Update: 20210720 04:35 AM 0400
episodhist.htm
by U Kyaw Tun (UKT) (M.S., I.P.C., USA),
Daw Khin Wutyi, Daw Thuzar Myint, Daw Zinthiri Han
and staff of Tun Institute of Learning (TIL).
Not for sale. No copyright. Free for everyone.
Prepared for students and staff of TIL
Research Station, Yangon, MYANMAR

http://www.tuninst.net ,
www.romabama.blogspot.com
 based on Episodic History of Mathematics :
by S. G. Krantz, 2006
in TIL PDF and SD libraries:

SGKrantzEpisodicHistMath<Ô> /
Bkp<Ô> (link chk 201030)
Preface  UKT
TOC to Krantz
01 Pythogoras  1
02 The Mystical Mathematics of Hypatia  69
03 Henri Poincaré, Child Prodigy  359
UKT 210701
S. G. Krantz, wrote in his Preface, "Together with philosophy, mathematics is the oldest academic discipline known to mankind." Academic  maybe. Language, in terms of vowels and consonants, as a human activity has been studied in the East much earlier by humans known as Rishis {ra.þé.} dedicated to the pursuit of an activity.
Rishis have found that the human mind can be developed to achieve "things" commonly thought impossible. We see such activity even today in Myanmar  e.g. the ability to "see" or gaze into the midday sun.
For most humans such ability has to be developed, but for a few such ability  such as the First Jhana {pa.hta.ma. Zaan} comes naturally at an early age. When a person enters into a Jhana state, he enjoys "pleasure" unknown to others and can remain immobile unaware of things happening around him.
In the case of Prince Siddhatha who eventually became the Gautama Buddha, such an extraordinary experience came at an early age during his father's Ploughing Ceremony.
The experience was "triggered" by a sight of a bird catching a worm for its meal. Eating is a common task every living one does from animals to humans, without an after thought. Clearly the bird is not even aware that he is "killing" the worm. He is just "eating". Would he be guilty of {paNati. pata.} ?
Now, how would the child be able to communicate with another about his experience? It is now known as the First Jhana {pa.hta.ma. Zaan}. It is an uncommon experience and the listener would not know what the child is talking about. If he were an ordinary person  the child of a commoner  he would surely be laughed at. I'm sure Prince Siddhatha, as a child, would not tell others of his experience. Yet, after he became the Buddha, he mentioned it to a Brahmin Poannar {braah~ma.Na. poaN~Na:}. See Questions and Answers of the Sixth Buddhist Council in {ma.haþic~sa.ka. þoat} included in {miz~Zi.ma. ni.kaèý}, Part 1, in PaliMyan and translation in BurMyan, p.130 of my personal collection.
The ability to communicate with each other of the species by use of Language involving syntax is a human achievement. We Communicate about the worldly affairs and also about philosophical questions. Another human achievement is the ability to count  the basis of Mathematics. Without Mathematics our communication on Science becomes very unreliable.
If language is just a skill for communication within one species, what about other species such as apes and dolphins. Since they do communicate with one another, their means of communication must be other than language. If they have the communicative skill, these nonhuman species might (my emphasis on might) have mathematical skills but unlike our mathematics.
S. G. Krantz, continues to write in his Preface, "Thus it is worthwhile to have a book that will introduce the student to some of the genesis of mathematical ideas. While we cannot get into the nuts and bolts of Andrew Wiles’s solution of Fermat’s Last Theorem, we can instead describe some of the stream of thought that created the problem and led to its solution."
Whatever the case may be a modern educated person must have some idea of modern mathematics. That might be impossible for many of us. But at least we should know how our human mathematics evolve.
Now read
•
Episodic History of Mathematics :
by S. G. Krantz, 2006 . Be prepared for some
surprises, such as one on
celebrated Pythagoras Theorem:

SGKrantzEpisodicHistMath<Ô> /
Bkp<Ô> (link chk 210702)
Krantz writes: "Mathematical history is exciting and rewarding, and it is a significant slice of the intellectual pie. A good education consists of learning different methods of discourse, and certainly mathematics is one of the most welldeveloped and important modes of discourse that we have.  p.roman04.
Krantz continues: "And in fact it [Pythagoras Theorem] is one of the most ancient mathematical results. There is evidence that the Babylonians and the Chinese knew this theorem nearly 1000 years before Pythagoras."  p003 [Note: Pythagoras, fl 6th century B.C.  AHTD ]
Gautama Buddha (563?483? B.C  AHTD), though not a mathematician, discovered by methodical thinking why human beings and other thinking beings (such as animals) suffer from "mental pains" such as fear and anxiety.
"Fear and anxiety often occur together but these terms are not interchangeable. Even though symptoms typically overlap, a person's experience with these emotions differs based on their context. Fear relates to a known or understood threat, whereas anxiety follows from an unknown or poorly defined threat."  https://www.verywellmind.com/fearandanxiety... 210701
The Buddha found the root cause of mental pain
{sait doak~hka.} "Suffering" is
{a.swè:} "Attachment". Buddha found if one were to achieve complete freedom
from all kinds of attachment (such as attachment to a person, to a material
object, or to an idea), you can be liberated.
Liberated from what? Traditional BurMyan Buddhists will say: "from the
cycle of beginning (birth), continuation of being, ending (death)" Than'tha'ra
{þänþa.ra}. "No!", SktDev speaking Brahmin Poanna
{braah~ma.Na. poaN~Na:} would say, "your pronunciation is wrong. It is
{Sþäm.SþaRa} संसार «saṃsāra»".
They place more emphasis on correct pronunciation that on meaning.
Straight away a debate between thibilant nonrhotic speakers and sibilant rhotic speakers began instead of on meaning of the words: BurMyanBuddhists' {þa.} is thibilant, whereas Hindus' Devanagari स «sa» is sibilant. Don't look up in IPA, which gives /θ/ without defining whether it is thibilant or sibilant. I've no choice but to invent a new glyph for Hindus' Devanagari sibilant: {Sþa.} स «sa».
Look for the meaning: and you get more confused. Wikipedia:
https://en.wikipedia.org/wiki/Saṃsāra 210701
"Saṃsāra is a Sanskrit/Pali word that means "world". [ref.1 and
2]. It is also the concept of rebirth and "cyclicality of all life, matter,
existence", a fundamental belief of most Indian religions. [ref.3 and 4]. In
short it is the cycle of death and rebirth. [ref.2 and 5]. Saṃsāra is
sometimes referred to with terms or phrases such as transmigration, karmic
cycle, reincarnation, and "cycle of aimless drifting, wandering or mundane
existence" [ref.2 and 6]. The concept of Saṃsāra has roots in the
postVedic literature; the theory is not discussed in the Vedas themselves.
[ref.7 and 8]. It appears in developed form, but without mechanistic details, in
the early Upanishads. [ref.9 and 10].
To be on safe ground, we'll just say that "if one were to achieve complete freedom from all kinds of attachment, you can be liberated." Liberated from mental pain {sait doak~hka.} "Suffering". Liberation is known as {naib~baan}. [Note: you achieve Mental Freedom {naib~baan} during lifetime, and BodilyFreedom {naib~baan} at death.]
Buddha's method is said to have been repeated by many resulting many achieving {naib~baan}. Whatever the case may be it is common experience that lessening {a.swè:} "Attachment" by Buddhist practice reduces mental suffering. It makes the practitioner living a peaceful life. Because of this, I claim Gautama Buddha to be a scientist. His thinking is methodical and his method can be practiced by anyone  even by nonBuddhists. However trying to make him divine, extolling his "magical attributes" {tänhko: tau}, we are making him into a religious figure. He is more than a mere "magician" and a "storyteller". His "insight", his "intellect" {ñaaN tau}, is more praiseworthy than {tänhko: tau}. To me he has achieved {ñaaNtau a.nûn~ta.} "infinite intellect".
Now let's get back to Krantz . I'll first give his TOC:
1 The Ancient Greeks 1
1.1 Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Introduction to Pythagorean Ideas . . . . . . . . 1
1.1.2 Pythagorean Triples . . . . . . . . . . . . . . . . 7
1.2 Euclid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 Introduction to Euclid . . . . . . . . . . . . . . . 10
1.2.2 The Ideas of Euclid . . . . . . . . . . . . . . . . . 14
1.3 Archimedes . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 The Genius of Archimedes . . . . . . . . . . . . . 21
1.3.2 Archimedes’s Calculation of the Area of a Circle . 24
2 Zeno’s Paradox and the Concept of Limit 43
2.1 The Context of the Paradox? . . . . . . . . . . . . . . . 43
2.2 The Life of Zeno of Elea . . . . . . . . . . . . . . . . . . 44
2.3 Consideration of the Paradoxes . . . . . . . . . . . . . . 51
2.4 Decimal Notation and Limits . . . . . . . . . . . . . . . 56
2.5 Infinite Sums and Limits . . . . . . . . . . . . . . . . . . 57
2.6 Finite Geometric Series . . . . . . . . . . . . . . . . . . . 59
2.7 Some Useful Notation . . . . . . . . . . . . . . . . . . . . 63
2.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 64
3 The Mystical Mathematics of Hypatia 69
3.1 Introduction to Hypatia . . . . . . . . . . . . . . . . . . 69
3.2 What is a Conic Section? . . . . . . . . . . . . . . . . . . 78
4 The Arabs and the Development of Algebra 93
4.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . 93
4.2 The Development of Algebra . . . . . . . . . . . . . . . . 94
4.2.1 AlKhowˆarizmˆı and the Basics of Algebra . . . . . 94
4.2.2 The Life of AlKhwarizmi . . . . . . . . . . . . . 95
4.2.3 The Ideas of AlKhwarizmi . . . . . . . . . . . . . 100
4.2.4 Omar Khayyam and the Resolution of the Cubic . 105
4.3 The Geometry of the Arabs . . . . . . . . . . . . . . . . 108
4.3.1 The Generalized Pythagorean Theorem . . . . . . 108
4.3.2 Inscribing a Square in an Isosceles Triangle . . . . 112
4.4 A Little Arab Number Theory . . . . . . . . . . . . . . . 114
5 Cardano, Abel, Galois, and the Solving of Equations 123
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 The Story of Cardano . . . . . . . . . . . . . . . . . . . 124
5.3 FirstOrder Equations . . . . . . . . . . . . . . . . . . . 129
5.4 Rudiments of SecondOrder Equations . . . . . . . . . . 130
5.5 Completing the Square . . . . . . . . . . . . . . . . . . . 131
5.6 The Solution of a Quadratic Equation . . . . . . . . . . . 133
5.7 The Cubic Equation . . . . . . . . . . . . . . . . . . . . 136
5.7.1 A Particular Equation . . . . . . . . . . . . . . . 137
5.7.2 The General Case . . . . . . . . . . . . . . . . . . 139
5.8 Fourth Degree Equations and Beyond . . . . . . . . . . . 140
5.8.1 The Brief and Tragic Lives of Abel and Galois . . 141
5.9 The Work of Abel and Galois in Context . . . . . . . . . 148
6 Ren´e Descartes and the Idea of Coordinates 151
6.0 Introductory Remarks . . . . . . . . . . . . . . . . . . . 151
6.1 The Life of Ren´e Descartes . . . . . . . . . . . . . . . . . 152
6.2 The Real Number Line . . . . . . . . . . . . . . . . . . . 156
6.3 The Cartesian Plane . . . . . . . . . . . . . . . . . . . . 158
6.4 Cartesian Coordinates and Euclidean Geometry . . . . . 165
6.5 Coordinates in ThreeDimensional Space . . . . . . . . . 169
7 The Invention of Differential Calculus 177
7.1 The Life of Fermat . . . . . . . . . . . . . . . . . . . . . 177
7.2 Fermat’s Method . . . . . . . . . . . . . . . . . . . . . . 180
7.3 More Advanced Ideas of Calculus: The Derivative and the
Tangent Line . . . . . . . . . . . . . . . . . . . . . . . . 183
7.4 Fermat’s Lemma and Maximum/Minimum Problems . . 191
8 Complex Numbers and Polynomials 205
8.1 A New Number System . . . . . . . . . . . . . . . . . . . 205
8.2 Progenitors of the Complex Number System . . . . . . . 205
8.2.1 Cardano . . . . . . . . . . . . . . . . . . . . . . . 206
8.2.2 Euler . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.2.3 Argand . . . . . . . . . . . . . . . . . . . . . . . 210
8.2.4 Cauchy . . . . . . . . . . . . . . . . . . . . . . . 212
8.2.5 Riemann . . . . . . . . . . . . . . . . . . . . . . . 212
8.3 Complex Number Basics . . . . . . . . . . . . . . . . . . 213
8.4 The Fundamental Theorem of Algebra . . . . . . . . . . 219
8.5 Finding the Roots of a Polynomial . . . . . . . . . . . . 226
9 Sophie Germain and Fermat’s Last Problem 231
9.1 Birth of an Inspired and Unlikely Child . . . . . . . . . . 231
9.2 Sophie Germain’s Work on Fermat’s Problem . . . . . . 239
10 Cauchy and the Foundations of Analysis 249
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 249
10.2 Why Do We Need the Real Numbers? . . . . . . . . . . . 254
10.3 How to Construct the Real Numbers . . . . . . . . . . . 255
10.4 Properties of the Real Number System . . . . . . . . . . 260
10.4.1 Bounded Sequences . . . . . . . . . . . . . . . . . 261
10.4.2 Maxima and Minima . . . . . . . . . . . . . . . . 262
10.4.3 The Intermediate Value Property . . . . . . . . . 267
11 The Prime Numbers 275
11.1 The Sieve of Eratosthenes . . . . . . . . . . . . . . . . . 275
11.2 The Infinitude of the Primes . . . . . . . . . . . . . . . . 278
11.3 More Prime Thoughts . . . . . . . . . . . . . . . . . . . 279
12 Dirichlet and How to Count 289
12.1 The Life of Dirichlet . . . . . . . . . . . . . . . . . . . . 289
12.2 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . 292
12.3 Other Types of Counting . . . . . . . . . . . . . . . . . . 296
13 Riemann and the Geometry of Surfaces 305
13.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 305
13.1 How to Measure the Length of a Curve . . . . . . . . . . 309
13.2 Riemann’s Method for Measuring Arc Length . . . . . . 312
13.3 The Hyperbolic Disc . . . . . . . . . . . . . . . . . . . . 316
14 Georg Cantor and the Orders of Infinity 323
14.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . 323
14.2 What is a Number? . . . . . . . . . . . . . . . . . . . . . 327
14.2.1 An Uncountable Set . . . . . . . . . . . . . . . . 332
14.2.2 Countable and Uncountable . . . . . . . . . . . . 334
14.3 The Existence of Transcendental Numbers . . . . . . . . 337
15 The Number Systems 343
15.1 The Natural Numbers . . . . . . . . . . . . . . . . . . . 345
15.1.1 Introductory Remarks . . . . . . . . . . . . . . . 345
15.1.2 Construction of the Natural Numbers . . . . . . . 345
15.1.3 Axiomatic Treatment of the Natural Numbers . . 346
15.2 The Integers . . . . . . . . . . . . . . . . . . . . . . . . . 347
15.2.1 Lack of Closure in the Natural Numbers . . . . . 347
15.2.2 The Integers as a Set of Equivalence Classes . . . 348
15.2.3 Examples of Integer Arithmetic . . . . . . . . . . 348
15.2.4 Arithmetic Properties of the Integers . . . . . . . 349
15.3 The Rational Numbers . . . . . . . . . . . . . . . . . . . 349
15.3.1 Lack of Closure in the Integers . . . . . . . . . . . 349
15.3.2 The Rational Numbers as a Set of Equivalence
Classes . . . 350
15.3.3 Examples of Rational Arithmetic . . . . . . . . . 350
15.3.4 Subtraction and Division of Rational Numbers . . 351
15.4 The Real Numbers . . . . . . . . . . . . . . . . . . . . . 351
15.4.1 Lack of Closure in the Rational Numbers . . . . . 351
15.4.2 Axiomatic Treatment of the Real Numbers . . . . 352
15.5 The Complex Numbers . . . . . . . . . . . . . . . . . . . 354
15.5.1 Intuitive View of the Complex Numbers . . . . . 354
15.5.2 Definition of the Complex Numbers . . . . . . . . 354
15.5.3 The Distinguished Complex Numbers 1 and i . . 355
15.5.4 Algebraic Closure of the Complex Numbers . . . 355
16 Henri Poincar´e, Child Prodigy 359
16.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . 359
16.2 Rubber Sheet Geometry . . . . . . . . . . . . . . . . . . 364
16.3 The Idea of Homotopy . . . . . . . . . . . . . . . . . . . 365
16.4 The Brouwer Fixed Point Theorem . . . . . . . . . . . . 367
16.5 The Generalized Ham Sandwich Theorem . . . . . . . . 376
16.5.1 Classical Ham Sandwiches . . . . . . . . . . . . . 376
16.5.2 Generalized Ham Sandwiches . . . . . . . . . . . 378
17 Sonya Kovalevskaya and Mechanics 387
17.1 The Life of Sonya Kovalevskaya . . . . . . . . . . . . . . 387
17.2 The Scientific Work of Sonya Kovalevskaya . . . . . . . . 393
17.2.1 Partial Differential Equations . . . . . . . . . . . 393
17.2.2 A Few Words About Power Series . . . . . . . . . 394
17.2.3 The Mechanics of a Spinning Gyroscope and the
Influence of Gravity . . . 397
17.2.4 The Rings of Saturn . . . . . . . . . . . . . . . . 398
17.2.5 The Lam´e Equations . . . . . . . . . . . . . . . . 399
17.2.6 Bruns’s Theorem . . . . . . . . . . . . . . . . . . 400
17.3 Afterward on Sonya Kovalevskaya . . . . . . . . . . . . . 400
18 Emmy Noether and Algebra 409
18.1 The Life of Emmy Noether . . . . . . . . . . . . . . . . . 409
18.2 Emmy Noether and Abstract Algebra: Groups . . . . . . 413
18.3 Emmy Noether and Abstract Algebra: Rings . . . . . . . 418
18.3.1 The Idea of an Ideal . . . . . . . . . . . . . . . . 419
19 Methods of Proof 423
19.1 Axiomatics . . . . . . . . . . . . . . . . . . . . . . . . . 426
19.1.1 Undefinables . . . . . . . . . . . . . . . . . . . . . 426
19.1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . 426
19.1.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . 426
19.1.4 Theorems, ModusPonendoPonens, and ModusTollens . . . 427
19.2 Proof by Induction . . . . . . . . . . . . . . . . . . . . . 428
19.2.1 Mathematical Induction . . . . . . . . . . . . . . 428
19.2.2 Examples of Inductive Proof . . . . . . . . . . . . 428
19.3 Proof by Contradiction . . . . . . . . . . . . . . . . . . . 432
19.3.1 Examples of Proof by Contradiction . . . . . . . . 432
19.4 Direct Proof . . . . . . . . . . . . . . . . . . . . . . . . . 434
19.4.1 Examples of Direct Proof . . . . . . . . . . . . . . 435
19.5 Other Methods of Proof . . . . . . . . . . . . . . . . . . 437
19.5.1 Examples of Counting Arguments . . . . . . . . . 437
20 Alan Turing and Cryptography 443
20.0 Background on Alan Turing . . . . . . . . . . . . . . . . 443
20.1 The Turing Machine . . . . . . . . . . . . . . . . . . . . 445
20.1.1 An Example of a Turing Machine . . . . . . . . . 445
20.2 More on the Life of Alan Turing . . . . . . . . . . . . . . 446
20.3 What is Cryptography? . . . . . . . . . . . . . . . . . . . 448
20.4 Encryption by Way of Affine Transformations . . . . . . 454
20.4.1 Division in Modular Arithmetic . . . . . . . . . . 455
20.4.2 Instances of the Affine Transformation Encryption . . . 457
20.5 Digraph Transformations . . . . . . . . . . . . . . . . . . 461
UKT: In general I'll leave out the drawings, and will not indicate their positions.
[p001begin]
Pythagoras (569–500 B.C.E.) was both a person and a society (i.e., the
Pythagoreans). He was also a political figure and a mystic. He was
special in his time because, among other reasons, he involved women as
equals in his activities. One critic characterized the man as “one tenth
of him genius, ninetenths sheer fudge.” Pythagoras died, according to
legend, in the flames of his own school fired by political and religious
bigots who stirred up the masses to protest against the enlightenment
which Pythagoras sought to bring them.
... ... ...
The Pythagorean society was intensely mathematical
in nature, but
it was also quasireligious. Among its tenets
(according to [RUS]) were:
• To abstain from beans.
• Not to pick up what has fallen.
• Not to touch a white cock.
• Not to break bread.
• Not to step over a crossbar [p001endp002begin]
• Not to stir the fire with iron.
• Not to eat from a whole loaf.
• Not to pluck a garland.
• Not to sit on a quart measure.
• Not to eat the heart.
• Not to walk on highways.
• Not to let swallows share one’s roof.
• When the pot is taken off the fire, not to leave the mark of it in the ashes,
but to stir them together.
• Not to look in a mirror beside a light.
• When you rise from the bedclothes, roll them together and smooth out the
impress of the body.
The Pythagoreans embodied a passionate spirit that is remarkable to our eyes:
Bless us, divine Number, thou who generatest gods and men.
and
Number rules the universe.
The Pythagoreans are remembered for two monumental contributions to mathematics. The first of these was to establish the importance of, and the necessity for, proofs in mathematics: that mathematical statements, especially geometric statements, must be established by way of rigorous proof. Prior to Pythagoras, the ideas of geometry were generally rules of thumb that were derived empirically, merely from observation and (occasionally) measurement. [UKT ¶]
Pythagoras also introduced the idea that a great body of mathematics (such as geometry) could be [p002endp003begin] derived from a small number of postulates. The second great contribution was the discovery of, and proof of, the fact that not all numbers are commensurate. More precisely, the Greeks prior to Pythagoras believed with a profound and deeply held passion that everything was built on the whole numbers. Fractions arise in a concrete manner: as ratios of the sides of triangles (and are thus commensurable — this antiquated terminology has today been replaced by the word “rational”) — see Figure 1.1.
Pythagoras proved the result that we now call the Pythagorean theorem. It says that the legs a, b and hypotenuse c of a right triangle (Figure 1.2) are related by the formula
a² + b² = c²
This theorem has perhaps more proofs than any other result in mathematics — over fifty altogether. And in fact it is one of the most ancient mathematical results. [UKT ¶]
There is evidence that the Babylonians and the Chinese knew this theorem nearly 1000 years before Pythagoras. In fact one proof of the Pythagorean theorem was devised by President James Garfield [(18311881)  the 20th US President]. ... ... ...
p069
[p069begin]
One of the great minds of the ancient world was Hypatia of Alexandria (370 C.E.–430
C.E.). Daughter of the astronomer and mathematician Theon, and wife of the
philosopher Isidorus, she flourished during the reign of the Emperor Arcadius.
UKT 210719: Timeline in Myanmar  https://en.wikipedia.org/wiki/Timeline_of_Burmese_history 210719
gives the year 200 AD as when the Pyus {pyu lumyo:) were converted to Buddhism, and the 7th century as Mons (mwûn lumya:} migrating to Lower Burma from Haribhunjaya and Dvaravati (presentday Thailand). Wikipedia further gives a specific date when Burmese lunisolar calendar was reset {þak~ka.raaz hpyo} "cutting down calendar years due to astrological reasons" in 640 CE. Ref. Maha Yazawin, vol. 1 2006:143.I wonder whether the father of Hypertia, Theon, as an astronomer and mathematician, would have heard of the Pyus, especially the PyuBur calendar which uses Metonic cycles to set the new year. I also wonder whether Hypertia the wife of philosopher Isidorus, could have been a Buddhist. For Metonic cycles, see https://en.wikipedia.org/wiki/Burmese_calendar 210719
Now, another source: https://www.ancientorigins.net/opinionguestauthors/buddhismancientegyptandmeroebeliefsrevealedthroughancientscript020931 210719
"Did Buddhism exist in Upper Egypt and the Lower Meroitic Empire? The answer appears to be yes. It was in Memphis that English Egyptologist and archaeologist W. M. Flinders Petrie found evidence of Buddhist colony. ... Flinders Petrie claimed these Buddhists date back to the Persian period of Egypt (c. 525405 BC) [UKT: Fatherkiller King Ajāthaśatru who became an ardent supporter of Buddhism, and who later hosted the First Buddhist Council some 50 years after the death of the Buddha, sent out many missionaries to the West. That was some 250 years later by King Asoka.]. Flinders Petrie wrote:
"on the right side, at the top is the Tibetan Mongolian, below that the Aryan woman of the Punjab, and at the base a seated figure in Indian attitude with the scarf over the left shoulder. These are the first remains of Indians known on the Mediterranean. Hitherto there have been no material evidences for that connection which is stated to have existed, both by embassies from Egypt and Syria to India, and by the great Buddhist mission sent by Ashoka as far west as Greece and Cyrene. We seem now to have touched the Indian colony in Memphis, and we may hope for more light on that connection which seems to been so momentous for Western thought".
If Petrie's dating is correct this puts Buddhists in Egypt two hundred years before Ashoka sent Buddhist missionaries to Egypt.
[UKT: more in the article]See also https://en.wikipedia.org/wiki/Pyu_citystate 210719
"The Pyu city states, {pyu mro.pra.neíngnän mya:} were a group of citystates that existed from c. 2nd century BCE to c. mid11the century in presentday Upper Burma. ... ... ... The Pyu realm was an important trading centre between China and India in the first millennium CE. Two main trading routes passed through the Pyu states. As early as 128 BCE, an overland trade route between China and India existed across the northern Burma. An embassy from the Roman Empire to China passed through this route in 97 CE and again in 120 CE. (Ref.33). But the majority of the trade was conducted by sea through the southern Pyu states, which at the time were located not far from the sea as much of the Irrawaddy {Érawa.ti} delta had not yet been formed, and as far south as upper Tenasserim {ta.nín~þari} coast towns such as Winga, HsindatMyindat, Sanpannagon and Mudon where Pyu artefacts have been found. (It is insufficient to conclude however that the Pyu had administrative and military control over these upper Tenasserim coastal towns.)(ref.34). The ports connected the overland trade route to China via presentday Yunnan.
Historians believe that Theon endeavored to raise the “perfect human being” in his daughter Hypatia. He nearly succeeded, in that Hypatia had surpassing physical beauty and a dazzling intellect. She had a remarkable physical grace and was an accomplished athlete. She was a dedicated scholar and had a towering intellect.
Hypatia soon outstripped her father and her teachers and became the leading intellectual light of Alexandria. She was a powerful teacher, and communicated strong edicts to her pupils. Among these were:
All formal dogmatic religions are fallacious and must never be accepted by selfrespecting persons as final.
Reserve your right to think, for even to think wrongly is better than not to think at all.
NeoPlatonism is a progressive philosophy, and does not expect to state final conditions to men whose minds are finite. Life is an unfoldment, and [p069endp070begin] the further we travel the more truth we can comprehend. To understand the things that are at our door is the best preparation for understanding those that lie beyond.
Fables should be taught as fables, myths as myths, and miracles as poetic fantasies. To teach superstitions as truths is a most terrible thing. The child mind accepts and believes them, and only through great pain and perhaps tragedy can he be in after years relieved of them. In fact men will fight for a superstition quite as quickly as for a living truth — often more so, since a superstition is so intangible you cannot get at it to refute it, but truth is a point of view, and so is changeable.
The writings of Hypatia have all been lost to time. What we know of her thoughts comes from citations and quotations in the work of others.
Hypatia was a pagan thinker at the time when Rome was converted to Christianity. Thus, in spite of her many virtues, she made enemies. Chief among these was Cyril, the Bishop of Alexandria. According to legend, he enflamed a mob of Christians against her. They set upon her as she was leaving her Thursday lecture, and she was dragged to a church where it was planned that she would be forced to recant her beliefs. But the mob grew out of control. Her clothes were rent from her body, she was beaten mercilessly, and then she was dismembered. The skin was flayed from her body with oyster shells. Her remains were then burned. The book [DZI] considers a variety of accounts of Hypatia and her demise. It is difficult to tell which are apocryphal.
Hypatia is remembered today for her work on Appolonius’s theory of conics, and for her commentary on Diophantus. All of these theories survive to the present time, and are still studied intently. She also did work, alongside her father, on editing Euclid’s Elements. The surviving presentation of Euclid’s classic work bears Hypatia’s mark.
Certainly Hypatia was one of the great thinkers of all time, and it is appropriate for us to pay her due homage. But we have no detailed knowledge of her work — certainly no firsthand knowledge. ... [p070end]
[p359begin]
Jules Henri Poincaré (1854–1912) is considered to have been one of the
great geniuses of twentieth century mathematics. Even while he was a
child his special gifts were recognized, and the entire country of France
watched in awe as he grew up to be a brilliant and creative man of
science. There were other distinguished individuals in Henri Poincaré’s
family. Poincaré’s father’s brother’s son Raymond was prime minister of
France several times and president of the French Republic during World
War I. The second son of that same uncle was a distinguished and high ranking
university administrator.
Young Henri Poincaré was so gifted that he was a hero in all of France. From a physical point of view, he was described in this way:
. . .ambidextrous and was nearsighted; during his childhood he had poor muscular coordination and was seriously ill for a time with diphtheria. He received special instruction from his gifted mother and excelled in written composition while still in elementary school.
Poincaré studied at the Lycée — today called the Lycée Henri Poincaré. He was the top student in every subject that he undertook. One of his instructors called him a “monster of mathematics”.
Young Henri enrolled at the École Polytechnique in 1873 and graduated in 1875. He was vastly ahead of all the other students in mathematics. But in other subject areas, such as athletics and art, he did [p359endp360begin] poorly. His deleterious physical coordination held him back in activities which were not cerebral. In fact his eyesight was so poor that he could not see what his teachers wrote on the blackboard. This failure helped him to develop his visual imagination.
After the École Polytechnique, Poincaré spent some time as a mining engineer at the École des Mines. At the same time he studied mathematics under the direction of Charles Hermite. He earned his doctorate in 1879. The examiners were not entirely happy with the thesis, for they found the presentation obscure and the organization confusing. Yet they acknowledged that this was a difficult subject area, and that the candidate had demonstrated great talent. So he was awarded the degree.
Poincaré’s first academic position was teaching mathematics at the University of
Caen. His lectures were criticized for their lack of organization.
He remained at Caen for only two years, and then he moved to a chair at the
Faculty of Sciences in Paris in 1881. In 1886 Poincaré was nominated, with the
support of Hermite, to the chair of mathematical physics and probability at the
Sorbonne. Hermite also promoted Poincaré for a chair at the École Polytechnique.
These were the two most prestigious professorships in all of France, and so a
measure of Poincaré’s prestige and recognition. He was to remain in Paris for
the remainder of his career, and he lectured on a different subject every year
until his untimely death at the age of 58.
Poincaré was remarkable for his work habits. He engaged in mathematical research each day from 10:00am until noon and from 5:00pm until 7:00pm. He would read mathematical papers in the evening. [UKT ¶]
Rather than build new ideas on earlier work, Poincaré preferred always to work from first principles. He operated in this fashion both in his lectures and in his writing. One expert described Poincaré’s method for organizing a paper as follows:
. . . does not make an overall plan when he writes a paper. He will normally start without knowing where it will end. . . . Starting is usually easy. Then the work seems to lead him on without him making a wilful effort. At that stage it is difficult to distract him. When he searches, he often writes a formula automatically to awaken some associa [p360endp361begin] tion of ideas. If beginning is painful, Poincaré does not persist but abandons the work.
Poincaré also believed that his best ideas would come when he stopped concentrating on a problem, when he was actually at rest:
Poincarée proceeds by sudden blows, taking up and abandoning a subject. During intervals he assumes . . . that his unconscious continues the work of reflection. Stopping the work is difficult if there is not a sufficiently strong distraction, especially when he judges that it is not complete . . . For this reason Poincaré never does any important work in the evening in order not to trouble his sleep.
UKT 210720: reading about this belief of Poincaré, I remember my days as a student studying for M.S. at IPC (Inst. of Paper Chemistry, Appleton, Wis, USA) when my roommate John Matter remarked about my study habits: "Joe, when are you going to study? I see you sleeping all the time." Whenever I came to a part of a problem which became too difficult for me to solve, I dropped into the easychair by my desk and went immediately to sleep. My unconscious mind was probably working  or, as I jokingly reply, my Goddess of Learning {þuraþ~pa.ti} was teaching me in my sleep. When I woke, the problem was already solved in my brain.
In 1894 Poincar´e published his important Analysis Situs. This seminal work laid the foundations for topology, especially algebraic topology. He defined the fundamental group — which is an important device for detecting holes of different dimensions in surfaces and other geometric objects. He proved the foundational result that a 2dimensional surface having the same homotopy as the sphere is in fact topologically equivalent to the sphere. [UKT ¶]
He conjectured that a similar result is true in 3 dimensions, and ultimately in all dimensions. This question has become known as the Poincaré Conjecture, and it is one of the most important questions of twentieth century mathematics. ... ... ... [p361endp362begin] ... Poincaré maintained his interest in physics, and made contributions to optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, relativity, and cosmology.
[p363begin] ... ... ...
Poincarée is arguably the father of topology (popularly known as “rubber sheet geometry”) and also of the currently very active area of dynamical systems. He made decisive contributions to differential equations, to geometry, to complex analysis, and to many other central parts of mathematics."
UKT: We'll stop here, because it may be boring to continue. Let's look what
Wikipedia has to say:

https://en.wikipedia.org/wiki/Topology 210720
"In mathematics, topology, is concerned with the properties of a
geometric object that are preserved under continuous deformation, such as
stretching, twisting, crumpling, and bending; that is, without closing holes,
opening holes, tearing, gluing, or passing through itself."
 UKT 210628 :
If you are a Christian, Hindu, Jew, or Muslim, you would be accepting that there is a Creator and of course an "afterlife" in Heaven or Hell. However, if willed by the Creator you will have another lifetime on what we call Earth or World as a human who may be mad, an idiot, a philosopher, or even buddha (the wisest of all beings). Worse, you may be born an animal or an insect. Buddhists too believe in an afterlife: the religionists also stipulated other unseen beings which my scientific training would not permit to believe.
Are these the Right View, you might be wondering? As a Buddhist, I must know: because the Fourth Law of Truth emphasises that a Buddhist must hold the Right View.
Studying the Questions and Answers of the Sixth Buddhist Council, Rangoon, 1954, has increased my knowledge of Theravada Buddhism. In one of the sessions {miz~Zi.ma. ni.kaaý} (vol 1), in one sermon named {a.pûN~Na.ka. þoat~tän} we find a partial list of nonBuddhist views, and Buddha's answer to them. First, Buddha lists the various views as follows: 1. , 2. , 3. , 4. , 5. , etc.
He sums them as either as "negative" {nût~ti.ka. a.yu} and "positive" {ût~ti.ka. a.yu}. He doesn't say anything about his Anatta {a.nût~ta.} view, in which he questions the "existence" of nonchanging, everlasting Self or Atta or Ātman आत्मन् which is related to the CreatorcumUniversal Judge .
Buddha's advice to those who do have either "negative" {nût~ti.ka. a.yu} or "positive" {ût~ti.ka. a.yu}, to side with {ût~ti.ka. a.yu}  for even if it were false, they have nothing to lose. On the other hand if they were to side with {nût~ti.ka. a.yu}, they would lose everything if the positive view proved to be true. This view of the Buddha is very pragmatic.
Now, we must ask if Buddha's view of Anatta {a.nût~ta.} were wrong, what then? It seems that Buddha's view on CreatorcumUniversal Judge is also pragmatic. According to Buddha, everyone, you and I, is responsible for our actions. You must not lay the blame or praise on the CreatorcumUniversal Judge. There is none to save you from your Sins. It is just a waste of time to pin your hope on God. I came to this view after listening and reading {a.pûN~Na.ka. þoat~tän} .
Let's now see what George Boole, a mathematician, said
in 1853, in his An Investigation of the Laws of
Thought p003. See TIL HDPDF and SDPDF libraries

GBooleLawsOfThought<Ô> /
Bkp<Ô> (link chk 210628)
"The general laws of Nature are not, for the most part, immediate objects of perception. They are either inductive inferences from a large body of facts, the common truth in which they express, or, in their origin at least, physical hypotheses of a causal nature serving to explain phenomena with undeviating precision, and to enable us to predict new combinations of them. They are in all cases, and in the strictest sense of the term, probable conclusions, approaching, indeed, ever and ever nearer to certainty, as they receive more and more of the confirmation of experience."
UKT 210629: What do we understand by "a large body of facts"? Scientific facts are borne out of physical experiments. If a current Scientific Theory can no longer explain observations of new physical experiments, the current theory must be modified or rejected. Here, what I have in mind is the birth of Quantum Theory. Read an interesting article in Scientific American by Scott Bembenek on March 27, 2018 :  https://blogs.scientificamerican.com/observations/einsteinandthequantum/ 210629
"By 1926, Albert Einstein had become completely unforgiving of quantum mechanics' probabilistic interpretation of the universe and would step away from it forever. In Einstein's mind, the universe must ultimately obey laws of physics that are fundamentally deterministic, and with respect to this, he would be uncompromising. Einstein made this most clear in response to a letter Marx Born (18821970) had written to him when he said: 'Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One [Christian God]. I am at all events convinced the He [God] does not play dice.' "
What would Gautama Buddha (Siddhartha Gautama, fl. bet. 6th and 4th century BC), whose First Four Laws, and Anatta Principle are the earliest scientific observations in the history of mankind, would say? I am not the one to answer my question, which means I will have to learn abut Buddhism from authentic sources such as the Questions and Answers of the Sixth Buddhist Council . But first, let's see what Wikipedia has to say:
From: https://en.wikipedia.org/wiki/The_Laws_of_Thought 210628
An Investigation of the Law of Thought on Which are founded the Mathematical Theories of Logic and Probabilities , George Boole, published in 1854, is the second of Boole's two monographs on algebraic logic. Boole was a professor of mathematics at what was then Queen's College, Cork (now University College), in Ireland.
The historian of logic John Corcoran wrote an accessible introduction to Laws of Thought [ref 1] and a point by point comparison of Prior Analytics and Law of Thought. [ref 2]. According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. [Aristotle: 384322 BC] Boole's goals were "to go under, over and beyond" Aristotle's logic by:
1. Providing it with mathematical foundations involving equations;
2. Extending the class of problems it could treat from assessing validity to solving equations, and;
3. Expanding the range of applications it could handle
 e.g. from propositions having only two terms to those having arbitrarily many.
More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. [UKT ¶]
First, in the realm of foundations, Boole reduced the four propositional forms of Aristotle's logic to formulas in the form of equations  by itself a revolutionary idea. [UKT ¶]
Second, in the realm of logic's problems, Boole's addition of equation solving to logic  another revolutionary idea  involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. [UKT ¶]
Third, in the realm of applications, Boole's system could handle multiterm propositions and arguments whereas Aristotle could handle only twotermed subjectpredicate propositions and arguments. For example, Aristotle's system could not deduce
"No quadrangle that is a square is a rectangle that is a rhombus"
from
"No square that is quadrangle is a rhombus that is a rectangle"
or from
"No rhombus that is a rectangle is a square that is a quadrangle".
Boole's work founded the discipline of algebraic logic. It is often, but mistakenly, credited as being the source of what we know today as Boolean algebra. In fact, however, Boole's algebra differs from modern Boolean algebra: in Boole's algebra A+B cannot be interpreted by set union, due to the permissibility of uninterpretable terms in Boole's calculus. Therefore, algebras on Boole's account cannot be interpreted by sets under the operations of union, intersection and complement, as is the case with modern Boolean algebra. The task of developing the modern account of Boolean algebra fell to Boole's successors in the tradition of algebraic logic (Jevons 1869, Peirce 1880, Jevons 1890, Schröder 1890, Huntington 1904).
• Mathematics of the Law of Karma
: A mathematical analysis of Law of Karma. Read: Mathematical Proof of the Law of Karma,
by Jargal Dorj , in American Journal of Applied Mathematics. Vol. 2, No.
4, 2014, pp. 111126, in TIL HDPDF and SDPDF libraries:
 JDorjMathLawKarma<Ô>
/ Bkp<Ô> (link
chk 200301)
Abstract: "... The existence of the Law of Karma will be proved and verified in
this article using the mathematical Set Theory. The incomprehension of the
“Self” and its emptiness is described in the Buddhist teachings as ignorant.
Herewith we shall explain the theory of the “Self’ and its emptiness founded on
the possession of the body and mind using the mathematical Set Theory. By
reading this article the reader will comprehend the “Self” and its emptiness and
overcome this ignorance.
Dolphins may be maths geniuses , Jennifer Viegas , Discovery News, 2012Jul18
 https://www.abc.net.au/science/articles/2012/07/18/3548573.htm 201030
Dolphins may use complex nonlinear maths when hunting, according to a new study that suggests they could be far more skilled than was ever thought possible before.
Inspiration for the new study, published in the latest Proceedings of the Royal Society A, came after lead author Tim Leighton watched an episode of the Discovery Channel's Blue Planet series and saw dolphins blowing multiple tiny bubbles around prey as they hunted.
"I immediately got hooked, because I knew that no manmade sonar would be able to operate in such bubble water," says Leighton, a professor of ultrasonics and underwater acoustics at the University of Southampton.
"These dolphins were either 'blinding' their most spectacular sensory apparatus when hunting  which would be odd, though they still have sight to reply on  or they have a sonar that can do what human sonar cannot …Perhaps they have something amazing," he adds.
Leighton and colleagues Paul White and student Gim Hwa Chua set out to determine what the amazing ability might be.
They started by modelling the types of echolocation pulses that dolphins emit. The researchers processed them using nonlinear mathematics instead of the standard way of processing sonar returns. The technique worked, and could explain how dolphins achieve hunting success with bubbles.
The math involved is complex. Essentially it relies upon sending out pulses that
vary in amplitude. The first may have a value of 1 while the second is 1/3 that
amplitude.
"So, provided the dolphin remembers what the ratios of the two pulses were, and
can multiply the second echo by that and add the echoes together, it can make
the fish 'visible' to its sonar," says Leighton. "This is detection
enhancement."
But that's not all. There must be a second stage to the hunt.
"Bubbles cause false alarms because they scatter strongly and a dolphin cannot
afford to waste its energy chasing false alarms while the real fish escape,"
explains Leighton.
The second stage then involves subtracting the echoes from one another, ensuring the echo of the second pulse is first multiplied by three. The process, in short, therefore first entails making the fish visible to sonar by addition. The fish is then made invisible by subtraction to confirm it is a true target.
In order to confirm that dolphins use such nonlinear mathematical processing, some questions must still be answered. For example, for this technique to work, dolphins would have to use a frequency when they enter bubbly water that is sufficiently low, permitting them to hear frequencies that are twice as high in pitch.
In order to confirm that dolphins use such nonlinear mathematical processing, some questions must still be answered. For example, for this technique to work, dolphins would have to use a frequency when they enter bubbly water that is sufficiently low, permitting them to hear frequencies that are twice as high in pitch.
"Until measurements are taken of wild dolphin sonar as they hunt in bubbly water, these questions will remain unanswered," says Leighton. "What we have shown is that it is not impossible to distinguish targets in bubbly water using the same sort of pulses that dolphins use."
UKT: More in the article.
Nonlinear system , https://en.wikipedia.org/wiki/Nonlinear_system 201020
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input.^{[1]}^{[2]} Nonlinear problems are of interest to engineers, biologists,^{[3]}^{[4]}^{[5]} physicists,^{[6]}^{[7]} mathematicians, and many other scientists because most systems are inherently nonlinear in nature.^{[8]} Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Dolphins may be maths geniuses , Jennifer Viegas , Discovery News, 2012Jul18
 https://www.abc.net.au/science/articles/2012/07/18/3548573.htm 201030
Dolphins may use complex nonlinear maths when hunting, according to a new study that suggests they could be far more skilled than was ever thought possible before.
Inspiration for the new study, published in the latest Proceedings of the Royal Society A, came after lead author Tim Leighton watched an episode of the Discovery Channel's Blue Planet series and saw dolphins blowing multiple tiny bubbles around prey as they hunted.
"I immediately got hooked, because I knew that no manmade sonar would be able to operate in such bubble water," says Leighton, a professor of ultrasonics and underwater acoustics at the University of Southampton.
"These dolphins were either 'blinding' their most spectacular sensory apparatus when hunting  which would be odd, though they still have sight to reply on  or they have a sonar that can do what human sonar cannot …Perhaps they have something amazing," he adds.
Leighton and colleagues Paul White and student Gim Hwa Chua set out to determine what the amazing ability might be.
They started by modelling the types of echolocation pulses that dolphins emit. The researchers processed them using nonlinear mathematics instead of the standard way of processing sonar returns. The technique worked, and could explain how dolphins achieve hunting success with bubbles.
The math involved is complex. Essentially it relies upon sending out pulses that
vary in amplitude. The first may have a value of 1 while the second is 1/3 that
amplitude.
"So, provided the dolphin remembers what the ratios of the two pulses were, and
can multiply the second echo by that and add the echoes together, it can make
the fish 'visible' to its sonar," says Leighton. "This is detection
enhancement."
But that's not all. There must be a second stage to the hunt.
"Bubbles cause false alarms because they scatter strongly and a dolphin cannot
afford to waste its energy chasing false alarms while the real fish escape,"
explains Leighton.
The second stage then involves subtracting the echoes from one another, ensuring the echo of the second pulse is first multiplied by three. The process, in short, therefore first entails making the fish visible to sonar by addition. The fish is then made invisible by subtraction to confirm it is a true target.
In order to confirm that dolphins use such nonlinear mathematical processing, some questions must still be answered. For example, for this technique to work, dolphins would have to use a frequency when they enter bubbly water that is sufficiently low, permitting them to hear frequencies that are twice as high in pitch.
In order to confirm that dolphins use such nonlinear mathematical processing, some questions must still be answered. For example, for this technique to work, dolphins would have to use a frequency when they enter bubbly water that is sufficiently low, permitting them to hear frequencies that are twice as high in pitch.
"Until measurements are taken of wild dolphin sonar as they hunt in bubbly water, these questions will remain unanswered," says Leighton. "What we have shown is that it is not impossible to distinguish targets in bubbly water using the same sort of pulses that dolphins use."
UKT: More in the article.
Nonlinear system , https://en.wikipedia.org/wiki/Nonlinear_system 201020
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input.^{[1]}^{[2]} Nonlinear problems are of interest to engineers, biologists,^{[3]}^{[4]}^{[5]} physicists,^{[6]}^{[7]} mathematicians, and many other scientists because most systems are inherently nonlinear in nature.^{[8]} Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.