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 (
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.
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