Darts on History of Mathematics Volume Ii

DARTS

ON

HISTORY OF MATHEMATICS

VOLUME II

SATISH C. BHATNAGAR

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ISBN: 979-8-7652-3559-1 (sc)

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Balboa Press rev. date: 11/15/2022

READERS’ COMMENTS

Visiting this museum is a part of my continuing approach of drilling into the nature of fuzzy mechanics of history as opposed to sharp linearity in mathematics.

In other words, any two mathematicians will either say that a proof/solution is correct or wrong – no question of any split!..........if two historians completely agree with each other, then, it may be reasonably assumed that one of them is taking a free ride on the other.

Hey Satish, I see that you offer two courses in the History of Math at UNLV. We offer only the undergrad as of now. Would you be kind enough to send me the two syllabi for my reading and information that might be helpful if we someday offer a graduate course in the History of Mathematics here at Lipscomb University? I enjoy reading your letters, essays, etc.

Dear Sir, Many thanks for your email. I must say you possess an art of storytelling and your experiences in life plus everyday encounters with math problems continue to enlighten me.

I certainly agree with all of that. I truly hope unlv appreciates the real gem they have in you to be able to bridge the two disciplines. Have a great Thanksgiving!

I wish we had this class when I was a student. I’ve always enjoyed your lectures. You are definitely one of the best lecturers in my many math courses. You encouraged me to think, not just memorize.

Once again, I think you are spot on. You describe precisely what a HoM should be and what you make it out to be. Not sure anyone else should even be allowed to teach it. Well, except maybe me if/when you do retire! Thanks for sharing!

Satish, A very nice personal tribute and some more history of you. Thank you for sharing. So, you could have been a number theorist, but instead chose government service.

Dear Dr. Bhatnagar, Thank you for sharing. Slowly I am trying to include more culture in my instruction. My students are doing a project on Ethnomathematics and they are enthused to explore how other cultures valued mathematics.

DISTRIBUTION OF CONTENTS

READERS’ COMMENTS

DEDICATION & FRONT COVER

DARTS ARE VECTORS TOO!

A HIST-O-MATH PREFACE (PART II)

GLOSSARY AND ABBREVIATIONS

I.       CLASSROOM CUTS

1. MINING MATHEMATICS IN MUSEUMS

2. SWITCHING THE HATS

3. FOCI ON HISTORY OF MATHEMATICS COURSE

4. A PITCH FOR HISTORY OF MATHEMATICS

5. FIRST ASSIGNMENT!

6. GLANCING AT DISCRETE MATHEMATICS

7. FACTS & FACTOIDS - HISTORY & HERESIES

8. CAUTION IN UNDERSTANDING HISTORY

9. VIGILANTES IN HISTORY OF MATHEMATICS

10. EXTREME STATES OF MATHEMATICS

11. UNIQUE TWINE OF MATH & HISTORY

12. IMPORTANCE OF CLASS PROJECTS

13. PEDAGOGY OF HISTORY OF MATHEMATICS

14. MINDS NEED FRESH AIR TOO!

15. BRICKS AND BOUQUETS ON DARTS

16. ORAL HISTORY AND MATHEMATICS

17. OPTIMIZING HISTORY OF MATHEMATICS!

18. PROBLEMS IN MATH PLUS AND ER!

19. SLICING HISTORY OF MATHEMATICS

20. LEARNING-WHILE TEACHING!!

21. PROJECTS DEFINE HISTORY OF MATH

22. UTILITY OF A COLLEGE COURSE

23. KNOWING THY HISTORY!

24. FRACTIONS - FRACTALS - FRICTIONS!?

25. SQUARING JOY IN PROJECTS

II.       HUMANISTIC SLICES

26. MAKING AND UNMAKING OF BENJAMIN BANNEKER

27. EXPANDING MATHEMATICAL HORIZONS

28. PUTTING THE TWO TOGETHER!

29. MEASURES OF GREATNESS…!

30. VON NEUMANN SPIKE!

31. BRICKS AND BOUQUETS ON THE DARTS

32. MATH THROUGH THE AGES - REVIEWED

33. MATHEMATICIAN IN MY BACKYARD!

34. WRITING TEXTBOOKS & PUBLISHING PAPERS

35. CELEBRATION - INSPIRATION!

36. MATHEMATICAL FOUNDATIONS OF MY CREATIVE WRITINGS

37. REUBEN HERSH PERISCOPE

38. ARUN VAIDYA - ‘FIRST’ GUJARATI MATH HISTORIAN!

39. DARK SPOTS ON THE MOON (PART III)

III.       INDIA SPICES

40. MATHEMATICS IN THE VEDAS & YOGA

41. HISTORY & PHILOSOPHY OF SCIENCE: A VEDIC PERSPECTIVE

42. FERMAT’S LAST THEOREM & MATHEMATICS IN THE VEDAS

43. HISTORY-PATIALA-MATHEMATICS

44. RP BAMBAH - AS I RECALL

45. TIME TO HONOR SOM DATT CHOPRA!

46. NAVIGATING MY STRAITS OF GEOMETRY

47. SHADOWS OF SCIENCE AND VIGYAN

48. SCIENCE AND VIGYAN ANALYZED

49. HISTORY OF MATH & HINDU RENAISSANCE

50. MADE IN MATHEMATICS!

51. ETHNOMATHEMATICS OF INDIA

IV.       SMORGASBORD BITES

52. SCIENCE & MATH IN STELAE

53. PROPOSAL - HoM SESSION AT THE AMS MEETING

54. A HUMANISTIC MATH REVIEW

55. EXPANDING HOM – HISTORY (PART I)

56. IS MATHEMATICS SELF-TAUGHT?

57. AN HOLISTIC BOOK REVIEW

58. MATHEMATICS - ARCHEOLOGICALLY!

59. OLD-FASHIONED STUDENT CHEATING DYING!?

60. LIMINALITY THROUGH A MATHEMATICAL PRISM

61. HISTORY OF MATH DEFINES NATIONS

62. IN SEARCH OF INFINITY

63. HUMANISTIC HISTORY OF MATHEMATICS

64. BLACK HISTORY MONTH & MATHEMATICS

65. WHO IS A MATHEMATICIAN?

66. (HISTORY OF MATHEMATICS) IN HISTORY

67. HISTORY-MATH-POLITICS

68. COVID-19 AND ONLINE MATH

69. RELIGION, MATH & POLITICAL POWER

70. WHAT RAMANUJAN MEANS TO YOU?

71. IGNITING HISTORY (OF MATH) IN INDIA!

72. MY MATHEMATICAL DEBTS!

V.       OTHER PERSPECTIVES

73. WHAT HOM MEANS TO YOU!

74. MY MATHEMATICAL JOURNEY & CORONAVIRUS

75. GYMNASTICS WITH HISTORY OF MATHEMATICS

76. HISTORY OF MATHEMATICS IN INDIA

77. A GLIMPSE OF SOVIET MATH (PART I)

78. A GLIMPSE OF SOVIET MATH (PART II)

COMMENTATORS & ANALYSTS EXTRAORDINAIRE

DEDICATION & FRONT COVER

Normally, I dedicate a book to a person who has inspired me to write it or to a person who has assisted me in an extraordinary manner. Sometimes, I dedicate a book to memorialize a deceased person who has left an indelible mark on my life. In the context of this book, there is no one person standing up in my mind.

Looking back to my dedication, ‘History for the Wise’ to Volume I, I realized that it was not completely correct as Volume I and Volume II of Darts on History of Mathematics are precisely on the History of Mathematics - not just about History in general - though, my approach to the History of Mathematics is getting highly humanistic.

Anyway, instead of leaving this page blank, I thought this note would fill a void. Who knows in near future someone or something may emerge who is worthy of the dedication of Volume II, or Volume III, which I am not going to rule out. So far, it seems that volumes on Darts have been evolving from my reflective writings alone.

As far as the cover design of Volume II is concerned, it is isomorphic to the one that I did design for Volume I. This is natural as that also reflects the similarity of their contents. Essentially, there is a permutation of colors. At the same time, Volume II has to look different in some respects. The world map, the darts and arrows, my picture, my bio and readers’ comments have the same placements and visual feel. However, I remain open to any change that may be suggested by the design department of the publisher.

Satish C. Bhatnagar

June 20, 2021

DARTS ARE VECTORS TOO!

(Modified from the Preface of Vectors in History, Volume I)

The word ‘Vector’, in the title of the book, Vectors in History (2012) is borrowed from mathematics, where vectors are not restricted to only two or three space dimensions, as generally used in physics and applied mathematics. Vectors represent quantities that need both magnitude and direction for their representations. Visually, vectors are shown by directed arrows or darts ‘freely flying around’.

Each reflection is like a vector or dart. Its magnitude is subjective, if measured by its impact factor. Vector is tangible, if it is measured by the number of words in it, varying from 600 to 6000. Its theme provides a direction; however, it may change its course during its flight. In other words, a particular reflection may have more than one attractor point. In mathematics, a zero vector has zero magnitude, but no direction. Naturally, it does not correspond to any reflection!

As far as the variety of directions of a vector is concerned, they are ‘doubly’ infinite in 2-dimensions and ‘triply’ infinite in 3-dimensions! These reflections do bring out a similar flavor, when it comes to the number of sub-topics that are touched upon. They are not bounded above - be that in the context of a country, person, religion, or region etc.

These reflections are truly like free vectors having any arbitrary initial and terminal points. They can fly off from anywhere and land anywhere. In the world of mathematics, two vectors are defined equal, when their magnitude and directions are the same. This is never a case in my reflections, as any two of them would differ both in words and themes! Besides, who would care to read ‘equal’ reflections?

In mathematics, two position vectors/forces can be added. In these reflections, there is generally no connection between the ending of one reflection with the beginning of the next. Consequently, one can read any reflection from anywhere without missing a beat from the previous one! Often there is more than one strand in a braid, and my style remains compact. Nonetheless, these reflections are snappy and penetrating - like darts and arrows - vectors!

A general structure of a reflection is that it spins off from a specific incident. I basically dive into a vortex with it, or look out through it as a window to the universe far and beyond. In the process, it is wrapped with other stories and concepts which are often cross-cultural and interdisciplinary. The delight they have given me on subsequent readings, I am sure, they would give in different measures to each reader. That is a kind of beauty and uniqueness of this book.

Satish C. Bhatnagar

Dec. 16, 2020

A HIST-O-MATH PREFACE (PART II)

A few words need to be said about how certain things happen in life without any planning. There is no preface with a title, A HIST-O-MATH PREFACE (PART I). Reason, there is no Darts on History of Mathematics, Volume I. When I finished the Darts on History of Mathematics in 2014, I did not foresee Volume II ever coming up. That is why its title did not have Volume I on it. Darts is a mathematical corollary of my book, Vectors in History, which I never thought of writing.

In order to bring closure to this thread, until the age of 60, it was not even in my dreams to see my name on any book. That is the beauty of life! Both volumes are independent in the sense one does not have to read Volume I before embarking upon Volume II. Therefore, this preface is modified from the preface of ‘Volume I’. Instead of putting the following texts in a series of quotes and italics, I am just adding an extra line space.

Writing the preface of a book is a crowning moment in an intellectual endeavor. The important things about a preface are the whys, hows, and whats about the book and its author. That is what follows:

How much mathematics is needed to understand this book? A little more than high school level! A history of math course can be taught with little math or with good doses of math - depending upon students’ background. No one is interested in the history of any one mathematics problem or a topic - it is boring. Yes, survey papers are possible. What is really needed is a mathematically mature and curious mind which can also navigate the choppy and muddy waters of math and history.

What are my credentials for writing this book? My inclination towards the History of Math (HoM) has been organic. In the 1980s, I started cutting away the umbilical cord that connected me with my PhD research. I was then groping around and trying to discover my new strengths. It is a thesis that before the age of 40, there is no meeting ground between one’s knowledge of mathematics and history - the two remain poles apart. Around the age of 50, one begins to sense a confluence of deductive thinking of mathematics and soft thinking of history. It started gaining traction in my mind.

The Darts on History of Mathematics is a corollary of my book, Vectors in History, Volume I (2012). Before writing a book on the HoM, I needed to establish my identity as an historian, as I approach history of mathematics as a subset of history in general. Also, if you browse standard textbooks on the HoM, they are stereotypical and isomorphic. The contents of early history are perpetuated from one author to the other without questioning. After all, who has the time and motivation for ‘revisionism’? The books are written by math professors who seldom developed a sense of history.

Long bibliographies and lists of references at the end of the books, and sometimes notes at the end of each chapter, overwhelm the readers. The facts go unchallenged - particularly, if they suit a particular academic culture. Years ago, when I questioned Morris Kline on mathematics in ancient India as written in his book, Mathematical Thought from the Ancient to Modern Times (1972), he wrote me back saying that he had only quoted from a book by Indian authors. Can a wrong proof of a theorem or a wrong solution of a problem ever be passed on by such referencing in mathematics? Never! Over the years, these padded bibliographies and references have lost my respect. They are not a part of any one of my books.

What prompted the writing of this book? The darts in the title of the book precisely point out towards such unverified, exaggerated and stretched out or ignored conclusions in their historical narratives. Such scenarios urge me to write these reflections on specific points. Therefore, Darts is not a typical textbook on the HoM, but it can be used as supplementary material along with a ‘regular’ textbook.

What is the incubation period of the book? The dates of some reflections do go back to more than 20 years. However, the writing of a book was not on the horizon until relatively recent. Also, not all old pieces of my writings were saved.

What is new in the book? Apart from its format; in brief, it has thought provoking angles of observation and deductive conclusions on many topics, which may look ordinary or rare.

Who will benefit from the book? Any lay person with an historical bent of mind on mathematical topics stands to gain from it. Both undergraduate and graduate students in history of mathematics courses would enjoy it. All reflections are independent - they are excellent bedtime reading too, which is never suggested in the learning of hardcore mathematics.

Is there any other book similar to this in format and style? Absolutely none! That is my moment of pride. It aligns with my disinclination for writing mathematics textbooks throughout my professional life of 50+ years. I do not see any intellectual challenge in the writing of one. It is uncreative to shuffle problems from other textbooks and change them by epsilons and deltas! A computer may be doing it.

Volume I has 71 reflections and write-ups in this book, but Volume II has 78 of them. They are unevenly divided into the same five sections. The divisions are not sharp as dictated by the very nature of reflections. Sometimes, the overlap between topics in some reflections is so large that their variants are also included in the Converging Matheriticles (2014) or/and Vectors in History (2021). I have realized that some divisions of reflections are better than no division, as it was done in my first book.

Reflections under the heading of Classroom Cuts are drawn from classrooms. The second section, named Humanistic Slices, contains reflections connected with the lives of mathematicians of both past and present. The third section, called Indian Spices, has assorted reflections dealing with mathematics of India - from ancient to the present. The fourth section is called, Smorgasbord Bites. As the title implies, it has an interesting mix of reflections. The fifth section includes the outlooks on the HoM of my friends, colleagues, and students of mathematics. It is rightly called the Other Perspectives.

A common feature of all my books is that they each can be read from anywhere, as the reflections are independent in contents. It fits in today’s fast life styles; no one has the time to start a book from its Page Number 1 and then wade through it to the very end. Brevity is the characteristic of the twitter age - using 140 characters or less. As a consequence, abbreviations are explained again, and certain references repeated as encountered.

The practice of dating each reflection is continued so that a reader may have a full perspective of its genesis in terms of time, place and my mindset. As a young reader, I paid no attention to it, but now, this is the first thing I look for in a book. The two dates on some reflections means that a significant revision was done on the second date. Another continuing feature is providing partial and full blank pages for the readers to scribble their comments, as they pop up while reading it. It comes from my compulsive habit of underlining and side-lining a significant part of a sentence or paragraph. Such markings become a source of quick reference in future.

Every book is a defining moment in the author’s life. For me, this Volume II firmly plants my feet in history. Increasingly, I look at a socio-political scenario through a lens of history. Consequently, my world of solid mathematics has shrunk. I have not volunteered to teach hard core graduate courses for nearly 20 years. Even my bread-and-butter courses on ODE have distanced from me.

The main reason for a change in my teaching load is due to the creation of Teaching Concentration (2002) in the MS program. My involvement has increased in the teaching of its three required courses. There are a few faculty members who can teach the hard-core math courses that I used to teach, but lately, I have been the only faculty member teaching these specialized courses in this concentration.

Acknowledgements. For writing Volume I, I got sabbatical leave for Spring-2014, and for writing Volume II, I got another sabbatical leave for Fall-2020. I thank UNLV and my department chairmen for their support. Also, I profusely thank Scotsman, Francis A. Andrew, a science fiction author and Professor of English Language (working in Oman for the last 15 years), for providing me feedback on the syntax and semantics on every reflection that I have written since 2009. With the result, along with my unabated passion for writing, I am beginning to enjoy the usage of the English language in a manner never ever done or thought before! Finally, any comments and suggestions on the book emailed at [email protected], would be greatly appreciated.

Satish C. Bhatnagar

June 22, 2021

GLOSSARY AND ABBREVIATIONS

Mantra is a set of supposedly energized syllables in Sanskrit – potent enough to affect material changes with right repetition and enunciation.

Sutra is a cryptic and condensed description of a principle or property. An example is of 18 sutras of Vedic Mathematics.

Tapa is combination of penance, meditation with austerities

Vedas refer to the most ancient four Hindu scriptures, namely; Rig, Yajur, Atharva, and Saam. Upvedas and Vedangas are ancillary treatises for a systematic study of the Vedas.

Rishi is an enlightened individual in terms of his/her cultivated powers of mind developed through Yoga over a long period of time.

Guru is far more than a high school and college instructor. There is an associated element of one-one-ness, loftiness and holistic nature - bordering spirituality.

Gurukul (a kind of Hindu seminary school going back to the Vedic period)

Shrimad Bhagwatam or Bhagwat is a holy scripture. It is often confused with Gita or Bhagwat Gita, which is a central part of a chapter in the epic of Mahabharata. Shrimad Bhagvat was compiled by Vyas after he had finished the Mahabharata. It is a great story of life.

advaita; non-duality - Principle of one-ness.

siddhis are the states of mind achieved after years of penance and yoga that one can materialize objects. Essentially, it is a reverse mass-energy equation. The late Satya Sai Baba of Puttaparthi had a reputation for pulling out jewelry items of thin air for his followers all over the world. Such a person is called Siddha.

IT: Information Technology

BTI: Bhatinda or Bathinda

DLH: for Delhi, the capital of India

UNLV: University of Nevada Las Vegas

JMM: Joint Mathematics Meetings

MAA: Mathematical Association of America

AMS: American Mathematical Society

NAM: National Association of Mathematicians (Founded by Afro-Americans)

AWM: Association of Women in Mathematics

SIAM: Society of Industrial and Applied Mathematics

IMS: Indian Mathematical Society

Math: Popular abbreviation for Mathematics - used interchangeably in the book

PDE: Partial Differential Equation(s)

ODE: Ordinary Differential Equation(s)

SECTION I

CLASSROOM CUTS

1. MINING MATHEMATICS IN MUSEUMS

(A Note to my students in the History of Mathematics course)

Our next class on Tuesday, 9/24/ 2013 will be held inside UNLV’s Barrick Museum - just east of the Lied Library. I have already spoken with a person-in-charge about it. Please be there before 4 PM, if you can, as the Museum closes at 5 PM.

Visiting this museum is a part of my continuing approach of drilling into the nature of fuzzy mechanics of history as opposed to sharp linearity in mathematics. You must be able to form some ideas as to how facts in history are created, communicated, perpetuated, constantly revised, and even challenged.

Before the museum visit, I would like you to review the first three chapters of the textbook – wherein, particularly various artifacts are over-emphasized for their historic values for mathematics.

You can get a lot out of this exercise. Observe the museum objects in silence – refraining from discussing anything with each other. Take individual notes. You will then understand subjectivity in the facts of history.

You are free to stand in front of any collection of artifacts for any length of time. In other words, you don’t have to check out every showcase in the Museum, or of any museum in the world for the same reason!

It is better if you don’t read the entire descriptions of the exhibits. However, after you have extracted and written out any mathematical or scientific information, then you may come back to it at your convenience, read the descriptions and draw some comparative conclusions.

I do not undermine the significance of any ‘ancient’ objects, but only caution you about drawing too many conclusions. We all know how human DNAs are loaded with information, but extracting that information requires incredibly sophisticated chemical analysis run in high-tech labs. So, you should scrutinize any mathematical findings in any textbook from this angle.

My suggestion on working during this museum visit is that you enter only one main ‘thing’ on one page of your notebook. Briefly scribble the ideas generated from visual examination. You are not touching any object, tasting it, smelling it, or hearing it! Your angle of vision is limited to 120 degrees.

You may check at the Museum office, if taking pictures of the items is allowed. In summary, approximate the age of an object(s), compare it with anything in the present. Derive mathematical awareness and knowledge of that society in terms of its geographical extent.

A report on mathematical findings will be your exclusive next weekly project. So, mull over this visit for a few days before putting down your thoughts. In case, you need to go and reexamine any exhibit, please do so out of the class hours.

Finally, I want this experience embedded into your psyche so that any time you visit any museum anywhere, you would examine its displays with as much open mind as possible. Your sharing any mathematical nugget with me will be appreciated.

Sep. 21, 2013

COMMENTS

Hi Satish, Your reflection reminded me of a book entitled On Stonehenge by Sir Fred Hoyle. This book had a large contingent of a discipline which Hoyle had devised - namely archeoastronomy. Hole had demonstrated in this book that Stonehenge was in fact an astronomical observatory designed mainly for the prediction of solar eclipses. Hoyle argued that conventional archeology was essentially funerary in nature and generally involved the retrieval of the remains of artifacts from the remote past. By showing how Stonehenge was designed as an astronomical observatory, Hoyle contended that the past could be brought to life. One could imagine the festivities associated with the astronomical calendar, the colourful costumes of the dancers, the melodious chanting of the singers and the general merrymaking of those attending these joyful performances.

Perhaps, you could extend Hoyle’s discipline of archeoastronomy to embrace archeomathematics. By observing the geometrical, trigonometrical etc. abilities of ancient artifacts, one could deduce the mathematical competence of these ancient peoples and so come to conclusions about their technical, architectural and artistic achievements. One could then determine how their mathematical abilities influenced their religious and philosophical systems. All best: Francis

Satish, very interesting. Is this the first time you are doing something of this sort? Noel

2. SWITCHING THE HATS

Teaching a course on