreciated just how absurd our notions of space and body are ought to be chastened when they turn to contemplate such Christian mysteries as the doctrine that there are three persons in one God or two natures, divine and human in the person of Christ. Such religious mysteries are less absurd than the things we take for granted, space and body, and we ought to confess our ignorance on all of these matters and look to see if we cannot find some greater guide who will reveal the truths our own knowing powers are unable to grasp. QUESTIONS
ON THE
1. Why could there be no
moment at which a moving arrow moves?
2. Why is it impossible
for a moving object to go from one extremity to the other? What important distinction between matter
and time is involved in this answer?
What absurd consequence would follow if an object could go from one
extremity to the other?
3. Why could an hour
neither begin nor end if time were composed of an infinite number of parts? Reading note: Bayle’s
reference to what will be said in the following remark concerening
the difficulty of determining the speed of motion is to Note G, the part
numbered “VI.”
4. Identify three things
that cannot be reconciled with the idea that a moving body might
simultaneously move with two different speeds relative to different
surrounding bodies.
5. What are the only three
conceivable types of composition of extension?
6. Why can extension not
be composed of mathematical points?
7. Why can extension not
be composed of extended but indivisible atoms?
8. What is the “sophism”
(fallacy or logical error) in the argument that if extension is not composed
of indivisibles (mathematical points or extended but indivisible atoms) then
it must be infinitely divisible? Reading note: Bayle’s
reference to professors needing to invent a jargon for students to use in
disputations. It was common practice
in the medieval universities for students to engage in public disputations
over such philosophical topics as infinite divisibility. Women being excluded, male relatives would
attend these disputations and follow the scholar’s progress with the same
sort of devotion modern parents lavish on their offsprings’
efforts in hockey or soccer. Many
medieval philosophical works were written up as a series of “disputed
questions” in which a question would be stated, a number of contrary
responses to the question canvassed, reasons for and against the different
responses surveyed, and a resolution proposed that did as much as possible to
capture what is correct in all the different responses while explaining the
grounds of the errors that led to the divergence of opinion. As the same questions were debated for
centuries and successive authors found it necessary to reference the views of
all their important predecessors the disputations became increasingly
scholarly, complex, and detailed. The
public in attendance at the disputation could make no sense of what was going
on, but they were nonetheless impressed by the show of erudition and the
unintelligible technical jargon.
9. What makes the
hypothesis of infinite divisibility the strongest of the three? 10.
What makes it as clear and evident as the Sun that
extension could not be infinitely divisible? 11.
What is required for an extended substance to exist? Why can this requirement not be met if
space is infinitely divisible? 12.
Why can extension exist only in the mind? 13.
What conclusion did Bayle draw from the fact that a cannon
ball, coated with paint and rolled along a table, will draw a line of paint
on the table? 14.
Would Bayle have accepted Locke’s distinction between type
(i) ideas of the primary qualities of bodies and type (ii) ideas of the
sensible qualities that bodies cause us to feel in virtue of the real
constitution of their insensibly small parts? 15.
Why are geometrical proofs of infinite divisibility
equally effective at disproving infinite divisibility? 16.
Give two reasons why infinite divisibility forbids the
beginning of motion. Give one why a
ball rolling down an inclined table could never roll off the edge of the
table. 17.
Why could one body not move faster than another? Why could we not suppose that when one body
moves faster than another that the slower one stops in its motion for longer
or shorter intervals? 18.
Given that motion does in fact undeniably exist, what is
the point of giving arguments to prove that it does not? NOTES
ON THE Note F — Zeno’s motion paradoxes The arrow. Note “F” of Bayle’s article on Zeno is devoted to a restatement of Zeno’s famous four paradoxes concerning motion: the “arrow,” the “racecourse,” the “Achilles,” and the “moving rows.” Beginning with the “arrow,” Bayle represented Zeno as arguing that there is no moment in time at which a flying arrow can actually move from one point in space to the next. This is because no body can be in two places at once. At every moment of time, therefore, the flying arrow must be in a place that is exactly “equal to itself” — it cannot occupy a space larger than it is. But a body is only in motion if it is in the process of leaving one space and passing into another. Since at each moment the arrow is in a space no larger than it is, it cannot be in motion at any moment. It would seem to follow that there is no time at which the arrow can move. An old answer to this argument, repeated by Aristotle but probably not original with him, is to grant that the arrow does not move at any single instant of time, but to insist that it nonetheless moves over time. Suppose an arrow travels at 60 kilometers an hour. In one minute it will travel 1 kilometer. In one second it will travel 16 2/3 meters. In a tenth of a second it will travel just under 2 meters. As the time intervals get shorter, so do the space intervals. At a bare instant of time no time passes, and so of course the arrow does not move over any space at an instant. But it does not follow that the arrow is not in motion over an interval. However, Bayle considered this answer to be ineffective. To work, the answer has to presume that time is infinitely divisible. Suppose, to pick an arbitrary measure, that there are no more than one billion parts in a thousandth of a second. Keeping with our previous example, it would then have to follow that in a smallest possible unit of time (ten trillionths of a second) the arrow would have to travel just under twenty trillionths of a meter — otherwise it would not manage to travel just under 2 meters in a ten trillion of these shortest units. But then, in a single instant of time, a ten trillionth of a second, it must occupy a space larger than itself, contrary to the axiom that no body can be in two places at once. If you don’t like the idea that a moment should last as long as a ten trillionth of a second, make it shorter. However short you make it, as long as it has any magnitude, the body will have to travel some distance over that interval and so be in a space larger than itself. The only way to escape this consequence is, as Bayle perceived, to insist that there is no end to the division of time. But Bayle claimed that time could not be infinitely divisible. Unfortunately, his argument for this crucial point is so abrupt as to be open to the charge of being a non sequitur. He asserted, plausibly enough, that no two parts of time can coexist. The parts of time can only exist one after another. And before a later one can come to exist, all the earlier ones in the series must first come to exist. From this fact, that each of the earlier parts of time must sequentially cease to exist before a later one can come to exist, Bayle leapt to the conclusion that time cannot be infinitely divisible. We might well ask why. His intuition may have been that in an infinitely divisible time, there is no moment that is the immediately prior moment to any given moment. Given any moment, and any earlier moment, however close to the first one, there will be no end to the number of intervening moments. From this it would seem to follow that, if the moments can only pass successively, one after another, and not multiple ones all together, it will be impossible to get from one moment to the next. Not only will you never reach the immediately prior moment that has to pass before the next moment can come to be (because there is no immediately prior moment), you will never be able to get beyond the moment you are at, because there is no immediately successive moment to the one you are at, but always one that would have to pass before any other one you might pick. Since time does pass (so Bayle supposed), it follows that it cannot be infinitely divisible. This is a very subtle argument. It might be answered by such sophisticated means as appealing to the theory of relativity to justify the claim that there is no absolute simultaneity, that temporal passage is merely “relative” to observers, and that from a neutral perspective time is merely a fourth temporal dimension with parts that all coexist, like those of space. But that was unthought of in the eighteenth century and even today it remains controversial whether it really solves the problem or just shunts it to the side (there still remains a problem of how time could even seem to pass for us). A further answer, more accessible to the 18th century mind, is that Bayle’s argument illegitimately supposes that time is composed of moments. It is not. It could only be composed of moments if it were finitely divisible after all, so that you could divide down to something that is not further divisible. Because it is infinitely divisible, it is not composed of moments but of times. However far down you divide a time, you will create only intervals that still contain a succession of further intervals within themselves and that are themselves preceded and followed by only finitely many intervals, because you have divided the time only finitely times. And because the time is infinitely divisible, you will never be able to finish dividing it, but will only go on forever creating more such intervals. You will never arrive at moments that cannot be further divided and that are the ultimate constituents of time. Moments are not parts of time but merely limits, marking the spot where one interval of time begins or ends. Not only does Bayle’s argument tacitly beg the question by supposing that time is not infinitely divisible, it tacitly supposes that there is more than one time: a time constituted by a series of moments, and a super-time over which these moments come to be and pass away. After all, if moments come to be and pass away, they could do so more quickly or more slowly. This gives rise to the thought that a thousand moments might take a ten thousandth of a second to successively arise and pass away, or, by being more snappy about it, might manage to do the job in only a hundred thousandth of a second. The rate at which moments pass might thus be faster or slower, on this view of the nature of time. But that is absurd. So is the alternative that moments take no time at all to arise or pass away. Were that the case all of time would pass in a flash. The true antidote to these absurdities is to reject the view that time is constituted by the arising and perishing of moments. There is only one time, the time with reference to which all other things arise and perish. If moments could pass more quickly or slowly then they would not be the constituents of this one, true time but of something else. The racecourse and the Achilles. More might be said on these matters, both pro and con. Granting that time is only finitely divisible, but supposing that space is infinitely divisible, Bayle had an easy job presenting Zeno’s racecourse and Achilles paradoxes. A runner would never be able to finish a race because each part of the race course would have to be crossed before crossing the next part. Given infinitely many parts in an infinitely divisible space, but only finitely many moments in a finitely divisible time, there would not be enough times available to be at each space. The only way around the problem would be to suppose the runner in more than one place at a time, which violates the dictum that no body can be in two different places at once. Similarly, a faster runner could never catch up to a slower one. Even supposing the faster runner could reach the spot the slower runner started from, doing this would take some time. Granting that the slower runner was not at rest over this time, it will have advanced some distance over this interval. Because the faster runner cannot be in two or more places at once, it will take some time to cross this remaining interval, over which time the slower runner, not being at rest, will have advanced some distance, and so on. Given the infinite divisibility of space, the slower runner will always be some incremental distance ahead. We might wonder why, since Bayle supposed that time is not infinitely divisible he would not have said the same about space. The answer is that Bayle had no particular commitment to the infinite divisibility of space. He would have been happy to let people choose which option they preferred to accept. Bayle’s restatement of Zeno’s motion paradoxes is for those who accept the infinite divisibility of space. Those who do not accept this view are referred to note “G” section VI, which raises equally serious problems for the supposition that motion might occur in a finitely divisible space, and to note “G” section I, which challenges the view that space could be only finitely divisible. The moving rows. It is more difficult to read Bayle’s restatement of the moving rows paradox sympathetically. As classically presented the paradox involves three rows of marchers in a stadium. One of the rows is stationary and serves as a reference point for the other two, which are moving past them in opposite directions at the same time. Time 0
Bayle’s restatement ignores the top row and simply envisions two bodies passing one another with reference to a surrounding space defined by a table or other ambient objects. The paradox proceeds by asking us to suppose that the B’s and C’s are moving at a rate of one position per moment of time and to envision what has happened after the first moment. The situation then should look like this: Time 1
As stated, the paradox is supposed to arise from the fact that in a single moment of time the B’s have simultaneously moved with two different speeds. Purportedly, they have crossed two intervals of space on their bottom side while at the same time only crossing one interval of space on their top sides. As thus described, the situation hardly seems paradoxical. The B’s only crossed two spaces on their bottom side because those spaces were simultaneously in opposite motion. Really the B’s only crossed one space in absolute space (as defined by the A’s) on either side. They just look like they crossed more than one on the bottom side because we are judging with reference to the C’s rather than the A’s. And there is nothing paradoxical in the claim that a body should move with one speed relative to one set of bodies and with a different speed relative to a different set of bodies if one of those sets of other bodies is itself in motion. Oddly, Bayle seems to recognize this objection only to obtusely insist that it does not constitute a full answer to the problem. I
admit that it is right to observe this difference [between motion with
relation to a space that is at rest and motion with relation to a space that
is itself in motion], but this does not remove the difficulty. For there still remains the problem of
explaining one item that seems incomprehensible: It is that in the same time that [the B’s]
traverse [two spaces] on [their] southern side, [they] only traverse [one] by
[their other] side. We might object that there is nothing incomprehensible about this and that it is completely explained by the point Bayle has conceded: that the motion only has different speeds when it is referred to different surrounding bodies. As long as we confine us to measuring relative to either one set or the other of the surrounding bodies, the motion on both sides will be the same. Both the top and bottom sides move one spot relative to the A’s and both the top and bottom sides move two spots relative to the B’s. There is something else Bayle said, however, that puts a different spin on his answer. …
these three essential properties of motion must be remembered: (1) A moving
body cannot touch the same part of space twice successively; (2) it can never
touch two parts of space at the same time; (3) it can never touch the third
before the second, nor the fourth before the third, and so on. Whoever can physically reconcile these
three things, with [the case of the moving rows] will not be an incompetent
man. To see what may have been in the back of Bayle’s mind, recall his dictum that time could not be infinitely divisible and suppose that the B’s cross one A on their upper side in the shortest time possible, a single moment. But then, in this single moment, they have passed over two C’s on their bottom side. This is incompatible with the three propositions cited above. The lead B must have passed over the first C before it passed over the second one, otherwise it would either have been in two places at once, or have touched a more remote place before touching a more proximate one. But if it crossed over the entire A either just in the time it passed over the first C or just in the time it passed over the second C, then it would have touched the same part of space twice in succession (it would have, in effect, have rested underneath the 2nd A from the left while completing its transit of the C’s, or have waited until it was across one C before starting its transit across the A). Accepting that this could not have happened (that is the first of Bayle’s numbered dictums above), there must therefore have been a prior stage in the process that looked like this: Time 0.5
At this stage, the lead B and the lead C would each have crossed only half of an A. This is not a problem if space is infinitely divisible. But it is a problem if time is not infinitely divisible. In that case there is no time (no 0.5 moment between time 0 and time 1) at which this arrangement could have come up. We cannot attempt to salvage the situation by saying that the B’s must in fact have used up two moments to cross one A. This is because the situation with the moving rows is infinitely reiterable. In the time it takes for the B’s to cross one C on their bottom side, they only get halfway across an A on their North side. This means, reciprocally, that both the A’s and the B’s must be divisible in half. The leading edge on each of the first three B’s marks the halfway point on an A, but reciprocally the edges of the A’s mark halfway points on the Bs’. This means that the lead C must have crossed two spaces, the first half of the lead B and the second half of the lead B, in one time. But no body can be in two places at once, or skip over an intermediate place to get to a later place. We cannot recognize a situation in which the lead C was only halfway across the lead B without correcting ourselves yet again and maintaining that it actually took two units of time for the lead C to cross the lead B, and hence two units for the lead B to get halfway across the lead A (so four to get all the way across, not one as we had originally maintained). But in this smaller unit of time the B’s will cross a yet smaller portion of the A’s, so will themselves be divided into yet further parts, requiring a yet shorter time for the lead C to cross just the first of these parts. There will be no end to this reiteration of considerations. Insisting on continuous motion of the two rows forces us to divide both the space and the time over which the motion occurs into infinitely many intervals. But time is not infinitely divisible according to Bayle. If we accept that there are indivisible moments in time, then something physically unaccountable happens during a single moment of the motion of the rows. The B’s cross just one part of an object on the upper side while crossing two parts of an object on the lower side. But this is impossible without either being in two places at once, or skipping over a place, It is worth reiterating that Bayle’s purpose in presenting and defending these arguments was not to seriously defend the proposition that motion does not exist. As he noted, those who attempt to refute the arguments by the simple expedient of getting up and walking up and down only prove that they missed the point. We all know that motion exists. What the arguments really prove is that even though motion is as obvious to us as anything can be, it defies all our powers to understand what is really going on when bodies move. The simple idea that a moving body passes over spaces over times conceals real problems that are grounded in the nature of temporal passage, the consequences of infinite divisibility, and our commitment to principles such as the principle that a moving body must be in motion at every moment of its motion, that it cannot be in two places at once, that no two moments of time can coexist, and that a moving body must cross each space in its path sequentially. Note G — other paradoxes of extension and
motion The composition paradox. In note “G” Bayle turned to consider a further set of problems. The first and most intricate paradox he presented is Zeno’s composition paradox, though Bayle did not present it as having originated with Zeno but spoke as if it was a point that Zeno could have brought up. The paradox addresses a question that was brought up concerning the earlier paradoxes: if we think that time is not infinitely divisible, why should we think that space is? The composition paradox shows why space cannot be taken to be composed of indivisible parts. But it does more than this. It then goes on to show why space cannot be supposed to be infinitely divisible. According to the composition paradox there is in fact no good option. If we pick one alternative it is not because it makes any sense but because we consider the problems with the other alternative to be even more insurmountable. In fact, all the alternatives are equally insurmountable and the only logical conclusion we can draw is that space cannot exist — or that if it does exist it is only as an “ideal” thing, like a line that has no breadth or a point that has no dimensions whatsoever. If we suppose that space is composed of indivisible parts, then there are two options. Those parts might either have no size whatsoever (they might be mathematical points), or they might have some size. If they have no size then they cannot add up to anything, and so could not, by being aggregated together, compose any extension. But if they have some size then they could not be indivisible. Anything that has size has a right side that is placed outside of the location of its left side. But whenever two things occupy two different places, one outside of another, it is possible to conceive of a division between them. It does not need to be the case that it is possible to actually separate them by means of any physical force. Separation involves moving the two from the locations they are at to yet other locations. But the bare fact that they are, even prior to separation, placed at distinct locations outside of one another proves that they are not truly one and the same thing but two things that, however closely proximate they may be to one another, are still two and not truly one. Nothing that is extended could therefore be a true atom. It would have to be something that is composed of parts. And then the question simply recurs concerning those parts: are they mathematical points with no extension (then they could not add up to anything extended in aggregation), or are do they already have some extension of their own (then they must be further divisible. We appear, therefore, to be forced to accept that the parts of space must always themselves be spaces that are further divisible into spaces, and hence that space must be infinitely divisible. But this option is no more tenable than the alternative. (Bayle remarked that the only thing in its favour is that it is more unintelligible than the alternative making it possible for those who opt for it to defend their positions by making use of obscure jargon, specious distinctions, and the claim that infinity is of its nature beyond the capacity for finite minds to grasp.) The problem here is that if space is infinitely divisible then it must have infinitely many parts. We have already seen that these infinitely many parts cannot have 0 extension. 0 added to itself, even infinitely many times, does not make anything more than 0. But if the parts have any extension at all, however little, and there are infinitely many of them, then they would have to add up to something infinitely large. There is no way, therefore, that a small space, like that occupied by a grain of sand, could be divisible into infinitely many infinitely small parts. However small the parts were, as long as there are infinitely many of them, it would take the whole universe to contain them all. They could not be squeezed into the volume of a grain of sand. A standard answer to this argument is that it begs the question by tacitly presupposing that space is divisible into indivisible parts, contrary to what is said by those who insist that it is not. Bayle’s argument does this when it asks us whether the infinitely many parts into which a space is divided are of no extension or of some extension. But if space is truly infinitely divisible, the answer continues, then you never arrive at infinitely many ultimate parts. It is the division that goes on to infinity, not the number of parts. At any point in the division you only ever produce finitely many parts of some finite size such that the aggregate sums to the exact extension you started off with. But you never produce infinitely many parts. You just go on producing smaller and smaller parts that are always further divisible, only ever producing finitely many at each stage, and never reaching the stage where you are finished and have actually separated infinitely many parts from one another. The question of whether these infinitely many parts are each of zero extension or of some extension is therefore illegitimate. Bayle considered and dismissed variants on this objection in his article. His core point was that if space exists, then it exists through being built from as many parts as it is in fact built up from. The parts have to be there from the start in order for there to be a whole. They don’t only come into existence through some process performed by us, such as cutting in half with a knife, or marking off bisecting points by ruler and compass constructions. They have to have been there and been what they are from the start in order for the space to exist quite independently of anything we might do to distinguish them from one another. It is a mere sophism, therefore, to claim that a space does not have infinitely many parts simply because our process of dividing it into parts must go on forever. If it exists at all it exists through the existence of as many of its parts as it contains. That number is either finite, in which case the space is only finitely divisible, or infinite, in which we can ask of each of those infinitely many parts whether it has some extension or has no extension. The only way we can viably take divisions between parts of space to be things that do not exist until we do something special to bring them into being is to consider space to be a merely ideal thing — a fiction of our imaginations rather than anything that has a real, independent existence. This is a debate that continues
to be a live one today (an excellent recent resource is our colleague, John
Bell’s The continuous and the
infinitesimal [ Bayle went on to note that there are other problems with infinite divisibility. Were space infinitely divisible then it would paradoxically be the case both that there could be no contact and hence no composite substances, and that there would have to be more than contact — bodies that touch would have to fall on top of one another and have infinitely many parts coexisting in the same space. To explain the first result in more detail, Bayle claimed that the difference between a composite substance such a piece of wood or iron, and an aggregate of discrete substances, such as a pile of gravel or a volume of ocean spray, is that the parts of the composite are stuck together and so in contact with one another whereas the parts of the aggregate are separated. (We no longer think that this is exactly the case, but early modern philosophers had only recently reconciled themselves to the possibility of action at a distance in the case of gravitation and were only beginning to countenance the possibility that cohesion might be due to some kind of attractive force that operates at a distance and not to interlocking and hence contact of rigid parts.) But if space is infinitely divisible then between any two purportedly adjacent parts there are infinitely many other parts. There can therefore be no contact of bodies. Any body is separated by an infinite space from all other bodies. Composites can therefore not exist. There can only be aggregates of individual bodies, swimming in an immense void. Yet, paradoxically, infinitely many of these bodies must overlap in the same space. According to Bayle, there is empirical proof of this. Coat a cannon ball with paint and roll it down a table. It leaves a line of paint behind it. This proves that the cannon ball must have touched the table despite what was proven in the previous paragraph. Notwithstanding that there is an infinite space separating the lowermost part of the cannon ball from the uppermost part of the table, our senses assure us that they touch, because the cannonball would not leave paint on the table otherwise. But if they touch, this touching cannot take the form of the lowermost point of the cannon ball being immediately adjacent to the uppermost point of the table. For there are no immediately adjacent points in an infinitely divisible space. All points are separated from all others by infinitely many intervening ones. So the contact of the cannon ball cannot take the form of immediate adjacency of parts. It must take the form of interpenetration. The lowermost point of the cannonball must exist in the same place as the uppermost part of the table. But now recall that in an infinitely divisible space there are no points. If one part of the cannonball coexists in the same space as a part of the table, then those two coexisting parts must themselves be infinitely divisible into infinitely many coexisting parts. Infinitely many table parts must coexist with infinitely may cannonball parts in the same space, in violation of the principle that no two bodies can be in the same place at the same time. There are yet further problems posed by the very geometrical demonstrations that are employed to prove that space must be infinitely divisible. These demonstrations are based on the principles that any two points define a unique straight line, and that, given any point, P, not on a line, AB another line drawn through P can intersect AB in at most one point. Granting that this is the case, consider two concentric circles with radii drawn from their common center out to their circumferences. Make the outer of the two circles as large as you want, and make the inner circle as small as you want. Now recall that any point on the diameter of the immense outer circle, F, and the center point of the circles, C, are two distinct points and so define a straight line, FC, the radius from the center, C, out to that point, F. And now consider any other point, G, on the circumference of the immense outer circle. This point does not fall on the line FC we previously drew. So any line drawn through G can intersect the line FC in at most one point. So consider the line from this second point, G, on the circumference of the large outer circle to the center, G. The one point of intersection of the lines FC and GC is C, the center of the two circles. But this means that the line FC must cross the circumference of the small inner circle in a distinct point, X, from the point where the line GC crosses the circumference of the inner circle. Otherwise the lines FC and GC would cross in more than one point. Now make the larger circle even larger, the smaller circle even smaller, and the distance between the points F and G even less. The same result will still have to follow. FC and GC will only be able to intersect at C, meaning they must cross the circumference of the small inner circle in distinct points. This demonstration and variations on it involving triangles or other geometrical figures were popular at the time. They were taken to prove that space must be infinitely divisible. This is the only supposition on which it would be possible for the inner circle to be crossed in a unique pair of points by the lines FC and GC regardless of how small their angle of incidence at C becomes. But Bayle noted that these demonstrations of the infinite divisibility of space have an untoward consequence that does as much to demonstrate the absurdity of the thesis of infinite divisibility as it does to prove that space cannot be only finitely divisible. The demonstration implies that for every point on the circumference of the larger circle there must be a point on the smaller circle. So there must be exactly the same number of points on each circle. But when two bodies are such that they can be laid on top of one another and for each part of the one there is a corresponding part of the other that it lies upon, we say that those two bodies are the same in size or length. So the inner circle has the same circumference as the outer one, in defiance of what our senses tell us. We see that could not be the case. And yet the demonstration shows us that for every unique part of the one there is a unique part of the other. Finally, repeating a point already made in the “Pyrrho” article, Bayle noted that if we accept the argument that bodies do not actually have any taste or colour because their taste and colour appears differently to different people, or to the same person at different times, then by parity of example we ought to accept that they could not be extended. After all, the same body can look large or small, or of one shape or another, depending on various circumstances. If it is a good inference to conclude that this sort of relativity means that all we are sensing when we sense colour or taste is some effect in us that the body may not possess, then it is a good inference that the same sort of relativity means that all we are sensing when we sense shape or size is some effect in us and that the body that causes this effect may well be in no way extended. I mention this objection at the end, though it is not where Bayle placed it, because it is properly considered to be an objection to the general claim that extension exists rather than an objection specifically directed against the thesis that this extension is infinitely divisible. The comparative motion paradox. Bayle closed remark “G” by stating a number of other paradoxical features of motion. A number of these paradoxes appeal to arcane details of scholastic and Cartesian metaphysics that make them less interesting and accessible than the paradoxes that have been discussed so far (II-III and V). One (IV) is simply a restatement of the racecourse paradox focused on the inverse possibility of starting to move in an infinitely divisible space, though it ends up restating the point about the impossibility of crossing the finish line as well. But one final paradox deserves mention. Bayle noted that whatever we might think of the composition of space and time, it seems impossible to conceive how there could be differences in the speed with which things move. It seems that all motions would have to occur at the same speed: one smallest possible unit of space in each smallest possible unit of time. No body could move faster than that on pain of being in two places at one smallest possible unit of time. No body could move more slowly than that on pain of not being in motion at all. So there could be no variations in the speed with which bodies move. This is an old problem that had a standard solution: people supposed that those bodies that appear to be in continuous motion at slower speeds must not be in continuous motion at all but in a kind of staccato motion punctuated by periods of rest. While the faster moving body moves four space units in four time units, the slower one moves one, rests for one, moves for a third, rests for a fourth, and so on. The problem with this view, Bayle noted, is that it cannot account for the motion of wheels. The part of a spoke attached to the rim of a large wheel will move at a much greater speed than the part attached to the hub. Were the parts closer to the hub to rest for increasingly long times (depending on how close they are to the hub) while the parts closer to the rim move, the spokes ought to visibly stretch and bend, which they do not. Even aside from this consideration it is an odd consequence that the only bodies that are actually in continuous motion should be those moving at the fastest possible speed, the speed of light, while the motion of all other bodies is continuously interrupted. ESSAY
QUESTIONS AND RESEARCH PROJECTS
1. Bayle’s restatement of
Zeno’s paradoxes was one of the most influential episodes in the history of
eighteenth century philosophy. It was
these paradoxes that likely motivated Leibniz’s subsequent “monadology” — his view that whatever exists must be
composed of simple and indivisible parts that cannot be construed as
occupying space and so must be considered to be a kind of mind or
spirit. Hume sought to respond to them
in the second part of the first book of his Treatise of human nature.
They also had a profound influence on Kant, who not only developed his
theory of “transcendental idealism” (affirming that everything that appears
in space or time is merely an appearance and not a thing in itself) in order
to address them, but was also inspired by them to articulate a force-based
physics, according to which “physical monads” do not fill space through
addition of their own extension but rather through exercising a repulsive
force that acts at a distance from the point the monad occupies. And Bayle’s paradoxes were at play in
disputes over how to understand the infinitesimal calculus independently
developed by
1. A widespread view that
the “paradoxes of the continuum” have been resolved by modern mathematics
notwithstanding, a number of scholars are of the opinion that they remain
paradoxical. Taking the work of Adolf Grünbaum (Modern
science and Zeno’s paradoxes [ |
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