Bergson, Time and Free Will, Chapter 2, §72 "This is Seen in the Definition of Velocity"

by Corry Shores
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Henri Bergson

Essai sur les données immédiates de la conscience

Time and Free Will: An Essay on the Immediate Data of Consciousness

Chapter II, "The Multiplicity of Conscious States," "The Idea of Duration"

Part XXV: Velocity and Simultaneity

§72 "This is Seen in the Definition of Velocity"

Previously we saw that physicists do not deal with durations when measuring motion. In the first place they are concerned with specific points where the body has moved. These points are correlations between our consciousness and the object's place. What happens in-between we experience as duration. But there is no extensive continuum of time in the objective world. There are just simultaneities. Hence physicists are interested in the differences of a body's place during distinct simultaneities.

Consider a physicist studying an iron ball rolling along a very long table. If the table were level, friction would cause the ball to slow down. Let's imagine instead that the table is tilted somewhat. So our ball is rolling down-hill. And by chance, the tilt compensates exactly for the friction. Thus the ball maintains a constant velocity.

The long table is extensive. So its distance can be divided. The scientist makes markings along the length of the table. These lines are spaced one meter apart.

The physicist also has a stop-watch. After the ball has reached a certain point, she begins to determine the time it takes for the ball to pass through each of the succeeding meter-marks. So she is watching the ball's location. And she is stopping-and-starting the watch at each meter-line, "click-click....click-click...click-click...click-click..." She notes the amount of time it takes for the ball to get through each meter-point.

She finds that the iron ball consistently takes 1 second to pass through each meter. We expect this. The ball is moving at a constant velocity. So it should take the same amount of time to pass a unit of distance, regardless of where the ball is along its course.

The physicist might presume that there is a continuum of time spanning between each click, and that the continuation of time corresponds continuously with the continuum of extensive space the ball traverses. Bergson is shattering this supposition. His argument is built on these notions that he has previously established:

1) Something cannot be in two places at once. Thus there is no time in space. [See §57 and §66]

2) Measures of motion involve comparing simultaneities between object locations and consciousness-moments [see §71].

The scientist's experiment nicely illustrates Bergson's perspective. For, she did not take note of the continuous correlation between the locations of the stop-watch's second hand and the ball's place along the table. Rather, she picked out determinate places along the object's course. The ball's placement at each of these points is simultaneous with the second-hand's

placement around the stop-watch dial. And she knows certain things about the stop-watch. It's mechanisms operate under very controlled conditions. So in one instance when we set the clock in motion, certain forces of the springs and gears are acting on the second-hand to make it move in a particular way. Then, in any other instance that we set the clock in motion, the second-hand will be acted-upon by the same forces. So we may expect each start of the watch to send the second-hand to the initial second-mark in the same amount of time as every other instance we start it.

So a regular amount of duration transpires between each instance that the second-hand crosses a second-mark. These crossings are simultaneous with the iron-ball's crossings through the meter-marks. The stop-watch hand moves at a constant velocity. And its passings through evenly spaced distance-measurements corresponds to the ball's passings through even spaces. Hence we know the ball moves at a constant velocity. And we know this without paying any heed to the time-intervals between the simultaneities. For, we are only comparing one simultaneity to another. We are not comparing every one of the infinity of points along a continuum of correspondence between a space-continuum and a supposed time-continuum.

With this example in mind, we turn to the text.

Bergson will explain how physics obtains the notions of constant velocity and also variable motion. And he will show that it does not suppose there to be a continuum of time. He will depict a scientific method that will determine velocities without using time-calculations. As well, he will examine the differential calculus used to determine the instantaneous velocities of objects in variable motion. Normally physicists use mathematical formulas to make these determinations. They seem to imply an objective temporality that the objects exist-in. Bergson breaks this conception. It requires extraordinary care to follow how he does this. The casual reader might not feel compelled to undergo this strenuous mental exercise. But those who do so will add weaponry to their critical war machinery. For, Bergson is undermining one of the basic foundations of physics: objective temporality.

Before, we noted the constant conditions of the stop-watch motion. Bergson considers something like the second-hand's movement as "a physical phenomenon which is repeated indefinitely under the same conditions, e.g., a stone always falling from the same height on to the same spot." (117bc)

Each rock has the same physical properties. And the environmental conditions are the same for each drop. Note also that just as one stone hits the ground, the next one begins its descent. Thus each fall of a stone will be like the second-hand moving through the space between second-markings.

Then Bergson has us imagine an object that moves along path AB [like our iron ball].

Now, we will roll the iron-ball near the falling stones. And we attend to the middle part of the ball's course. When we hear the crash of a fallen rock, we mark where the ball is along the line. We do that for three successive crashes. And we mark each point M, N, and P.

So after crash one, we mark M.

After the second crash, we mark N.

After the third crash, we mark P.

What we find is that the distance intervals AM, MN, and NP are all the same.

Now, because each of these distances are the same, and the durations for the falling rocks are also the same, we know that the iron ball was moving at a constant velocity.

[Someone might here object that scientists do not use falling rocks to measure velocity. This is why I suggested the stop-watch example at the beginning. Both the falling stones and the watch maintain constant conditions to bring about constant motion. The second-hand moves between each second-mark by means of the same causal conditions each time. And the rocks fall as well under the same conditions. So those who object can easily supplant the stop-watch example for Bergson's falling rocks.] What is crucial to note is that physicists measure the velocity of moving objects by pinpointing places in its movement. In the last section we saw how MIT physics professor Walter Lewin describes this procedure.

Presumably, physicists suppose a space-time continuum. Space is continuous. Time is continuous. And their correlation with each other is continuous. Along a continuum there are an infinity of points. So along the space-time continuum, there is also an infinity of points. But there is no way to determine a moving object's infinite number of space-time correlation points. So instead we merely demarcate certain points along the way. Then we fill in the rest in-between.

One way to do so is by means of a formulas. According to physics, there is the time-space continuum between our demarcated points along continuous motion. Something that moves fast covers a wide spatial extent in a short temporal extent. Likewise, something that moves slow covers a short spatial extent during a long temporal extension. The velocity is the ratio of distance to time.

One formula finds the average value for this ratio. Let's imagine that the scientist is using a stop-watch to measure the "time" between distance points A, M, N, and P. Velocity is distance over time. To find the distance between N and P, you subtract P - N. We will say that N is at the 2 meter mark, and P is at the 3 meter mark. Hence the change in distance is one meter. And let's say the ball reaches N at 2 seconds, and P at 3 seconds. So the change in time is 3 - 2. Both changes equal the value of 1. So one-over-one is one. That means the average velocity of the ball's movement through interval NP was one meter per second. We may render the formula this way.

So say the ball is moving at a constant velocity. We know that at any one point, it is tending to go the same velocity as it does at any other point in its movement. But now consider that there are warps in the table. So the ball speeds-up and slows-down at different parts. In this way, the ball exhibits "variable motion." We will first return to Bergson's example. We would know that there is variation if the ball travels different distances between the rock-crashes.

We see that the ball moved "faster" through MN than through NP.

Let's follow instead the professor's example where he shows continuous variation. Observe first how he plots what he admits to be the 'discrete' space-time points. Then watch how he fills-in those spaces. He has no objective grounds for creating such a continuum. He clearly presupposes that time is continuous.

The ball's motion does not move in two dimensions like the line shows. The vertical axis represents the distance points as correlating with the time points. This is an intensity, like Oresme's lattitude of forms. Really the ball moves on a line, and in the professor's example, it moves back-and-forth along the line. He better shows the one-dimensionality of the movement later at this segment.

Despite his representation being a curvy continuous line, the physicist needed to find determinate discrete points first, and then only afterwards did he show the supposed continuous motion. So he clearly has not demonstrated that there is a time continuum between each point. He merely presupposes it, and Bergson is removing the grounds for that supposition. He argues that indeed there is an extensional continuum between space-points. And yes, they correspond to moments of duration. But all we can ever do is compare discrete duration-space simultaneities, and use our imaginations or mathematical formulas to fill-in the values between.

If we want to know how fast the ball is tending to go at some part along a warp, we cannot find the average for an extent of space. For, within any such extent will be variation, no matter how small the extent. Physicists then use differential calculus to make this determination. To show how it works, we first undertake the procedure that finds the average velocity. But really we want to know about just one point. So first we pick a point. The professor picks time-point t2. We will eventually want to know what velocity the object is moving at that specific point. Yet, he will begin first by looking at an extent of change. And he chooses time point t3 as the end-point for that extent.

The difference between time-point t3 and time-point t2 will be the change in t, or Δt. To this change in time corresponds the change from distance point x2 and x3. Their change is Δx. We see that we now have a triangle. The rise-over-the-run for the diagonal (hypotenuse) line will tell us the average velocity for that interval. Yet really we wanted the velocity the object was tending at time-point t2. But we see now that this point makes-up an angle in the triangle. And its angular value will tell us the ratio of distance over time (Δx / Δt). We will call this angle, 'angle alpha,' or angle α.

We see that there is a curve instead of the straight line we imposed when determining the average velocity. And we also see that right at the point we want to measure, angle alpha, the line seems to be tending upward far more steeply than the overall rise-over-run average. The professor shows what that line might be. We call it the tangent.

There is a procedure that allows us to obtain that line. We begin with the change in time at its given extent. Then, we gradually reduce that extension until it reaches the "limit" at our 'zero-point' where angle alpha is. This reduces the extensive value to an inextensive value, or an intensive value. [Deleuze cites Leibniz' triangle demonstration for how this works. It is a fantastic illustration for how we obtain intensive values by reducing extensive ones. Also MIT professor David Jerison explains the differential method with remarkable clarity.] As the time value reduces, the distance value follows along the curvy path. To imagine how this process works, we could watch Kelly Liakos' wonderful animation.

When time approaches the limit at zero, it no longer has an extensive value. But it also is not yet 0. It has an inextensive or intensive value. [See this entry on the limit.] Likewise, the other variable will also not be zero, and it as well will not have an extensive value. Each of these extraordinarily small values has no value in comparison to the extensive values. However, one intensive value in relation to another does have a ratio value. In our application here, this value tells us the rise-over-run for how fast the object was tending at that instant. We know that in actuality it goes a different direction, because we see that the curvy line does not go where the professor's ruler is pointed. So this inner tendency is a virtuality and not an actuality. It really is there in the object. The object's tendencies of velocity-change are no less real that the places it travels-through. However, they are implicit internal intensive values that may not become explicated as extensive values. It all depends on the play of forces (and chance) which determine the actual course of the object.

So this is velocity at an instant. Physics calls it "instantaneous velocity."

We notice that this procedure involves us decreasing time continuously. So even though we are looking at motional tendencies at an instant, we obtain these values by presuming that there is a time continuum between simultaneities. Bergson will demonstrate that this same differential method can be conducted empirically using only simultaneities. This will show that the abstract differential method is a mathematical means to bring two simultaneities arbitrarily close to each other.

Let's return to Bergson's example where there was a variation in speed.

We will now find the instantaneous velocity at point M, but without using differential calculus. First we return to the way we determined a constant velocity. We began with the falling rocks. Physically identical rocks fall according to the same physical causes under the same physical conditions. So the falls of these rocks are equivalent to the ticks of a clock. We found that one ball crossed spaces spread this far apart for three given rock-falls.

But let's call this ball A1. Now imagine we have a slower iron ball. But we use the same rock-falling system (or we stay with seconds on the watch). Because it is moving slower, the distances will be less between each fall. We will call it A2, and we will trace its successive simultaneities with the rock-falls.

The first fall:

The second:

And the third:

Let's suppose that there is a mechanism which shoots the balls through A and B. This device has an analog gauge that permits it to throw the ball at any desired speed. So Ball A1 will have velocity v1. And Ball A2 will have velocity v2. And so on.

Now first recall the spacing of the varying ball motion.

We want to know the instantaneous velocity at M. And we want to obtain it using no more than comparisons of simultaneities. We will use an approximation method. We first find a rock-falling system that drops the rocks at a much faster rate. Then, we make markings real close to M. Note first that ball A goes faster from points A to M, and slower afterward from M to N. So probably it decelerates around M. We will now place new markings before and after M. We will place them according to the much faster rock-falling system. But the distance between M' and M will be longer than the distance between M and M''.

We will suppose that after 3 falls of the faster rock system, Ball A reaches point-M. So we know that Ball A's velocity changes around point-M. And we know that there is the same amount of duration between M', M, and M'', because each one demarks a fall of a rock. But we do not yet know how fast Ball A moved within interval's M'M and MM''. We may at least find the average velocity for those intervals. For example, we know that the ball moved faster during interval M'M. We could find a faster ball. Then we could apply the same rate of rock-fall. We want a ball whose constant velocity produces a series of markings whose length is the same as M'M. We will call it Ball Ah. This will tell us that Ball A moves in its M'M interval at the same velocity as Ball Ah moves throughout its whole course.

Ball Ah has its own steady velocity, which we will call Vh. Then we conduct the same procedure for Ball A's slower MM'' interval. We will find a slower ball. And we will set its markings to the same rock-fall rate. We want to adjust its velocity so that its markings are each as long as Ball A's MM'' marking. We will call it Ball Ap. And its constant velocity is Vp.

Thus we see that M'M = M'hMh. And MM'' = MpM''p. So we know that Ball A's variable velocity at point-M is in-between velocities Vp and Vh. But we want to make our approximation more precise. So we will now double the rock-fall rate. Then, at the sixth fall (instead of the third) we will mark that as M. We find again that for Ball A, M'M and MM'' are different lengths. This means that Ball A travels at different speeds for each extent. But the difference between them is relatively less than what we found with the slower rock-fall rate. So this repetition of the procedure using a faster rock-rate will give us two new velocities that better approximate the speed at M. They are balls An and Ai. And they have velocities Vn and Vi.

An's speed is close to our previous Ah velocity. Likewise for Ai and Ap. But because they measure the velocity closer to Ball A's point M, they are better able to approximate the velocity at that point. Now recall how in the differential calculus procedure, we slowly moved one variable down toward the limit at zero. Bergson does something analogous. He keeps increasing the rock fall rate, so that our M' and M'' marks get closer and closer to M. That means our approximations for the faster and slower constant velocities will get closer to each other as they both near point M. Eventually, both the faster and the slower will reach the limit at M. Then they both will have about the same velocity. Or put another way, at this "common limit," the differences between the two distances (before and after point M) "become smaller than any given quantity." [again, see this entry for more on limits.] Let's call this limit velocity the Vm of Ball A at point-M. It is the equivalent of the "instantaneous velocity" we obtain through differential calculus.

So we see that even when physics determines instantaneous velocities, they really want to know something that does not involve the continuous passage of time. They have an object whose velocity varies. They pick a point along its duration and wonder what the velocity is at that instant. But really this question asks about another object's motions. For, we are looking for another object that goes at a velocity which is the same as that of the varying ball only at this one point. They find it mathematically by increasing approximations. But throughout the process, the basis is always the comparison to constant velocities. And scientists always find constant velocities by means of simultaneities between steady repetitional motions and movements across distances (stop-watches measuring distances). So physicists might use differential calculus to obtain the instantaneous velocity. They want to know one speed at a point, despite all the variation before and after it. This one speed they find is the equivalent of a constant velocity in an object whose motion does not vary. So their mathematical method really only saves the difficulty of approximating empirically. They still are doing no more then finding another object whose simultenaities would match the varying object's motion had it continued its tendency at that given instant.

Thus mechanics and physics only uses simultaneities when it deals with time. And it only uses immobility when it measures motion. So it seems Bergson has undermined one of the primary suppositions of physics, that there is a passing of time that is objective and continuous. So any argument that physics proves there is an objective continuous time will not hold, given Bergson's refutation.

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Images from the pages summarized above, in the English Translation [click on the image for an enlargement]:

Images from the pages summarized above, in the original French [click on the image for an enlargement]:

Bergson, Henri. Time and Free Will: An Essay on the Immediate Data of Consciousness, Transl. F. L. Pogson, (New York: Dover Publications, Inc., 2001).

Available online at:

http://www.archive.org/details/timeandfreewill00pogsgoog

French text from:

Bergson, Henri. Essai sur les données immédiates de la conscience. Originally published Paris: Les Presses universitaires de France, 1888.

Available online at:

http://www.archive.org/details/essaisurlesdonn00berguoft

Video from:

Lewin, Walter. 8.01 Physics I: Classical Mechanics. Fall 1999. Video Lecture 2. MITOpenCourseWare. Creative Commons Licence.

Moving Secant Line animation from Kelly Liakos

available online at:

http://calculus7.com/id1.html