Reciprocity, or Action & Reaction

Hegel: Shorter Logic/Essence/Actuality/Reciprocity

§ 155

The characteristics which in Reciprocal Action are retained as distinct are [a] potentially the same. The one side is a cause, is primary, active, passive, etc., just as the other is. Similarly the presupposition of another side and the action upon it, the immediate primariness and the dependence produced by the alternation, are one and the same on both sides. The cause assumed to be first is on account of its immediacy passive, a dependent being, and an effect. The distinction of the causes spoken of as two is accordingly void: and properly speaking there is only one cause, which, while it suspends itself (as substance) in its effect, also rises in this operation only to independent existence as a cause.

§ 156

But this unity of the double cause is also [b] actual. All this alternation is properly the cause in act of constituting itself and in such constitution lies its being. The nullity of the distinctions is not only potential, or a reflection of ours (§155). Reciprocal action just means that each characteristic we impose is also to be suspended and inverted into its opposite, and that in this way the essential nullity of the ‘moments’ is explicitly stated. An effect is introduced into the primariness; in other words, the primariness is abolished: the action of a cause becomes reaction and so on.

Summary of Hegel’s Objective Logic (book 2.)

Hegel’s Second Book of the Objective Logic is summarised in the introductory part to his Subjective Logic

Subjective Logic
or
The Doctrine of the Notion

The Notion in General

§ 1279

What the nature of the Notion is, can no more be stated offhand than can the Notion of any other object. It might perhaps seem that, in order to state the Notion of an object, the logical element were presupposed and that therefore this could not in turn have something else for its presupposition, nor be deduced; just as in geometry logical propositions as applied to magnitude and employed in that science, are premised in the form of axioms, determinations of cognition that have not been and cannot be deduced. Now although it is true that the Notion is to be regarded, not merely as a subjective presupposition but as the absolute foundation, yet it can be so only in so far as it has made itself the foundation. Abstract immediacy is no doubt a first; yet in so far as it is abstract it is, on the contrary mediated, and therefore if it is to be grasped in its truth its foundation must first be sought. Hence this foundation, though indeed an immediate, must have made itself immediate through the sublation of mediation.

§ 1280

From this aspect the Notion is to be regarded in the first instance simply as the third to being and essence, to the immediate and to reflection. Being and essence are so far the moments of its becoming; but it is their foundation and truth as the identity in which they are submerged and contained. They are contained in it because it is their result, but no longer as being and essence. That determination they possess only in so far as they have not withdrawn into this their unity.

§  1281

Objective logic therefore, which treats of being and essence constitutes properly the genetic exposition of the Notion. More precisely, substance is already real essence, or essence in so far as it is united with being and has entered into actuality. Consequently, the Notion has substance for its immediate presupposition; what is implicit in substance is manifested in the Notion. Thus the dialectical movement of substance through causality and reciprocity is the immediate genesis of the Notion, the exposition of the process of its becoming. But the significance of its becoming, as of every becoming is that it is the reflection of the transient into its ground and that the at first apparent other into which the former has passed constitutes its truth. Accordingly the Notion is the truth of substance; and since substance has necessity for its specific mode of relationship, freedom reveals itself as the truth of necessity and as the mode of relationship proper to the Notion.

§  1282

The progressive determination of substance necessitated by its own nature, is the positing of what is in and for itself. Now the Notion is that absolute unity of being and reflection in which being is in and for itself only in so far as it is no less reflection or positedness, and positedness is no less being that is in and for itself. This abstract result is elucidated by the exposition of its concrete genesis; that exposition contains the nature of the Notion whose treatment it must have preceded. The chief moments of this exposition (which has been given in detail in the Second Book of the Objective Logic) can therefore only be briefly summarised here.

§ 1283

Substance is the absolute, the actuality that is in and for itself in itself as the simple identity of possibility and actuality, absolute essence containing all actuality and possibility within itself; and for itself, being this identity as absolute power or purely self-related negativity. The movement of substantiality posited by these moments consists in the following stages:

§ 1284

1. Substance, as absolute power or self-related negativity, differentiates itself into a relationship in which what were at first only simple moments are substances and original presuppositions. Their specific relationship is that of a passive substance, of the original immediacy of the simple inwardness or in-itself which, powerless to posit itself, is only an original positedness and of an active substance, the self-related negativity which as such has posited itself in the form of an other and relates itself to this other. This other is simply the passive substance which the active substance through its own originative power has presupposed for itself as condition. This presupposing is to be understood in the sense that the movement of substance itself is, in the first instance, under the form of one of the moments of its Notion, the in-itself, and the determinateness of one of the substances standing in relationship is also the determinateness of this relationship itself.

§ 1285

2. The other moment is being-for-self, which means that the power posits itself as self-related negativity, thereby sublating again what was presupposed. The active substance is the cause; it acts, that is, it now posits, whereas previously it only presupposed; so that (a) to the power is now added the illusory show of power, to the positedness the illusory show of positedness. What in the presupposition was original, becomes in causality, through the relation to an other, what it is in itself; the cause produces an effect, and that, too, in another substance; it is now power in relation to an other and thus appears as a cause, but is a cause only in virtue of this appearing. (b) The effect enters the passive substance, whereby it now also appears as a positedness, but is a passive substance only as such positedness.

§ 1286

3. But there is still more present in this than only this appearance, namely: (a) the cause acts on the passive substance and alters its determination; but this is positedness, there is nothing else in it to alter; the other determination, however, that it receives is causality; the passive substance therefore becomes cause, power and activity: (b) the effect is posited in it by the cause; but that which is posited by the cause is the cause itself which, in acting, is identical with itself; it is this that puts itself in the place of the passive substance. Similarly, with regard to the active substance, (a) the action is the translation of the cause into the effect, into the other of the cause, into positedness, and (b) the cause reveals itself in the effect as what it is; the effect is identical with the cause, is not an other-; thus the cause in acting reveals the posited being as that which the cause essentially is. Each side, therefore, in both its identical and negative relation to the other becomes the opposite of itself, so that the other, and therefore also each, remains identical with itself. But the identical and the negative relations are both one and the same; substance is self-identical only in its opposite and this constitutes the absolute identity of the substances posited as a duality. Active substance, through the act of positing itself as the opposite of itself, an act which is at the same time the sublating of its presupposed otherness, of passive substance, is manifested as cause or originative substantiality. Conversely, through being acted on, posited being is manifested as posited, the negative as negative, and therefore passive substance as self-related negativity, the cause meeting in this other simply and solely with its own self. Through this positing, then, the presupposed or implicit originativeness becomes explicit or for itself; yet this being that is in and for itself is such only in so far as this positing is equally a sublating of what was presupposed; in other words absolute substance has returned to itself and so become absolute, only out of and in its positedness. Hence this reciprocity is the appearance that again sublates itself, the revelation that the illusory being of causality in which the cause appears as cause, is illusory being. This infinite reflection-into-self, namely, that being is in and for itself only in so far as it is posited, is the consummation of substance. But this consummation is no longer substance itself but something higher, the Notion, the subject. The transition of the relation of substantiality takes place through its own immanent necessity and is nothing more than the manifestation of itself, that the Notion is its truth, and that freedom is the truth of necessity.

Hegel’s doctrine of notion

(Hegel’s introductionary remarks)

From:

Part One of the Encyclopedia of Philosophical Sciences: The Logic

Third Subdivision
IX. The Notion

§ 160

The Notion is the principle of freedom, the power of substance self-realised. It is a systematic whole, in which each of its constituent functions is the very total which the notion is, and is put as indissolubly one with it. Thus in its self-identity it has original and complete determinateness.

The position taken up by the notion is that of absolute idealism. Philosophy is a knowledge through notions because it sees that what on other grades of consciousness is taken to have Being, and to be naturally or immediately independent, is but a constituent stage in the Idea. In the logic of understanding, the notion is generally reckoned a mere form of thought, and treated as a general conception. It is to this inferior view of the notion that the assertion refers, so often urged on behalf of the heart and sentiment, that notions as such are something dead, empty, and abstract. The case is really quite the reverse.

The notion is, on the contrary, the principle of all life, and thus possesses at the same time a character of thorough concreteness. That it is so follows from the whole logical movement up to this point, and need not be here proved. The contrast between form and content, which is thus used to criticise the notion when it is alleged to be merely formal, has, like all the other contrasts upheld by reflection, been already left behind and overcome dialectically or through itself. The notion, in short, is what contains all the earlier categories of thought merged in it. It certainly is a form, but an infinite and creative form which includes, but at the same time releases from itself, the fullness of all content. And so too the notion may, if it be wished, be styled abstract, if the name concrete is restricted to the concrete facts of sense or of immediate perception. For the notion is not palpable to the touch, and when we are engaged with it, hearing and seeing must quite fail us. And yet, as it was before remarked, the notion is a true concrete; for the reason that it involves Being and Essence, and the total wealth of these two spheres with them, merged in the unity of thought.

If, as was said at an earlier point, the different stages of the logical idea are to be treated as a series of definitions of the Absolute, the definition which now results for us is that the Absolute is the Notion. That necessitates a higher estimate of the notion, however, than is found in formal conceptualist Logic, where the notion is a mere form of our subjective thought, with no original content of its own. But if Speculative Logic thus attaches a meaning to the term notion so very different from that usually given, it may be asked why the same word should be employed in two contrary acceptations, and an occasion thus given for confusion and misconception. The answer is that, great as the interval is between the speculative notion and the notion of Formal Logic, a closer examination shows that the deeper meaning is not so foreign to the general usages of language as it seems at first sight. We speak of the deduction of a content from the notion, e.g. of the specific provisions of the law of property from the notion of property; and so again we speak of tracing back these material details to the notion. We thus recognise that the notion is no mere form without a content of its own: for if it were, there would be in the one case nothing to deduce from such a form, and in the other case to trace a given body of fact back to the empty form of the notion would only rob the fact of its specific character, without making it understood.

Development
§ 161

The onward movement of the notion is no longer either a transition into, or a reflection on something else, but Development. For in the notion, the elements distinguished are without more ado at the same time declared to be identical with one another and with the whole, and the specific character of each is a free being of the whole notion.

Transition into something else is the dialectical process within the range of Being: reflection (bringing something else into light), in the range of Essence. The movement of the Notion is development: by which that only is explicit which is already implicitly present. In the world of nature it is organic life that corresponds to the grade of the notion. Thus e.g. the plant is developed from its germ. The germ virtually involves the whole plant, but does so only ideally or in thought: and it would therefore be a mistake to regard the development of the root, stem, leaves, and other different parts of the plant, as meaning that they were realiter present, but in a very minute form, in the germ. That is the so-called ‘box-within-box’ hypothesis; a theory which commits the mistake of supposing an actual existence of what is at first found only as a postulate of the completed thought. The truth of the hypothesis on the other hand lies in its perceiving that in the process of development the notion keeps to itself and only gives rise to alteration of form, without making any addition in point of content. It is this nature of the notion — this manifestation of itself in its process as a development of its own self which is chiefly in view with those who speak of innate ideas, or who, like Plato, describe all learning merely as reminiscence. Of course that again does not mean that everything which is embodied in a mind, after that mind has been formed by instructions had been present in that mind beforehand, in its definitely expanded shape.

The movement of the notion is as it were to be looked upon merely as plan: the other which it sets up is in reality not an other. Or, as it is expressed in the teaching of Christianity: not merely has God created a World which confronts him as an other; he has also from all eternity begotten a Son in whom he, a Spirit, is at home with himself.

§ 162

The doctrine of the notion is divided into three parts.

(1) The first is the doctrine of the Subjective or Formal Notion.

(2) The second is the doctrine of the notion invested with the character of immediacy, or of Objectivity.

(3) The third is the doctrine of the Idea, the subject-object, the unity of notion and objectivity, the absolute truth.

Hegel on Cognition

[The below excerpt up to § 233 was left out from my previous blog:]

[a] Cognition proper

§ 226

The universal finitude of Cognition, which lies in the one judgment, the presupposition of the contrast (§ 224) — a presupposition in contradiction of which its own act lodges protest — specialises itself more precisely on the face of its own idea. The result of that specialisation is that its two elements receive the aspect of being diverse from each other, and, as they are at least complete, they take up the relation of ‘reflection’, not of ‘notion’, to one another. The assimilation of the matter, therefore, as a datum, presents itself in the light of a reception of it into categories which at the same time remain external to it, and which meet each other in the same style of diversity. Reason is active here, but it is reason in the shape of understanding. The truth which such Cognition can reach will therefore be only finite: the infinite truth (of the notion) is isolated and made transcendent, an inaccessible goal in a world of its own. Still in its external action cognition stands under the guidance of the notion, and notional principles form the secret clue to its movement.

The finitude of Cognition lies in the presupposition of a world already in existence, and in the consequent view of the knowing subject as a tabula rasa. The conception is one attributed to Aristotle; but no man is further than Aristotle from such an outside theory of Cognition. Such a style of Cognition does not recognise in itself the activity of the notion — an activity which it is implicitly, but not consciously. In its own estimation its procedure is passive. Really that procedure is active.

§ 227

Finite Cognition, when it presupposes what is distinguished from it to be something already existing and confronting it — to be the various facts of external nature or of consciousness — has, in the first place, (1) formal identity or the abstraction of universality for the form of its action. Its activity therefore consists in analysing the given concrete object, isolating its differences, and giving them the form of abstract universality. Or it leaves the concrete thing as a ground, and by setting aside the unessential-looking particulars, brings into relief a concrete universal, the Genus, or Force and Law. This is the Analytical Method.

People generally speak of the analytical and synthetic methods, as if it depended solely on our choice which we pursued. This is far from the case. It depends on the form of the objects of our investigation, which of the two methods that are derivable from the notion of finite cognition ought to be applied. In the first place, cognition is analytical. Analytical cognition deals with an object which is presented in detachment, and the aim of its action is to trace back to a universal the individual object before it. Thought in such circumstances means no more than an act of abstraction or of formal identity. That is the sense in which thought is understood by Locke and all empiricists. Cognition, it is often said, can never do more than separate the given concrete objects into their abstract elements, and then consider these elements in their isolation. It is, however, at once apparent that this turns things upside down, and that cognition, if its purpose be to take things as they are, thereby falls into contradiction with itself. Thus the chemist e.g. places a piece of flesh in his retort, tortures it in many ways, and then informs us that it consists of nitrogen, carbon, hydrogen, etc. True: but these abstract matters have ceased to be flesh. The same defect occurs in the reasoning of an empirical psychologist when he analyses an action into the various aspects which it presents, and then sticks to these aspects in their separation. The object which is subjected to analysis is treated as a sort of onion from which one coat is peeled off after another.

§ 228

This universality is [b] also a specific universality. In this case the line of activity follows the three ‘moments’ of the notion, which (as it has not its infinity in finite cognition) is the specific or definite notion of understanding. The reception of the object into the forms of this notion is the Synthetic Method.

The movement of the Synthetic method is the reverse of the Analytical method. The latter starts from the individual, and proceeds to the universal; in the former the starting-point is given by the universal (as a definition), from which we proceed by particularising (in division) to the individual (the theorem). The Synthetic method thus presents itself as the development — the ‘moments’ of the notion on the object.

Definition, Division and Theorem
§ 229

[a] When the object has been in the first instance brought by cognition into the form of the specific notion in general, so that in this way its genus and its universal character or speciality are explicitly stated, we have the Definition. The materials and the proof of Definition are procured by means of the Analytical method (§ 227). The specific character however is expected to be a ‘mark’ only: that is to say it is to be in behoof only of the purely subjective cognition which is external to the object.

Definition involves the three organic elements of the notion: the universal or proximate genus (genus proximum), the particular or specific character of the genus (qualitas specifica), and the individual, or object defined. The first question that definition suggests, is where it comes from. The general answer to this question is to say, that definitions originate by way of analysis. This will explain how it happens that people quarrel about the correctness of proposed definitions; for here everything depends on what perceptions we started from, and what points of view we had before our eyes in so doing. The richer the object to be defined is, that is, the more numerous are the aspects which it offers to our notice, the more various are the definitions we may frame of it. Thus there are quite a host of definitions of life, of the state, etc. Geometry, on the contrary, dealing with a theme so abstract as space, has an easy task in giving definitions. Again, in respect of the matter or contents of the objects defined, there is no constraining necessity present. We are expected to admit that space exists, that there are plants, animals, etc., nor is it the business of geometry, botany, etc., to demonstrate that the objects in question necessarily are. This very circumstance makes the synthetic method of cognition as little suitable for philosophy as the analytical: for philosophy has above all things to leave no doubt of the necessity of its objects. And yet several attempts have been made to introduce the synthetic method into philosophy. Thus Spinoza, in particular, begins with definitions. He says, for instance, that substance is the causa sui. His definitions are unquestionably a storehouse of the most speculative truth, but it takes the shape of dogmatic assertions. The same thing is also true of Schelling.

§ 230

[b] The statement of the second element of the notion, i.e. of the specific character of the universal as particularising, is given by Division in accordance with some external consideration.

Division we are told ought to be complete. That requires a principle or ground of division so constituted that the division based upon it embraces the whole extent of the region designated by the definition in general. But, in division, there is the further requirement that the principle of it must be borrowed from the nature of the object in question. If this condition be satisfied, the division is natural and not merely artificial, that is to say, arbitrary. Thus, in zoology, the ground of division adopted in the classification of the mammalia is mainly afforded by their teeth and claws. That is so far sensible, as the mammals themselves distinguish themselves from one another by these parts of their bodies back to which therefore the general type of their various classes is to be traced. In every case the genuine division must be controlled by the notion. To that extent a division, in the first instance, has three members: but as particularity exhibits itself as double, the division may go to the extent even of four members. In the sphere of mind trichotomy is predominant, a circumstance which Kant has the credit for bringing into notice

Theorem
§ 231

[c] In the concrete individuality, where the mere unanalysed quality of the definition is regarded as a correlation of elements, the object is a synthetic nexus of distinct characteristics. It is a Theorem. Being different, these characteristics possess but a mediated identity. To supply the materials, which form the middle terms, is the office of Construction: and the process of mediation itself, from which cognition derives the necessity of that nexus, is the Demonstration.

As the difference between the analytical and synthetic methods is commonly stated, it seems entirely optional which of the two we employ. If we assume, to start with, the concrete thing which the synthetic method presents as a result, we can analyse from it as consequences the abstract propositions which formed the pre-suppositions and the material for the proof. Thus, algebraical definitions of curved lines are theorems in the method of geometry. Similarly even the Pythagorean theorem, if made the definition of a right-angled triangle, might yield to analysis those propositions which geometry had already demonstrated on is behoof. The optionalness of either method is due to both alike starting from an external presupposition. So far as the nature of the notion is concerned, analysis is prior, since it has to raise the given material with its empirical concreteness into the form of general abstractions, which may then be set in the front of the synthetic method as definitions.

That these methods, however indispensable and brilliantly successful in their own province, are unserviceable for philosophical cognition, is self-evident. They have presuppositions; and their style of cognition is that of understanding, proceeding under the canon of formal identity. In Spinoza, who was especially addicted to the use of the geometrical method, we are at once struck by its characteristic formalism. Yet his ideas were speculative in spirit; whereas the system of Wolf, who carried the method out to the height of pedantry, was even in subject-matter a metaphysic of the understanding.

The abuses which these methods with their formalism once led to in philosophy and science have in modern times been followed by the abuses of what is called ‘Construction’. Kant brought into vogue the phrase that mathematics ‘construes’ its notions. All that was meant by the phrase was that mathematics has not to do with notions, but with abstract qualities of sense-perceptions. The name ‘Construction (construing) of notions’ has since been given to a sketch or statement of sensible attributes which were picked up from perception, quite guiltless of any influence of the notion, and to the additional formalism of classifying scientific and philosophical objects in a tabular form on some presupposed rubric, but in other respects at the fancy and discretion of the observer. In the background of all this, certainly, there is a dim consciousness of the Idea, of the unity of the notion and objectivity — a consciousness too that the idea is concrete. But that play of what is styled ‘construing’ is far from presenting this unity adequately, a unity which is none other than the notion properly so called: a perception is as little the concreteness of reason and the idea.

Another point calls for notice. Geometry works with the sensuous but abstract perception of space; and in space it experiences no difficulty in isolating and defining certain simple analytical modes.

To geometry alone therefore belongs in its perfection the synthetic method of finite cognition. In its course, however (and this is the remarkable point), it finally stumbles upon what are termed irrational and incommensurable quantities; and in their case any attempt at further specification drives it beyond the principle of the understanding. This is only one of many instances in terminology, where the title ‘rational’ is perversely applied to the province of understanding, while we stigmatise as irrational that which shows a beginning and a trace of rationality. Other sciences, removed as they are from the simplicity of space or number, often and necessarily reach a point where understanding permits no further advance: but they get over the difficulty without trouble. They make a break in the strict sequence of their procedure, and assume whatever they require, though it be the reverse of what preceded, from some external quarter — opinion, perception, conception, or any other source. Its inobservancy as to the nature of its methods and their relativity to the subject-matter prevents this finite cognition from seeing that, when it proceeds by definitions and divisions, etc., it is really led on by the necessity of the laws of the notion. For the same reason it cannot see when it has reached its limit; nor, if it have transgressed that limit, does it perceive that it is in a sphere where the categories of understanding, which it still continues rudely to apply, have lost all authority.

§ 232

The necessity which finite cognition produces in the Demonstration is, in the first place, an external necessity, intended for the subjective intelligence alone. But in necessity as such, cognition itself has left behind its presupposition and starting-point, which consisted in accepting its content as given or found. Necessity qua necessity is implicitly the self-relating notion. The subjective idea has thus implicitly reached an original and objective determinateness — a something not-given, and for that reason immanent in the subject. It has passed over into the idea of Will.

The necessity which cognition reaches by means of the demonstration is the reverse of what formed its starting-point. In its starting-point cognition had a given and a contingent content; but now, at the close of its movement, it knows its content to be necessary. This necessity is reached by means of subjective agency. Similarly, subjectivity at starting was quite abstract, a bare tabula rasa. It now shows itself as a modifying and determining principle. In this way we pass from the idea of cognition to that of will. The passage, as will be apparent on a closer examination, means that the universal, to be truly apprehended, must be apprehended as subjectivity, as a notion self-moving, active, and form-imposing.

[b] Volition

§233

The subjective idea as original and objective determinateness, and as a simple uniform content, is the Good. Its impulse towards self-realisation is in its behaviour the reverse of the idea of truth, and rather directed towards moulding the world it finds before it into a shape conformable to its purposed End. This Volition has, on the one hand, the certitude of the nothingness of the presupposed object; but, on the other, as finite, it at the same time presupposes the purposed End of the Good to be a mere subjective idea, and the object to be independent.

§ 234

This action of the Will is finite: and its finitude lies in the contradiction that in the inconsistent terms applied to the objective world the End of the Good is just as much not executed as executed, the end in question put as unessential as much as essential, as actual and at the same time as merely possible. This contradiction presents itself to imagination as an endless progress in the actualising of the Good; which is therefore set up and fixed as a mere ‘ought’, or goal of perfection. In point of form however this contradiction vanishes when the action supersedes the subjectivity of the purpose, and along with it the objectivity, with the contrast which makes both finite; abolishing subjectivity as a whole and not merely the one-sidedness of this form of it. (For another new subjectivity of the kind, that is, a new generation of the contrast, is not distinct from that which is supposed to be past and gone.) This return into itself is at the same time the content’s own ‘recollection’ that it is the Good and the implicit identity of the two sides — it is a ‘recollection’ of the presupposition of the theoretical attitude of mind (§ 224) that the objective world is its own truth and substantiality.

While Intelligence merely proposes to take the world as it is, Will takes steps to make the world what it ought to be. Will looks upon the immediate and given present not as solid being, but as mere semblance without reality. It is here that we meet those contradictions which are so bewildering from the standpoint of abstract morality. This position in its ‘practical’ bearings is the one taken by the philosophy of Kant, and even by that of Fichte. The Good, say these writers, has to be realised: we have to work in order to produce it: and Will is only the Good actualising itself. If the world then were as it ought to be, the action of Will would be at an end. The Will itself therefore requires that its End should not be realised. In these words, a correct expression is given to the finitude of Will. But finitude was not meant to be the ultimate point: and it is the process of Will itself which abolishes finitude and the contradiction it involves. The reconciliation is achieved when Will in its result returns to the presupposition made by cognition. In other words, it consists in the unity of the theoretical and practical idea. Will knows the end to be its own, and Intelligence apprehends the world as the notion actual. This is the right attitude of rational cognition. Nullity and transitoriness constitute only the superficial features and not the real essence of the world. That essence is the notion in posse and in esse: and thus the world is itself the idea. All unsatisfied endeavour ceases, when we recognise that the final purpose of the world is accomplished no less than ever accomplishing itself. Generally speaking, this is the man’s way of looking; while the young imagine that the world is utterly sunk in wickedness, and that the first thing needful is a thorough transformation. The religious mind, on the contrary, views the world as ruled by Divine Providence, and therefore correspondent with what it ought to be. But this harmony between the ‘is’ and the ‘ought to be’ is not torpid and rigidly stationary. Good, the final end of the world, has being, only while it constantly produces itself. And the world of spirit and the world of nature continue to have this distinction, that the latter moves only in a recurring cycle, while the former certainly also makes progress.

§ 235

Thus the truth of the Good is laid down as the unity of the theoretical and practical idea in the doctrine that the Good is radically and really achieved, that the objective world is in itself and for itself the Idea, just as it at the same time eternally lays itself down as End, and by action brings about its actuality. This life which has returned to itself from the bias and finitude of cognition, and which by the activity of the notion has become identical with it, is the Speculative or Absolute Idea.

(c) The Absolute Idea

§ 236

The Idea, as unity of the Subjective and Objective Idea, is the notion of the Idea — a notion whose object (Gegenstand) is the Idea as such, and for which the objective (Objekt) is Idea — an Object which embraces all characteristics in its unity. This unity is consequently the absolute and all truth, the Idea which thinks itself — and here at least as a thinking or Logical Idea.

The Absolute Idea is, in the first place, the unity of the theoretical and practical idea, and thus at the same time the unity of the idea of life with the idea of cognition.

Schopenhauer on Hegel

Schopenhauer on Georg Wilhelm Friedrich Hegel

“If I were to say that the so-called philosophy of this fellow Hegel is a colossal piece of mystification which will yet provide posterity with an inexhaustible theme for laughter at our times, that it is a pseudo-philosophy paralyzing all mental powers, stifling all real thinking, and, by the most outrageous misuse of language, putting in its place the hollowest, most senseless, thoughtless, and, as is confirmed by its success, most stupefying verbiage, I should be quite right.
Further, if I were to say that this summus philosophus […] scribbled nonsense quite unlike any mortal before him, so that whoever could read his most eulogized work, the so-called Phenomenology of the Mind, without feeling as if he were in a madhouse, would qualify as an inmate for Bedlam, I should be no less right.[115]
At first Fichte and Schelling shine as the heroes of this epoch; to be followed by the man who is quite unworthy even of them, and greatly their inferior in point of talent — I mean the stupid and clumsy charlatan Hegel.”

Ueber die Grundlage der Moral, 1840 (On the Basis of Morality, p. 35.) Source: Wikipedia

Excerpts from Hegel’s Encyclopedia : The Logic

Part One of the Encyclopedia of Philosophical Sciences: The Logic

Third Subdivision: The Notion

C. The Idea

Organic Nature
Sensibility, Irritability, and Reproduction
§ 218

(1) The first is the process of the living being inside itself. …

The process of the vital subject within its own limits has in Nature the threefold form of Sensibility, Irritability, and Reproduction. As Sensibility, the living being is immediately simple self-relation-it is the soul omnipresent in its body, the outsideness of each member of which to others has for it no truth. As Irritability, the living being appears split up in itself; and as Reproduction, it is perpetually restoring itself from the inner distinction of its members and organs. A vital agent only exists as this continually self-renewing process within its own limits.

Objective Nature
The matter which it assimilates
§ 219

(2) But the judgment of the notion proceeds, as free… The dialectic by which the object, being implicitly null, is merged is the action of the self-assured living thing, which in this process against an inorganic nature thus retains, develops, and objectifies itself.

The living being stands face to face with an inorganic nature, to which it comports itself as a master and which it assimilates to itself. The result of the assimilation is not, as in the chemical process, a neutral product in which the independence of the two confronting sides is merged; but the living being shows itself as large enough to embrace its other which cannot withstand its power. The inorganic nature which is subdued by the vital agent suffers this fate, because it is virtually the same as what life is actually. Thus in the other the living being only coalesces with itself. But when the soul has fled from the body, the elementary powers of objectivity begin their play. These powers are, as it were, continually on the spring, ready to begin their process in the organic body; and life is the constant battle against them.

Birth, Death and Genus
§ 221

The living being dies, because it is a contradiction. Implicitly it is the universal or Kind, and yet immediately it exists as an individual only. Death shows the Kind to be the power that rules the immediate individual. For the animal the process of Kind is the highest point of its vitality. But the animal never gets so far in its Kind as to have a being of its own; it succumbs to the power of Kind. In the process of Kind the immediate living being mediates itself with itself, and thus rises above its immediacy, only however to sink back into it again. Life thus runs away, in the first instance, only into the false infinity of the progress ad infinitum. The real result, however, of the process of life, in the point of its notion, is to merge and overcome that immediacy with which the idea, in the shape of life, is still beset.

(b) Cognition in general

§ 223

This process is in general terms Cognition. In Cognition in a single act the contrast is virtually superseded, as regards both the one-sidedness of subjectivity and the one-sidedness of objectivity. At first, however, the supersession of the contrast is but implicit. The process as such is in consequence immediately infected with the finitude of this sphere, and splits into the twofold movement of the instinct of reason, presented as two different movements. On the one hand it supersedes the one-sidedness of the Idea’s subjectivity by receiving the existing world into itself, into subjective conception and thought; and with this objectivity, which is thus taken to be real and true, for its content it fills up the abstract certitude of itself. On the other hand, it supersedes the one-sidedness of the objective world, which is now, on the contrary, estimated as only a mere semblance, a collection of contingencies and shapes at bottom visionary. It modifies and informs that world by the inward nature of the subjective, which is here taken to be the genuine objective. The former is the instinct of science after Truth, Cognition properly so called — the Theoretical action of the idea. The latter is the instinct of the Good to fulfil the same — the Practical activity of the idea, or Volition.

[[a] Cognition proper

§ 226-232 are not excerpted here]

Volition

§233

The subjective idea as original and objective determinateness, and as a simple uniform content, is the Good. Its impulse towards self-realisation is in its behaviour the reverse of the idea of truth, and rather directed towards moulding the world it finds before it into a shape conformable to its purposed End. This Volition has, on the one hand, the certitude of the nothingness of the presupposed object; but, on the other, as finite, it at the same time presupposes the purposed End of the Good to be a mere subjective idea, and the object to be independent.

§ 234

This action of the Will is finite: and its finitude lies in the contradiction that in the inconsistent terms applied to the objective world the End of the Good is just as much not executed as executed, the end in question put as unessential as much as essential, as actual and at the same time as merely possible. This contradiction presents itself to imagination as an endless progress in the actualising of the Good; which is therefore set up and fixed as a mere ‘ought’, or goal of perfection.

While Intelligence merely proposes to take the world as it is, Will takes steps to make the world what it ought to be. Will looks upon the immediate and given present not as solid being, but as mere semblance without reality. It is here that we meet those contradictions which are so bewildering from the standpoint of abstract morality. This position in its ‘practical’ bearings is the one taken by the philosophy of Kant, and even by that of Fichte. The Good, say these writers, has to be realised: we have to work in order to produce it: and Will is only the Good actualising itself. If the world then were as it ought to be, the action of Will would be at an end. The Will itself therefore requires that its End should not be realised. In these words, a correct expression is given to the finitude of Will. But finitude was not meant to be the ultimate point: and it is the process of Will itself which abolishes finitude and the contradiction it involves. The reconciliation is achieved when Will in its result returns to the presupposition made by cognition. In other words, it consists in the unity of the theoretical and practical idea. Will knows the end to be its own, and Intelligence apprehends the world as the notion actual. This is the right attitude of rational cognition. Nullity and transitoriness constitute only the superficial features and not the real essence of the world. That essence is the notion in posse and in esse: and thus the world is itself the idea. All unsatisfied endeavour ceases, when we recognise that the final purpose of the world is accomplished no less than ever accomplishing itself. Generally speaking, this is the man’s way of looking; while the young imagine that the world is utterly sunk in wickedness, and that the first thing needful is a thorough transformation. The religious mind, on the contrary, views the world as ruled by Divine Providence, and therefore correspondent with what it ought to be. But this harmony between the ‘is’ and the ‘ought to be’ is not torpid and rigidly stationary. Good, the final end of the world, has being, only while it constantly produces itself. And the world of spirit and the world of nature continue to have this distinction, that the latter moves only in a recurring cycle, while the former certainly also makes progress.

§ 235

Thus the truth of the Good is laid down as the unity of the theoretical and practical idea in the doctrine that the Good is radically and really achieved, that the objective world is in itself and for itself the Idea, just as it at the same time eternally lays itself down as End, and by action brings about its actuality. This life which has returned to itself from the bias and finitude of cognition, and which by the activity of the notion has become identical with it, is the Speculative or Absolute Idea.

Finitude in Hegel

(From Book One, The Science of Logic)

§ 249

When we say of things that they are finite, we understand thereby that they not only have a determinateness, that their quality is not only a reality and an intrinsic determination, that finite things are not merely limited — as such they still have determinate being outside their limit — but that, on the contrary, non-being constitutes their nature and being. Finite things are, but their relation to themselves is that they are negatively self-related and in this they are negatively self-related and in this very self-relation send themselves away beyond themselves, beyond their being. They are, but the truth of this being is their end.

The finite not only alters, like something in general, but it ceases to be; and its ceasing to be is not merely a possibility, so that it could be without ceasing to be, but the being as such of finite things is to have the germ of decease as their being-within-self: the hour of their birth is the hour of their death.

[a] The Immediacy of Finitude

§ 250

The thought of the finitude of things brings this sadness with it because it is qualitative negation pushed to its extreme, and in the singleness of such determination there is no longer left to things an affirmative being distinct from their destiny to perish. Because of this qualitative singleness of the negation, which has gone back to the abstract opposition of nothing and ceasing-to-be as opposed to being, finitude is the most stubborn category of the understanding; negation in general, constitution and limit, reconcile themselves with their other, with determinate being; and even nothing, taken abstractly as such, is given up as an abstraction; but finitude is the negation as fixed in itself, and it therefore stands in abrupt contrast to its affirmative. The finite, it is true, lest itself be brought into flux, it is itself this, to be determined or destined to its end, but only to its end — or rather, it is the refusal to let itself be brought affirmatively to its affirmative, to the infinite, and to let itself be united with it; it is therefore posited as inseparable from its nothing, and is thereby cut off from all reconciliation with its other, the affirmative. The determination or destiny of finite things takes them no further than their end. The understanding persists in this sadness of finitude by making non-being the determination of things and at the same time making it imperishable and absolute. Their transitoriness could only pass away or perish in their other, in the affirmative; their finitude would then be parted from them; but it is their unalterable quality, that is, their quality which does not pass over into its other, that is, into its affirmative; it is thus eternal

§ 251

This is a very important consideration; but certainly no philosophy or opinion, or understanding, will let itself be tied to the standpoint that the finite is absolute; the very opposite is expressly present in the assertion of the finite; the finite is limited, transitory, it is only finite, not imperishable; this is directly implied in its determination and expression. But the point is, whether in thinking of the finite one holds fast to the being of finitude and lets the transitoriness continue to be, or whether the transitoriness and the ceasing-to-be cease to be. But it is precisely in that view of the finite which makes ceasing-to-be the final determination of the finite, that this does not happen. It is the express assertion that the finite is irreconcilable with the infinite and cannot be united with it, that the finite is utterly opposed to the infinite. Being, absolute being, is ascribed to the infinite; confronting it, the finite thus remains held fast as its negative; incapable of union with the infinite, it remains absolute on its own side; from the affirmative, from the infinite, it would receive affirmation, and would thus cease to be; but a union with the infinite is just what is declared to be impossible. If it is not to remain fixed in its opposition to the infinite but is to cease to be, then, as we have already said, just this ceasing-to-be is its final determination, not the affirmative which would be only the ceasing to be of the ceasing-to-be. If, however, the finite is not to pass way in the affirmative, but its end is to be grasped as the nothing, then we should be back again at that first, abstract nothing which itself has long since passed away.

(From Chapter C:Infinity

Transition

Remark 1: The Infinite Progress…

Remark 2: Idealism

§ 316

The proposition that the finite is ideal [ideell] constitutes idealism. The idealism of philosophy consists in nothing else than in recognising that the finite has no veritable being. Every philosophy is essentially an idealism or at least has idealism for its principle, and the question then is only how far this principle is actually carried out. This is as true of philosophy as of religion; for religion equally does not recognise finitude as a veritable being, as something ultimate and absolute or as something underived, uncreated, eternal. Consequently the opposition of idealistic and realistic philosophy has no significance. A philosophy which ascribed veritable, ultimate, absolute being to finite existence as such, would not deserve the name of philosophy; the principles of ancient or modern philosophies, water, or matter, or atoms are thoughts, universals, ideal entities, not things as they immediately present themselves to us, that is, in their sensuous individuality — not even the water of Thales. For although this is also empirical water, it is at the same time also the in-itself or essence of all other things, too, and these other things are not self-subsistent or grounded in themselves, but are posited by, are derived from, an other, from water, that is they are ideal entities. Now above we have named the principle or the universal the ideal (and still more must the Notion, the Idea, spirit be so named); and then again we have described individual, sensuous things as ideal in principle, or in their Notion, still more in spirit, that is, as sublated; here we must note, in passing, this twofold aspect which showed itself in connection with the infinite, namely that on the one hand the ideal is concrete, veritable being, and on the other hand the moments of this concrete being are no less ideal — are sublated in it; but in fact what is, is only the one concrete whole from which the moments are inseparable…)

Being and Nothing in Hegel

(Pertaining to the Heidegger texts cited before…)

From The Science of Logic

§ 111

Further, in the beginning, being and nothing are present as distinguished from each other; for the beginning points to something else — it is a non-being which carries a reference to being as to an other; that which begins, as yet is not, it is only on the way to being.

That which begins, as yet is not, it is only on the way to being. The being contained in the beginning is, therefore, a being which removed itself from non-being or sublates it as something opposed to it.

But again, that which begins already is, but equally, too, is not as yet. The opposites, being and non-being are therefore directly united in it, or, otherwise expressed, it is their undifferentiated unity.

§ 112

The analysis of the beginning would thus yield the notion of the unity of being and nothing — or, in a more reflected form, the unity of differentiatedness and non-differentiatedness, or the identity of identity and non-identity. This concept could be regarded as the first, purest, that is, most abstract definition of the absolute — as it would in fact be if we were at all concerned with the form of definitions and with the name of the absolute. In this sense, that abstract concept would be the first definition of this absolute and all further determinations and developments only more specific and richer definitions of it. But let those who are dissatisfied with being as a beginning because it passes over into nothing and so gives rise to the unity of being and nothing, let them see whether they find this beginning which begins with the general idea of a beginning and with its analysis (which, though of course correct, likewise leads to the unity of being and nothing), more satisfactory than the beginning with being.

I.

1.

i.

Chapter 1 Being

A Being

§ 132

Being, pure being, without any further determination. In its indeterminate immediacy it is equal only to itself. It is also not unequal relatively to an other; it has no diversity within itself nor any with a reference outwards. It would not be held fast in its purity if it contained any determination or content which could be distinguished in it or by which it could be distinguished from an other. It is pure indeterminateness and emptiness. There is nothing to be intuited in it, if one can speak here of intuiting; or, it is only this pure intuiting itself. Just as little is anything to be thought in it, or it is equally only this empty thinking. Being, the indeterminate immediate, is in fact nothing, and neither more nor less than nothing.

B Nothing

§ 133

Nothing, pure nothing: it is simply equality with itself, complete emptiness, absence of all determination and content — undifferentiatedness in itself. In so far as intuiting or thinking can be mentioned here, it counts as a distinction whether something or nothing is intuited or thought. To intuit or think nothing has, therefore, a meaning; both are distinguished and thus nothing is (exists) in our intuiting or thinking; or rather it is empty intuition and thought itself, and the same empty intuition or thought as pure being. Nothing is, therefore, the same determination, or rather absence of determination, and thus altogether the same as, pure being.

C Becoming

1. Unity of Being and Nothing

§ 134

Pure Being and pure nothing are, therefore, the same. What is the truth is neither being nor nothing, but that being — does not pass over but has passed over — into nothing, and nothing into being. But it is equally true that they are not undistinguished from each other, that, on the contrary, they are not the same, that they are absolutely distinct, and yet that they are unseparated and inseparable and that each immediately vanishes in its opposite. Their truth is therefore, this movement of the immediate vanishing of the one into the other: becoming, a movement in which both are distinguished, but by a difference which has equally immediately resolved itself.

Remark 1: The Opposition of Being and Nothing in Ordinary Thinking

§ 135

Nothing is usually opposed to something; but the being of something is already determinate and is distinguished from another something; and so therefore the nothing which is opposed to the something is also the nothing of a particular something, a determinate nothing. Here, however, nothing is to be taken in its indeterminate simplicity. Should it be held more correct to oppose to being, non-being instead of nothing, there would be no objection to this so far as the result is concerned, for in non-being the relation to being is contained: both being and its negation are enunciated in a single term, nothing, as it is in becoming. But we are concerned first of all not with the form of opposition (with the form, that is, also of relation) but with the abstract, immediate negation: nothing, purely on its own account, negation devoid of any relations — what could also be expressed if one so wished merely by ‘not’.

§ 136

It was the Eleatics, above all Parmenides, who first enunciated the simple thought of pure being as the absolute and sole truth: only being is, and nothing absolutely is not, and in the surviving fragments of Parmenides this is enunciated with the pure enthusiasm of thought which has for the first time apprehended itself in its absolute abstraction. As we know, in the oriental systems, principally in Buddhism, nothing, the void, is the absolute principle. Against that simple and one-sided abstraction the deep-thinking Heraclitus brought forward the higher, total concept of becoming and said: being as little is, as nothing is, or, all flows, which means, all is a becoming. The popular, especially oriental proverbs, that all that exists has the germ of death in its very birth, that death, on the other hand, is the entrance into new life, express at bottom the same union of being and nothing. But these expressions have a substratum in which the transition takes place; being and nothing are held apart in time, are conceived as alternating in it, but are not thought in their abstraction and consequently, too, not so that they are in themselves absolutely the same.

§ 137

Ex nihilo nihil fit — is one of those propositions to which great importance was ascribed in metaphysics. In it is to be seen either only the empty tautology: nothing is nothing; or, if becoming is supposed to possess an actual meaning in it, then, since from nothing only nothing becomes, the proposition does not in fact contain becoming, for in it nothing remains nothing. Becoming implies that nothing does not remain nothing but passes into its other, into being. Later, especially Christian, metaphysics whilst rejecting the proposition that out of nothing comes nothing, asserted a transition from nothing into being; although it understood this proposition synthetically or merely imaginatively, yet even in the most imperfect union there is contained a point in which being and nothing coincide and their distinguishedness vanishes. The proposition: out of nothing comes nothing, nothing is just nothing, owes its peculiar importance to its opposition to becoming generally, and consequently also to its opposition to the creation of the world from nothing. Those who maintain the proposition: nothing is just nothing, and even grow heated in its defence, are unaware that in so doing they are subscribing to the abstract pantheism of the Eleatics, and also in principle to that of Spinoza. The philosophical view for which ‘being is only being, nothing is only nothing’, is a valid principle, merits the name of ‘system of identity’; this abstract identity is the essence of pantheism.

From the Organon to the computer

Excerpts

Chris Dixon

https://www.theatlantic.com/technology/archive/2017/03/aristotle-computer/518697/?utm_source=msn

THE HISTORY Of computers is often told as a history of objects, from the abacus to the Babbage engine up through the code-breaking machines of World War II. In fact, it is better understood as a history of ideas, mainly ideas that emerged from mathematical logic, an obscure and cult-like discipline that first developed in the 19th century. Mathematical logic was pioneered by philosopher-mathematicians, most notably George Boole and Gottlob Frege, who were themselves inspired by Leibniz’s dream of a universal “concept language,” and the ancient logical system of Aristotle.

Mathematical logic was initially considered a hopelessly abstract subject with no conceivable applications. As one computer scientist commented: “If, in 1901, a talented and sympathetic outsider had been called upon to survey the sciences and name the branch which would be least fruitful in [the] century ahead, his choice might well have settled upon mathematical logic.” And yet, it would provide the foundation for a field that would have more impact on the modern world than any other.

The evolution of computer science from mathematical logic culminated in the 1930s, with two landmark papers: Claude Shannon’s “A Symbolic Analysis of Switching and Relay Circuits,” and Alan Turing’s “On Computable Numbers, With an Application to the Entscheidungsproblem.” In the history of computer science, Shannon and Turing are towering figures, but the importance of the philosophers and logicians who preceded them is frequently overlooked.

A well-known history of computer science describes Shannon’s paper as “possibly the most important, and also the most noted, master’s thesis of the century.” Shannon wrote it as an electrical engineering student at MIT. His adviser, Vannevar Bush, built a prototype computer known as the Differential Analyzer that could rapidly calculate differential equations. The device was mostly mechanical, with subsystems controlled by electrical relays, which were organized in an ad hoc manner as there was not yet a systematic theory underlying circuit design. Shannon’s thesis topic came about when Bush recommended he try to discover such a theory.

Mathematics may be defined as the subject in which we never know what we are talking about.”

Shannon’s paper is in many ways a typical electrical-engineering paper, filled with equations and diagrams of electrical circuits. What is unusual is that the primary reference was a 90-year-old work of mathematical philosophy, George Boole’s The Laws of Thought.

Today, Boole’s name is well known to computer scientists (many programming languages have a basic data type called a Boolean), but in 1938 he was rarely read outside of philosophy departments. Shannon himself encountered Boole’s work in an undergraduate philosophy class. “It just happened that no one else was familiar with both fields at the same time,” he commented later.

Boole is often described as a mathematician, but he saw himself as a philosopher, following in the footsteps of Aristotle. The Laws of Thought begins with a description of his goals, to investigate the fundamental laws of the operation of the human mind:

The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic … and, finally, to collect … some probable intimations concerning the nature and constitution of the human mind.

He then pays tribute to Aristotle, the inventor of logic, and the primary influence on his own work:

In its ancient and scholastic form, indeed, the subject of Logic stands almost exclusively associated with the great name of Aristotle. As it was presented to ancient Greece in the partly technical, partly metaphysical disquisitions of The Organon, such, with scarcely any essential change, it has continued to the present day.

Trying to improve on the logical work of Aristotle was an intellectually daring move. Aristotle’s logic, presented in his six-part book The Organon, occupied a central place in the scholarly canon for more than 2,000 years. It was widely believed that Aristotle had written almost all there was to say on the topic. The great philosopher Immanuel Kant commented that, since Aristotle, logic had been “unable to take a single step forward, and therefore seems to all appearance to be finished and complete.”

Aristotle’s central observation was that arguments were valid or not based on their logical structure, independent of the non-logical words involved. The most famous argument schema he discussed is known as the syllogism:

• All men are mortal.
• Socrates is a man.
• Therefore, Socrates is mortal.

You can replace “Socrates” with any other object, and “mortal” with any other predicate, and the argument remains valid. The validity of the argument is determined solely by the logical structure. The logical words — “all,” “is,” are,” and “therefore” — are doing all the work.

Aristotle also defined a set of basic axioms from which he derived the rest of his logical system:

• An object is what it is (Law of Identity)
• No statement can be both true and false (Law of Non-contradiction)
• Every statement is either true or false (Law of the Excluded Middle)

These axioms weren’t meant to describe how people actually think (that would be the realm of psychology), but how an idealized, perfectly rational person ought to think.

Aristotle’s axiomatic method influenced an even more famous book, Euclid’s Elements, which is estimated to be second only to the Bible in the number of editions printed.

Although ostensibly about geometry, the Elements became a standard textbook for teaching rigorous deductive reasoning. (Abraham Lincoln once said that he learned sound legal argumentation from studying Euclid.) In Euclid’s system, geometric ideas were represented as spatial diagrams. Geometry continued to be practiced this way until René Descartes, in the 1630s, showed that geometry could instead be represented as formulas. His Discourse on Method was the first mathematics text in the West to popularize what is now standard algebraic notation — x, y, z for variables, a, b, c for known quantities, and so on.

Descartes’s algebra allowed mathematicians to move beyond spatial intuitions to manipulate symbols using precisely defined formal rules. This shifted the dominant mode of mathematics from diagrams to formulas, leading to, among other things, the development of calculus, invented roughly 30 years after Descartes by, independently, Isaac Newton and Gottfried Leibniz.

Boole’s goal was to do for Aristotelean logic what Descartes had done for Euclidean geometry: free it from the limits of human intuition by giving it a precise algebraic notation. To give a simple example, when Aristotle wrote:

All men are mortal.

Boole replaced the words “men” and “mortal” with variables, and the logical words “all” and “are” with arithmetical operators:

x = x * y

Which could be interpreted as “Everything in the set x is also in the set y.”

The Laws of Thought created a new scholarly field—mathematical logic—which in the following years became one of the most active areas of research for mathematicians and philosophers. Bertrand Russell called the Laws of Thought “the work in which pure mathematics was discovered.”

Shannon’s insight was that Boole’s system could be mapped directly onto electrical circuits. At the time, electrical circuits had no systematic theory governing their design. Shannon realized that the right theory would be “exactly analogous to the calculus of propositions used in the symbolic study of logic.”

He showed the correspondence between electrical circuits and Boolean operations in a simple chart:

Shannon’s mapping from electrical circuits to symbolic logic (University of Virginia)

This correspondence allowed computer scientists to import decades of work in logic and mathematics by Boole and subsequent logicians. In the second half of his paper, Shannon showed how Boolean logic could be used to create a circuit for adding two binary digits.

Shannon’s adder circuit (University of Virginia)

By stringing these adder circuits together, arbitrarily complex arithmetical operations could be constructed. These circuits would become the basic building blocks of what are now known as arithmetical logic units, a key component in modern computers.

Another way to characterize Shannon’s achievement is that he was first to distinguish between the logical and the physical layer of computers. (This distinction has become so fundamental to computer science that it might seem surprising to modern readers how insightful it was at the time—a reminder of the adage that “the philosophy of one century is the common sense of the next.”)

Since Shannon’s paper, a vast amount of progress has been made on the physical layer of computers, including the invention of the transistor in 1947 by William Shockley and his colleagues at Bell Labs. Transistors are dramatically improved versions of Shannon’s electrical relays — the best known way to physically encode Boolean operations. Over the next 70 years, the semiconductor industry packed more and more transistors into smaller spaces. A 2016 iPhone has about 3.3 billion transistors, each one a “relay switch” like those pictured in Shannon’s diagrams.

While Shannon showed how to map logic onto the physical world, Turing showed how to design computers in the language of mathematical logic. When Turing wrote his paper, in 1936, he was trying to solve “the decision problem,” first identified by the mathematician David Hilbert, who asked whether there was an algorithm that could determine whether an arbitrary mathematical statement is true or false. In contrast to Shannon’s paper, Turing’s paper is highly technical. Its primary historical significance lies not in its answer to the decision problem,  but in the template for computer design it provided along the way.

Turing was working in a tradition stretching back to Gottfried Leibniz, the philosophical giant who developed calculus independently of Newton. Among Leibniz’s many contributions to modern thought, one of the most intriguing was the idea of a new language he called the “universal characteristic” that, he imagined, could represent all possible mathematical and scientific knowledge. Inspired in part by the 13th-century religious philosopher Ramon Llull, Leibniz postulated that the language would be ideographic like Egyptian hieroglyphics, except characters would correspond to “atomic” concepts of math and science. He argued this language would give humankind an “instrument” that could enhance human reason “to a far greater extent than optical instruments” like the microscope and telescope.

He also imagined a machine that could process the language, which he called the calculus ratiocinator.

If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, and say to each other: Calculemus—Let us calculate.

Leibniz didn’t get the opportunity to develop his universal language or the corresponding machine (although he did invent a relatively simple calculating machine, the stepped reckoner). The first credible attempt to realize Leibniz’s dream came in 1879, when the German philosopher Gottlob Frege published his landmark logic treatise Begriffsschrift. Inspired by Boole’s attempt to improve Aristotle’s logic, Frege developed a much more advanced logical system. The logic taught in philosophy and computer-science classes today—first-order or predicate logic—is only a slight modification of Frege’s system.

Frege is generally considered one of the most important philosophers of the 19th century. Among other things, he is credited with catalyzing what noted philosopher Richard Rorty called the “linguistic turn” in philosophy. As Enlightenment philosophy was obsessed with questions of knowledge, philosophy after Frege became obsessed with questions of language. His disciples included two of the most important philosophers of the 20th century—Bertrand Russell and Ludwig Wittgenstein.

The major innovation of Frege’s logic is that it much more accurately represented the logical structure of ordinary language. Among other things, Frege was the first to use quantifiers (“for every,” “there exists”) and to separate objects from predicates. He was also the first to develop what today are fundamental concepts in computer science like recursive functions and variables with scope and binding.

Frege’s formal language — what he called his “concept-script” — is made up of meaningless symbols that are manipulated by well-defined rules. The language is only given meaning by an interpretation, which is specified separately (this distinction would later come to be called syntax versus semantics). This turned logic into what the eminent computer scientists Allan Newell and Herbert Simon called “the symbol game,” “played with meaningless tokens according to certain purely syntactic rules.”

All meaning had been purged. One had a mechanical system about which various things could be proved. Thus progress was first made by walking away from all that seemed relevant to meaning and human symbols.

As Bertrand Russell famously quipped: “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

An unexpected consequence of Frege’s work was the discovery of weaknesses in the foundations of mathematics. For example, Euclid’s Elements — considered the gold standard of logical rigor for thousands of years — turned out to be full of logical mistakes. Because Euclid used ordinary words like “line” and “point,” he — and centuries of readers — deceived themselves into making assumptions about sentences that contained those words. To give one relatively simple example, in ordinary usage, the word “line” implies that if you are given three distinct points on a line, one point must be between the other two. But when you define “line” using formal logic, it turns out “between-ness” also needs to be defined—something Euclid overlooked. Formal logic makes gaps like this easy to spot.

This realization created a crisis in the foundation of mathematics. If the Elements — the bible of mathematics — contained logical mistakes, what other fields of mathematics did too? What about sciences like physics that were built on top of mathematics?

The good news is that the same logical methods used to uncover these errors could also be used to correct them. Mathematicians started rebuilding the foundations of mathematics from the bottom up. In 1889, Giuseppe Peano developed axioms for arithmetic, and in 1899, David Hilbert did the same for geometry. Hilbert also outlined a program to formalize the remainder of mathematics, with specific requirements that any such attempt should satisfy, including:

• Completeness: There should be a proof that all true mathematical statements can be proved in the formal system.
• Decidability: There should be an algorithm for deciding the truth or falsity of any mathematical statement. (This is the “Entscheidungsproblem” or “decision problem” referenced in Turing’s paper.)

Rebuilding mathematics in a way that satisfied these requirements became known as Hilbert’s program. Up through the 1930s, this was the focus of a core group of logicians including Hilbert, Russell, Kurt Gödel, John Von Neumann, Alonzo Church, and, of course, Alan Turing.

In science, novelty emerges only with difficulty.”

Hilbert’s program proceeded on at least two fronts. On the first front, logicians created logical systems that tried to prove Hilbert’s requirements either satisfiable or not.

On the second front, mathematicians used logical concepts to rebuild classical mathematics. For example, Peano’s system for arithmetic starts with a simple function called the successor function which increases any number by one. He uses the successor function to recursively define addition, uses addition to recursively define multiplication, and so on, until all the operations of number theory are defined. He then uses those definitions, along with formal logic, to prove theorems about arithmetic.

The historian Thomas Kuhn once observed that “in science, novelty emerges only with difficulty.” Logic in the era of Hilbert’s program was a tumultuous process of creation and destruction. One logician would build up an elaborate system and another would tear it down.

The favored tool of destruction was the construction of self-referential, paradoxical statements that showed the axioms from which they were derived to be inconsistent. A simple form of this  “liar’s paradox” is the sentence:

This sentence is false.

If it is true then it is false, and if it is false then it is true, leading to an endless loop of self-contradiction.

Russell made the first notable use of the liar’s paradox in mathematical logic. He showed that Frege’s system allowed self-contradicting sets to be derived:

Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves.

This became known as Russell’s paradox and was seen as a serious flaw in Frege’s achievement. (Frege himself was shocked by this discovery. He replied to Russell: “Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic.”)

Russell and his colleague Alfred North Whitehead put forth the most ambitious attempt to complete Hilbert’s program with the Principia Mathematica, published in three volumes between 1910 and 1913. The Principia’s method was so detailed that it took over 300 pages to get to the proof that 1+1=2.

Russell and Whitehead tried to resolve Frege’s paradox by introducing what they called type theory. The idea was to partition formal languages into multiple levels or types. Each level could make reference to levels below, but not to their own or higher levels. This resolved self-referential paradoxes by, in effect, banning self-reference. (This solution was not popular with logicians, but it did influence computer science — most modern computer languages have features inspired by type theory.)

Self-referential paradoxes ultimately showed that Hilbert’s program could never be successful. The first blow came in 1931, when Gödel published his now famous incompleteness theorem, which proved that any consistent logical system powerful enough to encompass arithmetic must also contain statements that are true but cannot be proven to be true. (Gödel’s incompleteness theorem is one of the few logical results that has been broadly popularized, thanks to books like Gödel, Escher, Bach and The Emperor’s New Mind).

The final blow came when Turing and Alonzo Church independently proved that no algorithm could exist that determined whether an arbitrary mathematical statement was true or false. (Church did this by inventing an entirely different system called the lambda calculus, which would later inspire computer languages like Lisp.) The answer to the decision problem was negative.

Turing’s key insight came in the first section of his famous 1936 paper, “On Computable Numbers, With an Application to the Entscheidungsproblem.” In order to rigorously formulate the decision problem (the “Entscheidungsproblem”), Turing first created a mathematical model of what it means to be a computer (today, machines that fit this model are known as “universal Turing machines”). As the logician Martin Davis describes it:

Turing knew that an algorithm is typically specified by a list of rules that a person can follow in a precise mechanical manner, like a recipe in a cookbook. He was able to show that such a person could be limited to a few extremely simple basic actions without changing the final outcome of the computation.

Then, by proving that no machine performing only those basic actions could determine whether or not a given proposed conclusion follows from given premises using Frege’s rules, he was able to conclude that no algorithm for the Entscheidungsproblem exists.

As a byproduct, he found a mathematical model of an all-purpose computing machine.

Next, Turing showed how a program could be stored inside a computer alongside the data upon which it operates. In today’s vocabulary, we’d say that he invented the “stored-program” architecture that underlies most modern computers:

Before Turing, the general supposition was that in dealing with such machines the three categories — machine, program, and data — were entirely separate entities. The machine was a physical object; today we would call it hardware. The program was the plan for doing a computation, perhaps embodied in punched cards or connections of cables in a plugboard. Finally, the data was the numerical input. Turing’s universal machine showed that the distinctness of these three categories is an illusion.

This was the first rigorous demonstration that any computing logic that could be encoded in hardware could also be encoded in software. The architecture Turing described was later dubbed the “Von Neumann architecture” — but modern historians generally agree it came from Turing, as, apparently, did Von Neumann himself.

Although, on a technical level, Hilbert’s program was a failure, the efforts along the way demonstrated that large swaths of mathematics could be constructed from logic. And after Shannon and Turing’s insights—showing the connections between electronics, logic and computing—it was now possible to export this new conceptual machinery over to computer design.

During World War II, this theoretical work was put into practice, when government labs conscripted a number of elite logicians. Von Neumann joined the atomic bomb project at Los Alamos, where he worked on computer design to support physics research. In 1945, he wrote the specification of the EDVAC—the first stored-program, logic-based computer—which is generally considered the definitive source guide for modern computer design.

Turing joined a secret unit at Bletchley Park, northwest of London, where he helped design computers that were instrumental in breaking German codes. His most enduring contribution to practical computer design was his specification of the ACE, or Automatic Computing Engine.

As the first computers to be based on Boolean logic and stored-program architectures, the ACE and the EDVAC were similar in many ways. But they also had interesting differences, some of which foreshadowed modern debates in computer design. Von Neumann’s favored designs were similar to modern CISC (“complex”) processors, baking rich functionality into hardware. Turing’s design was more like modern RISC (“reduced”) processors, minimizing hardware complexity and pushing more work to software.

Von Neumann thought computer programming would be a tedious, clerical job. Turing, by contrast, said computer programming “should be very fascinating. There need be no real danger of it ever becoming a drudge, for any processes that are quite mechanical may be turned over to the machine itself.”

Since the 1940s, computer programming has become significantly more sophisticated. One thing that hasn’t changed is that it still primarily consists of programmers specifying rules for computers to follow. In philosophical terms, we’d say that computer programming has followed in the tradition of deductive logic, the branch of logic discussed above, which deals with the manipulation of symbols according to formal rules.

In the past decade or so, programming has started to change with the growing popularity of machine learning, which involves creating frameworks for machines to learn via statistical inference. This has brought programming closer to the other main branch of logic, inductive logic, which deals with inferring rules from specific instances.

Today’s most promising machine learning techniques use neural networks, which were first invented in 1940s by Warren McCulloch and Walter Pitts, whose idea was to develop a calculus for neurons that could, like Boolean logic, be used to construct computer circuits. Neural networks remained esoteric until decades later when they were combined with statistical techniques, which allowed them to improve as they were fed more data. Recently, as computers have become increasingly adept at handling large data sets, these techniques have produced remarkable results. Programming in the future will likely mean exposing neural networks to the world and letting them learn.

This would be a fitting second act to the story of computers. Logic began as a way to understand the laws of thought. It then helped create machines that could reason according to the rules of deductive logic. Today, deductive and inductive logic are being combined to create machines that both reason and learn. What began, in Boole’s words, with an investigation “concerning the nature and constitution of the human mind,” could result in the creation of new minds—artificial minds—that might someday match or even exceed our own.”

Hegel’s Phenomenology of Mind (2)

Excerpts from a few selected pages

(Numbers from the Baillie translation have been inserted into the text.)

Φ 303. Observational psychology, which in the first instance states what observation finds regarding the general forms brought to its notice in the active consciousness, discovers all sorts of faculties, inclinations, and passions; and since, while narrating what this collection contains, the remembrance of the unity of self-consciousness is not to be suppressed, observational psychology is bound to get the length at least of wonderment that such a lot and such a miscellany of things can happen to be somehow alongside one another in the mind as in a kind of bag, more especially when they are seen to be not lifeless inert things, but restless active processes…

… Since, however, the process of apprehending it causes it at the same time to pass into the form of universality, to apprehend it is to find its law, and seems in this way to have a rational purpose in view, and a necessary function to fulfil.

Φ 305. The moments constituting the content of the law are on the one hand individuality itself, on the other its universal inorganic nature, viz. the given circumstances, situation, habits, customs, religion, and so forth; from these the determinate individuality is to be understood and comprehended. They contain something specific, determinate, as well as universal, and are at the same time something lying at hand, which furnishes material for observation and on the other side expresses itself in the form of individuality.

Φ 306. The law of this relation of the two sides has now to contain and express the sort of effect and influence these determinate circumstances exert on individuality. This individuality, however just consists both in being the universal, and hence in passively and directly assimilating and blending with the given universals, the customs, habits, etc., thus becoming conformed to them, as also in taking up an attitude of opposition towards them and thus transforming and transmuting them; and again in behaving towards them in its individual character with complete indifference, neither allowing them to exert an influence over it, nor setting itself actively against them…. If these circumstances, style of thought, customs, the whole state of the world, in short, had not been, then assuredly the individual would not have been what he is; for all the elements that find a place in this “ state of the world “ constitute this universal substance.

If the external element is so constituted in and for itself as it appears in individuality, the latter would be comprehended from the nature of the former. We should have a double gallery of pictures, one of which would be the reflexion of the other: the one the gallery of external circumstance completely encompassing, circumscribing, and determining the individual, the other the same gallery translated into the form in which those circumstances are in the conscious individual: the former the spherical surface, the latter the centre reflectively representing that surface within it.

Φ 307. But the spherical surface, the world for the individual, carries on the face of it this double meaning: it is in and for itself the actual world and situation, and it is the world of the individual. It is the world of the individual either in so far as this individual was merely fused and blended with it, had let that world, just as it is, pass into its own nature, and had taken up towards it merely the attitude of a formal consciousness; or, on the other hand, it is the world of the individual in the sense in which the given has been transformed and transmuted by that individual.

Since reality is capable of having this twofold meaning on account of this freedom of the individual, the world of the individual is only to be understood from the individual himself; and the influence of reality upon the individual, a reality which is represented as having a being all its own (an und für sich), receives through this individual absolutely the opposite significance — the individual either lets the stream of reality flowing in upon it have its way, or breaks off and diverts the current of its influence. In consequence of this, however, “ psychological necessity” becomes an empty phrase, so empty that there is the absolute possibility that what is said to have this influence could equally well not have had it.

Φ 308. Herewith drops out of account that existence which was to be something all by itself, and was meant to constitute one aspect, and that the universal aspect, of a law. Individuality is what its world, in the sense of , is. Individuality itself is the cycle of its own action, in which it has presented and established itself as reality, and is simply and solely a unity of what is given and what is constructed — a unity aspects do not fall apart, as in the idea of psychological law, into a world given per se and an individuality existing for itself. Or if those aspects are thus considered each by itself, there is no necessity to be found between them, and no law of their relation to one another.