Sunday, May 26, 2019

Aspects of nibbāna from a lecture by Bhikkhu Bodhi (1979)

[31 May 2019] Updated to try to clarify opening sentences.
[6 July 2019] Added an early version of the fable about the fish and the turtle.

In my exploration of intuition, I became fascinated by its role in Srinivasa Ramanujan's work in mathematics; he developed it in his spiritual practice with utmost commitment, which was rewarded by extraordinary discoveries.  Accounts of his animation philosophical discussions addressed the Absolute in Brahmanism, as the ultimate source for his results, thereby endowing a very positive outlook.

In Buddhism, there is the term nibbāna, which literally means 'without fuel [of craving]', and at first glance seems quite the opposite perspective, but its attainment may likewise be viewed positively as a transcendent state with supreme qualities - in which the Buddha is, for instance, 'Well-gone' and 'Knower of the worlds' (see e.g. the daily chant, iti pi so).

It can be a wonderful perspective to reflect on, to turn the mind to 'the deathless', especially when the path becomes difficult with its various obstacles.  So here I've transcribed a portion of a talk by Ven. Bhikkhu Bodhi, a most distinguished scholar monk, responsible for many authoritative translations, particularly for the Buddhist Publication Society and Wisdom Publications.  Bhikkhu Bodhi originally delivered this presentation as Lecture No. 6: nibbāna from a series in autumn 1979 at the Washington Buddhist Vihara and then, at the request of Bhante Gunaratna, recorded the series in the summer of 1981 for distribution on tape, which were subsequently digitised and the audio made available online by Bodhi Monastery.

This lecture has since been summarised and appeared in various places, sometimes with further editing, e.g. at Wisdom Quarterly, and portions have been  frequently quoted, e.g. on Dhamma Wheel.  But the summary paraphrases, omits some passages and contains some slight changes in meaning.   Whilst it makes the presentation more suitable for print, it still lacks references that were missing from the original lecture audio, though in his talk Bhikkhu Bodhi mentions a list of terms, which I guess was originally part of a handout that might have included the references.  So I decided to carry out a transcription of a chunk of about 20 minutes, and to make explicit as many of these references as I can, with links to translations, where available, by Bhikkhu Bodhi.


In his Dhamma talk, Bhikkhu Bodhi opens with the vandana, homage to the Buddha, by chanting "Namo tassa bhagavato arahato samma sambuddhasa”, as part of the customary veneration.  Then, briefly, he introduces this talk in the context of the series, by saying that so far the previous talks have dealt primarily with the problem of suffering, but suffering is only the starting point, and the truth of suffering represents the negative side.  In this talk and the following one he deals with the positive side, the third and fourth Noble Truth.

However, before doing so he reminds his listeners that we have to know about suffering to give us a reason to seek liberation and urgently, hence using the imagery of escaping a house on fire, the fire representing greed, hatred and delusion fuelled by flames of craving.

We now join shortly before the 12 minute mark, where Bhikkhu Bodhi relates this to nibbāna, discussing the psychological dimension.  (My insertions are generally in green inside square brackets [].)


[11.40] “The word nirvana [Sanskrit] or nibbāna [Pali] literally means 'the going out or extinguishing of a flame'.  And thus used figuratively by the Buddha, it means the extinguishing of the flame of craving, the extinguishing of the fires of greed, hatred and delusion.  The Pali writers also take the word to have the meaning of 'escape from the forest', that is escape from the forest of craving, or from the forest of Samsaric becoming.  The state of nibbāna is the ultimate goal of the Buddhist path, the end and the transformation of the entire practice.

The Buddha says that just as the waters of a river plunge into the ocean and merge with the ocean, so the spiritual path, the Noble Eightfold Path, plunges into nibbāna and merges with nibbāna.
[This might combine multiple references; the idea is expressed in slightly different words in the Daruka-khandha Sutta (the Simile of the Great Log), SN 35:200 (241) and the wording is from the Rādhasaṃyutta of the Saṃyutta Nikāya, SN 23:1(1), where the Venerable Rādha asks a series of incremental questions, finally concerning nibbāna, to which the Buddha states “the holy life is merged in nibbāna, its consummation is nibbāna, its culmination is nibbāna."]

Now when we try to talk about nibbāna, we immediately come up against the problem that nibbāna is said to be “beyond the range of speech and language”, [atakkāvacara – beyond logical reasoning].  nibbāna is a supermundane state, a reality which is to be seen, realized and experienced, not a concept to be conceptualised or an idea to be discussed.  Ultimately, nibbāna should be experienced and realized.  However, to make known the nature of nibbāna, we have to resort to words, we have to speak about it and therefore this lecture becomes possible.  If we were to really give a very direct communication of the nature of nibbāna, we just have to stop the lecture at this point, but then those who are listening would be disappointed.  So therefore I shall have to go on to speak, to fill up the rest of the tape.

nibbāna is an existing reality

Now the question comes up as to the nature of nibbāna and especially the question is asked: Does nibbāna signify only the extinction of the defilements and liberation from samsara or does it signify some reality existing in itself? According to the Theravada School of Buddhism, which I see to be solidly grounded in the actual word of the Buddha, nibbāna is not only the destruction of defilements and the ending of samsara but an actually existing reality, a reality which is transcendent to all the realms of phenomenal existence, to the entire empirical world of mundane experience.

There are certain reasons which can be offered in support of this view.  Here I don't wish to burden anybody with dry scholarship, but a look at the texts, at the suttas spoken by the Buddha, can help clarify for us our idea of what nibbāna is.  (This part might be a little bit dry, a little bit at times difficult to follow, but I ask the listeners to try to follow it, even if they have to repeat the tape.)

In the suttas we find that there are certain key words that the Buddha uses to designate existing realities.  These you can call 'ontological terms', terms with an existential meaning.  These key terms are the words dhamma, āyatana, dhātu, pada and sacca.  (The words are given on the list of Pali terms.)

We'll explain these words briefly and then show how each of them is applied to nibbāna.  First we'll take the word 'dhamma'.   We have dealt with this word and some of its more common meanings, such as 'the teaching of the Buddha', as 'the truth made known by the Buddha', and as 'the path that leads to the realization of truth.'

But this word 'dhamma' also has a more philosophical meaning. The word 'dhamma' signifies the basic actuality, the existing realities, those things which bear their own natures independent of our thinking, of our conceptual processing of them.  The dhammas are distinguished from conceptual entities – those things which do not exist in fact, but only as ideas or notions in the mind.  Now all the dhammas, the actual existences, are divided into two basic groups: the conditioned and the unconditioned, sankhata and asankhata.  A conditioned dhamma is an actuality which has come into being through causes and conditions, something which arises through the working of various conditions.  The conditioned dhammas include all the phenomena with which we are ordinarily familiar. These all fall into the five aggregates.  So any conditioned dhamma is either a material form, a feeling, a perception, a mental formation or an act of consciousness. 

Now all the conditioned dhammas go through three phases of becoming.  First there's the phase of arising [uppāda], then finally a phase of falling away, cessation [vaya], and in between the two there's a phase called ṭhitassa aññathatta, that is, the changing of that which stands, the transformation of that which persists, that is, while the conditioned dhamma lasts, while it persists, it undergoes constant change.  It doesn't remain static, but it undergoes transformation.  It's in a ceaseless process of becoming.   So the conditioned dhamma has these three phases: arising, transformation, and falling away.

Now in contrast to all of the conditioned dhammas, there is the class of the unconditioned, which is much simpler.  It contains only one actuality, that is: nibbāna.  In contrast to the conditioned, the unconditioned is not produced by causes and conditions.  And then, in contrast to the conditioned, the unconditioned has the three opposite marks, that is: it has no arising, it has no falling away, and it undergoes no transformation.

And the Buddha speaks distinctly of nibbāna as a dhamma.  He calls it 'the supreme dhamma', the 'uttamaŋ dhamma'.    And in one sutta he says, “Of all dhammas, conditioned and unconditioned, the most excellent dhamma, the supreme dhamma, is nibbāna.”  [Anguttara Nikaya, Sutta 4:34: ConfidenceSo nibbāna is definitely referred to by this key ontological term of Buddhism, the word, dhamma.

[There is also:
“To whatever extent there are phenomena conditioned or unconditioned, dispassion is declared the foremost among them, that is, the crushing of pride, the removal of thirst, … , the destruction of craving, dispassion, cessation, nibbāna.”  
Ratana Sutta, Sn 2.1]

Another important ontological term used by the Buddha is āyatana.  This word usually means 'realm', 'plane', or 'sphere'.  For example, the Buddha speaks about the different planes of existence as āyatanas and he also speaks about the six sense faculties as the six āyatanas.  In a famous passage from the text, the Udāna, the Buddha also speaks about nibbāna as an āyatana.  He says,

“Monks, there is a realm (atti tadāyatanaṃ) where there is neither earth, water, heat or air, neither the sphere of infinite space, the sphere of infinite consciousness, the sphere of nothingness, or the sphere of neither perception or non-perception [that is the four formless realms].  There is neither this world nor any other world, neither sun nor moon.  This I call 'neither arising, nor passing away, neither standing still, nor being born, nor dying.'  There is neither foothold there, nor development, nor any basis.  This is the end of suffering.”

So we see then in this passage the Buddha speaks of nibbāna as an āyatana, a realm or sphere.  It is a sphere where there is nothing at all that corresponds to our world of mundane common experience and therefore it has to be described entirely by way of negatives, as the negation of all finite properties.

Another word frequently used in the Buddha's discourses is the word 'dhātu'.  This word most often means 'element'. Thus the Buddha speaks of the four dhātus, the four material elements: earth, water, heat and air. At other times he speaks of 18 elements: the six sense organs, the six sense objects and the six kinds of consciousness.   But the Buddha also speaks of another dhātu, another element.  He calls this the amata dhātu, that is the deathless element and this deathless element is nibbāna.

Thus in one sutta he speaks about a monk who has reached the highest level in the development of insight, where he is seeing all of the five aggregates as impermanent, as dukkha, and as insubstantial.  Then, when he reaches the climax of insight, his mind suddenly turns away from all conditioned dhammas and he says that he focuses his mind upon the deathless element and that with his mind focused on the deathless element he reaches the destruction of the defilements. [MN 64: Mahamalunkyaputta Sutta]

In another sutta, the Buddha speaks of the nibbāna dhātu, the element of nibbāna.  And he compares it to an ocean.   He says that just as the great ocean remains at the same level no matter how much water pours into it from the rivers, it remains at the same height, without increase or decrease, so the nibbāna element remains the same no matter whether many or few people attain nibbāna. If many people attain nibbāna, the nibbāna element doesn't grow fuller; if few attain, the nibbāna element doesn't become diminished.  [Pahārāda Sutta, AN 8.19,  clause (5).]

And the Buddha speaks quite concretely about seeing nibbāna, the deathless element, almost as though it were the object of an act of vision. 

In another sutta he speaks of it as something that can be experienced by the body, an experience that's so vivid, so concrete and real that it can be described as 'touching the deathless element with one's own body'. [Cunda Sutta, AN 6.46]

The Buddha also speaks about nibbāna as a pada.  The word 'pada' means 'a state', and the Buddha calls nibbāna the amatapada, 'the deathless state'.  Thus he states in the Dhammapada, “Better than living a hundred years without seeing the deathless state is living one day seeing the deathless state.”
[Dhammapada 114]

Another word used in the texts is sacca.  This word means 'truth', not 'truth' as a statement, but truth as reality, as an existing reality.  There's a passage where the Buddha says, “That which the ignorant take to be true, that the Noble Ones, the Ariyans, know to be false.  That which the Ariyans know to be true, that the ignorant regard as false.  That which is of an imperishable nature, that is nibbāna, and that is the truth known by the Ariyans.”
[Perhaps this paraphrases verses in the Dvayatānupassanā sutta (Observation of Dualities), Sn 3.12].

So here in this passage that which the ignorant take to be truth, to be real, is a self, an ego entity and this the Ariyans know to be false since through their insight they have realized that all phenomena are without a self, that they're all insubstantial.  And that which the Noble Ones know to be truth, that is nibbāna, and this the ignorant take to be false, an imaginary thing or a vain notion.   But the Noble Ones, the Ariyans, have seen nibbāna, they've known through direct experience that it is real, the one ultimate reality that's imperishable. 

In another sutta, the Buddha says, “That which has a perishable nature, that is false, but that which is of an imperishable nature [accutapada], namely nibbāna, that is truth.  And then he says in the same sutta that this is the supreme noble truth, nibbāna, which is of an imperishable nature.  [ref.  ???]

So all of these textual sources, put together, I think, very clearly establish the view that nibbāna is an actual reality and not the mere destruction of defilements, the cessation of existence. 

Then there's also another famous passage, which, I think, also makes the matter very definitely clear.   This is the passage in the Udāna, where the Buddha says, addressing the monks:

“Monks, there is an unborn, an unoriginated, an uncreated, an unconditioned.  If there were not this unborn, unoriginated, uncreated, unconditioned, there would be no escape possible from the world of the born, the originated, the created, and conditioned.  However, since there is an unborn, unoriginated, uncreated, unconditioned, therefore escape is possible from the world of the born, the originated, the created, the conditioned.
[Nibbāna Sutta: Parinibbāna Ud. 8.3]

The Buddha here is saying that if there were no unconditioned reality, there would be no escape possible from the round of birth and death. The round of birth and death would go on forever; there would be no way at all to put an end to it.  But the Buddha adds the positive counterpart to it.  He says that there is, that there already exists an unborn, unoriginated, uncreated, unconditioned and therefore it is possible for the mind to know the unconditioned, to realize the unconditioned and by realizing the unconditioned to destroy the ignorance and craving which hold us in bondage, and thereby make an end to the round of becoming and reach deliverance from birth and death, deliverance from the world of the born, originated, created, and conditioned. 

Now since nibbāna is the precondition for this liberation to take place, since nibbāna must exist for it to be known and for liberation to take place, therefore it is evident that nibbāna cannot simply be reduced to the destruction of the defilements and liberation from the round.  Those events are conditioned events, they occur in time, while nibbāna is unconditioned, without any origination, timeless.

Is nibbāna conditioned by its path?

Now one particular problem that's sometimes raised over the statement that nibbāna is unconditioned.  It's said that it seems contradictory to say that nibbāna is unconditioned and yet by practising the path you attain nibbāna.  Doesn't this seem to make nibbāna something that's conditioned by the practice of the path, something that's produced by the path?   Doesn't nibbāna become an effect, something not unconditioned, not causeless?

Here the contradiction is only apparent: nibbāna itself – we have to make a distinction between nibbāna itself and the attainment of nibbānanibbāna itself is unproduced, unoriginated, it's always existent. But by following the path, by reaching Enlightenment, you realize nibbāna.  By practising the path you don't bring nibbāna into existence, but rather you discover something already existing, something always present. We can say the attainment of nibbāna, the realization of nibbāna, is produced by the practice of the path.  But this doesn't mean that nibbāna itself is brought into being by the path.

We can illustrate this by an analogy.  The city of New York is presently existing and there are highways leading into New York from all over the country.  By driving along the highway you can reach New York and enter the city.  We can't say that New York itself is produced by travelling along the highway.  Rather, the highway gives us entrance to New York; by travelling along the highway we can enter New York.  Similarly, the path leads to nibbāna; by following the path you reach Enlightenment and that brings the realization of nibbāna, but nibbāna itself is not created by the path."

END of transcription.

Bhikkhu Bodhi then proceeds to consider some of the terms and expressions used in the text as designations for nibbāna.  He also includes a nice story of a turtle and a fish (at about [34:00]) to indicate that nibbāna is not annihilation or non-existence.  I can't find any canonical reference.  It seems to be a version of a fable, originally presented in English by Bhikkhu Silacara in his book The Four Noble Truths, one of a series on Buddhism (at published in 1922.  I reproduce it here:
Once upon a time there was a fish. And just be cause it was a fish, it had lived all its life in the water and knew nothing whatever about anything else but water. And one day as it swam about the pond where all its days had been spent, it happened to meet a turtle of its acquaintance who had just come back from a little excursion on the land.
  "Good day, Mr. Turtle" said the fish; "I have not seen you for a long time. Where have you been?"
  "Oh!" said the turtle, "I've just been for a little trip on dry land."
  "On dry land!" exclaimed the fish. "What do you mean by on dry land? There is no 'dry land'. I never met such a thing. Dry land is nothing."
  "Well," said the turtle good-naturedly, "if you want to think so, of course you may; there's no one can hinder you. But that's where I've been, all the same."
  "O come," said the fish, "try to talk sense. Just tell me now: what is this land of yours like? Is it at all wet?"
  "No, it is not wet," said the turtle.
  "Is it nice and fresh and cold?" asked the fish.
  "No, it is not nice and fresh and cold," the turtle replied.
  "Is it clear, so that light can come through it?"
  "No, it is not clear; light cannot come through it."
  "Is it soft and yielding so that I could move my fins about in it and push my nose through it?"
  "No, it is not soft and yielding; you could not swim in it."
  "Does it move or flow in streams?"
  "No, it neither moves nor flows in streams."
  "Does it ever rise up into waves, then, with white foam on them? asked the fish, becoming just a little impatient at this string of "Noes".
  "No," replied the turtle truthfully, "it never rises up into waves that I have seen."
  "There now!" exclaimed the fish triumphantly. "Didn't I tell you that this land of yours was just nothing? 1 have just asked, and you have answered me that it is neither wet nor cool, nor clear nor soft, and that it doesn't flow in streams nor rise up into waves. And if it isn t a single one of these things and can t do any of these things, what else is it but nothing ? Don't tell me!"
  "Well, well," said the turtle, "if you are determined to think that dry land is nothing, I suppose you must just go on thinking so. But anyone who knows what is water and what is land would say you were just a very silly fish, for you think that anything you have never known, just because you have never known it, is therefore nothing."
And with that the turtle turned away and, leaving the fish behind in its little pond of water, set out on another excursion over the land that was "nothing".


I personally find these accounts of the positive aspects of nibbāna, as presented by Bhikkhu Bodhi, very inspiring.  And there's the prospect of more not just in the Pali canon; comparative analysis of Theravadin and Mahayana canons, which are derived from different schools, have each undergone different editorial processes involving revisions and omissions, and may well yield further examples.

Some scholars and practitioners, particularly Theravadins in the West, express a very minimal characterisation of nibbāna as the culmination of a process to eradicate defilements and uproot craving; they may say that all that subjectively remains on completion of the Path is experience.  They're not motivated to hear and even show antipathy to hearing more than that and argue that it would be a distraction.  I sense that underlying this resistance are perspectives confining attā and anatttā to the khandhas.  But the texts contain pointers beyond the khandhas; I have found, as did the Pali scholar, I. B. Horner, that this topic really merits further exploration.

I feel there is an imperative to explore these topics in the context of artificial intelligence, especially the 'strong AI hypothesis'.  I've been exploring aspects of nibbāna motivated by the wish to articulate the distinctness of human beings from machines.  Humans have the potential to attain to nibbāna, which is the source of ultimate wisdom, but machines, being conditioned, do not.  However, that distinction is hard to draw without a clear sense of transcendence being communicated in a constructive way.

Saturday, May 11, 2019

On Intuition in the Life and Work of Ramanujan

These past few weeks I have been engrossed in reading about Srinivasa Ramanujan (1887-1920), the brilliant mathematician from South India whose prolific work in number theory lit up the academic world from the early 20th century onwards.

In my studies in this subject I had heard the name, seen references, but didn't use his work.  Even though he's famous in the mathematical world and in India, I wasn't minded to give much attention until I happened to chat over lunch with Thomas Bewley, who described his experiences of playing Prof. H. F. Baker in the film, The Man Who Knew Infinity.  I promptly ordered a DVD and enjoyed watching the film, so I then ordered the substantial biography of the same name by Robert Kanigel.

I'm writing this having just finished Kanigel's book.  It's extensively researched and very detailed, covering Ramanujan's life and work; whilst aimed at the general reader, quite a lot of mathematical material has been presented, in a generally convincing way – Kanigel is numerate (he has a degree in mechanical engineering) and he has evidently spent considerable time grappling with the material in conversation with scholars.

I particularly like the way he shows how the relevancy of Ramanujan's work and various applications has ebbed and flowed; in his early years he struggled to make known his findings, but gradually some friends and associates tried to help promote his cause until they were able to tap into the British colonial networks.  Eventually there arose opportunity to write to the scholars at Trinity College, Cambridge, Britains' foremost centre for mathematical research.  Even then, Ramanujan had to keep persevering until he succeeded at the third attempt when his genius was recognised by Hardy, who nurtured Ramanujan's talent by instilling the rigours of proof and dissemination of the various results.  After Ramanujan's passing, Hardy continued to promote his cause through papers and continued reference.  Subsequent decades saw changing foci, but recently his work has become of great significance – his “mock theta functions” have been integral to the development of mock modular forms, which are now used in astrophysics, even to model singularities (black holes, etc.).

Personally, I might like to explore his work in partition functions (having a natural interest in combinatorics).  However, I am mainly interested at the moment in Ramanujan's spirituality, what we might learn about intuition.  Kanigel attempts to explore this area, knowing full well that members of the public like especially to know about how a human being can navigate the vicarious aspects of life and its innumerable obstacles, triumphing over adversity – the indomitable spirit.  Kanigel dutifully delves into this with cultural sensitivity – undertaking fieldwork in the foreign lands and cultures of the British Isles and India.  Through the information he gathers from interviewees, Kanigel recreates at some length daily scenes in which Ramanujan lived and breathed mathematics - in Kumbakonam, his home town, and in various other places such as Triplicane (now Tiruvallikkeni) (with its historical sites such as the Arulmigu Sri Parthasarathyswamy Temple), and other areas in the then Madras Presidency (now Tamil Nadu).

Even in a volume as extensively researched as this, the whys and wherefores as to Ramanujan's  mathematical discoveries can't be fully explained when it seeks to explicate an entire life story and indicate especially its mathematical import today.  Even so it's noticeable that whilst Kanigel appears comfortable explaining material facts, he finds it exceedingly difficult to fathom Ramanujan's spiritual inspiration.  Hence overall he writes sympathetically, but when it comes to religious aspects, he doesn't have much to say, and even occasionally strikes an incredulous tone.

For example, referring to a gathering that developed into philosophical discussion he writes (pp. 31-2):
Another time, when he was twenty-one, he showed up at the house of a teacher, got drawn into conversation, and soon was expatiating on the ties he saw between God, zero, and infinity - keeping everyone spellbound till two in the morning. It was that way often for Ramanujan.  Losing himself in philosophical and mystical monologues, he'd make bizarre, fanciful leaps of the imagination that his friends did not understand but found fascinating anyway. So absorbed would they become that later all they could recall was the penetrating set of his eyes.

I don't suppose Ramanujan felt lost; if anything, he was finding deeper relationships in what the author describes as "bizarre" and "fanciful".  Whilst it might have been tantalising to his audience, at the same time the culture readily accepted this kind of expression.

In another chapter Kanigel writes (p.66):
Later, in England, Ramanujan would build a theory of reality around Zero and Infinity, though his friends never quite figured out what he was getting at. Zero, it seemed, represented Absolute Reality. Infinity, or ∞, was the myriad manifestations of that Reality. Their mathematical product, ∞ x 0, was not one number, but all numbers, each of which corresponded to individual acts of creation. 

Kanigel's tone conveys shades of incredulity, but these kinds of views are taken seriously in many parts of the world.  At least in recent centuries, they seem to be more naturally appealing to Asians – from all over, whether the South, South-East, Far East, and the North.  So I'm interested to read accounts from their perspective, particularly Indian perspectives and interpretations – and how do they interpret Ramanujan today?

As Ramanujan is a national hero, there's no shortage of material, a fair amount being helpfully referenced in Kanigel's book.  Arriving as a newcomer, I try to find where possible sources with first-hand accounts, ideally published by authorities who have some historical connection.  My starting point has been a broad selection made available by The Institute of Mathematical Sciences in Tamil Nadu, a national research centre.

They list five books in a section on books, including Kanigel's volume.  Among the others, I've been looking at  'Ramanujan - The Man And The Mathematician' by S. R. Ranganathan, part of Great Thinkers of India Series, published by Asia Publishing House in 1967.  The publisher is still registered, based in Mumbai, but I can find no website for it.  Dr Ranganathan, who was a mathematician and library information professional in India; an endowment in his name is associated with another publishing company, Ess Ess Publications limited, and copies of the book are readily available from them.  (But it's also not hard to find a free version online.)

This book records some accounts by those who knew and met with Ramanujan.  One of the respondents is Dr. Mahalanobis, who was one of those who was there in the late night discussion that Kanigel refers to.  He recalls:
He sometimes spoke of “zero” as the symbol of the Absolute (Nirguna-Brahmam) of the extreme monistic school of Hindu philosophy, that is, the reality to which no qualities can be attributed, which cannot be defined or described by words, and which is completely beyond the reach of the human mind. According to Ramanujan, the appropriate symbol was the number “zero”, which is the absolute negation of all attributes.  He looked on the number “infinity” as the totality of all possibilities, which was capable of becoming manifest in reality and which was inexhaustible. 
(MN Reminiscences of Dr P C  Mahalanobis FRS,
Member of the Planning Commission of India: MP1, p.82)

Ramanujan starts with that philosophical position and then gives it mathematical expression, based on numerical properties that can exhibit the transcendent qualities.  These are not just made up fancifully, but rather there are references to a philosophical school.   Nirguna-Brahmam (or Para Brahman), is described in Hindu texts as the highest spiritual state, the formless Brahman, specifically in the sense of being absent of Maya, illusion.  It's a core belief in the Advaita Vedanta tradition.

Mahalanobis continued:
According to Ramanujan, the product of infinity and zero would supply the whole set of finite numbers.  Each act of creation, as far as I could understand, could be symbolised as a particular product of infinity and zero, and from each such product would emerge a particular individual of which the appropriate symbol was a particular finite number. I have put down what I remember of his views. I do not know the exact implication. 

Whilst Mahalanobis lacked understanding of the finer points, he could gain a general sense of what lay behind Ramanujan's words – there was valid and useful communication.  If they had been completely incomprehensible, then Ramanujan probably would not have sustained interest for so long.  Perhaps more significant still as an indication of the importance of this spiritual view, was the following reflection:
He seemed to have been perhaps emotionally more interested in his philosophical ideas than in his mathematical work. He spoke with such enthusiasm about the philosophical questions that sometimes I felt he would have been better pleased to have succeeded in establishing his philosophical theories than in supplying rigorous proofs of his mathematical conjectures.

This is a significant passage as it points to how important to him was his underlying spirituality of which mathematics was an expressions.  I think we see the deleterious effects of denying him support for this spirituality when Kanigel describes how to many Ramanujan appeared a much-changed man on his return to India in 1919.  Mentally and emotionally he was a different person: whereas previously he was full of fun and sociable in small groups, on his return he appeared withdrawn and angry.  It seems England was able to support his mathematics, but it came at the price of his Brahmin caste (at least for those who did not allow any exemptions to Samudrolanghana, the offence of crossing the sea) and his wellbeing.  There are areas that the book perhaps understates this sacrifice – which was more than the decline in his physical health.  Yet Ramanujan foresaw his own death (“I won't reach 35 years of age”), so the speculation around what might have been regarding alternative life paths and treatments of his tuberculosis should be set against that.

In modern times, we can still find views from India, especially religious teachers, who can give some indications of Ramanujan's spirituality.  Even though they might not have any formal background in mathematics and may lack rigorous language, they can express the 'inner voice', as it were.  For example, in his talk at SRCC College, available in a YouTube video, entitled The Secret of Ramanujan's Genius,  Sadhguru likens deities or, more specifically, murtis (forms) to energetic machines that are able to enhance particular faculties; unlike mechanical devices, such machines don't have moving parts, are easy to maintain, and are available all day and every day.  Ramanujan knew how to use the murti known as the goddess or deity Namagiri to receive mathematical insights and he seemed to be working continuously.  In the short excerpt, it's not explained how one cultivates practice of utilising these murtis, but in India it is typically through yogic or meditative training, and, as for most yogis, Sadhguru gives instruction in these, such as Isha Kriya.

Another perspective is shared in a presentation on teachings by Sri Aurobindo & The Mother: the quality of beauty is highlighted in a post where Sandeep, the author, asks: Where does Mathematics come from?   According to teachings in this tradition, having some correlation to the energetic machines, it as though humans have inner beacons of light that can be directed towards specific arts; an agile mind can shine the light in different directions.  But here, this longer article also emphasizes  development (I'd choose the word bhavana) of the capacity of attention and concentration.  Other posts on that informative site, including one that considers some views of Roger Penrose, describe how a prerequisite is knowing how to bring the mind into stillness (once the mind is at a standstill one can move easily in any direction); bringing the mind to a standstill is  key to allowing novel ideas to arise.

From my own Buddhist perspective, I would highlight that Ramanujan's superlative ability can only come through sustained kusala karma (skilful intentional actions), usually over many lifetimes.  In this way he would have generated puñña (merit), a kind of energetic fuel that with continued cultivation crystallizes as paramis (perfections) – puñña gives you the capacity to achieve, paramis enable that capacity to be readily and instantly available.  Perhaps Ramanujan refrained strictly from intoxicants leading to great clarity and receptivity of mind – certainly even in such a foreign environment he practised strictly as a Brahmin, so he retained that quality of mind seeking perfection.

Friday, April 26, 2019

Sharing some research in number theory

I have been exploring the life of Srinivasa Ramanujan, the brilliant Indian mathematician whose efforts in number theory from about a century ago are still captivating mathematicians today.  Reading about him in The Man Who Knew Infinity, Robert Kanigel's detailed biography, has prompted me to revisit the little bit of research in algebraic number theory that I undertook at the University of Glasgow. 

Under the direction of Professor Robert (“Bob”) Odoni, I submitted a thesis, Norms of Ideals in Direct Sums of Number Fields and Applications to the Circulants Problem of Olga Taussky-Todd, in April 1992.  Its originality was due to my supervisor; I barely understood the material and for various reasons felt that I couldn't continue towards a doctorate, but I wrote up what I had for a Master's and in September I duly underwent the viva with the external examiner, Dr. Stephen Wilson from Durham University.   Years later, I feel that I should in some way honour the memory of Prof. Odoni  by making the thesis available as a PDF file on Academia.

Taussky-Todd, who originally posed the problem about circulant matrices, was a versatile mathematician who explored many branches; in a long and productive life her output was considerable.  She initially studied chemistry because that her father was director of a vinegar factory, but then she switched to mathematics.  There is a parallel with Prof. Odoni, whose initial undergraduate studies were in chemistry, but he was not satisfied with the kinds of questions the subject addressed and so he turned to mathematics and never looked back.  I'm pleased to know that his contributions have been largely preserved in various papers, collated in a ResearchGate profile.

My thesis includes half a dozen references to Prof. Odoni's work (I have PDF versions that I'm happy to share).  He encouraged collaboration in research in many ways; when speaking of publications, I remember him praising in particular the Institute of Mathematics of the Polish Academy of Sciences, which is reflected in several of the listed publications.  All the papers he submitted there have been digitised and are being shared freely, reflecting this kind of openness. 

On a technical note, I prepared the thesis using (plain) TeX on an Atari ST, using GNOME   as a text editor.  I stored the files on a PC-compatible floppy and eventually transferred them off this obsolescent storage medium.   The sources I found a few weeks ago were from the time of the original submission, so they did not include any corrections.  Fortunately I still had these in the form of handwritten notes, which I could duly incorporate into the TeX sources.  And, wonderfully, on installing TeX years later I could still compile them to generate a DVI file and thence the PDF!  The final task was to modify the PDF to draw some straight lines to complete a couple of diagrams, which I achieved with PDFill.

As to why I'm reading about life of Ramanujan, I'm currently investigating the role of intuition in maths and computing for a little book I'm writing.

Thursday, January 10, 2019

A Rare Opportunity: On ‘Buddhism and Pali’

There's a new book that I'd like to introduce:

Buddhism and Pali (front cover)
Cover for 'Buddhism and Pali' (from

A little over 10 years ago I had the good fortune to study under Professor Richard Gombrich at the OCBS Pali Summer School of 2008.  I subsequently used the knowledge gained to read and translate from the Pali some passages in the Theravada Buddhist canon as part of my Master’s thesis.   So, I was pleased to see the publication of Prof. Gombrich’s latest book, Buddhism and Pali.

In this post I offer some thoughts about the material - they're more ad hoc responses to some ideas put forth than a book review because I’m not qualified to assess the finer points of linguistics.


The title of this post is quoted from page 8 of Gombrich’s ‘Buddhism and Pali’, where the author indicates why, despite being somewhat sceptical about its appeal to members of the public, he decided to go ahead and write this small volume: he sees its potential to raise awareness about the far-reaching significance of Pali.  I think he went ahead because he is a believer!  That is, he truly believes in the importance of the Buddha as a historical figure, in the value of his teachings, particularly for the depth of the philosophy, and in the necessity of knowing Pali as the means to properly access that knowledge and interpret it accurately.

I remember maybe 20 years ago hearing about the Jesus Seminar and its efforts to determine the historicity of Jesus and what could be reliably attributed to him.  It seems that in recent years this kind of approach has been attempted in respect of the Buddha Gotama, leading to similar results, where some academics deny that the canonical texts can be attributed to the Buddha himself and hence cast all kinds of doubt on the material.  As someone who is interested in Buddhist texts primarily as a practitioner, I’ve always felt that these kinds of scholarly views are rooted in a kind of hindrance, specifically sceptical doubt (Pali: vicikicchā - for the definition, I'm linking through to an online edition of the Pali Text Society's Pali-English dictionary).

So I naturally welcomed this new book, which constructs an argument, mainly linguistic, but backed up by cultural, historical and other observations, that shows how Pali could in fact be the language the Buddha actually used in his teachings across a vast swathe of what we now refer to as India.  As a scholar specialising in Indo-Aryan languages over his long career, Gombrich is sensitive to various issues that need to be addressed, especially concerning matters such as the integrity of texts and their preservation.

The book itself comprises four chapters and I’ll briefly introduce them and pick up on points I found interesting.

1. Pali in History

In the first chapter, Gombrich sets the scene, indicating the current use of Pali in the canon of the Theravada School of Buddhism, how it was essentially a language of recitation used by monks and nuns to maintain the teachings.  He then traces its linguistic history back to a few centuries BCE, showing in particular how it relates to Sanskrit.  The book is concise and succinct, making it easy to digest and follow the lines of thought.  Thus, we have a nice overview of the history of the Buddha, the geography of India and the modes of usage of Pali, especially among monastics.  Then the ancestry of Pali is outlined in three steps, altogether covering a single page.  So it’s a kind of primer, with some footnotes provided for further exploration.

Being condensed, many points are made only briefly. One of the first to catch my eye was that the word phāsa (p.11), which means ‘language’, was originally attached to ‘Pāli’ .  The same word has been brought into Thai (phāsa is rendered as ภาษา, hence ภาษาไทย). In fact, Thai has incorporated many such ‘loan words’ from Pali and Sanskrit, but also from many other languages (its influences are complex)!

The transmission of the Buddha’s teachings was initially an oral tradition, only later written down, typically on palm leaves.  We may take writing for granted now, but (I didn't see this mentioned by Gombrich) one doesn’t have to go back very far to find practitioners who did not make use of written texts.  For example, the founder of Wat Phra Dhammakaya on the outskirts of Bangkok was an illiterate nun, Khun Yay Upasika Chandra Khonnokyoong.

It's remarkable that the texts have been handed down over the centuries with hardly any corruption.  Gombrich uses his philological expertise to cleverly argue how Pali has been preserved and evolved for a variety of pragmatic and conventional reasons, usually under the influence of Sanskrit, but used in a distinct way; its distinction arising because it is an artifice - a vehicle for transmission that needs careful memorisation, but also reflects adaptation to local environments.  He makes the argument easier to follow by comparison with the use of the English language, which he furnishes with examples.

In passing, he also makes the point that the teachings tell us that all this is subject to impermanence and will eventually perish.  From a professional perspective, as another aside, I wonder about the digital context, where we can easily replicate data in the Cloud and ensure integrity using checksums.   Well, checksums are themselves subject to bit rot (corruption) and according to Buddhist cosmology our world and many realms above eventually get destroyed, so that would include all the data centres in the Cloud and in clouds!

I find conservative Gombrich’s statement that “as far as we know” Theravada Buddhism was confined in its first millennium to India and Sri Lanka (p. 22): I am persuaded by the thesis proposed by Lewis Lancaster in the Maritime Buddhism Project that traders carried religious practices across South and South-East Asia much earlier (e.g., across the Bay of Bengal, the Andaman Sea, through the Malacca Strait, and up to the South China Sea).  Collaborative research is ongoing to form an electronic atlas.  Whilst the picture is patchy, there is significant archaeological evidence - for example, in Thailand, as reported by Suchandra and Lipi Ghosh,  excavations have unearthed objects with Buddhist inscriptions (in Pali or Hybrid Pali and Sanskrit) dated to the Dvāravatī period that have strong associations with the sea: such as amulets, carried by seafaring merchants to ensure a safe voyage and even images of Buddha Dipankara calming the waves.  I also asked my uncle, Sean Trafford, who spent his working life at sea, much of it in and around SE Asia, and he found the thesis credible.

Perhaps the first intimation I received of such early connections for Thailand was a visit to Phra Pathom Chedi in 1988, which I subsequently wrote about as one of my first articles on the Web.   A traditional overview of how Buddhism came to Thailand was compiled by Ven. Dr. Saddhatissa  There is, of course, a large gap between the time of Emperor Ashoka and the mid first millennium C.E., but it’s narrowing - see e.g. Stephen A. Murphy and Miriam T. Stark’s introduction to period transitions. (Coincidentally, one of my cousins has just started her first year as an undergraduate in the Faculty of Archaeology at Silapakorn University, so I may get to hear about further developments!)

2. The Linguistic Character of Pali

The second chapter - on ‘The Linguistic Character of Pali’ - is the most technical.  It contains material that is second nature to the author, so it’s both easy for him to spell out and difficult for him to gauge how accessible it might be for the reader.  For anyone who has made an initial attempt to learn the language it’s a useful refresher of the main concepts, but others may well observe the guidance, thoughtfully inserted, to just read some of the more general remarks.

These include the use of verse, which I tend not to give much attention to, even though I frequently come across them in sutta study; although the texts are largely prose, verses do feature.  My appreciation and understanding of verse is limited, but some while ago I became keen to learn the meaning of a daily chant called 'A Buddha Prayer Song', so eventually I produced a translation, aided by Thai friends.  It was an example of prosody, so I was pleased to learn more about the metrical structure of such verse. 

3. Pali Prose Style

In the third chapter - on ‘Pali Prose Style’ - Gombrich discusses how its style was especially shaped by the oral tradition.  He also explores the question of how the oral transmission has been so efficient, with some corroboration from neuroscience and illustrations from other disciplines such as music.  There are practical strategies indicated for memorisation such as the division of labour and Gombrich indicates how the Buddhist Councils have helped ensure their preservation.

He goes on to use a linguistic argument based on an incremental style of prose, where words are successively augmented - in length and meaning.  When considered from the view of the practice, we find that the experience is often of gradual development, gradual evolution and progression, so to record this it is fitting to choose words that reflect this in structure and meaning.  This insight helps in the validation of texts, i.e. there is scope for validation through practice. This is particularly the case when undertaking a practice with a sequence of steps, as exemplified by the Ratha-vinita Sutta  (Relay Chariots, translation by Thanissaro Bhikkhu).

Gombrich takes as an example (pp. 57-61) the homage (vandanā) in the iti pi so formula, which is a standard recitation in Pali chanting.  I think it’s an ambitious choice because it’s very condensed with meaning at mundane and supramundane levels, but I think through meticulous analysis he arrives at a sensible interpretation, showing the value of a proper understanding of Pali - it makes rational sense.

However, the scholarly method follows a particular intellectual discipline, so the result is often carefully couched and sometimes speculative; perhaps its deeper meaning cannot be convincingly made by linguistics alone.  In particular, whilst Gombrich also makes the important point that the text carries with it an injunction, of ‘ought’, scholarship doesn’t necessarily bring about this ‘ought’ response, i.e. "Right, let’s undertake the practice!"  To me it seldom is that compelling.  Even from the perspective of practice, it still is a matter of interpretation dependent on view (diṭṭhi) and according to the path (magga) taken.  However, to some extent the words can help to validate practice, so it works both ways.  

Indeed, the linguistic reading can lead on to practice and I’ll try to illustrate this by focusing on the ‘ought’, drawing on the Dhammakaya tradition to which I belong.  I’ll start by making the assertion that the presentation style of a practitioner can be more persuasive about the ‘ought’.  I think especially of the late Chao Khun Phramongkolthepmuni, the re-discoverer of this practice.  When he was delivering sermons as Abbot of Wat Paknam Bhasicharoen in Thailand he spoke with a directness and assuredness that arose from meditation experience.  His audience was roused in large numbers over the years and it came about as the audience engaged a different mode of listening and interpreting, through the citta, a Pali word that I would render along the lines of “heart-mind”, to distinguish it from the brain alone. 

The Chao Khun explained every word in the iti pi so, and gave the formula a sense of direction reflecting investigation within what the Buddha referred to as “this fathom-long body” (Rohitassa Sutta).  I summarise the respective interpretations in a table as follows:

TermGombrichDhammakaya traditionComment (Dhammakaya)
BhagavatāBlessed OneBlessed OneBecause the Buddha is breaker of the wheel of Samsara
DhammoTeachingDhammaTeaching (mundane) and Truth as supramundane reality
sanditthikopracticalto be seenDirectly observable
akālikoimmediatetimeless and ever-presentWhether there are Buddhas in the world or not, the Path to Enlightenment lies open, all the same
ehipassikoCome and seeCome and see
opanāyikousefulleading inwardsthe operative mode is given by phrases such as kāye kāyānupassī, ‘contemplating body in body’ in accordance with the Middle Way
paccattamindividually, by oneselfindividually, by oneself
veditabboo be understoodto be realizeddirectly, not just intellectually
viññuhiby the intelligentby the wise

This practice-based approach was even reflected in the design of published materials.  For example, several of the terms, including opanāyiko, were employed on the front cover of a magazine of the Bhikkhus of Wat Paknam, surrounding the image of the Buddha in meditation.

Another area where the orientation of practice can prompt or fashion a different response concerns repetition (pp. 62-68). Certainly some formulaic repetition acts as little more than packaging, but the repetition in the ‘contents’ does need care.  As mentioned, repetition aids and reinforces memory, but it also establishes a rhythm, which can be cyclical in nature, like a spiral staircase, where the completion of each full 360 degree rotation brings a different aspect viewed from a series of successively higher windows (perhaps these are the “small changes” that Ven. Dr Walpola Rahula alluded to?).  Lists are usually intentionally ordered.  Hence the khandhas (pp. 64-5) are to be contemplated, each in turn (it’s not an intellectual exercise, it’s a sermon on practice, not about the practice).

But some cases of repetition do appear to be far less necessary: repetition that arises through reporting - the convention of repeating verbatim each time a new person is to hear what has been said.  Such preservation can make for excessive, even quite absurd, repetition as illustrated (p. 66) and I confess that in study groups we often skip reading out such repeated reports.

Gombrich makes the key point that repetition occurs at every level, which suggests that the Buddha used repetition in his speeches, thereby strengthening the case that the Pali does indeed record the Buddha’s own words.

4. Pali in Buddhist Ideology

Having laid the foundations, the final chapter presents the main thesis: that the Buddha gave his teachings in Pali as part of his ideology; the use of Pali emerged in opposition to Sanskrit as integral to the Buddha’s teachings being in contradistinction to the Vedas.  I hadn’t thought that the Buddha might so deliberately using language (or avoid the use of a particularly structured language) to take an ideological stance, but I can imagine it fits like this. 

Then Gombrich builds his case, showing how the Buddha gave his teachings to anyone who was prepared to receive them, instructing his disciples to make language accessible and adaptable to locality.  A particularly difficult area is establishing what language was used between the time of the Buddha and when it was first formally written down in Sri Lanka.  So Gombrich proceeds to develop a linguistic argument, inviting the reader to imagine the conditions in which the Buddha roamed far and wide and how Pali appears to have accommodated a wide range of dialects to match whilst being based on a predominant dialect, a ‘lingua franca’, becoming the ‘argot’ of the Buddha and his followers (rather than quoting Google, I suggest the entry in an online etymology dictionary).

Such language was used in a pragmatic fashion to convey meaning, so it can be argued that implies purging language of foreign terms.  I’m cautious about that; some terms need to be introduced with all nuances of the original meaning properly established or else much can be lost.  I think, for example, of the term “mindfulness”, which is typically used for sati, but in the process the sense of clear comprehension with an ethical basis has often been removed (think about the sequence of the Eightfold Noble Path).


In the epilogue, Gombrich leaves us with an open problem: how best to teach a basic course in Pali that will enable students to tackle the translation of texts with the aid of a dictionary and other tools.   From past experience, Gombrich has found that the course needs to be intensive over a short period, usually 2-3 weeks, but that has presented various challenges in terms of organisation and commitment irrespective of whether that is offered in person at a physical location or online. 

Efforts are currently being made to provide a more sustainable option through online courses with recorded sessions, which should improve retention, though it will mean reduced personal interaction.  Having been involved in developing and support e-learning platforms, and even a little research, I expect A.I. to offer considerable potential, especially assistance in understanding particular concepts being taught in drill practice.  Financial support would make things easier, remove some barriers, but overall I think that it still largely depends on the student's determination as to whether there will be a successful outcome.

Overall, this little volume has been stimulating and enjoyable and I'm pleased to recommend it.  It has reminded me how fortunate I was to learn the rudiments of Pali from an expert so that I could be in a position to explore the Buddha's teachings with more assurance; today, I mainly use that knowledge outside the academic context, in a local Buddhist group (where we follow chanting and meditation with sutta study, using a plethora of books and some electronic tools).

Postscript [12 January 2019]

After posting the article, I recalled having written about how the Buddha transmitted Dhamma based on knowing the receptivity of his audience and their kamma.  To explain how this worked, I tried to use analogies with concepts in physics (mainly holography, first introducing the basic idea and then elaborating with a further analogy of a radio set.