Mathematics and Consciousness




Boole was the first to systemise the Laws of Thought as a discipline, by applying  algebraic methods  to the processes of Thought. Boole may be considered the father  of the mathematics of the mind. In the latter half of the 19th century Mathematical Logic came into its own by the works of Peano, Frege, Pierce and Russell. The first three-quarters of the 20th century was the Golden Age of Mathematical  Logic. It was then that mathematicians intensely worked on the Grammar of Formal Language: Tarski on the Semantics of Formal Languages, Godel on the equivalence of Syntax and Semantics and also on the famous Incompleteness Theorem and Turing on the Theory of computation. There was success everywhere but the price paid for this success was that it all worked only for reasoning in mathematics. There was great  difficulty in  applying  it  elsewhere. Attempts to apply it to the programming of  computers to think,  produced the subject of Artificial Intelligence (AI). Attempts to use logic to analyze everyday language produced the subject of Mathematical Linguistics.  Attempts to mimick the steps through which brain responds to externally generated stimuli  and sends impulses to the rest of the body, resulted, with the help of engineers, in the subject of Robotics.  Applications multiplied: Chess playing, Control systems, Systems management, etc.   This is where we are now.


It is in the wake of all this  that we have arrived at  the modern thought that Mind is a computer and we think we are on the way to understanding the deeper workings of the mind. In the meantime during the thirties of the twentieth century , the physicists came up  with the Quantum Mechanical description of the micro-universe. This description, as we all know, has been so successful as to give us the laser, the transistor, the electron microscope, the super conductor and a host of other things. The philosophical implications of Quantum Mechanics, however, have been breathtaking. Any description of  Reality, says Quantum Mechanics, must always be subjective. The impact of the subject who observes will always persist. Without the interference of the Subject no object at the sub-atomic level can be observed with precision! This complexity of worldview where the observer himself cannot but be an integral part of the system under observation constitutes the most fundamental problem of science that we have brought as a knowledge-legacy into the twenty-first century.


 The wisdom generated by these implications has motivated us to the study of   Consciousness that  seems to be at the basis of everything claimed by the mind  as its own. But even here, it has taken  us almost half a century to come to grips with  the real necessity of the study. As early as the forties of the twentieth century two  mathematicians von Neumann,  and Norbert Weiner and one Biologist J.B.S. Haldane,  pointed out that Quantum Mechanical aspects of nature seemed tailor made for bringing Consciousness back into Science - a position which it had lost after  the advent of the Newtonian era. For almost three centuries, the emphasis continued to be on the pursuit of the understanding of the material universe. It is only since the eighties of the 20th century we have taken the suggestion and started on the quest of scientific understanding of Consciousness.  

All digital computers operate according to algorithms. But man is not just an algorithmic creature. He knows plenty of things which are not algorithmic. Mathematics itself is beyond just an algorithmic exercise.  Godel demonstrated that in any formal system there will be propositions which are true but cannot be demonstrated formally from the axioms of the system. In Penrose’s charter for the study of Consciousness he refers to the non-computational capabilities of the brain and so declares that modern computers and AI  cannot answer the needs. He seems to arrive at this via an application of the Godel methodology of thinking. But his use of Godel  has been seriously questioned by Mathematicians.   His resort to ‘microtubules’ (certain proteins found in all cells, that have useful properties for computation with individual neurons)  as the possible structures or loci for non-computational activity of the brain is not accepted by the biologists. According to them  microtubules can be disrupted by chemical agents without affecting the neural activity. All our evidence regarding consciousness depends upon reports of personal experiences and observation of our own perception, memories, imagery etc. So it appears to lean more towards the psychobiological field.   Understanding Consciousness scientifically seems to demand a good  knowledge, not superficial knowledge, of several disciplines - mathematics, physics, biology, psychology, philosophy and neurology.  The problem of the biologists however, is : How can neural activities in different locations in the brain be components of a single psychological entity?


Science researches into the constancy of Nature through its ‘laws’ that are continuously unravelled by its relentless research.  Reasoning reveals that side by side with the basic existence of the external universe there is a constant change and flux. This paradoxical union of being and becoming may be attributed to what one might call a universal ‘mind’,  the complete ramification of which is what science is all about.  The word ‘mind’ used here is only an imitation of the concept of  ‘individual mind’ which is the other feature of the twin existence of Man and Nature. While the individual mind has the facility of a subjective verification  and ‘experience’  of its own conclusions, the universal mind has no way of revealing itself to us except through human observation, experimentation (which is most often indirect), a rational analysis, a further speculation and further experimentation. The march of this absolutely honest science leads us to the concepts of causality, reductionism which seeks to understand the universe by fragmenting it, the seminal discovery that we are living in an expanding universe, the notion of eternality suggested by the concept of space-time, the possibly non-inert nature of the sub-atomic world, the blurring of the distinction between the observer and the observed in the micro-universe, and the opening out of genetic secrets through the gadgetry of molecular biology but which still is far away from the understanding of the origin of consciousness in life.  In passing we may mention that Hindu philosophy on the other hand starts by investigating the individual mind, but the complete holistic picture emerges only when the infinite character of the universal mind is brought in.


 Quantum Mechanics (QM)  is the crown jewel of 20th century achievements of Science. Penrose thinks that QM  will give the solution to the definition of Consciousness if physicists widen their axiom-base and develop new insights into the nature of the physical world. His  evidence for the non-classical nature of consciousness has to do with a time delay of one and a half seconds between external stimulus and consciously controlled response. But these, the neuroscientists say, can be explained by clever non-quantum classical physiology  and according to them, the difference between such explanations and Penrose’s is not significant. Penrose’s argument for QM, however, is more of a meta-argument. Classical Mechanics considers everything as a simple aggregate of local entities. The whole is just the sum of its parts. You put together all parts of an engine and there is the engine! The same with the computer. But the brain (mind?) is not just the aggregate of its cells. It seems to have an extra quality of ‘beingness’  as a whole. QM provides the framework for this two-level conception of  the intertwined aspects of brain-mind.


On the other hand QM shows extraordinary observer paradoxes. The moment something is observed  (at the micro-cosmic level) that something is not the same thing any more.  To those of us who find it difficult to digest this I usually give the following example: I have a cookie-tin at home with good solid brittle cookies in it. But the tin is so tightly closed that every morning I open it with effort, I disturb the contents of the tin and I never get a whole unbroken cookie. Every morning I take the broken cookies and hope for the best the next day. But the next day history repeats itself!  Something like this happens in QM observation. Observation disturbs the object observed. The principal conceptual difficulty therefore is that  Reality,  if it exists in a unique and determined state,  is only with reference to the observer and his instruments. When it applies to external objects it applies after they have been observed. This is where the observer paradoxes come in the picture. Thus Schrodinger’s cat is both alive and dead at the same time, until the box is opened and the cat is observed. So who is the observer? Does the cat count as one? Why not?  The Wave function of QM is the sum of all possible states or histories of the system. Observation collapses the wave function and brings out one actual unique state. But the transition from the possibilities to observation is not predictable. Any one of the possibilities may become actual. 


 In spite of all this, the metaphysics of QM, with its insight into the role of the observer,  is very relevant to the scientific studies of the subjective aspects of the mind.  There is an ‘observer’ of subjective awareness as all of us can experience. There is an observer of QM which is known by Modern Physics. “How do we associate the two?” is the million-dollar question. Is this association in the field of Science? This is what Henry P.  Stapp  (Berkeley) and Roger Penrose (Oxford) feel. Or is it in the realm of metaphysics? This is what  Stanley A. Klein (Berkeley) feels. The abstract conception is always possible if it is logically consistent. Non-euclidean geometries can be conceived though they cannot be imagined or visualized. Any synthetic proposition can be denied without contradiction. The opposite of any such proposition is conceivable. The fourth Cartesian axis that meets at 90 degrees with all of the other three can be conceived but not imagined. Thus the four dimensional analogue of a cube can be conceived. Mathematicians daily play with such conceptions far more complicated than any of these.  But even the conception that there is a probably infinite dimensional reality that is Absolute Reality, of which, each of us, including the Great Masters, sees only a projection in one or two of our axes, is very difficult, even as a conception - not to

speak of imagination or visualization! But this is exactly what the scriptures seem to say. How  can the cognizer be cognized?  --  says the Upanishad.


What is Reality?  Is it subjective or objective? Can it be both? Do we cognize the world because it is already there or do we create it by our cognition? Are there ever-increasing orders of Reality? Is there a mathematical model? Is Evolution only a biological phenomenon? By penetrating deeper and deeper towards the core of the human soul, does one get close to the comprehension of the Incomprehensible Reality? What kind of testimonies are valid in this  quest? Are Eastern and Western philosophies different in their handling of testimonies? What is Consciousness?  What is the seat of Consciousness? Does Physics  offer the slightest clue on how electro-magnetic vibrations are induced by Consciousness? What physical processes are involved here? In what sense are attempts like Chardin’s Thesis of The Omega Point  (1959) wherein  the idea of a non-physical entity (like radial energy) was used for purpose of guiding future evolution, analogous to the concept of Mahat in Indian Philosophy?  How logical is the conclusion (for example, as in Shadows of the Mind of  Roger Penrose, 1989) that ‘human insight lies beyond formal argument and beyond computable procedures’?


Research into elementary particles have led scientists to the conclusion that the ultimate particle of matter is probably a string as tiny as a length of 10-31 millimetres with no mass but with intrinsic tension and the capacity to oscillate as well as vibrate.  The electron is one mode of vibration of the string, the quark is another mode of vibration. Graviton is still another.  Interaction between particles is a matter of strings breaking into two or joining together or an open string forming a loop. The latest version of string theory is superstring theory. The world's top theoretical scientists are already deep into this field. But the deeper they probe the firmer is their conviction that many advanced tools of Mathematics are needed.  It appears that 'strings' live in a world of ten dimensions. Physicists are very comfortable with four dimensions - three of space and one of time.  But the extra six dimensions that the strings need seem to be lost for observation in a peculiar curling up which is where they need all the mathematics in the world. Much of this mathematics is yet to be developed. 


Is not all this scientific knowledge only paroksha jnana - i.e. knowledge gained by sense perceptioon and logical inference? Descartes, several centuries ago, helped usher in the age of science by asserting 'I think; therefore I am'.  Is it not more correct  now to reverse the words of Descartes and declare: 'I am; therefore I think'!


Whatever school of philosophy one believes in,  the goal of attaining a higher state of consciousness should be the dominant  purpose.  This higher state of consciousness  is not that of the physicist who ‘sees’ the microscopic universe as composed of myriads of ‘quarks’ and ‘strings’ but it is the Consciousness of the mystic  who ‘sees’ the macro and the micro  together in a holistic all-embracing Infinitude.  There can be Consciousness without the brain. To verify this, says the philosopher, it is necessary to isolate the principle of Consciousness from its contents and from its products. Such isolation can only be effected through some kind of mystical experience. This experience can be brought about by meditational practice. Disciplined training of the mind is necessary in order to banish the extrovert nature of the mind. When the individual mind gets into the habit of frequenting this  non-manifest field of existence through Meditation, automatically it also attains a heightened awareness for  its pursuits of material excellence.  To embark on this spiritual journey is the first step towards an enduring solution for the chronic problems afflicting mankind.


Can science deal with questions relating to human consciousness?  Penrose suggests four different viewpoints:


1.                     All thinking is computation; in particular, feelings of conscious awareness are evoked merely by carrying  appropriate computations.

2.                     Awareness is a feature of the brain’s physical action; whereas, any physical action can be simulated computationally, computational simulation cannot by itself evoke awareness.

3.                     Appropriate physical action of the brain evokes awareness, but this physical action cannot even be properly simulated computationally.

4.                     Awareness cannot be explained by physical, computational, or any other scientific terms.



#1 above is the strong view of AI. It assumes  consciousness is the result of the functioning of consciousness.  It is actually at the other extreme of the Vedanta view.  The latter is more aking to #4 above. Advaita Vedanta holds that the existence of pure consciousness (=Atman/Brahman) cannot be validated on the basis of testimonies that are invoked for validating scientific truths. Atman is the spiritual element which coexists with all other states of consciousness of our ordinary existence. Despite its proclaimed immanence by our shastras, there is no mention as to the location of this spiritual element except in a subtle way. According to Narayana Upanishad, the subtle spiritual heart is at the distance of one’s own finger span  (*adho nishhTyA vitasyAnte nAbhyAmupari tishhTati*)below the Adam’s apple and above the navel. The heart suspends like an inverted bud of a lotus (*lambatyAkosha-sannibhaM*). In it (or near it) there is a narrow space and in it ‘everything is supported’ (*tasmin sarvaM pratishhTitaM*). In the middle of that remains a Great Fire that permeates the whole body, there abides a tongue of Fire, of the colour of shining gold, which is the topmost among the subtle, dazzling like the flash of a lightning  that appears in the middle of a rain-bearing cloud, which is as slender as the awn of a paddy grain, and which serves as a comparison to illustrate subtleity.


*tasya madhe vahni-shikhA aNIyordhvA vyavasthitaH / nIlatoyada madhyasthA vidyullekheva bhAsvarA / nIvArashUkavattanvI pItA bhAsvatyaNupamA //*


The paramAtman dwells in the middle of that flame. (*tasyAH shikhAyA madhye paramAtmA vyavasthitaH*).  He is the material and efficient cause of tghe Universe and the Supreme Self-luminous Pure Consciousness. (*so’kshharaH paramaH svarAT*).


Penrose in discussing his four points of view concludes that the AI view (#1) is flawed because it rests its case totally on computational power and algorithmic capability.  However he is not inclined to embrace the other extreme (#4) which is the mystic viewpoint. So he says that a non-computational and non-algorithmic element has to be included in the study of consciousness by the discipline of physics.


Copyright © V. Krishnamurthy  1 Jan. 2007    Homepage