Showing posts with label Science of Logic. Show all posts
Showing posts with label Science of Logic. Show all posts

31 Jan 2009

Hegel Science of Logic Entry Directory


by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]


[Central Entry Directory]


Hegel


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
Section 2: Magnitude (Quantity)

Chapter 1 Quantity

A. PURE QUANTITY


B. CONTINUOUS AND DISCRETE MAGNITUDE

C. LIMITATION OF QUANTITY

Chapter 2 Quantum

B. EXTENSIVE AND INTENSIVE QUANTUM

(a) Their Difference
(b) Identity of Extensive and Intensive Magnitude
Remark 1: Examples of This Identity

C. QUANTITATIVE INFINITY

(c) The Infinity of Quantum

Remark 1: The Specific Nature of the Notion of the Mathematical Infinite


Remark 1: The Specific Nature of the Notion of the Mathematical Infinite, continued
And Remark 2: The Purpose of the Differential Calculus Deduced from its Application



Book Two: The Doctrine of Essence

Section Three: Actuality

Chapter 1 The Absolute

C The Mode of the Absolute

Remark: The Philosophy of Spinoza and Leibniz

§§1179-1183 Hegel's Critique of Spinoza in Science of Logic §§1179-1183, summarized


Subjective Logic, or The Doctrine of the Notion

Chapter 1: The Notion

The Notion in General




1 Jan 2009

Hegel, Science of Logic, Preface to the First Edition


by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]



[The following is summary. The full text of the summarized sections is at the end.]



Preface to the First Edition

§ 1

Germany's recent incredible philosophical advances have had "but little influence as yet on the structure of logic."

§ 2

It is remarkable that metaphysics, ontology, cosmology, and natural theology have been abandoned.

§ 3

Kantianism advises we do not take our understanding beyond experience, because then our cognitive faculty generates fantasies. Also, contemporary education teaches practical skills over theory. Hence the recent renunciation of speculative philosophy. Theology was once the guardian of "speculative mysteries." Now it favors "feelings" and historical erudition, so we no longer find monastics "exiled from the world" to contemplate the eternal.

So that having got rid of the dark utterances of metaphysics, of the colourless communion of the spirit with itself, outer existence seemed to be transformed into the bright world of flowers-and there are no black flowers, as we know.

§ 4

Logic was not neglected as much as metaphysics, because it has the semblance of practicality; nonetheless, it has been largely put aside in favor of far more practical studies.

§ 5

And yet, logic has retained a place among the sciences. But its content has become watered-down, and

logic shows no traces so far of the new spirit which has arisen in the sciences no less than in the world of actuality.

However, this is to be expected, because when spirit's substantial form comes to reconstitute its inner self,

all attempts to preserve the forms of an earlier culture are utterly in vain; like withered leaves they are pushed off by the new buds already growing at their roots.

§ 6

Even philosophy has recently resigned itself to the new antimetaphysical trends.

§ 7

But when new movements begin, they are anti-systematic so to rebel from the dominant systems they oppose. But after gaining ground, it becomes necessary for the new movements to systematize.

There is a period in the culture of an epoch as in the culture of the individual, when the primary concern is the acquisition and assertion of the principle in its undeveloped intensity. But the higher demand is that it should become systematised knowledge.

So even our recent anti-speculative movement is becoming more speculative as it comes to ground itself theoretically.

§ 8

On account of this neglect of metaphysics, "the science of logic which constitutes metaphysics proper or purely speculative philosophy, has hitherto still been much neglected."

Hegel will explain in the introduction what he means by this science of logic.

Anyone who might criticize the following effort to systematize the science of logic should first take into account the fact that this is the first such effort, and it is a profoundly difficult one at that.

The essential point of view is that what is involved is an altogether new concept of scientific procedure.

For philosophy to be a science, it cannot

1) adopt its method from a subordinate science like mathematics,

2) rely merely on inner intuition, or

3) use arguments grounded in external reflection.

On the contrary, it can be only the nature of the content itself which spontaneously develops itself in a scientific method of knowing, since it is at the same time the reflection of the content itself which first posits and generates its determinate character.

§ 9

The understanding determines, and holds the determinations fixed; reason is negative and dialectical, because it resolves the determinations of the understanding into nothing; it is positive because it generates the universal and comprehends the particular therein.

Dialectical reason is often distinguished from positive reason. But true reason is spirit and it is higher than positive reason or intuitive understanding.

True reason bears a quality shared by both understanding and dialectic:

it negates what is simple, thus positing the specific difference of the understanding; it equally resolves it and is thus dialectical.

Although it negates what is simple, it furthermore is positive, because it restores what originally was simple. For, through the dialectical movement, true reason restores what was initially simple by rendering it in a universal and concrete form.

And through this dialectical movement, the particular determines itself through the positing and resolving of difference. Hence the particular is not subsumed under the universal, because it determined itself rather than something determining it.

This spiritual movement which, in its simple undifferentiatedness, gives itself its own determinateness and in its determinateness its equality with itself, which therefore is the immanent development of the Notion, this movement is the absolute method of knowing and at the same time is the immanent soul of the content itself.

I maintain that it is this self-construing method alone which enables philosophy to be an objective, demonstrated science.

§ 10

So the spiritual movement involves external self-determination. This is how Hegel describes consciousness in his Phenomenology of Spirit. But this developmental process for consciousness and all spiritual life proceeds according to a logic.

Consciousness is manifested spirit, and it progresses through its own immediacy and external concretion. It finally attains a pure knowing that knows pures essences. These pure thoughts result when spirit thinks its own essence. The movement leading to this pure state is the spiritual life of thought, and philosophy exists through this movement by being what expresses these pure thoughts.

§ 11

Next, Hegel will explain the relation between logic and the Phenomenology of Spirit, and the whole of his Science of Logic should be considered a sequel to his Phenomenology of Spirit.


[The following is the original text that is summarized above.]

Preface to the First Edition
§ 1
The complete transformation which philosophical thought in Germany has undergone in the last twenty-five years and the higher standpoint reached by spirit in its awareness of itself, have had but little influence as yet on the structure of logic.
§ 2
That which, prior to this period, was called metaphysics has been, so to speak, extirpated root and branch and has vanished from the ranks of the sciences. The ontology, rational psychology, cosmology, yes even natural theology, of former times-where is now to be heard any mention of them, or who would venture to mention them? Inquiries, for instance, into the immateriality of the soul, into efficient and final causes, where should these still arouse any interest? Even the former proofs of the existence of God are cited only for their historical interest or for purposes of edification and uplifting the emotions. The fact is that there no longer exists any interest either in the form or the content of metaphysics or in both together. If it is remarkable when a nation has become indifferent to its constitutional theory, to its national sentiments, its ethical customs and virtues, it is certainly no less remarkable when a nation loses its metaphysics, when the spirit which contemplates its own pure essence is no longer a present reality in the life of the nation.
§ 3
The exoteric teaching of the Kantian philosophy — that the understanding ought not to go beyond experience, else the cognitive faculty will become a theoretical reason which itself generates nothing but fantasies of the brain — this was a justification from a philosophical quarter for the renunciation of speculative thought. In support of this popular teaching came the cry of modern educationists that the needs of the time demanded attention to immediate requirements, that just as experience was the primary factor for knowledge, so for skill in public and private life, practice and practical training generally were essential and alone necessary, theoretical insight being harmful even. Philosophy [Wissenschaft] and ordinary common sense thus co-operating to bring about the downfall of metaphysics, there was seen the strange spectacle of a cultured nation without metaphysics-like a temple richly ornamented in other respects but without a holy of holies. Theology, which in former times was the guardian of the speculative mysteries and of metaphysics (although this was subordinate to it) had given up this science in exchange for feelings, for what was popularly matter-of-fact, and for historical erudition. In keeping with this change, there vanished from the world those solitary souls who were sacrificed by their people and exiled from the world to the end that the eternal should be contemplated and served by lives devoted solely thereto — not for any practical gain but for the sake of blessedness; a disappearance which, in another context, can be regarded as essentially the same phenomenon as that previously mentioned. So that having got rid of the dark utterances of metaphysics, of the colourless communion of the spirit with itself, outer existence seemed to be transformed into the bright world of flowers-and there are no black flowers, as we know.
§ 4
Logic did not fare quite so badly as metaphysics. ®. That one learns from logic how to think (the usefulness of logic and hence its purpose, were held to consist in this — just as if one could only learn how to digest and move about by studying anatomy and physiology) this prejudice has long since vanished, and the spirit of practicality certainly did not intend for logic a better fate than was suffered by the sister science.
§ 5
Nevertheless, probably for the sake of a certain formal utility, it was still left a place among the sciences, and indeed was even retained as a subject of public instruction. However, this better lot concerns only the outer fate of logic, for its structure and contents have remained the same throughout a long inherited tradition, although in the course of being passed on the contents have become ever more diluted and attenuated; logic shows no traces so far of the new spirit which has arisen in the sciences no less than in the world of actuality. However, once the substantial form of the spirit has inwardly reconstituted itself, all attempts to preserve the forms of an earlier culture are utterly in vain; like withered leaves they are pushed off by the new buds already growing at their roots.
§ 6
Even in the philosophical sphere this ignoring of the general change is beginning gradually to come to an end. Imperceptibly, even those who are opposed to the new ideas have become familiar with them and have appropriated them, and if they continue to speak slightingly of the source and principles of those ideas and to dispute them, still they have accepted their consequences and have been unable to defend themselves from their influence; the only way in which they can give a positive significance and a content to their negative attitude which is becoming less and less important, is to fall in with the new ways of thinking.
§ 7
On the other hand, it seems that the period of fermentation with which a new creative idea begins is past. In its first manifestation, such an idea usually displays a fanatical hostility toward the entrenched systematisation of the older principle; usually too, it is fearful of losing itself in the ramifications of the particular and again it shuns the labour required for a scientific elaboration of the new principle and in its need for such, it grasps to begin with at an empty formalism. The challenge to elaborate and systematise the material now becomes all the more pressing. There is a period in the culture of an epoch as in the culture of the individual, when the primary concern is the acquisition and assertion of the principle in its undeveloped intensity. But the higher demand is that it should become systematised knowledge.
§ 8
Now whatever may have been accomplished for the form and content of philosophy in other directions, the science of logic which constitutes metaphysics proper or purely speculative philosophy, has hitherto still been much neglected. What it is exactly that I understand by this science and its standpoint, I have stated provisionally in the Introduction.
The fact that it has been necessary to make a completely fresh start with this science, the very nature of the subject matter and the absence of any previous works which might have been utilised for the projected reconstruction of logic, may be taken into account by fair-minded critics, even though a labour covering many years has been unable to give this effort a greater perfection. The essential point of view is that what is involved is an altogether new concept of scientific procedure.
Philosophy, if it would be a science, cannot, as I have remarked elsewhere, borrow its method from a subordinate science like mathematics, any more than it can remain satisfied with categorical assurances of inner intuition, or employ arguments based on grounds adduced by external reflection. On the contrary, it can be only the nature of the content itself which spontaneously develops itself in a scientific method of knowing, since it is at the same time the reflection of the content itself which first posits and generates its determinate character. ®
§ 9
The understanding determines, and holds the determinations fixed; reason is negative and dialectical, because it resolves the determinations of the understanding into nothing; it is positive because it generates the universal and comprehends the particular therein.
Just as the understanding is usually taken to be something separate from reason as such, so too dialectical reason is usually taken to be something distinct from positive reason. But reason in its truth is spirit which is higher than either merely positive reason, or merely intuitive understanding.
It is the negative, that which constitutes the quality alike of dialectical reason and of understanding; it negates what is simple, thus positing the specific difference of the understanding; it equally resolves it and is thus dialectical.
But it does not stay in the nothing of this result but in the result is no less positive, and in this way it has restored what was at first simple, but as a universal which is within itself concrete; a given particular is not subsumed under this universal but in this determining, this positing of a difference, and the resolving of it, the particular has at the same time already determined itself. This spiritual movement which, in its simple undifferentiatedness, gives itself its own determinateness and in its determinateness its equality with itself, which therefore is the immanent development of the Notion, this movement is the absolute method of knowing and at the same time is the immanent. soul of the content itself.
I maintain that it is this self-construing method alone which enables philosophy to be an objective, demonstrated science.®
§ 10
It is in this way that I have tried to expound consciousness in the Phenomenology of Spirit. Consciousness is spirit as a concrete knowing, a knowing too, in which externality is involved; but the development of this object, ®like the development of all natural and spiritual life, rests solely on the nature of the pure essentialities which constitute the content of logic.
Consciousness, as spirit in its manifestation which in its progress frees itself from its immediacy and external concretion, attains to the pure knowing which takes as its object those same pure essentialities as they are in and for themselves. They are pure thoughts, spirit thinking its own essential nature. Their self-movement is their spiritual life and is that through which philosophy constitutes itself and of which it is the exposition.
§ 11
In the foregoing there is indicated the relation of the science which I call the Phenomenology of Spirit, to logic. As regards the external relation, it was intended that the first part of the System of Science which contains the Phenomenology should be followed by a second part containing logic and the two concrete [realen] sciences, the Philosophy of Nature and the Philosophy of Spirit, which would complete the System of Philosophy. But the necessary expansion which logic itself has demanded has induced me to have this part published separately; it thus forms the first sequel to the Phenomenology of Spirit in an expanded arrangement of the system. It will later be followed by an exposition of the two concrete philosophical sciences mentioned. This first volume of the Logic contains as Book One the Doctrine of Being; Book Two, the Doctrine of Essence, which forms the second part of the first volume, is already in the press; the second volume will contain Subjective Logic or the Doctrine of the Notion.
Nuremberg, March 22, 1812.



Hegel. Science of Logic. Transl. A.V. Miller. George Allen & Unwin, 1969.
Text available online at:

29 Dec 2008

Hegel's Critique of Spinoza in Science of Logic §§1287-1288, summarized



Hegel, Science of Logic

Volume Two: Subjective Logic

The Doctrine of the Notion

§ 1287

Hegel explained previously that Spinoza's philosophy develops no further than the standpoint of substance. The primary defect of Spinoza's system is that it cannot return to substance after first moving through attributes then finally to modes.

To refute such a system, we firstly do not considered it as completely false. For, like Spinoza, we must begin with the relation between substance and its essence. The only thing false about Spinoza's system is its claim to be the highest standpoint. So the true system is not opposed to Spinoza's; the true system builds from it and subsumes it.

§ 1288

Our refutations also cannot take an exterior position to Spinoza's system; because it could just refuse to recognize our assumptions. Thus to refute Spinoza, we must first presuppose the self-conscious subject's freedom and self-subsistence. One such exterior position would reject Spinoza's attributing Thought to Substance.

The nerve, therefore, of the external refutation consists solely in clinging stubbornly to the antitheses of these assumptions, for example, to the absolute self-subsistence of the thinking individual as against the form of thought posited in absolute substance as identical with extension. The genuine refutation must penetrate the opponent's stronghold and meet him on his own ground; no advantage is gained by attacking him somewhere else and defeating him where he is not. The only possible refutation of Spinozism must therefore consist, in the first place, in recognising its standpoint as essential and necessary and then going on to raise that standpoint to the higher one through its own immanent dialectic.

What dialectically arises from substance is its opposite, the Notion; hence to refute Spinozism we must begin with substance and follow a line of reasoning that leads to the Notion.

Substance's unity is only an inner necessity. It becomes determinate when qualified by its attribute Thought. In this way, Substance posits its identity. But for substance to obtain this qualitative identity, the attribute Thought must stand against Substance and hence negate it. Thought and Substance sublate into the Notion. In this way, Substance and Thought are moments of reflection in a dialectical movement, and hence each is united with its other, "and consequently in its other is posited as simply and solely identical with itself."



Hegel. Science of Logic. Transl. A.V. Miller. George Allen & Unwin, 1969.
Text available online at:




26 Nov 2008

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

by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]


[Below is summary. At the end I cite the text in full. My interpretations not informed by a complete read of the text.]




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:

25 Nov 2008

2/7 is 0.285714..., and 1/(1 - a) as 1 + a + a2 + a3 etc. From Hegel Science of Logic § 556


by Corry Shores
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[Below is summary. At the end I cite the text in full. My interpretations not informed by a complete read of the text.]


from Hegel, Science of Logic, § 556:


The fraction 2/7 can be expressed as 0.285714..., 1/(1 - a) as 1 + a + a2 + a3 etc. As so expressed it is an infinite series; the fraction itself is called the sum, or finite expression of it.

2/7 can be expressed as 0.285714..., because:

2 divided by 7 long division



1/(1 - a) can be expressed as 1 + a + a2 + a3 etc., because we can represent it as a series of sums, and first we take out 1, that is, we subtract 1 from it (expanded as 1/1).

a1.sums

To obtain the same denominator, we multiply 1/1 by (1-a)/(1-a):

a2.sums

We combine them to obtain:

a3.sums

We distribute the negative in the numerator to get:

a4.sums

We subtract the ones to obtain:

a5.sums

We then pull the a out of the numerator:

a6.sums

This of course is equal to the original equation from which it was derived:

a8.sums

We then move the -1 over to the other side (and rotate the sides) to get:

a9.sums

But we see something interesting here, because we have the opportunity for a re-iteration. The left side is equal to a formula which includes that very same left side. So, we can substitute the whole right side in for the 1/(1-a) nested within the right side, to get:

a10.sums

We then distribute the a to both terms in parenthesis to obtain:

a11.sums

We may of course reiterate this process once again:

a12.sums

Thus we can see how 1/(1 - a) can be expressed as 1 + a + a2 + a3 etc.


From the original text:

    The fraction 2/7 can be expressed as 0.285714..., 1/(1 - a) as 1 + a + a2 + a3 etc. As so expressed it is an infinite series; the fraction itself is called the sum, or finite expression of it. A comparison of the two expressions shows that one of them, the infinite series, represents the fraction no longer as a ratio but from that side where it is a quantum as anaggregate of units added together, as an amount. That the magnitudes of which it is supposed to consist as amount are in turn decimal fractions and therefore are themselves ratios, is irrelevant here; for this circumstance concerns the particular kind of unit of these magnitudes, not the magnitudes as constituting an amount. just as a multi-figured integer in the decimal system is reckoned essentially as an amount, and the fact that it consists of products of a number and of the number ten and powers of ten is ignored. Similarly here, it is irrelevant that there are fractions other than the example taken of 2/7 which, when expressed as decimal fractions, do not give an infinite series; but they can all be so expressed in a numerical system based on another unit.


Second demonstration based on explanation in:
http://broodsphilosophy.wordpress.com/2006/06/28/hegel-and-infinite-series/


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