ISSUE No. 44

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.

This issue's article is the second half of an article by Andrew Nicholas who is a long-standing expert in Vedic Mathematics. The first part was in the previous newsletter and if you didn't read it it would best to do so before reading this. You can see all past newsletters at www.vedicmaths.org (click on newsletters). If you would like the contact Andrew about his article his email address is

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

PART 2: COMMENTARY

FURTHER INSIGHTS INTO THE FOUR PROVISIONS SYSTEM
There are certain ideals underpinning the Four Provisions system, and to do it justice we need to look at them. They are features which distinguish this from other systems of Euclidean geometry - as well as enabling us to look at the approach from a slightly different point of view. There are three such ideals:

IDEAL 1 Connect the formulation of the study with what is being studied, wherever possible. These meeting points may be called points of contact between the formulation and the object of study. They include definitions of point, line, plane, circle ,triangle, parallels ,right angles, etc.

This is one reason why it is helpful to define all words needed for the study. For almost every one is a point of contact, all of which need to be acknowledged.

Another important point of contact is Provision Four, which is the condition for space to be relative, i.e. to satisfy the principle of relativity. Each of these points of contact helps to locate the study fully in our world.

IDEAL 2 Acknowledge whatever is essential for the study. This is the role of the foundations. Carrying this part of the programme through led to the four provisions discussed earlier.

IDEAL 3 (1) Keep the number of assumptions to a minimum. And then (2) show that this is the minimum number of assumptions on which the study can be based, and (3) show that these assumptions are essential for the study. We then have grounds for describing the assumptions as axioms of the study.
All of these steps have been carried through successfully in this study. And in the process an idea has emerged concerning the nature of an axiom which may be new, and has certainly not been put into effect before: an axiom is an assumption which can be shown to be essential for the study.

An account of what may be called the 'Four Provisions approach' has now been given. Is there more to be said? Doubtless opponents of the system can think of a few things to say. Here is an example of what might be said by an advocate of the system widely regarded as leader in the field of Euclidean geometry,
David Hilbert's 'Foundations of Geometry' (1899):
Hilbertian Your definition of a straight line is not valid. You give,
Definition 13 That line which is uniquely specified given two points on it is said to be straight.
Now it is a well-established principle that a definition ought not to contain axioms and this definition consists of two axioms. As given by Hilbert they are:
Axiom1,1 For every two points A,B there exists a line 'a' that contains each of the points A,B
Axiom1,2 For every two points A,B there exists no more than one line that contains each of the points A,B
Need I say more? All your careful reasoning is worthless since you start off on the wrong foot.
Response To reply briefly, Hilbert gives 20 statements which he calls 'axioms', without proving that any of them are essential for the study. Nor does his approach allow them to be considered self-evident. Consequently his 20 statements are really so many assumptions, no reason being given for accepting them.

It sounds less impressive to say, 'Your definition is really a combination of two of the assumptions with which Hilbert begins his study' - which may be so but certainly does not amount to a flaw in the definition.

Furthermore, the use of 20 assumptions shows no respect for the principle that the number of assumptions should be kept to a minimum. It is a weakness in Hilbert's approach.

But there is more to be learnt from this example. Hilbert's first two axioms and the above definition of a straight line look at the same material, and present it in two slightly differing ways. We can use this material in proofs, whether we take it from the definition (Four Provisions approach) or from the two axioms (Hilbert's approach).

However, Hilbert pretends that we do not know what a point, a (straight) line, and a plane are (by leaving them undefined). In this way he turns his back on the origin of his axioms, turning them into assumptions: statements for which no justification is given.

But there is a further problem here. If point, (straight) line, and plane do not have their normal meanings, we cannot relate Hilbert's axioms to diagrams. This did not trouble Hilbert, who preferred not to have to deal with diagrams. In his system theorems are proved directly from the axioms, without reference to diagrams. He did use the latter 'for illustrative purposes', however, by assigning their ordinary meanings to point, (straight) line, and plane.

Hilbert's approach is heavy going, dispensing with diagrams as it does, and using a lot of axioms. But there is a more serious objection to it. For Euclidean geometry is a study of some properties of figures. Succinctly put, no figures, no Euclidean geometry. Thus, only when point, (straight) line, and plane have their normal meanings is this a study of Euclidean geometry. But it is an essential feature of Hilbert's system that point, (straight) line, and plane do not have their normal meanings. Either Hilbert's study is not a study of Euclidean geometry or else it is inconsistent.

Two objections to Hilbert's approach have now been pointed out, the first a serious one and the second a fundamental flaw. Both stem from his requirement that point, (straight) line, and plane be undefined and therefore unknown. But even apart from these, there is a fatal objection to his formulation: it is set in absolute space and is therefore anachronistic. It does not fit in with Newton's world-view, let alone Einstein's.

SOME OF NEWTON'S INSIGHTS ON GEOMETRY
Sir Isaac Newton's 'Principia' is set out in the same manner as Euclid's 'Elements', beginning with definitions and axioms and continuing with theorems. This is partly because of his subject-matter, dealing with objects (bodies) at rest and in motion. [The state of motion being the province of mechanics and the state of rest the province of geometry.] These two, including the study of forces acting on bodies, he calls 'universal mechanics'.

He points out that mechanics and geometry are interlinked, as the following two quotations from his Preface to the 'Principia' show:
(1)'…the description of right lines and circles, upon which geometry is founded, belong to mechanics.'
(2)'…geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.'
Newton explains that geometry deals with the state of rest. Effectively the 'Principia' extends the topic, embracing geometry (he took Euclid's theorems as given) and including the motion of objects (bodies). Thus it encompasses the states of rest and motion. This made it appropriate for him to write his book in the style of Euclid's 'Elements'.
One of the reasons for mentioning these points here is because since ancient times there has been some resistance to taking motion into account in geometry, as though it is not quite respectable. This is a pity, because it is vital to the subject. The states of rest and motion cannot be considered in total isolation from one another. Absolute space is one consequence of attempting to do so.
Another lesson from these two quotations, however, is that geometry relates to our experience of the world; it is not pure thought, unrelated to anything. Applied mathematics and physical sciences depend on their experiments, and topics of pure mathematics such as arithmetic and geometry draw on experience,

Finally, note that the 'Principia' deals with the relative motions of objects, and assumes that these are unchanged in shape or size by motion. That is to say, it contains Provision Four as a hidden axiom.
MORE ON SUFFICIENCY OF THE FOUR PROVISIONS
Are the Four Provisions sufficient to prove all valid theorems of Euclidean geometry?
Godel's Incompleteness Theorem tells us that there exist valid theorems which are not provable from the axioms. If it is applicable here, the answer is, no; the four provisions are not sufficient to prove all valid theorems of Euclidean geometry.
This raises the interesting prospect of finding a theorem of Euclidean geometry which cannot be proved using the Four Provisions, and which may turn out to be an illustration of Godel's Incompleteness Theorem.

REFERENCES
1) Mathematics: the Loss of Certainty, M.Kline, OUP 1980
2) Geometry for an Oral Tradition, A.P.Nicholas, Inspiration Books, 1999
3) The Thirteen Books of Euclid's Element's, Sir T.L.Heath, CUP, 1908
4) Essentials of English Grammar, O.Jespersen, Allen & Unwin, 1933
5) The Foundations of Geometry, translated from D.Hilbert's Grundlagen der Geometrie, Kegan Paul, 1st edition, London 1902
6) The Mathematical Principles of Natural Philosophy, translated by A Motte 1729, revised by F. Cajori, of Philosophiae Naturalis Principia Mathematica by Sir I Newton, 1686, University of California Press 1934

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## NEWS

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NEW CELEBRATORY WEB SITE

A message from Brian McEnery in Eireland who has been working for some time introducing Vedic Maths in Eireland:
To celebrate the 40th anniversary of the publication of Sri Bharathi Tirthaji's book on Vedic mathematics we have launched the Simple Sums OnLine web site, today, 17-04-05. The location is at present http://homepage.eircom.net/~ficheille/ss/

A NEW BOOK ON VEDIC MATHEMATICS

A book by Dr S. K. Cosmic Kapoor entitled "Vedic Mathematics Decodes - Space Book" has been published by Lotus Press, 4263/3 Ansari Road, Daryaganj, New Delhi -02. ISBN 81-89093-81-9. The book is sub-titled "Chase of space Plan for Existence on 6-Space format".

NEW VEDIC MATHS COURSE FROM DR KAPOOR

As announced in a recent Newsflash this internet course on Vedic mathematics, science & technology is now under way. You can enrol for the free course at www.learn-and-teach-vedic-mathematics.com. A new lesson is posted each day and all lessons are available for viewing.

A NEW BOOK

From Mr R P Jain of Motilal Banarsidass, the Indian publisher:
I'd like to report on the release of the above new arrival under running series India's Scientific Heritage # 11. The books is Ritual, Mantras & Science - An Integrated Perspective by Jayant Burde. The book is divided into 3 parts as follows :

a) Part 1 : Religious Rituals
b) Part 2 : Analytical Tools
c) Part 3 : Rituals & Science

In part 3 under Chapter (15): Science, Non-Science & Pseudo-Science from pages # 219 - 235, the writer has devoted nearly 3 pages on Vedic Mathematics from # 232 - 234. Further description about the book is available on the 1st page of our MLBD Newsletter, Feburary issue in the attachment. The publishers web site is at www.mlbd.com.

THE COSMIC CALCULATOR

This Vedic Mathematics course for schools has now been re-printed by Motilal Banarsidass. This printing is in colour and the five volumes in the set can now be purchased separately if desired.

JAIN

The Australian, Jain, is touring and giving presentations on a range of subjects including Vedic Maths:
Here is a quick update on my Tour to Europe.
May 10th: I leave New York to fly to Amsterdam
May 14th: Saturday: I lecture in the Morning at the Nexus Conference.
nb: The Post-Conference Seminars will not be in Amsterdam.
May 17th: I fly to Scotland to the Findhorn Community to have several Introductory Lectures and 5 days of Post-Conference Seminars. More details later to confirm all the dates. Any enquires now can be sent to: Keith at
Jain is giving courses in New York on May 5th, 6th, 7th, 8th. The first day is devoted to Vedic Mathematics. For more information: Download a 20+ page syllabus of Jains seminars in New York City http://www.fengshuiseminars.com/pdf/jain.pdf

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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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8th April 2005