Introduction
Hegel's philosophy is hailed and sometimes discredited because of the celebration over the same reason, as the ultimatum of the western philosophy. Hegel finalized a project which started at the very beginning of the Enlightenment to see the world as a logically definable system. Rene Descartes' system divided the world in to two portions, one is the consciousness and the rest is the extension. Two distinct entities wouldn't be granted as perfect for reasoning since an arbitrary relation or the in the case of no relation at all is not rational, so he had to usher a third party to synchronize worldly matters and it was the God. Spinoza, Leibniz to Wolff the attempt was to picture the most consistent rational model. Emmanuel Kant stepped forward with reconciling rationalism with its best competitor, empiricism, with redefining the prominent philosophical question 'What is the world?' to 'How do we perceive the world?', he omitted the God in its raw figure and resurrected as a priori intuitions - space and time followed by categories of human faculty. In other words, space and time is not empirical, they are innate registers which shape the human perception of the world. This step was a major turning point for the western philosophy, Kant brought the God onto the earth.
But the real revolutionary contribution comes with F.W.G Hegel. Before Hegel's all systems suggested were static, Hegel might have been skeptical about one certain philosophy for the world's being, because throughout the history it has been failed to prove itself as final. His revolutionary attempt was to define the change not the snapshot of the world's existence.
In early decades of twentieth century abundant changes were introduced into mathematics and logic. Classical set theory was mortally attacked by a simple yet startling question, named after its discoverer, Russel's paradox. Sets could be anything, it could be mangoes defined as elements of a set and even all mathematical theories could be defined as a set. When foundation of set theory shatters it literally affects everything. Later another simple paradox was brought into the world of mathematics and logic, called Godel's incompleteness theorems. Those theories pointed out that there is a logical impossibility to achieve such called 'the ultimate truth'. This halted the optimism awoken with scientific discoveries with a sharp existential drawback. What is the purpose of running after the truth? Philosophically it was the question which is risen with the Godel's Incompleteness Theorems.
My attempt here in this paper to apply Russel's paradox and Godel's Incompleteness Theorems to Hegelian Dialectic. I'll argue that the Hegel's Totality is contradictory.
Hegelian Dialectic
Hegel's three-valued logical model is famous in terms of thesis-antithesis-synthesis, but Hegelian original terms were abstract, negative and concrete. Hegel thought that every idea, theory or an explanation is wrong or incomplete, whole history is failure of ideas. For an example Aristotelean explanation for why things fall is that all things have innate tendency to go either upwards or downwards. The speed of the body moves is determined on two factors, the medium it is moving and the weight of it. Galileo later stated that the weight has no effect on the speed of fall. Based on Galileo's discoveries Newton formulated an abstract form of an attraction called gravity as the reason for why things fall. Einstein's General Relativity reformulated the answer with explaining it as a result of the space-time curvature. Such that, theories are not complete, they are either incomplete or false.
Thesis is the basic idea. Since it is false or incomplete there it rises an idea which points out incompleteness and contradictions of the basic idea. The new encounter is the anti-thesis, it contradicts the thesis. Hegel used the term 'Negate' to the anti-thesis. Then two opposite ideas merge and reconcile, omitting contradictions and filling gaps, there it becomes the synthesis. However, as Hegel sees ideas are not perfect in itself, so there should be another contradictory idea for the synthesis, so process continues.
Hegel pointed out that there are 3 laws of this Dialectic process.
1. The law of transformation of quantity to quality: This analogues to Sorites paradox. There is a heap of grains and we remove grains one by one. Each time we take the grains out what it left is a heap again. This continuous process halts leaving one grain or at the state of no grains at all, and at that moment what is there is not a heap at all. The gradual quantitative change suddenly has shifted the state.
Slavoj Zizek used the term Ptolemization(1) to explain a similar historic scenario in the book 'The sublime object of ideology'. Ptolemy proposed the Geocentric model of the universe based on the observation in his time. When new observations found mismatching with the Ptolemy's model they added additional factors to the same model to absorb the new findings. But it has its own threshold, sooner the geocentric model failed and Copernicus introduced a model where Sun is at the center and Earth is rotating around it. Quantity change made an upheaval on existing paradigm to change it to a completely different one.
2. The law of unity of opposites: Very basic rational relationship is the negation, the opposite. Day and night, light and dark are all opposite pairs. Hegel emphasized that one has no existence separately from its opposite. Day has no meaning without the night. Each pairs form unions outside of which neither can exist. The identity of each depends on the identity of the other.
3. The law of the negation of the negation: Thesis suffers with incompleteness and inner contradictions as its antithesis reveals. Thus it negates the thesis, and the synthesis being far more consistent and complete negates the antithesis too. So negation is itself negated.'The real is the rational and the rational is the real.' (Philosophy of Right, Preface) Hegel says and this statement to be rejected or to be accepted, commonly interpreted naively. This statement apparently sets up two sets, one is the 'real' context and the other is 'rational' context. What Hegel stated is that two contexts as equal. But this celebrated quote is loosely understood in that way. We can well define a set of rational- a theory could be stated with using the formal language. But the question is how the set of real would be defined? Real is neither a factual reference to the world nor contains a specific perceivable quality in itself. To clarify it further, mirage and a actual sight of water has no actual perceptional difference, what convince a sight of water in a desert as a mirage is a comparison between two actual perceptions, comparison is rational. Forequoted statement doesn't elaborate a logical relation between contexts of real and rational, it merely defines what is real. Whoever disagrees with this idea should have a different definition for the word 'real'. Then this opponent has been already trapped into fallacious usage of reason, if there is a definition it is indeed rational.
So there exists nothing beyond the rationality, according to Hegel. What exists, to be
specific what that could be experienced as being, is an idea. Ideas are incomplete, the are
constantly evolving to relatively consistent and complete stages. Hegel rejected ideas as
consistent and complete, he believed that only the Totality is true. totality comes with the
aggregation all small elements, but Hegelian view sounds other way around. Totality is
the real and it reveals itself with steps of emergence of ideas. So portions do not define
the totality but the totality defines the portions
.
Hegelian Totality is not an ultimate truth, what he emphasize is that everything could be
structured logically, to be precise everything is structured rationally and human subject
progresses to reveal the Absolute and to be itself.
Russell's paradox
Russell's paradox was introduced by Bertrand Russell in 1901 unleashing a great impact
on Foundations of Mathematics. This paradox exemplifies a contradiction of Naive set
theory created by Georg Cantor.
If we defines a set R containing all sets that are not members of themselves it leads to a
contradiction. If R is not a member of itself then by definition it should be a member of
itself. If R is a member of itself then contradicts with the definition. Symbolically:
This paradox is explained by Russell himself with a simple puzzle known as Barber
paradox.
The barber is the "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber
shave himself?(2)
Later in canonical ZFC set theory this paradox was avoided by introducing set of axioms,
where it doesn't derive that there is a set of all sets satisfying every property.
Godel's incompleteness theorems
Two theorems was proven by Kurt Godel in 1931, showing that Hilbert's program to find
the complete and consistent set of axioms for all mathematics is impossible.
This analogous to the famous Liar paradox. The statement 'I am a liar' is self
contradictory, since the statement is applicable to the statement itself, statement is false
by the meaning of the sentence itself.
Godel's first incompleteness theorem could be stated as follows.
Any effectively generated theory capable of expressing elementary
arithmetic cannot be both consistent and complete. In particular, for any
consistent, effectively generated formal theory that proves certain basic
arithmetic truths, there is an arithmetical statement that is true, but not
provable in the theory.(3)
Godel has proven that the rational systems cannot be both consistent and complete. If
each and every element in the system is related with properly defined relation to each
other some relations is to be found inconsistent. Vice versa if all relations are consistent
there should be undefined relations for some elements.
Godel's second incompleteness theorem
For any formal effectively generated theory T including basic arithmetical
truths and also certain truths about formal provability, if T includes a
statement of its own consistency then T is inconsistent.(4)
Loosely explaining, if a rational system proves that the system itself is consistent then the
system is inconsistent.
Hegelian dialectic, Russell's paradox and Godel's incompleteness
theorems
First question is, how plausible to apply mathematical theorems and concepts on
philosophy. We have to verify the task here as that we are not applying the mathematical
theorems to Hegelian philosophy itself, we are conducting a comparative analysis on
mathematical concepts and on what Hegel suggests. Hegelian dialectic is a meta-logic or
a formulation about logic itself, what we are analyzing here is logic.
Rational system is a string of signifiers, for an example axioms of Peano's arithmetic is a
rational system about Natural numbers, in formal logic axioms could be written as string
of symbols. When history progresses the knowledge at a time could be written as a string
of symbols, simply we take it as an encyclopedia.
- In the encyclopedia if one page shows the Tulip as a fruit and another page shows it as a
flower, system of knowledge is inconsistent.
- If there is a statement as that the car is a automobile and then if there's no definition for
automobile then the system of knowledge is incomplete
In Hegelian sense, the knowledge currently we possess is either inconsistent or
incomplete or both. Knowledge may contain several rational systems such as theories and
ideas. They could be labeled as set thesis and antithesis. They are not unified, that means
they are not referring the same context, one theory could satisfy the electromagnetic
activity and other could satisfy the gravity. In regards to the totality both theories are
incomplete. Holy grail of current physics is to reconcile Grand Unified Theories with
Gravity to explain the universe in one theoretical basis. As well as theories and ideas are
not complete they are mostly contradicting with each other. Again in physics Quantum
mechanics suffers a draw back at String theory and other types of theories are introducing
to the field resolving inconsistencies raised with String theory.
Likewise the knowledge is growing as the history proceeds and finally it should reconcile
everything once and for all and it's the achievement of the totality. As per the foregoing
picture of the encyclopedia the totality is the ultimate rational system, it could be stated as
a string of symbols placed to meet the completeness and the inconsistency.
There we find the logical impossibility of the completeness and the consistency of a
rational system.
- If the knowledge of totality is the set of all elements, subsets and relations, Russell's
paradox exemplifies that the totality set is contradictory with the definition.
- Godel's first incompleteness theorem states that if the system is complete then it's not
consistent and vice versa. If the totality achieves to the acknowledgment of revealing
everything Godel has proved logically that there will be mismatches among facts in the
totality.
- How the totality knows that the history has been ended and the final revelation is
achieved? Thesis contains own contradiction until it's revealed by antithesis. But the
totality is final final rational system, it contains no contradictions. In other words there's
no antithesis outside the total, if there is, then totality will not be the totality anymore. So
the totality has to have a statement in it proving that the totality is consistent in itself. But
Godel's second incompleteness theorem proves that a well formulated rational system
couldn't be included with a statement of its own consistent.
So far we have seen that the Hegelian concept of totality is a logical impossibility. If the
dialect progression converges to the totality and if the totality is contradictory dialectic
suffers an issue at its core. We have seen how ideas reconcile with others to engender
another. We value it as a progression or a development, is that could be right? Hegelian
history is a linear pan out. As Foucault defines the concept of history in a special term
'Archaeology', history is not continues rather it is a set of discontinuous disjunctive
discourses.
Other than that our new understanding stands against concept of omniscient and
omnipotent supreme being, logically there are things far beyond the grasp of every
subject even of an omniscient. Decades ago at the peak of classical physics, there was a
vaunted hope among several physicists of a machine which detects precise coordinates
and momentums of masses of universe and forecast the future wielding laws of classical
mechanics. The paradox is this, if machine literally knows everything machine itself
should be in the domain of it's own observation. Machine knows what it is currently
doing and what it will do in the future. Knowing is different from the object of knowing.
So it's paradoxical to state that one knows what he/she does at the moment. What is being
known is always objective, subject itself is always out of the grasp of the subject because
subject and object logically fails to be the same at the same time.
Conclusion
Hegelian totality is not the future, the idea or the thesis is the totality. If the totality
couldn't include a statement in itself about its own consistency then it is just a thesis.
Thesis is the totality because it is undisputed until it meets with its antithesis.
Bibliography
1.Phenomenology of Spirit, translated by A. V. Miller with analysis of the text and
foreword by J. N. Findlay (Oxford: Clarendon Press, 1977)
2.Bertrand Russell, Principles of Mathematics (1903)
3.http://plato.stanford.edu/entries/g...
බෝඩිමේ හැමෝම නිදි. අළු ගැහුණු තට්ටු ඇඳන් වල සීරුවට ගුලි ගැහුණු උන් නළල් රැලි ගස්වාගෙන බරපතල නින්දක ය, රාජකාරි නින්දකය. අළු පැහැ දුහුවිලි කෑ ෆ්ලොරසන්ට් බට ගැහි ගැහී දැල්වෙයි. මීයන්ගේ දත් පහරින් තුවාල වුණු පරණ විදුලි වයර් ඉඳ හිට ගිනි පුපුරු හලමින් දුමාරය නංවයි. එක සැනෙකින් විදුලි කම්බිවල රබර් ආවරණය ඉරා ගන්නට සමත් වන අවාසනාවන්ත මීයෝ කට කට ගා ගැහී දුම් දමමින් පරාල අග්ගිස්සෙන් බිම ඇද වැටෙති. යාන්තමින් පණ බේරෙන උන් වෙව්ලමින් ඇද වෙමින් ඇවිද ගොස් සපත්තු අස්සේ ගඳ ගසන උණුහුම් තෙත වාත කඳ තුළ සැඟවෙති. සමහරු සපත්තු කටින් එබෙන විට ම ඇතුළේ සිටින එකෙක් දත් විලිස්සා ගොරවයි, ඉතින් උන් අමාරුවෙන් නැවත සපත්තුව බැස අසල ඇති ඇඳ කණුවක තීන්ත තුළට නියපොතු බස්සමින් අතොරක් නැතිව එල්ලා ඇති පරණ කමිස, කලිසම්, යට ඇඳුම් කරා නැග ගියහ. ඔවූහු ඒවායේ සාක්කුවල සහ රැළි අතර දිගු සුසුමක් හලමින් වැතිර ගත්හ. තමා නැවතත් කෑම සොයාගෙන යා යුතු නැත, බෝඩිමේ බැළලියගෙන් බේරී දිව යන්නට වුවමනා නැත, අනෙක් උන් සමග දහදිය වැකුණු සපත්තු මේස් වලට විකාගත යුතු නැත, උන්ට සැබෑ විවේකය දැනෙමින් තිබිණි. බෝඩිමේ ඇඳුම් සාක්කුවලින් සහ සපත්තුවලින් නිතරම දුම් දැම්මේය, ඒවා පුළුටු ගඳකින් යුතු විය.
