Samsara is a bad dream, a nightmare, a painful illusion.
The purpose of our practice is to awaken from that bad dream,
as Shakyamuni did, and as many others have since. The mechanism
of that awakening has been analyzed for two and a half millenia.
There are many accounts, although I think there is only one awakening.
The fundamental technique is *samadhi.*

As a first approximation, we may say that the word "samadhi"
means "concentration." Right samadhi is the eighth component
of the Buddha's Noble Eightfold Path. Again, the Buddhist Triple Discipline
consists of *shila* (morality or discipline), *prajna* (wisdom
or insight), and samadhi. Thus samadhi is fundamental to Buddhism.

The Buddha Shakyamuni did not invent samadhi. He learned it from his teachers during the phase of his life prior to his enlightenment. The traditional account has the young Gautama, after leaving home, visit a series of yogic teachers who taught him a variety of meditative techniques. He is said to have achieved profound samadhis, mastering what each of these teachers had to offer. In each case, however, he remained unsatisfied. Finally, sitting all night under the Bodhi Tree, he went beyond what any of his teachers had been able to show him and had the experience that founded Buddhism. When he came to be a teacher in his own right, he taught many of the techniques that he had learned. Thus there are elements of traditional yoga that are included in the Pali Canon. In particular, there is a rather specific set of samadhis that are described in the Canon and that are said to have been practiced by the Buddha.

The discussion of samadhi in the Pali Canon is strikingly similar to
discussions of samadhi that we find in other traditions
from *the Common Path* ---
that is to say, the great Indian religious tradition that includes
Buddhism, Hinduism, Jainism, Yoga, Tantra, and so on. It is reasonable
to suppose that there was a single practice,
or collection of related practices, which were the common possession of
all of the teachers of the period around 500 BCE when the various
strands of the Common Path were emerging. These practices were referred to
as "samadhi." We find descriptions of these
practices in a number of ancient texts, some Buddhist and some not.

The clearest and most explicit ancient discussion of samadhi that I know
is found in the *Yoga Sutras* of Patanjali. Little is known
about Patanjali. He is traditionally identified with a grammarian of
the same name who flourished in the
second century BCE. Many scholars doubt this identification. Indeed,
it is sometimes claimed that the *Yoga Sutras* show the influence
of a fully developed Buddhism
which would not have existed until about the fifth century CE.
Patanjali himself was certainly not a Buddhist. Still, it is easy to
translate much of his approach to
enlightenment into Buddhist terms. At any rate, I will freely adapt some of his
terminology and of his conceptual apparatus for the discussion of samadhi.
In order to relate my discussion to the Buddhist context, I will also refer
to the *Visuddhimagga,* or "Path of Purification," of Buddhaghosa. This is the most authoritative
and systematic exposition of Theravada Buddhism. It was written in the fifth
century CE in Ceylon. I have used the translation by Bhikkhu Nanamoli.

In order to explain samadhi, let me take an example from my own
professional experience. Concentration is fundamental
to creativity. Most of our undergraduate students, unfortunately, have
never learned even the rudiments of concentration. For example, mathematics
has to do with objects which are not in the physical world. It is the
study of numbers, functions, geometrical
patterns, and a wealth of other abstract objects. If we want to think about
these objects, we have to be able to focus on them.
Our students are used to doing their mathematics homework while
watching television and talking on their cellphones. They are multi-tasking,
I suppose. That makes it impossible for them to do any serious mathematics,
because they cannot focus on the objects they are dealing
with. They can carry out a routine sequence of steps, an
*algorithm* in the jargon,
but they cannot do anything that is not prescribed by the rules.
Up to a point they can
compensate for this deficiency by drawing pictures. In a calculus course,
for example, we teach students how to draw graphs. Today there are
graphing calculators and computer
software that will produce such pictures automatically.
Such devices, however, are limited
in their applicability. They are not good at showing geometrical
patterns in three dimensions.
They are completely defeated by patterns in more than three dimensions,
because such patterns cannot be seen by the physical eye, only by
"the eye of the mind." Non-geometrical patterns can also
not be drawn by such devices. Moreover, looking
at a physical picture is nowhere near as good as looking at a mental picture.
A mental picture is completely ours, completely under our control.
A physical picture is external to us. We may not notice crucial details.
In order to use it, we have to focus on it, concentrate on it ---
in effect, turn it into a picture in our mind.

What is it like to work on a mathematical problem that cannot be solved algorithmically --- that is to say, which cannot be solved by a routine application of the rules? First we have to learn the relevant facts and terminology and computations. We have to become comfortable with the subject domain in which the problem arises. That is the easy part, but can of course be challenging if the mathematical domain is unfamiliar. Then we have to concentrate on the situation that the problem concerns. Often that amounts to visualizing or otherwise internalizing a pattern of some sort, although not necessarily a geometrical pattern. When we try to do that, we find that our mind wanders. All sorts of thoughts from our ordinary life come along that are irrelevant to the problem we are trying to solve. We have to keep bringing our mind back to the problem. Most of our students never get past that difficulty, and therefore they cannot succeed in any serious mathematics course beyond the sophomore level. The difficulty is that mathematics is abstract, which means in particular that it is emotionally barren. Students often ask things like, "How is this relevant to my life?" What they are asking for is some sort of hook to help them pull their attention back to the problem. I often taught engineering mathematics, where the technological application could be used as such a hook. In pure mathematics, the lack of such a hook is often crippling to students. The appeal of the material to the professional mathematician is aesthetic. But that is getting ahead of the story.

The yoga tradition recommends sitting down in an appropriate meditative
posture. That has the effect of making the musculature, the outermost sheath,
stable and comfortable so that it can be ignored.
My undergraduate students often had abysmal posture, which I
think made concentration more difficult.
Next, the yoga tradition recommends bringing our breathing under control.
This is *pranayama.* It has the effect of making the second sheath,
the prana body, steady and comfortable so that it can be ignored as well.

The next step in solving the mathematical problem is to get ourselves to
think about the problem --- that is to say, to produce a state of mind in
which the thoughts that arise are
about the problem. Patanjali refers to this as the first stage of samadhi.
In his jargon, which I will usually follow, this is
*savitarka samadhi*, or "samadhi with reasoning."
Achieving savitarka samadhi is an example of shamata. We
recognize irrelevant thoughts as irrelevant, and then let them go and do
not follow them. However here the
goal is not to be aware of the breath and the posture, but of the object
of the concentration --- the mathematical problem.

Buddhaghosa, writing Pali rather than Patanjali's Sanscrit, refers to
stages of *jhana* rather than stages of samadhi. The Pali word
"jhana" is cognate to the Sanscrit "dhyana," which
means meditation. I will ignore this minor terminological difference.
Buddhaghosa gives two different
classifications of the stages of samadhi. I will focus on the second,
which is closer to Patanjali.
Buddhaghosa's first stage is characterized by *vitakka*, corresponding
to the Sanscrit "vitarka" of Patanjali. Nanamoli translates this word as
"applied thought." What is characteristic
of this stage of samadhi, in my experience, is verbal or conceptual thought.
The mind is occupied with discursive reasoning about the object of the samadhi.
Buddhaghosa says that
vitakka "is the act of keeping the mind anchored ...
like the ringing of a bell,"
or "like a bird's spreading out its wings when about to soar into the air."
On the other hand, the thought characteristic of the next stage,
*vichara,* is "quiet, like the bird's planing
with outstretched wings after soaring into the air." (Visuddhimagga IV, 89)

In the first stage of samadhi we are resting in the conceptual mind. That sheath is active, but relaxed and focussed on the problem.

What is essential in any kind of concentration, whether solving a mathematical problem or sitting in meditation, is to let go of the inner monologue that maintains the emotionally charged dream that is the world of our ordinary life. Then the pain of samsara begins to recede, and there is room for other visions to appear.

When we do mathematics, it is often possible to solve the problem in this first stage of samadhi. We follow the words that come to us, and construct a rational argument that gives the answer to the problem. That rational argument is called a "proof," and all undergraduate mathematics that goes beyond algorithms amounts to the construction of proofs. Indeed, the same is true of much graduate mathematics and even some professional mathematics. However, not much creativity happens on this level. Let me assume, for the sake of my exposition, that the mathematical problem we are working on is sufficiently difficult that it cannot be solved just by reasoning about the situation that the problem concerns.

In the first stage of samadhi, there is still a stream of words.
The thoughts keep coming, although they are all thoughts about the object.
The next stage is to stop the thoughts,
so that our consciousness is fully absorbed in the object.
We are still aware of the object as being what it is.
But the words have stopped. (Actually, this is an oversimplification.
Words do still come, although there are long spaces between them.
But we are not attending to the words, and often they are not
particularly meaningful. My experience is that the words that come are
often some slogan that is repeated to absorb the functioning of the
discursive part of the brain. Such repetition of a slogan
is often done deliberately as a meditative technique. In that case it
is usually called *mantra*.)

Now we are in the second stage of samadhi, whether we are following
Patanjali's scheme or Buddhaghosa's. In Patanjali's jargon,
this is *savichara samadhi*, or "samadhi with reflection."
(What I am translating "reflection,"
Nanamoli translates "sustained thought." It means a non-verbal
contemplation of an object or pattern.)
It is here that creativity occurs. We are completely
absorbed in a geometrical pattern, let us say. Suddenly, the pattern shifts.
Now we see a possible solution to the problem. For example, consider a
classical Euclidean problem. It usually turns out that we have to
construct a new line, or something of the
sort. We are thinking about the picture, focussed on the picture, and
suddenly we see the line. The mathematician, utterly absorbed in his
trance, unaware of what is going on
around him (to the great annoyance of his wife), suddenly says, "Oh! I see!"
and the trance breaks. He quickly makes a sketch or writes a note
on his yellow pad, so that he does not forget what he saw, and now
he is willing to pay attention to his surroundings.

In order to understand samadhi, as we see it discussed in the yogic and Buddhist texts, it is crucial to understand what happens when the mathematician sees the new line in his diagram, or more generally when he sees what he hopes is the solution to the problem he is working on. His mind is completely focussed on some sort of abstract pattern. He is completely absorbed --- entranced. Moreover, in his mind, the words have largely stopped. Suddenly, the new object appears. He sees it or hears it. It comes from nowhere. He does not consciously create the new object. It is just suddenly there. That is the appearance which is the product of his trance. People say vague things like, "it was the product of his unconscious mind." The fact is that no one knows where such ideas come from. They emerge, magically, from the trance.

In the second stage of samadhi we are ignoring the conceptual mind and are resting in the perceptual mind. The conceptual mind, so to speak, is idling. The perceptual mind is working to decipher the pattern we are trying to understand.

But a trance has to be interpreted. It is not its own intepretation, contrary to what some people think who write about divine revelation. The idea comes from nowhere, and it comes without instructions about how it is to be interpreted or used. That is the function of the mathematician's training. He can apply his algorithmic skills, which do not involve samadhi, or his reasoning skills, which at most involve the first stage of samadhi. One way or another, he tries to make the idea solve the problem. Professional mathematicians often say, "The idea is ...," and then say a few words. They expect that their hearer can do the routine work that leads to the explicit solution, which may be many pages of computation or reasoning. But the idea is not routine. It emerges from the samadhi.

Often it turns out that the idea does not work. That is the role of what the mathematician calls "rigor." His training has given him a critical apparatus to distinguish between a correct proof or computation, and one that only seems correct. Since the trance seems to dictate its own interpretation, it often happens that the mathematician thinks the idea works, but is wrong. Traditionally, he goes down the hall and shows his work to a colleague, who points out the error if there is one. Mathematicians often say, "I have a solution, but no one has checked it."

Now suppose that no idea comes, or that the idea that came turned out
not to work, and
no new idea comes. The mathematician is fully entranced, completely
absorbed by the
pattern. Now, if he is lucky, the pattern becomes his consciousness,
so that there is
no longer any conceptualization whatever. Patanjali's metaphor is that
the mind of the yogin is like a piece of glass sitting on a colored cloth.
The glass takes on the color. If you look through the glass, you do not see
the glass, but only
the color. In that case, there is bliss. The state is joyous.
This is the third stage of samadhi, *sananda samadhi*, or "samadhi
with bliss." (Buddhaghosa separates this stage into two stages.
He does not use a word cognate to the
Sanscrit *ananda,* which I am translating as "bliss." Instead,
he speaks of a third stage characterized by *piti,* which
Nanamoli translates "happiness," and
a fourth stage characterized by *sukha,* which Nanamoli
translates "bliss." I do not
know what distinction Buddhaghosa, following the Pali tradition, is
trying to make. I do not find such a distinction
in my practice. I will follow Patanjali in not making the distinction.)
The professional pure
mathematician thinks of this state as the reward for his hard work.
This state is
what makes it all worthwhile. He is usually not able to say much about
the state as
such. He talks instead about the "enormous aesthetic appeal" of the subject,
or "the amazing beauty of the ideas in this area," or something of the sort.
That is, he falsely attributes his bliss to the object of the trance, rather
than to his own mind.
This is called "seeking the Buddha outside yourself." It is a characteristic
human mistake.

In the third stage of samadhi, we are ignoring the perceptual mind and are resting in the emotions. In the mathematical case, that sheath cannot usually make a contribution to the solution of the problem. Nevertheless, at this stage it is active and quite prominent in our experience.

I should point out that although both Patanjali and Buddhaghosa, as well as
much of the rest
of the tradition, describe the third stage of samadhi as blissful ---
that is to say, euphoric ---
there are important meditative techniques which lead to other emotional
flavors in this stage of samadhi. An example is the Tibetan technique
called *vipashyana.*

The bliss that we experience in the third stage of samadhi is a distraction
from the object of the trance. In the mathematical context, it does not
contribute to
solving the problem. The mathematician, if he is very accomplished, learns to
let go of the bliss. What then happens is that his mind lets go of the
specific object he has been studying and opens to a larger vista in which
that object is
located. He understands the subject as a whole, seeing broad patterns which,
perhaps, illuminate the larger context of his problem. In Patanjali's jargon,
this is the fourth
stage of samadhi and is
called *sasmita samadhi*, or "samadhi with self-consciousness."
Buddhaghosa says that this fourth stage of samadhi is characterised by
*ekagatta*, which
Nanamoli translates "unification of mind." This word is also often
translated "one-pointedness."
Buddhaghosa emphasizes that the fourth stage is characterized by equanimity
and mindfulness. The meditator is now free from all clinging.

In the mathematical setting, it is in the fourth stage of samadhi that we most strongly encounter the phenomenon that mathematicians call "intuition". It is the highest achievement of the professional. Colleagues say, "His intuition is wonderful. He was able to see that such and such had to be true, although it took ten years before anyone was able to prove it." His insight emerged from the trance. It was based on nothing, but was correct.

In the fourth stage of samadhi all of the sheaths are quiet except the innermost Emptiness, which manifests itself as a bare consciousness of the object of our meditation.

Finally, if we can reach the fourth stage of samadhi, there is a way beyond.
We must let go of the sense of self,
and with it of the object. Then there is nothing at all: no reasoning,
no conceptualization, no bliss, no sense of self. That is what Patanjali calls *nirbija samadhi*, or "samadhi
without a seed". (The "seed" is the object of the concentration.)
To my knowledge, mathematicians do not reach
this stage. Or at least, not many of them do.

Lodrö Chödrak

4/5/2012

Lodrö Chödrak can be reached by e-mail at

Lodro@LodroChodrak.com