Bergson, Time and Free Will, Chapter 2, §69 "Two Elements in Motion..."

by Corry Shores
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Bergson, Time and Free Will

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

Part XXII: Is Motion Measurable?

§69 "Two Elements in Motion: (1) the Space Traversed, which is Homogeneous and Divisible; (2) the Act of Traversing, Indivisible and Real only for Consciousness"

Space is a homogeneous medium. So when we spatialize duration, we homogenize it. We think of time then as a homogeneous movement in an ideal space or "fourth dimension." The concept of motion, Bergson says, is the "living symbol" of the homogenized duration. Previously we analyzed time in order to distinguish duration from space. Now we will analyze movement so to differentiate our motion-sensation's qualitative intensity from the moving object's quantitative extensity.

We think of space as homogeneous and divisible. And there are objects in space. These objects move from place to place. But no object can be in two places at once. Rather, things may be in only one place at once. But then nothing in space is ever changing place [see Zeno's paradoxes, II.B] In a sense, there is no motion in space. But we do perceive motion. So movement is possible by means of some conscious act. (111a) We perceive the places that the object traverses. And we keep those places in our mind. Then, we synthesize them together.

The question is, how does this synthesis occur?

One possibility is that we retain the past positions. Then, we abstract those points and place them in ideal space. Finally, we connect the lines so to represent the motion. But then, all we have done was create something that is merely spatial. A line does not move. But could it be that we then transpose the points of this line to another ideal dimension somehow intersecting with the first? No, because as soon as we synthesize together those new points extending through the new dimension, we again obtain a stagnant line (or plane). So our sense of motion cannot just be a sense of extension or quantity. There must also be a qualitative component, or a "feel" of motion that we obtain as well. (111bc)

Bergson invites us to conduct a demonstration. We close our eyes. Then we dart our closed eyes quickly and briefly. We had a sensation. But we did not perceive any distance of motion. Now we recall the last time we saw a shooting star. Here again our eyes darted. Likewise we obtained a qualitative sensation from that movement. But we also perceived the space that the shooting star crossed in the sky. Bergson says that when we see motion, we instinctively distinguish the extensive (non-temporal) space traversed from the "absolutely indivisible sensation of motion or mobility." (111-112)

So we distinguish two elements in motion:

1) the extension of [a-temporal] space that the moving body crosses. This is a homogeneous quantity that exists in real space. And,

2) our mental synthesis of the points crossed. This element only has reality in our consciousness. We may consider it a quality or an intensity (112b)

Now, we might think that we can consider half of the shooting star's motion. But here we are really just thinking of half the distance it travelled. For, the shooting star's motion comes about through a mental action. And mental acts inter-permeate, and they do not extend in spatial-like way. So they are not divisible. But this means that when we divide motion, we incorrectly intermingle the intensity or quality of our motion-sensation with the extent of the motion.

Also, we tend to place time and space together in an interlaced continuum, so that the motion seems to make a continuous line through time-space. To do this, we spacialize different moments, and place them along-side each other. But then a progress in space would involve the past co-existing with the present. We saw that temporal continuity is only possible through a mental synthesis. So when we think of motion this way, we are erroneously letting the extent of the motion intermingle with the intensity or quality of our movement-sensation.

So in other words, intensity and extensity seep together in these above two ways. Hence there is an endosmosis of sorts.

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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

Bergson, Time and Free Will, Chapter 1, §45 "Break-down of the Assumption that the Sensation is a Sum, and the Minimum Differences Quantities"

by Corry Shores
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[The following is summary; my commentary is in brackets.]

Bergson, Time and Free Will

(Essai sur les données immédiates de la conscience)

Chapter I, "The Intensity of Psychic States"

Part XIV: "Psychophysics"

§45 "Break-down of the Assumption that the Sensation is a Sum, and the Minimum Differences Quantities"

Bergson questions the postulate [explicated in the previous section] that a change from one sensation level to another can be quantified by finding the sum of minimal sensations leading up to that second level [for more on this technique, see this entry and the entries for §41; §42; §43; and §44]. Bergson has us imagine that we experience a sensation S. We then increase the stimulus gradually and slowly until we first perceive an increase in sensation. Then, we consider the difference between the first state S and the second state S' as a difference of one unit, which is an arithmetical difference: a difference of one. (65-66)

However, Bergson argues that the only way the transition from S to S' can be considered an arithmetical difference is if we were conscious of an interval between S and S'. We would need to feel the sensation rise from S to S' by means of the addition of something. (66a)

We gave this transition a name: ΔS. By doing so, we made it a thing. We first give it identity, which means we thereby give it reality. Then, after considering it as something real, we regard it as a quantity, that is, as a unit of numerical value. (66b)

But Bergson argues that the transition from S to S' is not a reality, because the only realities we experienced were the S and S' that we passed-through (66b.c).

So if on the other hand, S and S' were numbers, then we could assert that there is some reality between them, because there is a series of smaller values making-up the arithmetical difference between them.

Yet, if S and S' prime are just simple states of sensation, with none felt between them, then what would constitute the interval between them? It would seem that the transition only can be conceived when we abstractly think of S and S' as numbers, and then impose on them arithmetical properties. But really, S and S' are unique singular states with no numerical values between them. [for Deleuze's application of Bergson's idea to Spinoza's affection, see the middle of the entry on Deleuze's Cours Vincennes: 20/01/1981.]

[Next entry in this series.]

[Directory of other entries in this series.]

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

Leibniz "Justification of the Infinitesimal Calculus by that of Ordinary Geometry"

by Corry Shores
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The following will explain Leibniz' "Justification of the Infinitesimal Calculus by that of Ordinary Algebra." Deleuze reads from this letter in his Leibniz courses [Cours Vincennes - 22/04/1980], and he reproduces in The Fold this image from the letter:

[Below is a summary of the letter, taken in incremental steps.]

Gottfried Leibniz

Justification of the Infinitesimal Calculus

by that of Ordinary Algebra

in

Philosophical Papers and Letters

Leibniz constructs the above geometrical diagram:

Straight line AX

and straight line EY

meet at point C

Straight line e forms between E and A, and it is perpendicular to AX.

Likewise, line y forms between X and Y, and it is also perpendicular to AX.

Lines e and y are parallel, because they are perpendicular to the same line.

Line AC is called c,

And all of line AX is called x

We know by Euclid's Proposition 15 that the triangles' angles at C share equal value, because they are opposite or vertical angles.

And we know by Proposition 27 that the other angles must be equal as well

Because all corresponding angles are the same in both triangles, they are similar triangles

This means that the corresponding sides maintain their proportions to each other. We obtain the length of the large triangle's vertical side by subtracting the small triangle's vertical side c from the full line x; so Leibniz describes the proportionality of the triangles' sides by writing

(x - c) / y = c / e

In other words, the large triangle's vertical side is to its horizontal side as the small triangle's vertical side is to its horizontal side.

Now we move line EY closer to point A, always maintaining the angle at C, and hence also at E

Because the angles remain the same, so too will the proportions of their sides, so we can expect the ratio of c to e to remain constant. Leibniz has us assume that this ratio is not 1, for then there would not be a relation of two different values. Thus for this reason, the C angle cannot be 45-degrees; for then the E angle would be as well, which would mean that as an isosceles right triangle, its sides would be the same value. We will see that it is important for this demonstration that there be a difference in value between the terms of the ratio.

Now we imagine that line EY continues toward A and then passes through it.

We see that in the process, points E and C will fall on point A as lines e and c vanish:

We notice it more easily when we magnify the diagram:

And because c has vanished, the vertical side of the larger bottom triangle is no longer defined as x -c, but is rather just x

But we see that as the diagonal line progressed to A, the proportions between the sides of the similar triangles remained the same:

(Animation above is my own, made with Open Office Draw and Unfreeze.)

Looking specifically at the triangle comparisons,

What we see is that the smaller triangle vanished, but as it was vanishing, the larger triangle maintained the proportional relations between c and e. So even though on the one hand we no longer take c and e to have any extensive finite values, we still know that even in their infinitely small size, we can determine their relationship to one another. For, we know that their ratio always remains the same as the larger triangle whose terms retained their extensive finite values. So, we recall that before the contraction, the formula was:

(x - c) / y = c / e

But now it can be rendered:

x / y = c / e

Because now, x - c = x; for c no longer has a finite extensive value to take from x, even though its relation to e still may be represented as

x / y.

So e and c are not nothing, because they preserve the ratio of CX to XY.

Leibniz also says that the ratio of the infinitely small c to e preserves the ratio of the radius and tangent of the angle at C, which would be the same as x and y:

So we know that c and e do not have extensive finite values, but we also know that they are not nothing. For, if they both equaled zero, then x / y would equal 0 / 0, which equals 1 [0r 1/1]. Hence in that case, x and y would have the same value, and thus x would equal y. However, this is absurd, because it could only be so if C's angle were 45-degrees. But we began by presuming that it was not.

So c and e are not taken to be zero, except in their relation to x and y. But when they are in a ratio to each other, c and e have an algebraic relation to one another.

And so they are treated as infinitesimals, exactly as are the elements which our differential calculus recognizes in the ordinates of curves for momentary increments and decrements.

(545d)

Although some reject the notion of the infinitesimal, it proves advantageous in solving certain algebraic problems that otherwise would be insoluble. As well in physics this principle may take the form of Leibniz's Law of Continuity, by which one may regard

equality as a particular case of inequality, rest as a special case of motion, parallelism as a case of convergence, etc., assuming not that the difference of magnitudes which become equal is already zero but that it is in the act of vanishing; and similarly in the case of motion, not that it is already zero in an absolute sense but that it is on the point of becoming zero.

(546b)

This sort of method was used by Archimedes; and anyone who criticized his use of infinitely small values would yet be unable to show how we may designate some magnitude for them.

Thus,

rest, equality, and the circle terminate the motions, the inequalities, and the regular polygons which arrive at them by a continuous change and vanish in them. And although these terminations are excluded, that is, are not included in any rigorous sense in the variables which they limit, they nevertheless have the same properties as if they were included in the series, in accordance with the language of infinites and infinitesimals, which takes the circle, for example, as a regular polygon with an infinite number of sides. Otherwise the law of continuity would be violated, namely, that since we can move from polygons to a circle by a continuous change and without making a leap, it is also necessary not to make a leap in passing from the properties of polygons to those of a circle.

(546c.d)

Leibniz, Gottfried. "Justification of the Infinitesimal Calculus by that of Ordinary Algebra." Philosophical Papers and Letters. Ed. & Transl. Leroy E. Loemker. Dordrecht: D. Reidel Publishing Company, 1956.