Mathematics the Vedic Approach to Mathematics

7/30/2019 Mathematics the Vedic Approach to Mathematics

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The Vedic Approach to Mathematics

Forty years ago, as a machinists helper, with no thought that mathematicswould become my lifes work, I discovered the classic, gali What is Mathematics?by Richard Courant and Herbert Robbins. They never answered their question; orrather, they answered it by showing what mathematics is, not by telling what itis. After devouring the book with wonder and delight, I was still left asking, Butwhat is mathematics, really?

Thus begins Reuben Hershs book What is Mathematics, Really? Afterthis start, Prof. Hersh reviews the major schools of the philosophy of

mathematics from the Greeks to the present. He somewhat modestlydevelops an argument for his own philosophy, which he calls humanism,mathematics as a human activity, a product, and a characteristic ofhuman culture and society.

The odd thing about this book, which has enjoyed much popularityin recent years, is that clearly the author really wants to know whatmathematics is, yet equally clearly is not very convinced himself that theanswer he proposes is the right one.

Like Prof. Hersh, I have also spent many years wondering about thenature of mathematics. Like him, I found many delightful examples ofmathematics, but no one who really seemed to know what it all meant.

I spent a lot of time thinking about this, starting from the age of 14.I burned with a desire to know the answers. I did most of this thinkingoutdoors, in woods and fields, on hills and near streams. I let myselfthink without limits. I came to some conclusions that surprised myself.They seemed so at odds with what everyone else said, yet they seemed

so natural to me. They were so different from everthing I had ever heardthat I didnt even have words to describe them. Aside from the occasionalenigmatic remark that brought a stare or a laugh, for a long time I didntspeak these thoughts.

I now know that I was thinking about consciousness. I was thinkingabout the self-referential nature of consciousenss, and how this expressesitself through mathematics. I now know too that I was not the firstone to think in this way about consciousness, that the knowledge ofconsciousness as self-referral is central to the Vedic tradition of India.

Most importantly, this knowledge contained not only a philosophy ofself-reference, but also techniques for experiencing pure consciousness,

Copyright c2001--2002 Kevin Carmody


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the experience of consciousness as self-reference in its pure state. Thisexperience is liberating and expanding. It is not limiting or isolating.

I discovered too that the Vedic tradition contained a system of arith-metic and an understanding of the experience of pure consciousness as akind of mathematics.

What remains for me is to describe this to you, the interested reader,along with my own thoughts about the connection of Vedic mathematicsto Western mathematics and how it resolves outstanding issues in thephilosophy of mathematics.

Reuben Hershs arguments andmy responses from the Vedic point of view

Preface

I show that, from the viewpoint of philosophy [emphasis his] mathematics

must be understood as a human activity, part of human culture, historicallyevolved, and intelligible only in a social context. I call this viewpoint humanist.

There is a basic confusion here between mathematical truths and theexpression of these truths. The expressions are a human activity, butthe truths are true regardless of how we express them. That is whymathematics is a science. It is a useful science. It is useful to study it andit is useful to apply it. Because it expresses the truth of nature in a usefulway, the public supports the teaching of mathematics at all levels in our

educational system.

Hershs humanism fails to grasp this essential nature of mathematics.It equates mathematics with composition. It does not do enough topromote the search for truth in mathematics, and thus allows error. Thisdoes not make for a satisfactory philosophy of mathematics.

However, Hershs humanism has one bright spot: it promotes theviewthat human consciousness is involved in the expression of mathematics.

We will see later how this is a crucial insight.



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Chapter One: Survey and Proposals

Round Trip to the Fourth Dimension. Is there a 4-cube?

Does a 4-cube really exist?

We might as well ask, Does a 3-cube exist? If so, where? The Vedicapproach answers these questions, through the principle Knowledge isstructured in consciousness.

Quick Overview

No matter how isolated and self-sufficient a mathematician may be, thesource and verification of his work goes back to the community of mathematicians.

The mathematical community may help, but it is not the source ofmathematical ideas, nor is it their ultimate authority. If what Hersh isclaiming were true, then we could not so much as balance our checkbookswithout some mathematical authority pronouncing its approval. To thecontrary, all mathematical expressions can be verified independently. Thisis because mathematics is not merely theoretical---it is a science.

The view that mathematics is in essence derivations from axioms is back-ward. In fact, its wrong. An indispensible partner to proof is intuition. This tellsus what to try to prove.

An excellent point. Without any feeling from experience, we have nobasis for deciding what is right. Intellectual decisions are ultimately basedon some feeling of rightness, or else they are undependable. Without

dependability, there is no science.

Formalism: A First Look

Two principal views of the nature of mathematics are prevalent amongmathematicians---Platonism and formalism. Platonism is dominant, but its hardto talk about in public. Formalism feels more respectable philosophically, but itsalmost impossible for a working mathematician to really belielve it.

Platonism is a view that the basis of the universe is a transcendentreality, which is often characterized as objective. Who knows how to



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talk about a transcendent reality? Yet the Vedic approach allows us todo just this in an intelligent and responsible manner. It shows how andwhy an ultimate transcendent reality is the source of both objective and

subjective, which is experienced as pure consciousness.

Formalism ignores or denies the role of experience. In so doing, itignores or denies feelings of rightness. It is a great contradiction that thosewho cling to this as a philosophy of mathematics, or perhaps even as away of life, so adamantly deny emotion, when the very act of denial isrooted in fear, anger, and ignorance. This state is the result of incompleteexperience, which the Vedic approach easily rectifies.

The formalist philosophy of mathematics if often condensed to a short slogan:

Mathematics is a meaningless game. (Meaningless and game remainundefined.)

The sad part about this slogan is that if meaningless and gameremain undefined, then the statement itself is meaningless. The statementdestroys itself. It offers nothing to explain the scientific efficacy ofmathematics, nor its beauty, nor its rightness.

For a game, . . . two things are needed:

(2) people to play by the rules

(1) rules.

. . .[But] themakingof rules doesnt follow rules!

. . . People often make rules deliberately. . . . These rule-making tasks dontfollow rules. But that doesnt make them arbitrary. Rules are made for a purpose.

The purpose of rules is to ensure smooth functioning of a system.Rule-making must be performed from a level of consciousness whichcomprehends how to accomplish this.

However, mathematics is not just a system or scheme---it is a science.It incorporates a system, but is not just merely a system. A science isa system which expresses the laws of nature in a systematic and usefulmanner. The key difference between a science and another type of systemis the status of the laws of nature in a science. The purpose of a science isto express the laws of nature in a way which is useful. Other systems are

based on laws of nature, and they may accomplish something useful---butthey do not necessarily expressthe laws of nature.



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The laws of nature cannot be changed by any human being or groupof human beings. Science therefore ultimately rests on a reality which is

beyond the ability of human beings to manipulate. This does not mean

such a reality cannot be experienced or that it is beyond our knowledge.It simply means that science rests on a foundation which is the same foreveryone, and is therefore far from arbitrary.

Some formalists in the philosophy of mathematics say discovery is lawless---has no logic---while proof or justification is nothing but logic.

This point of view fails to recognize that a law may exist without beingexpressed. The laws of nature do not depend on any human intellect fortheir existence. The human intellect can express them, but does not create

them. They are in fact not created at all---they exist solely within andby virtue of the self-sustaining existence of the unmanifest side of nature.There are laws of nature which govern the phenomenon of discovery.Cognitive psychology has discovered many of these laws.

These considerations on games and rules in general show that one cantunderstand mathematics (or any other nontrivial human activity) by simply

finding rules that it fillows or ought to follow. Even if that could be done, it wouldlead to more interesting questions: Why and whence those rules?

True. Ultimately, all rules and phenomena come from nature. Eventhe intellect comes from nature. Intelligence comes from nature. Thisrecognition is the basis of the spiritual approach, which alone makes senseof all the facts, since it alone is all-encompassing.

Must We Be Platonists?

The standard version [of Platonism] says mathematical entities exist outsidespace and time, outside thought and matter, in an abstract realm independent ofany consciousnes, individual or social.

While the Vedic point of view agrees with this in some respects,this kind of Platonism is nevertheless mystical and unsatisfying. The keything that the Vedic approach brings to Platonism is that this abstractrealm can be experienced and that the nature of this experience is pureconsciousness. Pure consciousness is the home of all the laws of nature,including mathematical objects and the laws which govern them.


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Mathematics the Vedic Approach to Mathematics