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dimension of a logical brane, with apparently the same result but achieved by different means. Treating logical branes as factorized terms of the fundamental product of matrix logic, we determine that the dimension of a brane is 2n-l , n = 1, 2,..., where n is the number of atomic products in a fundamental reduction:

Fundamental Product <ll> <1i><11> . . . . . <11><11><11> '"

,,,<11><ii><11><11> <11><11><11><11><11> <i I><1 I><11 >< I I><1 i:><l I>

i

Matrix dimension . | |

1 i

3 5 7

i

9 i

11 . i

Spins 4 8 12 16 26 24~

The dimension of the branes are odd and each new atomic product adds two dimensions, a pair, to a brane. Coherently, the number of spins needed to form a logical brane is even, with a nibble of spins being added to each new level. If the spins are the electrons of an atom, a 3-dimensional matrix brane can be realized by the oxygen atom 80 ~6, an l 1-dimensional one by calcium 2oCa 4~ Because the covariant logical inferences involve the octet of spins, the 3-branes are important for experimental verification of matrix logic.

THE CAT AND THE LIAR

Those who play with cats must be expected to be scratched

Miguel de Cervantes

THE INTELLECTUAL CATASTROPHES

Science is a relentless assault on human intuition. Common sense has failed us on many occasions in the past, and will most certainly fail us in the future. How often have the ideas and theories we cherish, the feelings and beliefs we hold so dear had to be abandoned and as we have been forced to make a U-turn because the experimental facts reveal that were entertaining a

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wrong idea or premiss. In order not to be trapped in controversies such as whether, and if so, how, human thought can be represented by physical processes, let us consider two fundamental intellectual catastrophes, one occurring in logic and the other in quantum physics, which can help to untangle the paradox of the brain.

We enter the world with an exaggerated belief in truth, na'fvely taking for granted that things can be divided into true or false, and that it is within the power of the human mind to distinguish the true from the false. But as soon as the faculty of logical thinking is developed we come across the Liar paradox, known for centuries. If history is to be believed it was first discovered by a Creatan thinker in ancient Greece, proclaiming

"ALL CRETANS ARE LIARS". Everybody lies, said the Liar. If everybody lies, the statement of the Liar is false. In other words, the Liar, a Creatan, is lying. But if he is lying, then the Creatans are not liars. Then the Liar tells the truth, then all Creatans are liars, and the vicious circle begins to revolve all over again, never ending.

The Liar paradox has a long and distinguished curriculum vitae. Such is the power of this intellectual catastrophe in undermining the fundamental axiom of the mind that it still reverberates to the present day. There has been great deal of discussion of the Liar in the literature. Numerous papers and books have been written about the paradox [Ref 48]. A countless number of mathematicians, logicians and philosophers, fools and geniuses, have thought about it with amazement or indignation, claiming from time to time that a solution has been found. Russell's paradox in the set theory and G6del's incompleteness theorems are technical mathematical versions of the Liar, and in fact have been inspired by it. GOdel paraphrased the paradox with the statement "This sentence cannot be proved true". If the sentence is correct, then it cannot be proved to be correct. Patrick Grim was first to analyze the Liar in terms of fuzzy set and chaos theory [Ref 27]. We have made an attempt to describe the paradox in terms of negative logical values, treating it as a dynamical oscillation process in time where self-referential deduction periodically enforces two mutually excluding and incompatible conclusions [Ref 88, 91 ].

Counterintuitive as it is, we have no choice but to conclude that the Liar must be both true and false! The paradox forces consciousness into an alternating loop of reasoning which does not halt, oscillating indefinitely between the true and the false. Consciousness fluctuates. Finite reasoners such as ourselves are generally smart enough to break out of such a vicious circle rather than to continue in it forever. This, however, is not a solution but a defeat. The paradox forcefully tells us that our intuition is fundamentally at fault. Different answers will be given to the same question in different instants of time. Binary intelligence fails to accommodate the Liar. In a major contradiction to the basic axioms of classical logic it is

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possible to deduce from the same initial data a proposition that is both true and false. But even if one abandons the Law of Excluded Middle and pursues the intuitionistic approach, the question of finding a fundamental explanation of the paradox remains unanswered.

The events take an unexpected turn when it is realized that the mystery of the Liar can be explained by another mystery from the seemingly unrelated field of quantum mechanics, the SchrOdinger cat paradox. The latter concerns the coherent superposition of macroscopically distinguishable states, known as cat states. Progress in science is often based on connecting different fields previously considered to be unrelated. Such unification is particularly relevant in brain science, presenting physics with not only formidable technical difficulties, but also a major conceptual challenge. While naturally a physical system, the brain is at the same time essentially an information-logical system. Therefore any meaningful study of the brain cannot be based on physical considerations alone, be they classical or quantum, but necessarily has to include the logical cognitive degrees of freedom, fundamentally entwined with its physical mechanism. In quantum theory unitary time development can result in macroscopic quantum superposition or SchrOdinger's cats, which must then somehow be explained away. The cats have been of considerable interest, but the quantum superposition of macroscopically distinguishable states is very difficult to obtain. We can push up the classical-quantum border slightly to form for example mesoscopic superposition of atoms but in macroscopic reality we never observe Schr6dinger cats. Schr6dinger cats are made of a very large number of molecules. The interactions with the macroframe causes quantum coherence to decay into a statistical mixture very rapidly, which is used to explain why one cannot obtain a macroscopic coherent superposition of an alive and a dead cat. The difficulties with the Cat paradox are so significant that they have prompted proposals to modify quantum theory, despite all its unquestionable experimental success. Wigner suggested that the failure to obtain macroscopic quantum interference is due to a nonlinear term unaccounted for in the Schr6dinger equation, which cancels out the coherence between macroscopic states. Mesoscopic coherent interference effects between distinct states of an individual atom have been monitored experimentally but an effective control over the coherent superposition of macroscopic states remains as elusive as ever. From Bohm's hidden variables approach to Gell-Mann- Hartle's consistent histories many unsuccessful attempts have been made to make the cats disappear, and one begins to wonder whether physics is on the right track. Instead of trying to get rid of the embarrassment of Schr6dinger's helpless cat, wouldn't it be more logical, without taking the unfortunate cat too literally, to look, instead of Hilbert space, for other spaces where cat states can actually exist? What if thought is a cat state? A Schr6dinger's cat in every head? For seventy years theorists and experimentalists have been searching for Schr6dinger's cats in the outer world of quantum mechanics, missing the fact

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the cats are a permanent fundamental feature of the inner world of cognitive logic.

The principal thesis of this section is to show that the Liar paradox is a macroscopic coherent superposition realized by the brain, indirectly reflecting the quantum nature of the elements of which it is made up. The Schr0dinger Cat paradox not only explains the Liar paradox, with both paradoxes seeming to be one and the same thing, but a more general notion of truth is needed, one which allows for partial inconsistency and self-reference. This notion comes from topology. The logical rules of inference are the classical limit of probability rules in which conditional probabilities are reduced to definite Boolean true or false [Ref 89]. The existence of the coherent superposition of incompatible events, persisting up to the point at which a measurement is made, is the major conceptual difficulty of quantum theory. It is commonly believed that this is an exclusively quantum phenomenon without a classical macroscopic analogue. But, as the Liar shows, classical logic itself is beset with conceptual difficulties of superposition, the interpretation of which remains controversial. Because both the Liar and the Cat have an important bearing on the theory of consciousness, it would be wrong to dismiss these issues as merely philosophical. The Cat is an unavoidable consequence of the principle of linear superposition which is central to quantum mechanics. The wavefunction of a quantum system is a coherent superposition of all possible states, including the orthogonal states of the system, which can be mutually exclusive, one the negation of the other. For example, if spin-up and spin-down are the two possible states of a spin-l/2 particle, the particle will also exist in a coherent superposition of both states lup> + Idown>. Projecting such superposition to the macroscopic level leads to absurd situations. In Schr6dinger's cruel gedanken experiment a poor cat is in a box together with a cyanide capsule which may release its content depending on the state of a quantum detector measuring the radioactive decay of some atoms. The laws of classical logic break down in the quantum realm, because both alternatives the cat alive and the cat dead must be present in the state of quantum superposition. For the outside observer, the cat is in a linear superposition of both altematives, which must coexist and interfere with each other. The absence of quantum superposition at the classical level has been a serious problem in quantum mechanics" quantum interference at the microscopic level implies a superposition between distinguishable macroscopic states. Quantum measurement not only detects the pre-existing properties of a measured system, it in part creates in self-referential fashion the properties it reveals. In the macroworld, when observing a nice little cat we can determine that the cat is either perfectly alive and playful with probability p(alive) or unfortunately has played out all her games and dead with probability p(dead). Probability theory demands that

p(alive) + p(dead) = 1.

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However, the Schr0dinger cat, subjected to the strange demands of quantum theory behaves in quite a weird manner, existing in a bizarre coherent superposition of both dead and alive. Laplace, who viewed probability as an improvement on common sense would be disappointed to see the rules of classical probability break down. The complete sample space must include p(alive+dead), which would give meaningless value, no longer confined to the unit interval:

p(alive) + p(dead) + p(alive+dead) > 1.

The C a t cannot be accommodated by classical probability, which cannot exceed unity. Quantum theorists get around the problem by making use of a wavefunction which does not stand for actual physical properties but is merely a tool for calculating probability. Likewise, the treatment of the Liar in terms of classical probability leads to a violation of familiar probability laws. Suppose we characterise the Liar by the conditional probability p(AIB) where A and B are the two intertwined true and false outcomes of the paradox. Then the Liar's conditional

p(A^B) p(AIB) = p(B) ' p(B) , 0,

is not defined for p(B) = 0, yielding the uncertainty 0 . . = .

0" Infinite reality and finite theory differ. One can be sceptical and doubt

the possibility of macrophysical Cats but the Liar is clearly an actual macroscopic coherent superposition. We need just one small but bold logical step to connect the two paradoxes into a single one. To achieve this goal we have to turn for help to topology, which provides the adequate theoretical framework. Consider a bilateral manifold and some system which can reside on either side of the manifold, with probability p(up) on one side and with the

probability p(down) on the opposite side, hence

p(u) + p(d) = I. Suppose now that our system is the Liar (or the Cat) in true state (alive) when found on a given side of the manifold, and in the false state (dead) when found on the opposite side. Without informing the Liar or the Cat, let us twist the edges of the manifold and glue them into a MObius unilateral topology. Since the manifold has 'lost' sides and become nonorientable, the question whether the Liar is truthful or dishonest, likewise whether the Cat is dead or alive, cannot be answered classically. We have changed nothing in

the Liar and in the Cat states. But we have a strange and very real manifold in which the Cat and the Liar are definitely on both 'sides' at the same time, because these now form a convex. We arrive at absurd situations, uninterprctable in terms of classical probability:

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p(true) + p(false)= 1 + 1 = 2. o r

p(alive) + p(dead)= 1 + 1 = 2. But a nonorientable MObius strip is not absurd and does actually exist in the macroscopic world. A new reality, quantum in essence but macrophysical in form, emerges on it. This reality requires for its description neither classical probabilities nor complex probability amplitudes but improbability instead.

Fig. 16 The Liar or the Cat residing on a unilateral MObius strip. For odentable topology the probability of false 0 is complement of the true 1. For nonorientable topology -1 is a complement of

the improbability 2.

The MObius strip, that darling of mathematicians, has been known for century and a half. What we didn't know is that we can use it to describe a macroscopic Schr/Sdinger cat state and the Liar, actually realized as a cognitive thought. A topological approach to consciousness indicates that a very different form of actualization occurs in a self-measuring system. The process of

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conversion from quantum truth to classical truth is tampered with because the observer becomes an integral part of the system and the distinction between the observing and the observed vanishes. In technical terms this entails that consciousness no longer needs to convert quantum amplitudes into certainties, but instead, swaps improbabilities and probabilities. Outside the framework of wavefunction, classical (topo)logical states can get coherently mixed as if they were quantum superpositions. Since wavefunction is not needed to account for such superposition, a fruitful way forward would be to look for other physical principles that can. Such a new principle we find in topology, which provides new clues and ultimately an access to the intractable physics of consciousness, without relegating the mind to mechanical interactions. The affinity of the Cats and the Liar suggests that Schrtidinger cat states exist in the thinking brain, and one paradox can be used to explain the other.

These results also provide a new and unconventional framework for the study of Schrtidinger cat states. Quantum truth is very different from classical truth, lndeterministically it describes often mutually contradictory possibilities. Quantum systems are spread out theoretically over all spacetime. A particle can be both here and there, it is both a particle and a wave. Different quantum truths all coexist simultaneously in wavefunction until the act of measurement selects one of them to be a classical truth. Quantum measurement connects the realm of quantum possibilities into the real of classical certainty. In reverse manner to the decoherence of a quantum system through interaction with the macroworld, a macrosystem can emit coherences, thus creating a cat state. The road between classical and quantum is two-way, with topology providing a framework both for the design of the microphysical Cat and the macrophysical Liar.

THIS IS ABSOLUTELY TRUE+FALSE

Who does not enjoy Shakespeare's drama and poetry? Luckily the great writer lived in more simpler times than ours. In modem culture Shakespeare is replaced by SchrOdinger. "To be OR not to be" gives way to the absurdity of "to be AND not to be". The Liar is the fundamental example of the true+_faise states of logical consciousness. In the context of the two dual theories of the brain, the geometrical and the topological, one can explain the Liar as a SchrOdinger cat which entails the microphysical quantum basis of consciousness. But no less meaningfully one can explain the Liar as a cognitive nonorientable MObius which entails the macroscopic topological foundation of consciousness. The fact that there exist coherent superpositions not necessarily represented by the wavefunction has an important bearing on cognitive logic. MObius states are macroscopic coherent superpositions true_+false. We have used the SchrOdinger cat to explain the Liar paradox. But topology solves both. In the final analysis the new effects, neither classical

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nor quantum, must exist in the topological brain. Topology is critical for the theory of consciousness. Under the usual assumptions of classical logic true and false are separable. Under the usual assumptions of quantum theory quantum information cannot be cloned. An object cannot be observed in two places at once. However, if not all natural processes are governed by unitary evolution, the laws of quantum mechanics will need revision, and consciousness provides a compelling reason for a modification of quantum theory. In topological consciousness a single percept can be cloned into two, signifying the emergence of a cognitive MObius.

Fig. 17 Cognitive two-in-one 'two-slit experiment'. A finger will be cognized as two fingers, if placed between the crossed middle and index fingers.

QUANTUM MATHEMATICS

Truth is stranger than fiction.

George Byron

We equate truth with mathematics. The incredible, almost mystical power of mathematical deduction has puzzled many thinkers. In Maxwell's field equations, Einstein's mass-energy relation, Schr0dinger's equation of motion, Wigner's irreducible group representations and many other instances physics celebrates the magic of mathematical consciousness. Whether mathematical truth is in the outer world or the inner world of abstractions, once the axioms are correctly guessed, the mathematical structures built around them obtain a kind of life of their own. Symbols can be trusted, and we rely on the ability of formalisms to describe the known effects and to predict the unknown. When a mathematical description of some phenomenon is written down, we expect it to tell us things we did not envisage initially.

Physics is digital. A powerful numerology is built into physical systems. A proton weighs 1836 times as much as an electron; there are 6 different flavours of quarks, the second orbit of an atom is saturated with precisely 8 electrons, the charge of a quark can be in absolute value only 1/3 or 2/3 of the charge of an electron. This list can go on and on. The interaction of

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fundamental particles such as protons, neutrons, quarks and leptons can be quantified digitally in terms of a certain set of quantum numbers, changing to another number set. The quantum world is almost like a computer which, receiving an input set of data, processes them to give us an output number. At the fundamental level, the world, which we perceive as classical and continuous, is in fact discrete and essentially numerical. It is possible, if only in theory, to assign to each and every particle a distinct set of quantum numbers, and this is what quantum physicists actually do to identify a particle, and in doing so reduce the whole complex physical theory of the universe to a digital computation. There will be, however, a catch if objects and states exist which refuse such straightforward digitizing.

In searching for absolutes, mathematics provides us with a powerful abstract means of acquiring knowledge about the real world. There is a sense of mystery about this strange ability of mathematics to tackle and reveal the laws of physics, sometimes without any actual physical tests. Even more so because serious difficulties exist in the foundations of mathematics, since Cantor gave his famous diagonal proof concerning the properties of infinite sets. Classical intuition often breaks down when we enter the domain of the infinite. In a sense mathematics is macrophysical science. The true results of mathematics, as opposed to those of physics, cannot be ambiguous. It is a common belief that mathematical statement necessarily ought to be definite, and whenever an undecidable result occurs we should discard it as invalid and inconsistent.

Is there a new lesson for us to learn? Open any standard textbook on mathematics, and find the following innocent-looking series, known to mathematicians since 1703:

1 - 1 + 1 - 1 + 1 - 1 + . . . This infinite series is a sequence of units alternating in sign. The simplicity of the series, however, is misleading. If we place brackets the following way

( 1 - 1 ) + ( 1 - 1 ) + ( 1 - l ) + . . . = 0 the sum of the series will be nil. But if we place brackets differently, the sum is unity"

1 + ( - 1 + 1) + (-1 + 1 ) + ... = I. The textbook will tell us that the series does not converge, but oscillates back and forth between 0 and 1, never reaching a limit. A question which naturally presents itself is how it is possible that one and the same sequence of numbers can yield a different result, simply because one chooses a different way of placing the brackets. How can the result of mathematics depend on one's mental choice? How it is possible that the result of a precise mathematical computation depends on how one decides to look at it? Is mathematics physical or mental? Is it deterministic or nonclassical? What would happen if the strict provisions of two-valued classical logic were not built into our reasoning?

Consumed by the overriding concern for the macrophysical nature of mathematical results, we lost track of the possibility of a quantum mathematics, a mathematics where mutually excluding possibilities may exist in perfect

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harmony in the domain of the infinite. Indeed, if we abandon the finite classical mode of reasoning, we can see the series as converging to a quantum limit of coherent superposition, like the SchrOdinger cat. From the classical standpoint the series is in a nonconverging endless process of oscillation, but in quantum mathematics, classical in form but due to the infinity factor quantum in result, the series converges to a different kind of dual number which is defined by at least two ordinary numbers. Instead of saying that the series does not converge, we can say that it does converge but to a superposed (dual) quantum number:

1 + ( - 1 + 1) + ( - 1 + 1)+ ... =~ s. In this conceptual switch we treat a number not just as a value but also as a process. The terminology suggests that superposed numbers are essentially quantum-mechanical. Although the quantum-mechanical level seems to be suitable, the superposed numbers exist in the macroworld as one easily obtains coherent superposition macroscopically on a MObius strip without wavefunction. The series does not converge in the orientable world, but can converge in a nonorientable topology. An adequate mathematics is needed to gain an understanding of the outer world of physics or the inner world of the mind. The exact sciences are based on the commonly reaffirmed premiss that physics obeys certain mathematical rules, and does not simply conform to mathematics because we have been able to make increasing use of it. The theory of the brain is no exception. The study of consciousness is becoming less and less the subject of philosophy and more and more the subject of exact science. The problem of consciousness can be only grasped with appropriate math. There was a time when the whole of mathematics was just the few first natural numbers, until today in French the number eighty is denoted as quatre-vingts, the 4 times 20. But then the notion of negative numbers evolved, leading later to the notion of zero number. Many centuries passed before the concept of a complex number, then the quatemion, then the octonion and the matrix emerged. This prompts the question whether consciousness has sufficiently evolved mathematically to understand itself or do we still lack a critical mathematical insight?

Is there an actual reality corresponding to quantum mathematics? What kind of reality can correspond to numbers which combine mutually excluding values? If one had time to carry out infinite tests, one could not exclude the possibility that consciousness would converge to a quantum limit of coherent superposition of the two classical states. Topology indicates that quantum mathematics is relevant to the real world. For example, there seems to be no essential difference between odd and even numbers, but if numbers are counted as actual physical particles, bosons or fermions, the difference between the odd and even becomes fundamental. Is there a supersymmetric theory which is in a position to treat both sort of numbers in one unified framework? On a MObius strip there is no problem with quantum mathematics. We exist on 3-brane, but there is no guarantee that it is not part of some hyperdimensional topology we unaware of.

I

Mental brackets

Superposed numbers exist not only in quantum mechanics but also can be found in the macrophysical world of consciousness. Consider an examples from psychology indicating the reality of superposed numbers. Suppose we want to answer a question: whether the line in the following figure outlines a face or a section of a vase:

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Seeing is believing. Depending on how we place attention's mental brackets, we may see the face or the vase. At the same time we are well aware that a single line runs through the picture. We have one gestalt but with two different meanings, attending to one or to the other, one at a time. Although at lower cognitive levels we can separate the two percepts, at higher levels it can be difficult or even impossible. Our cherished true or false is not enough when consciousness oscillates between orthogonal states, producing interference patterns out of phase with itself. Describing cognitive states by superposed numbers, we can combine mutually excluding alternatives into superposition.

LOGICAL ROTORS

TOPOLOGICAL PHASE AND ATOMIC ORBITS

We suggested that topology is essential in enabling the brain to harness cognitive effects. What are the links of this idea to quantum physics? The topological phase is acquired in logical differentiation and can be quantified as a multiple of the fermionic half-twists, which determines the (topo)logical potential"