**Essay outside of the NMT (No-Me Teaching) series 23**

Algebra of Logistic Curve

Some Ramana Maharshi quotes:

*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*

**then see if the question of others arises afterwards.**^{st}*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."*

*The nature of bondage is merely the rising, ruinous thought "I am different from the Reality." Since one surely cannot remain separate from the Reality, reject that thought whenever it rises.*

*You speak of memory & oblivion of the fullness of the Self. Oblivion & memory are only thought-forms. They will alternate so long as there are thoughts. But Reality lies beyond these. Memory or oblivion must be dependent on something. That something must be foreign to the Self as well, otherwise there would not be oblivion. That upon which memory & oblivion depend is the idea of the individual self.*

*When one looks for it, this individual "I" is not found because it is not real. Hence this "I" is synonymous with Illusion or Ignorance [maya_, , or ajnana].*

*To know that there never was Ignorance is the goal of all the spiritual teachings. Ignorance must be of one who is aware. Awareness is jnana. Jnana is eternal & natural, ajnana is unnatural & unreal.*

*In Deep Sleep man is devoid of possessions, including his own Body.*

*Instead of being unhappy he is quite happy. Everyone desires to sleep soundly.*

*The conclusion is that Happiness is inherent in man & is not due to external causes.*

*One must realize the Self in order to open the store of unalloyed Happiness.*

*lf a man thinks that his Happiness is due to external causes & his possessions, it is reasonable to conclude that his Happiness must increase with the increase of possessions & diminish in proportion to their diminution. Therefore if he is devoid of possessions, his Happiness should be nil. What is the real experience of man ? Does it conform to this view ?*

*:*

__Quantum Reality 5__"

*In a few years, all the great physical constants will have been approximately estimated, & the only occupation which will then be left to the men of Science will be to carry these measurement to another place of decimals.*" James Clerk Maxwell 1871

[less than 3 decades before Max Planck’s announcement of the

*Quantum*]

Albert Einstein derived important new physical concepts from pure Mathematics, & yet as a

*Realist*he & his World

**–**wide

*Realist*admirers (

*among others*) never doubted that the value of Relativity theories lies in their description of a

*Mind-Independent*Reality. Somewhat the same (

*to a less mathematical degree*) could be said for the breakthrough Physics of Newton & Galileo before him. Herein lies a contradiction explicated in the very heart of Nobel laureate. Eugene Wigner’s “

*Unreasonable Effectiveness of Mathematics in the Natural Sciences*” which spoke both to

*Scientific Realism*& to the nature of Mathematics: "It is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an

*uncanny*number of cases to an amazingly accurate description of a large class of phenomena." The Law of Gravity used to model freely falling bodies on the surface of the Earth, for instance, was extended on the basis of "very scanty observations" to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations."

Pointing to James Clerk Maxwell’s equations for elementary Electrical & Magnetic phenomena inspired Maxwell’s concept of EMR

*electro-magnetic*radiation, starting with visible Light & continuing to describe then-recently discovered radio waves, etc. Wigner characterizes such examples as illustrating that "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious & that there is no rational explanation for it."

According to Wigner, these & numerous other examples (

*including String Theory*) cited by himself & others “put a heavy strain on our faith in our theories & on our belief in the Reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called the

*Ultimate Truth*. The reason that such a situation is conceivable is that, fundamentally,

*we do not know why our theories work so well*. Hence, their accuracy may not prove their truth & consistency.”

Richard Hamming, Mathematician & Computer Science pioneer, agreed with Wigner to the extent that: “

*humans see what they look for*”; “

*humans create*[like vectors, tensors, quaternions, etc.]

*& select the Mathematics that fit a situation*”; “

*Mathematics addresses only a part of human experience*”; & “

*Evolution has primed humans to think mathematically*”. He further analyzed the pivotal work of Galileo as being, not an experimentalist revolution beyond Aristotle’s often-faulty arm-chair physics. Rather Galileo’s revolution lie in experimentally testing & conforming his own more-refined arm-chair physics.

Somewhat similarly, Newton’s “brilliance” lies in proposing linear proportions (

*to 1*) for contributing factors in Gravity & Force in general, such as Mass (

^{st}order*of each body for General Gravitation*) & proportionality constants (

*like the Gravitational constant & Earth-surface Gravity acceleration*). The Inverse Square with distance derives, for instance, from a sphere’s surface area 4π r

**for Conservation of Energy in an isotropic force in 3-D Space. Heisenberg’s mysterious Uncertainty Principle derives from properties of the Fourier integrals used in the Mathematical formulation of Quantum Mechanics.**

^{2}Sir Arthur Eddington, who experimentally verified General Relativity, suggested the parable: "Some men went fishing in the sea with a net, & upon examining what they caught they concluded that there was a minimum size to the fish in the sea."

Some more contemporary Physicists, like Max Tegmark, actually go further when they attempt to reconcile such issues with the

*Realism*they still adhere to. Tegmark does not credit the human Mind with imposing Mathematics

*onto*the Universe, but rather claims that the Universe is not mental, but is

*Mathematical*in nature. Thus humans only re-discover that inherent

*Realist*structure. Similarly, but rather than in Math, Philosopher of Mathematics, Ivor Grattan-Guinness, sees the close correspondence between

*Realism*& Mathematical theory in the

*linguistic*nature & structure of Analogy, Generalization, & Metaphor. Our typical conjectures about correspondences between a Model & the Reality, between the Map & the Territory, make the form-fitting of Mathematics to Physics inevitable.

*We rejoice (actually we are relieved of a need) when, just as if it were a lucky chance favoring our aim, we do find such systematic unity among merely empirical laws.*Imannuel Kant

*The most incomprehensible thing about the Universe is that it is comprehensible. How can it be that Mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of Reality? As far as the laws of Mathematics refer to Reality, they are not certain; and as far as they are certain, they do not refer to Reality.*Albert Einstein

*Physics is Mathematical not because we know so much about the physical World, but because we know so little; it is only its Mathematical properties that we can discover.*Bertrand Russell

*There is only one thing which is more unreasonable than the unreasonable effectiveness of Mathematics in Physics, and this is the unreasonable ineffectiveness of Mathematics in Biology.*Israel Gelfand

Some more selected verses from the Ramana Maharshi disciple Master Nome:

If one is immersed in

*Samsara*, the repetitive Cycle of Illusion, Birth, & Deaths, filled with Suffering, the way to liberate oneself is by Self-Knowledge. Liberation, which is the goal of spiritual practice, is one’s own if, inquiring to know the Self, he discards all notions of “me” & “mine” attains complete certitude as to the Space-like nature of Reality, & “abides” devoid of physical & mental forms, & the assumption of an Ego-entity.

If the Self would change states, such as states of Ignorance & Knowledge, of Bondage & Liberation, that Self would be destructible, & Liberation itself would be artificial or unreal. Liberation is not a change of state from one state into another state. It is not reasonable to imagine a separation & later union in relation to the Self, for then both would be transitory. Transience cannot be attributed to the Real, just as the unreal cannot be attributed to the Real, or the dual to the Non-Dual. Since there truly can be neither later union nor prior separation, Liberation cannot consist of an Individual entering into

*Brahman*or

*Brahman*coming to the Individual. The True Nature of the Self is never destroyed, is changeless, is uncaused, & cannot be obtained or lost. Any new appearance, or coming into being of any state would be the Effect of a precedent Cause. Such would be changeful, transitory, & not self-existent. Such cannot be permanent Liberation. Self-Knowledge alone is Liberation. Knowing the Self to be oneself is the greatest attainment. To wrongly assume the non-Self to be the Self is Ignorance.

The removal of the superimposed mis-conceptions of what the Self alone is constitutes the path to Liberation. No other view is reasonable, since such always involves some Dualism, some belief in an existent individual experiencer & a self-existent

*objective*thing. There is also then the conception that Reality becomes other than what it is, & that the unreal actually comes to be. Liberation cannot be a change of condition, because such involves mutability & thus destructibility, parts or divisions, & a change in its nature.

Any belief that superimposition occurs on some substrate of non-existence, or that belief that Illusion can actually create itself, or that there is no Absolute Self, all such beliefs should be abandoned. This is because of the existence of Being itself is irrefutable. Furthermore, it is not reasonable that something could come out of “nothing”. Superimposition (

*in Illusion*) occurs on some real thing, & this for “someone” who knows the Ignorance. What is it Ignorance of ? And who knows is the Ignorance for ? If one so inquires, one find that Being alone exists, Consciousness alone exists.

__Calculus for Yogis, part 7__Now in a certain version of

*e*in what's called "complex space" describes a Circle even though we don't see that in ordinary

^{x }*x*&

*y*coordinates, where we see a gradually rising & then rapidly rising curve. The special "complex space" version is graphed as a Circle, often with Radius 1. If we look, for instance, at a Radius heading to the upper right direction, & at the point where it intersects the Circumference, we can drop a Vertical line & create a Right Triangle that includes as a Base, the

*x*axis.

The angle that the Radius line made with that x-axis Base can be called Angle A . To some extent Angle A is characterized by the ratio of that Vertical line length to the length of X axis Base. This "Opposite" side over "Adjacent" side ratio, called the Tangent of A is one of the ways to calculate the Slope, in fact the Derivative of the slanting line of the Radius. This tangent of a, abbreviated as "tan A" & can in turn be considered the ratio of 2 other functions of A.

The vertical line for this unit Radius Circle equals has a length equal to Sine A, abbreviated as "sin A". The portion of the x-axis making up the Base has a length equal to "Cosine A". [

*If the radius were more or less than 1, that proportion would be multiplying these functions, sin A & cos A, but have no effect on tan A*.]

So whereas tan A is definable as "Opposite over Adjacent", sin A can be defined as "Opposite side over the Radius", the hypotenuse. That = 1 in this case. The cos A can be just found as the "Adjacent side length over the radius", again just equal to that length because the Radius = 1.

More meaningful than defining Sine & Cosine in terms of right triangles is to consider them as the Coordinates of that point of intersection of the Radius & the unit Circle formed by a

*complex space*version of

*e*.

^{x }In that sense some of this complicated version of

*e*is composed of the Sine of

^{x }*x*& Cosine

*x*. We see this showing up in the (

*Power Series*) Polynomial Infinite Series for sin

*x*& cos

*x*.

Recall that:

∞

*e*

^{x }**=**Σ

*x*

^{n }**/**n

**!**

**=**1

**+**

*x*+*x*

^{2 }**/**2 !

**+**

*x*

^{3 }**/**3 ! +

**….**

n

**=**0

& now we compare that to:

∞

sin

*x*

**=**Σ (

**–**1)

^{n}*x*

^{2n+1 }**/**(2n+1)

**/**!

**=**

*x*

**–**

*x*

^{3 }**/**3

**!**

**+**

*x*

^{5 }**/**5

**!**

**–**

*x*

^{7 }**/**7

**!**.

n

**=**0

∞

cos

*x*

**=**Σ (

**–**1)

^{n }*x*

^{2n }**/**2n

**!**

**=**1

**–**

*x*

^{2 }**/**2 !

**+**

*x*

^{4 }**/**4 !

**–**

*x*

^{6 }**/**6 !

**….**

n

**=**0

Again, the alternating Minus signs arise out of the "complex space" relationship.

Thus we reduce more complicated Exponential & Trigonometric functions to more ordinary Polynomial terms & compare the Polynomial series for

*e*with the Polynomial series for sin

^{x }*x*& cos

*x*because we know how to take the Derivative of those terms.

We previously took the Derivative of that expression for

*e*, term by term, to show that the Derivative the of each of terms numerically goes back to, is equal to our initial

^{x}*e*.

^{x }Doing the same for sin

*x*& cos

*x*, we find that the Derivative each of terms goes back to terms of the other Series, sin

*x*to cos

*x*, &

*vice versa*. except that the Minus signs are reversed in the 2

**case. That is, the exercise shows that: the Derivative of sin**

^{nd}*x*is

**=**cos

*x*, & the Derivative of cos

*x*is

**= –**sin

*x*.

We saw that we can look at the

*inverse*of the Derivative, in a matter speaking, which is called the Integral, with the so-called "indefinite" Integral. We truly have that inverse to within a added constant, leaving that added constant out, or making it be a 0, Then we can treat the Indefinite Integral as if it was the

*inverse*of the Derivative.

So looking at a couple of Polynomials like 7

*x*

**+ 4**

^{2}*x*+ 3, we saw that the Derivative is 14

*x*+ 4 , & for Integral we go the other way to:

^{7}/_{3}*x*

**+**

^{3}

^{4}/_{2}*x*+ 3

*x*+ a constant.

Up until now we delayed introducing any version of the symbol for the integral because it's unduly strange looking, scary, and off putting to the uninitiated. But will simplify it by saying that the symbol is based on a big "S". In fact since it's a Roman alphabet "S", in a way. it's less exotic than the Greek capital "S", Sigma that we saw for a generic term representation of an Infinite Series. In any case, that statement just above can be rewritten symbolically as:

d

**/**d

*x*[7

*x*

**+ 4**

^{2}*x*+ 3]

**=**14

*x*+ 4 (Derivative) &

∫ [7

*x*

**+ 4**

^{2}*x*+ 3] d

*x*

**=**

^{7}/_{3}*x*

**+**

^{3}

^{4}/_{2}*x*+ 3

*x*+ a constant (Integral)

The Analogy we drew was that the Derivative is the

*projected*Perspective toward Happiness, viewing toward the estimated Future, based on the immediate past average; & the Integral is an accumulated

*memory*or record by which we define Reality & Identity , instead of the Happiness. Looking at these rather complex looking structures created on paper, we van so peak at the Mind, much as elsewhere we compare the Mind to Mathematical Coordinate Spaces. Math mirrors Mind.

Now there are special reasons for seemingly idle curiosity about sin

*x*& cos

*x*. For one thing, these 2 functions are the building blocks of the many "wave" phenomena that we see in the Sciences. Back in the area of Mathematics, but Mathematics very applicable to Science, the sin

*x*& cos

*x*are also building blocks for the Fourier transforms. Fourier transforms & Laplace transforms, Quantum Mechanical Correlation, & similar topics will also be introduced by taking a closer look at the Integral in general.

Derivatives & Integrals can often be determined by systematic rules, like the Power Rule such as is the case for Polynomials. But in other cases when we take the Integral, solving is more of an art form, Detective intuition, trial & error, & so forth yield some basic processes, one being Integration by Parts. We pause in our discussion to look at Integration by parts because it will serve as a foundation for the idea of a Transform.

What we will see is actually closer in form to Conjugate Variables in Quantum Mechanics, which are the basis for the Heisenberg Uncertainty Principle. Quantum Mechanics also uses the Fourier transform to move from ordinary Distance Space to a Transform of that, called Momentum Space. And so with these motivations we will look at Integration by Parts.

The above themes & 1600 pages more are freely available as perused or downloaded PDF’s, the sole occupants of a Public Microsoft Skydrive “Public Folder” accessible through:

or with Caps-sensitive:

Duplicates have been available on

**:**

__jstiga.wordpress.com/__
[But from now on, they will be different & still usually daily.]

"school of Gaudapada , Shankara, Ramana, Nome

"

*There is no Creation, no Destruction, no Bondage, no longing to be freed from Bondage, no striving for Liberation, nor anyone who has attained Liberation. Know that this to be Ultimate Truth.*"**–**the "no creation"**–***Ajata Vada**for very succinct summary of the teaching & practice, see:*

**www.ajatavada.com/**
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