ISSUE No. 43

A warm welcome to our new subscribers.
Vedic Mathematics is becoming increasingly popular as more and more people are introduced to the beautifully unified and easy Vedic methods. The purpose of this Newsletter is to provide information about developments in education and research and books, articles, courses, talks etc., and also to bring together those working with Vedic Mathematics. If you are working with Vedic Mathematics - teaching it or doing research - please contact us and let us include you and some description of your work in the Newsletter. Perhaps you would like to submit an article for inclusion in a later issue or tell us about a course or talk you will be giving or have given.
If you are learning Vedic Maths, let us know how you are getting on and what you think of this system.

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This issue's article is by Andrew Nicholas who is a long-standing expert in Vedic Mathematics. The article being quite long, and in two parts, so we are including the first part here and the second part with the next newsletter. If you would like the contact Andrew about his article his email address is

Following Andrew's article (i.e. in Newsletter 45) we will have an article by Anand Pattabiraman, an eleven year old sixth grader enrolled in the ROGATE Program (Resources Offered for Gifted And Talented Education) of the National Talent Network, in Tenafly, New Jersey, USA, who has conducted research on Vedic Math.

## GUESS WHICH SUBJECT LACKS SOUND FOUNDATIONS (AND WHAT CAN BE DONE ABOUT IT)

PART 1: THE PROBLEM AND THE SOLUTION

It is widely believed that the foundations of mathematics were sorted out long ago. This is not so. The truth is that not one of the established systems can be shown or even trusted to have sound foundations. But theorems depend on the foundations; how far can we trust them if the foundations are unsound?
Morris Kline drew attention to this state of affairs in a book published in 1980, 'Mathematics: the loss of certainty'. Two quotations from it suffice to make the point.
(1) 'The disagreements about the foundations of "the most certain science" are both surprising and, to put it mildly, disconcerting. The present state of mathematics is a mockery of the hitherto deep-rooted and widely reputed truth and logical perfection of mathematics.'
(2) 'According to Homer, the gods condemned Sisyphus, king of Corinth, after his death to roll a big rock uphill, only to see it fall back to the bottom each time he neared the summit. He had no illusion that some day his labors would end. Mathematicians have the will and the courage that comes almost instinctively to complete and solidify the foundations of their subject. Their struggle too may go on forever, they too may never succeed. But the modern Sisyphuses will persist.'

Kline evidently suspected that the problem may never be solved - and may even be insoluble. I disagree, and show in this paper how the problem can be solved, giving as an example the foundations of a recently formulated system of Euclidean geometry.

The essence of the solution is this. It is the job of the foundations to state everything necessary for the study. If we can show that the proposed foundations are necessary for the study we are on firm ground. To complete the job it is also desirable to show that they are sufficient for the study, so far as is possible.

Surprisingly enough, the key first step has never been taken for any of the established systems of mathematics. It will now be shown how the above-mentioned new system of geometry satisfies the first of these steps fully, and the second in part.

But prior to taking a look at this, a discussion on the nature of Euclidean geometry helps to set the scene. Some of these simple points have far-reaching consequences, as this paper shows.

WHAT IS EUCLIDEAN GEOMETRY?
It arises in our experience of space. Objects around us have shapes, with surfaces and boundaries, and the figures of geometry can be thought of as simplified representations of these. Furthermore there are measurable aspects of figures: lengths, angles, areas and volumes. We call them magnitudes because what they have in common is magnitude (size). Euclidean geometry is a study of magnitudes in figures.

Geometry works with theorems, these being statements
which are then proved. Mostly, these statements relate directly to figures. Typically a theorem tells us that, e.g., Angle A equals angle B (angles A and B being illustrated in a figure), and a proof follows. Note that the underlined statement can be spoken. It can also be written in mathematical short-hand. That statements in mathematical short-hand can be spoken is often over-looked.

THE NEW SYSTEM
A good way of introducing this new system is by asking the question:
What do we really need in order to study Euclidean geometry?

The study is set in the context of an oral tradition. The original reason for doing so is discussed in 'Geometry for an Oral Tradition', but an immediate consequence is that it draws attention to the role of language in the study. Since we cannot do without it, formal acknowledgement is in order. And the same applies to anything else we cannot manage without. So a good first step is to note what else is indispensable.
Investigating this, I have found three requirements without which the study of Euclidean geometry is not possible in the context of an oral tradition. They are:
A language, in use.
A means of drawing figures.
The ability to recognize valid reasoning.

Lacking any one of these three the study cannot proceed. E.g. without a language it can neither be formulated nor explained. Indeed, without a language there can be no oral tradition. Again, figures are needed because geometry is a study of some of their properties.

But these three requirements are not sufficient for the study. For they make no commitment to any type of geometry, Euclidean or non-Euclidean. That is to say, they provide nothing to help get the study of geometry under way.

To get started we need at least one assumption in the form of an axiom. For theorems are proved using earlier theorems, and we need a starting point somewhere. Effectively, an axiom is a theorem which we do not prove.

Where can we find such an axiom? A helpful consideration is that Euclidean geometry has an everyday role as the geometry of our everyday experience. Indeed, this is where the study originates. It describes the spatial properties we are familiar with, as well as some we habitually overlook. This gives a clue as to where we might look for an axiom.

Euclidean geometry deals with space, and today we recognize that the latter is relative. It is no longer adequate to treat physical space as absolute, as is done by the accepted formulations of Euclidean geometry. This was acceptable in Euclid's day, but not now.

Now we are on the track of what is needed, for we will only be dealing with relative space if the principle of relativity is satisfied by our formulation of Euclidean geometry. And as it happens there is just one way of satisfying the principle of relativity which yields Euclidean geometry. Expressed physically, it is for objects to be unchanged in shape or size by motion. Expressed more mathematically it is for magnitudes to be unchanged on being moved around.

Thus, an axiom required for Euclidean geometry is:
magnitudes are unchanged by motion (i.e. on being moved around).
This is the fourth requirement of the study. Thus all four have been shown to be essential. There is also some evidence that they are sufficient for the study, as will be explained next.

SUFFICIENCY
Evidence that the Four Provisions are sufficient to prove the theorems of Euclidean geometry is provided in two ways:
[1] In 'Geometry for an Oral Tradition' they are used to develop theorems as far as elementary properties of a circle. [2] Equivalents or near-equivalents to Euclid's five Common Notions and five Postulates follow from the four provisions (see the Commentary to 'Geometry for an Oral Tradition'). This shows that the system is capable of proving some of the theorems, and suggests that it may be capable of as much as or more than the 'Elements'.

THE CONCEPT OF A PROVISION
In the book 'Geometry for an Oral Tradition' the term provision is introduced for what is provided. This is a concept of greater generality than that of an axiom. Each of the above-mentioned four requirements is a provision, but only the fourth is an axiom.

One might ask, what is the difference between an axiom and other provisions? Note that any proof whatever makes use of the first and third provisions, and the second comes in wherever there is a figure. But the axiom proposed here is needed only for Euclidean geometry. E.g. 'Geometry for an Oral Tradition' shows how it can be used to start off the theorems of Euclidean geometry. This points to an axiom being an assumption made for the purpose of a particular study. [The 'Shorter Oxford Dictionary' gives, for axiom 'a self-evident principle', pointing out that this goes back to Aristotle]

THE NEED FOR A COMMENTARY
Since 'Geometry for an Oral Tradition' does not follow a standard approach, a Commentary (of 51 pages) is included to discuss the issues arising, and to show that the approach is a valid one.

DEFINITIONS
The system will not work efficiently - or even not at all - unless the definitions are well-handled.

In everyday use there are lots of words used which are not normally defined because all concerned know what they mean. This system follows suit: words in common use need not be defined.

Definitions are given, however, of terms needed for the study, words such as theorem and angle and line, even though some of them are in common use. Two important definitions are those of magnitude and equality. With their aid three theorems can be proved which Euclid gives as axioms (Common Notions 1-3). For a suitable definition gives us something we can work with in proofs.

In general, if we do not define terms necessary for the study, more axioms are needed to fill the gap [definitions of words such as theorem and postulate excepted]. This study sets out to give all relevant definitions, so minimizing the number of axioms required.

There are a number of words in common use which have a bearing on geometry, such as circle, line, straight, angle, length, square, etc. As this list shows, when we begin the study we do have some knowledge of geometry; it makes sense to begin there, and then deepen and extend what we know. Giving a definition is part of the process.

This brings us to some illuminating points concerning definitions. There is a tradition in mathematics, passed on by Euclid, of giving definitions in sequence. Euclid probably presented the definitions in this way as a means of introducing them one at a time. This would explain why he avoids using a word until it is defined. Consequently, words defined earlier in the sequence may be used in later definitions, but not vice-versa.

Amongst mathematicians this procedure has resulted in a pretence (for mathematical purposes) that we do not know what a word means until it has been defined. This is unrealistic, and often untrue. In fact, knowing what an angle or a straight line is proves useful, enabling us to judge whether definitions proposed for them are acceptable or not.

But note further, that whereas later definitions can make use of earlier ones, the first definition has no other definition to draw on. Consequently, we need to start with something undefined.

The obvious source of undefined words is, of course, words in common use; for this is how speech works. But words not on the list of definitions are usually ignored, as though they do not exist. Mathematicians make use of them, and know what they mean, but do not define them. There is no thought that they are unknown because they are undefined. Language does not receive formal acknowledgement in mathematics today, but there are nevertheless rules or conventions governing its use.

A SUMMARY OF THE FOUR PROVISIONS APPROACH
That leading grammarian of the English language, Otto Jespersen, makes a telling comment:
'Educated people are apt to forget that language is essentially speech.' (Essentials of English Grammar)

By setting the study in the context of an oral tradition, the focus on language is a focus on speech. And in the light of Jespersen's comment this makes it relevant today.
So that is the first point: whereas mathematicians have always tacitly assumed a language to be available, in this study language is overtly acknowledged to have a key role, in its spoken form.
In everyday speech words in common use are rarely defined. This study follows suit: words in common use need not be defined.
Definitions are given, however, of words essential for the study (mathematical terms). Words such as theorem and axiom apart, every relevant definition gives us something to work on in proofs, making it possible to keep the number of axioms to a minimum.
A clear statement of the role of the foundations is required in order that we can ensure that the foundations proposed are necessary and sufficient for the purpose. The foundations are a statement of what is needed for the study, once the teacher and one or more willing pupils are assembled in a suitable place. [Teacher and pupil may be the same person, as in the case of a researcher.]
A generalization of the axiom concept, the provision, is introduced. It is used to include formally what has previously been tacitly assumed, including implicit acknowledgement of our relevant powers and faculties. Without these the study would not be possible, and axioms cannot provide for them.
The key steps are: (1) a statement of the four provisions proposed, and (2) demonstration that they are essential for the study. The Four Provisions are: [1] A language, in use. [2] A means of drawing figures; specifically, a plane, a pen, a straight edge, and a pair of compasses. [3] The ability to recognize valid reasoning. [4] The sole axiom; magnitudes are unchanged by motion (i.e. on being moved around).
The demonstration that these four are essential for the study can be summarized as follows: [1] In the absence of any one of the first three provisions the study is clearly not possible. [2] To ensure that the study is set in relative space the principle of relativity needs to be satisfied. There is only one way of doing so which yields Euclidean geometry, and that is the fourth provision.
That the Four Provisions are to some extent sufficient has been shown.
Finally, this being a non-standard approach, a Commentary is needed to demonstrate its underlying unity. (A 51-page Commentary is given in 'Geometry for an Oral Tradition')

CONCLUSIONS
The necessary and sufficient condition is a well-known one, but no attempt appears previously to have been made to apply it to the foundations in any branch of mathematics. The consequence is that the establishing of sound foundations has appeared to be beyond reach.

This paper shows how the situation can be rectified, giving foundations which are demonstrably essential for Euclidean geometry as an example.

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## FORTHCOMING EVENTS

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NEW COURSE FROM DR S K COSMIC KAPOOR

Dr Cosmic Kapoor will shortly be launching a new course on Vedic mathematics. It will be an online course similar to the ones previously given and again free. More details later.

JAIN IN NEW YORK CITY

Jain (of Australia) is giving courses in New York on May 5th, 6th, 7th, 8th. The first day is devoted to Vedic Mathematics. Further details below.

Phone: 1-917 608 6913
Email:
Fees:
Early booking by 1st April: \$150 per day
Late Booking after 1st April: \$175 per day
Venue: Manhattan, New YORK CITY
Download our 20+ page syllabus of Jains seminars in New York City
http://www.fengshuiseminars.com/pdf/jain.pdf
DAY 1 May 5th, Thursday
VEDIC MATHEMATICS
How to multiply large numbers...in your head! Find out how to accomplish rapid calculations - without a calculator.

Do your children really understand the fragmented and confusing mathematical data taught to them? Would you be interested in another method that boosts confidence, increases memory skills and is presented in a fun and exciting manner?

After this course of study, you will never multiply, divide or add in the same manner that you are currently using to solve arithmetic problems. You will walk out of Jain's class confidently multiplying double digit numbers without a calculator. You do not need to know any mathematics to do this course, in fact, which has been proven many times, children who have had learning problems with the way we teach them numbers, can perform these exercises such faster than the adults in the class!

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## OTHER NEWS

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VEDIC MATHS RESEARCH

A research scholar is seeking funding and grant to continue his work in researching the Vedic Maths sutras. He would also like to know of Institutions where VM is being researched. If you are doing research or can advise Mohan on this please contact him at

Clive Middleton, the webmaster for this site has recently completed the massive task of updating the whole site. This involves checking many links, facts and finding what is new since the last update. Since this gives an insight into how the Vedic Maths scene has changed and developed in recent years I asked Clive to tell us what he had found. Below are his comments.
Every now and then we spend the time updating the list of links to web sites and book lists related to Vedic Maths as they get out of date. Web sites disappear or change address and occasionally people create new web sites that we were not aware of before. Also new books get published. Therefore we updated the web site book and links lists over last Christmas, as I felt that it had been left too long since they were last updated.
Generally checking the web sites and books available on Vedic Maths is quite time consuming and takes over a couple of days to complete. Having done this 3 times now over the last few years, certain trends are becoming more apparent.

1) There are a range of amateur sites on Vedic Mathematics, or sites that mention Vedic Mathematics in passing. A few years ago lots of people seemed to be adding content to their web sites about Vedic Mathematics. This trend seems to have stopped and there are less new amateur sites on Vedic Mathematics. Although all the old sites on Vedic mathematics still seem to be present.

2) The list of people and organisations that are more serious about Vedic Mathematics also seems to have settled down to a core list. This mainly consists of the following
Kenneth Williams
James Glover
The world Academy of Vedic Mathematics
MatheVedics
Dr. S.K.Kapoor
Jain
What appears to be good is that these people/organizations appear to be updating and occasionally creating new course material on Vedic Mathematics.

3) There are new books on Vedic Mathematics published on a regular basis, with electronic material becoming more available over time.

4) Mostly the Material on Vedic Mathematics is just a reworking of the original material by Tirthaji, with the better people and organisations trying to improve the presentation and understandability of the material available, although there are people actually developing new material, out there. Seeing this made me realise how few people actually create something new or even attempt to improve on what is already available. This was proved to me by seeing sites that had just copied sections of our web site without our permission. As I also saw other material copied several times, so I assume the same is happening to other authors.

5) The major driving force behind Vedic Mathematics seems to be the Indian CAT examination system. As this encourages students to need to acquire practical mental arithmetic skills to pass their exams. This seems to be helped along slightly by Hindu's having an affinity to learning practical skills related to their religious traditions.

6) Most people learning Vedic Mathematics are doing it outside of the traditional educational establishments and are doing so because of their own desires to improve themselves (or at least their parent's desires for them to improve). Certain people claiming that Vedic Mathematics has become part of the core curriculum in western countries, know that this is only the case in certain schools that specialise in providing education with a more spiritual framework i.e. these people should know better, exaggerating the truth helps no-one.

7) Vedic mathematics appears to have become a political football, with different groups fighting to promote and prevent Vedic Mathematics depending on their own agendas i.e. religious groups promoting Vedic Mathematics to further a Hindu nationalist agenda and secular groups (communists, scientists etc) being against it as they do not like spiritual concepts in general. This conflict is currently preventing the spread of Vedic Mathematics through the traditional education system and will continue to do so until it is resolved.
Generally Vedic Mathematics is becoming more established with a greater range of material becoming available. Currently Vedic Mathematics is being learnt outside of the mainstream educational system. I would expect this to change over time, as more people learn Vedic mathematics and it becomes more accepted. This change will probably come in India first, due to revival of cultural knowledge and the pressures of India's exam system.

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WORLD ACADEMY FOR VEDIC MATHEMATICS - contact details

Registered Address: Vishwa Punarnirman Sangh, Raval Bhawan, Near Telankhedi Garden, Nagpur-440 001, India.
Contacts in other Cities in India :
Delhi R.P. Jain, MLBD bookstore
91(011) 2385-2747 / 2385-4826 / 2385-8335 / 2385-1985.
&
Varanasi 91 (0542) 2352331
Kolkata MLBD bookstore 91 (033) 22824872
Mumbai MLBD bookstore 91 (022) 24923526 / 24982583
Nagpur Alka Sahani 91 (0712) 2531363 / 2550906 / 2545637
/
Pune MLBD bookstore 91 (020) 24486190
Dr. Bhavsar 91 (020) 25899509 / 21115901
Bangalore School of Ancient Wisdom - Devanahalli, 91 (080) 768-2181 / 7682182 / 558-6837
/
MLBD bookstore 91 (080) 6533729 / 6542591
Chennai 91 (044) 24982315

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If you would like to send us details about your work or submit an article or details about a course/talk etc. for inclusion, please let us know on

Previous issues of this Newsletter can be copied from the Web Site: www.vedicmaths.org
Some articles from previous newsletters are:
Issue 2: "So What's so Special about Vedic Mathematics?"
Issue 3: Sri Bharati Krsna Tirthaji: More than a Mathematical Genius
Issue 4: The Vedic Numerical Code
Issue 6: The Sutras of Vedic Mathematics
Issue 7: The Vedic Square
Issue 8: The Nine Point Circle
Issue 11: Is Knowledge Essentially Simple?
Issue 14: 1,2,3,4: Pythagoras and the Cosmology of Number
Issue 16: Vedic Matrix
Issue 17: Vedic Sources of Vedic Mathematics
Issue 18: 9 by 9 Division Table
Issue 19: "Maths Mantra"
Issue 20: Numeracy
Issue 21: Only a Matter of 16 Sutras
Issue 22: Multiplication on the Fingertips
Issue 23: India's System of Mental Mathematics
Issue 24: The Sign of Nine
Issue 25: Maharishi's Vedic Mathematics
Issue 26: Foreword
Issue 27: Mathematics with Smiles: the Vedic Way
Issue 28: The Absolute Number
Issue 29: Report on India Tour
Issue 30: Vedic mathematics - excerpts from research paper
Issue 31: Why Vedic Mathematics?
Issue 32: Kolkata Workshop - an Overview
Issue 33: Report on Vedic Mathematics Workshop
Issue 34: LETTER TO NCERT DIRECTOR
Issue 35: LETTER FROM NCERT
Issue 36: WAVM Brochure
Issue 37: VM PROJECT
Issue 38: The Evolution of Simple Sums
Issue 39: The Cosmic Trust Mathematics Cause

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Editor: Kenneth Williams

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24th February 2005