Essay outside of the NMT (No-Me Teaching) series 21

e^-x

Some   Ramana Maharshi quotes:

Even now you are that.  You are not apart from it.

The degree of the absence of concepts is the measure of your progress towards Self- realization. But Self-Realization itself does not admit of progress, it is ever the same. The Self remains always in Realization. The obstacles are concepts. Progress is measured by removal of obstacles to understanding that Self is always realized. So thoughts must be checked by seeking to whom they arise. Go to their source & they will not arise.

When one daily practices more & more abiding in the heart, the Mind will become extremely pure due to the removal of its defects, & the practice will become so easy that the purified Mind will plunge into the heart as soon as the Inquiry is commenced.

When you enter the inner stillness of Being,  the heart-going Mind is called the resting Mind.

When unity is replaced by a variety of perceived phenomena, it's called outgoing Mind.

Know that the Consciousness always shines as the formless Self, the true “I”.

Your duty is to be, & not to be this or that.

All that is required to realize the Self is to be still.

The Self is not now understood to be Truth, the one Reality.

The Truth of your Self alone is worthy to be scrutinized & known.  Taking it as the target of your attention, you should keenly seek to know it in your spiritual heart. This Knowledge of yourself will be revealed only to the Consciousness which is silent, clear, & free from the activity of the agitated & suffering Mind.

Whenever you are disturbed by thoughts you need merely withdraw within to the Self This is not concentration or destruction of the Mind but withdrawal into the Self.

The Mind, turned outwards, results in thoughts & objects. Turned inwards, it becomes itself the Self. To ask the Mind to kill the mind is like making the thief the policeman. He will go with you & pretend to catch the thief, but nothing will be gained.

So you must turn inward & see from where the Mind rises & then it will cease to exist.

The Mind is only a bundle of thoughts. How can you extinguish it by the thought of doing so, or by a desire ?  Your thoughts & desires are part & parcel of the Mind. The Mind is simply enhanced by new thoughts rising up. Therefore it is foolish to attempt to kill the Mind by means of the Mind. The only way of doing it is to find its source & hold on to it. The Mind will then fade away of its own accord.

  

It is with the inward-going Mind that you eliminate the outward-going Mind.

You do not set about saying there is a Mind & I'm going to kill it, but you seek the Source of the Mind. Then you will find that the Mind does not exist at all.

All that you need do is to find out the origin of the "I-thought" & abide there. Your efforts can extend only thus far. Then the Beyond will take care of itself.

The Ego-Self appears & disappears & is transitory, whereas the real Self is permanent.

You wrongly seem to identify the real Self with the Ego-Self.  See if that mistake has come about. The Ego-Self does not exist at all.

To whom is the trouble ?  The trouble also is imagined. Trouble & pleasure are only for the Ego.

Take care of yourself.  Let the World take care of itself.  See your Self.  If you are the Body there is the gross World also.  If you are spirit all is Spirit alone.

Do it yourself 1^st  then see if the question of others arises afterwards. 

There is no goal to be reached, nothing to be attained.  You are the Self.  You exist always.  No more can be predicated of the Self than that it exists.

Seeing God or the Self is only being the Self or yourself.

Seeing is being. You, being the Self, want to know how to attain the Self. 

It is something like a man being somewhere & asking how many ways there are to reach the place & which is the best way for him. All that is required of you is to give up the thought that you are this Body & to give up all thoughts of the external things or the not-Self.     

People say they aren't able to know the all pervading Self.

Even the smallest child says, "l exist. I do. This is mine."

Everyone understands that the thing "I" is always existent.  Only when the "l" is there, is there feeling you are the Body.

Knowing one that is always "visible" is one's own Self, is it necessary to search with a light ?

To say that we do not know the atma swarupa [the real nature of the Self] which is not different but which is in one‘s own Self is like saying, "l do not know myself."

Quantum Reality – Time 6 - metaphysical C:

Egocentric Presentism – other persons can be conscious, but their experiences are simply not present. Similarly, in related work, Hare argues for a theory of Perspectival Realism in which other perspectives do exist, but the present perspective has a defining intrinsic property. In one example that Hare uses to illustrate his theory, you learn that you are 1 of 2 people, named A & B, who have just been in a train crash; and that A is about to have incredibly painful surgery. You cannot remember your name. Naturally, you hope to be B. The point of the example is that you know everything relevant that there is to know about the objective world; all that is missing is your position in it, that is, whose experiences are present, A's or B's. This example is easily handled by egocentric Presentism because under this theory, the case where the present experiences are A's is fundamentally different from the case where the present experiences are B's. Hare points out that similar examples can be given to support theories like Presentism  in the philosophy of time. Perspectival Realism, there is a defining intrinsic property that the things that are in perceptual awareness have. Consider seeing object A but not object B. Of course, we can say that the visual experience of A is present to you, and no visual experience of B is present to you. But, it can be argued, this misses the fact that the visual experience of A is simply present, not relative to anything. This is what Perspectival Realism attempts to capture, resulting in a weak version of metaphysical Solipsism.

Same type of argument is often used in the philosophy of time to support theories such as Presentism. Of course, we can say that A is happening on [insert today's date]. But, it can be argued, this misses the fact that A is simply happening (right now), not relative to anything.

Theory of Relativity:  the conceptual Observer is at a geometric point in both Space & time at the apex of the “light cone” which observes events laid out in Time as well as Space. Different Observers can disagree on whether 2 events at different locations occurred simultaneously depending if the observers are in relative motion. This theory depends upon the idea of Time as an extended thing & has been confirmed by experiment, thus giving rise to a philosophical viewpoint known as 4-dimensionalism. However, although the contents of an Observation are time-extended, the conceptual Observer, being a geometric point at the origin of the Light Cone, is not extended in Time or Space. This analysis contains a paradox in which the conceptual Observer “contains nothing”, even though any real Observer would need to be the extended contents of an Observation to exist. This paradox is partially resolved in Relativity theory by defining a “frame of reference” to encompass the measuring instruments used by an Observer. This reduces the Time separation between instruments to a set of constant intervals.

Some of the difficulties & paradoxes of Presentism can be resolved by changing the normal view of Time as a “container” or thing unto itself & seeing Time as a measure of changing spatial relationships among objects; thus observers need not be extended in Time to exist & be aware, but rather they exist & the changes in internal relationships within the Observer can be measured by stable countable events.

According to the Growing Block Universe theory of Time (or the growing block view), the Past & Present exist & the Future does not exist. The Present is an objective property, to be compared with a “moving spotlight”. By the passage of Time more of the World comes into being, therefore the Block Universe is said to be growing. The Present is supposed to be the place where this is supposed to happen, a very thin slice of Space-Time, where more of Space-Time is coming into being.

The Growing Block View is an alternative to both Eternalism (according to which Past, Present, & Future all exist) & Presentism (according to which only the Present exists). It is held to be closer to common-sense intuitions than the alternatives.

Recently several philosophers have said that if the Growing Block View is correct we have to say that we don't know whether Now is Now. (The first occurrence of "Now" is an indexical & the 2^nd occurrence of "Now" is the “objective tensed property.” The term implies the sentence: "This part of Space-Time has the property of being Present".)

If people are talking in the Past, & at the same time thinking that “this” (their discussion) is occurring “Now”, the according to the Growing Block View, Tense is a real property of the World so their thought is about “Now” – they think, tenselessly, that their thought is occurring on the “edge of being” – their own objective Present. But we know they are wrong, because they are in the Past. They don't know that Now is Now. But how can we be sure we are not in the same position? Therefore we don't know whether Now is Now.

However, some have argued that there is an Ontological distinction between the Past & the Present. For instance, they argue that although there exists a Past, it is lifeless & inactive. Consciousness, as well as the Flow of Time is not active within the Past & can only occur at the boundary of the Block Universe in which the Present exists.

Eternalism is a philosophical approach to the Ontological nature of Time, which takes the view that all points in Time are equally "real", as opposed to the Presentist idea that only the Present is real. Modern advocates often take inspiration from the way Time is modeled as a Space-Time dimension in SR (Special-Relativity), giving Time an Ontology (property of existence) similar to that of Space. But the basic idea of Eternalism dates back at least to McTaggart's “B-theory” of “untensed time”.

This would mean that Time is just another dimension, that Future events are "already there", & that there is no objective Flow of Time. It is sometimes referred to as the "Block Time" or "Block Universe" theory due to its description of Space-Time as an unchanging 4–dimensional "Block", as opposed to the view of the World as a simply being a 3–dimensional Space modulated by the passage of Time.

Conventionally, Time is divided into three distinct regions; the "Past", the "Present", & the "Future". Using that Representational model, the Past is generally seen as being immutably fixed, & the Future as undefined & nebulous. As Time passes, the moment that was once the Present becomes part of the Past; & Part of the future, in turn, becomes the new Present. In this way, Time is said to pass, with a distinct present moment "moving" forward into the Future & leaving the Past behind.

Within this intuitive understanding of Time is the philosophy of Presentism, which argues that only the Present exists. It does not travel forward through an environment of Time, moving from a real point in the Past & toward a real point in the Future. Instead, the Present simply changes. The Past & Future do not exist & are only concepts used to describe the real, isolated, & changing present.

This conventional model presents a number of difficult philosophical problems, & seems difficult to reconcile with currently accepted scientific theories such as SR the theory of Special-Relativity.

Special-Relativity suggests that the concept of simultaneity is not universal. Observers in different frames of reference can have different perceptions of whether a given pair of events happened at the same time or at different times, with there being no physical basis for preferring one frame's judgments over another's (though in a case where one event A happens in the Past Light Cone of another event B), all frames will agree that A happened in the Past of B. So, in Special Relativity there can be no physical basis for picking out a unique set of events that are all happening simultaneously in "the Present".

Presentists have responded in the way that a Presentist could deny Naturalism. Such denial could take different forms. One could, claim that SR is not a theory about Time but about something else instead. Alternatively, one could retort by accepting that SR speaks to the geometry of space-time but reject that this has any Ontological import. Then, a Presentist might reject SR-Realism, simply asserting that SR is not approximately true of the World. Also, considerations from Quantum Mechanics can be invoked in an attempt to establish that SR is false or incomplete insofar as it lacks an absolute, privileged Frame of Reference.

Presentist might simply accept that SR offers a perfectly empirically adequate theory, but to insist that Absolute Simultaneity still exists. It is just that we cannot possibly detect the privileged Frame of Reference which determines the Present. In other words, Absolute Simultaneity is not empirically accessible. Metaphysics fully relies on postulated extra-structure that can't even in principle be observed. It violates Ockham's Razor so that the move cannot be justified by putting some post-verificationist Philosophy of Science on one's flag.

However, there are some, argued that it is possible to accept the physical predictions of Special Relativity while adopting an alternative interpretation of the theory in which there is a single privileged Frame whose judgments about Length, Time, & Simultaneity are the "true" ones, even though there would be absolutely no empirical way to distinguish this Frame from other frames, & no real experience could identify it.

When appealing to findings from empirically well-grounded disciplines, philosophers face a strong temptation to overstate their case — especially if their philosophical opponents can be relied on to be relatively innocent of new developments in the relevant science. I fear that some B-theorists have succumbed to the temptation, judging by the relish with which they often pronounce a verdict based on Relativity. They can practically hear the crunch of the lowly metaphysician’s armor giving way, as they bring the full force of incontrovertible physical fact down upon our A-theoretically-addled heads. But what actually hits us, and how hard is the blow? SR is false; GR’s future is highly uncertain; and the Presentist’s conflict with either version of Relativity is shallow, since the Presentist’s manifold can satisfy the same geometrical description as a B-theorist’s manifold, and afford explanations of all the same phenomena in precisely the same style. In these circumstances, how could appeal to SR or GR justify the frequent announcements that the A-theory–B-theory dispute has been “settled by physics, not philosophy”?

While the present is intuitively understood as the object that moves through the environment of time, it is common to also describe time as an object that moves, in the same way that a passenger on a train perceives the environment passing by. This perception of the passage or flow of time can can be confused with the previous idea of the present moving through time, leading to the misunderstanding that time is moving through time, i.e., that it is moving through itself. This illogical premise can lead to circular questions asking how fast time travels per unit of Time.

The concept of "Time passing" can be considered to be internally inconsistent, by asking "how much time goes by in an hour?" The question "how fast does Time pass" seems to have no satisfactory answer, in which answers such as "a second per second" would be, as some would argue, circular and thus false. In addition, even if we do accept the above answer, then the statement "a second per second" can be expressed as a fraction which is always equal to "one". But this "one" has no meaning beyond being a number and is thus also the wrong kind of answer. Therefore, the argument goes, the rate of the passage of Time is nonsensical.

Calculus for Yogis, part 4

There are many ways to approach  e  but one easy way is to look at how simple increase occurs, how, as afore mentioned above, the Derivative, the rate of growth   d y / d x  for  e^x  happens to numerically equal the value of   y  =  e^x   itself.  This describes that common situation of "the more you got, the more you get."  Wealth grows faster & faster the larger the amount or principal grows as  e^t   also.

A savings account, compounded continuously, grows that  e^t   way which Einstein called "the most amazing thing in the Universe", which was kind of a joke because he was privy to many amazing things about the Universe.  Some say that we "cannot get our heads around" the Exponential function because a Brain–Mind neuro-circuit "functions" in Exponential functions & like an eyeball, cannot see itself, but only a reflection or photo of an eyeball.

Weatherby way out on the high steep artwork early on in the flat part and reset the scale we get the same appearance so it's always rising the same way but it always looks like from your present point that the pass was very flat and slow in the future getting incredibly steep this is the rise in CO2 in the atmosphere the rising temperature of the planet in average temperature of the planet in climate change the rising population the rising various pollutants and so on as mentioned increasing bank accounts rise that way and decreasing bank accounts descend according to E to the X into the minor sex which is also the dissent of radioactivity over time in a sample of radioactive material or if that is a pure element or pure isotope

So if we know that  d y / d x  happens to be numerically equal  y  when   y  =  e^x  we must wonder about this curious number   e   which, as it happens can be determined in various mysterious ways, such as the following:

e =   lim  (1 +  1/ x)^x

      x => ∞

Both a "limit", like the preceding, & infinite series to be soon mentioned, "converge", rather than "blow up", for x  smaller than 1.  When we evaluate e  itself by setting x  = 1  in   e^x ,  we get:  e ~   2.72,   e =  2.718281828 …  and then no other repetition occurs right away.  But that early repetition of  18281828  allows us to actually remember e  to great accuracy because it's got that 18281828  pattern. 

The succeeding numbers vary and are not generally repeating in such a Transcendental number where the number the different numerals goes on forever.  And yet we can say the number in itself has a distinct value, as in saying that  e   is the value of parenthesis one plus one over  x , all to the  x   in the Limit that  x  goes to Infinity.  Now that's all more abstract than "the more you got, the more you get."  & that the rate of increase  numerically equals the value of the function itself.

Then optional notation to augment the Infinite series – Polynomial approach that here follows will be veritably "opaque" to the uninitiated but it deserves mention because of its elegance.  The "factorial" for 5, for instance, is  5! = 5×4×3×2×1  =  120.  That same kind of  "count dawn"  works generally, with the quirky Zero case here being: 

0!  =  1!  =  1

Much as for the Zeroth Power, the initiated might appreciate that for m  =  n :

1  =  m ! (m – n) !   =    n ! (n – n) !    =    n ! (0) !  =   n ! /  1    =   n !

This behavior or definition for  0 !  is necessary for calculating Probability with factorials.

The "factorial" is about the most rapidly growing function of an Integer, & it simplifies even the first few terms of an Infinite series, & more so a generic term.

That's the easy part, because the other useful optional notation to augment the Infinite series – Polynomial approach for use with the generic term is a "Summation" indicated by "Sigma", the Greek capital letter "S", namely Σ :

 4

Σ n   =   4  +  3  +  2  + 1  =  10.    [like it or not, take it or not – optional]

n=0

Both a "limit" above & the Infinite series to be soon mentioned, "converge", rather than "blow up", for x  smaller than 1.  In the case of infinite series, an approximate value results after a limited number if terms.

For example & for practice, one Series of simple terms suggests modification of the shortcut approach to estimating "uncertainties" in Arithmetic. 

                      

1/(1 – x)   =    1  +   x   +    x ²   +  x ³   +   x ⁴   ….    for small  x

                      

The same pattern holds for decimals. but is perhaps most easily seen for fractions of .99  & 1.01.  The approximate rule for "uncertainties" states that we keep the same number of "significant figures" as in  1.0/.99  ~  1.0   whereas  1.0/.99  ~  1.01  is more meaningful, even though a ratio of  2  figures  goes to an approximation with 3 figures.  But this treatment follows:

                      

1/ .99   =   1/(1 – .01)    ~    1  +  .01  +  .0001    ~    1  +  .01     =   1.01              

Likewise, for a fraction of 1.01:

       

Correspondingly, the "same number of significant figures" yields as in  1.00/1.01  ~  .990   whereas  1.0/1.01  ~  .99  is more meaningful, even though a ratio of  3  figures  goes to an approximation with 2 figures.  But again, this treatment follows:

1/(1 + x)   =    1   –   x   +    x ²    –  x ³   +   x ⁴   ….    for small  x

1/1.01  =  1/(1 + .01)    ~    1   –  .01  +  .0001    ~    1   –  .01     =   .99

Aside from the series treatment, the consistency in both cases follows a  1%  "uncertainty"  &   % –uncertainty is the truer rule.

In a similar vein, we can familiarize ourselves with Polynomial series with 2 similar approximations:

                        

1/(1 – x)²   =     1  +   2 x   +  3 x ²   + 4 x ³   +  5 x ⁴   ….    for small  x    &               

1/(.99)²    =  1/(1 – .01)²     ~    1  +  .02    =   1.02

                        

1/(.99)³    =  1/(1 – x)³   =     1  +  3 x   +  6 x ²   + 10 x ³   +  15 x ⁴   for small  x   &

1/(1 – .01)³      ~    1  +  .03    =   1

To indicate these 2 Polynomial series in the above-mentioned  "Summation"  "Sigma" notation:

                          ∞

1/(1 – x)  =       Σ  x ⁿ / n   =   1  +   x   +    x ²   +  x ³   +   x ⁴   ….    for small  x

                  n  = 0

                          ∞

1/(1 + x)  =  –    Σ  x ⁿ / n   =    –  x    –  x ² / 2    –  x ³ / 3   –   x ⁴ / 4    ….    for small  x

                  n  = 0

                           ∞

1/(1 – x)²    =     Σ  n x ^{n –1}   =   1  +   2 x   +  3 x ²   + 4 x ³   +  5 x ⁴     ….    for small  x

                   n  = 1

                             ∞

1/(1 – x)³  =  ¹/₂   Σ  (n–1) n x ^{n –2}   =   1  +  3 x   +  6 x ²   + 10 x ³   +  15 x ⁴ ….  small  x

                   n  = 1                      =   0  +  ¹/₂  x  1  x  2  x  x⁰   +   ¹/₂  x  2  x  3  x  x¹

                        +   ¹/₂  x  3  x  4  x  x ²   +   ¹/₂  x  4  x  5  x   x ³   +   ¹/₂  x  5  x  6  x   x ⁴

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