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<div align="center">
<h1>Incompleteness and the Other</h1>
<br>
<h3>Adriraj Talukdar</h3>
<br>
<p>
<b>Incompleteness Theorem:</b> The famous mathematician Kurt Godel demonstrated the
proofs of two very important theorem known as the Incompleteness Theorems. They
are as follows:
</p>
<p>
(1) Any consistent formal system F within which a certain amount of elementary
arithmetic can be carried out is incomplete; i.e., there are statements of the language
of F which can neither be proved nor disproved in F
</p>
<p>
2) For each formal system F containing basic arithmetic, it is possible to canonically
define a formula Cons(F) expressing the consistency of F. Under general
assumptions, this canonical consistency statement Cons(F) will not be provable in F
</p>
<p>
The second theorem is the stronger one since it shows that no formal system can
prove its own consistency and thus it is of more importance to us.
</p>
<p>Godel also wrote on what can be called some sort of a philosophical corollary of the
incompleteness theorem. The first theorem suggests that there is an arithmetic
statement of natural numbers known to be true by intuition or real life experience
which is not provable or disprovable under any consistent system of axioms dealing
with arithmetic of natural numbers. What this suggests is that human mind is itself
capable of solving this arithmetic problem. So is human consciousness a formal
system? If it is considered complete then is it consistent?</p>
<p>
What Godel himself said in 1951 is that <b>either ... the human mind (even within the
realm of pure mathematics) infinitely surpasses the power of any finite machine, or
else there exist absolutely unsolvable diophantine problems</b>
</p>
<p>
Now Godel thought that human consciousness will be fatally irrational if it could ask
questions that it can not answer. So human mind surpasses infinitely the capabilities
of any finite machine using any formal logic. Now the question is if human
consciousness is a formal system, can it prove its own consistency?
</p>
<p>
<b>Consistency of Consciousness</b>: By the second incompleteness theorem, no formal
logical system can prove its own consistency. To show if a system is consistent or not
we have to investigate from outside that system. So how can we think about the
consistency of consciousness? To do that consciousness has to make reference to
itself outside itself, that is, it has to establish a vantage point and project itself from
there so that consciousness can observe its own operations. Do not we do thatalways? Humans are not only capable of thinking, rather they can think about
themselves thinking, they can think about the processes involving thought. Our
consciousness can use itself as an object of its own investigation. But then what is the
vantage point consciousness uses? This investigation can be continued if we look at
the moment when we became conscious, that is, when we realize ourselves as self-
conscious beings.
</p>
<p>
<b>What is I</b>: Now let us dive into a totally different area, that is of psychoanalysis.
Psychoanalyst Jaques Lacan described what he calls the Mirror Stage as the first
instantiation of Self-consciousness. Well, Lacan was not the first person to suggest
such a stage. Charles Darwin observed his 8 months old son to refer his reflection by
his own name. He realised not only that his son recognised his own reflection on the
mirror but even looked at the mirror from distance when called upon by name.
Darwin also observed higher apes could also be surprised by their own reflections on
mirrors but were quickly disinterested by it in contrast to human child who were very
curious. Lacan suggested that children beforehand can not realise their finitude - that
is they think other people and everything else is the same thing and that is themselves
only. But upon seeing its reflection a child realises that it has a finite body and that it
is different from other people and things. And it realises it can think and so can other
human beings. The child realises not only it exists, it can think about its existence -
or in fact the child exists because it can think making its existence distinguished from
others. A child doesn't literally need a mirror. A child knows it has limbs and organs
just like other humans (in contrast to other others - chairs, trees, cats, balls) and can
see its action getting reflected in the actions of other children and adults too. Even a
blind child also goes through same realisation - but this time it does not reflect itself
upon a visual image (on the mirror or on other humans) but by touch and sound.
</p>
<p>
But what this realisation actually needs is not the 'other' to reflect on, it requires the
ability to think about its existence. A child realises its self-consciousness by virtue of
thought. We have the innate ability to think and all thoughts have intentionality - that
is it is always about something. And when we can think about something we can
actually distinguish between things and between me and a thing that is not me. But on
what is thought grounded on? What is the structure of thought? In what form do
thoughts pop out in our mind? That is precisely the role of language. Language bears
the form and the content of thought and thus is the logic of thoughts.
</p>
<p>
<b> Language as the innate logic</b>: Let us first start with the word 'innate'. Linguist and
philosopher Noam Chomsky put out in his linguistic theory that humans are born
with an innate system called the Language Acquisition System. This system allows a
child to learn how to use language. Though society and environment plays a greater
role in learning words and syntax, it is due to L.A.S. that a child can actually learnlanguage. It is widely debated if the ability to acquire language is innate (nativism) or
is it purely learnt from environment (nurturism). But in support of L.A.S. we can
bring up how deaf children who are otherwise unable to learn languages by hearing,
can still develop sign languages and can efficiently communicate with others. A great
example is that of Nicaraguan Sign Language. In Nicaragua deaf children of various
schools developed a completely new creole-type sign language among themselves,
mixing gestures from their homes and classrooms and adding more signs by
themselves. It is argued from this that the ability to grasp language is innate in
childrens' brains. Another argument for innate L.A.S. is that how children can learn
their first language very quickly ( in 2-4 years after birth) without negative example
and poverty of stimulus. They can draw words from a finite lexicon and form
sentences by them in infinite number of possibilities without much external help.
</p>
<h3>Non-contradiction and Negativity:</h3>
<p>
In this sense, that can not be investigated in a purely formal setup. Nevertheless, we
are always using logic, i.e. we are invoking necessary connections between concepts,
notions, statements as we speak or think. For an example, when we say yellow flower
we necessarily imply that the flower can not be uncoloured. Through our logic we set
the rules of reasoning and inference so that more can be deduced from given sets of
notions and informations.
</p>
<p>All of formal logic holds to be true the Law of Non-contradiction: A statement p and
its negation ~p can not be simultaneously true, otherwise everything follows. In
formal symbols, p&~p = False.</p>
<p>We always try to make our arguments consistent or free from contradictions.
However, language, due to its infinite reproduction, can not be confined by this law.
Let us look at an example given by Chomsky to differentiate syntax and semantics:
Colourless green ideas sleep furiously.</p>
<p>We always try to make our arguments consistent or free from contradictions.
However, language, due to its infinite reproduction, can not be confined by this law.
Let us look at an example given by Chomsky to differentiate syntax and semantics:
Colourless green ideas sleep furiously.</p>
<p>Take for another example: I know one thing that I know nothing.Here the contradiction is obvious. But again in use the speaker implicitly clarified
that this very statement is not excluded from knowledge even when they declared
their set of knowledge to be empty. Empty set is implicitly a singleton here.</p>
<p>So our thoughts do not adhere to the Law of Non-contradiction necessarily.
However in our reasoning we try to remove contradiction by added context or other
means. But thoughts are always in motion, they are changed, updated and pondered
over more continuously. Where is that driving force? Here comes the role of
Negativity.</p>
<p>Let us look back at the instance of mirror stage. The consciousness of a child is
undivided, totalising everything that is there for the child. The instance it reflects on
itself its consciousness as if gets split in self (that is reflected) and other (that reflects)
Mind it, self and other are not two different physical objects. These words are simply
used to denote the split, which is fundamental. The split is immanent i.e. that emerges
from within and can not sustain itself. So consciousness must develop in a way that
subsumes, or rather sublates the split. That is because the split is a negation to the
undivided preconscious and thus arises a contradiction. Now we call this emergence
of split as negativity as it begins with a negation of a pre-notion (here consciousness).
If we look at consciousness and its functionality, majorly at the movements of
thought - there arises contradictions - in perceptions or experiences - that is to be
resolved. And this resolu</p>
<h3>The Science of Cognition:</h3>
<p>
Cognition refers to "the mental action or process of acquiring knowledge and
understanding through thought, experience, and the senses". When we observe say a
physical phenomenon - we keep ourselves distant from the sytem itself and the
phenomenon presents itself to us independent of our interpretation of it i.e.
objectively. But when we are going to study cognition or the processes of thoughts
say in another person, their thoughts as they are are not accessible to me and in
whichever way we observe them, they will be subject to our own thoughts and thus
subjective. So a science of cognition can not naively appeal to a logical positivism of
the sciences of nature.
Let us get where we started: if human mind surpasses infinitely the abilities of any
finite machine it is only through eternally continuous self-update by negativity, by
continuous development of thoughts. So the study of cognition has to take into
account a more careful reading of phenomena and conceptions regarding cognitive
processes. In a science of cognition the notions of subjectivity and objectivity loosetheir vivid distinction as the observations are at the same time objective insofar that
the observer observes from a vantage point and also subjective as the observation is
interpenetrated with the observer ( as they think of thought in their thoughts).
So we have to develop a cognitive science whose methodology will take into account
these nuances and will be systematic but not mechanistic. It will be structural but
descriptive as well and above all, a science of cognition will adopt a methodology
that is reflexive rather than finite and reductive and will deal with questions in a more
meta level.
</p>
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