The Interval of Motion
On Wednesday Andy described Leibniz's efforts in Pacidius Philalethi to offer a satisfactory account of continuous motion. Intuitively, we are inclined to think that continuous motion involves a body traversing successive points in space in successive instants of time. But Leibniz argues that this has to be a mistake: space doesn't consist of preexistent minima through which a body passes (this is the argument James discussed). But neither do we want to say that motion is discontinuous, in the sense that it involves "leaps" between discrete regions of space. So how can we uphold continuity in way that doesn't commit us to an unacceptable account of the structure of space?
Leibniz's answer is given at (c) on Andy's handout: "Bodies receive 'instantaneous impulses' (from everything, at all times), and the changes of state occur across the aggregates of two neighboring points (defined by a division)." Levey's presentation of the argument in sec. 4 ("Charinus's Solution") of his paper is quite lucid. But most of us still had trouble seeing exactly what Leibniz was up to. What's so special about non-uniform motion? How do points arise in the continuum? Why is Leibniz entitled to claim that these points are densely packed if, as he also insists, motion is between adjacent points (i.e. points between which there are no intermediate points)?
As I suggested, Leibniz's key move is to reject an explanation of motion as passage through a space-time continuum, conceived as a uniform succession of points and a uniform succession of instants. Instead, motion--or change of state--is taken as the basic fact, in terms of which the continuity of passage can be understood.
"Change of state" is by nature discontinuous, or non-uniform: it consists in a pair of incompatiable states between which there is no intermediate state. Change of this sort is ascribed to action on a body, and Leibniz assumes (as an undefended premise) that such action is always occurring, so that any body is constantly changing its state. Given this, he claims (i) that any finite interval of apparently uniform motion is actually infinitely divided into smaller intervals of motion (i.e. parts), and (ii) that this structure of infinitely many smaller intervals of motion shows how it is possible to conceive of motion as continuous, or motion across a densely ordered set of points.
How are we to understand this? Let d be any finite interval of motion (in the simplest case, B appears to move in a straight line at 1 m/s through 1m). Leibniz's analysis of this motion posits the following
Undefended premise:
Given any finite interval of motion d, for any rational n > 1, there is an instantaneous action on B such that (i) B’s state of motion changes from S to S’, and (ii) this change of state partitions B’s motion into the subintervals d/n and d-d/n.
The partitioning of what we took to be a uninform interval of motion has the effect of "designating" (or "assigning") two adjacent points (the right extremum of the subinterval d/n and the left extremum of the subinterval d-d/n) across which B is said to move.
Pictorially, for n = 3/2, we have something like this:
--------------------------------------••--------------------
Since Leibniz claims that any body is constantly changing its state, any finite subinterval we choose will be similarly partitioned (for any rational n):
---------------------------••---------------------------
d/2
--------------••-----------••---------------------------
d/4
------••------••-----------••---------------------------
d/8
Because any interval anywhere in the line can be further divided into smaller subintervals, in each of which there are designated endpoints (extrema), it follows both that points in the line are densely ordered, and that motion always occurs between adjacent points (i.e. locus proximus motion).
Where does leave us? Which questions has Leibniz at least tried to address, and which questions has he left unaddressed?
Leibniz's answer is given at (c) on Andy's handout: "Bodies receive 'instantaneous impulses' (from everything, at all times), and the changes of state occur across the aggregates of two neighboring points (defined by a division)." Levey's presentation of the argument in sec. 4 ("Charinus's Solution") of his paper is quite lucid. But most of us still had trouble seeing exactly what Leibniz was up to. What's so special about non-uniform motion? How do points arise in the continuum? Why is Leibniz entitled to claim that these points are densely packed if, as he also insists, motion is between adjacent points (i.e. points between which there are no intermediate points)?
As I suggested, Leibniz's key move is to reject an explanation of motion as passage through a space-time continuum, conceived as a uniform succession of points and a uniform succession of instants. Instead, motion--or change of state--is taken as the basic fact, in terms of which the continuity of passage can be understood.
"Change of state" is by nature discontinuous, or non-uniform: it consists in a pair of incompatiable states between which there is no intermediate state. Change of this sort is ascribed to action on a body, and Leibniz assumes (as an undefended premise) that such action is always occurring, so that any body is constantly changing its state. Given this, he claims (i) that any finite interval of apparently uniform motion is actually infinitely divided into smaller intervals of motion (i.e. parts), and (ii) that this structure of infinitely many smaller intervals of motion shows how it is possible to conceive of motion as continuous, or motion across a densely ordered set of points.
How are we to understand this? Let d be any finite interval of motion (in the simplest case, B appears to move in a straight line at 1 m/s through 1m). Leibniz's analysis of this motion posits the following
Undefended premise:
Given any finite interval of motion d, for any rational n > 1, there is an instantaneous action on B such that (i) B’s state of motion changes from S to S’, and (ii) this change of state partitions B’s motion into the subintervals d/n and d-d/n.
The partitioning of what we took to be a uninform interval of motion has the effect of "designating" (or "assigning") two adjacent points (the right extremum of the subinterval d/n and the left extremum of the subinterval d-d/n) across which B is said to move.
Pictorially, for n = 3/2, we have something like this:
--------------------------------------••--------------------
Since Leibniz claims that any body is constantly changing its state, any finite subinterval we choose will be similarly partitioned (for any rational n):
---------------------------••---------------------------
d/2
--------------••-----------••---------------------------
d/4
------••------••-----------••---------------------------
d/8
Because any interval anywhere in the line can be further divided into smaller subintervals, in each of which there are designated endpoints (extrema), it follows both that points in the line are densely ordered, and that motion always occurs between adjacent points (i.e. locus proximus motion).
Where does leave us? Which questions has Leibniz at least tried to address, and which questions has he left unaddressed?
- Most obviously, he has offered an account of the structure of the interval of motion, which he conceives as composed of an infinity of contiguous parts--and not as a composite of inassignable minima. On this account, points exists only as extrema of determinate finite intervals of motion.
- Blurred somewhat here is the distinction between the ideal and the actual. In both cases, I think, Leibniz wants to say that points only exist as extrema. In ideal continua such as space and time, there are no points prior to our designating some "cut." In and of themselves, such continua are perfectly uniform, or indeterminate, wholes. In actuals such as motion, by contrast, the divisions are given by nature, hence points are also given, but only as extrema of parts.
- It might be objected that Leibniz does not adequately address Zenonian concerns about how motion is possible. Instead, as Arthur suggests, he takes the reality of motion for granted and attempts to explicate what the structure of the continuum must be for us to have a coherent conception of it. But does the Zeno problem ever really go away? Although Leibniz defines motion as passage between two adjacent points, for any body to get to the next pair of adjacent points, it will have to pass through an infinity of vanishingly small finite intervals of motion. If so, can we coherently think of it as reaching those points?
- Alternatively, does Leibniz's account imply that "in the limit" one ends up with just a succession of adjacent points? If that is right, then we also seem to have a problem, since we are back at the position that he wants to reject (this is Levey's objection).
posted by Don at 5:49 PM
2 Comments:
I have a different question, but one that is hopefully related: just what is the partitioning of the continuum supposed to be? Your choice of phrase, Don - that partitioning "has the effect of "designating" (or "assigning") two adjacent points" - sharpens a concern I had.
Previous proposals about the nature of the continuum were rather intuitive, even if they did turn out to be problematic. We're accustomed to thinking about stuff as being composed of smaller stuff, and explaining the properties of stuff by recourse to smaller stuff.
But this partitioning of the continuum is weird, in that it seems points are only generated by or result from some kind of action. For the ideal continuum, this isn't terribly surprising, since we already knew that the whole was supposed to be prior to its parts. And I guess it makes sense to think of partitioning as an act of mind, performed on an ideal thing or concept. Does this mean that we can just straight-up directly designate contiguous points in the continuum? If so, it seems like cheating that we're able to do so when we conceive of points as extrema, but not as minima. It's as if some conceptual slight-of-hand allows us to do something that was once impossible; namely, find contiguous points on the continuum.
At the risk of being redundant: When we thought of points as minima, designating two adjacent points was a pursuit always foiled by the density of the continuum - some point between the two could always be found, as long as the minima had any extension. Since unextended minima weren't admissible (Th. Abstract Motion), we could never find adjacent points. But all we have to do is stipulate some act of "partitioning," call our points 'extrema' generated by that partitioning, and then we get adjacent points for free? Without independent motivation for thinking we've got a hold on this partitioning thing, the whole solution seems circular, or a skyhook.
So I'd like to know more about partitioning. I guess I can imagine a story where it's a peculiar mental act, which in the ideal realm gets us to a dense ordering of points, and in the actual realm gets us actually infinitely divided matter (thanks to God, and monads, and the actual's resulting from monads). But the story is extremely sketchy.
Any thoughts? What is partitioning?
Adam: You're right. The idea of "partitioning" the interval, in such a way as to produce two adjacent points seems pretty murky. Not only are we given no physical explanation how this happens, it is not even clear how it could be defined mathematically. It is interesting that when Levey introduces his fractal analysis he begins to talk of "singularities" and "sharp corners," which so I far as I can see delivers only a single point, not two adjacent points.
Andy: I find Levey a little confusing on this point. Sometimes he raises the worry that any finite interval reduces "in the limit" to a "powder of points" (402-3). At other times, however, he limits himself to the claim that Leibniz has two competing analyses, one of which is framed in terms of finite intervals, the other of which posits a densely ordered series of adjacent points. The problem is that Leibniz also wants to explain the latter in terms of the former, i.e. points as extrema of intervals. Moreover, as you suggest, without the reduction of intervals to points, we have no explanation of motion within the interval (however small it is taken to be). Concerning your last suggestion, I would say this. What division to the limit gives us is not simply a "succession of adjacent points" (as I supposed) but a densely order series of pairs of adjacent points. I think this is what Levey is saying in the second paragraph of p. 401. If so, then I don't see any advantage in connection with the Zeno problem. But doesn't the problem, in fact, disappear with the appeal to transcreation? God can always re-create bodies at adjacent points, even if there is an inassignable distance between any such pair of points?
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