§ 1. The complete transformation of philosophical thought in Germany has had little influence on logic.
§ 2. When a nation loses its metaphysics, its own pure essence is no longer a present reality in the life of the nation.
§ 3. Kantian philosophy was a justification for the renunciation of speculative thought.
§ 4. As if one could only learn how to digest and move about by studying anatomy and physiology.
§ 5. Once the spirit has reconstituted itself, all attempts to preserve an earlier culture are utterly in vain.
§ 6. Even those who are opposed to the new ideas have become familiar with them and have appropriated them.
§ 7. A new creative idea displays a fanatical hostility toward the entrenched systematisation of the older principle.
§ 8. Philosophy, if it would be a science, cannot borrow its method from a subordinate science like mathematics.
§ 9. Reason is negative and dialectical, because it resolves the determinations of the understanding into nothing.
§ 10. The development of all natural and spiritual life, rests solely on the content of logic.
§ 11. Logic is the first sequel to the Phenomenology, to be followed by the Philosophy of Nature and the Philosophy of Spirit.
§ 12. I have tried to remedy the imperfection of its treatment in the first edition.
§ 13. To exhibit the realm of thought philosophically, that is, in its own immanent activity.
§ 14. The forms of thought are, in the first instance, displayed and stored as human language.
§ 15. In physics,the category of force has become predominant, but more recently polarity has played the leading part.
§ 16. While logical objects may be thoroughly familiar to educated people it does not follow that they are intelligently apprehended.
§ 17. The forms of thought have been freed from the material in which they are submerged.
§ 18. The need to occupy oneself with pure thought presupposes that the human spirit must already have travelled a long road.
§ 19. It is customary, to include logic in the curriculum of youth, not yet involved in the practical affairs of life.
§ 20. In life, the categories are degraded to serve in the creation and exchange of ideas.
§ 21. It is our thinking that must accommodate itself to notions and our caprice ought not to want to mould them to suit itself.
§ 22. We cannot go outside our subjective thought, and just as little can we go beyond the nature of things.
§ 23. To focus attention on this logical nature which animates mind, moves and works in it, this is the task.
§ 24. Here and there in this mesh there are firm knots which give stability and direction to the life.
§ 25. The loftier business of logic is to clarify the categories and raise mind to freedom and truth.
§ 26. In its negation, the true is associated with limitation and finitude, its untruth and unreality.
§ 27. The merely formal categories concern only the correctness of the knowledge of facts, not truth itself.
§ 28. Content, divorced from form, cannot in fact be formless.
§ 29. No subject can be expounded with a strict immanent plasticity as thought in its own necessary development.
§ 30. This restriction to what is simple gives scope for the free play of caprice.
§ 31. Plato revised his Republic seven times over.
§ 32. The noisy clamour of current affairs and the deafening chatter leave little room for the passionless calm of knowledge of pure thought.
§ 33. The need to begin with the subject matter itself, without preliminary reflections, is felt most strongly in logic.
§ 34. Logic cannot presuppose any of these laws of thinking, for these constitute part of its own content.
§ 35. When logic is taken as the science of thinking in general, it is understood that its matter, must come from somewhere else.
§ 36. As thinking and the rules of thinking are supposed to be the subject matter of logic, these constitute its content.
§ 37. It is time that Logic received a completely changed shape.
§ 38. It is assumed that the material of knowing is present on its own account as a ready-made world apart from thought.
§ 39. Thinking is supposed to adapt and accommodate itself to the object.
§ 40. Thinking does not go out of itself to the object; this, as a thing-in-itself, remains a sheer beyond of thought.
§ 41. These views on the relation of subject and object bar the entrance to philosophy and must be discarded at its portals.
§ 42. Ancient metaphysics had a higher conception of thinking than is current today.
§ 43. Reflective understanding took possession of philosophy.
§ 44. This turn, which appears as retrograde step, is based on the elevation of reason into the loftier spirit of modern philosophy.
§ 45. If the forms of the understanding cannot be determinations of the thing-in-itself, still less can they be of the understanding.
§ 46. The forms of objective thinking have been removed only from the thing, but have been left in the subject.
§ 47. ... the nothingness of the thing-in-itself, this abstract shadow divorced from all content, and intended to destroy it.
§ 48. They are dead forms and the spirit which is their living, concrete unity does not dwell in them.
§ 49. The point of view from which logic is to be considered, how it differs from previous modes of treatment.
§ 50. I have exhibited consciousness in its movement from the immediate opposition of itself and the object to absolute knowing.
§ 51. The Notion of pure science is here presupposed as the Phenomenology of Spirit is the deduction of it.
§ 52. As science, truth is pure self-consciousness in its self-development.
§ 53. This objective thinking then, is the content of pure science.
§ 54. Logic is not a thinking about something — the necessary forms of thought are the content and the ultimate truth itself.
§ 55. One must discard the prejudice that truth must be something tangible.
§ 56. The determinations of thought have objective value and existence.
§ 57. The critical philosophy turned metaphysics into logic but the logical determinations were given an essentially subjective significance.
§ 58. The utter unworthiness of logic in comparison with other spheres cannot fail to strike the most superficial observer.
§ 59. Logic, as exhibited in the text-books, may be said to have fallen into contempt.
§ 60. The additions of psychological, pedagogic and even physiological material which logic received have been recognised as disfigurements.
§ 61. Logic is so dull and spiritless ... a childish game of fitting together the pieces of a coloured picture puzzle.
§ 62. Hitherto philosophy had not found its method.
§ 63. It is the inwardness of the content, the dialectic which it possesses within itself, which is the mainspring of its advance.
§ 64. The divisions and chapter headings are only of historical value. They do not belong to the content.
§ 65. Such preliminary definitions and divisions are in themselves nothing else but such external indications.
§ 66. The divisions which appear in this system are not intended to have any other significance than that of a list of contents.
§ 67. The negative which the Notion possesses within itself constitutes the genuine dialectical moment.
§ 68. Dialectic was held to be merely the art of practising deceptions and producing illusions.
§ 69. Dialectic is the grasping of opposites in their unity or of the positive in the negative.
§ 70. It is one thing for him who comes to logic for the first time, but another for him who comes back to it from these sciences.
§ 71. Only after acquaintance with the other sciences does logic cease to be a merely abstract universal and reveal itself as the universal which embraces the wealth of the particular.
§ 72. The system of logic is the realm of shadows.
§ 73. Thought becomes at home in abstractions and develops a power of assimilating in rational form all the various sciences.
§ 74. The general division here can only be provisional; the main distinctions which will emerge in the development.
§ 75. We are presupposing that the division must be implicit in the Notion itself.
§ 76. The quality of being right-angled, etc., by which triangles are classified, is not implicit in the Notion of triangle.
§ 77. In Logic the opposition in consciousness between a subject and an object is known to be overcome
§ 78. The determinations used on the pathway to truth, have lost their independent character and are now in their truth.
§ 79. Accordingly, logic should be divided primarily into the logic of the Notion as being and of the Notion as Notion.
§ 80. There results a sphere of mediation. This is the doctrine of essence.
§ 81. Objective logic would correspond in part to what with Kant is transcendental logic.
§ 82. Kant’s chief thought is to vindicate the categories for self-consciousness as the subjective ego.
§ 83. Consciousness embraces within itself the opposition of the ego and its object which is not present in that original act.
§ 84. If philosophy was to make any real progress, a consideration of the ego, of consciousness as such was necessary.
§ 85. The objective logic takes the place of metaphysics which meant to construct the world in terms of thoughts alone.
§ 86. The subjective logic is the logic of the Notion, no longer external but the subject itself.
§ 87. Logic has three parts: Being, Essence and Notion.
§ 88. What philosophy begins with must be either mediated or immediate, and it is easy to show that it can be neither the one nor the other.
§ 89. The principle of a philosophy does, of course, also express a beginning.
§ 90. What is the first for thought ought also to be the first in the process of thinking.
§ 91. The logical beginning appears either as a mediated result or as an immediacy.
§ 92. There is nothing in heaven, or in nature or in mind or anywhere else which does not equally contain both immediacy and mediation.
§ 93. In logic, the presupposition is that which has proved itself to be the result of that phenomenological consideration.
§ 94. Logic is pure science, that is, pure knowledge in the entire range of its development.
§ 95. All that is needed to rid oneself of all other reflections and opinions whatever, simply to take up, what is there before us.
§ 96. Pure knowing is only simple immediacy.
§ 97. Simple immediacy is itself an expression of reflection and contains a reference to its distinction from what is mediated.
§ 98. If no presupposition is to made, all that is present is that we propose to consider thought as such.
§ 99. The beginning therefore is pure being.
§ 100. Preliminary prejudices must be disposed of within the science itself where their treatment should be awaited with patience.
§ 101. Absolute truth must be a result, and conversely, a result presupposes a prior truth.
§ 102. The advance is a retreat into the ground, to what is primary and true.
§ 103. It is equally necessary to consider as result that into which the movement returns as into its ground.
§ 104. Hence the line of the scientific advance becomes a circle.
§ 105. This progress in knowing must be determined by the
nature of the subject matter itself and its content.
§ 106. The beginning is neither arbitrary nor a provisional assumption, but is subsequently shown to have been properly made.
§ 107. If it were determinate, it would have been taken as something mediated.
§ 108. Here at the start, where the subject matter itself is not yet to hand, philosophy is an empty word.
§ 109. The determination of being so far adopted for the beginning could also be omitted, so that the only demand would be that a pure beginning be made.
§ 110. The beginning therefore contains both, being and nothing, is the unity of being and nothing
§ 111. In the beginning, being and nothing are present as distinguished from each other.
§ 112. Let those dissatisfied with being as a
beginning, begin with the general idea of a beginning.
§ 113. This takes it for granted that everyone has roughly the same general idea of it.
§ 114. The beginning cannot be made with anything
concrete, anything containing a relation within itself.
§ 115. What the subject matter is, that will be explicated in the development of the science, cannot be presupposed.
§ 116. Let those who are still dissatisfied with this beginning avoid these defects by beginning in some other way.
§ 117. Some begin with the ego, because the first truth must be something with which we are acquainted.
§ 118. But the ego is the most concrete of all things.
§ 119. The ego which formed the starting point is still entangled in the world of appearance.
§ 120. In logic we are concerned the determinate reality in thought of what is inner.
§ 121. This simple determination which has no other meaning, this emptiness, is therefore the beginning of philosophy.
§ 122. These preliminary were not so much intended to lead up to it as rather to eliminate all preliminaries.
§ 123. Being is first, as against another in general; second, immanently self-determining; third, abstract indeterminateness.
§ 124. Being is distinct from essence, for later in its development it proves to be in its totality, only one sphere of the Notion
§ 125. Being will posit itself in three determinations:
quality, quantity and measure.
§ 126. With Kant these are supposed to be only titles for his categories though they are, in fact, themselves categories.
§ 127. Hitherto the determination of quantity has been made to precede quality, but the beginning is made with being as such.
§ 128. Measure is a relation, the specific relation between quality and quantity.
§ 129. Relation falls within the section Quality.
§ 130. Being is the indeterminate immediate.
§ 131. The first being is in itself determinate, and therefore, secondly, determinate being, thirdly, being-for-self.
§ 132. Pure Being has no diversity within itself nor any with a reference outwards.
§ 133. Nothing is the same determination, or rather absence of determination, and thus altogether the same as, pure being.
§ 134. But it is equally true that they are not undistinguished from each other, that, on the contrary, they are not the same.
Remark 1: The Opposition of Being and Nothing in Ordinary Thinking.
§ 135. Nothing is usually opposed to something; but is already thereby distinguished from a determinate something.
§ 136. Heraclitus brought forward the concept of becoming and said: being as little is, as nothing is, or, all flows, all is a becoming.
§ 137. Those who maintain that nothing is just nothing, are unaware that they subscribe to the pantheism of the Eleatics and Spinoza.
§ 138. Nowhere in heaven or on earth is there anything which does not contain within itself both being and nothing.
§ 139. What is in question is not such concrete something but only the pure abstractions of being and nothing.
§ 140. It is not a matter of indifference to it whether a certain other content with which it is in relation is, or is not.
§ 141. Kant said that being is not a determination of the content of a thing.
§ 142. Ordinary thinking transforms the abstract being and nothing which should be apprehended into a definite being and nothing.
§ 143. The concept of a determinate existence, says Kant, gains nothing by its being perceived.
§ 144. What is the first in the science had of necessity to show itself historically as the first.
§ 145. The reference back from particular finite being to being in its abstract universality is the first theoretical demand and the first practical demand too.
§ 146. The abstract definition of God, on the other hand, is precisely that his Notion and his being are unseparated and inseparable.
Remark 2: Defectiveness of the Expression 'Unity, Identity of Being and Nothing'
§ 147. The proposition: “being is the same as nothing” enunciates both determinations, being and nothing, and contains them as distinguished.
§ 148. The proposition contains the result, but the result is not itself expressed in the proposition.
§ 149. The deficiency is made good in the first place by adding the contrary proposition: being and nothing are not the same.
§ 150. The result which we have here before us is becoming, which is not merely the one-sided or abstract unity of being and nothing.
§ 151. The third in which being and nothing subsist must also present itself here, and it has done so; it is becoming.
§ 152. The challenge to distinguish between being and nothing also includes the challenge to say what, then, is being and what is nothing.
Remark 3: The Isolating of These Abstractions
§ 153. Transition is the same as becoming.
§ 154. We shall consider some of the results which appear when being and nothing are postulated in isolation.
§ 155. With Parmenides as with Spinoza, there is no progress from being ot absolute substance to the negative, to the finite.
§ 156. Jacobi recognised very clearly the insubstantial nature, the non ens, of abstraction.
§ 157. This abstract purity of continuity is the same as what the Indian calls Brahma, sitting for years on end looking at the tip of his nose.
§ 158. In this void, Jacobi says, he experiences the opposite of what Kant assures him he should experience.
§ 159. Is this still a synthesis if Jacobi omits precisely that which makes the unity a synthetic unity?
§ 160. Consciousness can make empty space, empty time, its object , but it does not stop at that; it goes beyond it.
§ 161. It is the thought of such indeterminates that is to be demonstrated as null.
§ 162. Because being is devoid of all determination whatsoever, it is not being but nothing.
§ 163. Being is posited only as immediate, therefore nothing emerges in it. But all the subsequent determinations are more concrete.
§ 164. That nothing is the result and should make the beginning (as in Chinese philosophy), need not cause us to lift a finger.
§ 165. The dialectic employed by Plato in treating of the One in the Parmenides is also a dialectic of external reflection.
§ 166. Being, taken as it is immediately, belongs to a subject, is something enunciated.
§ 167. Nothing, taken in its immediacy, shows itself as affirmative, as being.
§ 168. Just as little is seen in pure light as in pure darkness.
§ 169. The transition of being and nothing into each other is to be understood as it is without any further elaboration of the transition by reflection.
Remark 4: Incomprehensibility of the Beginning
§ 170. Let us consider the Kantian antinomy which proves that a beginning of the world, or of anything, is impossible.
§ 171. If the world, or anything, is supposed to have begun, then it must have begun in nothing.
§ 172. In this proof nothing is brought forward against becoming, or beginning and ceasing, against this unity of being and nothing.
§ 173. With the absolute separateness of being from nothing presupposed, then of course beginning or becoming is something incomprehensible.
§ 174. Tthere is nothing which is not an intermediate state between being and nothing.
§ 175. We call dialectic the higher movement of reason in which such seemingly utterly separate terms pass over into each other spontaneously.
§ 176. Becoming is the unseparatedness of being and nothing, not the unity which abstracts from being and nothing.
§ 177. Becoming therefore contains being and nothing as two such unities, each of which is itself a unity of being and nothing.
§ 178. Becoming is in this way in a double determination.
§ 179. They are not reciprocally sublated, but each sublates itself in itself .
§ 180. Becoming is an unstable unrest which settles into a stable result.
§ 181. Becoming is the vanishing of being and nothing; but at the same time it rests on the distinction between them.
§ 182. This stable oneness is being, yet no longer as a determination on its own but as a determination of the whole.
§ 183. Becoming, as this transition into the unity of being and nothing, is determinate being.
Remark: The Expression ‘To Sublate’
§ 184. To sublate constitutes one of the most important notions in philosophy which repeatedly occurs throughout the whole of philosophy.
§ 185. ‘To sublate’ has a twofold meaning in the language: on the one hand it means to preserve, to maintain, and equally it also means to cause to cease, to put an end to.
§ 186. It is remarkable to find that a language has come to use one and the same word for two opposite meanings.
§ 187. Being and nothing are the same; but just because they are the same they are no longer being and nothing, but now have a different significance.
§ 188. Through its quality, something is determined as opposed to an other, as alterable and finite.
§ 189. Determinate being falls into three parts: Determinate being as such, Finitude, Qualitative infinity.
§ 190. Determinate being as such, is first of all quality, as reality and negation.
§ 191. From becoming there issues determinate being, which is the simple oneness of being and nothing.
§ 192. It is not mere being, but determinate being [Dasein], there-being.
§ 193. Only that which is posited in a Notion belongs in the dialectical development of that Notion to its content.
§ 194. Determinate being is concrete.
§ 195. Determinateness has not yet severed itself from being.
§ 196. Determinateness thus isolated by itself in the form of being is quality.
§ 197. Quality, taken in the distinct character of being, is reality.
§ 198. Negation taken as mere deficiency would be equivalent to nothing; but it is a determinate being.
Remark: Quality and Negation
§ 199. Reality may seem to be a word of various meanings because it is used of different, indeed of opposed determinations.
§ 200. God was defined as the sum-total of all realities, and of this sum-total it was said that no contradiction was contained in it.
§ 201. Reality as thus conceived is assumed to survive when all negation has been thought away.
§ 202. If reality is taken in its determinateness, then the sum-total of all realities becomes a sum-total of all negations, all contradictions.
§ 203. Negation is as little an ultimate for philosophy as reality is for it truth.
§ 204. Spinoza grasped thought and being as attributes, as not having a separate existence, a self-subsistent being of their own.
§ 205. Negation stands directly opposed to reality.
§ 206. Quality is especially a property only where, in an external relation, it manifests itself as an immanent determination.
§ 207. Jacob Boehme's 'Qualierung' or 'Inqualierung' signify quality's own internal unrest by which it produces and maintains itself only in conflict.
§ 208. In determinate being its determinateness has been distinguished as quality, but there is distinction of reality and negation.
§ 209. This sublatedness of the distinction is determinate being's own determinateness, a something.
§ 210. Determinate being, life, thought, etc., determine themselves to become a determinate being, a living creature, a thinker (ego).
§ 211. Something is the negation of the negation in the form of being.
§ 212. Something as a becoming is a transition, the moments of which are themselves somethings.
§ 213. In this section, the negative determination contained in determinate being is developed.
§ 214. 1. Something and other are, in the first place, both determinate beings or somethings.
§ 215. Otherness thus appears as a determination alien to the determinate being
§ 216. Both are determined equally as something and as other, and are thus the same.
§ 217. The other is to be taken as isolated, as in relation to itself, abstractly as the other.
§ 218. The otherness, which is at the same time a moment of it, is distinct from it and does not appertain to the something itself.
§ 219. 2. Something preserves itself in the negative of its determinate being.
§ 220. Determinate being as such is immediate, without relation to an other.
§ 221. Being-for-other and being-in-itself are, therefore, posited as moments of one and the same something.
§ 222. Non-being is, in this unity of being and non-being, not negative determinate being in general, but an other.
§ 223. Being-in-itself has also present in it non-being itself, for it is itself the non-being of being-for-other.
§ 224. Being-for-other is negative determinate being which points to being-in-itself as to its own being.
§ 225. 3. Both moments are determinations of what is one and the same, namely, the something.
§ 226. People fancy that they are saying something lofty with the 'in itself'; but what something is only in itself, is an abstract determination.
§ 227. Things are called ‘in themselves’ in so far as abstraction is made from all being-for-other.
§ 228. In Being, the meaning of each opposite is complete even without its other; in Essence, the opposites are meaningless without one another.
§ 229. Determinateness reflected into itself is, therefore, again in the simple form of being, a Quality.
§ 230. The in-itself into which something is reflected into itself out of its being-for-other is no longer an abstract in-itself.
§ 231. 1. The quality of the essential unity of the in-itself in the simple something with its other moment, is the determination of the in-itself.
§ 232. The determination of man is thinking reason; by it he is distinguished from the brute.
§ 233. 2. The filling of the in-itself with determinateness is distinct from the determinateness which is only being-for-other.
§ 234. It is the quality of something to be open to external influences and to have a constitution..
§ 235. In so far as something alters, the alteration falls within its constitution.
§ 236. Determination and constitution are thus distinguished from each other.
§ 237. The first was only an implicit alteration belonging to the inner Notion; now alteration is also posited in the something.
§ 238. The transition of determination and constitution into each other is sublation of their difference, resulting in the positing of something in general.
§ 239. There is a single determinateness of both: this determinateness is limit..
§ 240. 3. Being-for-other is the indeterminate, affirmative community of something with its other.
§ 241. [a] Something, therefore, is immediate, self-related determinate being, and has a limit, in the first place, relatively to an other.
§ 242. The limit is simple negation or the first negation, whereas the other is, at the same time, the negation of the negation.
§ 243. Something is the limit relatively to another something, but the limit is present in the something itself.
§ 244. [b] The negative determinate being and the determinate being of the something fall outside each other.
§ 245. In accordance with this difference of something from its limit, the line appears as line only outside its limit, the point.
§ 246. [c] But further, something as it is outside the limit, the unlimited something, is only a determinate being in general.
§ 247. Something with its immanent limit posited as the contradiction of itself, directed and forced out of and beyond itself, is the finite.
§ 248. The being of something is determinate; something has a quality and in it is not only determined but limited.
§ 249. When we say of things that they are finite, we understand that non-being constitutes their nature and being.
§ 250. The thought of the finitude of things brings this sadness with it because it is qualitative negation pushed to its extreme.
§ 251. No philosophy or opinion will let itself be tied to the standpoint that the finite is absolute.
§ 252. This nothing is supposed to be only nothing.
§ 253. This contradiction is abstractly present simply in the circumstance that the something is finite, or that the finite is.
§ 254. Determination and constitution showed themselves as sides for external reflection.
§ 255. In order that the limit in something should be a limitation, something must at the same time in its own self transcend the limit.
§ 256. The ought therefore contains the determination in double form.
§ 257. The finite has thus determined itself as the relation of its determination to its limit.
§ 258. What ought to be is, and at the same time is not. If it were, we could not say that it ought merely to be.
§ 259. The limitation of the finite is not something external to it; on the contrary, its own determination is also its limitation.
§ 260. The finite as the ought transcends its limitation; its limit is also not its limit.
§ 261. Something has a limitation in so far as it has negation in its determination.
Remark: The Ought
§ 262. The ought has recently played a great part in philosophy, especially in connection with morality.
§ 263. 'You can, because you ought'.
§ 264. In the Ought the transcendence of finitude, that is, infinity, begins.
§ 265. The very fact that something is determined as a limitation implies that the limitation is already transcended.
§ 266. The sentient creature, in the limitation of hunger, thirst, etc., is the urge to overcome this limitation and it does overcome it.
§ 267. Leibniz said that if a magnet possessed consciousness it would regard its pointing to the north as a determination of its will.
§ 268. Reason and Law are not in such a bad way that they only ought to be.
§ 269. The ought as such contains limitation, and limitation contains the ought.
§ 270. The infinite is a fresh definition of the absolute; as indeterminate self-relation it is posited as being and becoming.
§ 271. Even so, the infinite is not yet really free from limitation and finitude.
§ 272. The infinite is (a) affirmative as negation of the finite, (b) the abstract, one-sided infinite, (c) the genuine infinite.
§ 273. The infinite is the negation of the negation, affirmation, being which has restored itself out of limitedness.
§ 274. It is the very nature of the finite to transcend itself, to negate its negation and to become infinite.
§ 275. The immediate being of the infinite resuscitates the being of its negation.
§ 276. Finitude is limitation posited as limitation; infinity is the nothing of the finite.
§ 277. The infinite posited over against the finite, in a relation wherein they are as qualitatively distinct others, is the spurious infinite.
§ 278. The finite remains as a determinate being opposed to the infinite; there are two worlds, one infinite and one finite.
§ 279. This contradiction develops its content into more explicit forms.
§ 280. As thus separated they are just as much essentially connected by the very negation which separates them.
§ 281. The infinite only emerges in the finite and the finite in the infinite.
§ 282. In this void beyond the finite, again there arises the void, in which a new limit, is encountered - and so on to infinity.
§ 283. The finite is finite only in its relation to the ought or to the infinite, and the latter is only infinite in its relation to the finite.
§ 284. The progress is a contradiction which is not resolved but is always only enunciated as present..
§ 285. The progress to infinity is only the perpetual repetition of one and the same tedious alternation of finite and infinite.
§ 286. The infinity of the infinite progress remains burdened with the finite as such, is thereby limited and is itself finite.
§ 287. In this alternating of the finite and the infinite, their truth is already implicitly present.
§ 288. The finite is only that which must be transcended, the negation of itself in its own self, which is infinity.
§ 289. The infinite and the finite viewed as connected with each other.
§ 290. This yields the decried unity of the finite and the infinite.
§ 291. The unity of the infinite which each of these moments is, is differently determined in each of them.
§ 292. The simple unity of the infinite and finite was falsified by the understanding; so too is the double unity.
§ 293. The unity of the finite and infinite is not an incongruous combination alien to their own nature; but each is in its own self this unity.
§ 294. Infinity is the negative of finitude, and hence of determinateness in general; the sublating of itself in the finite is a return from an empty flight.
§ 295. This is in itself self-relation, affirmation, but as return to itself, through the mediation which the negation of negation is.
§ 296. The infinite progress expresses the connection of terms which are also distinct from each other.
§ 297. Starting from the finite, the limit is transcended. We now have its beyond, the infinite, but in this the limit arises again.
§ 298. In the infinite, the beyond of the limit, there arises another limit which has the same fate, namely, that as finite it must be negated.
§ 299. Both finite and infinite are this movement in which each returns to itself through its negation.
§ 300. Since both the finite and the infinite are moments of the progress they are jointly the finite, and are equally together negated in it.
§ 301. As the infinite, they are the finite and the infinite, which are themselves in process of becoming.
§ 302. The image of true infinity becomes the circle, the line which has reached itself, closed and present, without beginning and end.
§ 303. It is not the finite which is the real, but the infinite. Thus reality is further determined as essence, Notion, Idea, and so on.
§ 304. The negation is thus determined as ideality; ideal being [das Ideelle] is the finite as it is in the true infinite.
§ 305. Ideality can be called the quality of infinity; but it is essentially the process of becoming, and hence a transition.
Remark 1: The Infinite Progress
§ 306. The infinite and the progress to infinity are the expression of a contradiction which is itself put forward as the final solution.
§ 307. Infinite progress is seen when one remains fixed in the contradiction of the unity of two determinations and their opposition.
§ 308. Infinite progress, the developed infinite of the understanding, is so constituted as to be the alternation of the two determinations.
§ 309. The resolution is not recognition of the equal correctness and incorrectness of the two assertions, but the ideality of both.
§ 310. Some placed the essence of philosophy in aswering the question: how does the infinite go forth from itself and become finite?
§ 311. To understand how to put questions presupposes a certain education, if one wants a better answer than that the question is idle.
§ 312. Expressions of sensuous conception arouse the suspicion that they spring from the level of ordinary conception.
§ 313. In the infinite this is expressed; it is the not-finite. The unity of the finite and infinite thus seems to be directly excluded.
§ 314. The infinite no more is than pure being is.
§ 315. The unity of the finite and infinite and the distinction between them are just as inseparable as are finitude and infinity.
Remark 2: Idealism
§ 316. The proposition that the finite is ideal [ideell] constitutes idealism.
§ 317. By the ideal [dem Ideellen] is meant the form of figurate conception and imagination, and what is simply in my conception.
§ 318. Determinate being is the sphere of difference, of dualism, the field of finitude.
§ 319. Being-for-self is first the One, secondly repulsion and attraction, thirdly, the alternation of repulsion and attraction.
§ 320. We have arrived at the general Notion of being-for-self. All that is now necessary to justify our use of the term.
§ 321. being-for-self is infinity which has collapsed into simple being.
§ 322. This moment expresses the manner in which the finite is present in its unity with the infinite, or is an ideal being.
Remark: The German Expression, 'What For a Thing' (Meaning 'What Kind of a Thing')
§ 323. “Was für ein Ding” — in other words that which is, and that for which it is, are one and the same.
§ 324. To call thought, spirit, God, only an ideal being, presupposes the standpoint from which finite being counts as the real.
§ 325. Eleatic Being or Spinoza's substance is the abstract negation of all determinateness. The idealism of Malebranche is more explicit.
§ 326. The Leibnizian idealism lies more within the bounds of the abstract Notion.
§ 327. The idealisms of Kant and Fichte, do not go beyond the ought, and remain in the dualism of determinate being and being-for-self.
§ 328. Being-for-self is the simple unity of itself and its moment, being-for-one.
§ 329. 1. negation in general, 2. two negations, 3. two that are the same, 4. sheer opposites, 5. identity as such, 6. relation which is negative and yet to its own self.
§ 330. The one is the simple self-relation of being-for-self in which its moments have collapsed.
§ 331. The one is a process of determining - and as self-relation it is an infinite self-determining.
§ 332. In its own self the one simply is; the one is not capable of becoming an other: it is unalterable.
§ 333. It is indeterminate but not, however, like being; its indeterminateness is the determinateness which is a relation to its own self.
§ 334. In this simple immediacy the mediation of determinate being, and all difference and manifoldness, has vanished.
§ 335. The one is the void as the abstract relation of the negation to itself.
§ 336. Being-for-self determined in this manner as the one and the void has again acquired a determinate being.
Remark: Atomism
§ 337. The one in this form appeared as the atomistic principle, according to which the essence of things is the atom and the void.
§ 338. The void is the ground of movement only as the negative relation of the one to its negative.
§ 339. Physics with its molecules and particles suffers from this principle of extreme externality, just as does that theory of the State which starts from the particular will of individuals.
§ 340. The one and the void constitute the first stage of the determinate being of being-for-self.
§ 341. The one is a becoming of many ones.
§ 342. But this is not really a becoming, for becoming is a transition of being into nothing: the one, on the other hand, becomes only one.
§ 343. This repulsion is the positing of many ones but through the one itself. The second is the mutual repelling of ones presupposed as already present.
§ 344. The one repels only itself from itself, therefore does not become but already is.
§ 345. The ones are presupposed relatively to one another - supposed or posited by the repulsion of the one from itself.
§ 346. Plurality appears not as an otherness, but as a determination completely external to the one.
§ 347. The plurality of ones is infinity as a contradiction which unconstrainedly produces itself.
Remark: The Monad of Leibniz
§ 348. The Leibnizian idealism takes up the plurality immediately as something given and does not grasp it as a repulsion of the monads.
§ 349. The many ones have affirmative being external to them - the abstract void. But they are infinity posited in the immediacy of being.
Remark: The unity of the One and the Many
§ 350. The plurality is, in the first place, non-posited otherness, the limit is only the void, only that in which the ones are not.
§ 351. This mutual repulsion is the posited determinate being of the many ones.
§ 352. The being-for-self of the many ones shows itself as their self-preservation through the mediation of their mutual repulsion.
§ 353. The ones not only are, but they maintain themselves through their reciprocal exclusion.
§ 354. This reflection that the ones show themselves to be one and the same and indistinguishable, is a comparison made by us.
§ 355. The negative relationship of the ones to one another is only a going-together-with-self.
§ 356. Self-subsistence pushed to the point of the one as a being-for-self is abstract, formal, and destroys itself.
§ 357. It is an ancient proposition that the one is many and especially that the many are one.
§ 358. Repulsion is the self-differentiating of the one.
§ 359. Attraction belongs equally to each of the many ones as immediately present; none has any precedence over another.
§ 360. There is thus in the one one, the unity of repulsion and attraction in general.
§ 361. The difference of the one and the many is now determined as repulsion and attraction.
§ 362. Repulsion is indifferent to attraction which is externally added to it as presupposed. Attraction is not presupposed by repulsion.
§ 363. Repulsion is, although negative, still essentially relation.
§ 364. Not merely is repulsion presupposed by attraction, but equally, too, there is a reverse relation of repulsion to attraction.
§ 365. Each being a self-mediation, evident after closer consideration of them and brings them back to the unity of their Notion.
§ 366. That each presupposes itself is already implied in the relationship between repulsion and attraction, initially relative.
§ 367. Relative repulsion is the mutual repelling of the present many ones which are supposed to be immediately given.
§ 368. Attraction is the positing of the real one, in contrast to which the many in their determinate being are only ideal and vanishing.
§ 369. This self-presupposing of the two determinations each for itself, means that each contains the other as a moment within it.
§ 370. This being, in the determination it has now acquired, is quantity.
§ 371. The fundamental determination of quality is being and immediacy, in which determinateness is so identical with the being of something, that with its alteration the something itself vanishes.
§ 372. This unity is: [a] being only as immediacy; [b] determinate being; [c] being-for-self.
Remark: The Kantian Construction of Matter from the Forces of Attraction and Repulsion
§ 373. Attraction and repulsion, as we know, are usually retarded as forces.
§ 374. Kant, constructed matter from the forces of attraction and repulsion, or “set up the metaphysical elements of this construction”.
§ 375. Matter as it exists for sense perception is no more a subject for logic than space. But attraction and repulsion are also based on the determinations here considered.
§ 376. Later philosophers of nature gave the name construction to the most baseless concoction of unbridled imagination.
§ 377. Kant's method is analytical, not constructive. He presupposes the idea of matter, then asks what forces are required to maintain it.
§ 378. Kant remarks that the force of attraction just as much belongs to the concept of matter 'although it is not contained in it'.
§ 379. Kant from the start one-sidedly attributes to the concept of matter only the determination of impenetrability.
§ 380. Kant is chiefly concerned to banish the vulgar mechanistic way of thinking which stops short at impenetrability.
§ 381. These forces are only moments which pass over into each other.
§ 382. He defines attraction as a penetrative force by which one bit of matter acts directly on the parts of another even beyond the contact.
§ 383. It therefore follows, quite tautologically, that where repulsion is assumed to be not, there no repulsion can take place.
§ 384. Kant assumes further that 'through the force of attraction, matter only occupies space but does not fill it'.
§ 385. Forces through which bodies act on one another and are set in motion, are something quite different from forces supposed as [constitutive] moments of matter.
§ 386. The same opposition of attractive and repulsive forces is made by their more developed form of centripetal and centrifugal forces.
§ 387. Quantity is the determinateness which has become indifferent to being, a limit which is just as much no limit.
§ 388. The indifference of the limit and of the something to the limit, constitutes quantitative determinateness.
§ 389. Firstly, pure quantity is a compact, infinite unity which continues itself into itself.
§ 390. Secondly, this develops a determinateness which only an external one, quantum.
§ 391. Thirdly, quantum in a qualitative form is quantitative ratio.
§ 392. The ratio is only a formal unity of quality and quantity. Its dialectic is its transition into their absolute unity, into Measure.
Remark: Something's Limit
§ 393. In something, its limit as quality is essentially its determinateness.
§ 394. A magnitude is usually defined as that which can be increased or diminished; magnitude would thus be that of which the magnitude can be altered.
§ 395. Quantity is sublated being-for-self.
§ 396. Continuity is, therefore, simple, self-same self-relation, which is not interrupted by any limit or exclusion.
§ 397. In continuity, magnitude immediately possesses the moment of discreteness - repulsion.
§ 398. Quantity is the unity of these moments of continuity and discreteness.
Remark 1: The Conception of Pure Quantity
§ 399. Pure quantity has not as yet any limit or is not as yet quantum.
§ 400. It is this externality of continuity for the ones to which atomism clings and which ordinary thinking finds it difficult to forsake.
§ 401. It is the notion of pure quantity as opposed to the mere image of it that Spinoza has in mind.
§ 402. More specific examples of pure quantity are space and time, also matter as such, light, and so forth.
§ 403. Time is an absolute coming-out-of-itself, a generating of the one, and immediately the annihilation of it.
Remark 2: The Kantian Antinomy of the Indivisibility and the Infinite Divisibility
§ 404. It is the nature of quantity that gives rise to the conflict or antinomy of the infinite divisibility of space, time, etc.
§ 405. This antinomy consists solely in the fact that discreteness must be asserted just as much as continuity.
§ 406. The second of Kant's four (cosmological) antinomies deals with the antithesis constituted by the moments of quantity.
§ 407. Kantian antinomies remain an important part of the critical philosophy; they brought about the downfall of previous metaphysics .
§ 408. Kant wanted to give his four cosmological antinomies a show of completeness by the classification which he took from his schema of the categories.
§ 409. Kant did not take up the antinomy in the Notions themselves, but in the concrete form of cosmological determinations.
§ 410. Kant's conception of the antinomies is that they are 'contradictions which reason must necessarily come up against'.
§ 411. The Kantian antinomies contain nothing more than the categorical assertion of each of two opposed moments taken in isolation from the other.
§ 412. The relevant antinomy here concerns the so-called infinite divisibility of matter.
§ 413. To the simple, the atom, there is here opposed the composite, which is a very inferior determination compared to the continuous.
§ 414. Like the Kantian proofs of the rest of the antinomial propositions, this proof makes the detour of being apagogic.
§ 415. The main point, and in face of which all that precedes is completely superfluous, is mentioned by the way, in a parenthesis.
§ 416. The assertion that the parts are simple is only a tautology.
§ 417. The apagogical detour thus contains the very assertion which should result from it.
§ 418. We see the externality, that is contingency, of composition put forward as a consequence after it had already been introduced.
§ 419. The laboured, tortuous complexity serves no other purpose than to produce the merely outward semblance of a proof.
§ 420. The apagogical form is a groundless illusion.
§ 421. The apagogical proof begins with a proposition, but oddly enough immediately forgets it.
§ 422. The Kantian distinction between intuition and concept has given rise to a deal of nonsense.
§ 423. There is also involved here a clash between the continuity of space and composition; the two are confused with each other.
§ 424. Even the best microscopes and the keenest knives have not enabled us to come across anything simple.
§ 425. Neither taken alone has truth; this belongs only to their unity. This is the true dialectical consideration and the true result.
§ 426. Infinitely more ingenious and profound than this Kantian antinomy are the dialectical examples of the ancient Eleatic school.
§ 427. The solutions propounded by Aristotle of these dialectical forms merit high praise.
§ 428. The Kantian solution of the antinomy consists solely in the supposition that reason should not soar beyond sensuous perception and should take the world of appearance as it is.
§ 429. 1. Quantity contains the two moments of continuity and discreteness.
§ 430. Quantity is a concrete unity only in so far as it is the unity of distinct moments.
§ 431. 2. Immediate quantity is continuous magnitude.
Remark: The Usual Separation of These Magnitudes
§ 432. Space, time, matter, etc. are continuous magnitudes which possess the absolute possibility that the one may be posited in them.
§ 433. Continuous and discrete magnitude can be regarded as species of quantity.
§ 434. Discrete magnitude has first the one for its principle; secondly, it is a plurality of ones; and thirdly, it is essentially continuous.
§ 435. This limit, which is related to the unity and is the negation in it, is also, as the one, self-related.
§ 436. Since the one which is a limit includes within itself the many ones of discrete quantity, it equally posits them as sublated within it.
§ 437. Quantum, which to begin with is quantity with a determinateness or limit in general is, in its complete determinateness, number.
§ 438. Quantity is quantum, or has a limit, both as continuous and as discrete magnitude.
§ 439. The very nature of quantity is to be indifferent to its limit. But equally, quantity is not unaffected by the limit.
§ 440. This one is thus the principle of quantum, but as the one of quantity.
§ 441. Quantum completely posited in these determinations is number.
§ 442. Amount and unit constitute the moments of number.
§ 443. The number is not a plurality over against the enclosing, limiting one, but itself constitutes this limitation.
§ 444. The contradiction of number or of quantum as such within itself is the quality of quantum.
Remark 1: The Species of Calculation in Arithmetic; Kant's Synthetic Propositions a priori
§ 445. Spatial magnitude and numerical magnitude are regarded as two species, the former being a determinate magnitude just as much as the latter.
§ 446. Arithmetic considers number and its figures; or rather does not consider them but operates with them.
§ 447. Number has for its principle the one and is, therefore, simply an aggregate externally put together, devoid of any inner connectedness.
§ 448. The qualitative difference which constitutes the determinateness of number is that of unit and amount.
§ 449. Numbers can be produced in two ways, either by aggregation, or by separation of an aggregate already given.
§ 450. 1. After these remarks we proceed to indicate the species of calculation.
§ 451. The number produced by counting are in turn themselves counted.
§ 452. Kant considers the proposition: 7 + 5 = 12 to be a synthetic proposition.
§ 453. Counting is not determined by sensation which, according to Kant's definition of intuition is all that remains over for the a posteriors.
§ 454. Kant's assertion of the synthetic nature of the foundations of pure geometry is equally without any solid basis.
§ 455. Kant's notion of synthetic a priori judgements belongs to what is great and imperishable in his philosophy.
§ 456. The negative species of calculation corresponding to addition, subtraction, is the wholly analytical separation into numbers.
§ 457. 2. The next determination is the equality of the numbers which are to be counted.
§ 458. Division is the negative species of calculation with the same determination of the difference.
§ 459. 3. The two numbers related to each other as unit and amount are still immediate with respect to each other and therefore unequal.
§ 460. A graded instruction based on a logically formed division of the subject treats of powers before it treats of proportions.
§ 461. Philosophy must be able to distinguish what is an intrinsically self-external material.
Remark 2: The Employment of Numerical Distinctions for Expressing Philosophical Notions
§ 462. Pythagoras represented rational relationships (or philosophemata) by numbers.
§ 463. Number is the absolute determinateness of quantity.
§ 464. Number is at the same time the abstraction of the manifoldness of sense.
§ 465. The mind which rises above the world of the senses and contemplates its own essence, may therefore happen on number.
§ 466. When what is most alive are transposed into this element of pure self-externality, they become dead, inert determinations.
§ 467. The richer in relationships thoughts become, the more confused and meaningless becomes their representation in numbers.
§ 468. Thought is set its hardest task when the movement of the Notion through which alone it is the Notion, is denoted by numbers.
§ 469. In symbols the truth is dimmed and veiled by the sensuous element; only in the form of thought is it fully revealed.
§ 470. In the concrete philosophical sciences philosophy must take the logical element from logic, not from mathematics.
§ 471. Machines can perform arithmetical operations. So much for the idea of making calculation the means for educating the mind and stretching it on the rack in order to perfect it as a machine.
§ 472. 1. Quantum has its determinateness as limit in amount.
§ 473. The direct opposite of Extensive magnitude is not discrete but intensive magnitude.
§ 474. 2. The determinateness of something in terms of number does not require it to be distinguished from another numerically determined something.
§ 475. The limit of quantum passes over into simple determinateness.
§ 476. The degree is thus a specific magnitude, a quantum.
§ 477. 3. In number, quantum is posited in its complete determinateness; but as intensive quantum.
§ 478. None of the various distinct degrees is separate from the others but each is determined only through them.
§ 479. Degree is not external to itself within itself.
§ 480. The determinateness of intensive magnitude is, therefore, to be considered from two sides.
§ 481. Extensive and intensive magnitude are thus one and the same determinateness of quantum.
§ 482. The qualitative something makes its appearance, for the identity is the unity which is self-related through the negation of its differences.
Remark 1: Examples of This Identity
§ 483. Extensive and intensive quantum are usually distinguished in the ordinary conception of them as kinds of magnitude.
§ 484. In the conversion of the mechanical into the dynamic view, there occurs the concept of separately existing, independent parts.
§ 485. The other determinateness which occurs here is the quantitative as such.
§ 486. Number itself necessarily has this double form immediately within it.
§ 487. In the circle the one is called degree because the determinateness of any part of the circle derives from the many parts outside it.
§ 488. The magnitude of a more concrete object exhibits its dual aspects of being extensive and intensive.
§ 489. A higher degree of temperature expresses itself as a longer column of mercury.
§ 490. A larger surface can be coloured with a more intensive colour than with a weaker colour used in the same way.
§ 491. In the spiritual sphere, high intensity of character is bound up with a correspondingly far-reaching reality in the outer world.
Remark 2: The determination of degree as applied by Kant to the soul
§ 492. The determinateness of intensive quantum has been applied by Kant in a peculiar way to a metaphysical determination of the soul.
§ 493. The difference between extensive and intensive quantum is indifferent to the determinateness of quantum as such.
§ 494. A quantum, therefore, in accordance with its quality, is posited in absolute continuity with its externality, with its otherness.
§ 495. The one is infinite or self-related negation, hence the repulsion of itself from itself.
§ 496. Thus quantum impels itself beyond itself.
§ 497. Quantum alters and becomes another quantum.
§ 498. Finitude and infinity each acquire in themselves a dual, and indeed, an opposite meaning.
§ 499. Qualitative determinateness, as an immediacy, is related to otherness essentially as to an alien being.
§ 500. The progress to infinity is the expression of contradiction, here, of quantum as such.
§ 501. The infinite progress is only the expression of this contradiction, not its resolution; .
§ 502. The continuity of quantum with its other produces the conjunction of both in the expression of an infinitely great or infinitely small.
§ 503. This infinity which is perpetually determined as the beyond of the finite is the spurious quantitative infinite.
Remark 1: The High Repute of the Progress to Infinity
§ 504. The spurious infinite is commonly held to be something sublime, while in philosophy it has been regarded as ultimate.
§ 505. Kant at the close of the Critique of Practical Reason, .
§ 506. This exposition deserves praise mainly on account of truthfulness.
§ 507. Haller's description of eternity, called by Kant terrifying.
§ 508. This heaping and piling up of numbers is regarded as what is valuable in a description of eternity.
§ 509. Astronomers pride themselves on the sublimity of their science because it has to deal with an innumerable host of stars.
§ 510. To the infinity of outer, sensuous intuition, Kant opposes the other infinite.
§ 511. The ego in being thus alone with itself is, it is true, the reached beyond.
§ 512. Kant declares those exaltations to be unsatisfying for reason, which cannot stop at them.
§ 513. The ego in its self-determining proceeds to determine nature and liberate itself therefrom.
§ 514. Ego and non-ego or the pure will and the moral law, are presupposed as completely self-subsistent and mutually indifferent.
§ 515. The standpoint is powerless to overcome the qualitative opposition between the finite and infinite.
§ 516. In Fichte's Theory of Science, the first axiom is ego = ego.
§ 517. Only in the qualitative opposition does the posited infinitude emerge and the quantitative determination itself pass over into the qualitative.
Remark 2: The Kantian Antinomy of the Limitation and Nonlimitation of the World
§ 518. The Kantian antinomies are expositions of the opposition of finite and infinite in a more concrete shape.
§ 519. This antinomy concerns the limitation or non-limitation of the world in time and space.
§ 520. The antinomy is two simple opposite assertions: there is a limit, and the limit must be transcended.
§ 521. The other part of the proof which concerns space is based on time.
§ 522. The basis of the proof itself is the direct assertion of what was to be proved.
§ 523. The assumed limit of time is a now as end of the time already elapsed.
§ 524. In truth, time is pure quantity.
§ 525. The antithesis.
§ 526. The assumption is made that the world as an existence presupposes another conditioned existence in time, and so on to infinity.
§ 527. The proof regarding the infinity of the world in space is the same.
§ 528. Thesis and antithesis are nothing but the opposite assertions, that a limit is, and that the limit equally is only a sublated one.
§ 529. The solution of these antinomies, as of those previously mentioned, is transcendental.
§ 530. 1. The infinite quantum as infinitely great or infinitely small is itself implicitly the infinite progress.
§ 531. Quantum as degree is unitary, self-related and determinate within itself.
§ 532. But this is what quantum as such is in itself.
§ 533. Let us take the progress at first in its abstract determinations as we find them.
§ 534. The first sublation, the negation of quality as such whereby quantum is posited, is in principle the sublating of the negation.
§ 535. The infinite, which in the infinite progress has only the empty meaning of a non-being, is in fact nothing else than quality.
§ 536. Quantum is sublated quality; but quantum is infinite.
§ 537. Quantum as self-related as an indifferent limit in its externality and therefore posited as qualitative, is quantitative ratio.
Remark 1: The Specific Nature of the Notion of the Mathematical Infinite
§ 538. The introduction of the mathematical infinite has led to important results, but mathematics has not succeeded in justifying its use.
§ 539. Mathematics, being unaware of the nature of its instrument, is unable to determine the scope of its application.
§ 540. The science of mathematics can only defend itself by denying the competence of metaphysics.
§ 541. Mathematics finds a radical contradiction to that very method which is peculiar to itself and on which as a science it rests.
§ 542. Mathematics shows that results obtained by the use of the infinite agree with those found by the strictly mathematical method.
§ 543. Consideration of justifications and characteristics of the mathematical infinite I shall undertake at some length in this Remark.
§ 544. The usual definition of the mathematical infinite is that it is a magnitude than which there is no greater.
§ 545. It is the reflection that quantum is sublated, which is usually not made, and which creates the difficulty for ordinary thinking.
§ 546. Kant's finds the said definition does not accord with what is understood by an infinite whole.
§ 547. Kant objects to infinite wholes being regarded as a maximum.
§ 548. Kant's concept of infinite he calls truly transcendental.
§ 549. The character of the mathematical infinite and the way it is used in analysis corresponds to the Notion of the genuine infinite.
§ 550. The infinite quantum contains within itself first externality and secondly the negation of it.
§ 551. The Notion of the infinite as abstractly expounded here will show itself to be the basis of the mathematical infinite.
§ 552. A fractional number, 2/7 for example, is not a quantum like 1, 2, 3, etc., but is directly determined by two other numbers.
§ 553. The representation of infinity by a fractional number is still imperfect.
§ 554. Letters, the next universality into which numbers are raised, are only general symbols and indeterminate possibilities.
§ 555. The ratio first is a quantum, secondly, this quantum is not immediate but contains qualitative opposition.
§ 556. The fraction 2/7 can be expressed as 0.285714..., 1/(1 - a) as 1 + a + a2 + a3 etc..
§ 557. The series is itself infinite.
§ 558. The nature of this infinity of the series is the spurious infinity of the progression.
§ 559. In the infinite series, inexactitude is actually present, whereas in the mathematical infinite there is only appearance of inexactitude.
§ 560. The infinite series contains the spurious infinity.
§ 561. In the infinite series the negative is outside its terms which are present only qua parts of the amount.
§ 562. The existence of infinite series which cannot be summed is external and contingent with respect to the form of series as such.
§ 563. The terminological inversion with the fraction and its expression as a series, also occurs in the mathematical infinite.
§ 564. Spinoza opposes the concept of true infinity to that of the spurious.
§ 565. With Spinoza, substance and its absolute unity has the form of an inert unity.
§ 566. The mathematical example with which Spinoza illustrates the true infinite is the space between two unequal eccentric circles.
§ 567. The incommensurability which lies in Spinoza's example embraces in general the functions of curved lines.
§ 568. The expression 'variable magnitudes' is very vague and ill-chosen.
§ 569. In 2/7 or a/b, 2 and 7 are each independent determinate quanta and the relation is not essential to them.
§ 570. In an equation in which x and y are determined primarily by a power-relation, x and y as such are still supposed to signify quanta.
§ 571. In this concept of the infinite, the quantum is genuinely completed into a qualitative reality.
§ 572. This concept has been the target for all the attacks on the mathematics of this infinite, i.e. differential and integral calculus.
§ 573. Such an intermediate state, as it was called, between being and nothing does not exist.
§ 574. What is infinite is not comparable as something greater or smaller.
§ 575. The originators did not establish the thought as Notion and had to resort to expedients which conflict with their better cause.
§ 576. The thought cannot be more correctly determined than in the way Newton has stated it.
§ 577. Newton did what the scientific method of his time demanded, he only explained what was to be understood by an expression.
§ 578. Newton points out that final ratios are not ratios of final magnitudes, but limits.
§ 579. Carnot: by virtue of the law of continuity, the vanishing magnitudes still retain the ratio from which they come, before they vanish.
§ 580. The empirical reality is thereby raised above itself.
§ 581. The other form of Newton's exposition of the magnitudes in question is equally interesting, namely, as generative magnitudes.
§ 582. The conception of infinitesimals which is implicit in the increment or decrement itself, is inferior to the above determinations.
§ 583. In the science of mathematics there cannot be any question of such empirical accuracy.
§ 584. Euler insists the differential calculus considers the ratios of the increments, but the infinite difference as such is wholly nil.
§ 585. Lagrange's opinion is that the ratio of two magnitudes does not present any clear concept as its terms become simultaneously zero.
§ 586. In the actual application of the method of infinitesimals, the genuine Notion of the infinite cannot exercise any influence.
§ 587. The older analysts had little scruples but the moderns wish differential calculus to attain to the rigour of the proofs of the ancients.
§ 588. Some have attempted to dispense with the concept of the infinite, and without it to achieve what seemed bound up with its use.
§ 589. Fermat, Barrow, Leibniz and Euler, frankly believed that they were entitled to omit the products of infinitesimal differences.
§ 590. Newton had an ingenious device to remove the arithmetically incorrect omission of the products of infinitesimal differences.
§ 591. Other forms Newton employed in derivations are bound up with concrete meanings of the elements relating to motion.
§ 592. Newton made the mistake because he omitted the term of the series containing that power on which the specific problem turned.
§ 593. The procedure is made to depend on the qualitative meaning.
§ 594. Carnot gives a most lucid exposition of what is essential in the ideas referred to above.
§ 595. Lagrange reverted to Newton's original method of series to be relieved of the difficulties inherent in the idea of the infinitely small.
§ 596. The demonstrated qualitative character of the form of magnitude in the infinitesimal, is found in the category of limit of the ratio.
§ 597. The very expression 'limit' implies that it is a limit of something.
§ 598. The idea of limit is supposed to have the advantage of avoiding the inconsistency here involved.
§ 599. [Continuous or fluent magnitude enters with the consideration of the external and empirical variation of magnitudes.]
§ 600. We shall discuss the confusion which the conception of approximation currently used in expositions of the calculus.
§ 601. The so-called infinitesimals express the vanishing of the sides of the ratio as quanta.
§ 602. The limit here does not have the meaning of ratio; it counts only as the final value.
§ 603. The increments or infinitesimals have been considered only from the side of the quantum which vanishes in them.
§ 604. Analogous is the assumption that infinitely small parts of the same whole are equal to each other.
§ 605. Propositions are put forward as results of calculus, without enquiry whether by and in themselves they have a real significance.
§ 606. Much has been accepted as proof for no other reason than that the result was always already known beforehand.
§ 607. The empty scaffolding of such proofs was erected in order to prove physical laws.
Remark 2: The Purpose of the Differential Calculus Deduced from its Application
§ 608. The so-called application, presents greater difficulties, but also the more interesting side; the elements of this concrete side are to be the object of this Remark.
§ 609. The method of the differential calculus shows on the face of it that it was not invented and constructed for its own sake.
§ 610. The nature of the infinitesimal is the qualitative nature of determinations of quantity which are related to each other as quanta.
§ 611. The specifically qualitative character of quantity is first indicated in the quantitative relation as such.
§ 612. The core of the whole business is the actual procedure in the mathematical solution of a certain group of problems.
§ 613. (a) Equations in which any number of magnitudes (here we confine ourselves to two) are combined into a qualitative whole.
§ 614. The magnitudes have simply and solely the character of variables such as occur in the problems of indeterminate analysis.
§ 615. The difference between variables in calculus, is that at least one of those variables is found in a power higher than the first.
§ 616. (b) We have now to indicate what is the interest on which the treatment of the equation is focused.
§ 617. The sole point of importance here is the qualitative determinateness of the terms.
§ 618. The binomial: xn = (y + z)n = (yn + ny(n-1)z + ... ).
§ 619. The object is an equation, ym = axn.
§ 620. The representation of these functions of potentiation of a variable is taken as a sum complex within itself.
§ 621. The increment is supposed to be not a quantum but only a form, the whole value of which is that it assists the development.
§ 622. For what purpose are such functions sought.
§ 623. The answer follows directly and automatically from the nature of the matter.
§ 624. The appearance of arbitrariness of the differential calculus is clarified by awareness of where its application is permissible.
§ 625. For the determination of its moment we shall take the simplest example from equations of the second degree.
§ 626. The equations between these lines and the co-ordinate are linear equations.
§ 627. The first discoverers could only record their findings in a wholly empirical manner.
§ 628. The power forms in the equation are reduced to their first functions. But the value of the terms of the equation is thereby altered.
§ 629. Lagrange rejected this pretence and took the genuinely scientific course.
§ 630. I must also mention the tangential method of Descartes, if only for its beauty and its fame.
§ 631. The final equation obtained in this way is the same as that obtained by the method of the differential calculus.
§ 632. The omission of the constant has the meaning that the constant plays no part in the determination of the roots if these are equal.
§ 633. To differentiate denotes that differences are posited. In integration, on the other hand, the constant must be added in again.
§ 634. Another important sphere in which the differential calculus is employed is mechanics.
§ 635. The erroneous assumption that 2at is part of the motion regarded as a sum, gives the false appearance of a physical proposition.
§ 636. The motion represented by the equation s = at2 we find, says Lagrange, empirically in falling bodies.
§ 637. The application of the differential calculus to the elementary equations of motion does not of itself offer any real interest.
§ 638. The aim has been to make prominent and to establish the simple, specific nature of the differential calculus and to demonstrate it.
§ 639. The other part of the problem of the calculus appears in connection with its formal operation, namely the application of the latter.
§ 640. The problem now is to determine which of the moments determining the subject matter is given in the equation itself.
§ 641. The differentiation of an equation of several variables yields the differential coefficient, not as an equation but only as a ratio.
§ 642. The usual method makes the matter easy for itself by using the idea of the infinitesimal difference.
§ 643. The rectified arc stands to a certain function given by the equation of the curve, in the relation of the original function to its derivative.
§ 644. It is the derived function which in integral calculus is given relatively to the original, which has first to be found by integration.
§ 645. It is superficial to say that the integral calculus is simply the converse problem of the differential calculus.
§ 646. Lagrange did not smooth over the difficulties of its problems simply by making those direct assumptions.
§ 647. Lagrange's rectification of curves provides an insight into the translation of the Archimedean method into modern analysis.
§ 648. The idea of the infinitesimal occurs in Archimedes' method, as well as later in Kepler's treatment of stereometric objects.
§ 649. In these methods the affirmative aspect as such which is veiled by the merely negative determination fails to be recognised.
§ 650. Each mode of calculation has as its subject matter a specific determinateness; so too has the differential and integral calculus.
§ 651. Differential and integral calculus has a more particular interest in common with series: to determine the coefficients of the terms.
Remark 3: Further Forms Connected With the Qualitative Determinateness of Magnitude
§ 652. The infinitesimal of the differential calculus is, in its affirmative meaning, the qualitative determinateness of magnitude.
§ 653. From the analytical side, the different power determinations appear in the first place as only formal and quite homogeneous.
§ 654. The sole interest of this procedure is to determine the points and the lines into which the lines and planes have been resolved.
§ 655. It is the need to have recourse to the infinitely small, which constitute the greatest difficulty.
§ 656. It will now be evident how the qualitative element here considered differs from the subject of the previous Remark.
§ 657. The alleged pure summation does in fact include a multiplication.
§ 658. Representing planes as sums of lines is also often employed when multiplication as such is not used to produce the result.
§ 659. The area of a circle bears the same proportion to the area of an ellipse, as the major axis does to the minor axis.
§ 660. The criterion for Cavalieri's method of indivisibles referred to above equally is justified by it.
§ 661. This leads us to reflect on the difference which exists with respect to that feature into which the determinateness of a figure falls.
§ 662. Cavalieri means to distinguish what belongs to the outer existence of the continuous figure from its determinateness.
§ 663. This form of coincidence is a childish aid for sense perception.
§ 664. With parallel lines and with parallelograms there enters another factor.
§ 665. Tacquet asked which line, in the calculation of conical and spherical surfaces, should be taken as the basis.
§ 666. These objections have their origin in the idea of the infinite aggregate of points of which the line is supposed to consist.
§ 667. The affirmative meanings, in the various applications of the infinitely small in mathematics, remain in the background.
§ 668. The introduction of the infinite which is meant to remove the difficulty only serves to aggravate it and prevent its solution.
§ 669. The infinite quantum as the unity of both moments, of the quantitative and qualitative determinateness, is a ratio.
§ 670. In the ratio, quantum is no longer merely indifferent but is qualitatively determined as simply related to its beyond.
§ 671. This relation is itself also a magnitude; the quantum is not only in a ratio, but it is itself posited as a ratio.
§ 672. Ratio as such is direct ratio; indirect or inverse ratio; and in the ratio of powers, becomes measure.
§ 673. It only remains therefore to expound the abstract Notion of these ratios.
§ 674. 1. In the direct ratio the determinateness of either quantum lies reciprocally in the determinateness of the other.
§ 675. 2. The exponent is any quantum; but it is self-related in its own externality and a qualitatively determined quantum.
§ 676. The exponent is this difference as a simple determinateness, i.e. it has immediately within it the significance of both.
§ 677. 3. The two therefore constitute strictly only one quantum.
§ 678. The exponent ought to be the complete quantum, since the determination of both sides coincides in it.
§ 679. The ratio as now before us is the sublated direct relation.
§ 680. In the inverse ratio the exponent as quantum is likewise immediate and is any quantum assumed as fixed.
§ 681. The ratio is so determined that the amount as such is altered relatively to the other side of the ratio, to the unit.
§ 682. 2. We have now to consider more closely this qualitative nature of the inverse ratio, more particularly in its realisation.
§ 683. In conformity with these determinations, each of the two moments has its limit within the exponent.
§ 684. This continuity of each in the other constitutes the moment of unity through which they are in ratio.
§ 685. The exponent is a limit of the sides of its ratio within which they increase and decrease relatively to each other.
§ 686. 3. The outcome of this, however, is the transition of the inverse ratio into a different determination from that which it had at first.
§ 687. The ratio is now specified as the ratio of powers..
§ 688. 1. The quantum has reached the stage of being-for-self.
§ 689. The exponent of this ratio is no longer an immediate quantum as it is in the direct ratio and also in the inverse ratio.
§ 690. 2. The ratio of powers appears at first to be an external alteration to which any quantum can be subjected.
§ 691. The quality of quantum as the posited difference of itself from itself is simply this: to be a ratio.
§ 692. 3. But with the positing of quantum in conformity with its Notion, it has undergone transition into another determination.
§ 693. At first, then, quantity as such appears in opposition to quality.
§ 694. This is the truth of quantum, to be Measure.
Remark
§ 695. The remark to be made here concerns the intrusion of quantitative forms into the pure qualitative forms of thought in philosophy.
§ 696. There is just as much to be said against powers as against all symbolism whatever philosophy.
§ 697. Philosophy needs no help from the world of sense or products of the imagination in its own peculiar province.
§ 698. The use of numbers, the mathematical infinite, and suchlike is nothing more than a convenient means of evading the task of grasping the Notion.
§ 699. Abstractly expressed, in measure quality and quantity are united.
§ 700. Thirdly, we now have self-related externality.
§ 701. Kant did not apply triplicity to the genera of his categories, but only to their species which, alone he called categories.
§ 702. With Spinoza, the mode is likewise the third after substance and attribute.
§ 703. In Indian pantheism, Brahma (abstract thought) progresses through Vishnu, particularly in the form of Krishna, to a third form, Siva.
§ 704. The mode itself is declared to belong essentially to the substantial nature of a thing.
§ 705. Mode has the specific meaning of measure. Spinoza's mode, like the Indian principle of change, is the measureless.
§ 706. Measure in its more developed, more reflected form is necessity.
§ 707. Measure, having realised its own Notion, has passed into essence.
§ 708. At first, measure is only an immediate unity of quality and quantity.
§ 709. Natural science is still far from an insight into the connection between quantities and the organic functions on which they depend.
§ 710. In the realm of spirit there is still less to be found a characteristic, free development of measure.
§ 711. Qualitative quantity in the first place an immediate, specific quantum.
§ 712. 1. Measure is the simple relation of the quantum to itself, its own determinateness within itself; the quantum is thus qualitative.
§ 713. All that exists has a measure.
§ 714. Measure as a standard is a quantum arbitrarily assumed as the intrinsically determinate unit relative to an external amount.
§ 715. Immediate measure is a simple quantitative determination.
§ 716. Quantitative determinateness is twofold: that to which the quality is tied and that which can be varied without affecting the quality.
§ 717. Because quantum is posited as the external limit which is by its nature alterable, and so alteration requires no explanation.
§ 718. 2. The sudden conversion into a change of quality of a change which was apparently merely quantitative attracted the attention of the ancients.
§ 719. Does the pulling out of a single hair from the head produce baldness?
§ 720. The contradiction which results is not a sophism, for such contradiction is not a sham or a deception.
§ 721. What is refuted is the error of one-sidedly holding fast to the abstract determinateness of quantum.
§ 722. Quantum, as an indifferent limit, is the aspect of an existence which leaves it open to unsuspected attack and destruction.
§ 723. 3. Measure in its immediacy is an ordinary quality with a specific magnitude attaching to it.
§ 724. This is first a rule, a measure which is external with reference to mere quantum.
§ 725. Comparison is an external act, the unit itself being an arbitrary magnitude.
§ 726. Measure is a specific determining of the external.
§ 727. Something, in so far as it is a measure within itself, has the magnitude of its quality altered from outside itself.
§ 728. Intensive, and extensive quantum is the same quantum , once in the form of intensity and again in the form of extension.
§ 729. The strictly immanent qualitative form of the quantum is solely its determination as a power.
Remark
§ 730. Remperature is a quality in which these two sides of external and specified quantum are distinguished.
§ 731. 1. The qualitative, intrinsically determinate side of the quantum exists only as a relation to the externally quantitative side.
§ 732. 2. In measure there enters the essential determination of variable magnitude, for measure is quantum as sublated.
§ 733. Magnitude, simply as magnitude, is alterable.
§ 734. The two sides thus related have, in keeping with their abstract aspect as qualities generally, some particular significance.
§ 735. The direct ratio is reduced to the merely formal determination which has no existence except as an intellectual abstraction.
Remark
§ 736. These kinds of motion, no less than their laws, rest on the development of the Notion of their moments, of space and time.
§ 737. It is a great service to ascertain the empirical numbers of nature, but an infinitely greater service to make them moments of a law.
§ 738. 1. In the form of specified measure, the quantitative element of both sides is qualitatively determined.
§ 739. The immediate qualities also belong to measure.
§ 740. 2. It is still with reference to the specific measure an externally given quantum.
§ 741. 3. Measure has now acquired the character of a specified quantitative relation which, as qualitative, has in it the ordinary external quantum.
§ 742. Measure is now determined as a correlation of measures which constitute the quality of distinct self-subsistent somethings — or things.
§ 743. Self-exclusive and self-subsistent measure are one with each other, and self-subsistent measure enters into a negative relation with itself.
§ 744. Measures are no longer merely immediate but self-subsistent, because they have become relations of measures which are themselves specified.
§ 745. Something is immanently determined as a measure relation of quanta which also possess qualities.
§ 746. The exponent is the specific quantum of the something.
§ 747. As purely quantitatively determined, the compound would be a mere addition of the two magnitudes.
§ 748. Not only is one of the qualitative sides posited as alterable but measure itself.
§ 749. (1) If two things forming a compound body owed their natures to a simple qualitative determination, they would destroy each other when combined.
§ 750. (2) This combination with a number of others which are likewise measures within themselves, yields different ratios.
§ 751. It seems that a self-subsistent measure which forms a series of exponents with a series of such measures, is distinguished from another measure.
§ 752. Those measures which yield a series of exponents of the ratios between the members of that series, are in themselves self-subsistent measures.
§ 753. 3. In this form there is a return to the particular way in which quantum is posited as self-determined, i.e., as degree.
§ 754. The individual note is the key of a system, but again it is equally an individual member in the system of every other key.
§ 755. In elective affinity as an exclusive, qualitative correlation, the relationship is rid of this quantitative difference.
§ 756. Neutralisation is not only the form of intensity; the exponent is essentially a measure determination and therefore exclusive.
Remark: Berthollet on Chemical Affinity and Berzelius's Theory of it
§ 757. Chemical substances are the most characteristic examples of measures which are characterised solely by their relationship to other measures.
§ 758. The law of the chemical affinities of acids and alkalis states that if two neutral solutions are mixed resulting the products are neutral.
§ 759. Berthollet modified the general conception of elective affinity by the concept of the activity of a chemical mass.
§ 760. Berzelius in his Textbook of Chemistry.
§ 761. Berzelius, himself makes use of the conception of degrees of affinity.
§ 762. Affinity is reduced to a quantitative difference.
§ 763. Chemical affinity has been distinguished from elective affinity.
§ 764. If the experimental method has been the guiding star in the theory of proportions, then mixing with the corpuscular theory, a desert lying away from the path of experience, forms all the greater contrast with it.
§ 765. Every chemical action is at bottom an electrical phenomenon.
§ 766. It is just this kind of metaphysics which is proclaimed and echoed too with the greatest pretension.
§ 767. The problem would be to recognize the exponents of the ratios of the series of specific gravities as a system based on a rule.
§ 768. When substances are combined, there occurs also a neutralisation of the specific gravities.
§ 769. The last determination of the measure relation was that being specific it is exclusive.
§ 770. The relation to itself of the measure relation is distinct from its externality which represents its quantitative aspect.
§ 771. They form in this way a nodal line of measures on a scale of more and less.
§ 772. Here we have a measure relation, a self-subsistent reality which is qualitatively distinguished from others.
§ 773. People fondly try to make an alteration comprehensible by means of the gradualness of the transition.
Remark: Examples of Such Nodal Lines; the Maxim, ‘Nature Does Not Make Leaps’
§ 774. The system of natural numbers already shows a nodal line of qualitative moments which emerge in a merely external succession.
§ 775. Qualitative nodes and leaps occur in chemical combinations when the mixture proportions are progressively altered.
§ 776. It is said, natura non facit saltum [there are no leaps in nature].
§ 777. In thinking about the coming-to-be of something, it is assumed that what comes to be is already sensibly or actually in existence.
§ 778. With the expansion of the state and an increased number of citizens, the laws and the constitution acquire a different significance.
§ 779. Magnitude is that side of determinate being through which it can be caught up in a seemingly harmless entanglement which can destroy it.
§ 780. The abstract measureless is the quantum as such which lacks an inner significance.
§ 781. This transition of the qualitative and the quantitative into each other proceeds on the basis of their unity.
§ 782. The alteration is only change of a state, and the subject of the transition is posited as remaining the same in the process.
§ 783. Measure is, in the first instance, only the immediate unity of quality and quantity as an ordinary quantum which is, however, specific.
§ 784. This process is equally the progressive determination of measure in its realisation and the reduction of measure to the status of a moment.
§ 785. Being is the abstract equivalence in which there is supposed to be as yet no determinateness of any kind.
§ 786. what has thus been determined as qualitative and external is only a vanishing determinateness.
§ 787. This determination of indifference is posited within the indifference itself and how the latter is therewith posited as being for itself.
§ 788. 1. The reduction of measure relations which at first ranked as self-subsistent measures, establishes their common substrate.
§ 789. At first it is essentially the merely quantitative external difference which is present in it.
§ 790. The difference is present, further, as two qualities, one of which is sublated by the other.
§ 791. each side is in its own self an inverted relation. As formal, this relation recurs in the two distinct sides.
§ 792. 2. As this indifference, being is now the specification of measure no longer in its immediacy, but measure as developed.
§ 793. The determinations come into immediate opposition and this develops itself into a contradiction.
§ 794. 3. Each quality enters within each side into relation to the other.
§ 795. There can be no question of a quantitative difference or of a more of the one quality.
§ 796. This unity has for its result not the unity which is merely indifferent, but that immanently negative and absolute unity called essence.
Remark: Centripetal and Centrifugal Force
§ 797. The relationship of quantitative difference of two factors determined qualitatively, is applied to the elliptical motion of the celestial bodies.
§ 798. All that can properly be required of a theory has been accomplished; but to reflective understanding this did not appear sufficient.
§ 799. It is evident that it would be an alien force which effected this reversal.
§ 800. The same has been applied to the forces of attraction and repulsion to explain the different densities of bodies.
§ 801. With Spinoza, attributes, thought and extension, then the modes too, the affections are introduced quite empirically.
§ 802. It is the dissolution of measure, in which both moments were directly posited as one.
§ 803. This reflection of differences into their unity is not the product of the subjective thinker, but it is their very nature to sublate themselves.
§ 804. The process is not a transition, nor an external alteration, but its own self-relating which is the negativity of itself.
§ 805. The determinations are no longer simply affirmative as in the sphere of being, but a sheer positedness.
§ 806. Implicit being has vanished and its unity is this simple self-relation only as a result of the sublating of this presupposition.
§ 807. The truth of being is essence.
§ 808. When this movement is pictured as the path of knowing, then this beginning, appears to be an activity of knowing external to being.
§ 809. But this path is the movement of being itself.
§ 810. Essence is in this way only a product, an art