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Buddhism and Stereotyping Perceptions
An answer has prodded me to completely reconsider my understanding of Buddhism and emptiness [this answer]. I never realized that Buddhism involves the dismantling of preconceptions and false certainty. Due to some spiritual experiences, I feel it's clear that the path to a happier life for me is changing this.
However, I know for a fact that I continuously generate such preconceptions and approximations almost pathologically. I feel that doubt greatly arises from this, in my mind; I keep making inexact and hasty conclusions, and assuming things. I cannot merely not know.
My question is: What are the ways to dismantle such biases of mind? I specifically mean those of speculation, stereotyping, jumping to conclusions, and so on.
Simply put, I'm wondering how one trains or comes to accept uncertainty.

The habit of mind to Speculate and Stereotype is developed over thousand of years of psychological evolution. It was at some point in history, very important for humans to classify the objects, they saw and and hence speculation and stereotyping gave advantage, when the world around humans was not very complex.
The point I am making is, these are habit patterns of mind created over thousands of past lives.
As against 'jumping to conclusions', as we mean it in a judgemental sense, is some kind of recent phenomena of fast paced life.
May be, even the stone age humans must have been jumping to conclusions at sight of some thing akin to snake.
So, I am trying to say, these are habit patterns of everybodies mind.
Now answering your question, how one trains to accept the uncertainity
The point of Buddhism is not to train the mind (there are therepies in modern psychology to do just that), instead we try our best to see things as they are. So the way to dismantle the bias is, to be aware to the present moment and be watchful of your mind as much as possible.
This gives rise to intelligence, and through that intelligence you dismantle the habit pattern.
When you fail, don't be judgemental towards yourself and feel bad. Instead make a note and understand that as all your intelligence was available what you did was the best you can do.
Through such repeated feedback your biases will be dismantled without specifically trying to train the mind which is a superficial and phony solution.
To put it simply, Stay Aware to the Present Moment.

I don't have any better advice than this song:
Mahamudra transcends all words
And symbols, but for you, Naropa,
Earnest and loyal,
I have to say this:

The Emptiness doesn’t need support,
Mahamudra rests on nothing.
Without making any effort,
One can break the yoke
Thus - reaching Liberation.

If one sees nothing when staring into space,
If with the mind one then observes the mind,
One destroys distinctions and reaches Buddhahood.

The clouds that wander through the sky
Have no roots, no home; nor do the distinctive
Thoughts, which float through the mind.
Once the nature of mind is seen,
Discrimination stops.

In space shapes and colours form,
But neither by black nor white is space tinged.
From the nature of mind all things emerge,
The mind is not tainted by virtues or vices.

The darkness of ages cannot hide the bright sun
Nor the long kalpas of samsara
can hide the splendid light mind.
Though words are spoken to explain the Emptiness,
Emptiness for itself can never be expressed.

This world is transient,
like ghosts and dreams, without any substance.-Tilopa

You can find it full here

You are asking how to train in overcoming mind's tendency to stereotype and to reify superficial perceptions. Generally speaking, entire Buddhist path is about training in this. Much of the philosophical study (Madhyamaka) and practice of Zen and Vajrayana's higher tantras (Mahamudra and Dzogchen) are aimed at transcending the limits of superficial mind.
Working with a live contemporary teacher adds a practical aspect that is not readily explained in texts. This aspect has to do with application of Buddhist insights to everyday life. The superficial mind with its tendency for stereotyping keeps creating reoccurring patterns in one's personal life. It is by exploring and challenging these patterns that we get to see our superficial mind objectively. This leads to gradual disidentification and eventual freedom from stereotypical thinking.
In my personal experience undergoing this training, there are two parts to it. The first one is explained conceptually, for the student to practice on his own, between the periods s\he meets with the teacher. The second part is traditionally taught by the teacher in-person, without much explanation.
The part that's explained conceptually is a standard instruction on non-attachment. In my "lineage" it goes something like this:

REMEMBER: every time whenever you feel a negative emotion in response to a real life situation,
assume the emotion comes from an "attachment" (a traditional word for preconceptions and stereotypes), then
engage in 5-second introspection to quickly identify the specific attachment, then
make a silent symbolic gesture of admonishing your ego for trying to protect its solidity, and then
mercilessly take the attachment away from your ego,
FINALLY, watch the emotion subside and deal with the situation properly.

The act of giving up your attachment is designed to leave you highly vulnerable - but also much more open to other perspectives and frames of reference. By definition, the more a certain idea is important to you, the harder it is to let go of attachment. Ego is a master of rationalization and to be honest some of its reasons are actually pretty logical and often valid, but in context of this practice all these rationalizations must be ignored, and all attachments must be let go. The exercise gets seriously difficult when it involves dealing with other people over different interests and perspectives, especially when this is a real life or work situation, especially when you are right - but this is exactly what makes this practice so effective.
The second exercise consists of your teacher occasionally pushing you to operate outside of your comfort zone. The normal student's reaction is to indulge in one's egoistic defenses until one day the student learns to get over them and open up to non-preconceptualized activity. This involves enormous creativity on the teacher's part, setting up all kinds of embarrassing situations in which the student is put on the spot and does not know how to react. It's hard to give specific examples since they tend to be very personal. A good Zen master or Vajra master has a talent for endless improvisation of these. The practice is both very painful and very effective for driving the student outside his or her preconceptualization.
In the absence of live teacher my advice is to get creative yourself and start finding ways to put yourself in situations when you do not have a good plan of action. Luckily, modern social life gives endless opportunities for trying one's luck outside one's comfort zone and watching oneself stumble.
Both of these exercises work by forcing the student to lay aside one's  preconceived notions, and practice operating in the preconceptual space of "beginner's mind".
It's seems awkward to explain all this in writing, as it is usually presented in a much more pithy and casual style in the middle of real life situations. Despite my limited ability to explain it with clarity, I can't overemphasize the importance of this practice. In my experience, it makes night-and-day difference in one's ability to comprehend and realize true Dharma. Hope it helps.

I've bumped into various prosaic and spiritual messages that have cultivated in me a certainty that most certainties are unjustified.
Studying debate in high school they drilled how to detect and avoid logical fallacies into us. It's easier to detect in the writing of others than your own, I find. If you can reliably detect them in your own thinking, you will cultivate a mental awareness that can help avoid these things. I could be wrong, but I think there's important elements of mindfulness involved with this process. I've found fallacies are used the most frequently when it comes to truths people don't want to face.
I'd not try to avoid speculation. I'd actually encourage it, especially when it comes to criticizing what's going on in your head. What information could you acquire, at least in principle, that would prove you wrong? Employ any criticisms you might offer to someone else to yourself. I suspect this facilitates taming Ego. The other side of the coin has similar effects. Have compassion for yourself as you have compassion for others.
Meditate. Watch your thoughts very carefully. What topics arise frequently in what order? Do you notice any patterns? You're likely to find there are some things you want to be true that might not be. That I suspect is the source of a lot of jumping to conclusions.
Do not seek perfection. Perfection is Ego lying to you. Don't get discouraged from meditating because your thoughts take a long time to settle. Don't think that you just can't do it.
These suggestions are heavily science/logic influenced, very linear. There's some problems stuck in thinking that way. I find Thich Nhat Hanh has a lot of great advice deviating from that pattern while cultivating understanding, especially in Zen Keys. He is very good at clarifying what information is being ignored when we try to contain things in our head with concepts.
Hope that helps!

The cause to your biases, speculation, stereotyping, jumping to conclusions,.....are your strong attachments to Impermanence.
To cure this, you must Recognize the "Nature of impermanence". Once you did, your biases, speculation, stereotyping....will cease to exist, for they are also  impermanent.


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How do I practice ' see things as they are'?
Further to the answers to this question, I want to understand what literally is 'see things as they are' or as the sutta states, Yathā-bhūta-ñāna-dassana.
My questions are,

I understand that if I am looking at a beautiful woman, I am obviously not seeing as she is, but what when I am looking at a clear blue sky, not clouded by ignorance, lust, and greed, am I seeing things as they are? Or I should necessarily have to have insight into Three marks of existencece? Can I see things as they are without Enlightenment?
If so, can only an enlightened being see things as they are?
Sometimes when we look at a flower we can see something transcendental, something which is greater than the sum of the parts. (Not a concept from Buddhism) but some people have an aura around them, even the Buddha in pictures is shown with an aura around His head. So does seeing this metaphysical reality constitute seeing thins as they are?

You can see things as they are, everyone has direct perception, but it is typically rare in beginner's mind. Awareness of the delusion once the formation arises points consciousness directly at the seed of the manifestation, rather than the mind defilement itself that arises from it. Meditation and mindfulness train the mind to deliver wisdom (panna) of insight quick enough so one doesn't get caught in the defilement unknowingly.
See this explained here, from Thich Hnat Hanh - "Understanding Our Mind: 51 Verses on Buddhist Psychology":

When seeds of consciousness manifest themselves in our mind
  consciousness, either we perceive them directly or we do not.
(...)
The first field of perception is the perception of
  things-in-themselves, perceiving directly without distortion or
  delusions. This is the only one of the three modes of perception that
  is direct. This way of perceiving is in the realm of noumena, or
  suchness. Suchness (tathata) means “reality as it is.”
(...)
Are we capable of touching reality-in-itself? The Buddhist teachings
  say that we can. A flower can be the manifestation of the world of
  suchness, if we perceive it directly. It all depends on our mode of
  perception whether we touch the suchness of the flower or only an
  image of it that our minds have created. Our perceptions rarely reach
  the mode of things-in-themselves, however. We usually perceive things
  in the other two modes, as representations or mere images.
(...)
We are unable to reach the mode of perceiving
  things-as-they-are-inthemselves because our distorted image is a
  “representation,” not a direct perception.
(...)
The first five consciousnesses—the sense consciousnesses of eye, ear,
  nose, tongue, and body-are capable of touching the realm of
  things-inthemselves, especially when they contact their objects of
  perception without the participation and intervention of mind
  consciousness. When mind consciousness gets involved, however, there
  will always be some thinking and imagination, and the image brought to
  it by one of the sense consciousnesses will become distorted. (...)
  Anything touched in the realm of
  things-inthemselves by a sense consciousness and then processed by
  mind consciousness becomes a representation. Yet even our mind
  consciousness can touch the realm of things-in-themselves from time to
  time. When we have a strong intuition, our mind consciousness is in
  touch with the realm of suchness. Intuition is a form of knowing that
  is not based in thinking and imagining.
(...)
We are capable of reaching the field of things-in-themselves, the
  world of suchness, but because we think and discriminate we don't
  usually perceive things as they truly are. The nature of our mind is
  obstructed (parikalpita). This means that we build a world full of
  illusion for ourselves because of the distorted way we perceive
  reality. Meditation is to look deeply in order to arrive at
  reality-first the reality of ourselves and then the reality of the
  world. To get to that reality, we have to let go of the images we
  create in our consciousness and our notions of self and other, inside
  and outside. Our practice is to correct this tendency to discriminate
  and think dualistically, so that reality will have a chance to reveal
  itself.

Note that this is Mahayana perspective.

In my understanding of the explanations I received:
As many Buddhist concepts, "seeing things as they are" is a pointer to something happening in real life, but not necessarily in a sense we assume. Its meaning is more practical than strictly technical.
Most of the time we are "in our heads". We have so many preconceptions, thoughts, and inner chatter going on, that we can barely see what's going on in the real world. Instead, our vision is limited by our thoughts and preconceptions to such a degree that we barely notice things outside of our frame of reference. We tend to ignore them for two reasons: either
A) in our frame of mind they are not cohesive phenomena, they seem illogical (because we don't understand their inner logic) - and so we tend to ignore them as noise, or
B) they go against our emotional attachment to some sort of theory or ideology, and so we semi-subconsciously dismiss them as fake / illusory / insignificant.
When we learn to suspend our judgement and lay aside our attachments and preconceptions (this is what Mahayana's realization of Emptiness entails in practice), and stop our never-ending inner-monologue, and lose solidity of this heap of overgeneralizations we call "ego" -- then and only then we start seeing things "as they are".
In my experience with it, this vision liberated from preconceptions can be allegorically called "quantum" or "multidimensional". Because we are not limited to one theory and one interpretation of what's going on, we see them all, or rather, we directly see the entire "space of interpretations". There is an element of ambiguity, but it's not regular ambiguity which is hazy vision - instead it is a very clear kind of ambiguity.
It's a very practical, very direct kind of vision. We are simply not in our heads, are eyes are open and we really look around, and see things we have not seen before.
This is the critical point of the explanation.
And then on top of that, as we develop prajna, we begin to "see" "hidden connections" between things. These are not as much some magic properties as abstract relationships of (latent) influence and causation.
Combination of this "quantum" vision and seeing the latent influences is called "seeing things as they are" - in my understanding.

In this context, the word 'multidimensional' sounds more befitting than 'quantum'. thanks for the answer.

Thanks, fixed the spelling. Although, to me, quantum fits pretty well too. It's like we are in multiple places at once. Thanks!

Craving, delight & seductive qualities beguile the mind; including when seeing radiant monks & Buddhas. To see things as they are requires a pure mind without defilements; without craving.
Its a chicken vs egg scenario. Enough defilements must be cleansed so seeing things as they are can put an end to defilements.
,
Mentality ('nama') is just as or even more beguiling than physicality ('rupa'). For example, the so-called 'beauty' of a female form is more than mere physicality. The mind of the woman is also exuding an enticing quality.
Thus, to see things as they are, both internal & external bodies & minds must be seen as merely that: mental (nama) elements & physical (rupa) elements.
The elements are listed in MN 115. Viewing external objects as mere mentality phenomena (nama) and physical phenomena (rupa) is described in SN 12.19.

You know black hole because you fluently understand the knowledge of the relation of physical&chemical properties about black hole. It doesn't mean you have seen black hole.
"Seeing things as they are" is the same. You know their paṭiccasamuppāda (causes) in every time before you know them (effects) directly.
You think some one mad because of their properties, so you never think the dead body is mad. The same way, the practitioner think clinging-aggregates are "impermanent, suffering, anattā" because they aggregates depending on clinging-paṭiccasamuppāda. So, without paṭiccasamuppāda 12, the practitioner can not see impermanent-characteristic, suffering-characteristic, anattā-characteristic"  of "impermanent-aggregates, suffering-aggregates, anattā-aggregates".
This is the reason why the practitioner can meditate sammasana-ñāṇa(realizing everything have 3 characteristics) after paccayapariggahañāṇa (realizing paṭiccasamuppāda/paṭṭhāna).
So, to practice "Seeing things as they are", you have to practice to see everything in paṭiccasamuppāda style. But before that step, you have to make sure that you having good enough adhi-sīla & adhi-samādhi (sīlavisuddhi&cittavisuddhi).
See, KN Paṭisambhidāmagga Dhammaṭṭhitiñāṇaniddesa & Visuddhimagga Paccayapariggahañāṇaniddesa. Both path of them describing yathābhūtañāṇa, directly:

The knowledge that has been established by the overcoming of doubt about
  the three periods of time by discerning the conditions (paccayapariggaha[ñāṇa]) of
  mentality-materiality according to the various methods should be
  understood as “purification by overcoming doubt (kaṅkhāvitaraṇavisuddhi).” Other terms for it
  are “knowledge of the relations of states (dhammaṭṭhiti)” and “correct knowledge (yathābhūtañāṇa)” and “right vision (sammādassana).”
For this is said: “Understanding of discernment of conditions thus, ‘Ignorance is a condition, formations are conditionally arisen,
  and both these states are conditionally arisen,’ is knowledge of the
  causal relationship of states” (Paþis I 50). And:

...
“When he brings to mind as impermanent, he correctly knows and sees
  the sign. Hence ‘right seeing’ is said. Thus, by inference from that,
  all formations are clearly seen as impermanent. Herein doubt is
  abandoned. When he brings to mind as painful, he correctly knows and
  sees occurrence. Hence ... When he brings to mind as not-self, he
  correctly knows and sees the sign and occurrence. Hence ‘right seeing’
  is said. Thus, by inference from that, all states are clearly seen as
  not-self. Herein doubt is abandoned.

Another, I also describe yathābhūta in this answer:
https://buddhism.stackexchange.com/a/26699/10100

It is not ''to see things as they really are'', contrary to the stock translation.
it is:

once you want to stop experiencing once and for all misery,
to see once and for all ''how things arise and go away''
which means to see once and for all they are mediocre, pathetic and unreliable, ie dukkha
which means seeing once and for all they are not ''I nor mine'', ie not your business not your story, ie anatta
which means experiencing once and for all dissatisfaction with them and not to even touch them
which means experiencing once and for all disinterest with them
which means nirvana.

The first step to be sucked into this spiral is

to ''''''''be moral'''''' like some puthujjana say, meaning not causing drama over mundanities with other people until you have no regrets nor remorse nor worry about their mundane activities,
once there is no worry over the mundane activities created and cared by those people, you are left alone to have time to experience the ''tireless energy'' and joy of the jhanas,
once there is this joy, energy, relaxation, you become concentrated and have most of the ''''seven factors of Enlightenment''''''
which I copy paste here

** - Mindfulness (sati).
 - Investigation of the nature of reality (dhamma vicaya).**
 - Energy (viriya) also determination
 - Joy or rapture (pīti)
 - Relaxation or tranquility (passaddhi)
 - Concentration, clear awareness (samādhi)
 - Equanimity (upekkha).

Now to get to the point of no return, you must develop the first two, otherwise you are stuck at the jahnas like the hindus.
It turns  out what remains is precisely remembering the doctrine and investigating it, which exactly means the first list above:

once you want to stop experiencing once and for all misery,
to see once and for all ''how things arise and go away''
which means to see once and for all they are mediocre, pathetic and unreliable, ie dukkha
which means seeing once and for all they are not ''I nor mine'', ie not your business not your story, ie anatta
which means experiencing once and for all dissatisfaction with them and not to even touch them
which means experiencing once and for all disinterest with them
which means nirvana.

you said 'otherwise you are stuck at the jahnas like the hindus', I am also of somewhat similar opinion that Hindu texts are stuck in Jhnanas. If it is possible can you provide me more info on this? for my further research.

Under Alara Kalama and Uddaka Ramaputta, Buddha discovered that such states of concentration are impermanent, bringing merely momentary ease and do not lead to liberation. Therefore, Buddha's teaching was mindfulness with breath, that is, bringing body and mind together as one, and not mindfulness of breath without being attentive to the body.

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The "awakened" outside of a monastic/retreat setting?
Most people cannot live in the secluded setting of a monastery or retreat. Some Buddhist traditions nonetheless affirm that laypeople can become enlightened. Are there (inspiring) examples of such laypeople?
How (well) do so-called awakened spiritual/meditation masters function outside the seclusion of the monastic/retreat environments? Please give examples. (Only one comes to mind: Patrul Rinpoche).

Conversely, how many enlightened people do we ever hear about, not living a monastic or renunnciate life? Could there be a reason for that?

I think to develop deep meditation, deep vipassana (insight), jhana, etc, a monastic setting is required so meditation can be practised full-time.
However, a person with keen insight, especially into impermanence, can easily attain stream-entry outside of a monastic setting.
Note: This does not apply to believers in reincarnation since they avoid the natural reality of the impermanence of life.

only the mundane, wordly parts of the Buddha's teachings (e.g. kamma, rebirth, mundane right view), not the teachings leading to total liberation, deal with "functioning" in the layperson's life.
the Buddha taught the path leading to total liberation (which supramundane right view leads to)  - which does not necessarily have to translate into worldly 'functioning' for material ends.

Daniel Ingram claims (or claimed) to be an arahant.
His site: http://integrateddaniel.info/
More on him in this thread, which mentions other enlightened people.
https://www.reddit.com/r/Buddhism/comments/1tiy7o/dan_ingram_arahant_or_raving_manic_lunatic/
I can't say whether they are or aren't enlightened. From what I read, I doubt it.

Milarepa
...His mother, bitter, sent Milarepa to train in black magic, to wreak revenge on those who had blighted her life...
Then later in life he came to meet his teacher Marpa the translator
...he came to regret his actions, and looked for help in shedding the bad karma he had acquired during his vengeful adolescence...
Tilopa
He practiced this teaching at Somapuri, but when the monastery saw him take a female consort for the practice of union yoga, he was forced to quit the community.
Tilopa profited from his expulsion by travelling throughout India, searching out many teachers and learning their methods.
Naropa
Several years later, at twenty-eight, his interests drew him to Nalanda University, near Pullahari, famous for its Buddhist philosophers...But then a dakini appeared to him, telling him that meditation practice was more important than philosophizing, and that he should search out a certain Tilopa for instructions.Leaving the monastery...
This is an entire lineage Tilopa->Naropa->Marpa->Milarepa
As they say :The truly enlightened are not necessarily very religious. Religon is a convienent description to take care of our fears, to passify us.

Answer would be better with links to sources or describing these persons.

Ignition--starting of engine--running of the motor car. So if this ignition happens due to leaving incomplete in past life, issues making life uninterested, then be you are in monastery or at your home ,the engine starts and the motorcar runs towards destination of enlightenment. Now if ignition is absent ,then go on rubbing ,rubbing,rubbing until spark is produced. it can produce in a day or cant produce throughout life. In monastery ,where deliberation of practice is there, the rubbing process goes on . At least there is hope that some or other day the spark/ignition will produce.And when it happens ,the engine starts and the car runs towards enlightenment unless stopped by meting some accident or b/d of the engine.This i elaborated that the enlightenment can be attained anywhere but required is the above prerequisite.

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Does anyone read the part 2 of Nagarjuna's karika as an argument about infinite divisibility?
Is Nagarjuna is arguing that time cannot really be both infinitely divisible and extended, so is a conceptual construction?
I'm asking because verse 2.23 suggested that to me, taken completely out of context, anyway.

[A going] which is other than the going by which a goer is made evident does not [enable a goer to] go.  Because it is impossible for
  going to be twofold within a single goer.

Emphasis, obviously, mine. I really like the idea that Nagarjuna was saying that time must be infinitely divisible, but time as we experience it (evidently) is not infinitely divisible, because if it was it would take forever to compose it.

As I think it was pointed out in an answer, it is often I think said that Nagarjuna means time cannot inherently exist because it is infinitely divisible. I'm just adding that it can neither be infinitely divided nor not.
I would not conclude that time does not flow, but that anything that arises has already passed, radical impermanence, suggesting that extinction and so on is already present.

Here is the translation that by Jay L. Garfield of that same verse:

2.25 One that both is and is not a goer does not go in any of the three ways. Therefore, going, goer and that which is to be gone over are non-existent.

This is from the second chapter of Nagarjuna's Fundamental Treatise where Nagarjuna refutes the inherent existence of motion: going and that which goes. This verse is the last of the chapter and Je Tsongkhapa in his Ocean of Reasoning explains (by adopting Buddhapalita's earlier explanation) the three ways refer to:

That which is gone
That over which one is going
That which over which one has yet to go

ie., the three times. Je Tsongkhapa says there is another way to understand the three aspects here (by adopting Chandrakirti's explanation) as referring to:

That which is a goer
That which is not a goer
That which is and is not a goer

In sum, the verse can be read as refuting the inherent existence of the goer, the place one is to go, or the going. All of these exist as mere conventions. There is no essence of motion.
Anyway, maybe you can update your question to indicate how you arrive at the conclusion that time cannot be both infinitely divisible and extended? That's basically saying that time cannot be regarded as modeled by the real numbers... but rather must be the infinite integers?? I don't think this is what Nagarjuna is saying because even if time is infinitely divisible and extended it would still be proper to model time by the past, present and future.
Let's say that the past, present, and future are modeled by real numbers. All we have to do is identify the present with a specific real number and say all those numbers less than, but not equal to the present model the past and all those numbers greater than, but not equal to the present model the future. Even with this, Nagarjuna's argument in the chapter would still accomplish the goal: refuting the inherent existence of the goer, the act of going, and the place where one goes...
You also refer to verse 23 in your update, but this verse according to Je Tsongkhapa is just a refutation to the idea that there is more than one action or "going" that makes a goer go. Nagarjuna's position is that only one action is tenable because a two-fold action makes no sense. Although it is obvious that it takes time to perform an action, these verses and this chapter do not explicitly deal with time. The chapter is about motion and how motion does not inherently exist.
However, there is a chapter in Nagarjuna's Treatise that explicitly deals with time. That is chapter 19 and there is one verse in particular that deals with whether or not time "endures":

19.5 A nonenduring time is not grasped.
  Nothing one could grasp as time
  Could exist as enduring.
  If time is not grasped, how it is known?

You can read more about this verse and Je Tsongkhapa's explanation in Ocean of Reasoning, but I would just add that it is clear from the first phrase that we experience moments of time as enduring. That is directly contradictory to the way I used the real numbers above to model time. We perceive the present moment as enduring for some length of time. If we look at the present moment and analyze it we'll find it is made of parts. For anything that has an extent must be made of at least three parts: the beginning, the middle, and the end. Thus, if the present moment is enduring then it must be made of parts. If it is made of parts, we can analyze further and ask if an inherent entity that is this present moment can be found in the beginning of this present moment? In the middle ? In the end? By looking and analyzing like that it can be known for sure that the present moment cannot be made of any inherent entity. The present moment does not exist inherently.
So, I'd conclude and say that it is precisely through analyzing an enduring moment of time and breaking it up and dividing it infinitely we can know that no enduring moment whatsoever truly exists. Time does not truly exist. It only exists as mere label. It only exists conventionally.
That is emphatically not to say that time does not exist at all! That way lies nihilism and we can be sure that if we look at the above explanation as somehow indicating nihilism, then we have grasped the snake of emptiness incorrectly and ought to put it down right away!

good answer, thanks. i hope the edit helps?

@user3293056 Hi, I've updated the answer accordingly with your edit. Hope this helps!

the section on real numbers i didn't like, i felt that it didn't understand the question. i did like the section on time being an infinite regress, and that's something quite like what i was saying... i might edit the question again

Not sure how you arrived at that interpretation, care to elaborate?
To me, this reads as a typical application of Buddhist co-dependency principle: an agent of an action is identified as such only by the fact that he is the subject of that very action. While an action must have something undergoing it, so without an agent it is meaningless either.
Therefore, any notion of "change", as in something undergoing change while still staying itself in some sense - is a conceptual fiction.
I suppose you assumed that "three aspects" refer to past/present/future moments of going? In my intuition instead, they refer to the three possible interpretations of the agent given in the text above: agent as the goer, agent as not a goer (yet or anymore), and agent as not a goer before and after and a goer in between.
This doesn't mean your insight is wrong, even if that's not what Nagarjuna meant, so I be interested in hearing more about it.

hey, thanks for the reply, hope the edit is OK!


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Difficulty with Meditation Posture
I am able to meditate when on a bus, or when I'm walking, or sometimes standing; however, no matter how hard I try I seem to avoid formal sitting. I can't help but have the feeling that formal sitting would be most effective, and that I'm avoiding something about a formal posture that is also responsible for growth. I think this is merely effort.
What can I do to engage in formal sitting, and avoid always only meditating when it is effortless? Should I: 1) diminish the time of sessions 2) change my meditation type 3) change the consistency of my meditations 4) apply something else?
In terms of consistency, I wonder if consistent yet short sessions are better than long and inconsistent ones?
Thank you.

hey eggman I see that you have asked this queston multiple times,could you define your question a little better what exatly you looking for?did my answer help you in any way?

As per my teacher, this happens when we train ourselves to hate meditation. We do that by turning meditation into torture and punishment.
Instead, meditation should be something natural and (yes) pleasant.
-- 1) diminish the time of sessions?
Yes. You can start from something as short as 30 seconds. Do not ramp up too fast, only ramp up when you really WANT to sit more.
-- 2) change my meditation type?
Yes. Do not weigh your emotional mind with any notion of the correct target state. Do not blame yourself for not conforming to some ideal style, posture, or state of mind. Allow your meditation to be a natural discovery process that it is. Like a child playing a game, completely natural and spontaneous.
-- 3) change the consistency of my meditations?
Yes. Better meditate regularly, twice a day, 30 seconds each time - than randomly for an hour.
-- 4) apply something else?
Yes. Make it pleasant for your emotional mind. Beautiful setting, nice scenery, little visual noise in your peripheral field, comfortable pants, etc.

I am inner heat practitioner and I have had the same difficulties as you!
I could not understand the importance of why should I take the lotus posture when I am way more comfortable sitting in a chair or even laying in the ground. Know that my type of Yoga works with internal energies and posture is very important!
Milarepa used a belt:

Milarepa can often be seen depicted in paintings with round white
  earrings along with a bright red meditation belt, extending from the
  right shoulder to the waist on the left side, used to hold the body in
  particular and sometimes difficult to perfect yoga postures.

Practice after practice I started to see some changes,the more I became proficient the more the more I started to give importance to posture,energy alignment,breath length and relaxation.
Soon enough(6 months) I found staying in the lotus posture for hours without moving at all,totally relaxed and emerged into the fire!
I hope that this helps you understand that relaxation is the key in the posture position and I would encourage you to find that relaxation stop,still as a mountain!

You are having aversion towards formal sitting and that is something to work on in mindfulness practice. Start to note this aversion more often, don't ignore it as you have to transform it into eagerness and diligence.
It is the desire of only pleasant experience of meditation that you are most likely developing and that is to be let go of.
It (from experience) is also subconsciously rooted fear of unpleasant meditation session experiences and negative outcome of a session. That too is state of clinging to. It is something that meditation and especially Insight of meditation is designed to work with by concentrating on unpleasant sensations, pain, and hinderances such as desires, aversions and restlessness.

There is the mud, and there is the lotus that grows out of the mud. We
  need the mud in order to make the lotus.
Thich Nhat Hanh

Buddhism and Eightfold path is focused on alleviating suffering, how else can you train yourself to deliver this knowledge if not from letting go of unpleasant, miserable feelings? Let dukkha in, acknowledge it, and then let it out.
I would advise on seated meditation 20 minutes in the morning after waking up and 20 minutes in the evening for the first 6 months or until you are fully comfortable with it. You can then start increasing it by small increments.


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How to Sustain Alertness
I am wondering if alertness is best sustained in terms of:

Attempting to sustain alertness in meditation, not meditating when too lethargic. In other words, always exerting oneself and keeping sharpness when in meditation, and stopping when the mind becomes unclear.
Or, engaging with meditation even when lethargic, continuing one's meditation despite the mind being unclear and hazy. In other words, waiting for the mind to clear up.

Which is more accurate?

Your meditation method cannot to be too comfortable, nor too uncomfortable, that is why people mostly choose to sit in half or full lotus with straight back so they don't slouch and sloth. Be careful not to slouch, keep back upright, work on posture and correct it.
Question 1:

Attempting to sustain alertness in meditation, not meditating when too
  lethargic. In other words, always exerting oneself and keeping
  sharpness when in meditation, and stopping when the mind becomes
  unclear.

Sustain, but gentle mindfulness or awareness, I wouldn't say alertness as it implies stress as in fight or flight type of thing. Note that you are feeling lethargic. Do not stop, note that mind becomes unclear. Eventually correct your posture, straighten the back and continue. It is generally to be mindful of it just to the extent of eating, or waking up (both in meditation and daily life)

Again, bhikkhus, a bhikkhu is one who acts in full awareness when
  going forward and returning; who acts in full awareness when looking
  ahead and looking away; who acts in full awareness when flexing and
  extending his limbs; who acts in full awareness when wearing his robes
  and carrying his outer robe and bowl; who acts in full awareness when
  eating, drinking, consuming food, and tasting; who acts in full
  awareness when defecating and urinating; who acts in full awareness
  when walking, standing, sitting, falling asleep, waking up, talking,
  and keeping silent. 9. “In this way he abides contemplating the body
  as a body internally, externally, and both internally and externally …
  And he abides independent, not clinging to anything in the world. That
  too is how a bhikkhu abides contemplating the body as a body

Question 2:

Or, engaging with meditation even when lethargic, continuing one's
  meditation despite the mind being unclear and hazy. In other words,
  waiting for the mind to clear up.

Engage, yet gently noting haziness and confusion, lack of clarity. Eventually correct your posture, straighten the back and continue.
On mental hindrances, such as:

Sensual Desire
Ill-will
Sloth and Torpor
Restlessness and Remorse
Doubt

https://www.accesstoinsight.org/lib/authors/nyanaponika/wheel026.html
Bonus: "Is there such a thing as bad meditation?":

questioner has asked regarding sustenance of alertness------you are sitting in a coffee shop aside road, lot of traffic noises are running, you are engrossed in reading your whatsapp/msgs on your cell  and suddenly you hear a blonde laughing similar to ringing of small bells. You focus your attention/awareness on it to listen still clearly and minutely. Now what does this indicate? .This implies that you see,listen,smell,touch,think as driven by desire. So here master is not you who is practising awareness on any object(kasinas), but the deep rooted kileshas,desires,asavas. So all  deliberate efforts to maintain/sustain awareness goes in vain. so these roots are to be uprooted. humble suggestion to refer "Pure Dhamma " site on net and read how to remove asavas.


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Did the Buddha intend for everyone to be enlightened?
Was it the Buddha's intention that all beings to be free from suffering i.e. enlightened?

As the Dalai Lama acknowledges, “Buddhism has evolved differently
  in different times and places and yet the essential Dharma remains the
  same.  The Buddha’s prime concern was that all beings should find
  peace and freedom from suffering.

Did he say whether it was possible for everyone to be enlightened, so that there was no one left unenlightened?
Would there then be no further (re)birth?

The Buddha does not have the intention for everyone to be enlightened.
However, the Buddha has compassion, which is his expression of alleviating the suffering of others. He is willing to teach everyone who is willing to listen to him.
So, the end goal of enlightening everyone is not his goal. However, his goal is to teach others who are willing to listen, and hope that they will also help others.
The teaching of others is his goal, and not the end goal of enlightening them. That's up to the people whom he teaches. It's in their hands, not his.
The Desanaa Sutta shows the 3 types of people he is willing to teach, to various degrees:

"Does not the Blessed One dwell in compassion for all living beings?"
"Indeed, headman, the Tathaagata does dwell in compassion for all
  living beings."
"Well then, Lord, does not the Blessed One teach Dhamma in full to
  some, but not so fully to others?"
"I will reply to this question, headman, with another. Answer as seems
  proper to you. What do you think? Suppose a peasant farmer has three
  fields, one excellent, one middling, and one poor, sandy, salty, with
  bad soil. Tell me: when the farmer wants to sow his seed, which field
  would he sow first: the excellent one, the middling one or the poor
  one that is sandy, salty and with bad soil?"
"Lord, the farmer who wanted to sow his seed would sow the excellent
  field first. Having done that, he would sow the middling field next,
  and the one that was poor, sandy, salty, with bad soil he might or
  might not sow. Why? Well it might do for cattle-food."
"Well, headman, that excellent field is like my monks and nuns. To
  them I teach the Dhamma which is lovely in its beginning, lovely in
  its middle and lovely in its ending, in spirit and in letter, I
  display to them the holy life, perfectly fulfilled and purified. Why?
  Because these people adhere to me as their island, their shelter,
  their resort, their refuge.
"The middling field is like my male and female lay-followers. To these
  too I teach the Dhamma which is lovely in its beginning, lovely in its
  middle and lovely in its ending, in spirit and in letter, I display to
  them the holy life, perfectly fulfilled and purified. Why? Because
  these people adhere to me as their island, their shelter, their
  resort, their refuge.
"The poor field that is sandy, salty and with bad soil is like my
  wandering recluses and Brahmans of other sects. To them I also
  teach the Dhamma which is lovely in its beginning, lovely in its
  middle and lovely in its ending, in spirit and in letter, I display to
  them the holy life, perfectly fulfilled and purified. Why? Because if
  they only understand a single phrase, it would long be for their
  profit."

The Kesi Sutta shows that, as an excellent teacher, he has various techniques to teach others, but he may give up on stubborn students:

"Kesi, I train a tamable person [sometimes] with gentleness,
  [sometimes] with harshness, [sometimes] with both gentleness &
  harshness.
"In using gentleness, [I teach:] 'Such is good bodily conduct. Such is
  the result of good bodily conduct. Such is good verbal conduct. Such
  is the result of good verbal conduct. Such is good mental conduct.
  Such is the result of good mental conduct. Such are the devas. Such
  are human beings.'
"In using harshness, [I teach:] 'Such is bodily misconduct. Such is
  the result of bodily misconduct. Such is verbal misconduct. Such is
  the result of verbal misconduct. Such is mental misconduct. Such is
  the result of mental misconduct. Such is hell. Such is the animal
  womb. Such the realm of the hungry shades.'
"In using gentleness & harshness, [I teach:] 'Such is good bodily
  conduct. Such is the result of good bodily conduct. Such is bodily
  misconduct. Such is the result of bodily misconduct. Such is good
  verbal conduct. Such is the result of good verbal conduct. Such is
  verbal misconduct. Such is the result of verbal misconduct. Such is
  good mental conduct. Such is the result of good mental conduct. Such
  is mental misconduct. Such is the result of mental misconduct. Such
  are the devas. Such are human beings. Such is hell. Such is the animal
  womb. Such the realm of the hungry shades.'"
"And if a tamable person doesn't submit either to a mild training or
  to a harsh training or to a mild & harsh training, what do you do?"
"If a tamable person doesn't submit either to a mild training or to a
  harsh training or to a mild & harsh training, then I kill him, Kesi."
"But it's not proper for our Blessed One to take life! And yet the
  Blessed One just said, 'I kill him, Kesi.'"
"It is true, Kesi, that it's not proper for a Tathagata to take life.
  But if a tamable person doesn't submit either to a mild training or to
  a harsh training or to a mild & harsh training, then the Tathagata
  doesn't regard him as being worth speaking to or admonishing. His
  knowledgeable fellows in the holy life don't regard him as being worth
  speaking to or admonishing. This is what it means to be totally
  destroyed in the Doctrine & Discipline, when the Tathagata doesn't
  regard one as being worth speaking to or admonishing, and one's
  knowledgeable fellows in the holy life don't regard one as being worth
  speaking to or admonishing."

The Maghadeva Sutta shows that the Buddha does not want his disciples to stop practising his teachings, and wants them to continue practising and teaching throughout the generations:

But now I have founded a good practice that does lead to
  disillusionment, fading away, cessation, peace, insight, awakening,
  and extinguishment. And what is that good practice? It is simply this
  noble eightfold path, that is: right view, right thought, right
  speech, right action, right livelihood, right effort, right
  mindfulness, and right immersion. This is the good practice I have now
  founded that leads to disillusionment, fading away, cessation, peace,
  insight, awakening, and extinguishment. Ānanda, I say to you: ‘You all
  should keep up this good practice that I have founded. Do not be my
  final men.’ Whatever generation is current when such good practice is
  broken, he is their final man. Ānanda, I say to you: ‘You all should
  keep up this good practice that I have founded. Do not be my final
  men.’”

Finally, it's up to people whether they want to become enlightened or not, according to the Mahaparinibbana Sutta:

"Therefore, Ananda, be islands unto yourselves, refuges unto
  yourselves, seeking no external refuge; with the Dhamma as your
  island, the Dhamma as your refuge, seeking no other refuge.
"And how, Ananda, is a bhikkhu an island unto himself, a refuge unto
  himself, seeking no external refuge; with the Dhamma as his island,
  the Dhamma as his refuge, seeking no other refuge?
"When he dwells contemplating the body in the body, earnestly, clearly
  comprehending, and mindfully, after having overcome desire and sorrow
  in regard to the world; when he dwells contemplating feelings in
  feelings, the mind in the mind, and mental objects in mental objects,
  earnestly, clearly comprehending, and mindfully, after having overcome
  desire and sorrow in regard to the world, then, truly, he is an island
  unto himself, a refuge unto himself, seeking no external refuge;
  having the Dhamma as his island, the Dhamma as his refuge, seeking no
  other refuge.
"Those bhikkhus of mine, Ananda, who now or after I am gone, abide as
  an island unto themselves, as a refuge unto themselves, seeking no
  other refuge; having the Dhamma as their island and refuge, seeking no
  other refuge: it is they who will become the highest, if they have the
  desire to learn."

+1 but I understand a slightly different emphasis -- i.e. in the first quote he says he teaches everyone (all three fields) in the hope that everyone may benefit. And in the last quote, the "if" may be an artefact of the translation (it's not present in this translation) -- I see that as intended to encourage the assembly that the Dhamma (and their own effort) will be sufficient after he has passed (I don't see it as specifically intended to exclude other people).

@ChrisW This Sujato translation states, "Whether now or after I have passed, any who shall live as their own island, their own refuge, with no other refuge; with the teaching as their island and their refuge, with no other refuge—those mendicants of mine who want to train shall be among the best of the best.”" where "those who want to train" gives them a choice - it's up to them.

@ChrisW This Anandajoti translation states, "Ānanda, whether at present or after my passing, lives with himself as an island, himself as a refuge, with no other refuge, with the Teaching as an island, the Teaching as a refuge, with no other refuge, those monks of mine, Ānanda, will go from darkness to the highest—whoever likes the training." where it ends with "whoever likes the training" gives them a choice - it's up to them.

@ChrisW This Thanissaro translation states, "For those who, now or when I am gone, live with themselves as their island, themselves as their refuge, with no other as their refuge; with the Dhamma as their island, the Dhamma as their refuge, not with another as their refuge, will be my foremost monks: those who are desirous of training.” where "those who are desirous of training" gives them a choice - it's up to them.

Yes I don't doubt it (that the many translations are similar): I think the option is being presented, as there, available, and continuing to be available even though the Buddha was about to pass. I just don't read into that sentence an explicit message (as might be implied by "if") that there will necessarily be those who never benefit.

This answer appears to show that the Buddha does aspire to lead all beings to Nirvana.

@avatarKorra Diamond Sutra is a Mahayana text, while my answer is based on Theravada texts.

Answer to the original question:
The Buddha doesn't have wantings per se. He has uprooted craving. He simply helps beings to achieve freedom from suffering out of compassion. Beings are infinite, so any thought of making everyone enlightened is absurd. The Buddha certainly had no such ambition.
Answer to later additions:
I do not follow Dalai Lama and the use of the word 'concern' is not appropriate as enlightened beings are not 'concerned' per se. But there's a big difference between saying what everyone should do and having an ambition that everyone will do it. Ex: It's fine to say "everyone should eat healthy food at all times". Because that's just good advice to all. But having that as an ambition is not realistic.

Please see the revised qutesion, with quote from the Dalai Lama. Why would the Buddha have the ambition for only some to be free from suffering, and not everyone? It has been said that in Buddha's teaching is the idea that every human being has the potential for awakening.

added more to the answer

Please explain how this answer does not contradict the Buddha's teaching on the Four Immeasurables which is said to be an infinite mind precisely because beings are infinite. Are the Four Immeasurables unrealistic?

sending Metta to all beings is a meditation technique used to counter aversion in the mind towards anyone. It doesn't mean that you are going to have an unrealistic ambition of freeing everyone from suffering.

Metta is nothing more than the wish for sentient beings to have happiness. Happiness is nothing more than liberation from samsara. Immeasurable means it has no measure and is limitless. One wishing to develop good qualities should train like the Buddha: cultivate the mind wishing for all sentient beings to have happiness and the causes of happiness. You label this unrealistic and imply that it is an unworthy endeavor, in so doing you label the Buddha's wish unrealistic and unworthy.

Try to understand the difference between having an ambition to make all beings enlightened and simply wishing everyone to be enlightened to develop the quality of Metta in the mind.

The wish for all beings to be enlightened is quite meaningless if paired with the unwillingness to help them do so. In other words, your emphasis on the impossible or unrealistic goal of a Bodhisattva is not congruent with the wish for all beings to have happiness and its causes. This emphasis on seeing the wish as unrealistic is limiting and remember the four immeasurables are limitless! Why would the Buddha advise us to train our minds to wish for something that is unrealistic? Isn't that absurd in your view?

It's meaningful and useful as it subdues cruelty and aversion in the mind. Buddhism is not about chasing after unrealistic goals. If you can't understand the difference between well wishing and having an ambition, no more illumination can come.

A Buddha does not want what is not possible.

59. Even so, on the rubbish heap of blinded mortals the disciple of the Supremely Enlightened One shines resplendent in wisdom.
174. Blind is the world; here only a few possess insight. Only a few, like birds escaping from the net, go to realms of bliss.
Dhammapada

Then, having understood Brahma's invitation, out of compassion for beings, I surveyed the world with the eye of an Awakened One. As I did
  so, I saw beings with little dust in their eyes and those with much,
  those with keen faculties and those with dull, those with good
  attributes and those with bad, those easy to teach and those hard,
  some of them seeing disgrace & danger in the other world.
'Open are the doors to the Deathless to those with ears. Let them show their conviction. Perceiving trouble, O Brahma, I did not tell
  people    the refined,    sublime Dhamma.'
MN 26

And, Master Gotama, when having directly known it, you teach the Dhamma to your disciples for the purification of beings, for the
  overcoming of sorrow & lamentation, for the disappearance of pain &
  distress, for the attainment of the right method, & for the
  realization of Unbinding, will all the cosmos be led [to release], or
  a half of it, or a third?
When this was said, the Blessed One was silent.
AN 10.95

I think that some Mahayana answers suggest that it is possible to participate in, to partake of, to benefit from the Buddha's enlightenment -- see for example here, or here.

Instead of writing "it's possible to participate" above, I thought of writing "it's possible for everyone or for anyone to participate" (or, "it's possible for anyone who wants to, to participate"). However, introducing the concept of "everyone" or "each person" may not conform to the  doctrine of anatta or sunyata -- see also the distinction between lokuttara and lokiya.
And it's exactly that (i.e. asking about "everyone") might be a/the problem inherent in the question.
I mean, you ask whether "all beings" can be free from suffering i.e. enlightened -- but maybe it's identifying as "a being" that is (or is a cause of) the actual problem which is an antithesis of enlightenment -- see for example Vajira Sutta:

Why now do you assume 'a being'?
  Mara, have you grasped a view?

It is possible, though, to infer other answers from the canon -- e.g. there's the story of the weaver's daughter, which is in the commentary associated with Dhp 174 quoted in Dhammadhatu's answer.
That story is quite a person-centred view (it says that this person became a stream winner before she died, that most of the villagers weren't so enlightened, that the father became an arahant afterwards) -- and (especially relevant to the question) it implies that the Buddha saw she had the capacity to become enlightened, and delayed his Dhamma talk specifically to wait for her.

Also note that, at least according to the Pali canon, there are different stages of enlightenment.

To clarify the question, it's asking if a world where everyone is enlightened was conceived by the Buddha.

I think that the principle message from Pali canon, on that subject, is in MN 26 -- see the passage there which starts with Then the thought occurred to me, 'This Dhamma that I have attained is deep

Thanks that is a great point. Don't we find out though, that the Buddha decides to teach the Dhamma? And for those who have heard the Dhamma, he later says Go forth, O Bhikkhus, for the good of the many, for the happiness of the many, out of compassion for the world..

A Buddha is born via generating the uncontrived Mind of Enlightenment. This is an immense compassion and wish to free all sentient beings from suffering completely. It is such an uncommon mind that only Arya Bodhisattva's and fully enlightened Buddha's can even imagine what this mind would be like to experience.
As for the other answers... I think it is quite silly to say that the Buddha is not concerned or does not wish to help unlimited sentient beings or that this is somehow not realistic. I think this is not so subtle way of criticizing the Mahayana for being somehow unreasonable or unrealistic. This is  rather ironic given that even the Pali tradition itself contains teachings on the Bodhisattva vehicle. Consider the Four Immeasurables which can be found in the Pali canon. What are the Four Immeasurables? They are a form of mind training for Bodhisattva's wishing to generate the Mind of Enlightenment:

May all sentient beings have happiness and its causes;
May all sentient beings be from suffering and its causes;
May all sentient beings never be separated from limitless bliss;
May all sentient beings abide in equanimity, free from bias, attachment, and hatred.

These thoughts are said to be limitless precisely because sentient beings are limitless. To say that generating this mind is unrealistic in order to criticize the Mahayana is silly and unproductive. It is profoundly beneficial to train your mind to generate these uncontrived thoughts. Even more, one cannot become a Buddha without doing so! How silly to say that because the goal is impossible that generating this mind is not worthwhile!
The quintessential training in compassion is exemplified by Shantideva's Guide to the Bodhisattva Way of Life. It is essentially an expansion and elaboration of the very same attitudes and minds expressed by the Buddha in the Pali Canon's sutta's on the Four Immeasurables. Shantideva's guide is in my opinion one of the most beautiful and sacred religious texts on earth. I would happily dare anyone to read it completely and come away with a different opinion :)

Yes, as the Buddha states in the Diamond Sutra:

All living beings, whether born from eggs, from the womb, from
  moisture, or spontaneously; whether they have form or do not have
  form; whether they are aware or unaware, whether they are not aware or
  not unaware, all living beings will eventually be led by me to the
  final Nirvana, the final ending of the cycle of birth and death.

@avatarKorra Diamond Sutra is Mahayana. The Buddha never said such a thing according to Theravada.

@SankhaKulathantille sorry you feel that way, but the question was not about a debate between Mahayana and Theravada. And I have no interest in disparaging Theravada, or any other religion for that matter.

@jacknad I was not stating a feeling. I was stating the fact "The Buddha never said such a thing according to Theravada" for the benefit of the OP.

@SankhaKulathantille Thank you for the clarification. I apologize for my misunderstanding.


text:
How does *Buddhist* meditation differ from others and lead to awakening?
The practice  of meditation is central to certain Buddhist traditions, e.g. Vajrayana, Dzogchen, Zen, important for recognizing Buddha nature.
Furthermore, Vajrayana and Theravada traditions assert that the meditation serves as a basis for realizing selflessness:

the purpose of meditative stabilization is to serve as a basis for
  achieving supramundane special insight realizing selflessness, the
  emptiness of inherent existence, through which afflictive emotions can
  be removed completely and forever
-- Dalai Lama
So what is the purpose of satipaṭṭhāna? The purpose is to see
  anattā, that there is no self, no me, nor anything that belongs to a
  self.
-- Ajahn Brahm

But dhyana and samadhi are not unique to Buddhism, also found in schools such as Hinduism which do not lead to realization of no self.
And even highly accomplished Buddhist meditators have found that with a radiant mind in meditation, the sense of self can still fail to be uprooted.

At that stage the mind was so radiant that I came to marvel at its
  radiance...
“If there is a point or a center of the knower anywhere, that is an agent of birth.”
-- Ajahn Maha Boowa

How exactly does Buddhist meditation practice lead to realizing selflessness, versus other forms other forms of meditation which do not? What exactly makes Buddhist meditation different?
What made the Buddha's meditation technique differ from others' (e.g. Alara Kalama and Uddaka Ramaputta) which did not lead to enlightenment?  (Please be more specific than "Middle Way").

Very good question.
Buddhist meditation is based on the understanding of the mechanism of arising and cessation of suffering. Suffering arises whenever there is a conflict between "is" and "should". Craving for any "should" because of attachment to that (usually to a concept within a framework) is the generator of suffering. Letting go of the "should", or even better, transcending the boundaries of a framework that served as the foundation for the concept, leads to cessation of suffering.
Buddhist meditation involves methodically applying the above principle from coarse to progressively finer "shoulds".
As you thus let go of these attachments, preconceptions, prejudices, and overgeneralizations - the notion of self dissolves as the hidden counterpart of these overgeneralizations.
However, I think you got the relationship wrong way. It's not that the peace of no-attachment (no-should, no-overgeneralization) is a step that leads to selflessness which then leads to Enlightenment. It is the other way: freedom from attachments, preconceptions, prejudices, and overgeneralizations -- including the overgeneralization known as "Self" - leads to Enlightenment and Peace.
Non-Buddhist meditation is either trial and error, or something trivial like gazing at the candle or running water, or it is brute-force stopping the thoughts, or it is visualization of some images that has useful side effects, or working directly with psychosomatic energies. The last two types were later repurposed by Tibetan Buddhists and integrated into Buddhist framework of progressive attainment of suchness.

the story I've heard sounds like the Buddha spontaneously found a superior meditation that led to his enlightenment. But, what was the key difference from where his teachers had failed? They both seem like highly concentrated meditative states of absorption. What did he do differently?

He spontaneously discovered letting go and then worked through all the ramifications.

"Letting go" -- this is what the non-Buddhist meditations did not have. Now I see. To "let be" is a recurring part of Dzogchen instruction, so I've read.

"You got it, Kaundinya!" ;))

..."Non-Buddhist meditation is either trial and error, or something trivial like gazing at the candle or running water, or it is brute-force stopping the thoughts..." Really? Does nobody meditate effectively except Buddhists? Is this what you meant to say?

In context of stopping dukkha, non-Buddhist schools are not effective, because they (by definition) do not operate within the framework of dukkha and cessation of dukkha. They may achieve their other objectives but not the Buddhist objective as effectively as the Buddhist methods.

How a doctrine influences the outcome of meditation is both powerful & subtle.
For example, it can be observed on Buddhist chatsites how much certain individuals egoistically attach/cling to delusions about 'jhana'. These delusions about personal attainments of (imaginary) jhana arise from their emphasising of 'jhana' and their de-emphasising of abandoning 'self-clinging'.
Similarly, if a religion or doctrine teaches about 'True Self', 'Brahman' or 'God', then follower will develop an inclination or bent to regard meditative experiences as 'God' or 'Self' or 'Brahman'.
In short, the underlying doctrine or culture is extremely influential.
In genuine Buddhist doctrine, the Buddha strongly emphasised the abandoning of self-clinging & self-views. Therefore, if a beginner meditator attempts to do their very best to abandon self-clinging & self-views, this will affect the quality of concentration, which will be much more lucid, sensitive & free.
As such meditation develops, when subtle clinging arises, the mindful meditator (who is properly instructed) will quickly learn & act to dissolve that clinging.
Therefore, the more selfless the basic instruction, the more selfless will be the fruition of meditation and the more selfless will enlightenment be.
In MN 118 (final paragraphs) and SN 48.10, for example, the Buddha directly taught that right mindfulness and right concentration have the quality of 'letting go' ('vossagga').
This quality of 'vossagga' is what distinguishes genuine Buddhist meditation from non-genuine Buddhist meditation, which includes the non-Buddhist techniques found in the most famous later-day Buddhist commentaries, such as the Visuddhimagga, which emphasis force and suppression rather than letting go.
Buddhist meditation is the abandonment of craving, attachment and every type of self-conceit (I making & my making) without exception; including as much as possible from the very beginning of practise. About this, AN 4.34 says:

Monks, among things conditioned and unconditioned, dispassion is
  reckoned to be the best of them all: the crushing of all infatuation,
  the removal of thirst, the uprooting of attachment, the cutting off of
  the round, the destruction of craving, dispassion,
  Nibbāna. Those who have faith in the Dhamma of dispassion have
  faith in the best; and for those who have faith in the best, the best
  result will be theirs.

You have already answered this question yourself ---> "such as Hinduism which do not lead to realization of no self."
To be specific, Satipatthana is the key difference.
The whole technique of Satipatthana, along with observing the three marks of existence (Impermanence, Suffering, and no-self), is the key difference between Buddhism meditation with Tirthika that practices meditation.

The question asks why does Buddhist meditation lead to realization of no self, whereas others e.g. Hinduism do not. How does the quote already answer the question?

Perhaps you may focus on the latter part of my answer, too.

In the context of Buddhist meditation, there are 40 subjects of meditation (Kammaṭṭhāna). These are divided into

Samatha for concentration and mastery over the mind
Vipassanā for wisdom

All the Samatha techniques were borrowed from contemporary meditation techniques. What is unique to Buddhism is Vipassanā which results in wisdom (Paññā).
In Buddhism, the practice has 3 folds:

morality (Śīla) - developed though recollection of morality (Śīla)
mastery over the mind (Samadhi) - developed through Samatha
wisdom (Paññā) - developed through Vipassanā

Many contemporary teachers and even before and after the Buddha taught morality (Śīla) and mastery over the mind (Samadhi). Using these techniques you can develop morality to some degree and gain certain spiritual development. In religions like Christianity, there are many cases of pious people who have been able to perform miracles. The Hindus also had practices which resulted in Samadhi upto the highest Jhana. They reaped the benefits of developing Jhana. Alara Kalama has the 7th Jhana and Uddaka Ramaputta has the 8th Jhana. The Bodhisatta was not satisfied with these Jhana as the ultimate realisation investigated further found there still is a step further and found the way to develop Paññā. These 3 fold training, without any extremities and taking out any vital elements, in totality codified in the Noble Eightfold Path was the middle way.

Buddhist meditation practice leads to realizing selflessness since that is one of the targets of meditation. The purpose of meditating on another object, such as the breath, is to train the mind to focus on the object of meditation. Once the practitioner has the ability to pick and hold an object of meditation, the practitioner is able to choose selflessness or emptiness as the object, and eventually have a direct realization.

The Visuddhimagga (english translation by Bhikku Nanamoli) chapter XXII.46 draws our attention to 'balancing the two powers of serenity and insight in such a manner that neither exceeds the other'. The canonical source from the 'Path of Discrimination' is also given. If serenity or samatha, based on breathing is equated to form/body, the power/energy  behind the body is said to be atomic.  If insight or vipassana based on thoughts and contemplating the three characteristics is equated to mind/consciousness, the power/energy behind consciousness is said to be cosmic. In Abhidhamma, the Higher Doctrine, we understand that the cosmic power is 17 times more powerful than the atomic. If therefore, it is necessary to 'balance these two powers' it means that the power of cosmic needs to be reduced and the power of atomic needs to be increased, in such a manner that 'neither exceeds the other'? The merger of these two powers presumably leads to a state of vipassana samadhi, which is clearly more powerful than the serenity or samatha samadhi. The yin/yang, shiva/shakthi, name/form.

cosmic power is 17 times more powerful than the atomic. Do you have a reference for this?


text:
Was the Buddha a bodhisattva (for the benefit of all) before his awakening?
Was the Buddha a bodhisattva before his awakening? Are there references from the suttas which point to when (possibly in past lives)?

Bodhisattva is the Sanskrit term for anyone who has generated
  Bodhicitta, a spontaneous wish and compassionate mind to attain
  Buddhahood for the benefit of all sentient beings.

The following appears to suggest that the four immeasurables may have been present in the Buddha's past lives:

The early Buddhist texts assert that pre-Buddha ancient Indian sages
  who taught these [four immeasurables] were earlier
  incarnations of the Buddha.

I apologize if the answer is obvious, but I haven't found a direct answer from the suttas.

Absolutely he was! In fact, there is no way to become a Buddha without becoming a Bodhisattva first. Numerous Mahayana Sutras attest to this. Without developing the immeasurable compassion of a Bodhisattva there is no way for a sentient being to become a Buddha.
As for the Theravada who do not regard the Mahayana Sutras as authentic Buddhavacana: the Pali canon still has many references to the past lives of the Buddha where he was a Bodhisattva. And they are not all part of the so-called Jataka tales either. See here for many examples.
You can find an example here - AN 5.196 - which is not one of the Jataka tales:

“When the Tathagata—worthy & rightly self-awakened—was still just an
  unawakened bodhisatta, five great dreams appeared to him. Which five?
“When the Tathagata—worthy & rightly self-awakened—was still just an
  unawakened bodhisatta, this great earth was his great bed. The
  Himalayas, king of mountains, was his pillow. His left hand rested in
  the eastern sea, his right hand in the western sea, and both feet in
  the southern sea. When the Tathagata—worthy & rightly
  self-awakened—was still just an unawakened bodhisatta, this was the
  first great dream that appeared to him.

Emphasis mine.
And here is the original Pali:

“Tathāgatassa, bhikkhave, arahato sammāsambuddhassa pubbeva sambodhā
  anabhisambuddhassa bodhisattasseva sato pañca mahāsupinā pāturahesuṃ.
  Katame pañca? Tathāgatassa, bhikkhave, arahato sammāsambuddhassa
  pubbeva sambodhā anabhisambuddhassa bodhisattasseva sato ayaṃ
  mahāpathavī mahāsayanaṃ ahosi, himavā pabbatarājā bibbohanaṃ ahosi,
  puratthime samudde vāmo hattho ohito ahosi, pacchime samudde dakkhiṇo
  hattho ohito ahosi, dakkhiṇe samudde ubho pādā ohitā ahesuṃ.
  Tathāgatassa, bhikkhave, arahato sammāsambuddhassa pubbeva sambodhā
  anabhisambuddhassa bodhisattasseva sato ayaṃ paṭhamo mahāsupino
  pāturahosi.

Emphasis mine again.
Here is another example from the Pali-canon which is not one of the Jataka tales:

“So it is, Ananda. So it is. Even I myself, before my Awakening, when
  I was still an unawakened Bodhisatta, thought: ‘Renunciation is good.
  Seclusion is good.’ But my heart didn’t leap up at renunciation,
  didn’t grow confident, steadfast, or firm, seeing it as peace. The
  thought occurred to me: ‘What is the cause, what is the reason, why my
  heart doesn’t leap up at renunciation, doesn’t grow confident,
  steadfast, or firm, seeing it as peace?’

Emphasis mine.
And here is the original Pali:

“Evametaṃ, ānanda, evametaṃ, ānanda. Mayhampi kho, ānanda, pubbeva
  sambodhā anabhisambuddhassa bodhisattasseva sato etadahosi: ‘sādhu
  nekkhammaṃ, sādhu paviveko’ti. Tassa mayhaṃ, ānanda, nekkhamme cittaṃ
  na pakkhandati nappasīdati na santiṭṭhati na vimuccati etaṃ santanti
  passato. Tassa mayhaṃ, ānanda, etadahosi: ‘ko nu kho hetu ko paccayo,
  yena me nekkhamme cittaṃ na pakkhandati nappasīdati na santiṭṭhati na
  vimuccati etaṃ santanti passato’?

And here is yet another one:

“Bhikkhus, before my enlightenment, while I was still a bodhisatta,
  not yet fully enlightened, it occurred to me: “What now is feeling?
  What is the origin of feeling? What is the way leading to the
  origination of feeling? What is the cessation of feeling? What is the
  way leading to the cessation of feeling? What is the gratification in
  feeling? What is the danger? What is the escape?”

Here is another one:

‘Both the quotations you have made, O king, are accurate. But when he
  spoke of Āḷāra Kālāma as his teacher, that was a statement made with
  reference to the fact of his having been his teacher while he (Gotama)
  was still a Bodisat and before he had attained to insight and to
  Buddhahood; and there were five such teachers, O king, under whose
  tuition the Bodisat spent his time in various places—his teachers when
  he was still a Bodisat, before he had attained to insight and to
  Buddhahood. And who were these five?

And another:

At Sāvatthī. “Mendicants, before my awakening—when I was still
  unawakened but intent on awakening—I thought: ‘What’s the path and
  practice for developing the bases of psychic power?’

Now, why did I emphasize "intent on awakening?" Because, this translation chose to use "intent on awakening" as a substitute for the original Pali which has this:

Sāvatthinidānaṃ. “Pubbeva me, bhikkhave, sambodhā anabhisambuddhassa
  bodhisattasseva sato etadahosi: ‘ko nu kho maggo, kā paṭipadā
  iddhipādabhāvanāyā’ti?

So I hope you can clearly see that according to both the Mahayana and the Theravada that the Buddha was an unenlightened Bodhisattva before he became the Buddha.
Now, was the Buddha as a Bodhisattva for the benefit of all? Here is a passage from a canonical Pali sutta describing the Bodhisattva Gotama being born into the world for the benefit of all:

“The Bodhisatta, the foremost jewel, unequaled, has been born for
  welfare & happiness in the human world, in a town in the Sakyan
  countryside, Lumbini. That’s why we’re contented, so wildly elated.
  He, the highest of all beings, the ultimate person, a bull among men,
  highest of all people, will set turning the Wheel [of Dhamma] in the
  forest named after the seers, like a strong, roaring lion, the
  conqueror of beasts.”

And from that same sutta here is the testament to his motivation:

He, seeing the utmost purity, will set rolling the Wheel of Dhamma
  through sympathy for the welfare of many.

So, to sum up: have no doubt The Buddha was a Bodhisattva who manifested the Mind of Enlightenment - pure Bodhicitta - for the benefit of all before his awakening.

Wonderful quotes. Thank you. So just to make sure - when Theravadans use the term Bodhisattva, they don't just mean "the Buddha prior to his enlightenment", but also the aspiration to liberate all sentient beings, correct? The bodhisattva definition is fairly uniform across the traditions, right?

I am no expert on Theravada but I suspect some would agree and some would disagree. Theravada is just a word after all that different people with different minds decide to use to identify different things :). The Jataka tales for sure contain exposition on the seeds of bodhicitta. The Four Immeasurables do as well. I think anyone who continues to develop the four immeasurables to perfection will not fail to generate pure bodhicitta. However, you should know that the Mahayana regards ultimate bodhicitta to be exactly the same as the wisdom realizing emptiness.

Just found this answer, that bodhisatta and bodhisattva may be understood differently by different traditions, and while it seems not definitive in Theravada whether bodhicitta was there prior to the Buddha's awakening, we can say that it was definitely there from the view of Mahayana (and Vajrayana). Does this sound about correct?

Yes, there is no doubt at all that prior to his awakening the Buddha developed pure bodhicitta

Would you happen to have any quotes from the Pali where Buddha developed pure bodhicitta or that show he had compassionate aspirations? I mean the bodhisatta/bodhisattva quotes are very good, but Theravadans appear  to argue on (1) the term's meaning, and (2) that the Buddha was only interested in saving himself, not others. Such quotes would help to remove any doubts of bodhicitta from (1) and (2).

@avatarKorra the last quotes in the answer are from canonical pali canon and I think point to seed of bodhicitta. I'm not aware of bodhicitta the word being used in pali canon. In the Jakata tales there are numerous references to what amounts to bodhicitta if not the word itself. Jataka tales are part of pali canon, but some do not consider them authoritative. Finally, I am sure that you will always find some Theravada practitioners who deny Bodhisattvas are motivated by bodhicitta. Then again, you'll find plenty who say the opposite. Theravada and Mahayna are just words after all.

The Jataka Tales (which is in the fifth Nikaya) say "Bodhisattva".
See also Buddha's past lives in the Pali Suttas outside of Jataka and Where is the description of the vow and the Bodhi tree?

"Bodhisattva" is a reification, just like "a Buddha". To say that "Bodhisattva is a reification" - is, too, a reification. "For the benefit of all" is a reification. "For" and "benefit" and "all" - are reifications as well.

Śāriputra:  How then should the Bodhisattva, the great being, course
in perfect wisdom?
The Lord: Here the Bodhisattva, the great being, coursing in the
perfection of wisdom, truly a Bodhisattva, does not review a Bodhisattva, nor the word "Bodhisattva", nor the course of a Bodhisattva, (nor the perfection of wisdom, nor the word "perfection of wisdom". He does not review that "he courses", nor that "he does not course"). He does not review form, feeling, perception, formative forces, or consciousness. And why? Because the Bodhisattva, the great being, is actually empty of the own-being of a Bodhisattva, and because perfect wisdom is by its own-being empty. And why? That is its essential original nature. (For it is not through emptiness that form etc. is empty.) Nor is emptiness other than form, etc.
And why? The very form, etc., is emptiness, the very emptiness is
form, etc. And why? Because "Bodhisattva", "perfect wisdom", "form",
etc. are mere words. Because form, etc., are like an illusion. Illusions and
mere words do not stand at any point or spot; they are not, do not come
into being, are false to behold. For of what the own-being is seen to be an
illusion, of that there is no production or stopping, no defilement or
purification. Thus a Bodhisattva, a great being who courses in the perfection of wisdom, also does not review the production (of any dharma);
nor its stopping (or abiding, its decrease or increase), defilement or
purification. (He does not review form, etc., nor "enlightenment", nor
what is called an "enlightenment-being".) And why? Because words are
artificial. People have constructed a counter-dharma. They express it conventionally by means of an adventitious designation (which is
imagined and unreal, and they settle down in that conventional
expression). A Bodhisattva, a great being who courses in perfect wisdom,
does not review (that which is said to correspond to) all those words,
(does not get at them). Not reviewing them, (not getting at them, he does
not mind them), does not settle down in them.
Furthermore, a Bodhisattva, a great being who courses in the perfection
of wisdom, does not consider the fact that these are mere words, i.e.
this "Bodhisattva", this "enlightenment", this "Buddha", this
"perfection of wisdom", this "coursing in the perfection of wisdom",
this "form", etc. Just as one speaks of a "self, and yet no self is got at,
and no being, soul, personality, person, individual, or man, etc., on
account of unascertainable emptiness. And why? Because there a
Bodhisattva does also not review that by means of which he would settle
down. Coursing thus, a Bodhisattva, a great being courses in perfect
wisdom.

(from The Large Sutra on Perfect Wisdom, translated by E.Conze)

Additional detail on the bodhisattva Gautama’s reflection that motivated
him to set out on his quest can be gathered from another occurrence of
the ‘before awakening’ phrase, found in the Discourse on the Noble Quest,
the Ariyapariyesanā-sutta, and in its Madhyama-āgama parallel. The two
discourses contrast an average person’s quest for worldly things that are
subject to decay and death to the noble quest for what is not subject to decay
and death. Both reveal that this noble quest motivated the bodhisattva
to go forth in search of awakening, presenting his reflection in nearly identical terms:

“Being myself subject to old age … and death, now suppose I were to
search for what is free from old age … and death, for the unsurpassable
peace from bondage, Nirvāṇa.”
“Being myself truly subject to old age and death … suppose I were to
search … for what is free from old age and death … for the unsurpassable
peace from bondage, Nirvāṇa.”

This appears to be the only passage in the Pāli Nikāyas that explicitly formulates what motivated the bodhisattva Gautama to set out in search of
awakening. Notably, this formulation does not in any way reflect a concern
for others. Rather, according to this autobiographical report the bodhisattva
Gautama’s motivation was to find a solution for the problem of being “himself”,
attanā/自, subject to old age and death etc.
The same trait recurs in the description of the successful completion of
the bodhisattva’s quest, given in the Ariyapariyesanā-sutta and its parallel.
According to both versions, having arrived at the unsurpassable peace
from bondage that is free from old age and death, Gautama realized that he
had fully liberated himself from the prospect of future birth and existence.
Here, too, there is no reference at all to being able to save others. Instead,
the way the Buddha perceived his own awakening – according to early Buddhist discourse – is formulated entirely in terms of having freed himself.
The conspicuous lack of any concern for others becomes even more prominent
with the next episode recorded in the Ariyapariyesanā-sutta, according
to which the newly awakened Buddha was disinclined to teach others and decided to rather remain content with having reached liberation himself.
...
Such reluctance on the part of the newly awakened Buddha is not easily
reconciled with the notion common to all Buddhist traditions that he had
prepared himself over immense periods of time with the sole intention of
executing the task of leading others to liberation. Judging from the account
given in the Ariyapariyesanā-sutta, the reason for the Buddha’s disinclination
to teach was that it would be fatiguing (klamatha) and vexing (viheṭhā)
for him if others should fail to understand the profound truth he had
realized. Similar reasons recur in other reports of his initial hesitation to
teach, found in an Ekottarika-āgama discourse, in the Catuṣpariṣat-sūtra, and in the Vinayas of the Dharmaguptaka, Lokottaravāda-Mahāsāṃghika,
Mahīśāsaka and (Mūla-)Sarvāstivāda traditions.
That is, when reflecting on the possibility of sharing his discovery with
others, according to a range of sources the Buddha considered the matter
entirely from the perspective of how it would affect himself. This ties in
with the observations made above regarding his motivation and his reaching
of the final goal. Throughout, according to these texts the Buddha’s predominant
concern is with himself, when he forms his initial motivation to
set out for awakening, when he successfully completes his quest, and even
when he reflects about what course of action is to be taken next.
This certainly does not imply that from the perspective of the early discourses
the Buddha was not compassionate. [...] However, the early
discourses do not give any indication that a concern for others was part of
the motivation of the bodhisattva Gautama to set out on his quest for awakening,
nor does the successful reaching of this goal show any immediate
relationship to teaching activity.
...
In sum, in early Buddhist thought the compassionate impulse to become
active for the sake of others was associated with the Buddha as well as with
arhats and those who aspire to become arhats, but was not seen as a quality
that motivated the bodhisattva’s quest for awakening.

(from The Genesis of the Bodhisattva Ideal, by Analayo Bhikkhu)

Man I wish some talented translator like Thubten Jinpa would have a go at this sutra. Preferably the one in 8000 words...

you can find much the same passage in the 8000 line version here: https://www.maitreya.nl/pdf/Both-8000-verse-prajnaparamita.pdf at the end of page 40

As for the second long quoted portion of your answer, I don't agree :) Consider... can a first motivation be found? Is it possible to manifest the motivation to liberate countless sentient beings without first manifesting the desire or motivation to free one self? I'm not sure this is what @avatarKorra is asking, but that second quote sounds like it is trying to answer the question, "Did the Buddha ever have the motivation to liberate countless sentient beings before he became enlightened?" which I think is obviously affirmative.

Agreeing and disagreeing is papanca. The non-papanca is to see it as is.

@YesheTenley right, that second question, "Did the Buddha ever have the motivation to liberate countless sentient beings before he became enlightened?" is closer to what I'm asking -- but not placing anyone's enlightenment first (i.e. before his) so much as just wanting to know if the aspiration to liberate all beings was present at all leading up to the Buddha's awakening. To your first question, maybe the Buddha's insight was that in the big picture, there is not even a question of "first" or separation at all between one's awakening and others'.

Yup, I think that's close

@AndreiVolkov do you think that the Buddha had past lives where he accumulated merit? If so, then what was his motivation in those past lives (possibly bodhicitta)? Was this part of the conditions which led to his discovery of 'letting go'?

@YesheTenley in regard to the second quote, while the question arose in him on whether or not to teach others, it seems clear that the conditions were already there for the Buddha to even consider teaching others, and furthermore, he finally decided to.

Well, I think past lives are a reification. And I think merit is a reification too. But there were certainly causes in the past lives that contributed to Buddha's enlightenment. The wish to pursue NOT the ways of the world or any tradition, but the intuitive drive to find the Truth by himself, was there for a while... probably for at least ~1000 years as far as I can see - how many lifetimes is that?

According to the Dalai Lama, "The master of these teachings [The Way of the Bodhisattva], the Buddha, began by generating the wish for enlightenment. He then accumulated positive actions through many lifetimes over a period of three great uncountable kalpas. " In those past lifetimes, did the Buddha accumulate those positive actions knowing it would only help himself, or others, too?

He didn't think like that. He was just doing what he thought was right, not thinking about himself or others. Just doing what he thought was right.

@AndreiVolkov your comment about agreeing/disagreeing caused a very big grin to appear on my face :) it also made me think of this: http://bigthink.com/Think-See-Feel/salinger-buddhist still, I think it inspired a conventional question I will ask in another post :)

@avatarKorra I think the Buddha reached perfect enlightenment long long before he was born as Gotama. This is commonly accepted in Mahayana. Also, I don’t think there was any real hesitation or confusion on the part of the Buddha in teaching the Dharma... manifesting hesitation was a skillful means.

@AndreiVolkov The only new word the Buddha used that wasn’t just common Indian religious language was bodhisattva—that one word.. Why do you think this is, that Buddhism introduced the new word 'bodhisattva'?