user-1: If this was bone-obvious to everyone else, then silly me, but I just thought of it because I heard these two words separately long ago and connected them while answering another question...
Are "the Observer" and the process of being Mindful, in essence the same thing? Does one involve the other inherently?
I have used them differently and never connected them before, because I thought:  

The Observer is a developed system within "me" (which is a collection of points of view of varying levels of awareness) and not everyone has an Observer or is aware of it all the time.  
Mindfulness is something I do and is not the activity of another point of view within "me".

But, trying to be mindful when I have only one point of view is not very possible, and who would be mindful, other than the Observer?
Finally: is transcending the Observer (and the need for Mindfulness) the same as Nonduality? I think it is. (Don't worry about answering this question, it is just an idea.)
EDIT: My Healing Teacher said that I have to get people to "define their terms", so here are my definitions:
The Observer is the experience of being aware of myself. It feels like I am witnessing my own thoughts and actions.
Mindfulness is the process of being aware of what I am doing. (To me, this does not require an Observer, but that is what I am asking.)
"Being aware of myself" means... Well, that I know what I am doing right now. Different from the other two.
A Mind - is something that could do otherwise, it exercises choice on some level.
Awareness is the action of a sufficiently developed mind.
Consciousness is awareness of being a self.
A Self is something that knows it is a self, and that other selves know, etc.
(Don't even get me started on the idea of Mindfullness being a way to empty the mind!)
Has this made it any clearer what I am asking? Please try to use words such as these for an answer, supplementing with the appropriate Buddhist terms.
ADDITION: here are some links which I hope can help people understand what I am asking:  

This RYUC one is a bit of a muddle, but the Observer corresponds with what I am asking about, and also is similar to what I call a Neo state. The Witness is more like how I would describe nonduality.  
This one about Energy Healing says that "In Buddhism, developing the witness/observer is a foundational piece of their teachings." Ha!  
In this one, they use the words observer and witness interchangeably.


user295: What is realised at the experiential level is the Dhamma for you. What someone else experiences is not the Dhamma for you. In this sense there is no nonduality as people experience reality in different way (as for what I have learned.) There is nothing similar between experiences of disciples, though perhaps experience of among Buddha's may be similar since they experience all what is conditioned. The main goal of experiencing "reality as it is" to develop revulsion to the 5 aggregates. So what you you experience should be to the extent to avoid unwise attention of attention tained by perverted perceptions, thoughts and views. In this context there is always an observer who experiences phenomena though consciousness aided by the faculties, or one that who feels1.

1

“It is to one that feels that I teach Dhamma,
  not to one that does not feel.”
  —The Awakened One

Also see: To one that feels by Luangpor Teean Juttasubho 

user-1: Sati (mindfullness) has the agenda of remembering, keeping in mind, so its quite proper to give it the name "observer" on the level of being an "overseer". Samma sati (right mindfulness) keep the right things in mind. It is often compared with a door-keeper (of the fortress of practice), a good one.
Some short but very usefull essays, to get out of common misunderstandings are:

The Agendas of Mindfulness
Mindfulness Defined
Untangling the Present: The Role of Appropriate Attention

(Note: this answer has not been given with the agreement to be means of trade or the purpose of/for trade and/or keep people trapped and bound. How you handle it lies in your sphere, but does not excuse the deed here either.)

user8157: 'Mindfulness' ('sati') means to 'remember' rather than to 'observe' ('anupassi').
In the context of basic meditation, 'sati' remembers to observe rather than forgets to observe. 
'Mindfulness' & 'observing' are two distinctly separate functions & activities of mind, even though they function together in mediation. 

And what is the faculty of mindfulness? There is the case where a monk, a disciple of the noble ones, is mindful, highly meticulous,
  remembering & able to call to mind even things that were done & said
  long ago. SN 48.10

"What is sammasati? Sati means to bear in mind or bring to mind. Sati is the state of recollecting, the state of remembering, the state of
  non-fading, the state of non-forgetting. Sati means the sati that is a
  Spiritual Faculty, the sati that is a Spiritual Power, Sammasati, the
  Sati that is an Enlightenment Factor, that which is a Path Factor and
  that which is related to the Path. This is what is called sammasati."
  [Vbh.105, 286]

The Blessed One said, "Suppose, monks, that a large crowd of people comes thronging together, saying, 'The beauty queen! The beauty
  queen!' And suppose that the beauty queen is highly accomplished at
  singing & dancing, so that an even greater crowd comes thronging,
  saying, 'The beauty queen is singing! The beauty queen is dancing!'
  Then a man comes along, desiring life & shrinking from death, desiring
  pleasure & abhorring pain. They say to him, 'Now look here, mister.
  You must take this bowl filled to the brim with oil and carry it on
  your head in between the great crowd & the beauty queen. A man with a
  raised sword will follow right behind you, and wherever you spill even
  a drop of oil, right there will he cut off your head.' Now what do you
  think, monks: Will that man, not paying attention to the bowl of oil,
  let himself get distracted outside?"
"No, lord."
SN 47.20


user6994: According to Mahayana abhidharma,

Mindfulness is a non-forgetfulness of the mind with respect to a familiar object. It has the function of non-distraction.


It is one the five object-ascertaining mental factors, and it is called 'object-ascertaining' because:

These mental factors hold [an object] through apprehending the
  individual features of the object. They are said to "individually
  ascertain objects."

As it is a mind, it is by nature clear and knowing, and it is a basis of designation of the person but is not the person. It is an apprehender, an object-possessor, but it is not a 'observer' because an observer is a person.
A mind, a cognizer such as mindfulness and so forth is possessed by a person (the agent), it has a function (the action) and apprehends its object.
There are various levels of nonduality. In general, a realization that occurs in a non-dualistic manner is free from the appearances of objects and subjects. So it is necessarily free from the apprehension of the object and the subject as being different substances.

In addition, mindfulness is not a process, it is a mind. As such:

It has the nature of being clear
It performs the function of knowing

That it is clear refers to the fact that it has the ability to take the aspect of its object. An eye-consciousness seeing blue takes the [subjective] aspect of blue.
That it is knowing refers to the fact that it apprehends its object. This point is quiet complicated and there is much debate. By nature a mind experiences its object. I hope this answer helps you, as you defined your "Observer" as an experience.

user43: As far as I know, Observer (kṣetra-jñaḥ - "the knower of the field [of experience]" or Vedagu - "the one who enjoys [the objects of senses]") is a Hindu concept. It's an ancient expression for the subject of experience, the spectator in the Cartesian Theater:

The king said: ‘Is there, Nāgasena, such a thing as The Knower?’  
[Nagasena:] ‘What is this, O king, The Knower (Vedagu)?’  
[King Milinda:] ‘The living principle within which sees forms through the eye, hears sounds through the ear, experiences tastes through the tongue, smells odours through the nose, feels touch through the body, and discerns things (conditions, “dhammā") through the mind—just as we, sitting here in the palace, can look out of any window out of which we wish to look, the east window or the west, or the north or the south.
  ...
  Then the Elder [Nagasena] convinced Milinda the king with discourse drawn from the Abhidhamma, saying: ‘It is by reason, O king, of the eye and of forms that sight arises, and those other conditions—contact, sensation, idea, thought, abstraction, sense of vitality, and attention —arise each simultaneously with its predecessor. And a similar succession of cause and effect arises when each of the other five organs of sense is brought into play. And so herein there is no such thing as The Knower (Vedagu).’

Mindfulness in Buddhism is entirely different. It is not a thing, it's a quality you develop. In my opinion and in contrast with modern trends (DBT etc.), mindfulness in Buddhism should not be understood as open-ended awareness. It is mindfulness of something specific you are trying to cultivate, habituate, get used to - in order to change your patterns.
EDIT:
From Buddhist perspective, we misinterpret our experience of awareness to implicate an entity that is aware. "I am aware" we say -- and this recipient of data, the point of view, the source of light, is what ancient Hindu's used to call Vedagu. Buddhists say such "point of view" is merely a perceptual illusion - like a bush that looks like a dog until you look closer. If you look very carefully, they say, you'll clearly see that there is no "you" that is aware. Instead (this is a Yogacara explanation) there is a simple feedback loop. The current dharma (state of mind - citta) serves as an input into the associative perception mechanism (sanjna) which finds matching imprints of previous experiences (samskaras) among those recorded in memory (alaya-vijnana). These imprints become the new state of mind, which serves as an input into the next iteration of the cycle and so on.
If you look at it this way, it is easy to see why we are fooled into thinking there is a subject of experience, "the point of view" that does the watching. As yet another thought takes shape, the gap between it and the next thought seems like there is "I am watching", then from this gap comes the association - this seems like "I think", but all it really is, is the associative perception mechanism (sanjna) that closes the loop by connecting the mind with the memory. "I am this thing that perceives" / "I am the observer" - is the mistake of identification with sanjna.
I guess you could shift the emphasis from the associative mechanism to the repository of past imprints that serves as the source of the next thought - and alternatively say that it is our memory that is The Observer, that it is our memory that "pays attention" to our thoughts/experiences and comes up with associations - which can lead some people to identify with samskara-skandha (the imprints group, basically memory) - another view that Buddha rejected.
The subjective experience, the virtual life-of-I created by our mind that we mistake for reality, is called "vijnana". This is the whole story that includes The Observer, the observed, and everything that happens between them.
What looks to us as self-awareness, with its sense of depth, is merely the recursive nature of the associative loop. It is a similar kind of effect you get when you point a video camera at TV that shows its own output - with a filter in the middle that adds matching elements of past memories. The same basic principle is at work behind Google's DeepDream:

We ask the network: “Whatever you see there, I want more of it!” This creates a feedback loop: if a cloud looks a little bit like a bird, the network will make it look more like a bird. This in turn will make the network recognize the bird even more strongly on the next pass and so forth, until a highly detailed bird appears, seemingly out of nowhere.
  ...
  The results are intriguing—even a relatively simple neural network can be used to over-interpret an image, just like as children we enjoyed watching clouds and interpreting the random shapes.
  ...
If we apply the algorithm iteratively on its own outputs and apply some zooming after each iteration, we get an endless stream of new impressions, exploring the set of things the network knows about.


Glossary:

Cartesian theater
state of mind (Citta)
associative perception mechanism (Saṃjñā)
imprints of previous experiences (saṃskāra)
recorded in memory (ālaya-vijñāna)
the imprints group, basically memory (samskāra-skandhas)
subjective experience, the virtual life-of-I created by our mind that we mistake for reality (Vijñāna)


user-1: 
The Observer is the experience of being aware of myself. It feels like I am witnessing my own thoughts and actions.

In Buddhism, "myself" as an entity would be a mind-object: a concept, something one conceived / imagined, which receives attention -- a 'self' is taught to not be in the range of actual experiences such as a touch, a smell or primitive mental phenomena (such as a mental image, or mental sound) that can be contacted.
In Buddhism, one is "aware" of the detailed micro experiences (that, among other things, give rise to a concept of self). In buddhism, this "detailed awareness", or "attention" to a specific "primitive experience" is called consciousness (vinanna), which is formulated in buddhism as six-fold: eye-consciousness, ear-consciousness, nose-consciousness, tongue-consciousness, body-consciousness, mind-consciousness1.
Then, it's said that if there's  eye organ and visual forms,  eye-consciousness arise. The coming of the three is contact. For example in my visual field now there are bed sheets, wall frames and a TV. But while I write this, I'm not conscious of any of these forms, only of this visual character I'm typing right now.
For example, I just realized there was a lamp in the ceiling with lights on in my visual field, even though it has been reachable all the time. Until that moment, there was no contact (my consciousness was not directed) to this particular visual object.
The same occurs with mind as a sense: there are mind objects (e.g. thoughts) a "mind organ", and a mind-consciousness arise; the coming of the three is contact (e.g. when one gives attention to a particular thought). With contact, pleasant, unpleasant (or neither one) occur -- and so on.

Mindfulness is the process of being aware of what I am doing. (To me, this does not require an Observer, but that is what I am asking.)

I think this is close to what  mindfulness (sati) is in buddhism. It has different/related meanings depending on context but without going into details, I would summaryze as Thanissaro Bhikku put it: "What does it mean to be mindful of the breath? Something very simple: to keep the breath in mind."

"Being aware of myself" means... Well, that I know what I am doing right now. Different from the other two.

I'm not sure how your "being aware of what I am doing" is different than your "know what I am doing".

A Mind - is something that could do otherwise, it exercises choice on some level.

In buddhism, I think this could be mano, sankhara, or even possibly cetana.

Awareness is the action of a sufficiently developed mind.

I don't quite understand what you mean on this one.

Consciousness is awareness of being a self.

I would not say one is aware of a self (or 'myself', see first point) -- e.g. as if 'self' was an experienced object grasped by any of the 6 sense faculties. But one does elaborate a (by buddhism, incorrect) understanding of being or having 'a self'.
Now, to the question...

Is Mindfulness the same as having an Observer?

In Buddhism (and if you understand mindfulness to be sati), no. 
For a detailed investigation on the meanings of mindfulness I suggest  Tse-fu Kuan's Mindfulness in Early Buddhism

1: The point, I think, is not to say that there is such a thing as consciousness and it can be perfectly divided in six part (an ontological statement). Rather, it is that understanding the mind in this way is useful / helps with training and understanding of dhamma, and consequently, with final liberation.

-1: <p>If this was bone-obvious to everyone else, then silly me, but I just thought of it because I heard these two words separately long ago and connected them while answering another question...</p>

<p><strong>Are "the Observer" and the process of being Mindful, in essence the same thing? Does one involve the other inherently?</strong></p>

<p>I have used them differently and never connected them before, because I thought:  </p>

<ol>
<li><em>The Observer</em> is a developed system within "me" (which is a collection of points of view of varying levels of awareness) and not everyone has an Observer or is aware of it all the time.  </li>
<li><em>Mindfulness</em> is something I <em>do</em> and is not the activity of another point of view within "me".</li>
</ol>

<p>But, trying to be mindful when I have only one point of view is not very possible, and who would be mindful, other than the Observer?</p>

<p>Finally: is transcending the Observer (and the need for Mindfulness) the same as Nonduality? I think it is. (Don't worry about answering this question, it is just an idea.)</p>

<p><strong>EDIT:</strong> My Healing Teacher said that I have to get people to "define their terms", so here are my definitions:<br>
<strong>The Observer</strong> is the <em>experience</em> of being aware of myself. It feels like I am witnessing my own thoughts and actions.<br>
<strong>Mindfulness</strong> is the <em>process</em> of being aware of what I am doing. (To me, this does not require an Observer, but that is what I am asking.)<br>
<strong>"Being aware of myself"</strong> means... Well, that I know what I am doing right now. Different from the other two.<br>
<strong>A Mind</strong> - is something that <em>could do otherwise</em>, it exercises choice on some level.<br>
<strong>Awareness</strong> is the action of a sufficiently developed mind.<br>
<strong>Consciousness</strong> is awareness of being a self.<br>
<strong>A Self</strong> is something that knows it is a self, and that other selves know, etc.<br>
(Don't even get me started on the idea of Mind<em>full</em>ness being a way to <em>empty</em> the mind!)</p>

<p>Has this made it any clearer what I am asking? Please try to use words such as these for an answer, supplementing with the appropriate Buddhist terms.</p>

<p><strong>ADDITION:</strong> here are some links which I hope can help people understand what I am asking:  </p>

<ol>
<li><a href="http://ryuc.info/common/creativity_perspective/detached_witness.htm" rel="nofollow">This RYUC one</a> is a bit of a muddle, but the Observer corresponds with what I am asking about, and also is similar to what I call a Neo state. The Witness is more like how I would describe nonduality.  </li>
<li><a href="http://scienceofenergyhealing.com/developing-the-witness-observer/" rel="nofollow">This one about Energy Healing</a> says that "In Buddhism, developing the witness/observer is a foundational piece of their teachings." Ha!  </li>
<li><a href="http://onewithnow.com/awareness-awakening-to-the-witness-within/" rel="nofollow">In this one</a>, they use the words observer and witness interchangeably.</li>
</ol>
 295: <p>What is realised at the experiential level is the Dhamma for you. What someone else experiences is not the Dhamma for you. In this sense there is no <a href="https://en.wikipedia.org/wiki/Nondualism" rel="nofollow noreferrer">nonduality</a> as people experience reality in different way (as for what I have learned.) There is nothing similar between experiences of disciples, though perhaps experience of among Buddha's may be similar since they experience all what is conditioned. The main goal of experiencing "reality as it is" to develop <a href="http://www.themindingcentre.org/dharmafarer/wp-content/uploads/2009/12/20.1-Nibbida-piya.pdf" rel="nofollow noreferrer">revulsion</a> to the <a href="https://en.wikipedia.org/wiki/Skandha" rel="nofollow noreferrer">5 aggregates</a>. So what you you experience should be to the extent to avoid unwise attention of attention tained by <em><a href="http://www.themindingcentre.org/dharmafarer/wp-content/uploads/2009/12/16.11-Vipallasa-S-a4.49-piya.pdf" rel="nofollow noreferrer">perverted perceptions, thoughts and views</a></em>. In this context there is always an observer who experiences phenomena though consciousness aided by the faculties, or one that who feels<sup>1</sup>.</p>

<hr>

<p><sup>1</sup></p>

<blockquote>
  <p>“It is to one that feels that I teach Dhamma,
  not to one that does not feel.”
  —The Awakened One</p>
</blockquote>

<p>Also see: <a href="http://www.ahandfulofleaves.org/documents/To%20One%20That%20Feels_Teean.pdf" rel="nofollow noreferrer">To one that feels</a> by Luangpor Teean Juttasubho </p>
 -1: <p>Sati (mindfullness) has the agenda of remembering, keeping in mind, so its quite proper to give it the name "observer" on the level of being an "overseer". Samma sati (right mindfulness) keep the right things in mind. It is often compared with a <a href="http://zugangzureinsicht.org/html/tipitaka/an/an07/an07.063.than_en.html#mindfulness" rel="nofollow">door-keeper</a> (of the fortress of practice), a good one.</p>

<p>Some short but very usefull essays, to get out of common misunderstandings are:</p>

<ul>
<li><a href="http://zugangzureinsicht.org/html/lib/authors/thanissaro/agendas_en.html" rel="nofollow">The Agendas of Mindfulness</a></li>
<li><a href="http://zugangzureinsicht.org/html/lib/authors/thanissaro/index_en.html" rel="nofollow">Mindfulness Defined</a></li>
<li><a href="http://zugangzureinsicht.org/html/lib/authors/thanissaro/untangling_en.html" rel="nofollow">Untangling the Present: The Role of Appropriate Attention</a></li>
</ul>

<p><em>(Note: this answer has not been given with the agreement to be means of trade or the purpose of/for trade and/or keep people trapped and bound. How you handle it lies in your sphere, but does not excuse the deed here either.)</em></p>
 8157: <p>'Mindfulness' ('sati') means to 'remember' rather than to 'observe' ('anupassi').</p>

<p>In the context of basic meditation, 'sati' <em>remembers</em> to observe rather than <em>forgets</em> to observe. </p>

<p>'Mindfulness' &amp; 'observing' are two distinctly separate functions &amp; activities of mind, even though they function together in mediation. </p>

<blockquote>
  <p><em>And what is the faculty of mindfulness? There is the case where a monk, a disciple of the noble ones, is mindful, highly meticulous,
  remembering &amp; able to call to mind even things that were done &amp; said
  long ago. SN 48.10</em></p>
  
  <hr>
  
  <p><em>"What is sammasati? Sati means to bear in mind or bring to mind. Sati is the state of recollecting, the state of remembering, the state of
  non-fading, the state of non-forgetting. Sati means the sati that is a
  Spiritual Faculty, the sati that is a Spiritual Power, Sammasati, the
  Sati that is an Enlightenment Factor, that which is a Path Factor and
  that which is related to the Path. This is what is called sammasati."
  [Vbh.105, 286]</em></p>
  
  <hr>
  
  <p><em>The Blessed One said, "Suppose, monks, that a large crowd of people comes thronging together, saying, 'The beauty queen! The beauty
  queen!' And suppose that the beauty queen is highly accomplished at
  singing &amp; dancing, so that an even greater crowd comes thronging,
  saying, 'The beauty queen is singing! The beauty queen is dancing!'
  Then a man comes along, desiring life &amp; shrinking from death, desiring
  pleasure &amp; abhorring pain. They say to him, 'Now look here, mister.
  You must take this bowl filled to the brim with oil and carry it on
  your head in between the great crowd &amp; the beauty queen. A man with a
  raised sword will follow right behind you, and wherever you spill even
  a drop of oil, right there will he cut off your head.' Now what do you
  think, monks: Will that man, not paying attention to the bowl of oil,
  let himself get distracted outside?"</em></p>
  
  <p><em>"No, lord."</em></p>
  
  <p><em>SN 47.20</em></p>
</blockquote>
 6994: <p>According to Mahayana abhidharma,</p>

<blockquote>
  <p>Mindfulness is a non-forgetfulness of the mind with respect to a familiar object. It has the function of non-distraction.</p>
</blockquote>

<p><br>
It is one the five object-ascertaining mental factors, and it is called 'object-ascertaining' because:</p>

<blockquote>
  <p>These mental factors hold [an object] through apprehending the
  individual features of the object. They are said to "individually
  ascertain objects."</p>
</blockquote>

<p>As it is a mind, it is by nature clear and knowing, and it is a basis of designation of the person but is not the person. It is an apprehender, an object-possessor, but it is not a 'observer' because an observer is a person.</p>

<p>A mind, a cognizer such as mindfulness and so forth is possessed by a person (the agent), it has a function (the action) and apprehends its object.</p>

<p>There are various levels of nonduality. In general, a realization that occurs in a non-dualistic manner is free from the <strong>appearances</strong> of objects and subjects. So it is necessarily free from the apprehension of the object and the subject as being different substances.</p>

<p><br>
In addition, mindfulness is not a process, it is a mind. As such:</p>

<ol>
<li>It has the nature of being clear</li>
<li>It performs the function of knowing</li>
</ol>

<p>That it is clear refers to the fact that it has the ability to take the aspect of its object. An eye-consciousness seeing blue takes the [subjective] aspect of blue.</p>

<p>That it is knowing refers to the fact that it apprehends its object. This point is quiet complicated and there is much debate. By nature a mind <strong>experiences</strong> its object. I hope this answer helps you, as you defined your "Observer" as an experience.</p>
 43: <p>As far as I know, Observer (<em>kṣetra-jñaḥ</em> - "the knower of the field [of experience]" or <em>Vedagu</em> - "the one who enjoys [the objects of senses]") is a Hindu concept. It's an ancient expression for the subject of experience, the spectator in the Cartesian Theater:</p>

<blockquote>
  <p>The king said: ‘Is there, Nāgasena, such a thing as The Knower?’  </p>
  
  <p>[Nagasena:] ‘What is this, O king, The Knower (Vedagu)?’  </p>
  
  <p>[King Milinda:] ‘The living principle within which sees forms through the eye, hears sounds through the ear, experiences tastes through the tongue, smells odours through the nose, feels touch through the body, and discerns things (conditions, “dhammā") through the mind—just as we, sitting here in the palace, can look out of any window out of which we wish to look, the east window or the west, or the north or the south.<br>
  ...<br>
  Then the Elder [Nagasena] convinced Milinda the king with discourse drawn from the Abhidhamma, saying: ‘It is by reason, O king, of the eye and of forms that sight arises, and those other conditions—contact, sensation, idea, thought, abstraction, sense of vitality, and attention —arise each simultaneously with its predecessor. And a similar succession of cause and effect arises when each of the other five organs of sense is brought into play. And so herein there is no such thing as The Knower (Vedagu).’</p>
</blockquote>

<p>Mindfulness in Buddhism is entirely different. It is not a thing, it's a quality you develop. In my opinion and in contrast with modern trends (DBT etc.), mindfulness in Buddhism should not be understood as open-ended awareness. It is mindfulness of something specific you are trying to cultivate, habituate, get used to - in order to change your patterns.</p>

<p>EDIT:</p>

<p>From Buddhist perspective, we misinterpret our experience of awareness to implicate <em>an entity</em> that is aware. "I am aware" we say -- and this recipient of data, the point of view, the source of light, is what ancient Hindu's used to call Vedagu. Buddhists say such "point of view" is merely a perceptual illusion - like a bush that looks like a dog until you look closer. If you look very carefully, they say, you'll clearly see that there is no "you" that is aware. Instead (this is a Yogacara explanation) there is a simple feedback loop. The current dharma (state of mind - citta) serves as an input into the associative perception mechanism (sanjna) which finds matching imprints of previous experiences (samskaras) among those recorded in memory (alaya-vijnana). These imprints become the new state of mind, which serves as an input into the next iteration of the cycle and so on.</p>

<p>If you look at it this way, it is easy to see why we are fooled into thinking there is a subject of experience, "the point of view" that does the watching. As yet another thought takes shape, the gap between it and the next thought seems like there is "I am watching", then from this gap comes the association - this seems like "I think", but all it really is, is the associative perception mechanism (sanjna) that closes the loop by connecting the mind with the memory. "I am this thing that perceives" / "I am the observer" - is the mistake of identification with sanjna.</p>

<p>I guess you could shift the emphasis from the associative mechanism to the repository of past imprints that serves as the source of the next thought - and alternatively say that it is our <em>memory</em> that is The Observer, that it is our memory that "pays attention" to our thoughts/experiences and comes up with associations - which can lead some people to identify with samskara-skandha (the imprints group, basically memory) - another view that Buddha rejected.</p>

<p>The subjective experience, the virtual life-of-I created by our mind that we mistake for reality, is called "vijnana". This is the whole story that includes The Observer, the observed, and everything that happens between them.</p>

<p>What looks to us as self-awareness, with its sense of depth, is merely the recursive nature of the associative loop. It is a similar kind of effect you get when you point a video camera at TV that shows its own output - with a filter in the middle that adds matching elements of past memories. The same basic principle is at work behind Google's <a href="http://googleresearch.blogspot.co.uk/2015/06/inceptionism-going-deeper-into-neural.html" rel="nofollow">DeepDream</a>:</p>

<blockquote>
  <p>We ask the network: “Whatever you see there, I want more of it!” This creates a feedback loop: if a cloud looks a little bit like a bird, the network will make it look more like a bird. This in turn will make the network recognize the bird even more strongly on the next pass and so forth, until a highly detailed bird appears, seemingly out of nowhere.<br>
  ...<br>
  The results are intriguing—even a relatively simple neural network can be used to over-interpret an image, just like as children we enjoyed watching clouds and interpreting the random shapes.<br>
  ...<br>
  <em>If we apply the algorithm iteratively on its own outputs and apply some zooming after each iteration, we get an endless stream of new impressions, exploring the set of things the network knows about</em>.</p>
</blockquote>

<hr>

<p>Glossary:</p>

<ul>
<li><a href="https://en.wikipedia.org/wiki/Cartesian_theater" rel="nofollow">Cartesian theater</a></li>
<li>state of mind (<a href="https://en.wikipedia.org/wiki/Citta" rel="nofollow">Citta</a>)</li>
<li>associative perception mechanism (<a href="https://en.wikipedia.org/wiki/Sa%E1%B9%83j%C3%B1%C4%81" rel="nofollow">Saṃjñā</a>)</li>
<li>imprints of previous experiences (<a href="https://en.wikipedia.org/wiki/Sa%E1%B9%85kh%C4%81ra" rel="nofollow">saṃskāra</a>)</li>
<li>recorded in memory (<a href="https://en.wikipedia.org/wiki/Eight_Consciousnesses#.C4.80layavij.C3.B1.C4.81na" rel="nofollow">ālaya-vijñāna</a>)</li>
<li>the imprints group, basically memory (<a href="https://en.wikipedia.org/wiki/Skandha#The_five_skandhas" rel="nofollow">samskāra-skandhas</a>)</li>
<li>subjective experience, the virtual life-of-I created by our mind that we mistake for reality (<a href="https://en.wikipedia.org/wiki/Vij%C3%B1%C4%81na" rel="nofollow">Vijñāna</a>)</li>
</ul>
 -1: <blockquote>
  <p>The Observer is the experience of being aware of myself. It feels like I am witnessing my own thoughts and actions.</p>
</blockquote>

<p>In Buddhism, "myself" as an entity would be a mind-object: a concept, something one conceived / imagined, which receives attention -- a 'self' is taught to not be in the range of actual experiences such as a touch, a smell or primitive mental phenomena (such as a mental image, or mental sound) that can be contacted.</p>

<p>In Buddhism, one is "aware" of the detailed micro experiences (that, among other things, give rise to a concept of self). In buddhism, this "detailed awareness", or "attention" to a specific "primitive experience" is called <code>consciousness</code> (vinanna), which is formulated in buddhism as six-fold: eye-consciousness, ear-consciousness, nose-consciousness, tongue-consciousness, body-consciousness, mind-consciousness<sup>1</sup>.</p>

<p>Then, it's said that if there's  eye organ and visual forms,  eye-consciousness arise. The coming of the three is <code>contact</code>. For example in my visual field now there are bed sheets, wall frames and a TV. But while I write this, I'm not <code>conscious</code> of any of these forms, only of this visual character I'm typing right now.</p>

<p>For example, I just realized there was a lamp in the ceiling with lights on in my visual field, even though it has been reachable all the time. Until that moment, there was no <code>contact</code> (my consciousness was not directed) to this particular visual object.</p>

<p>The same occurs with mind as a sense: there are mind objects (e.g. thoughts) a "mind organ", and a mind-consciousness arise; the coming of the three is <code>contact</code> (e.g. when one gives attention to a particular thought). With <code>contact</code>, pleasant, unpleasant (or neither one) occur -- <a href="http://www.accesstoinsight.org/tipitaka/sn/sn12/sn12.044.than.html" rel="nofollow noreferrer">and so on</a>.</p>

<blockquote>
  <p>Mindfulness is the process of being aware of what I am doing. (To me, this does not require an Observer, but that is what I am asking.)</p>
</blockquote>

<p>I think this is close to what  mindfulness (sati) is in buddhism. It has different/related meanings depending on context but without going into details, I would summaryze as Thanissaro Bhikku put it: <a href="http://www.accesstoinsight.org/lib/authors/thanissaro/mindfulnessdefined.html" rel="nofollow noreferrer">"What does it mean to be mindful of the breath? Something very simple: to keep the breath in mind."</a></p>

<blockquote>
  <p>"Being aware of myself" means... Well, that I know what I am doing right now. Different from the other two.</p>
</blockquote>

<p>I'm not sure how your "being aware of what I am doing" is different than your "know what I am doing".</p>

<blockquote>
  <p>A Mind - is something that could do otherwise, it exercises choice on some level.</p>
</blockquote>

<p>In buddhism, I think this could be <em>mano</em>, <em>sankhara</em>, or even possibly <em>cetana</em>.</p>

<blockquote>
  <p>Awareness is the action of a sufficiently developed mind.</p>
</blockquote>

<p>I don't quite understand what you mean on this one.</p>

<blockquote>
  <p>Consciousness is awareness of being a self.</p>
</blockquote>

<p>I would not say one is aware of a self (or 'myself', see first point) -- e.g. as if 'self' was an experienced object grasped by any of the 6 sense faculties. But one does elaborate a (by buddhism, incorrect) understanding of being or having 'a self'.</p>

<p>Now, to the question...</p>

<blockquote>
  <p>Is Mindfulness the same as having an Observer?</p>
</blockquote>

<p>In Buddhism (and if you understand mindfulness to be <em>sati</em>), no. </p>

<p>For a detailed investigation on the meanings of mindfulness I suggest <a href="http://rads.stackoverflow.com/amzn/click/0415437377" rel="nofollow noreferrer"> Tse-fu Kuan's Mindfulness in Early Buddhism</a></p>

<hr>

<p><sup>1</sup>: The point, I think, is not to say that there is such a thing as <code>consciousness</code> and it can be perfectly divided in six part (an ontological statement). Rather, it is that understanding the mind in this way is useful / helps with training and understanding of dhamma, and consequently, with final liberation.</p>
