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VEDIC MATHEMATICS NEWSLETTER Newsletter of the World Academy of Vedic Mathematics (part of the International Research and Resource Foundation for Indian heritage)
A warm welcome to our new subscribers. **************************** 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 apnicholas9@fastmail.fm 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 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? 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? Geometry works with theorems, these being statements THE NEW SYSTEM 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.
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. 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: SUFFICIENCY
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 DEFINITIONS 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 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. CONCLUSIONS This paper shows how the situation can be rectified, giving foundations which are demonstrably essential for Euclidean geometry as an example. ****************************
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 (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 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!
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 cmu01@rediffmail.com
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. 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 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.
WORLD ACADEMY FOR VEDIC MATHEMATICS - contact details Registered Address: Vishwa Punarnirman Sangh, Raval Bhawan, Near
Telankhedi Garden, Nagpur-440 001, India. **************************** Your comments about this Newsletter are invited. Previous issues of this Newsletter can be copied from the Web Site:
www.vedicmaths.org To subscribe or unsubscribe to this Newsletter simply send an email
to that Visit the Vedic Mathematics web site at 24th February 2005
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