Science of the Absolute Chapter 5 - Prologue
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Created on Monday, 20 October 2008 19:02
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Published on Monday, 20 October 2008 19:02
Written by Patrick Misson
AN INTEGRATED SCIENCE OF THE ABSOLUTE
The neutral vision of the Absolute should not be tarnished even with the slightest taint of negativity. In the last chapter one attained to the extreme limits of negativity in the notion of nature, revealing the potentialities of a universe full of possible varieties. There is an element of surprise within it reminiscent of the Big Bang theory of modern cosmology. This active and expanding aspect of the universe, though revealed as a mere possibility and not as a fact, touched the extreme limit where visible manifestation is potentially or kinetically implied between pradhana and prakriti respectively. This limit belongs to perceptual physics and not to conceptual metaphysics.
We are now going to balance this one-sidedness in the present chapter. Such a balancing implies a double correction enabling the extreme negativity to cancel itself out into normality by finding within the Absolute a corresponding numerator factor. It is in this sense we have called this chapter Normalization. Further, this normalization is to be understood as taking place, as in pure mathematics, within the core of pure reasoning or consciousness. Thought here moves within its own positive and negative amplitude. No outward fact, thing or event is important. The mathematical formalism of Hilbert agrees with the contents of this chapter. Two pure aspects of the Absolute interact here, abolishing the duality of each other so as to reveal the fully normalized Absolute. Normalization and re-normalization refer respectively to the double correction implied. We have seen how Eddington recognized this approach, as revealed by Whittaker (see page 7).
Although Narayana Guru's title for this chapter is Bhana-Darsana (vision through consciousness), this normalization peculiar to Eddington has unified and somewhat subjective status very similar to the epistemology proper to Vedanta. We have also seen how scientists like Schrodinger recommend a new synthetic way for understanding reality (see pages 63, 67-68 & 284). Selectionism, structuralism and subjectivism belong to the new epistemology of science.
Bergson's metaphysics paves the way for such a new epistemology in the context of a Unified Science where physics and metaphysics could coexist, Bergson says that one has to enter into the situation and not take snapshots from outside. He further states that perceptual and conceptual time, sitting together as it were back to back, can exist in relation to a common parameter. Such a time is given transparently to the view of the future man of research and the double correction which does violence to the natural inclination of thought is to be applied. Bergson is undoubtedly putting his finger on some of the subtler aspects of normalization. We have already quoted these points from him profusely. They help to reveal the fact that physics and metaphysics can be treated together under one unified discipline. When this is done, Bergson inevitably asks: "Can we then not attain the Absolute?"
In Vedanta this double-sided reference is also familiar. The Katha Upanishad (III.1) mentions two selves drinking of the truths pertaining to the world order in the sukrita-loka (world attained through good works):
"There are two that drink of righteousness (rita) in the world of good deeds; Both are entered into the secret place (of the heart), and in the highest upper sphere.
Brahma-knowers speak of them as 'light' and "shade', and so do householders who maintain the five sacrificial fires, and those too who perform the triple Nachiketas-fire." (1)
These two selves exist together as twin entities, inseparable like sunlight and shade. Also in the Svetasvatara Upanishad (IV.6) is the familiar example of the two birds of equally beautiful plumage sitting together in the same tree. One eats sweet fruit, while the other looks on passively. These striking examples refer to the double correction to be applied to the notion of the Absolute.
We read as follows:
"Two birds, fast-bound companions,
Clasp close the self-same tree.
Of these two, the one eats sweet fruit;
The other looks on without eating." (2)
Even the four classical examples used in Vedanta each involve two aspects of reality or truth. The examples of wave-water, rope-snake, pot- clay and mother-of-pearl and silver are meant to be used in different contexts where normalization of error is intended. In the last example of mother-of-pearl and silver, the interplay of right and wrong between the two sides is so subtle and natural that it can be used most profitably in explaining the need for normalizing the physical and metaphysical prejudices hindering the vision of a fully neutralized Absolute. In the snake-rope example the frightened man suffers in a more crudely and unphilosophical form the conflict of the yes-no of the situation. The clay-pot example should interest the logician who gives primacy to the material cause. The wave-water example is the most inevitable one for Advaita Vedanta where all paradox and contradiction are abolished by a fully verticalized view. Even the bubbles and foam in water produced by horizontal disturbances are not admitted as having any reality other than water. These two broad divisions in consciousness exist in Vedanta as well as in modern thought.
1. MATHEMATICAL AND MYSTICAL LANGUAGE
In this chapter we are dealing with these twin aspects of consciousness, interacting as it were by an infinitesimal difference. The ambivalent and antinomian factors involved come into a delicate interplay which is to be understood as belonging to either in a purely mathematical or highly mystical language.
Occasionalism as known to Descartes, whereby the will of God makes for psycho-physical interaction, is the nearest metaphysical notion related to what we are concerned with here. Gestalt psychology with configurations emerging to view as integrated entities and not merely as eidetic presentiments also help us.
We have to do here also with a world of one-to-one correspondence. Even the one-sided formalism of Hilbert is left behind here and semiotic or logistic movements in consciousness, unilaterally having their perceptual or conceptual contents, have to be bypassed in terms of two-sided intuitionist mathematics algebraically and geometrically at once. Such consciousness as implied here is neither inside nor outside and yet seems to be in both these places at one and the same time. The alternation between the twin aspects is so rapid as to make it unnoticeable. Electromagnetic pulsations and not biological alternations are analogies that apply here. Manifested objects can emerge to view as if from pure transparent nothingness, or else exist in the form of more subtle inner experience, leaving only a faint impression of a schematic order in consciousness. We are thus in a world of mathematical 'things' or entities and these should not be confused with the mere abstractions of pure mathematics, as pointed out by Hilbert. Only these subtle structural features of intuitionist mathematics, which are only now beginning to be precisely formulated, can lend us any frame of reference for discussing this twin emergence of generic and specific things occupying consciousness in its purest form.
We have already referred to two striking examples from the Upanishads where a mystical rather than mathematical picture-language is adopted. The implications are so subtle that they have eluded the analysis of even the best critical minds.
Sankara in his various commentaries on the Upanishads does not even attempt a structural explanation of them but rather treats them all as revealed versions of scriptural truth not originating in man and therefore to be treated as valid on face value. The modern scientifically-minded person has now the task of making his choice. He can treat this kind of lingua mystica as referring to some sort of "secret doctrine" and therefore impossible to analyze or alternately he can apply to the subject-matter the latest methods of intuitionist mathematics. The structured elements present in this chapter can be treated as a frame of reference with which the further analysis of the characteristics of consciousness would be possible. Although we are here on highly a priori synthetic ground where the Schematismus of Kant holds good, mathematical analysis of its component elements into broad categories in terms of pure thought is not impossible. This is where the lingua mystica and the language of mathesis universalis can step in conjointly to take over the difficult task of the analysis of pure consciousness The very purity of such consciousness makes the participation of mind and matter possible on homogeneous ground with only an infinitesimally small separation between them. When this separation is abolished the Absolute is revealed. Such is the limiting point suggested in the last verse of this chapter.
In the first verse we have to note that nothing of the phenomenal or physical world referring to events or things is allowed to enter the pure and transparent domain of consciousness where the contents of this chapter prove themselves by a double verification taking place within the most abstract of parameters whose plus and minus sides interlace or interpenetrate almost indistinguishably.
2. EPISTEMOLOGICAL REVISION OF SCIENCE
In the Preliminaries we have mentioned the notion of complementarity and indeterminism. Niels Bohr tries to avoid all mysticism, suggesting instead the idea of complementarity. Such a notion is supposed to regulate the relations between physics and metaphysics in the same way as psychology and physics complement each other. While he insists on avoiding mysticism he is inclined to go further than most modern psychologists in suggesting a drastic revision of physics, so as to make it capable of facing the new problems of relativity and quantum mechanics.
Heisenberg enters more deeply into the nature of the paradox. He says that modern progress in science has made man face himself as a problem to himself because the positive direction of progress reveals certain limitations beyond which it cannot go. Just as the compass of the ship refuses to point north, but instead turns towards the iron of the vessel, Heisenberg imagines humanity caught between the horns of a paradox or dilemma whereby, as he puts it, "the image of the universe, according to the science of Nature, ceases to be, in its proper sense, the image of the universe, according to the sciences of Nature" (see page 132 of Volume I) Heisenberg also attempts elsewhere a structural classification of the various branches of physics. (see page 131 of Volume I). They all belong to four epistemological orders, requiring a "fifth" class or ensemble which would include all of them together under one grand and inclusive ensemble, or class of classes. These classifications however remain very vague in their outlines, and Heisenberg is not able to find a place for Schrodinger's equations or structural models. He is thus almost on the same vague ground as mysticism, and the help we can derive from Heisenberg is not very definite.
Schrodinger on the other hand boldly seeks out new epistemological avenues to save science from its present impasse. Regarding the "hopeless conflict" between Berkeleian idealism and its uselessness for understanding the real world, he says:
"The only solution to this conflict, insofar as any is available to us at all, lies in the ancient wisdom of the Upanishads." (see page 204)
Schrodinger's position seems to go further than most of his fellow scientists in the matter of agreeing with the overall position found in this chapter. It is pointed out by J.P.Vigier that De Broglie places matter and mind together on neutral and common ground in the context of the substance of Descartes (see pages 289- 290). One common feature of the position taken by all modern scientists is the adherence to Eddington's subjectivism, selectionism and structuralism. De Broglie's own position is not very clear and so far we have been unable to find in any of his writings anything to shed light on this. All these leading scientific thinkers are for a thoroughgoing revision of the epistemology of science. The normalization of this chapters refers directly to this problem.
3. THE STRUCTURE OF INTUITIONIST MATHEMATICS
Whatever the status of ultimate physical reality is according to the revised epistemology of physics, an irreducible paradox is bound to persist at its core. The particle and wave have to be reconciled by the correct structural forms which are neither matter nor mind. Complementarity might give a central place to cause as an all-comprehensive concept. The principle of incertitude is capable of dissolving this paradox into less ambiguous terms. In the Cartesian notion of "Substance" the two aspects of cognition and extension do not get fully abolished except when treated as a mathematical Absolute. In Spinoza's philosophy this unity of Substance in spite of its structural duality, is accomplished at a higher axiomatic level.
The Monadology of Leibniz similarly presupposes a special variety of units called monads where the paradoxical implications are transcended by "sufficient reason" and "pre-established harmony." Liebniz's Monadology is ultimately covered by the all inclusive notion of the monas monadum. Kant's ding-an-sich or thing-in-itself is where the epistemological status of ultimate reality attains to its pure limits. The thing-in-itself cannot be known except through the a priori synthetic approach. The categories which are its components are capable of being subjected to a schematic treatment. Spinoza sees the possibility of subjecting his Absolute Substance to the correlated reference of a mesh system where we find the same correlating elements of Cartesian structuralism. We have already covered these matters in detail in other writings. (3)
All we wish to underline here is that a paradox remains irreducibly at the core of the thing-in-itself of Kant and the Absolute Substance of Spinoza. Only a mathematical treatment can accomplish its abolition and dissolution. Intuitionist mathematics or axiomatic thought now makes it possible for us to see how perceptible and conceptual entities could inhere, reciprocally abolishing all taint of duality in one and the same integrated normative notion of the Absolute.
In the philosophy of Kant there is an abstract and pure ground common to mathematics and philosophy. The a priori synthetic and the a posteriori analytic are movements in pure reason and its mathematical and philosophical criticisms. Pure reason does not need any support or confirmation from things, entities, percepts or even concepts. The thing-in-itself has a status having nothing to do with the world of physical experience.
The characteristic of Kantian thought in relation to mathematical thinking is brought out in the following way:
"Kant's answers to our test questions about the nature of pure and applied mathematics can now be roughly formulated ... The propositions of applied mathematics are a posteriori insofar as they are about the empirical material of perception and a priori in so far as they are about space and time. Pure mathematics has for its subject-matter the structure of space and time free from empirical material. Applied mathematics has for its subject matter the structure of space and time together with the material filling it." (4)
We notice here that between the a priori and a posteriori there is a pure thing-in-itself, giving room for analytic as well as synthetic processes of thought to take place in it with a character which is the same in intuitionist mathematics. Kant's position is further clarified, where rational dynamism can operate without the need of direct sense experience which only serves as the occasion for the acquisition of the knowledge of the implied national dynamism.
We read the following:
It is important to emphasize that Kant does not regard, say, rational dynamics as merely one of many thinkable alternative theories, but as part of that one natural science which is synthetic and a priori - i.e. which is true of the world and independent of sense-experience. Sense-experience in this view is in no way the ground of our knowledge of rational dynamics but merely the occasion of acquiring it. Just as a child learns that a certain answer to a certain sum is correct on the occasion of experimenting with the beads of an abacus, so Galileo acquired the knowledge of the law of freely falling bodies on the occasion of his experiments at Pisa. (5)
Purer mathematical intuition is without any logical or ratiocinative content and could be free from even semantic processes. It consists only of pure semiosis which is outside the scope of the consciousness even of the eternal moment, i.e. the present, which touches the empty thing-in-itself and with which pure intuition is directly concerned. Between the aspects of memory and of the given data of events possible within this pure reasoning process, there is still room for a two-fold reference; the first having its seat deeper in memory, and the second of a more prospective character. Together they make up a closely-linked couple, alternating quickly to blend their respective mathematical content so as to make for the emergence alternately or together of forms or ideas. The horizontal elements entering into the composition of such an interplay between two levels in pure consciousness tend to be eliminated to the extent that mathematical intuition attains to a higher and higher abstraction. The double-sided vestige of ambivalence or antinomy is retained in the form of a subtle mathematical factor referred to as continuity and this is retained in the form of a subtle mathematical factor sometimes referred to as "two-ity". We read the following from L.E.J. Brouwer's paper entitled "Historical Background, Principles and Methods of Intuitionism" appearing in the Oct.-Nov. 1952 issue of South African Journal of Science. Sections from this paper are seen quoted by Kroner, as follows:
"The first act of Intuitionism completely separates mathematics from mathematical language, in particular from the phenomena of language which are described by theoretical logic, and recognizes that intuitionist mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time, i.e. of the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the two-ity thus born is divested of all quality, there remains the empty form of the common substratum of all two-ities. It is this common substratum, this empty form, which is the basic intuition of mathematics." (6)
The further implications of "two-ity" are strikingly brought out by Kroner, where this idea is seen to be the basis of a one-one correspondence that Frege attributes to numbers in general.
We read as follows:
"Just as perceptual objects which resemble each other must be positive or neutral candidates for inexact concepts, so mathematical objects which stand in one-one correspondence must be positive candidates for exact concepts. Resemblance or empirical similarity is quite different from that one-one correspondence or mathematical similarity, which Frege uses in defining number." (7)
We have already cited Poincaré who points out in connection with Cantor's theory of ensembles where one-one correspondence between similar ensembles is to be included as basic idea (see page 130). Here we come up against a subtle structural feature within modern intuitionist mathematics, where algebraic and geometric languages meet to reveal a paradox hiding in the heart of pure mathematics.
Although we have not the space to enter fully into all the implications pertaining to the structure of pure mathematical elements, it is nonetheless fairly simple for the lay reader to see the general plan revealed by the following interaction between so-called proper and improper elements. We now turn to G. Verriest who has this to say:
"In order to give, in all possible cases, a significance to the sum and the product of two elements, one introduces two new elements: the element zero V (which means absence of points, lines and planes) and the universal element U (or the totality of elements, that is to say, space). By definition, the element zero V is contained in any element whatsoever and it is recognized that it has a negative number of dimensions; equally by definition the universal element U contains any element whatsoever and has three dimensions. These two new elements are called improper elements, and the others are proper elements. Let us now suppose that A and B represent, for instance, two straight lines which do not meet in space; one would then have A+B = U and A.B = V and one sees that the sum and the product of the two elements always represent an element which is either proper or improper." (8)
It is not difficult to discover a similar structural plan in Kroner´s book also bringing light on the same distinction between vertical and horizontal aspects called here "perceptual" or "mathematical" objects. They are treated as having no direct participation with each other, but must necessarily belong to the same ground in consciousness where alone mathematics could be possible. This participation cannot be any other than at the element zero V. The sum and the product are both resolved in favour of proper (horizontal) or improper (vertical) elements.
We read from Kroner once again:
"It is natural and easy to extend the unconnectedness-thesis from concepts to objects, statements and theories. We define a perceptual object as one which has only perceptual characteristics, a mathematical object as one which has only mathematical characteristics; and two objects as 'unconnected' if their characteristics are unconnected. Thus mathematical and perceptual objects are unconnected. We define a statement as perceptual, if and only if to assert it is to assign or refuse a perceptual characteristic to one or more objects; and we define it as purely exact, if and only if the concepts which are assigned or refused in asserting the statement are purely exact. We call the assigned or refused concepts the 'constituent concepts' of the statement; and define two statements as unconnected, if and only if their constituent concepts are unconnected. Mathematical and perceptual statements are thus unconnected. Lastly, we call a theory purely exact, if and only if all the statements and all the constituent concepts of the statements are purely exact; and we call it perceptual if one or more of its statements and, therefore, one or more of their constituent concepts are perceptual. Thus mathematical and perceptual theories are unconnected." (9)
Thus the pure philosophy of mathematical intuition tallies with its own experienced or experimental counterpart. The former tends to be axiomatic and descends through various stages of postulates, theorems, riders, lemmas, etc. in order to give logical results of descending certitude. Only hypothetical certitudes are derived from experimental observations which are most certain at the pole of actual things and events having a horizontal reference.
The hypothetical constructions find their place in that intermediate zone where reasoning, descending from axioms, is able to meet the ascending hypothetical constructions. Axiomatic and experimental thinking thus meet and yield an ambiguous certitude in the middle zone where both types of thinking join. The various grades of certitude involved are distinguished as follows:
"To sum up our discussion of applied mathematics: the application to perception of pure mathematics, which is logically disconnected from perception, consists in a more or less strictly regulated activity involving (i) the replacement of empirical concepts and propositions by mathematical, (ii) the deduction of consequences from the mathematical premises so provided and (iii) the replacement of some of the deduced mathematical propositions by empirical. One might add (iv) the experimental confirmation of the last-mentioned propositions – which, however, is the task of the experimental scientists rather than the theoretical." (10)
The present state of intuitionist mathematics is superseding even the formalism of Hilbert as is sufficiently clear from the above-mentioned quotations. It is interesting to note that in the quotation to follow the use of the word "belief", which term is rather unusual for the ordinary scientist to use, as he more usually prefers to be called a sceptic or a man of systematic doubt. An element of faith is necessarily involved when one goes from perceptual physics to conceptual metaphysics. It is intuitionist mathematics alone that can accomplish this transition without violating the requirements of human understanding. In a later quotation we will see Bergson referring to the "faith of the physicist" because even for the theoretical physicist it is necessary to go beyond what is strictly perceptual.
Faith is necessary in order to have a normative reference outside of the perceptual for the adequacy or regularity of thought. We conclude this section with a final quotation from Kroner:
"It is indeed likely that intuitionist mathematics on the lines of Brouwer's programme will continue to flourish, whether his theses are accepted as self-evident insights or not. Many mathematicians are profoundly interested in its problems without being noticeably interested in its privileged status. Belief in the satisfiability of the intuitionist programme has not been shaken." (11)
4. THE PERCEPTUAL AND MATHEMATICAL REALITIES OF RELATIVITY
In the Chapter on Methodology we examined in detail Bergson's revaluation of Einstein's two theories of relativity. These stages of revaluation represent the broad methodological outlines that can apply equally well to relativity in general. When relativism and plurality are subjected to methodic revaluation we pass through more and more radically revised visions of reality where all particulars can be included under many generic terms. What is perceptually visible in the data of the senses can be made more and more abstract to the extent that generalization is pushed backwards to its innermost sources where pure synthetic reason attains a unitive absolutist status.
Bergson has here applied his genius to the revision and revaluation of Einstein's theory which gives primacy to "observation as the final arbiter." Starting from this limiting point where the observable features of space and its shrinking are seen in the light of the hypothesis of the Lorentz contraction, Bergson takes his inquiry through what he calls demi-relativity to a more complete one where there is a bilateral rather than a unilateral application of the equations of Lorentz.
Then he also brings in the notion of reciprocity, taking the place of mere relativity. Two physicists have an equal epistemological status by virtue of this principle of reciprocity applied to each of them. The referred and the referent status of two rival systems are finally shown to be capable of being epistemologically interchangeable.
Bergson's revaluation goes still further. He is able to bring into one and the same structural scheme a perceivable time and an inner-experienced duration, where between them two simultaneities and two successions are involved. Time and space come to have a subtler form of reciprocity. In his final states of revaluation he is able to effect the transition from the physical to the metaphysical through an appeal to the mathematical way of thinking, whereby the same physicist entering a ball and fully convinced of physics when finally shot into outer space becomes convinced of the truth about the unique and absolute philosophical status of time. It is not necessary for us to review again the precise arguments, supported by valid and correct mathematical equations, by which Bergson has reduced multiple and relativistic times to one unique absolute Time given to both common sense and the highest mathematical reasoning. Although Time is the central factor chosen by him for such a methodological treatment, the same treatment in its broad features can also be applied to any other basic pluralistic category. Such an application reduces it to unity and gives it the same unique and unrivalled absolutist status. We have already pointed out how Platonic Truth and Beauty are amenable to this treatment.
We have also seen that phenomenological reduction also permits this kind of reduction by cancellation of counterparts. In trying to apply his absolutist methodology to Einstein's relativity Bergson´s task was highly complicated because he had to keep in mind two theories of relativity. Einstein formulated these two theories at different times and in his unified field theory he went into further mathematical abstractions. For the purposes of our present study we cannot afford to get lost in the complications and ramifications found in these formulations and in Bergson's revaluation. In spite of the complexity of the situation it is still possible for us to look into the four versions of reality with which Einstein and Bergson were concerned. Both use pure mathematics in their effort to establish a unity where diversity prevails. While Einstein prefers to remain an orthodox physicist treating observation as the final arbiter, Bergson prefers to be both a philosopher and a physicist with a revised epistemology.
The mathematical convention acceptable to both of them remains the same. Einstein's prejudice in favour of observables makes him take an altogether different course from Bergson. We shall not attempt to compare or contrast them. As Bergson represents our own point of view in the matter of treating unitively physics and metaphysics we shall try to recognize his four stages of revaluation. In this revaluation he attains fully to an absolutist vision of reality.
5. ELIMINATION OF UNNECESSARY STRUCTURAL ASPECTS
In the present chapter we have pointed out how thought moves in a highly purified mathematical atmosphere where even semantic and logistic aspects of structuralism can be eliminated under the principle of Occam's Razor. The structural details of the colour solid when dealing with phenomenological and other aspects belonging to the Absolute can now be dispensed with.
We are here concerned with more or less hypothetical and theoretical notions, as well as mathematical and subjective abstractions. The greater part of the scaffolding can now be removed, with just enough left for the analysis of the subtler characteristics we are dealing with.
We can forget all about the three dimensions of length, breadth and depth and reduce the world of pluralistic things in space so as to make them refer to one horizontal axis. Time on the other hand is best studied from the inner experience of duration. We leave off the piecemeal approach proper to space where mechanistic division and subdivision are permitted. If we review again the various stages whereby Bergson accomplished the reduction of the multiplicity of time and plurality of space, arriving at a Cartesian frame of reference consisting of two correlates, as adopted even by Einstein, we shall see what is essential for further discussion in this chapter. The fourfold quaternion of complex mathematical elements will suffice for discussing the implications of the similar fourfold division used by Narayana Guru in this chapter.
In the context of Vedantic thought this same fourfold basic structure belonging to the absolute Self is fully recognized in the Mandukya Upanishad. They are referred to as sthula (gross or perceptual), sukshma (subtle or virtual) karana, (causal) and turiya (completely absolute beyond any relativity); All this will be discussed later on. These fourfold aspects refer respectively to three states of consciousness, waking, dreaming and deep sleep. The fourth state (i.e. turiya) attains the Absolute. We also find that in the present chapter Narayana Guru takes these fourfold features in duplicate form from the original fourfold pattern found in the previous chapters on the ontological side. This fourfold structure now tends to have a one-one correspondence with its own counterpart representing the side of the metaphysical. Thus physics and metaphysics become unified.
Bergsonian reduction follows lines similar to what Narayana Guru has adopted. We see however that Bergson employs his own unique picturesque language where he not only talks of microscopic and sub-microscopic beings, but also used the literary device of two-dimensional and uni-dimensional flat animals carrying on imaginary conversations between themselves. His systems of reference are many, where some are called "referring" and others "referred to". There are also privileged systems with fuller and fuller mathematical reciprocity wherein a mythical Paul and Peter are supposed to exist with different degrees of self-consciousness, sometimes falling within the limits of physics while at other times transcending such limits and arriving at the world of metaphysics. Bergson also gives vivid descriptions of models with or without colours, progressively reducing their horizontalized structural implications into a fully verticalized version. In such a version we find even helicoidal structures and vermicular forms, all of which can be safely forgotten in this chapter. Consciousness with its two-sided implications is presented here in pure transparency.
Difference of perspectives in space tend to expand or contract it, while differences in time, experienced within consciousness, lengthen or shorten its duration. What is more, time can eat into space and vice-versa. It is here we have to think of Langevin's famous example of a man shot into outer space who finds himself aging slower than men on earth. This is one of the famous scientific myths now exploded and discredited. Bergson is able to discuss the causes of this fanciful yarn and finally establish a unique and single Time for both common sense and unified Science.
We have however to keep all these matters in mind before we can undertake to distinguish Bergson's fourfold epistemological grades of reality. This is very important to remember. In this same connection it what O.F. Sutton said about mathematicians being no respecters of convention should not be forgotten. Instead such people seem to be more eager to break away from convention than to adhere to it. On the other hand somebody like Bergson fully respects mathematical convention when he is able to equate two privileged systems of reference. He considers both systems to be mathematically interchangeable without violating any convention. He opens up the way to integrating physics and philosophy with the help of mathematics, which must also be given the status of an integrated discipline. It is in this sense that we have characterized the present work as having a fully scientific status.
6. BERGSON´S REVALUATION OF EINSTEIN
Einstein is a scientist who gives primacy to observables. His continuator, Eddington, definitely states that concepts matter. While scientists are making up their minds about this we find Bergson entering the same field as a full-fledged philosopher who wants to be treated neither as a dialectician nor a metaphysician, but as one who respects all the claims of reality under the notion of the Absolute. Even the reality of colour is not outside his scope. We have already explained these matters in Chapter Two. For the purposes of this chapter we have just pointed out that much of what Bergson has already said is not now needed. We cannot however, even at this stage, abolish the basic aspects of the equations and structural features involved here, without which methodology would remain esoteric and merely mystical. Let us therefore say that the Cartesian coordinates, together with the equations that answer to them, constitute the irreducible element of protolinguistic or metalinguistic features that we have still to retain in this chapter.
It is true that the mathematics employed by Einstein in developing his unified field theory, which involves elaborate considerations such as orthogonal n-legs and other complicated structural features of space, is also not strictly necessary for us. A simple quaternion having a duplicated or double aspect with a subtle reciprocity between the physical and metaphysical is all we need for our purposes.
The physical corresponds to what Narayana Guru calls bhanasraya or the perceptual ground of consciousness, and to bhana which is its conceptual and therefore metaphysical counterpart. To abolish the duality between them while still retaining its ontological reality is, as we have said, the terminating limit of this chapter.
7. THE FOURFOLD ASPECTS OF BERGSON´S REVALUATION
In this long and laborious revaluation of relativity in terms of a unique absolutist Time, we have to recognize four distinct stages in which categorical and structural realities within consciousness emerge into the field of contemplative vision. The status of each of these four orders of reality are either actually observable, subjectively or virtually experienceable or have a fully verticalized status in the profounder sense found in pure intuitionist mathematics. The vertical axis of reference is the synthetic a priori within whose amplitude pure mathematical processes move up or down between the limbs of analysis and synthesis All this we have already explained.
We can characterize the first category as consisting of visible lines of light when reduced to its ultimate implications as Bergson does. These lines of light moreover lengthen or shorten according to the Lorentz contraction. The figures that they make with their lines of light are varied according to Pythagorean distances or what Whitehead calls "separation."
The distinctive feature of this first category is its visibility. The possibility of its absorption into its own horizontal negative counterpart with a virtual status as more subjective, and having a mathematical, or abstracted and generalized form, distinguishes the second limb or category.
Coming to the third category, we have to think of the limitation of the instruments used in the Michelson-Morley experiment, as also the equations of Lorentz which may be said to answer to the same frame of reference. Instead of mobility, visibility or elasticity we have to suppose immobility, rigidity and mathematical invisibility. It has thus a virtual or hypothetical status, although it can be included as belonging to physics in that it can be considered, in principle at least, to be perceptual in an extended sense though not capable of being actually perceived. Categories one and two can be placed as having an epistemological reciprocity and interchangeability along a horizontal axis, while the third and fourth would occupy the negative vertical.
We have now to recognize that categories three and four belong to a pure context resembling intuitionist mathematics. Simultaneity and succession of events coexist here without contradiction, because both of them are inner or outer realities moving within the direct and immediate inner experience of a Peter or a Paul. If Peter insists on being a physicist he has the freedom to choose this system in category three.
Category four is on the plus side of the vertical axis and we can think of it as belonging to Paul who has the freedom to treat his own system as a privileged one. Finding himself in a similar situation as his fellow physicist Peter does in his rival system, whatever paradoxical elements might be present in his experience he attributes to the error of judgement of his counterpart, thinking himself to be right from his own standpoint.
Mathematically, however, the two systems are interchangeable. When this last fact is recognized the connection between philosophy and science will also be fully recognized. The eventuality of Paul being converted from his faith as a physicist to the more inclusive one of a philosopher takes place when he enters the ball which will shoot him into outer space where he is supposed to see the inhabitants of the earth age a hundredfold. The interchangeability of the privileged system of Peter and Paul effectively abolishes this myth of early relativity, and as Bergson points out, Paul becomes a philosopher while he is in outer space. The transition between his faith in physics and in philosophy thus takes place en route while he is in the process of change in movement. The subtle epistemological implications of the conversion of Paul are best stated in Bergson's words as follows:
"But once again one cannot express oneself mathematically except in a hypothesis of a privileged system; and the physicist feeling himself free from the hypothesis of reciprocity, once he has paid homage to it in choosing as he liked a system of reference, abandons it to the philosopher and expresses himself already in the language of the privileged system. Taking his stand on the faith of such a physics Paul would enter into the ball and will see en route that philosophy was right." (12)
8. STRUCTURALISM IN THE MANDUKYA UPANISHAD
The resemblance between the fourfold structuralism found in the co-ordinates of Descartes and in the equations of Lorentz and which is also present in Bergson's reduction of relativity into absolutism, is so striking that we now present in full the Mandukya Upanishad, because it is fully pertinent to the same structural problem at hand.
The first verse of the Upanishad begins with the mystical syllable AUM, and in a later verse (the eighth) there begins a psychological analysis involving a non-objective inner substance. The structure, although proposed in a vague mystical way rather than in mathematical language, nonetheless stands out quite distinctly. Narayana Guru's own ideas in this chapter may be said to have directly or indirectly been inspired by this Upanishad. We read (our own translation) as follows:
1. That (Eternal) Syllable, AUM, is all this: Its further elaboration past, present and future, all is this AUM indeed; even what is beyond transcending the three times, that too is AUM.
2. All here is the Absolute (brahman) indeed; this Self (atma) is the Absolute; this same Self (He is four-limbed (catushpad).
3. In the waking state (He is) overtly conscious (bahirprajnah) having seven parts (13) and nineteen faces. (14) Nourishing (Himself) on the concrete (sthulabhuk) the Universal Man ( vaisvanara), the first limb.
4. In the DREAM state (He), the inwardly conscious (antahprajnah) with seven parts and nineteen faces nourishing (Himself) on the well-selected (praviviktabhuk) is the luminous one (taijasah), the second limb.
5. That (state) wherein., on falling asleep, one desires nothing at all, sees no dream at all, that is the WELL-DORMANT (sushuptam) (which) attaining to a unitive status (eki-bhutah), filled even with a knowing, content (prajnanaghana) made of bliss ( anandamaya), nourishing (Himself) on bliss (anandabhuk), of a sentient mouth (cetomukha) is the knower (prajna), the third limb.
6. This is the LORD OF ALL (sarvesvara), the All-Knower (sarvajna). This is the inner Negation-Factor (antaryamin). This is the Source (yoni) of everything and the beginning and end of beings
7. As not inwardly conscious (antaprajna), not outwardly conscious (bahirprajna), not conscious both-wise (ubhayatah-prajna), as not filled with a knowing content (prajnanaghana), not conscious ( prajna), not unconscious (aprajna), unseen, non-predicable, ungraspable, bereft of quality, unthinkable, indeterminate, as the substance of the certitude of a unitive Self (ekatmaprathyayasaram), as the calmer of the manifested (prapancopasamam), tranquil (santham), numinous (sivam), non-dual (advaitam) is the fourth limb considered to be.
He is the Self (atma) that is to be realized (vijneyah).
8. The same Self treated as the AUM is substance (matra); state is Substance and Substance State; under the letters A, U and M.
9. The A stands for the WAKING state where the Universal Man (vaisvanara) is the first substance because of obtaining (apti) or being the first (adimatva). He obtains all he wants and becomes first too, who understands thus.
10. The U stands for the DREAMING state which is the Luminous One (taijasa), the second substance because of superiority ( utkarsha) or from being intermediate (ubhayatva). He leads wisdom-generations (jnanasanthathi) and becomes one of sameness (samana) too. None ignorant of the Absolute could be born in the family of him who understands thus.
11. M stands for the WELL-DORMANT state, the knower (prajna), which is third because of ascent (miti) or from descent (apiti). He verily ascends (minoti) or descends (into) everything here who understands this.
12. Free from substantiality (amatra) the Fourth is outside discussion (avyavaharya), calmer of the manifested (prapancopasamam), numinous (sivam) is this non-dual (advaitam) One which is even the AUM the Self itself. He enters the Self by the Self who knows thus.
9. DOUBLE CORRECTION AND SCIENTIFIC CERTITUDE
In the Preliminaries we have in a general way examined the question of normalization and certitude. Even simple forms of certitude, such as determining the length of an object by measurement though the use of arbitrarily chosen units of measure, involve the acceptance of the a priori although this is highly repugnant to scientists who insist on giving primacy to observation. All axioms and postulates, without which no formulation of a scientific theory in mathematical terms is possible, involve the a priori which has become inseparable from any scientific method, whether complex or simple. This is not generally admitted by most scientists and they often seem surprised when told that essentially all axiomatic thinking is the same as a priorism. A priorism is akin to dogmatism and therefore is a kind of belief or faith. When such dogmatism is based on religious scripture it becomes highly objectionable and the scientific thinker prefers disbelief.
Where theories involve subtle factors that cannot be directly verified, such as the lines of light in the contraction of Lorentz, they belong less and less within the context of visible verification. This is how myths and superstitions become possible in theoretical science. Some kind of normalized reference becomes necessary for giving certitude a scientific character. A double correction from the sides of the theoretical as well as the practical has to be applied together or alternatively so as to achieve scientific certitude. We have seen how Eddington refers to the example of Procrustes in trying to explain his idea of a normalized science. We have ourselves referred to the more simple yet striking example of the Pythagorean theorem where certitude is reached from both sides of the knowledge situation.
Certitude does not reside in the operation of mathematical symbols nor in mere visible verification. Both have to meet on a neutral middle ground. An interplay of ascending and descending dialectical processes must meet and neutralize each other. This will yield degrees of apodictic certitude in theories or hypothetical constructions. Moderns like Karl Popper, Vincent Edward Smith, and Paul Freedham seem to accept in one degree or another a scientific method implying a kind of dialectical process (see page 43-49).
Statistical and probability methods which are largely employed in population counting and in defining economic trends are as full of uncertainty as any statement in a fairy tale. Still they pass as respectable scientific verities. Newspapers seem to respect outmoded and exploded theories such as those of Malthus. Superstitions die hard once established. The witch-hunting of the Middle Ages is perhaps the only form of injustice comparable to this superstition in modern scientific thinking which seems to influence so easily both the popular mind and politicians who shape the policies of their country.
All these evils arise from the non-recognition of the need for normalization. One-sided loyalties either to the scientific or the religious attitude spell the same disaster from either end of scepticism or belief.
In the present chapter which is highly mathematical and purely epistemological in status, the paradox is seen to be at its thinnest, yet still persisting at the core of philosophical thinking. It is when perfect balance is maintained between the ontological negative and the teleological positive that a neutral outlook is possible. It is only on the basis of a plain and practical sense that any idea of absolutist normalization can be possible, without religious exaggerations or arbitrary rejection of the a priori. It is with this central neutral normative notion of the Absolute that a unified and scientific character is given to the subject-matter and object-matter of the present work. A direct reference to this method of double correction or normalization is found in Verse 8 of this chapter.
10. MATHEMATICAL AND SCIENTIFIC STRUCTURALISM
Modern writings on the Philosophy of Science concerned with the latest developments in non-classical mathematics are seen more and more to use the term "structure" in a special context of their own. Phenomenology and operationism have adopted the same term in a very extended sense. After the formalism of Hilbert, the intuitionist mathematics of Borel and Brouwer are the latest development in new mathematics. The laws of physics have been classified on the basis of a structural discipline. The possibilities of such a structural comparison or analysis of disciplines more widely distinct than physics is now considered to be a great and interesting possibility.
The "Grand Larousse" has the following description of this new development in mathematical thinking:
"Every representation of a discipline by another puts into evidence a structural aspect common to the two disciplines. It is in part the occasion for a formalizing transcription. The Hilbertian project has exercised an accelerating influence on a complete recasting of mathematics, having for its aim the making out of it a general theory of structures.
It is here a matter of evolution towards abstraction which is in the very nature of mathematical activity. It has above all progressed by the function which has been operative between the theory of groups (ensembles) and the structural studies realized by the algebras (other than the ordinary algebra such as the algebra of quaternions, groups, etc.) Present day topology offers us a striking example of this new tendency. The merit for this general recasting of mathematics accrues most particularly to the French School called Bourbakis." (15)
We see from this how structuralism offers us an instrument wherein different disciplines claiming to be scientific can all be classified, compared and referred to a central structural norm. When we remember together with this feature of structuralism that it necessarily reflects the laws of physics and the logical operations and functions normal to it, extending over the whole domain of natural laws, including even language and logic, we easily see what an important instrument of research it constitutes.
We have already seen how Bergson becomes very enthusiastic about certain aspects of structuralism that are explained in his study on relativity. His enthusiasm sufficiently shows itself when he says:
"We shall then have in our hands a powerful means of investigation, a principle of research of which one could predict even from the present, that the human spirit will not renounce ..."
Even if one aspect of structuralism inspires a philosopher like Bergson, it is not an exaggeration to any that intuitionist mathematics structurally understood holds the key to many an intricate problem.
We ourselves feel strongly in favour of structuralism and we have worked out in greater detail some of the structural implications in our monograph on protolinguistic structuralism with the hope that this will eventually serve, not only as a regulative factor in logic, but also as a framework for linguistic or semantic characteristics. In the monograph we have also referred to the experimental foundations for such a structural pattern. Logistic and propositional calculus have implicit in them certain structural features called matrices, but the rest of logic still uses a language constituting as it were a language to explain language, sometimes referred to as a metalanguage. The possibilities for such a language have been explained by Russell and Whitehead under the sonorous and misleading title of "Principia Mathematica". Leibniz's great dream of a universal mathematics must have been the inspiring factor behind this attempt.
The future of such a mathematical and logical metalanguage is not bright at present. One can even predict from all indications that it is heading towards a blind alley. This is because of its complexity and of innate technical difficulties in the matter of typing and printing a complete set of new symbols never before used. What we have presented as an alternative to this is a simpler approach based on the geometry of algebra, rather than on any form of new algebraic symbolism. When Leibniz spoke of a universal mathematics it was not certain if he visualized the kind of development that Russell and others have in mind. The key to Leibniz's characteristica universalis we have reason to guess lay in another direction. He must have seen in the algebra of geometry the foundations for a universal mathematics.
Instead of a metalanguage it is simpler and more legitimate to think of a structural language or visual version of symbolism constituting a protolinguism at the source of the primary movements in thought, where, as Bergson says, 'the human mind naturally feels at home.'
The formulations of theories can be made to answer to the structural pattern. When the full implications of such a one-to-one correspondence becomes elaborated as we might hope for in the near future, we have a powerful method of double correction and verification. When speculation goes off the mark it can quickly catch itself violating the first principles of the structural pattern and in a reciprocal manner the algebraic language will act as a corrective to the geometric one.
The science of structuralism is completely new and its possibilities and features are still at the stage of trial and error. This is seen from P. Destouches-Fevrier's "La Structure des Theories Physiques". The eminent physicist, De Broglie, who has written an introduction to the work does not undertake to appraise its value and excuses himself as one not essentially qualified as a logician. Nonetheless he does not hesitate to recognize the soundness of the efforts made by Destouches-Fevrier and vouches for the comprehensive criticism and inflexible vigour which throws light on the relations between diverse theories by which it succeeds in bringing out the essentially new traits found in contemporary quantum physics. About the other aspects of this structuralism he says:
"Such studies necessarily involve an attentive examination of the logical structure of these new theories and on this point the preoccupation of Destouches-Fevrier joins up with those of the eminent representatives of mathematical physics and the philosophies of science notably von Neumann, Birkhoff and Reichenbach" (17)
There are many unfamiliar terms developed for the purpose of this new structural approach. Expressions such as composibility, adequacy, universal function, operator, properties of fact, properties of right, subjective logic and four types of schemas, are all of special significance in this new branch of physiological mathematics. We cannot do any more than indicate how the terms are used and also how the general lines of structuralism are presented. It is of particular interest to us to note the fourfold schema at the end. We read the following from this original and thought-provoking work:
"It results from this that the experimental calculus of propositions in the case of subjectivity is altogether different from the calculus of classical logic (valid in the case of objectivity) and constitutes a new logic of complementarity and of subjectivity. Logical calculus constitutes a trellis (or "lattice") that is ortho-complemented, which satisfies the rules of a projective algebra of geometry with an infinite number of dimensions with an ortho-complementarity." (18)
"The examination of other diverse properties of the structure of physical theories furnishes the principles of their classification."
"They are classified in relation to their sizes, a neutral size being a size that is comparable with any other size whatsoever."
"The theories could equally well be classed in relation to evolutionary properties of the operator: 1. non-linear theories; 2. linear theories; 3. theories that are completely linear."
After referring to a normal and general concept of a universal function (f), and how such a normal function could be one and only one for any given physical theory except in an objectivist context, the author continues:
"We have afterwards distinguished between properties of fact and properties of right and have studied in particular the incomposability of right with sizes. In considering sizes in an extended sense one can complete the theory of decomposition of an element of precision (forecasting) in a linear combination of fundamental elements. This has led us to classify subjectivist theories into four types: theories of a vectorial schema, theories of a quasi-vectorial schema, theories having a closed schema and theories having an open schema." (19)
11. TOTAL SPECULATIVE GROUND REVEALED BY STRUCTURALISM
Any philosophy claiming to be both scientific and absolutist at one and the same time, has to face many basic epistemological and methodological difficulties. The Vedanta philosophy is primarily concerned with the Absolute and also claims to use critical or reasoned speculation together with tacit acceptance of certain texts to support its claims. We have within Vedanta widely divergent varieties such as those represented by the schools of Sankara, Ramanuja and Madhva. In this work we are not limited to a philosophical tradition or scriptures like the Vedas, representing only one cultural growth. There are other philosophies and scriptures of the world which claim our equal respect. In India there are also the Buddhist, Jain and South Indian traditions which are not orthodox because they are not unquestionably loyal to the Vedic word.
Of the three main representatives of Vedanta, Sankara's philosophy is the most outstanding because of its uncompromising attitude regarding the Absolute whereby it can hold its own in the light of critical and speculative reasoning. He is not unaware of the use of semantic secrets such as the polyvalence of direct and indirect meanings. Some of his examples and analogies seem to reveal a knowledge of the subtleties of syntactics as well as advanced mathematical concepts like the theory or ensembles.
Sankara has the difficult task of applying his own exegetics to explain the various enigmas found in the Upanishads. His efforts to explain them and sometimes to explain them away reveal a peculiar difficulty inasmuch as some of his analogies and clarifications are themselves at least as vague or mysterious. There is often not much difference between the original enigma and the equally enigmatic clarifications.
How can a good God be held responsible for the evil found in His creation? Is God an instrumental or material cause of the universe? Or is He both causes in one? How are actualities derived from thin and pure epistemological mental factors where the Absolute belongs, without heterogeneity between God the creator and the gross world created by Him? Is the principle of contradiction to be admitted in speculations about the nature of God? Or is this to be transcended by a higher dialectical approach, bypassing contradiction? Are we to give to argumentative speculative reasoning as much importance as to the axiomatic approach, treating the Word of the scriptures as the only arbiter for deciding all problems? Was the world created at one stroke or was it a slow evolutionary process? Is God a person hypostatically seated above His creation in His loneliness and glory? And if so, how does He come to be seated also in the hearts of men and at the core of material creation? How can the hypostatic and hierophantic elements of the sacred be correctly fitted into a unitive overall scheme if God is not to contradict Himself or be reduced to a mere tautological entity? When contradiction or tautology are not dissolved by the strongest of speculative reasonings, how is the basic paradox in the total knowledge-situation finally to be resolved? How is the Absolute, whose last vestiges might refuse to be abolished, to be attained without any taint of ambiguity? These are some of the problems that a Science of the Absolute has to face.
Anyone who reads Sankara's commentary on the Brahma Sutras will see that he uses certain methods which, strictly speaking, are untenable. He says for example that the exception proves the rule and that glaring contradictions found in the scriptures have to be accepted in the name of loyalty to the Vedic word which could never make a mistake. Even outside Sankara´s polemics, we very often hear in discussions of such texts opinions that are clearly contradictory. Again proverbs such as "nothing succeeds like success" and "the proof of the pudding is in the eating" are so fundamental in character that one does not know if a valid proof has been advanced at all. The saying "the exception proves the rule" can be turned around and we can ask if many exceptions do not abolish the rule. Infinite regression is treated as objectionable by all philosophy, yet we do not know why it should be so. There is a basic paradox behind all these questions which is clearly enunciated in the Bhagavad Gita II, 16.
"What is unreal cannot have being and non-being cannot be real; the conclusion in regard to both these has been known to philosophers." (20)
Stated this way the paradox means almost nothing we can lay hold on. It has the vague form of a double assertion of being at the expense of the double negation of non-being. Analyzed in this way we can already see a positive and a negative aspect, where double assertion and double negation can operate in the general context of the quaternion which is the innermost or nuclear form of subjectivist structuralism. This is where mathematics and logic have a common ground.
Subtle semiosis can also move through a priori synthesis and a posteriori analysis in the domain of a double corrective necessary for arriving at absolutist certitude, resulting both from the axiomatic and the experimental sides of the situation.
Even Sankara fails to resolve the residual paradox. Of Sankara's method of dealing with the two aspects of the Absolute, one from the relativistic side and the other from the fully absolutist side, we have in the following selection from his commentary on the Brahma Sutras (III.3.17) a good example. By Sankara´s approach from both sides at once it is seen that the paradox is not completely abolished by his laboured exegetics. The term upadhi (conditioning adjunct) introduced by Sankara does not specially help in the matter of making the paradox altogether disappear. We read the following:
"There is a Self called the living one (the individual soul), which rules the body and the senses, and is connected with the fruits of action. With regard to that Self the conflict of scriptural passages suggests the doubt, whether it is produced from Brahman like ether and the other elements, or if, like Brahman itself, it is unproduced. Some scriptural passages, by comparing it to sparks proceeding from a fire and so on, intimate that the living soul is produced from Brahman; from others again we learn that the highest Brahman, without undergoing any modification, passes, by entering into its effects (the elements), into the condition of the individual soul. These latter passages do not thus record an origination of the individual soul." (21)
Sankara now brings in a questioning sceptic who, after showing his arguments, concludes by saying the individual soul is a product of Brahman. Sankara answers this as follows:
"...For we know from scriptural passages that the soul is eternal, that it has no origin, that it is unchanging, that what constitutes the soul is the unmodified Brahman, and that the soul has its Self in Brahman. A being of such a nature cannot be a product. The scriptural passages to which we are alluding are the following:
The living Self dies not. (Chandogya Upanishad VI.11.3); This great unborn Self undecaying, undying, immortal, fearless is indeed Brahman. (Brihadaranyaka Upanishad IV.4.25); The knowing Self is not born, it dies not. (Katha Upanishad 1.2.18); The Ancient is unborn, eternal, everlasting. (Katha Upanishad 1.2.18) Having sent forth that, he entered into it. (Taittiriya Upanishad 11.6); Let me now enter those with this living Self and let me then evolve names and forms. (Chandogya Upanishadnails. (Brihadaranyaka Upanishad 1.4.7); Thou art that. (Chandogya Upanishad VI.8.7); I am Brahman. (Brihadaranyaka Upanishad 1.4.10); This Self is Brahman, knowing all. (Brihadaranyaka Upanishad 11.5.19).
All these texts declare the eternity of the soul, and thus militate against the view of its having been produced. But it has been argued above that the soul must be a modification because it is divided, and must have an origin because it is a modification. It is not, we reply, in itself divided; for scripture declares that 'there is one God hidden in all beings, all-pervading, the Self within all beings.' (Svetasvatara Upanishad VI.11); it only appears divided owing to its limiting adjuncts, such as the mind and so on, just as the ether appears divided by its connection with jars and the like. Scripture (viz. Brihadaranyaka Upanishad IV.4.5, 'that Self is indeed Brahman, made up of knowledge, mind, life, sight, hearing,' etc.) also declares that the one unmodified Brahman is made up of a plurality of intellects (buddhi), etc. By Brahman being made up of a mind and so on is meant that its nature is coloured thereby, while the fact of its being entirely separate from it is non-apparent.
Analogously we say that a mean, cowardly fellow is made up of womanishness. The casual passages which speak of the soul's production and dissolution must therefore be interpreted on the ground of the soul's connection with its limiting adjuncts; when the adjunct is produced or dissolved, the soul also is said to be produced or dissolved. Thus scripture also declares, 'Being altogether a mass of knowledge, having risen from out of these elements it again perishes after them. When he has departed there is no more knowledge.' (Brihadaranyaka Upanishad IV,5.13). What is meant there is only the dissolution of the limiting adjuncts of the Self, not the dissolution of the Self itself. The text itself explains this, in reply to Maitreyi's question "Here, Sir, thou hast landed me in utter bewilderment. Indeed I do not understand him, that when he has departed there is no more knowledge'), in the words, 'I say nothing that is bewildering. Verily, beloved, that Self is imperishable and of an indestructible nature, but it enters into contact with the sense organs.' Non-contradiction moreover of the general assertion (about everything being known through one) results only from the acknowledgment that Brahman is the individual soul. The difference of the attributes of both is also owing to the limiting adjuncts only. Moreover the words 'Speak on for the sake of final deliverance' (uttered by Janaka with reference to the instruction he receives from Yajnavalkya about the vijnanamaya atman) implicitly deny that the Self consisting of knowledge (i.e. the individual soul) possesses any of the attributes of transitory existence, and thus show it to be one with the highest Self. From all this it follows that the individual soul does not either originate or undergo destruction." (23)
Even the most critical of readers of the above will not be completely free from the same kind of bewilderment expressed above by Maitreyi to Yajnavalkya. If we search for any explanation from Sankara we find two specific factors which he relies on. The first refers to upadhi or conditioning adjunct. This must be distinguished from adhyasa or superimposition. The favourite example of upadhi is the one of a clear crystal seemingly red because of being placed on red silk. If one presses the question further and asks what outside factor other than the all-inclusive Brahman constitutes such a conditioning adjunct, the answer relies on a second factor referred to as darkness or ignorance, sometimes also further qualified as anadi (without beginning). In our terminology this factor of ignorance does not have anything but a negative and horizontal reference., The adhyasa on the other hand being of a more subjective order refers to the vertical axis. Since the two references persist, to that the paradox must be taken to persist also. The more adding of more technical terms by Sankara does not totally abolish the paradox with its implied contradiction.
Another famous speculative argument is contained in the overall reliance on non-contradiction by scripture. Non-contradiction does not by itself constitute sufficient reason enabling us to arrive at a proper conclusion. Many open questions remain non-contradicted. As a result of all these weaknesses in the arguments at least some residual paradox must be said to linger on in the mind of the critical reader. The gap between axiomatic thinking and simple everyday experience remains to be bridged and it is in the region of this hiatus in the total knowledge-situation that structuralism must come to the rescue. It can render great service by fixing the limits of improbabilities and impossibilities.
When something is impossible it is also not probable; but probability can still be expected to be valid unless something is proved impossible. General ideas belonging to the world order distinguished as ritam descend from the more axiomatic pole.
However, satyam or ontological reality is built up from below and so when one meets the other absolute certitude results. These structural peculiarities have first to be kept in mind together with all the other features we have referred to. As in the case of pure numbers in the theory of ensembles neither events nor things need be supposed in a world belonging to the order of the schematismus. Both can co-exist, implying a qualitative vertical succession and a quantitative horizontal contradiction till both are abolished on fully attaining the Absolute. When Sankara in the above quotation takes it for granted that the individual soul can exist without contradiction within the totality of the Absolute, the only plausible reason for the co-existence of two such different things in one is the principle of a single ensemble pertaining to another more inclusive ensemble. Modern mathematical thinking is therefore the only way whereby the various claims in the Upanishads can all be accommodated together. Such schematic thinking also seems to be present in some of the other analogies and subtle reasonings found elsewhere in Sankara´s commentary on the Brahma Sutras.
12. STRUCTURALISM AS IMPLIED IN SANKARA
In the above quotations from Sankara we already notice from his use of semantic analogy to the predicative adjunct of femininity of a man who is cowardly and weak, the attempt to reduce into subjective and qualitative linguistic or mathematical terms what is otherwise objective and quantitative. It is only after such a reduction to a homogeneity between the various factors involved in the visible world that the Absolute as the cause of the gross world can be derived. In Vedanta cause and effect are interchangeable terms as Sankara declares in many places.
Primacy of cause is adopted in Vedantic methodology for the purposes of a stage-by-stage reduction of two correlated conjugates or factors until perfect homogeneity between them is established. It is only when such a mutual transparency has been established between the vertical and horizontal aspects that the lingering paradox can be resolved and fully abolished. This involves a double correction and recognition of full inner reciprocity or complementarity in a perfect homogeneous matrix. The interaction or osmosis between the qualitative and quantitative aspects can take place only when equality of the epistemological status between them has been first established.
The process of further reduction into absolute unity takes place at one and the same time by double negation and double assertion in a more fully logical, semantic or mathematical sense. This kind of stage-by-stage reduction is exactly what is found in Bergson's methodology where the time of common sense agrees with a unique and absolute Time. Passing through stages of demi-relativity to full relativity, and then establishing horizontal reciprocity before attempting to abolish finally the duality between physics and metaphysics, Bergson accomplishes the task of giving an absolutist status to relativity. A detailed methodology, similar to Bergson's has also to be adopted by the Science of the Absolute, so as to bridge the gap between experiments and axioms.
Substituting terms like nescience (avidya) and giving it an eternal status side by side with the Absolute does not finally abolish the duality implied in the paradox. The Bhagavad Gita (III-39) refers to this as "the eternal enemy of the wise", comparing it to an ever-burning fire of passion. In spite of the difficulty of fully abolishing paradox, Sankara in certain places makes bold speculative attempts to resolve the paradox by the use of analogies taken from common experience.
It is not hard to see in this the same intense wish to abolish all vestiges of duality in the context of a fully epistemological version of the Absolute. This is not the same as the Vedanta of Ramanuja and Madhva who are more willing to dilute Vedanta for popular consumption. We will now select a few striking examples from Sankara´s commentary on the Brahma Sutras. We see in them a strong suggestion of schematic thinking on the lines of modern mathematical developments.
I. THE JUGGLERS
In I.1.17 Sankara labours much to establish a unity between the Self consisting of knowledge and the Self consisting of bliss. The Taittiriya Upanishad (II.7) is quoted where it says: 'It (i.e. the Self consisting of bliss) is a flavour; for only after perceiving a flavour can this (soul) perceive bliss.' The problem is that one who perceives cannot be the perceived. Sankara admits that the Upanishads permit the belief in two Selfs, one to be searched for and the other given to natural experience. In trying to reconcile this, he resorts to a device highly reminiscent of mathematical structuralism. We read:
"Nor can it be said that the Lord is unreal because he is identical with the unreal individual soul; for the Lord differs from the soul which is embodied, acts and enjoys, and is the product of Nescience, in the same way as the real juggler who stands on the ground differs from the illusive juggler, who, holding in his hand a shield and a sword, climbs up to the sky by means of a rope…." (24)
The ontological and teleological versions of the same Absolute having their respective positions at the minus and plus poles of the vertical axis are clearly indicated here.
II. THE UMBRELLA MEN
In I.2.11 there is a delicate problem between two grades of paradox found in two analogies in the Upanishads. In one, reference is made to two birds in a tree. The other analogy is where two souls entering the same cave are given equal status by drinking the results of their good deeds. In answer to a questioner who points out that both these examples cannot reveal the same unity, Sankara uses an ingenious example. Though it is taken from common usage in language, in the light of modern mathematics we can recognize the same theory of numbers and notion of ensembles implied in this analogy where Sankara employs mathematical abstraction and generalization of groups or sets. The exact quantitative number of people holding the umbrella is brushed aside as irrelevant to the situation in the analogy below. It refers to the umbrella-men which is just meant to refer to a group of persons under one heading or as a class.
We read as follows:
"Just as we see that in phrases such as 'the men with the umbrella (lit. the umbrella-men) are walking,' the attribute of being furnished with an umbrella which properly speaking belongs to one man only is secondarily ascribed to many, so here two agents are spoken of as drinking because one of them is really drinking." (25)
III. MENDICANTS AND BRAHMINS
In I.4.16 Sankara is again faced with the difficulty of referring to two grades of created entities with a double aspect implied between them. They are to be traced to the same cause (which is the Absolute) where there is no duality. Sankara resorts to another device where two ensembles have a one-to-one correspondence between them.
"The passage therefore sets forth the maker of the world in a double aspect, at first as the creator of a special part of the world and thereupon as the creator of the whole remaining part of the world; a way of speaking analogous to such every-day forms of expression as, 'The wandering mendicants are to be fed and then the Brahmanas." (26)
The theory of ensembles found in the previous example is further elaborated here by Sankara.
IV. THE FALCON
In I.1.27 Sankara explains the apparent contradiction between passages in the Upanishads. One passage puts the heavenly world beyond the earthly one, while the other passage suggests a continuity with the earthly world. Sankara resorts again to common language. Here the visual and structural geometric aspects are more evident. A vectorial space and direction is involved. We read:
"Just as in ordinary language a falcon, although in contact with the top of a tree, is not only said to be on the tree but also above the tree so Brahman also, although being in heaven, is here referred to as being beyond heaven as well." (27)
V. PARROTS AND CAGES
In I.2.7 Sankara is faced with the problem of limited space occupied by the Absolute. Space according to him can be viewed in two distinct ways. In the contemplative context of the Absolute space is not a physical actuality but a universalized mathematical entity where limited space, as in the eye of a needle, is its dialectical counterpart. The worldly way of treating space as physical actuality with local and fixed duration is non-mathematical.
This has to be reduced into a mathematical unity having a universal concrete status. The reference to parrots and cages has the purpose of emphasizing the universalization of the concrete. Absolute space has no limitations and only a schematically revised version of universal concrete space can be accommodated within it without any inner conflict. We read:
"The case is, moreover, to be viewed as analogous to that of the ether. The ether, although all-pervading, is spoken of as limited and minute, if considered in its connection with the eye of a needle; so Brahman also. But it is an understood matter that the attributes of limitation of abode and of minuteness depend, in Brahman's case, entirely on special forms of contemplation, and are not real. The latter consideration disposes also of the objection that if Brahman has its abode in the heart, which heart-abode is a different one in each body, it would follow that it is affected by all the imperfections which attach to beings having different abodes, such as parrots shut up in different cages, viz., want of unity, being made up of parts, non-permanency, and so on." (28)
The same principle of universality of reality is described in the same sutra where a man is referred to as being both king of Ayodhya and King of the World.
Before passing on from the Prologue to the Bhana Darsana we have to note that the "consciousness" of this chapter has a dynamic and epistemological grade of its own. The chapter begins with an interchange of two aspects of consciousness, each having a four-fold character. Even Gestalt Psychology recognizes that a configuration such as that of music is distinct from the instrument, its mechanistic base. There is a quick and almost imperceptible succession whereby essences in consciousness are interchanged as if by a subtle osmotic process where participation is possible between the ontological and the teleological aspects of the Absolute.
By this kind of interaction between the generic and the specific classes, systems, sets or ensembles belonging to each of the four limbs of the double structural pair of references, four orders of reality emerge which will become apparent after scrutinizing the text. What should be further noted in advance is that "consciousness", here does not become so positive as to be the basis for the active ratiocination treated of in the seventh Chapter (Jnana Darsana). Although primacy here is slightly in favour of the ontological side, the attempt is to give both these sides an equality of status as proper to the central position occupied by this chapter in relation to the work taken as a whole. In the last verse of this chapter we have to note especially that this participation is openly declared. The mahavakya (great dictum) sat-eva-tat is stated in the last verse for this purpose. This is done in order to show the transition from the negative to the positive side, which becomes fully accomplished only after we have passed the neutral point between the first and second half of the work. If exosmosis is a process taking place from the negative to the positive, endosmosis is the reciprocal process of the same. These subtle distinctions have to be kept in mind.
 See our "Search For a Norm in Western Thought," Values, Oct 1965 to Nov. 1966.
 S. Kroner, "The Philosophy of Mathematics", Hutchinson University Library, London, 1960, p. 29.
 "Les Nombres et les Espaces", Verriest, Armand Colin, Paris 1956 pp.180-181. Our translation.
 Bergson, Dur. et Sim. p. 109. Our translation.
 "Bright heaven is the head of this Universal Man, the sun is the eye, the wind is the breath, space is the water-making part of the body, supporting earth is the foot and fire is the heart", as suggested in the Chandogya (V, 18.2), where we find two more items: hair compared to sacrificial grass and chest compared to sacrificial space. Contemplative correspondences are here implied, rather than actual entities.
 The five senses of perception (jnanendriyas) viz. hearing, touch, sight, taste, smell, five objective or actual aspects of the organic functioning of the ego (karmendriyas) called speech, handling, locomotion, generation and evacuation; five vital urges (pranas) together with that centre where the impressions meet ' which is called in the mind (manas); the discriminating intellect (buddhi); the sense of individuality (ahamkara) and the relational sense (citta) - thus making nineteen in all.
 Paris, 1963. Our translation
 This monograph is to be appended to Vol. III of this work.
 Intro. To "La Structure des Theories Physiques" by Destouches-Fevrier, P.U.F., Paris 1951 p. ix. Our translation.
 Destouches-Fevrier, p. 330. Our translation.
 Destouches-Fevrier, pp. 330-331.Our translation
 Bhagavad Gita, p.124.
 Ved.Sut. Comm. Sank., Vol.II, pp. 29- 30.
 Ibid. Vol. II, pp. 31-33
 Ved. Sut. Comm. Sank., Vol.II, p. 70.
 Ved.Sut. Comm.Sank., Vol.II , p. 119