Articles discussing connections between Maharishi Vedic Science and foundational aspects of mathematics and computer science.
Articles on Maharishi Vedic Science and Mathematics
For an introduction to Maharishi Vedic Science, see the following unpublished article, originally written as part of the paper "Vedic Wholeness and the Mathematical Universe," listed below. Introduction to Maharishi Vedic Science.
Magical Origin of the Natural Numbers, Extended Version
The paper suggests a new version of the Axiom of Infinity that incorporates the intuition that the underlying reality of "the infinite" is self-referral dynamics of an unbounded field. Taking this step, it is shown, leads naturally to a solution to the Problem of Large Cardinals. Original power point slides are here.
ABSTRACT: A turning point in the history of mathematics was Cantor's discovery that infinite sets exist. Some time after this discovery, when the foundational axioms for all of mathematics---the ZFC Axioms---were being developed, Cantor's discovery took the form of a fundamental axiom, now known as the Axiom of Infinity. This axiom expresses Cantor's discovery with extreme economy, asserting nothing more than that the natural numbers 1, 2, 3, . . . can be collected together to form a single set (an infinite set). Because of this economical formulation, the Axiom of Infinity provides little intuition about the nature of ''mathematical infinity.'' Lacking a sufficiently clear idea about the nature of the infinite, mathematicians have floundered as they have attempted to come to grips with very strong and unusual forms of the infinite, known now as large cardinals, which have emerged in research in the past century. These notions of the infinite cannot be proved to correspond to ''real'' infinite objects in the mathematical universe, but nevertheless seem quite real. A question for which there is, to this day, no universally accepted answer, is, Do large cardinals exist?
In this article, we suggest a new form of the Axiom of Infinity, which provides much richer intuition about the mathematical infinite, and which points the way toward an account of large cardinals. This new axiom is based on a deep insight about the true nature of the infinite. This insight is drawn both from the ancient wisdom of several traditions of knowledge, concerning the origin of the natural numbers, and also from the paradigm provided by quantum field theory for understanding the ultimate constituents of the physical universe. Both perpectives suggest to us that a collection of discrete objects, like the set of natural numbers, should be understood as precipitations of the dynamics of an unbounded field. What is important about the set of natural numbers, therefore, is the field that gives rise to them. In this spirit, we show that the sequence of natural numbers ''arises from'' the transformational dynamics of a Dedekind self-map. We show that a deep understanding of Dedekind self-maps, whose value at first seems only to generate the natural numbers, reveals that large cardinals themselves arise as ''precipitations'' of Dedekind self-maps. Following this logic to its natural conclusion, we conclude that, mathematically speaking, ''everything'' arises from ''unmanifest'' transformational dynamics that move the totality of the universe within itself.
Addressing the Problem of Large Cardinals with Vedic Wisdom
ABSTRACT: Shortly after Cantor's discovery of the existence in mathematics of an endless hierarchy of different sizes of infinite sets, a new challenge concerning the nature of infinity in mathematics arose. Enormous infinities, known as large cardinals, have turned out to be the key to solving many mainstream problems in mathematics, but because of their extraordinarily strong properties, large cardinals cannot be proven to exist at all. The Problem of Large Cardinals is to find a natural way to enrich the standard axioms of set theory so that large cardinals can be derived. To accomplish this goal, a much deeper intuition about the nature of the infinite than has been available so far is needed. We suggest that precisely such intuition can be extracted from the ancient Vedic wisdom. We formulate a new axiom of set theory, strongly motivated both by this ancient wisdom and by mathematical considerations, which provides a solution to the Problem of Large Cardinals.
Mathematics of Pure Consciousness
ABSTRACT: Adi Shankara, the foremost exponent of Advaita Vedanta, declared "Brahman alone is real, the world is mithya (not independently existent), and the individual self is nondifferent from Brahman." A fundamental question is, How does the diversity of existence appear when Brahman alone is? The YogaVasistha declares, "The world appearance arises only when the infinite consciousness sees itself as an object." Maharishi Mahesh Yogi has elaborated on this theme: Creation is nothing but the dynamics of pure consciousness, which are set in motion by the very fact that pure consciousness is conscious; being conscious, it assumes the role of knower, object of knowledge and process of knowing. To help clarify these issues, we offer a mathematical model of pure consciousness. We show that in a natural expansion of the universe of mathematics by ideal elements, there is a unique set Ω whose only element is itself, and which is equal to the set of all possible transformations from itself to itself. All "real" mathematical objects can be seen to arise from the internal dynamics of Ω. All differences among numbers, and among all mathematical objects, are seen to be ghostly mirages, hiding their true nature as permutations of one set, Ω.
Recent Progress in the Mathematical Analysis of the Infinite
The next two articles have appeared in P. Corazza, A. Dow (Eds.), Consciousness-Based Education: A Foundation for Teaching and Learning in the Academic Disciplines: Vol. 5. Consciousness-Based Education and Mathematics, 2011, MUM Press.
This is a very detailed discussion of Maharishi Vedic Science and the set theory of the infinite. The article gives a survey of ideas of set theory and the mathematics of the infinite, leading up to the modern-day theory of large cardinals. The article shows how the Problem of Large Cardinals -- the problem of determining how these enormous large cardinals arise and how they can be accounted for (since they cannot be proven to exist from the known foundational axioms) -- can be addressed, and even solved, through an application of Maharishi Vedic Science.
This is a presentation of Maharishi Vedic Science and the mathematical infinite written originally for mathematicians, but is reasonably readable by anyone. The article offered fundamental intuitions about the universe, which expand upon known intuitions derived from the work of Cantor and other more modern researchers, as the basis for introducing a new axiom in mathematics, the Wholeness Axiom. The Wholeness Axiom embodies, in mathematical language, the essential features of the dynamics of wholeness in Maharishi Vedic Science: Wholeness is an all-inclusive totality that moves within itself, knows itself, remains unchanged by its own self-transformations, unfolds on the basis of a collapse of its unbounded nature to its point, and that is present at every point in creation. The paper shows how these principles are given expression in mathematics, and how this expression, in the form of the Wholeness Axiom, solves the Problem of Large Cardinals.
Articles on Maharishi Vedic Science and Computer Science
The next two articles have appeared in K. Levi, P. Corazza (Eds.), Consciousness-Based Education: A Foundation for Teaching and Learning in the Academic Disciplines: Vol. 8. Consciousness-Based Education and Computer Science, 2011, MUM Press.
This introduction to a graduate course on Algorithms begins with the question "What is computation?" We ask whether Nature's way of computing all that it accomplishes has the same algorithmic character as the mathematical tools that are used in science to model Nature. We review the perspectives offered by Maharishi Vedic Science and recent discoveries in quantum field theory, both of which conclude that Nature's computational dynamics are based on self-referral performance at a hidden, unmanifest level. We then show how the very mathematics of computation has its basis in self-referral dynamics by showing that every computable function may be defined as the fixed point of an operator on a vast (and therefore, in a sense, unmanifest) function space.
This introduction to a graduate course on Algorithms shows how the class of functions that are actually used in Computer Science arises from a much vaster class of number-theoretic functions through a sequence of "collapses" to successively narrower function classes. The vast majority of number-theoretic functions are essentially indescribable. By contrast, it is possible to know in full detail the behavior of the functions belonging to the much more restricted class of definable functions. Restricting further to the class of computable functions, it becomes possible to describe the behavior of each function in terms of a computer program; that is, one can describe how to compute each function in this class. And the narrowest class of functions - the class of polynomial time bounded functions - consists of functions that can be computed in a feasible way, efficiently enough to be of practical value in real applications. We discuss how this sequence of collapses parallels the creative dynamics of pure consciousness itself, as described by Maharishi's Vedic Science, as it brings forth creation through the collapse of the abstract, indescribable infinite value of wholeness to the concrete, specific point value within its nature.