We have been dreaming for a long time to create a machine
(read Artificial Intelligence) which will be able to encapsulate human
thinking, which will be able to think in a manner we humans are able to think.
But what if some aspects of human thinking is non computational, by non
computational I mean that something which cannot be achieved in a series of
computational steps or to put it in the language Computer Science, something
for which no algorithm can be written down. Now what are such aspects which
cannot be algorithmically achieved? The answer is any such thought processes
which advances in leaps and bounds instead of in as series of sequential steps,
to think of it any intuitional thought process which comes to your mind is non
algorithmic. In many cases of problem solving scenarios we see that intuition
takes precedence over thoughtful logical approach. Ask any mathematician about
his problem solving approach and he will say that more often than not the
starting point is some conviction or hunch before the systematic approach to a
problem is taken.

The mathematician Roger Penrose in his book Shadows of the
Mind has outlined the proof for non computational process of human mind. What
Penrose has done is, he has used Gödel’s theorem and Turing’s theorem for logic
theory to show that elements of logic as encapsulated in mathematical problem
solving is non computational. Gödel’s theorem basically says that given a
sufficiently complex mathematical system with a set of predefined axioms and
postulates its always possible to create a mathematical statement(read problem)
which is nether provable nor un-provable within the given mathematical system.
Note that this process will go on ad-infinitum, for example you might expand
the given system to include some postulates which will lead to either proving
or un-proving the given problem, however the adding of an additional postulate
will also give us the flexibility to create another problem statement which can
neither be proved nor unproved within the expanded system. Turing’s theorem
answers on of the longstanding problems in computer science – Can any algorithm
be designed which can tell us whether another algorithm with an input can give
us the result in finite number of steps or not. For example if the there is a
algorithm for solving some algebraic equation will the given algorithm will
ever be able to solve a given equation fed to it or not, can an algorithm predict
this.(this is called the halting problem – whether an algorithm will ever halt
or not). The answer to this is “No”; no such algorithm can be made.

Combining these two and with some clever use of proof by
contradiction Penrose has shown that our problem solving process might be non
algorithmic. The proof however covers a narrow domain of mathematical logic and
not all experts agree with it. This conjecture if correct will mean that there
will always be certain aspects of human thought which a machine will never be
able to mimic, since no algorithm can be created for such processes.

But how does the human brain achieve such a feat?? There are
no clear cut explanations for the fact. However some understanding comes from
Neuroscience. It has been observed that human brain consists of a large amount
of cross wiring between different sensory processing centres. For example a
sharp object might not only create a visual representation in the brain but
also create an auditory impression (high pitch) even though no actual auditory
input was there. Think for a moment what comes to your mind when you see the
image of a spike! Because of this cross wiring human mind might be able to
translate a problem from one domain to another. For example a problem statement
might talk about a series of numerical steps, but we might be able to visualize
the problem as that concerning a geometrical figure (say staircase) and voila!!
The solution to the problem turns out to be the properties of that geometrical
figure. Such type of transformations to solve familiar problem are common in
science. These types of transformations may be one of the causes for Non
computational problem solving in human brain. Some others including Penrose
suggest that the non computability arises due to some deeper mechanisms in
human brain which include the effects attributed to quantum physics. This
hypothesis is however not widely accepted as there is a large gap between the
scale of human cells and the scale at which quantum effects have been observed
so far.

In summary if non algorithmic algorithms are indeed proved
to be an integral part of human mind them it would pose a serious limit on how
far the current artificial intelligence programs can go in mimicking the human
mind. Also it’s a serious challenge in the current understanding of the
mechanisms that drive the human brain.

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