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Geometry of Consciousness

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Geometry of Consciousness
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  1 Geometry of Consciousness Meera Chakravorty The character of a fiction, Lady Thomasania Covertly of the novel ‘Arcadia’ by Tom Stoppard, asks her tutor Septimas, “  Each week I plot your equations dot for dot, xs against ys in all manner of algebrical relation and every week they draw themselves on commonplace geometry, as if, the world of forms were nothing but arcs and angles. God’s truth, Septimas, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell why not a rose?” Interestingly, this view of Thomasania, however fictitious a character she might be, got reflected in the writing of the biologist D’Arcy Wentworth Thompson whose mathematical research analysis of the world published in 1917 titled ‘On   Growth and Form’ evoked a lot of excitement among scholars, scientists including Alan Turing and the biologists like John Tyler and Stephen Jay Gould but failed to be a part of mainstream study (Moore, 31). In this context, it is important to mention that Realist schools of Indian philosophy, the Sankhya and the Nya y a-Vaisheshika probably had begun to see mathematical / geometrical dimensions to explain consciousness and the world. The Sa n khya school, according to one view, is believed to have derived it’s nomenclature from the term ‘Sankh y a  ̄ ’ which means ‘ number ’ and implies that we have lost the philosophy of numbers that must have excited the scholars to explore the world through numerical research. The Indian logicians (the Nyaya-Vaisheshikas) viewed the cosmos by labeling it into Category/ Substance counting from ‘One to Seven’ and describing this grand design as the ultimate way to understand the mystery of creation. Continuing to explore from the context of Indian philosophy, the insights that describe Consciousness with the help of mathematical or geometrical principles as one may note, emphasize on an open question attempting to grapple with what is the phenomenon of Consciousness which is termed as ‘Brahman’ in the philosophical tradition, what is it ( Ko’ayam  ̇  Brahmah), in what sense can we talk about it and what can we expect to find if we look for it? The philosophy of the Upanishad-texts examines these issues bringing in paradoxes so that the readers can make up their own minds. Some of the analyses in this source lead logically to particular constructs while others are ‘a logical ’  and are new way of constructing the understanding of Consciousness. One such explanation goes to formulate the ‘definition’ , if    we may use the term, of Consciousness as ‘Infinite’ in the language of the Upanishad, the terms used are ‘Anantam Brahma’, the intuitive concept of the Infinite as that which is endless, unlimited, un-surveyable, immeasurable and so on. If this definition is accepted functionally, then the proposition Geometry of Consciousness’ becomes difficult to explain because from the point of view of this definition if something is immeasurable then it cannot be defined. Therefore, can we say that all these paradoxes used here in the context of Consciousness are designed to either refine or to circumvent or to accommodate to create a sort of definition which in turn needs a lot of explanations though.  2 Yet, not to lose sight of the starting point of ‘geometry of Consciousness’, attention may be drawn to understand this position through Zeno’s paradox, for instance. Zeno of Elea in the early Greek tradition is known to have formulated a paradox which mentions time and space having time points and space points respectively. According to a story, Achilles, who runs much faster than the tortoise, lets it start a certain distance ahead of him in a race. The paradox is that Achilles seems never to be able to overtake the tortoise, no matter how great the difference in their speeds. For, in order to do so, he must first reach the first (space) point at which the tortoise starts, by which time the tortoise will have advanced a fraction of the distance initially separating them. Achilles must then make up this distance, by which time the tortoise will have advanced again. He must then make up this new distance, by which time the tortoise will have advanced yet again. And so on ad infinitum (Moore: 27). Thus keeping Zeno’s paradox in mind if Consciousness is viewed as infinitely divided space-time points of geometrical description and if these space- time points are measurable then there will be a fallacy in the definition of “Infinite’ which is earlier mentioned as immeasurable. Consciousness then will be understood as a collection of the innumerable space-time points besides being measurable. By contrast, scientists like Ian Stewart say that these are guidelines to exploit the deep rules, and not the rules themselves. It may not be out of context to remind here that the deep rules are also what the Indian sources like Upanishads want the investigators to try to fathom, for instance, when it is suggested in the phrase ‘Satyam Jnanam Anantam Brahma’, in which the last term is ‘Ananta’ , it must be noted that while the first two term s stand for Truth and Validity associated with the term ‘Ananta’, it also shows the details of the process of evolution symbolically that appears to explain the Macrocosm via microcosm. Ananta also means Infinity. While exploring the matrix of Consciousness, it is interesting to find that an example from the study of plant growth shows that the use of abstract formal system showing the relationship between large-scale structure and the form of smaller parts: the uncanny resemblance between the overall shape of a tree and the pattern of one of its twigs. Lindenmayer-system ( L-systems) , in the name of Astrid Lindenmayer the mathematical biologist, is defined by an alphabet of symbols, an initial string of symbols from that alphabet and a collection of production rules, which tell you how to replace individual symbols with other symbols or group of symbols. This contributes to the production rules. How is this related to serious biology? Lindenmayer says that it tries to give substance to a ‘developmental programme’. We should think of the initial string as standing for some particular type of cell or cluster of cells. The production rules tell us that there are biological processes that will replace particular kinds of cells or cluster of cells, with other cells and clusters--- thus, one of the symbols might stand for a node and a rule might tell us that, whenever there is a node, it is replaced, at the next developmental stage, by a branch. L-systems abstract completely from the details of how these process work, revealing how the recursive sequence of process generates the patterns found in algae, ferns, leaves, flowers and trees ( Kitcher: 31).  3 At the centre of this argument lies the perception which can be compared with the philosophical position of Ramanuja, the follower of Vishishtadwaita school of philosophy. His premises in the proposition of the concept of Brahman, a term in Indian metaphysics used for Consciousness, though appear to be a height of abstraction yet appears to be supportive of ‘L - System’ as he explains that the Consciousness is associated with the development of the whole of the living world too, and hence, to be known as in his terms as ‘Chidachit vishishta Brahman’ when translated it means that both Consciousness and Matter ( chit and achit respectively) are one reality and cannot be seen in isolation. The Sankhya school on the other hand holds that both the biological and psychological developments can be understood as emerging from Matter/Prakriti ( the term used by the Sankhya for Matter) though the presence of Consciousness is an imperative factor. This is one dimension. The other is the theory proposed by Ian Stewart who says how general studies of symmetry-breaking (as when you pile weights on a vertical spring, which eventually buckles on one side or the other) can illuminate our understanding of biological processes and explores the idea that shapes of cells, tissues or even whole organisms — can sometimes be understood as consequences of minimization principles ( in the simplest case, already studied by Thompson, the emergent structures are those that require minimum energy) (Ibid). The idea of a circle in geometry with infinite space- points is given it’s fullest expression by transplanting these relentless process into a meaningful concept in the Indian tradition as something eternal on one hand and non-eternal on the other. 1  In the same way time or Kala (in Sanskrit) also is conceived both as time relative and time eternal. 2  In kalachakra, circular time as eternal is associated with Consciousness metaphorically known as Mahakala yet another name of Shiva who is himself the ‘Great time’ where time is interminable. One may be rem inded of Eliot’s ‘Four Quartrets’ in which he frames time both ways: ‘If   all time is eternally present / all time is unredeemable. / Time past and time future / What might have been and what has been / all is always now…If time is viewed as past, present, future etc. or as hours, years, millennium and so on it is so shaped as ‘chronicity’ for day to day work for convenience of routine transaction. Otherwise, time is elusive hence conceptualizing it as Consciousness becomes the crux of the matter, Indian log icians referred to this as khandakala or ‘anityakala’ the time -flux while ‘nityakala’ is eternal and ever present therefore the logicians claim that all thing / events happen in time and further define time as ‘Kalah Sarvadharah’ or time is the foundation for everything possible. Consciousness as ‘Time - Consciousness’ can be geometrically seen in the shape and significance of a circle where Consciousness as kalachakra show explicity through the form of the circle, in mathematical calculations or deduction, remainder etc. yet it does not represent any hierarchy in the following expression fundamentally mentioned in the Sanskrit source: ‘Om pornamadah purnamidam purnat purnam udachyata   Purnasya purnamadaya purnameva avashishyate’ .  4 When translated means: ‘That  is whole. This is Whole. When you deduct Whole from the Whole the remainder is Whole ‘. If this is drawn as circle metaphorically the implications become meaningful showing how the time linear, time chronological, or time circular are two sides of the same coin, here in this context we should read the terms ‘This’ / ‘That’ as  terms used for Consciousness. The idea that geometry of Consciousness is possible cannot give so much as a bird’s eye view of the interpretative variations that have taken place in the conceptualization of reality. However, the various interpretations in a way democratize the understanding of Consciousness since no theory can pursue a particular chronicity and claim as the ultimate one which alone can explain in terms of eternity the fact of Consciousness. The geometry of consciousness may create timeless patterns attributed to the idea of evolution of the cosmos or cosmic world and the admission of its indestructibility and universality may be explained as a continuous series of modifications in matter (Prakriti), but it cannot exhaust an understanding of Consciousness through these patterns alone. Any equation of theorizing Consciousness founded on an assumption that it will bestow upon the subject to be known the whole meaning and will attempt to counter-balance all other presuppositions and identifications may appear overbearing. Aristotle, well aware of the problems that afflict the infinite, but also reluctant to abandon the concept completely, responded to the dilemma by drawing distinction between the ‘actual infinite’ and the ‘potential infinite’. The infinitude of t he actual infinite exists at same point in time. The infinitude of the potential infinite exists over time. Imagine a clock endlessly ticking. Its ticking is potentially, but never actually, infinite. All the objections to the infinite, Aristotle insisted, are objections to the actual infinite. They are objections to the idea of an infinitude that exists all at once. The potential infinite, by contrast, is a fundamental feature of reality. It is there to be acknowledged in any process that can never end: in the process of counting, for example (Moore: 27). The philosophy behind this argument is the great intuitive appeal, the basic idea towards understanding Consciousness. For over two thousand years Aristotle’s view of Infinity prevailed but later thinkers construed the references of time in the actual/ potential distinction metaphorically. Late in the 19 th  century, Canter presented a coherent, rigorous, systematic mathematical theory of infinite by taking the paradoxes in the stride. He showed, for example, that the set of positive integers is not unlimited (in size): it has fewer members than it has subsets. He also showed that it is not immeasurable: we can give a precise mathematical measure to how big it is. There is a sense, then, in which he establishe d that what is ‘really’ infinite is something of an altogether different kind. In a curious way, his work served, in the end, to corroborate the Aristotelian orthodoxy that ‘real’ infinitude can never be actual (Ibid). This is the crux of the problem. Consciousness as real infinitude can never be actual. As the Upanishads suggest that the dynamics of any system are governed by the play of Consciousness whatever may be the object or process. Alan Turing showed that the dynamics of the system are governed by (partial differential) equations, and, by looking at the solutions as the parameters in those equations take on different values, its possible to see quite different patterns and this is known as the effects of ‘ diffusion-  5 driven’ instability ( Kitcher: 31). Influenced by Turing, Hans Meinhardt has shown how his analyses of ‘reaction - diffusion’ does not identify the particular molecules play a role in forming ‘stripes’ but will show pigmentation pattern distinctive of particular species. While James Murray, yet another mathematical biologist is much inspired by Turing when he formulates a remarkable theorem that demonstrates that the same equations that yield stripes when the domain of the diffusion is smaller give rise to spots when the size of that domain increases sufficiently (Ibid). It is hard to imagine how Consciousness too needs analyses at its various expressions. It appears to be full of enigmas because there seem to be an endlessness of both its simple and complicated regulations sometimes journeying across the non- abstract space and sometimes through ‘zones of emptiness’ which might give one the idea of viewing it from the architectural or any other dimension calling it the ‘architecture of Consciousness’ as it happened with the idea of ‘geometry of Consciousness ‘ and may equally bring out its ‘ungraspable density’ influencing yet another new sense of logic to explain the contrasts that may be again astounding. The researchers in the Neurobiological brain-behaviour have come a long way in their study in Consciousness. Dr. Saibal Gupta, the Cardiothoracic surgeon and cultural historian mentions of two important papers out of many, one by Francis Crick and Christoff Koch and the other by Christoff Koch and Klaus Hepp. They explain that , although we have hopefully convinced our physics colleagues that classical physics is superior framework for explaining HBF ( higher brain function ), we hurry to stress that on molecular and membrane level there are beautiful biophysical problems where the border between quantum and classical physics has to be drawn (Gupta: 678-9. Other scientists like Sir Roger Penrose and Henri Poincare have envisaged and analysed Consciousness as mathematical in nature. As a mathematical physicist, Penrose sees the world as a structure precisely governed by according to timeless mathematical laws whose concepts are essential to describe the physical world. Penrose is not the first mathematician to be aware of the inexplicable role that the mind plays in what may be called mathematical intuition. Henri Poincare, a late nineteenth century French mathematician, wrote a book named ‘Foundation of Science’. In it he described how solutions to mathematical problems suddenly came to him when he was not thinking about them. Once he was on a geological expedition and a solution occurred to him while looking intently at a rock face. Another time he solved a problem when he had coffee at night and was tossing in bed. He described this as the subliminal self, from where hidden ideas float up to the conscious mind (Ibid: 679). If there is a dialogue between Wittgenstein and the idealist Advaitin regarding the position of Self/ Consciousness to the world, Wittgenstein may find it difficult to rub shoulders with the latter as his series of widely negated ideas about the world vis a vis Self appears obscure. Because, according to Wittgenstein, the Advaitin is a Solipsist and the Solipsist’s predicament is that when he denies the existence of everything except himself and the world of his own experiences, he is unable to point to what it is that, according to him, does not exist, because it lies outside his world… But what is this unique self, of whose existence he feels assured…It is only the metaphysical sub  ject, which is a kind of
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