Bergson, Time and Free Will, Chapter 2, §55 "Number in Process of Formation is Discontinuous, but, When Formed, is Invested with the Continuity ..."

by Corry Shores
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Bergson, Time and Free Will

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

Part XVI: Numerical Multiplicity and Space

§55 "Number in Process of Formation is Discontinuous, but, When Formed, is Invested with the Continuity of Space"

Previously we discussed how we conceive units in simple intuitions that take them only provisionally as indivisible. For, we also presuppose that any unit may be divided infinitely. And if units can be infinitely divided, then they are continuous magnitudes.

However, we noted that we do conceive numbers provisionally as indivisible, because we think them in a single intuition. Thus we need to clarify what we mean by the discontinuity of number.

Even when we infinitely subdivide a number, we do so into discrete units. When we consider them, we "fix our attention successively" on each unit making-up the number. Thus "it is always by jerks, by sudden jumps, so to speak, that we advance from one to the other." (82d)

In geometry, we consider a point to be an indivisible location in extensive space. So if two points are different, then they do not touch. Thus we conceive there to always be an extensive interval of space between points, and we must "jump" from one point to the next. Likewise, when we conceive of a unit in an indivisible mental action, we also think that we must "jump" to arrive-at the next unit. But each unit by itself is indivisible and hence non-spatial. And, just as there is an absolute difference between the non-extensive point and the space it is located-between, so too do we believe that the units themselves are non-spatial. (83a)

Now also consider what happens when we are imagining the points in space. Our attention sometimes wavers, and we then imagine the continuous lines that 'connect the dots,' so to speak. Now take for example when we are conceiving the number 4. We might think that there are four different indivisible units that make-up the 4, just like four discrete points in space. But, we also consider 4 in its "finished state" as a synthesis of its constituent units. In this case its discrete parts as well have been united by a continuity, "this union is an accomplished fact: the points have become lines, the divisions have been blotted out, the whole displays all the characteristics of continuity." (83b) And, we also believe that we may divide 4 in an infinite number of ways. Hence we must think of it also as a continuous magnitude.

So we must distinguish two types of unity:

1) the unity that we are thinking, for example, the unity of the number 4, which is taken to itself be an indivisible unit.

2) the unity that is the object which we may subsequently divide again any way we please.

Hence we may also distinguish the two in terms of

a) the number in process of formation through unification, and

b) the number formed through unification, which may then be divided.

So while we are conceiving a unit, it is indivisible. And while building-up a number from discrete unitary parts, it is discontinuous. However, when we consider number in its "finished state," we objectify it. And as an object, we may divide it infinitely.

Now let's consider the difference between subjective and objective knowledge.

1) [We know our own body to belong to us. There is really no extra data that could tell us otherwise. So,] when our knowledge of something is subjective, then we know it completely and adequately.

2) [We know that apples are red, green, or yellow. But it is conceivable that we might someday see orange ones, blue ones, and purple ones. Hence,] when our knowledge of something is objective, "a constantly increasing number of new impressions could be substituted for the idea which we actually have of it." (83-84)

[Our knowledge of our own body is our inherent feeling of it. We are always feeling every part of our body, even though we are not always aware of every part.] So complex feelings contain large numbers of simple parts. [And if we did become aware of the feeling we have in some part of our body, we have a different sort of state of consciousness than when we are unconsciously aware of our bodies. So,] as long as the simpler elements of a complex feeling do not stand out with perfect clarity in our consciousness, then they are not completely realized. And, the complex feeling results from a synthesis of all these constituent parts. Yet, if we were to become aware of these simpler elements, then there will be a simple part of our conscious state that has changed to a different sort of awareness. And since some of the totality of parts have changed, so too will the whole synthesis change. Thus when our consciousness attains a distinct perception of one of the simpler parts of our psychic state, the whole of that psychic state changes.

[Now consider instead when we imagine the way an apple looks. When we attend more to the appearance of its stem, that does not add any new information or awareness to what we know about the apple. For, the whole of the mental image contains everything that we already know about how the apple looks. Thus, ] no matter how our thinking analyzes a mental image, this will not change the way it appears to us, "because these different analyses, and an infinity of others, are already visible in the mental image which we form of the body, though they are not realized." (84)

[In the case of our bodily awareness, we are not actually aware of all the many constituent sensations of our body. But we can become explicitly aware of any of them. And in a sense, we are already aware of all of them, but in an implicit way. So we might say that our consciousness of our body's simpler parts are virtual awarenesses.

In a sense they are there, but if they were actualized, they would change the whole awareness. However, when imagining the apple, all the simpler parts of our consciousness of the image are fully actualized, because to attend to any of them more closely does not change the appearance of the apple. Thus, ]

this actual and not merely virtual perception of subdivisions in what is undivided is just what we call objectivity. (84b)

cette apperception actuelle, et non pas seulement virtuelle, de subdivisions dans l'indivisé est précisément ce que nous appelons objectivité. (64b)

So, the parts of subjective knowledge are only virtually divisible, but the parts of objective knowledge are actually divisible. Thus, when the mind conceives a number, its conception is subjective: it thinks the parts in one indivisible process. But in this process it thinks the parts as falling successively in some given space. But even though this indivisible mental process unites the parts, they have not disappeared. After we finish conceiving the number, it becomes an object that we are objectively aware-of. In this way, its parts have not vanished. They remain, which is why we are able to subsequently divided the unified number.

Hence numbers are parts of space. And thus space is the "material" that our mind uses to build-up numbers, and it is the "medium" into which the mind places them as well. (84d)

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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