"Is this mathematics or magic? It's both. It is magic while you don't understand it, and it is mathematics when you understand it" (Sri Bharati Krsna Tirthaji).
"At its deepest level, Vedic mathematics is the very structure of pure consciousness itself."
(Maharishi Mahesh Yogi)
"The Vedas are the constitution of the universe" (Maharishi Mahesh Yogi).
The Vedas
The Vedas are the oldest sacred texts of the Indian literature of the extinct Vedic religion, which predated the Hindu religion. They are among the oldest written documents of mankind. They are written in Sanskrit. It is not known exactly when they were written, but it is believed that they were written between 1500 and 900 B.C. About 5 centuries after the Vedas, the Upanishads, of mystical orientation, were composed.
The Vedas are considered to have a divine origin, being revelations of Brahma, the Hindu god of creation. The word "veda" literally means "knowledge" or "wisdom", which in turn comes from the Indo-European term "weid", meaning "to see". The Vedas are the source of knowledge that links man with the cosmos. Being the source of all human knowledge and all sciences, they deal with all branches of knowledge: medicine, architecture, mathematics, grammar, psychology, astronomy, etc., including spiritual matters.
The Vedas consist of "mantras" and "sutras".
Mantras are Sanskrit words or phrases that are recited (aloud or internally) repetitively as an object of meditation to transcend the mind. The sound of the mantras has a meaning of its own, in addition to the meaning of the words. The word "mantra" comes from the Sanskrit "man" (mind) and "tra" (instrument). The chanting of Vedic mantras produces beneficial vibrations in those who chant them and in those who listen to them.
The sutras are aphorisms, formulas, rules, patterns, general principles or universal laws that explain both the workings of consciousness and nature. The word "sutra" is the root of the word "suture," the operation performed by physicians to close a wound. The sutras are "threads" of knowledge, formulas, rules or aphorisms that are easy to remember. Their rhyme and coherence are designed to easily connect and activate the mechanisms of our consciousness.
In traditional Hinduism, the Vedas were studied only by the upper castes. The lower castes (Sudras) − also called "untouchables"− were forbidden to read, recite and even listen to them.
The Vedic texts were discovered and studied in the West at the end of the 19th century. In 2003, UNESCO declared the tradition of Vedic chanting - alive to this day in India - an intangible heritage of humanity.
The Vedas are made up of four texts:
Rig-Veda. a set of mantras (hymns and verses) that are recited aloud.
Yajur-Veda. a set of prose mantras for practical purposes.
Sama-Veda. set of verses to be recited during ceremonies of soma, a psychotropic drug that could only be taken by priests.
Atharva-Veda. set of hymns in metre and prose.
The first three Vedas are considered "the threefold sacred science". The Atharva-Veda, the fourth veda, has two parts. The first part consists of prayers aimed at curing diseases, attaining long life, protecting against disasters and attaining the desires of life. The second part contains speculative and philosophical hymns. This second part is really considered an appendix (Parishishta).
The Vedas consist of an enormous number of documents −of the order of thousands−, many of which have not yet been translated. with the passage of time they have become dispersed and deteriorated. They are highly structured documents in themselves and in relation to each other. Scholars of the Vedas have found some documents called "ganita sutras" ("ganita" means mathematics) that deal with many mathematical topics: demonstrations of the theorems of Pythagoras and Apollonius, areas and volumes of geometrical objects, permutations and combinations, irrational number, and so on. It is surprising to see how advanced Hindu mathematics was thousands of years before the development of European mathematics.
At a deep level each sutra can be considered as a highly refined formula for producing a high level of coherence and order in the brain, facilitating flexible, fast and accurate problem solving.
The traditional way to solve problems is:
Rigidity. There is only one way to solve a problem.
Sequentiality. Solving a problem is done step by step, sequentially. Analytical consciousness, the consciousness associated with the left hemisphere of the brain, is used.
By contrast, the Vedas provide:
Flexibility. There are many ways to solve a problem, giving free rein to personal creativity.
Parallelism. Many work steps of the traditional system are grouped together to produce the same result, but simultaneously, all at once. Synthetic consciousness, the consciousness associated with the right hemisphere of the brain, is used.
Vedic Mathematics
Vedic mathematics (VM) is the name given to the ancient mathematical system of the Vedas, a system that was rediscovered between 1911 and 1918 by Sri bharati Krsna Tirthaji − a scholar in Sanskrit, English, mathematics, history, philosophy and science − analyzing the second part, the appendix (Parishist) of the Atharva-Veda. His findings he disclosed in his book "Vedic Mathematics", published in 1965 [Tirthaji, 1989].
According to his research, all mathematics is based on 16 sutras and 13 sub-sutras (corollaries or sub-formulas). These formulas describe the natural functioning of the mind and consciousness and can be applied in many ways to solve mathematical problems. The sutras are easy to understand, apply and remember.
Mathematics is the science that provides the most abstract language for understanding the various phenomena of nature. VM unites this abstract language with the contribution of the knowledge of the laws of the mind, intelligence and consciousness.
With VM, mathematical problems are solved in a simpler, more direct and faster way than with conventional mathematical methods. The branches of mathematics to which it is applied are mainly arithmetic and algebra, but also geometry, differential calculus, integral calculus, matrix calculus, etc. It is the foundation of all branches of mathematics.
Titthaji divides the methods of VM into two classes: general ones, valid for all kinds of expressions, and specific ones, valid for when the expressions (numerical or algebraic) satisfy certain conditions, for example: numbers ending in a certain digit, numbers of the same number of digits, numbers close to a certain power of 10, etc.
The world owes much to India in the realm of mathematics. India was the country that invented the decimal system of numeration and introduced zero, two fundamental contributions to the history of mankind. It is also believed that they invented the binary numbering system, thousands of years before it was discovered in the West.
The Indian notational system spread to the Western world through the Arabs and has been accepted as universal. it consists of 9 symbols called "anka" (meaning "mark") for the digits 1 through 9, and the zero symbol called "sunya" (meaning "void"). There was also a symbol for the unknown, called "varna".
VM is considered a great contribution to mathematical science, in the same traditional Indian line of providing fundamental concepts associated with mind and consciousness.
The sutras
According to Tirthaji, in the Vedas the numbers are replaced by syllables (devanagaris) of Sanskrit, in order to make them easier to read, learn and remember.
The 16 VM sutras are as follows:
1
By one more than the above.
2
All of 9 and the last of 10.
3
Vertically and crosswise.
4
Transpose and apply.
5
when the sum is the same, the sum equals zero.
6
If one is proportional, the other is zero.
7
By addition and by subtraction.
8
By completion and non-completion.
9
By differences and similarities.
10
By deficiency.
11
Part and all.
12
Remainder by the last digit.
13
The last and twice the penultimate.
14
For one less than the previous one.
15
The product of sums.
16
The factors of the sum.
In addition to these sutras, there are 13 sub-sutras or corollaries:
1
Proportionally.
2
The rest remains constant.
3
The first by the first and the last by the last.
4
For 7 the multiplicand is 143.
5
For osculation.
6
To decrease by deficiency.
7
Whatever the deficiency, decrease by that amount and establish the square of the deficiency.
8
The last total is 10.
9
Last terms only.
10
The sum of the products.
11
By alternate elimination and retention.
12
By mere observation.
13
The product of the sum is the sum of the products.
Characteristics of Vedic mathematics
It is a coherent system.
The sutras constitute a structured, interrelated and unified system. For example, the general method of division is exactly the opposite method of multiplication, and the method of calculating the square root is the opposite method of squaring.
It is a mental system.
Calculations can be performed mentally, although they can also be written down.
It is an open and creative system.
Calculations can be performed in multiple ways, with diverse methods, so the system is open to the free creativity of the students or practitioners.
It is an efficient system.
Calculations are reduced to a minimum. They are straightforward and the result can be obtained in a single line. Problems where many operations are needed to obtain the solution can be solved with a simple sutra in one operation. For example, calculating 1/129 requires 18 steps with the traditional system. With the MV only one step is required.
It is a simple system.
The sutras are simple and easy to learn. Because of this, mathematical problems are simplified because as little effort as possible is expended.
It is a pattern recognition system.
connects to the semantics of numbers, what they mean. Instead of calculating blindly and mechanically, it is about perceiving or detecting patterns or general expressions that lead to simpler and more direct calculations.
For example, the traditional calculation of 123×999 involves doing 14 operations, to which carry operations (adding 1 to a previous result) would have to be added. Recognizing the pattern 999 = 1000 − 1, the result is almost immediate
123×999 = 123×(1000 − 1) = 123000 − 123 = 122977
operation that can practically be done on a mental level.
It is a system that facilitates verification.
One can easily verify that a result is correct by using an alternative way of solving the problem. Or there are formulas to verify, simply, quickly and in a single operation, that the problem is correctly solved.
It is a system that produces greater intellectual enjoyment.
VM is more enjoyable and pleasurable than traditional mathematics because it activates awareness, produces mental clarity, awakens creativity and intuition, facilitates discovery, increases mental agility, and improves memory. It also provides confidence and security. Traditional mathematics, in general, has been rejected by students, creating anxiety and uneasiness. MV offers a new approach to solve the crisis in mathematics education.
It uses negative digits.
VM uses a technique called "vinculum" or positional sign, which is applied to a digit d, symbolized by d, and stands for −d, the negative of d. For example,
11 = 10 − 1 = 9
2153 = 2000 − 100 + 50 − 3 = 1947
When there are several negative digits in a row, they equal a negative number formed by those digits. Examples:
1234 = 1200 − 34 = 1166
123456 = 120000 − 3400 + 56 = 116656
This system provides greater flexibility of representation than classical notation. Thanks to this technique, a number can be expressed in several ways, for example,
167 = 247 = 173
68 = 72 = 132 = 1932 = 19932
The process of moving from a number with one or more ties, to normal decimal format is called "normalization". For example, 3344 = 2736.
A number can be expressed by other numbers and using the positional representation of powers of 10. The different component numbers are separated by the character "|". For example:
12|−3|5 equals 12×102 −3×10 + 5
This also allows you to represent numbers in different ways. For example, the above number can be represented, for example, as 117|5 and 11|7|5. And many more ways using negative numbers.
Maharishi Vedic Science
Maharishi Mahesh Yogi has been the introducer in the West of the so-called "Transcendental Meditation". His method is based on using a mantra to transcend the mind and thoughts to reach "pure consciousness", the field of all possibilities and the source of all laws of nature. That state is identified with the unified field of consciousness, with the deep level that underlies all existence.
In 1988, Maharishi revitalized and advanced VM by connecting the ancient tradition of pure consciousness and modern science, specifically unified field theory (from physics) and category theory (from mathematics). He called it "Maharishi Vedic science" (cVM). Its essential points are as follows:
Since all the fundamental frequencies of creation are alive in the Veda, Maharishi refers to the Veda as "The Constitution of the Universe" [Maharishi, 1994].
Vedic Science is the science of the Veda, the infinite organizing power inherent in the structure of pure knowledge. Pure knowledge is the state of consciousness in which it knows itself, that is, when consciousness is completely self-referent, when it has nothing in its structure that is not itself. At its deepest level, the VM is the very structure of pure consciousness.
Each sutra can be considered as a refined formula that produces a high degree of coherence and order in brain functioning, which clarifies the mind and facilitates the discovery of solutions to mathematical problems.
A coherently functioning brain is a "cosmic computer". Through proper programming it is possible to achieve all that is desired. The sutras are the "cosmic software", which compute faster and more accurately.
VM is the mathematics of the Veda, the structure of pure knowledge, the level of indivisible unity. Within this unity there are two values: existence and intelligence. Intelligence knows itself and conceptually creates 3 values: the knower, the process of knowing and the known. Thus 1 becomes 2 and 2 becomes 3. From the interaction of these 3 values the diversity of the universe is created. All mathematics is described as the unfolding of this field of pure intelligence.
"Vedic Mathematics is the mathematics of the absolute, self-referential field of pure consciousness, where everything is simultaneous, where everything is simultaneously administered on the level of perfect order" [Maharishi, 1996].
All areas of mathematics become illuminated and clarified by being connected to the source from which they all arise.
The field of application of VM extends from mere numerical computation to the more abstract aspects of the dynamics of intelligence.
CVM solves the problem posed by Wigner in his famous article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences": "The enormous usefulness of mathematics in the natural sciences is something that borders on the mysterious and has no rational explanation" [Wigner, 1960].
It explains the connection that exists between the subjective (mental) realm and the objective (physical) world that science examines. It explains the nature of consciousness. The mind and the physical world are not separate entities, but are two aspects of the same reality, the unmanifest and manifest world, respectively, of creation. These two aspects are structured within the consciousness of each person. The mind is more abstract and subtle than the physical world, but both exist simultaneously and inseparably.
Science and mathematics are closely related. Mathematics studies abstract patterns, the structure of laws governing the subjective values of consciousness and intelligence. Science investigates the underlying structure of objective phenomena. But both are expressions of the same underlying field of consciousness, and both are governed by the same laws. Mathematics and science study the same laws of nature.
Vedic Mathematics and Mental Arithmetic
Traditional arithmetic is based on calculations that are performed from right to left. It is a mechanical arithmetic, analytical (digit by digit), rigid, without alternatives, formal, closed, requiring pencil and paper to write down intermediate results, subject to errors, slow, monotonous and boring.
Mental arithmetic, on the other hand, is based on calculations that are performed from left to right. It is a flexible arithmetic (it admits several paths to reach the final result), synthetic, conceptual, which does not require pencil and paper for intermediate results (the calculation is performed in a direct way, on a single line), open, fast, creative (it allows discovering new algorithms or methods), it is enjoyable, it opens awareness and reduces complexity. Possible errors can be detected by alternative ways.
In a calculation, the first digit to the left of the result is the most significant, the second is the next most significant, and so on. Therefore, the calculation from left to right is more conceptual. First an approximate result (or the order of magnitude of the result) is obtained, which is progressively refined until the final result is reached. It is a process that we can call "top-down". Examples:
36×3 is conceptually 3 times 30 + 3 times 6 = 90 + 18 = 108.
17×99 is 100 times 17 minus 17 = 1700 - 17 = 1683.
On the other hand, in the right-to-left calculation it is required to perform the complete calculation to obtain the most significant part of the result. It is the opposite of the previous process: "bottom-up".
The MV basically uses mental arithmetic, performing calculations from left to right, although we are free to perform calculations from right to left as well.
MENTAL and Vedic Mathematics
Critique of Tirthaji's Vedic Math
VM has generally been widely accepted in the mathematical community. However, criticisms have arisen in several respects:
There is no evidence that the contents of Tirthaji's book are of Vedic origin. The alleged appendix (Parishishta) to the Atharva-Veda appears not to be written in Sanskrit and its style, according to scholars, is rather contemporary.
The sutras do not have a general or universal character so that they can be considered archetypes of consciousness, i.e., of mind and nature. There are sutras that are specific (applying under certain conditions) and others that are general. In both cases, they are mainly a collection of general formulas applicable as recipes or mathematical tricks of arithmetic and algebra. The tricks are based on well known and established mathematical principles. They have no mystery and make no theoretical contribution to the field of mathematics, but are contributions of a practical kind, stimulating the semantics and creativity of the practitioner, rather than operating in a blind, mechanical way. It has been suggested that it was Tirthaji himself who invented these tricks, as he had a strong mathematical background, and that they do not come from the Vedas, which does not detract from their merit.
In addition to the 16 sutras, there are 13 sub-sutras (or corollaries) that are similar in nature to the sutras, so there would really be 29 sutras in all.
There are mathematical tricks that do not contemplate the sutras or the sub-sutras. There are also many others to be discovered.
There is no basis for claiming that all mathematics derives from the 16 MV sutras because the sutras are not generic. For them to be the foundation of mathematics they should be universal in character.
There are a great variety of mathematical fields (group theory, topology, algebraic geometry, etc.), and it is very doubtful that they can apply the MW to all of them.
Decimal fractions were unknown in India before the 17th century, so Vedic mathematicians could not calculate the decimal fraction of 1/19.
Critique of Maharishi Vedic Science (CVM)
CVM is at the right philosophical level, but it still lacks development. It is in the line, not of the sutras, but of modern mathematics, namely in category theory, including topos theory. But the theory of categories is complex, which goes against the conception of consciousness, which is simple and universal in nature.
Category theory is a theory that goes against Einstein's razor ("Everything has to be as simple as possible, but not simpler"). It is based on a single primitive (the morphism or arrow),and trying to build mathematics with a single primitive leads to great complexity.
Mathematics (and all formal sciences) must be constructed with the primal archetypes, which are the dimensions of consciousness (or degrees of freedom) of the mind.
MENTAL, the mathematics of consciousness
The message of the VM is to look for patterns, which activate the synthetic consciousness, the consciousness associated with the right hemisphere of the brain. To avoid, as much as possible, the mechanical. By recognizing patterns, calculations are more effective. But his methods are not universal. They are applied mainly in special cases, when the expressions belong to a previously detected pattern. But there is no doubt that VM is useful, it opens consciousness and is effective. It is a very interesting and promising field of research.
MENTAL is the closest to the philosophy of Vedic science, as its primitives are archetypes of consciousness, common to the internal (mental) and external (physical) world. They are also structured as a universal language.
According to the VM, the sutras are the foundation of mathematics. But the sutras are, in general, of a mathematical type, and mathematics cannot ground itself.
For Maharishi, MV is the constitution of the universe. The primary archetypes of MENTAL are the Magna Carta, not only of the universe, but of possible worlds.
The primary archetypes are not "the cosmic software", but the instruction set of the universal computer.
The recipes or tricks of the VM correspond to properties that are specified by the generic (parameterized) expressions of MENTAL.
MV is oriented to solve problems. With MENTAL, problems are solved more easily because the resources are more powerful.
MV does not provide generic or basic mechanisms to build expressions. MENTAL is oriented to build expressions through semantic primitives.
In MENTAL you can create imaginary expressions. Note that two imaginary expressions are the foundation of differential and integral calculus, and of complex numbers: ε^2 = 0 and i^2 = −1, respectively.
The term "mental" refers to operations that are performed internally, at the mental level. In MENTAL the mental refers to something deeper: the archetypes of consciousness.
The true sutras are the archetypes of MENTAL, which are universal patterns of clear semantics. In general, the VM sutras are ambiguous and it is not known when I can be applied.
Sub-sutras can be made to correspond in MENTAL to generic relations between semantic primitives, which are considered formal axioms. For example, the product of a sum is the sum of the products.
MENTAL is the foundation of everything, including human knowledge.
MENTAL better explains, with more foundation, the question posed by Wigner. [see Mathematical Applications Wigner's Question].
Addenda
On Tirthaji, the founder of Vedic mathematics
Tirthaji was a great scholar and a great scholar. By the age of 20 he had completed several degrees. He obtained top honors in the studies of Sanskrit, philosophy, English, mathematics, history and science.
When he learned that parts of the Vedas contained mathematics, he decided to investigate these scriptures to locate the contents and study them. For 8 years (between 1911 and 1918) he was dedicated to this task, virtually alone in the forests of India. He finally succeeded in deciphering the mathematical system. It seems that the system was deliberately encrypted by means of a code in which the numbers were replaced by Sanskrit syllables. Once Tirthaji discovered this code, he was able to bring the entire system to light.
It was in 1916 that he presented the 16 sutras that he claimed he had "rescued" from the Vedas, the ancient sacred texts of India. He went on to write a lengthy treatise in which he devoted a volume to each sutra. But this original 16-volume work - which was kept in the house of one of his disciples - was lost in a fire. In his last years he wrote the only MV treatise, which was published in 1965, 5 years after his death.
Tirthaji visited the West (the U.S., specifically) for the first time in 1958, two years before his death. He was invited by the "Self Realization Fellowship" (Los Angeles) to spread the message of Vedanta (the teachings of the Vedas). He also gave a talk to mathematics students at the california Institute of Technology (Pasadena, california). His book includes some of his lectures.
Tirthaji was Jagadguru Shankaracharya (great religious leader) of Govardhan Peath (Puri state, India), from 1925 until his death.
Since Tirthaji's publication, the VM has been expanding, with numerous works by many other authors having been published. There are many publications on the Internet, including videos.
The spread and development of Vedic Mathematics
Symposiums, seminars, courses and workshops on VM have been held in various countries around the world.
VM is spreading in the field of education. Today it is being taught in many schools and colleges, in India and outside India. It is also being taught to students of economics. The Maharishi Schools incorporate it in their teaching programs.
VM has been developed especially in the UK, with applications in trigonometry, 3D geometry, differential equations, matrices, determinants, logarithms, exponentials, etc. These developments have been published in several books. At Skelmersdale School (Lancanshire, England) they developed a course called "Cosmic Computer" written by students between 11 and 14 years old. It was published in 1988. This course is part of the National curriculum for England and Wales.
"The Vedic Mathematics Research Group" published 3 new books on VM in 1984, the centenary year of Tirthaji's birth.
Bibliography
Bolling, George Melville; Negelein, Julius von. Atharva Veda. Harrassowitz, 1909-1910.
Bathia, Dhaval. Vedic Mathematics Made Easy. Jaico Publishing House, 2006.
Chandler, K. Modern Science and Vedic Science. An Introduction. Modern Science and Vedic Science, I (1), pp. 5-26, MUM Press, 1987.
Dani, S.G. Myths and reality, on “Vedic mathematics”. Frontline, 5 Nov. 1993.
Dow, M. Anne. A Unifying Principle. Describing How Mathematical Knowledge Unfolds. Internet.
Gorini, catherine A. Maharishi’s Vedic Mathematics. The Fulfillment of Modern Mathematics. Internet.
Gorini, catherine A. How Maharishi Vedic Science Answers the Questions of the Unreasonable Effectiveness of Mathematics in the Sciences. Internet.
Griffiths, A.; Hoube, J.E.M. (eds.). Vedic Studies: Texts, Language and Ritual. Proceedings of Third International Vedic Workshop. Groningen: Egbert Forsten, 2004.
Gupta, Atul. The Power of Vedic Maths. Jaico Publishing House, 2005.
Hagelin, J. Is consciousness the Unified Field? A Field Theorist Perspective. Modern Science and Vedic Science, I (1), pp. 28-87, MUM Press, 1987.
Kandasamy, W.b. Vasantha; Smarandache, Florentin. Vedic Mathematics. “Vedic” or “Mathematics”: A Fuzzy & Neutrosophic Analysis. Automaton, 2006. Disponible en Internet: Scribd.
Maharishi Mahesh Yogi. Vedic knowledge for everyone. Maharishi Vedic University Press, 1994.
Mayers, Lester. High-Speed Math. Krieger, 1975.
Nader, T. Human physiology: expression of Veda and the Vedic literature. Maharishi Vedic University, 1995.
Nasser, Vali. Speed Mathematics Using the Vedic System. The Marketplace for Digital content, 2004.
Pratishtan, Sri Sathya Sai Veda. Vedic Mathematics. Internet.
Price, J. Maharishi’s Absolute Number: The Mathematical Theory and Technology of Everything, 1997.
Puri, Narinder. An Over View of Vedic Mathematics. Workshop on Vedic Mathematics, March 25-28, 1988, The University Rajasthan, Jaipur.
Puri, Narinder. Ancient Vedic Mathematics: Magic Speed Answers to All Mathematical Problems Using Sixteen Simple Sutras from the Vedas. Pushp 1 & 2. Spiritual Science Series, 1988. (Estos dos libros han sido traducidos a más de30 idiomas.)
Visnu, Swami b.b. Physics to Metaphysics. The Vedic Paradigm. Disponible en Internet.
Waerden, b.L. van der. Science awakening. Oxford University Press (USA), 1971.
Wigner, Eugene. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. En communications in Pure and Applied Mathematics, vol. 13, No. I, Febrero 1960. Disponible en Internet.
Williams, Kenneth R. The Natural calculator. Motilal banarsidass, 2003.
Williams, Kenneth R. Discover Vedic Mathematics. Motilal banarsidass, 2005.
Williams, Kenneth R. Triples: Applications of Pythagorean Triples. Motilal banarsidass (India Scientific Heritage), 2003.
