A Theory of Qualities at Boundaries including a solution to Zeno's Flying Arrow Paradox
I. General Introduction
In this paper, I propose a theory of boundaries within a continuum. It is a rather simple piece of metaphysics, but one which - to my knowledge - has not been worked out in detail to date although this is desirable. It is developed from the following assumptions:
I) There are several sorts of continua: periods, parts of space, surfaces and, as will be seen later on, a kind of entities I call qualitative spans.
II) There are also several sorts of boundaries: instants as boundaries of periods, lines as boundaries of surfaces or surfaces as boundaries of parts of space, specific determinate values of a determinable quality as boundaries of qualitative spans.
III) Boundaries are no parts of continua with infinitesimal extension (such things do not exist). They are extensionless. They fall within or bound continua. They do not make up any continuum.(1) Boundaries are empirically inaccessible.
IV) Continua are wholes which are irreducible to sets of boundaries that fall within them. Thus, a period is not a set of instants, a surface or part of space is not a set of space-points and a qualitative span is not a set of determinates.(2)
II. A theory of instantaneous states (3)
I start by developing a theory of instantaneous states. A theory of instantaneous states is a special case of a theory of qualities at boundaries. I would first like to point out why I think we need a theory of instantaneous states. I will then point towards a certain intuitive requirement which a good theory of instantaneous states should fulfil. I will do so by using a particular example: the spatial position of a clockhand in motion. Still using the example of the clockhand I will then propose a theory of instantaneous states which I think is really good. When introducing it, I will present it in a relatively simple form which already assumes the principle "Natura non facit saltus" - nature does not jump. I then want to focus on a really interesting side-effect of my theory: It solves the problem of Zeno's flying arrow.
b) Why do we need a theory of instantaneous states?
On the one hand we want to talk about instantaneous states. E.g., a great deal of physics is basically talk about instantaneous states and physics has turned out to be a useful thing. Of course, while talking about instantaneous states we want our talk to have some empirical content.
But, on the other hand, we never observe any instantaneous states. Every observation takes time. So observations take place during periods, not at instants. Instants are merely extensionless boundaries of periods. Instants are not infinitely short periods - such things do not exist. So instants are empirically inaccessible. (This is a matter of principle and not a matter of psychology - a purely mechanical measuring device can do no better on this point than a conscious observer. Neither can a conscious observer do any better than the measuring device: just as there is no instantaneous measurement, there is no instantaneous introspection, since even every thought takes time.)
So, we need a theory which ties statements about instantaneous states to observation in order to secure some empirical content for statements about instantaneous states. Let us start looking for one by fixing an intuitive requirement which it should fulfil.
c) Intuitive requirements for a good theory of instantaneous states
There are several intuitive requirements for a good theory of instantaneous states. They can be illustrated by the example of a clock with a continually proceding clockhand:
(1st requirement) If the clock is working allright, the clockhand should point upwards at 12:00.
(2nd requirement) If the clock has stopped from 12:00 to 2:00, it should point upwards at 1:00.
(3rd requirement) If the clock is working allright up to 12:00 and then remains in the same position for two hours, it should point upwards at 12:00.
(4th requirement) If it remains in the upward position for some from 12:00 to 2:00 and the clock is working again after 2:00, it should point upwards at 2:00.
The first and the second requirement are self-evident. In addition to them, it can be argued that the third and the and fourth requirement are desirable results concerning the old problem of the moment of change between opposite states.(4) This is why: If the clockhand (concerning the third requirement) did not point upwards at 12:00 it would have to proceed a little further in order to reach the upward position. We may assume that this would take some time, i.e. some time immediately after 12:00. This would contradict the assumption that at least for a short time immediately after 12:00 the clockhand does in fact point upwards.(5) The argument concerning the fourth requirement is analogous with being away from the target position instead of having reached it.
d) A bad theory of instantaneous states
Already the first requirement shows that the following theory of instantaneous states is too simple to be good:
"The clockhand points upwards at instant t iff t bounds a period of the clockhand pointing upwards to either side."
It is true that the definiens implies the definiendum. But unfortunately, the definiendum does not imply the definiens. For according to the first requirement, the clockhand points upwards at 12:00. Now if it does and if the definiendum implies the definiens, then there would have to be a period of time before and after twelve a clock during which the clockhand remains pointing upwards. This is clearly not the case. So a more sophisticated theory of instantaneous states is needed.
e) A good theory of instantaneous states
There is a better theory which fulifils alle the intuitive requirements stated. The theory works with a new kind of theoretical entities: I call them qualitative spans. They exist with respect to a determinable quality like the "spatial position of the clockhand". E.g., if the clock is working allright, the position-span of the clockhand between 12:00 and 6:00 is the righthand half of the dial.
Just as periods are not made up of instants or parts of space are not made up of space-points, spans are not made up of instantaneous states. Just like periods and parts of space they are true continua. They are irreducible wholes.
A span can be contained in another span in an intuitively clear sense of containment. E.g., the position-span of the clockhand between 12:00 and 3:00 is contained in the position-span of the clockhand between 12:00 and 6:00. Spans within periods can be observed. Spans at instants cannot. Nevertheless, they exist according to the theory I want to propose. For the special case under consideration it goes like this:
s is the instantaneous position-span of the clockhand at instant t iff
s is that span which is contained in all position-spans of the clockhand from all periods within which t falls.
This agrees with the intuition of construing an instantaneous state as an empirical "limes" of photographs of ever shorter exposure. The shorter the exposure the smaller the position-span of the clockhand in motion. But it is always more extended than on a picture of the clockhand at rest. Look here:
This area is exactly that span which is contained in all position-spans of the clockhand from all periods within which the instant 12:00 falls. So according to our definition this is the instantaneous position of the clockhand at 12:00. That is, concerning the first requirement the clockhand does in fact point upwards at 12:00 noon (we don't observe this, though!).
It is easily seen that the proposed theory fulfils the other requirements, too. Here is the situation concerning the second requirement:
All the results are just what was required.
Of course, the theory presented by talking about clockhands can be extended to physical objects in general:
s is the instantaneous position-span of a physical object A at instant t iff s is that span which is contained in all position-spans of A from all periods within which t falls.
g) A solution to Zeno's problem of the flying arrow
The proposed theory of instantaneous states allows to reconcile a description of motion as being at different places at different instants with the intuition of motion as a genuine flux. It solves Zeno's flying arrow paradox at its very core. I think this is what Zeno essentially argued:
(Starting point) Whenever an object is in motion and on no other occasion it is (in a certain manner) at different places at different instants.
P1) All which is given are instantaneous positions.
P2) You cannot arrive at motion as a genuine flux by putting together instantaneous positions
C) Therefore there is no motion as a genuine flux. Motion as a genuine flux is only an illusion inside us.
(So the core of the flying arrow problem, as I see it, does not involve the claim that the arrow is at rest at each instant of its flight. If it did, Aristotle would have solved the problem in Physics VI,9 and nobody would have had to worry about it later on.)
If Zeno's problem is what I think it is, then Zeno's starting point is highly plausible. Furthermore, if this is what Zeno meant, then his argument is only the earliest expression of a long tradition in the philosophy of space and time: Very similar statements are found in Reid (6), Bergson (7) and Russell. Especially Russell is in agreement with Zeno when he says: "Motion consists nerely in the occupation of different places at different times..." (8) I call this the reductionist view of motion. We are faced with a problem: Although Zeno's starting point is plausible, the reductionist view of motion is highly implausible for two reasons:
(1) It goes against our intuition of motion as a genuine flux. It is an "at-at-theory" which, in a way, chops up motion.
(2) The reductionist view is incompatible with the fact that instants are empirically inaccessible. For if motion is nothing but being at different places at different instants, an object should disappear from our sight as soon as it moves, since it is in each position of its motion for just one single instant at which we cannot observe it there.
In this situation we have to see what has gone wrong, and we should be content to have something better than the reductionist view. Here the theory of instaneneous states I proposed is really of great help:
First, it allows us to see what has gone wrong: The idea behind the theory of instantaneous states I proposed very much suggests that the second premise of Zeno's argument is true. It also endorses Zeno's starting point. But it can do without the first premise. Since this is perfectly consistent, Zeno is simply unjustified in concluding from the starting point to the first premise. The "therefore" in Zeno's argument is a non sequitur. So let's cross it out. In fact, according to the idea behind the theory of instantaneous states proposed, Zeno's first premise is wrong: What is given, are not instantaneous positions but spans within periods. So let's cross this out, too. Of course, with the first premise, the whole argument fails.
Furthermore, the theory of instantaneous states proposed is itself something better than the reductionist view. It allows to describe motion as being at different places at different instants without a reductionist aim in mind. It avoids the problems of the reductionist view: If we use it, we need neither worry about chopping up motion nor about blind spots. The answer to Zeno, according to the proposed theory of instantaneous states is:
What is given, is motion as a genuine flux, and we only construe instantaneous positions from it. Zeno failed to see the abstractive work to be done in order to arrive at instantaneous positions. He (and Reid and Bergson and Russell) failed to see that talk about instantaneous positions goes beyond immediate sense experience. However, it does: it is everyday metaphysics.
f) Extensions of the theory
The proposed theory of instantaneous states can be generalized in several ways:
aa) Other determinable qualities
Firstly, the theory can be easily extended to just every determinable quality, not just position in space. It works just as well for temperatures, weights, shades of colour or pitches. 10° - 35° is a temperature span, 5kg - 10kg is a weight span etc. Let us call spans with regard to determinable qualities D-spans. So we can generalize the theory as follows:
s is the instantaneous D-span of a physical object A at instant t iff s is that span which is contained in all D-spans of A from all periods within which t falls.
Consider, e.g., the following little piece of music. Let D stand for "pitch".
There is a long downward glissando from E to B in bar 1; it ends in a long B which takes up bar 2 and 3, before there is an upward glissando in bar 4. (9) Let us ask ourselves, what pitch we should assign to the four bar-lines, not as sounding then (for this would take time) but as being played.
Bar-line between measures 1 and 2: The pitch span of measures 1 and 2 together is E-B. The pitch span of the second half of measure 1 and the first half of measure 2 is a little less, say D-B. The pitch span of the last quarter of measure 1 and the first quarter of measure 2 is still less, say C-B. The only pitch span which is contained in all periods within which the bar-line falls is B-B. So B is the pitch we assign to the bar-line. This corresponds to the third requirement concerning the clockhand. (10)
Bar-line between measures 2 and 3: The pitch span of measures 1 and 2 together is B-B. The pitch span of the second half of measure 1 and the first half of measure 2 is equally B-B. The pitch span of the last quarter of measure 1 and the first quarter of measure 2 is again B-B. The only pitch span which is contained in all periods within which the bar-line falls is B-B. So B is the pitch we assign to the bar-line. This corresponds to the second requirement concerning the clockhand.
Bar-line between measures 3 and 4: The pitch span of measures 1 and 2 together is B-E.(11) The pitch span of the second half of measure 1 and the first half of measure 2 is a little less, say B-D. The pitch span of the last quarter of measure 1 and the first quarter of measure 2 is still less, say B-C. The only pitch span which is contained in all periods within which the bar-line falls is B-B. So B is the pitch we assign to the bar-line. This corresponds to the fourth requirement above.
In this example, the spans were taken from a kind of "pitch-space". It is easy to imagine a "temperature space" or a "weight space" as well and think of analogous examples for these cases.
bb) Natura (non) facit saltus
Secondly, the theory can be formulated in a more general way, without assuming the principle that nature doesn't jump. This is done by splitting up the periods within which our instant t falls, namely by splitting them up into lefthand and righthand halves.
a) A jump in nature and how to cope with it
The more general formulation is desirable, because the principle that nature does not jump is not plausible all the way. Let us, e.g., look at another little piece of music:
Here, there is a discontinuous change of pitch from measure 1 to measure 2. So here nature does jump with respect to pitch. Intuitively, we will not want to assign a pitch to the bar-line in this case (although we did in the other examples). The most intuitive result would in fact be that there is no pitch to be assigned here. So we need a version of our theory which yields all the results from the previous version which was formulated under the assumption that nature does not jump. But the extended version should in addition to that yield plausible results for the cases in which nature does jump (which were not accounted for in the previous version). In this case we are justified in calling the previous version a special case of the new version which would be a more general version theory of instantaneous states than the one I proposed first. Here it is:
s is the instantaneous D-span of a physical object A at instant t iff s is both identical with that span which is contained in all D-spans of A from all periods which are bounded by t towards the future and with that span which is contained in all D-spans of A from all periods which are bounded by t towards the past. (12)
For all cases in which nature does not jump this version is clearly result-conservative with respect to the first version, which can be easily seen by checking the examples again applying the new version. (13) So let us have a look at how the definition works in a case where nature does jump, our second little piece of music:
The pitch span with respect to measure 1 is E-E. So is the pitch span with respect to the second half of it, the last quarter of it a.s.o. So the only pitch span which is contained in all pitch spans from all periods which are bounded by the bar-line (14) towards the future is E-E.
The pitch span with respect to measure 2 is B-B. So is the pitch span with respect to the second half of it, the last quarter of it a.s.o. So the only pitch span which is contained in all pitch spans from all periods which are bounded by the bar-line towards the past is B-B.
There is no pitch span which is identical with both B-B and E-E, for B-B and E-E are different pitch spans. So there is no instantaneous pitch span of the musical instrument on which the piece is played (a viola, by the way) with respect to the instant denoted by the bar-line. And this is just what we are content with.
b) The weird clock
The more general definition has an interesting consequence for position in space (an similar determinable qualities). To see why, try to imagine that the clock has stopped in the upward position from 12:00 to 2:00 and then, after performing an instantaneous jerk, remained in the 3 o'clock position for another two hours. Where is the clockhand at 2:00, the instant of the jerk? That position span of it which is contained in all position spans from all periods which are bounded by 2:00 towards the future is the 12 o'clock position. That position span of it which is contained in all position spans from all periods which are bounded by 2:00 towards the past is the 3 o'clock position. Clearly, there is no position span which is both identical with the 12 o'clock position and the 3 o'clock position. Therefore, according to the definition, if the clockhand performs a jerk at 2:00, it is nowhere at 2:00. Now for any macroscopic object we will hold the intuitive principle that it must be somewhere at any instant of its existence. From this principle plus the theory of instantaneous states, it follows that there can be no jerks in nature with respect to position in space . This means that the old principle "Natura non facit saltus" (which seemed to be unable of any further justification) turns out to be a mere a consequence of the theory of instantaneous states plus an intuitive additional assumption.
III. Extension of the theory to boundaries in space
The more general version of theory can be easily extended from instants, i.e. boundaries in time to lines as boundaries of surfaces and to surfaces as boundaries of parts of space. Those spatial entities, too, are two-sided. Such a theory is desirable, since, just like instants, t least lines without any bredth, and arguably also surfaces without any thickness are, like instants, empirically inaccessible. If we call both instants and lines simply "boundaries", the extended version of the theory can be formulated like this:
s is the D-span of A at instant at a two-sided boundary b iff s is both identical with that span which is contained in all D-spans of A from all continua which are bounded by b towards one side of b and with that span which is contained in all D-spans of A from continua which are bounded by b towards the other side of b.
Regard this drawing which corresponds to the first little piece of music:
This time the continua are two-dimensional surfaces. Now is the line which bounds the area of fading grey on the one side and the area of white on the other side? The colour span of the left half of the drawing is dark grey - white. The colour span of the right half of this half is grey - white. The only colour span contained in all those colour spans The colour span of the right half of this half is the colour span light grey - white etc. So the colour span which is contained in all these spans is white - white. The colour span of the right half is white, and so is the colour span of the left half of it, and the colour span of the left half of the left half of it etc. So the colour span contained in all these colour spans is white - white, too. Therefore, the line should be assigned the colour white.
Just as there are jumps, there are spatial discontinuities. Look at this drawing which corresponds to the second piece of music.
The colour span of the left half is black - black, and so is the colour span of the left half of it, and the colour span of the left half of the left half of it etc. So the colour span contained in all these colour spans is black - black, too.
The colour span of the right half is white, and so is the colour span of the left half of it, and the colour span of the left half of the left half of it etc. So the colour span contained in all these colour spans is white - white, too. Since no span is both identical with both the span white - white and the span black - black, no colour will be assigned to this line.
Surfaces as boundaries of parts of space should be treated analogously. Lines in space and space points on surfaces or in space, which are likewise empirically inaccesible, should be assigned the quality of a surface they fall within (if there is any such quality).
IV) Concluding Remark
In how many positions can Asterix's left foot be seen in this picture?
The typical, wrong answer would be: in infinitely many instantaneous positions. The correct answer is: in one position.
(1) A set of boundaries may be used to identify, i.e. definitely describe a continuum. But that does not mean that this set is the continuum any more than the parents of a single child are the child (they, too, may be used to identify the child as "the only child of A and B").
(2) A formal tool for handling continua and boundaries, which takes seriously the fundamental difference between them as well as continua as irreducible entities in their own right is a modified version of C.Hamblin's "Interval Semantics" as stated in his "Instants and Intervals". Cp. "The moment of change" passim, esp. Introduction2.
(3) A shortened version of section II was delivered as a talk on the ECAPII conference in Leeds in September 1996. Much of the material of it is also contained in "The Moment of Change" part III.
(4) Cp. "The Moment of Change" passim for a comprehensive discussion of literature on this topic from antiquity on.
(5) Cp. Aristotle's Physics book VI, 235b26ff on this point (perhaps together with my interpretation and formal reconstruction of the passage in my "The Moment of Change", ch.I,2 and appendix B) as well as Richard Sorabji's articles on Aristotle and the moment of change.
(6) Thomas Reid: Essays on the Intellectual Powers of Man, Glasgow 1785, III,v. "...though in common language we speak with perfect propriety and truth, when we say that we see a body move, and that motion is an object of sense, yet when as philosophers we distinguish accurately the province of sense from that of memory, we can no more see what is past, though but a moment ago, than we can remember what is present: so that speaking philosophically, it is only by the aid of memory that we discern motion, or any succession whatsoever. We see the present place of the body; we remember the successive advance it made to that place. The first can then only give us a conception of motion, when joined to the last." I have taken the reference to this passage from Michael Inwood: Aristotle on the Reality of Time, In: Lindsay Judson (ed.): AristotleŽs Physics - A Collection of Essays, Oxford 1991 pp.151-178.
(7) Henri Bergson: Essai sur les données immédiates de la conscience. In: uvres [1 vol.], Paris 1959 p.72 ff.:"En dehors de moi, dans lŽespace, il nŽy a jamais quŽune position unique... [dŽun objet mouvant], car des positions passées il ne reste rien. Au dedans de moi un processus dŽorganisation ou de pénétration mutuelle des faits de conscience se poursuit, qui constitue la durée vraie... [L]a succession existe seulement pour un spectateur conscient qui se remémore le passé... le mouvement, en tant que passage dŽun point à un autre, est une synthèse mentale, un processus psychique et par suite inétendu. ...[E]n quelque point de lŽespace que lŽon considère le mobile, on nŽobtiendra quŽune position. Si la conscience perçoit autre chose que des positions, cŽest quŽelle se remémore les positions successives et en fait la synthèse."
(8) Russell, Principles of Mathematics §447.
(9) In fact, bar-lines are the best illustrations of instants I can think of. They just bound bars. They cannot contain any piece of music however short. Measures, of course, correspond to periods.
(10) It also agrees with Aristotle's very plusible view on ends of processes mentioned above.
(11) Or E-B. The order in which the boundaries of a span are stated does not matter.
(12) The spans involved in this definition resemble very obviously right-hand and left-hand differential coefficients in differential calculus.
(13) One example may suffice here: If the clock is working allright, then (by now: clearly) that position span of the clockhand which is contained in all position spans from all periods which are bounded by 12:00 towards the future is the twelve o'clock position. And that position span of the clockhand which is contained in all position spans from all periods which are bounded by 12:00 towards the past is the twelve o'clock position, too. Clearly, that position span which is identical with both the twelve o'clock position and the twelve o'clock position is the twelve o'clock position. Thus, if the clock is working allright, the clockhand is in the twelve o'clock position at 12:00, QED.
(14) Or - to be very exact - "...which are bounded by the instant which is denoted by the bar-line..."