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

Bergson, Time and Free Will, Chapter 1, §2 "Such Differences Applicable to Magnitudes but not to Intensities"

by Corry Shores
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Bergson, Time and Free Will

Chapter I, "The Intensity of Psychic States"

Part I, "Intensity and Extensity"

§2 "Such Differences Applicable to Magnitudes but not to Intensities"

One number or physical object is greater-or-lesser than another. Quantification applies to such extensive magnitudes. Here we compare unequal "spaces." A greater space contains the smaller one.

But how can a more intense sensation contain one of less intensity?

(2)

Mais comment une sensation plus intense contiendra-t-elle une sensation de moindre intensité?

(1-2)

Common sense tells us that we arrive at higher sensations after first passing through lower ones. However, this conception of intensity produces vicious circular reasoning.

The reason we may arrange numbers in ascending order is because they relate to each other as containers and contents. We superpose one quantity upon another to see which contains the other.

But intensities, however, cannot be superposed upon each other. So we have no grounds to say that intensities increase or decrease. For, we have no cause to consider them yet as magnitudes.

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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.

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

Hegel, Science of Logic, Remark 4 §§170-175

by Corry Shores
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Hegel

Science of Logic

Volume One: The Objective Logic

Book One: The Doctrine of Being

Section 1: Determinateness (Quality)

Chapter 1 Being

C Becoming

1. Unity of Being and Nothing

Remark 4: Incomprehensibility of the Beginning

§ 170

In the following, Hegel will prove that a beginning of the world — or of anything at all — is impossible.

§ 171

It is impossible for anything to begin, either in so far as it is, or in so far as it is not; for in so far as it is, it is not just beginning, and in so far as it is not, then also it does not begin.

For something to have begun, there must be a point of transition. But if at that point the beginning has begun, then it already has being, and thus it is no longer beginning. But if at that point the beginning has not yet begun, that it does not yet have being, and so it also is not beginning. Thus it is inconceivable that something might begin, because there is no way to account for the precise point of transition between being and non-being.

If the world, or anything, is supposed to have begun, then it must have begun in nothing, but in nothing — or nothing — is no beginning; for a beginning includes within itself a being, but nothing does not contain any being. Nothing is only nothing.

For the same reason, something cannot cease to be, because then being would have to contain nothing, "being is only being, not the contrary of itself."

§ 172

We are not here disproving becoming or the unity of being and nothing.

§ 173

So when we posit an absolute separatedness of being from nothing, we cannot account for beginning or becoming.

§ 174

This ambiguity of becoming as the unity of being and nothing is found also in infinitesimal magnitudes (treated later in more depth).

These magnitudes have been defined as such that they are in their vanishing, not before their vanishing, for then they are finite magnitudes, or after their vanishing, for then they are nothing.

Some object that these infinitesimal magnitudes must either be something or nothing, because there is no "intermediate state between being and non-being." (Hegel here parenthetically objects that the term 'state' is an "unsuitable, barbarous expression"). But Hegel has already shown that there is "nothing which is not an intermediate state between being and nothing." By adopting such a stance, mathematics has attained "its most brilliant successes."

§ 175

Reasoning that makes the false absolute separation between being and non-being is sophistry, not dialectic, because "sophistry is an argument proceeding from a baseless presupposition which is uncritically and unthinkingly adopted." However, dialectic is

the higher movement of reason in which such seemingly utterly separate terms pass over into each other spontaneously, through that which they are, a movement in which the presupposition sublates itself.

Through their "dialectical immanent nature," being and nothing "manifest their unity, that is, becoming, as their truth."

From the translation:

Remark 4: Incomprehensibility of the Beginning

§ 170

What has been said indicates the nature of the dialectic against the beginning of the world and also its end, by which the eternity of matter was supposed to be proved, that is, the dialectic against becoming, coming-to-be or ceasing-to-be, in general. The Kantian antinomy relative to the finitude or infinity of the world in space and time will be considered more closely under the Notion of quantitative infinity. This simple, ordinary dialectic rests on holding fast to the opposition of being and nothing. It is proved in the following manner that a beginning of the world, or of anything, is impossible:

§ 171

It is impossible for anything to begin, either in so far as it is, or in so far as it is not; for in so far as it is, it is not just beginning, and in so far as it is not, then also it does not begin. If the world, or anything, is supposed to have begun, then it must have begun in nothing, but in nothing — or nothing — is no beginning; for a beginning includes within itself a being, but nothing does not contain any being. Nothing is only nothing. In a ground, a cause, and so on, if nothing is so determined, there is contained an affirmation, a being. For the same reason, too, something cannot cease to be; for then being would have to contain nothing, but being is only being, not the contrary of itself.

§ 172

It is obvious that in this proof nothing is brought forward against becoming, or beginning and ceasing, against this unity of being and nothing, except an assertoric denial of them and an ascription of truth to being and nothing, each in separation from the other. Nevertheless this dialectic is at least more consistent than ordinary reflective thought which accepts as perfect truth that being and nothing only are in separation from each other, yet on the other hand acknowledges beginning and ceasing to be equally genuine determinations; but in these it does in fact assume the unseparatedness of being and nothing.

§ 173

With the absolute separateness of being from nothing presupposed, then of course — as we so often hear — beginning or becoming is something incomprehensible; for a presupposition is made which annuls the beginning or the becoming which yet is again admitted, and this contradiction thus posed and at the same time made impossible of solution, is called incomprehensible.

§ 174

The foregoing dialectic is the same, too, as that which understanding employs the notion of infinitesimal magnitudes, given by higher analysis. A more detailed treatment of this notion will be given later. These magnitudes have been defined as such that they are in their vanishing, not before their vanishing, for then they are finite magnitudes, or after their vanishing, for then they are nothing. Against this pre notion it is objected and reiterated that such magnitudes are either something or nothing; that there is no intermediate state between being and non-being ('state' is here an unsuitable, barbarous expression). Here too, the absolute separation of being and nothing is assumed. But against this it has been shown that being and nothing are, in fact, the same, or to use the same language as that just quoted, that there is nothing which is not an intermediate state between being and nothing. It is to the adoption of the said determination, which understanding opposes, that mathematics owes its most brilliant successes.

§ 175

This style of reasoning which makes and clings to the false presupposition of the absolute separateness of being and non-being is to be named not dialectic but sophistry. For sophistry is an argument proceeding from a baseless presupposition which is uncritically and unthinkingly adopted; but we call dialectic the higher movement of reason in which such seemingly utterly separate terms pass over into each other spontaneously, through that which they are, a movement in which the presupposition sublates itself. It is the dialectical immanent nature of being and nothing themselves to manifest their unity, that is, becoming, as their truth.

Hegel. Science of Logic. Transl. A.V. Miller. George Allen & Unwin, 1969.

Text available online at:

http://www.marxists.org/reference/archive/hegel/index.htm

Hegel, Science of Logic, Section 2: Magnitude (Quantity) §479-§481

by Corry Shores
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Hegel

Science of Logic

Volume One: The Objective Logic

Book One: The Doctrine of Being

Section 2: Magnitude (Quantity)

B. EXTENSIVE AND INTENSIVE QUANTUM

(b) Identity of Extensive and Intensive Magnitude

§ 479

Intensive magnitude is a “unitary one of a plurality.” An intensive magnitude may come in a variety of different degrees, but that magnitude is neither a simple one with no numerical value, nor is it a plurality, because its value is taken together. It obtains its value through it relations to other degrees. Hence something of the 10^th degree is half that of something that is of the 20^th degree. The 10 degrees of a 10-degree-intensive-magnitude are contained within the whole value, but as a continuity.

§ 480

Intensive magnitudes are determined in two ways. They are determined by other intensive quanta that it is continuous with. So the 10^th degree is 10 because it stops before reaching what lies beyond that value. Thus in this way it obtains its determinateness through its exclusion of other values.

Intensive magnitudes also receive their determinate value by being a value in itself. So the 20^th degree contains 20 within itself which determines its value.

But as soon as we consider the 20^th degree as having 20 degrees within it, that value of 20 taken on its own is an amount, making the degrees it describes an extensive quantum rather than an intensive one, because intensive quanta are continuous.

§ 481

Extensive and intensive magnitude are thus one and the same determinateness of quantum; they are only distinguished by the one having amount within itself and the other having amount outside itself.

An extensive magnitude passes over into an intensive one when its internal plurality collapses into oneness, leaving plurality to lie outside it. Conversely, the intensive magnitude passes over into an extensive one when its “differently determined intensities” are considered in terms of an external amount.

From the original text:

(b) Identity of Extensive and Intensive Magnitude

§ 479

Degree is not external to itself within itself. Nevertheless, it is not the indeterminate one, the principle of number as such, which is not amount save in the negative sense of not being any particular amount. Intensive magnitude is primarily a unitary one of a plurality; there are many degrees, but they are determined neither as a simple one nor as a plurality, but only in the relation of this self-externality, or in the identity of the one and the plurality. if, therefore, the many as such are indeed outside the simple, unitary degree, nevertheless the determinateness of the degree consists in its relation to them; it thus contains amount. just as twenty, as an extensive magnitude, contains the twenty ones as discrete within it, so does the specific degree contain them as the continuity which this determinate plurality simply is; it is the twentieth degree, and is the twentieth degree only by virtue of this amount, which as such is outside it.

§ 480

The determinateness of intensive magnitude is, therefore, to be considered from two sides. Intensive magnitude is determined by other intensive quanta and is continuous with its otherness, so that its determinateness consists in this relation to its otherness. Now in the first place, in so far as it is a simple determinateness it is determinate relatively to other degrees; it excludes them from itself and has its determinateness in this exclusion. But, secondly, it is determinate in its own self; this it is in the amount as its own amount, not in the amount as excluded, nor in the amount of other degrees. The twentieth degree contains the twenty within itself; it is not only determined as distinguished from the nineteenth, twenty-first, and so on, but its determinateness is its own amount. But in so far as the amount is its own — and the determinateness is at the same time essentially an amount — the degree is an extensive quantum.

§ 481

Extensive and intensive magnitude are thus one and the same determinateness of quantum; they are only distinguished by the one having amount within itself and the other having amount outside itself. Extensive magnitude passes over into intensive magnitude because its many spontaneously collapse into oneness, outside which the many stand. But conversely, this unitary degree has its determinateness only in the amount, and that too in its own amount; as indifferent to the differently determined intensifies it has within itself the externality of the amount; and so intensive magnitude is equally essentially an extensive magnitude.

§ 482

With this identity, the qualitative something makes its appearance, for the identity is the unity which is self-related through the negation of its differences; these differences, however, constitute the determinate being of the quantitative determinateness; this negative identity is therefore a something, and a something which is indifferent to its quantitative determinateness. Something is a quantum; but now the qualitative determinate being, as it is in itself, is posited as indifferent to quantum. Quantum, number as such, and so forth, could be spoken of without any mention of its having a something as substrate. But the something now confronts these its determinations, through the negation of which it is mediated with itself, as existing for itself and, since it has a quantum, as something which has an extensive and an intensive quantum. Its one determinateness which it has as quantum is posited in the differentiated moments of unit and amount; this determinateness is not only in itself one and the same, but its positing in these differences as extensive and intensive quantum is the return into this unity which, as negative, is the something posited as indifferent to it.

Hegel. Science of Logic. Transl. A.V. Miller. George Allen & Unwin, 1969.

Text available online at:

http://www.marxists.org/reference/archive/hegel/index.htm