Einstein Algebra and General Theory of Relativity Preserve the Empirical Structure of the Theories

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In general relativity, we represent possible universes using relativistic spacetimes, which are Lorentzian manifolds (M, g), where M is a smooth four dimensional manifold, and g is a smooth Lorentzian metric. An isometry between spacetimes (M,g) and (M,g′) is a smooth map φ : M → M′ such that φ ∗ (g′) = g, where φ∗ is the pullback along φ. We do not require isometries to be diffeomorphisms, so these are not necessarily isomorphisms, i.e., they may not be invertible. Two spacetimes (M,g), (M′,g′) are isomorphic, if there is an invertible isometry between them, i.e., if there exists a diffeomorphism φ : M → M′ that is also an isometry. We then say the spacetimes are isometric.

The use of category theoretic tools to examine relationships between theories is motivated by a simple observation: The class of models of a physical theory often has the structure of a category. In what follows, we will represent general relativity with the category GR, whose objects are relativistic spacetimes (M,g) and whose arrows are isometries between spacetimes.

According to the criterion for theoretical equivalence that we will consider, two theories are equivalent if their categories of models are “isomorphic” in an appropriate sense. In order to describe this sense, we need some basic notions from category theory. Two (covariant) functors F : C → D and G : C → D are naturally isomorphic if there is a family ηc : Fc → Gc of isomorphisms of D indexed by the objects c of C that satisfies ηc ◦ Ff = Gf ◦ ηc for every arrow f : c → c′ in C. The family of maps η is called a natural isomorphism and denoted η : F ⇒ G. The existence of a natural isomorphism between two functors captures a sense in which the functors are themselves “isomorphic” to one another as maps between categories. Categories C and D are dual if there are contravariant functors F : C → D and G : D → C such that GF is naturally isomorphic to the identity functor 1C and FG is naturally isomorphic to the identity functor 1D. Roughly speaking, F and G give a duality, or contravariant equivalence, between two categories if they are contravariant isomorphisms in the category of categories up to isomorphism in the category of functors. One can think of dual categories as “mirror images” of one another, in the sense that the two categories differ only in that the directions of their arrows are systematically reversed.

For the purposes of capturing the relationship between general relativity and the theory of Einstein algebras, we will appeal to the following standard of equivalence.

Theories T1 and T2 are equivalent if the category of models of T1 is dual to the category of models of T2.

Equivalence differs from duality only in that the two functors realizing an equivalence are covariant, rather than contravariant. When T1 and T2 are equivalent in either sense, there is a way to “translate” (or perhaps better, “transform”) models of T1 into models of T2, and vice versa. These transformations take objects of one category – models of one theory—to objects of the other in a way that preserves all of the structure of the arrows between objects, including, for instance, the group structure of the automorphisms of each object, the inclusion relations of “sub-objects”, and so on. These transformations are guaranteed to be inverses to one another “up to isomorphism,” in the sense that if one begins with an object of one category, maps using a functor realizing (half) an equivalence or duality to the corresponding object of the other category, and then maps back with the appropriate corresponding functor, the object one ends up with is isomorphic to the object with which one began. In the case of the theory of Einstein algebras and general relativity, there is also a precise sense in which they preserve the empirical structure of the theories.

Odyssey. Note Quote.

In tracing an interpretation of the Odyssey it is not necessary to follow the order of the poem as arranged by Homer or by whoever compiled the Homeric legends; it is easier and more rewarding to take the simple narrative in the natural sequence of events. After leaving the battlefield of Troy, Odysseus embarks for his native isle, “Ithaca the Fair,” expecting to arrive there quickly; but at the very outset a tempest drives the fleet off its course, and a great fight impedes his progress. The destruction of all his ships but one, and of many of the sailors, follows quickly. One of the strangest incidents in this introductory part is the encounter with Polyphemus, the Cyclopean giant with a single eye in the middle of his forehead. Before we shrug him off as a creature of early man’s distorted imagination, we should hold in mind the possibility of a symbolic meaning. Creatures of a similar type were mentioned by many archaic peoples far removed from each other; possibly they referred to some past event in human history, now forgotten.

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After their escape from this one-eyed monster and some further perilous adventures, Odysseus and his companions soon reach the island of the enchantress Circe, who represents the fascination of sensual delights. Odysseus is unaffected by the gross enticements which overwhelm his fellows, now turned into swine by the goddess; and his boldness and “confidence in heaven” finally conquer the temptress and compel her to serve him. She restores the men to human form and instructs Odysseus how to find the way to the Underworld.

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The entrance into Hell or the Underworld, the “open tomb,” has more than one meaning in ancient allegories, and is always introduced in some form in myths of initiation; Orpheus, Aeneas and many others had to make the dread “descent.” In the legend of Perseus and Andromeda, the hero aided by the gods flies to the hideous regions of cold and darkness and destroys the deadly Medusa before he can rescue the princess of Ethiopia from the monster. For Odysseus the event is an ordeal of terror. Circe has warned him that before he may go further, he must gain information about the future from Tiresias, a venerable prophet, who lives with the inhabitants of Hades, though he himself is not dead. The approach to the entire experience is surrounded by fearful dangers and to pass safely through the multitudes of vengeful shades calls for the highest physical and moral courage. Like all heroes of epics of the soul, he has to traverse the Valley of the Shadow of Death: to face the shades, the lingering remains of past sins and errors; then to learn what is necessary for his further progress.

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The tone of the poem changes at this point; the lightness and gaiety with which Odysseus has related his adventures is replaced by a deep solemnity, and the scenes of Hades are described with intense vividness and many touches of realism. Are these portrayals actual revelations of postmortem life? Leaving the more impure regions, Odysseus moves on, sees stern Minos, the Judge of the Dead, and even gets a passing view of the heavenly world or the Elysian fields, where the higher and immortal parts of man are held to exist between incarnations on earth. (Plato and Plutarch give valuable insights into the Greek teachings on this mysterious subject which are found to be practically identical with the Egyptian, Indian, and other ancient views of these after-death states.)

At length, having interviewed the sage Tiresias, Odysseus returns to Circe who outlines the perils still lying ahead on his homeward journey. Then come the hazardous Straits between Scylla and Charybdis, and the subtle allurements of the Sirens. Exquisitely fair, they offer him the satisfaction of the pride of knowledge, telling him they know “Whate’er beneath the sun’s bright journey lies,” and singing with all the charm of celestial music:

Blest is the man ordain’d our voice to hear,
The song instructs the soul, and charms the ear.
Approach! thy soul shall into raptures rise!
Approach! and learn new wisdom from the wise.

Having passed successfully through the trials of the Underworld, will he be overcome by pride and rash self-confidence? Knowing well the overwhelming power of this temptation, the hero takes every precaution, has himself tied to the mast and stops the ears of his crew with wax against the Sirens’ songs. They steer safely through the Straits — only to plunge again into difficulty when his men, to satisfy their gluttony, kill and devour Apollo’s sacred oxen. This so arouses the wrath of the god that he sends a great tempest to destroy the last of Odysseus’ crew, and the brave man is left with nothing but his own strength and the favor of Athena, his guide.

In his desperation and loneliness he meets with a temptation that almost proves his undoing. He succumbs for seven years to the blandishments of the lovely nymph Calypso in her enchanted Atlantean island. Calypso even offers him “immortal life, exempt from age or woe.” But with the help of Athena, the personification of divine wisdom, he summons the strength to resist.

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This is one of the passages in the Odyssey that reveals the high understanding of the poet and the profound quality of his teaching. For here is shown the wide gulf between any artificial prolongation of the life of the personality with its selfish cravings and that genuine immortality born of steadfast aspiration and self-control which leads to union with one’s inner god. Such a philosophy nourishes the roots of our being and reminds us of the words of the Nazarene:

If any man will come after me, let him deny himself, and take up his cross, and follow me. For whosoever will save his life shall lose it: and whosoever will lose his life for my sake shall find it. –– Matthew, 16:24-5

Paul, the wise master builder, in common with the great teachers of antiquity, refers to the same principle when he speaks of being changed “in the twinkling of an eye.” This is a cryptic saying suggesting the spontaneous springing into activity of the power of intuition which sees the difference between a nobler life and the delusions of sensual gratification.

When Odysseus makes his decision, the irresistible power of the Olympian deities is exerted in his favor, Calypso abandons her inducements and, like Circe, is transformed from a temptress into a helper. Odysseus builds a new vessel with his own hands and sets out joyfully for home, a voyage still not without its risks. Upon his arrival, he discovers the terrible conditions to which his wife and son have been reduced by the outrageous conduct of her admirers and soon perceives that his greatest battle is yet to come. His wife, Penelope, who stands for the climax of his endeavors, his goal, does not immediately throw herself into his arms. Ragged, worn, and disguised as an old man, he is not easily recognized by her, though his aged nurse and faithful dog know him quickly. Even when Athena restores him to the prime of life, and to greater dignity and beauty than before, he has to prove his identity to Penelope before she will accept him. This hesitation on her part is not, as some have thought, a blemish on the story; it could not be otherwise. It is traditional that anyone wishing recognition by the higher self must make a clear demand; he must unmistakably recognize and call upon his inner god before it can help him.

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Ask, and it shall be given you; seek, and ye shall find; knock, and it shall be opened unto you (Matthew., 7:7).

Odysseus’ supreme opportunity comes when he finds his palace invaded and his wife surrounded by a mob of suitors, all trying to persuade her that he is surely dead and that she should choose a second husband from among them. Repugnant as they are, they have no power over Odysseus, but he must destroy them before he can regain his rightful place. They represent lingering traces of lower desires which must be slain forever if he will be master in his own household. At last the battle is won, the evil forces overpowered, and Odysseus, calm, purified, asserts his noble identity to Penelope and is joyously received by her.

From a practical point of view, the scene of this last struggle and the method adopted by Odysseus in challenging the suitors may appear singular, but there is good warrant for these in the mystical symbolism obviously familiar to Homer. The contest takes place at close quarters, in the confined space of the palace hall, yet the hero has to depend upon his mighty bow for success — the bow that none other can wield — instead of the more logical weapons of sword or spear. The bow is the weapon of Apollo, god of light, and the day of Odysseus’ victory is sacred to that deity. In Hindu philosophy also, the bow, or in some cases the arrow, stands for man himself who must be strong enough in texture to stand the strain. In one of the Upanishads, it says:

Having taken the bow, the great weapon, let him place on it the arrow, sharpened by devotion. Then, having drawn it with a thought directed to That which is, hit the mark, O friend — the Indestructible. . . . It is to be hit by a man who is not thoughtless; and then, as the arrow becomes one with the target, he will become one with Brahman. — Mundaka,II ii, 3-4

The Odyssey closes with the hero, now triumphant as the rightful king and leader, going forth and subduing the few remaining rebels after which, the poet says, the “willing nations knew their lawful lord.” His future reign is left to the imagination, but it is secure in peace and wisdom for, having conquered the enemies in his own house, he cannot fail.

Paradox of Phallocentrism. Thought of the Day 34.0

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The paradox of phallocentrism in aIl its manifestations is that it depends on the image of the castrated woman to give order and meaning to its world. An idea of woman stands as lynch pin to the system: it is her lack that produces the phallus as a symbolic presence, it is her desire to make good the lack that the phallus signifies. The function of woman in forming the patriarchal unconscious is two-fold. She first symbolises the castration threat by her real absence of a penis, and second thereby raises her child into the symbolic. Once this has been achieved, her meaning in the process is at an end, it does not last into the world of law and language except as a memory which oscillates between memory of maternal plenitude and memory of lack. Both are posited on nature (or on anatomy in Freud’s famous phrase). Woman’s desire is subjected to her image as bearer of the bleeding wound, she can exist only in relation to castration and cannot transcend it. She turns her child into the signifier of her own desire to possess a penis (the condition, she imagines, of entry into the symbolic). Either she must gracefully give way to the word, the Name of the Father and the Law, or else struggle to keep her child down with her in the half-light of the imaginary. Woman then stands in patriarchal culture as signifier for the male other, bound by a symbolic order in which man can live out his phantasies and obsessions through linguistic command by imposing them on the silent image of woman still tied to her place as bearer of meaning, not maker of meaning.

Triangulations on Linguistic Phenomenology: Austin, Seattle and Derrida

Even if the main motive behind How to do Things With Words is a description of total speech situations, Austin admits to his work as linguistic phenomenology in order to investigate the uses of everyday linguistic practices in situations where for some reasons, these practices could prove defective. This stance only opens up Austin’s position to re-description of total speech situations if any implausibility sets in.

A speech act could be infelicitous, happy or unhappy, and Austin attaches a parasitic tag to the speech act, when it can mean either non-serious usage falling outside the proper context of linguistic use, or an abnormal placement within the context of linguistic use. Searle wholeheartedly agrees with the parasitic tag when it is employed within language in a facetious manner. Parasitic feeding upon the speech acts could either be pretentious or metaphorical, yet real illocutionary acts. In short, a status of dependency is accorded to such parasitic acts. If one looks at the fictional discourse, it is the reference that becomes parasitic, for even if the failure of the reference happens to be error prone, in case of pretentious speech acts, this error is simply replaced by a parasitic reference. If for Austin, non-serious utterances infect the speech acts, thus impacting its normal use on the way, and thereby falling under the doctrine of etiolations (to cause, make pale) of language, then this clearly entails Austin’s faith in the theory of performative utterances that include non-serious, abnormal utterances that impact the normal usage of speech act. Now in order to discover any nexus between these non-serious, abnormal discursive practices with infelicities, even if abnormalities are culpable of misinvoking conventional references and commands/orders, one has to fall back onto Searle’s invocation of intentionality of the speaker to sort out the said connection. The problem, however, in Austin lies in the derogatory sense attached to these abnormalities with the use of the word ‘infection’, thus aligning these pejorative senses even to the parasitic instances of speech acts. Searle breaks free from this pejorative sense, and even argued that Austin himself never meant any derogatory sense for ‘parasitic’ manner.

Derrida criticizes this positionality of Austin’s parasitic discourses, while Searle comes to Austin’s defense, in turn opening up the gates for Derrida criticizing Searle’s more refined theory of parasitic discourse. Derrida’s criticism of Austin rests on the former’s use of iterability, dissemination and citationality. Even if utterances are repeatable, they carry seeds of alterations, and hence iterability. Derrida’s analysis of speech and writing clearly exhibits that utterances are irreducibly polysemic thus underlining any particular/singular/univocal meaning attached with them, and hence dissemination. Citationality refers to every utterance being called upon during repetition to unveil sameness and difference with the previous such utterance thus throwing open the field for further modification. Derrida’s reading of Austin takes on two implications, a principle of citationality in Austin, and an impossibility of determining the performative act that is either normal or parasitic. In other words, Derrida criticizes the notion of felicity conditions, even if he holds true beliefs towards the potency of illocutionary acts to enable language undergo transformations. He never believes in the success of a performative utterances as rule-based upon conventions.

The real problem begins with Searle’s understanding of Derrida, which is nothing short of poor interpretation. Searle’s type/token distinction gets muddled up in Derrida’s iterability. For the former, when one says that an element in linguistics is iterable, is just to say that logicians’ type-token distinction must apply to all rule-governed elements of language in order that the rules can be applied to new occurrences of phenomenon specified by the rules. Without this feature of iterability, there could not be the possibility of producing an infinite number of sentences with a finite list of elements. A cursory reading is enough to show the paucity of this logic.

Non-self Self

Philosophy is the survey of all the sciences with the special object of their harmony and of their completion. It brings to this task not only the evidence of the separate sciences but also its special appeal to the concrete experience – Whitehead

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Vidya and Avidya, the Self and the not-Self, as well as sambhūti and asambhūti, Brahman and the world, are basically one, not two. Avidya affirms the world, as a self-sufficient reality. Vidya affirms God as the Other, as a far away reality. When true knowledge arises, says the Upanishads, this opposition is overcome.

The true knowledge involves comprehension of the total Reality, of the truth of both Being and Becoming. Philosophic knowledge or vision cannot be complete if it ignores or neglects any aspect of knowledge or experience. Philosophy is the synthesis of all knowledge and experience, according to the Upanishads and according also to modern thought. Brahmavidya, philosophy, is sarvavidyapratishthā, the basis and support of all knowledge, says the Mundaka Upanishad. All knowledge, according to that Upanishad, can be divided in to two distinct categories – the apara, the lower, and the para, the higher. It boldly relegates all sciences, arts, theologies, and holy scriptures of religions, including the Vedas, to the apara category. And that is para it says, yayā tadaksharam adhigamyate, by which the imperishable Reality is realized.’

The vision of the Totality therefore must include the vision of the para and the apara aspects of Reality. If brahmavidya, philosophy, is the pratisthā, support, of sarvavidyā, totality of knowledge, it must be a synthesis of both the aparā and the parā forms of knowledge.

This is endorsed by the Gita in its statement that the jnana, philosophy, is the synthesis of the knowledge of the not-Self and the Self:

क्षेत्रक्षेत्रज्ञयोर्ज्ञानं यत्तज्ज्ञानं मतं मम ।

kṣetrakṣetrajñayorjñānaṃ yattajjñānaṃ mataṃ mama |

The synthesis of the knowledge of the not-Self, avidya, which is positive science, with that of the Self, vidya, which is the science of religion, will give us true philosophy, which is the knowledge flowering in to vision and maturing into wisdom.

This is purnajñāna, fullness of knowledge, as termed by Ramakrishna. The Gita speaks of this as jñānam vijñāna sahitamjñāna coupled with vijñāna, and proclaims this as the summit of spiritual achievement:

बहूनां जन्मनामन्ते ज्ञानवान्मां प्रपद्यते ।
वासुदेवः सर्वमिति स महात्मा सुदुर्लभः ॥

bahūnāṃ janmanāmante jñānavānmāṃ prapadyate |
vāsudevaḥ sarvamiti sa mahātmā sudurlabhaḥ ||

‘At the end of many births, the wise man attains Me with the realization that all this (universe) is Vasudeva the indwelling Self); such a great-souled one is rare to come across’