Thursday, November 30, 2006

Shankara's Introduction to his Commentary on Gita

On Gita Jayanthi, by some strange coincidence, I happened to start the English translation of the Srimad Bhagavad Gita by Swami Nikhilananda. The learned Swami -- from an institution which I immensely respect -- has done an excellent English translation of the commentary of Shankara. Shankara, when he wanted to start, some can say reinstate, the philosophy of Advaita in the pavithra bhoomi (sacred land) of India, wrote a commentary on the Bhagavad Gita. This was a part of his commentary on each of the text of prasthana-traya, the other two being The Upanishads and The Brahmasutras. The Bhagavad Gita is a part of Mahabharata and technically should be considered a smriti (remembered) text. It is however considered a sruti (revelatory) text due to its source (Lord Krishna, an avatar) and the influence upon Indians of generations.

The Bhagavad Gita, being such a great source of daily-inspiration for millons of Indians spanning across centuries, has been called by some scholars as a book that is (1) not amenable to Advaitic interpretation and (2) has many inconsistent thoughts. The scholars -- including the great Sarvepalli Radhakrishnan himself -- had to go through great pains in writing translations of Gita. They had to use classifications like (1) the first six chapters (called karma-shatkam) talk about the concept of renunciation of the deeds of karma as a method of liberation, (2) the next six chapters (called bhakti-shatkam) talk about the love of the personal God as a method of liberation and (3) the last six chapters (called gyana-shatkam) talk about the way of knowledge as a method of liberation.

Another great scholar, Eliot Deutsch, the learned scholar who has written the books exposing the philosophical content on Advaita and source books on Advaita, himself had to use the terms "progressive teaching of Gita" for explaining the "inconsistency" of the Gita.

All this may confuse a spiritual-student -- including the author of this post -- to mistakenly conclude about the message of the Gita. This is particularly true when: (1) if the student is mature enough to search for message in the Gita, but not mature enough -- as the author of the post was -- to understand the message that was clear (hind sight is always 20/20). (2) Also, in students who have a reasonable amount of maturity and thirst for knowledge, the words "The gita is not considered an Advaitic Text" can lead one away from Gita, when it is known that Advaita is the crux of indian philosophical systems. If such a student searches for a message in Upanishads, he is bound to be more confused, as the Upanishads are too experiences of seers. The Upanishads themselves being experiences of different seers in different times and situations would surely confuse any such student.

The way out of that confusion is, as it always should have been, the well known axiom: "go to the source". Shankara, being a brilliant philosopher himself does not have an inch of confusion and dispels all confusions from any such students hearts. The commentary of Shankara on Gita, nay Shankara's introduction itself to the Gita itself, is enough to dispel any such doubts on any spiritual practitioner. Before beginning such a reading, let us begin an old prayer that explains the significance of each Gita in the context of Upanishads:

The Upanishads are as a herd of cows; Krishna the Son of a cowherd, is their Milker. Arjuna is the calf, the supreme ambrosia of the Gita the milk, and the wise man the drinker.

Here is the introduction:

Of the two kinds of dharma dealt with in the Vedas: the one characterized by activity and the other by renunciation. This twofold Dharma, the cause of the stability of the world order and also the direct means by which men attain prosperity and the Highest Good [Liberation], was followed by members of the different castes -- the brahmin, kshatriya, and the rest -- and of the different dharmas, desirous to secure their welfare.

People parctised the Vedic dharma for a long time. Then lust arose among them; discrimination and wisdom declined. Unrigheousnedd began to outweigh righteousness. Thus, when unrighteousness prevailed ine world, Vishnu [the all pervading one], the First Creator, also known as Narayana, wishing to ensure the continuance of the universe, incarnated Himself, in part, as Krishna. He was born to Devaki and Vasudeva for the protection of the brahmins on earth and their spiritual ideal. By the protection of the brahmin ideal, the dharma of the Vedas is preserved, since all different castes and ashramas are under its control.

The Lord, the eternal Possessor of Knowledge, Soveignty, Power, Strength, Energy, and Vigour, brings under His control maya -- belonging to Him as Vishnu -- the primordial Nature, characterized by the three gunas. And then, through the maya, He is seen as though born, as though endowded with a body, and as though showing compassion for men; for He is, in reality, unborn, unchanging, the Lord of all created beings, and by nature eternal, pure, illuminated, and free.

Though the Lord had nor purpose of His own to serve, yet, with the sole desire of bestowing favour on men, He taught this twofold Vedic dharma to Arjuna, who was deeply sunk in the ocean of grief and delusion; for a dharma spreads and grows when accepted by high-minded persons.

It is this dharma taught by the Lord that the omniscient and venerable Vyasa, the compiler of Vedas, embodied in seven hundred verses under the name of the Gita.

This scripture, the Gita, is a compendium of the essential teachings of the whole of the Vedas; its meaning is extremely difficult to grasp. Many commentators desiring to present a clear idea of that meaning, have explained the words, and the meaning of the words of the sentenses, and also the arguments. But, I find that, to the people of ordinary understanding, these explanations convey diverse and contradictory meanings. Therefore, I intend to write a brief commentary on the Gita, with a view to determining precisely what it signifies.

The ultimate aim of the Gita is, in a word, the attainment of the Highest Good, characterized by the complete cessation of relative existence and its cause. This is realized by means of that dharma whose essence is devotion to Self-knowledge attained through the renunciation of all action. With reference to this dharma laid down in the Gita, the Lord says in the Anugita:

"That dharma is quite sufficient for the attainment of Brahman." (Mahabharatha Chapter on Ahsvamedha, xvi 12)
In the same treatise it is said:
"He who is righetousness and without unrighteousness -- he who is absorbed in one Goal, silent and without thinking."
"Knowledge is characterized by renunciation."

In the concluding chapter of the Bhagavad Gita, Sri Krishna says to Arjuna: "Abandon all dharmas and come to Me alone for shelter." (XVIII 66)

The dharma characterized by activity and prescribed for the different castes and ashramas is, no doubt, a means of securing worldly welfare and also of attaining the regions of the gods; but when it is practised in a spirit of self-surrender to the Lord, and without desire for fruit, it leads to the purification of the mind. A man of pure mind becomes fit to acquire devotion to the path of knowledge and attains Knowledge. Thus by means of the dharma of activity, one ultmately realizes the Highest Good. With this view in mind the Lord says in the Gita: "He who works without attachment, resigning his actions to Brahman." (V. 10) "The yogis act, without attachment, for the purification of the heart." (V. 11)

The purpose of the two fold dharma described in the Gita is the attainment of the Highest Good. The subject-matter is the Supreme-Reality known as Vasudeva, the Ultimate Brahman. It expounds both in a specific manner. Thus the Gita treats of a specific subject, with a specific end in view, and there is a specific relation between the subject-matter and the object.

Knowledge of the Gita enables one to attain the goal of all human aspiration. Hence my attempt to explain its teachings.

May we all mature enough to understand the real message in Gita.
Om Tat Sat! Read the rest of this entry >>

Wednesday, November 15, 2006

Link to FAQ in Polyhedral Computation by Fukuda

Here is a link to the extremely useful and well written FAQ in polyhedral computation by Fukuda. Here is a PDF version.

See the sections on face-lattice, polarity aka. duality and Minkowski-Weyl. The number of facets of a d-dimensional, n-vertex polytope grows linearly with n. However, its slope is so high that it grows intractable within no time (fascinating!).

Note: See the sub-sections where many problems, simple and hard, are discussed.

11/25: Probably the polylib-page on Polyhedra , with material taken from Schrijver's book is useful for definitions. Read the rest of this entry >>

Monday, November 06, 2006

LibraryThing: fun stuff with books

I recently became a member of LibraryThing, which allows people to keep an online catalogue of books. It has a simple interface to add books (with a search engine backed up my Amazon power!) and allows one to see some fun stats. Here are some: My author gallery, my author cloud and my tag cloud. Read the rest of this entry >>

Sunday, November 05, 2006

Matrix Cook Book and parody of P = / != NP.

The Matrix Cook Book looks very useful and has ben added to the quick links. It was a quick link in Suresh's Geomblog. The Geomblog also has a parody of a typical P=/!=P conversaton in comp.theory. Thanks for both Suresh! Read the rest of this entry >>

Saturday, November 04, 2006

Ramakrishna Mission and TTD

Being an admirer of expositions of Vedanta by great men, many of whom happen to be from Ramakrishna Mission, it fills up by heart with immense joy when I see the following on the backcover of a translation of Upanishads book:

The words say "Tirumala Tirupathi Devasthanam is supporting Ramakrishna Mission in bringing these books at low prices." Tirupathi Venkateshwara supporting Ramakrishna Mission is the manifestation of the statement from Bhagavad Gita: "Dharma Samsthapanarthaya Sambhavaami Yuge Yuge", where Naarayana supports Dharma and the Vedic truths in many forms.

Some of the books on Vedanta I have read by people of the great order:

  1. Upanishads and Bhagavad Gita by Swami Prabhavananda
  2. Self-Knowledge etc. Swami Nikhilananda
  3. Upanishads by Swami Sarvananda (both English and Telugu)
  4. Lectures on Mandukya by Swami Ranganathananda
  5. The Upanishads by Swami Gambhirananda

I have an intuitive feeling that, in the last century, Ramakrishna Mission has atleast matched the peethams set up by Shri Adi Shankara in propagating Vedanta. Read the rest of this entry >>

Thursday, November 02, 2006

Convex Polytopes and notes from Develin's works

Mike Develin is a Mathematician who has written a compendium (PDF link) to accompany Ziegler's wonderful book. Also, the first two chapters of his thesis (PDF link) are a great read for anyone interested in convex polyhedra. Some notes:

On homogenization and equivalence:

Given a polytope P\subset \mathbb{R}^d, we can form a cone associated to it by considering the cone of all points {(1,v)} where v \in V. Two crucial points: (1) The shape of the polytope can be recovered by intersecting this cone with the hyperplane x = 1. (2) You might notice that the shape of the polytope we obtain is dependent on the orientation of the polyhedral cone in R^d. This leads to the very important concept of projective equivalence. Two polytopes are defined to be projectively equivalent if they can be obtained as cross-sections of the same polyhedral cone one dimension higher; this notion of equivalance is stronger than the notion of combinatorial equivalence, where two polytopes are equivalent if their faces have same combinatorial structure, and weaker than the notion of affine equivalence, which relates polytopes which are affinely isomorphic to each other.

On polarity:

the face-lattice L(P) is just a partially ordered set, or poset with elements being the faces of the polytope and F < G if F \subseteq G.
One key property of polytopes is that the intersection of any two faces is itself a face, which corresponds to the fact that any two elements of the poset have a unique maximal lower bound. The aforementioned notion of combinatorial equivalence corresponds to two polytopes having the same face lattice.

With face lattice, it is easy to give a combinatorial description of the polar polytope P^\Delta. The polar polytope realizes the full power of the duality between the two formulations of polytopes, in terms of vertices and in terms of inequalities. Assuming that P is full-dimensional (embedded in R^d, where d is the dimension of P), P^\Delta is the object in the dual space V^* consisting of those linear functionals f for which f(x) \le 1 holds everywhere on P. To do this, we need to pick the position of the origin inside P, but once we have done this, the entire combinatorial and indeed projective type of P^\Delta is determined. Furthermore, the face-lattice of P^\Delta is precisely the opposite poset of L(P).

... the Farkas lemma implies that the polar polytope is the convex hull of the facet-defining functionals, which provides a natural correspondence between the vertices of P^\Delta and the facets of P. In fact, the lattices are isomorphic under this correspondence; the face lattice is completely determined by which subsets of {1,2,...,n} are facets, so this, along with the fact that the vertices of P correspond to the facets of P (as constraint f(x) <= 1 on P is equivalent to the intersection of half-spaces f(v) <= 1 for v \in V) exhibits explicit correspondence.

Thanks Mike!

Postscript (11/05): The chapter "Basic Properties of Convex Polytopes" is available via the following link: thanks to Prof. Ziegler). The chapter is from the book: Handbook of Discrete and Computational Geometry with the following table of contents. Read the rest of this entry >>