Gödel's Incompleteness Theorem: The Most Important Mathematical Theorem in the Twentieth Century

Below is a photo the late mathematical genius Kurt Gödel (left) with Albert Einstein. Gödel's Incompleteness Theorem was the most important mathematical theorem in the twentieth century (its significance is explained in simple terms below). Among other things, Gödel's theorem proves no "Theory of Everything" in physics can ever achieve the status of certainty (this was only recently recognized by Stephen Hawking, as explained by Physicist and Philosopher of Science Stanley Jaki here.



The following material, which explains the mind-bending implications of this crucial theorem in simple terms, is reproduced from http://www.miskatonic.org/godel.html

From Jones and Wilson, An Incomplete Education:

"In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms... of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.

"Gödel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths... It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself."

From Boyer, History of Mathematics

"Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions ... It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced."

From Nagel and Newman, Gödel's Proof

"He proved it impossible to establish the internal logical consistency of a very large class of deductive systems - elementary arithmetic, for example - unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves... Second main conclusion is... Gödel showed that Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set... Even if the axioms of arithmetic are augmented by an indefinite number of other true ones, there will always be further mathematical truths that are not formally derivable from the augmented set."

From Rucker, Infinity and the Mind

"The proof of Gödel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate. His basic procedure is as follows:
1. Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
2. Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.
3. Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."
4. Now Gödel laughs his high laugh and asks UTM whether G is true or not.
5. If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
6. We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").
7. "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."

Think about it - it grows on you... With his great mathematical and logical genius, Gödel was able to find a way (for any given P(UTM)) actually to write down a complicated polynomial equation that has a solution if and only if G is true. So G is not at all some vague or non-mathematical sentence. G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics...

Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final ultimate truth... But, paradoxically, to understand Gödel's proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel's name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it.

From Douglas Hofstadter, Gödel, Escher, and Bach

"All consistent axiomatic formulations of number theory include undecidable propositions... Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved... How can you figure out if you are sane? ...Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no means inevitable. Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is "peculiar' or not, given that you have only your own logic to judge itself? I don't see any answer. I am reminded of Gödel's second theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent.

The other metaphorical analogue to Gödel's Theorem which I find provocative suggests that ultimately, we cannot understand our own mind/brains... Just as we cannot see our faces with our own eyes, is it not inconceivable to expect that we cannot mirror our complete mental structures in the symbols which carry them out? All the limitative theorems of mathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally."

G. W. F. Hegel's Paradigmatic Revolution: An Introductory Outline and Critique

© 2006 by sirhemlock@yahoo.com

I. Hegel in Context: An Age of Revolution

1776 American Revolution

1789 French Revolution spoke eloquently for "fraternity, equality, and liberty." Their voice was raised in unison for human alleviation from the insufferable of the cult of the past. The blood that ran in Paris streets ran because of the tension between Conservativism and Liberalism. The 19h century revolution was heralded by such prophets as Darwin, Comte, and Marx. By 1848 the intoxicating nectar of industrial victory had inebriated all of "Christian" Europe. The strongholds of reaction had fallen! New social structures were to arise.

1804-1815 Napoleonic Wars

1830-1871: Bourgeoisie Liberalism triumphs though Industrial Revolution. No conservative society had ever produced so much in such short time as did 19th century. The First Industrial Revolution produced a wave of opti­mism which heavily influenced developing philosophic, scientific, sociological and other paradigmatic revolutions (progress/development/evolution). Individual man's rational freedom became enthroned not only as an end in itself, but also as the master key to society's achievement. But in front of this silver lining was a cloud. The effect of the Industrial Revolution on man was terrifying. S/he was dehumanized. The new economic and political man was alienated by his act of production. The First Industrial Revolution occurred primarily in England; the Wesleyan revival alone saved England from the moral and spiritual decadence created by the coming of technology and a new power class, the laborer (cf. Timothy Smith, Revivalism and Social Reform).

The potency of the Age of Revolutions produced not only optimism, and terror, but a variety of countercurrents on the Left and on the Right. “For over twenty-five years, from the beginning of the French Revolution in 1789 to the Battle of Waterloo in 1815, Europe was in turmoil. France beheaded her king and queen; many revolutionaries in their turn went to the scaffold too. Spain, Italy, Germany, Austria, and Russia became battlefields. The British Isles were in danger of being invaded, and Britain's fleet fought at Trafalgar the tyrant who had sprung up from from the revolutionary army. After 1815 there was a universal desire for peace and tranquility. The Holy Alliance was organized; Europe sank into reaction, England into a state of conservatism. The abortive revolutionary wave of 1830 did not reach the British Isles. No wonder that in the climate of reaction to the eruptions of revolution and the Napoleonic Wars the theory of uniformity became popular and soon dominant in the natural sciences." –Vel

II. Vital Data:

1770 (August 27): Georg Wilhelm Fredrich Hegel (1770-1831). Oldest child of a minor state official; early scholarship is not remarkable. Interested in history, Greek and Latin literature and theology; began a notebook of copious quotes.

1788 Entered Tubingen (18 years old). It was the wish of his parents that the family should produce at least one preacher.

1790 PhD in Philosophy. Graduation requirement: to write a book and defend it.

1793 Worked in the area of theology. Very unsuited. Certificate from this period indicates that Hegel was talented, but was "mediocre in industry, knowledge, and speaking and philosophy." Holderlin and Schelling - read Kant and Plato together. Known as "the old man” (students entered university at 16 or 17; most did not make it further).

1793-1796 Tutored in Bern­

1797-1800 Tutored in Frankfort. Hegel did much writing and studying during this period; many partially completed manuscripts on religion and economics; attacked Christian orthodoxy; wrote Spirit of Christianity, published in 1917. History and religion dominate in Hegel.

1799 Father died and left him a small inheritance

1800 First systematic statement of his philosophy.

1800 Joins Schelling at Jena (golden period of Jena is over); becomes Privatdocent at the university. Dissertation "On the Orbits of the Planets” (Celestial mechanics; Newton/LePlace) German awareness begins.

1801 Late afternoon 1ecturs on logic and metaphysics, eleven students.

1803 Journal of Critical Philosophy – collaboration with Schelling; Hegel seems at first to be following Schelling, but makes a brak with him.

1805 Appointed extraordinary professor with first stipend (full professor)

1806 Napoleonic campaign and the battle of Jena – ends academic career

1807 The Phenomenology of the Spirit.

1807-1808 Edited the Bamberger zeitung, but disliked journalism

1808-1816 Headmaster at Aegidien gymnasium (high school) in Nuremberg

1811 Married and had two sons (another “natural” son was born in Jena).

1812 Science of Logic (2 volumes)

1816-1818 Professorship at Heidelberg

1817 Encyclopedia of Philosophical Sciences.

1818-1831 Professor at Berlin; recognized leader of philosophic thought in Germany; popularity continually increased during these years.

1821 The Philosophy of Right; from this point on he dedicated himself to his lectures which were later published.

1830 Became rector of the University and was decorated by William III of Prussia.

HEGEL AND HEGELIAN HISTORIOGRAPHY: A SECOND PARADIGMATIC REVOLUTION

I. Three Methods of Writing History (pagination from Hegel’s Phenomenology)

1. Original History. Characterized by brief intervals between event and historical record. Often contemporaneous with the historian. History of this nature carries with it the spirit and life of both event and contemporaneous author.

2. Reflective History. An effort to contextualize history in a greater whole, to visualize one’s part in history.

a. Universal History. Survey of a people, country, or the world (pp. 5,6; 6-9). Larger than original history (e.g. Israel’s history).

b. Pragmatic History. For Hegel the past has no moral basis for the present (pp. 7-9, 10). Cf. collapse of biblical canon and authority in European thought within 50 years.

c. Critical History. Evaluation of truth/reliability of narrative; not history per se (pp. 9;11).

d. Conceptual History. Theme histories (art, religion, law) came close to universal history, but not large enough. Themes artificially imposed on events/actions.(pp. 9-10, 12; cf. Hegel was the fountainhead of comparative religion, history of religion, critical historiography; Wellhausen et al were Hegelian, etc.; cf. trajectory of academic anti-Semitism from Luther to German historical criticism).

3. Philosophical History. Meaning and interrelation of events is not external but internally woven by the intrinsic movement of the Idea in history (pp. 10, 13; removes supernatural).

II. The Course of World History

1. The Principle of Development

a. The purpose of history is development (pp. 68, 95, 70, 99).

b. Meaning in history is found in development (pp. 69, 70, 98). For Hegel anything new was immanent in the past (removes supernatural).

c. Development in history is optimistic, towards “perfectibility.” This is in contrast to a mechanistic view of nature that runs incessantly with no creativity and newness. Creativity is found in the Spirit’s movement in new cultures (pp. 68, 96).

d. The end of history is contained in the possibility of the Spirit’s essence according to dialectic. “That which contains at as urge within itself” (pp. 70, 71; 100, 101). Compare classical liberalism’s inevitability of progress contra sin interferes with progress. Progress demonstrated was used to take the sting out of the Christian doctrine of sin/salvation.

2. The Origin of History:

a. No history exists before the actualization of Spirit in the State (pp. 74, 104).

b. Culture of history of peoples prior to statehood is “pre-history” (pp. 74, 105).

c. History is not the record of familial, patriarchal events (pp. 75, 105).

d. A people’s knowledge (Vedic, Buddhist, Hebrew) of its original knowledge of God before intellectual dimming and perversion do not constitute history (pp. 71-74, 103, 103).

3. Course of Development-Dialectic

a. The World-Spirit is exhibited in the spirit of people as states reaching their development, then perishing once the “goal” is reached. They are but one epoch in the Spirit’s movement (pp. 78, 79, 111).

b. As the national spirit reaches its purpose, another “higher” universal epoch is reached by the Spirit. Each epoch is as a Phoenix – following its destruction, another arises out of the ashes (pp. 87, 120).

III. Reason as a Basis for History

1. Reality is reasonable. Reality and therefore history are intelligible

2.Reality has its being and substance in Absolute Reason (pp. 11, 14).

3. Reason dictates the content and form of reality. Reason also organizes phenomena (pp. 11,14)

4. Reason bridges the gap between the infinite and finite. Hegel attempts to bridge the gap between the subject and object (pp. 13-15, 18, 19).

5. Reason rules the world through "religious truth." e.g. Providence (pp. 14, 15, 19).

6. Providence is non-demonstrable. It occurs in non-universal events. Reason supplants Providence in its shortcomings (pp. 15-19, 20)

IV. The Idea of History and Its Realization:

1. History takes place in Nature/Spirit. Hegel differentiates between.physical time and time of consciousness. (pp. 20, 26).

2. The Idea wills. Idea is the same substance as the will of God. (pp. 21,22,30).

V. The Idea of Freedom. The End of World History

1. Freedom is dependence upon nothing else. (pp. 22, 23, 32).

2. Freedom is the essence of the Spirit. (pp. 22, 23, 32).

3. As Freedom (dependent on nothing else), Spirit has essence being in itself, self contained existence. It "comes to itself, produces itself, makes itself." (pp. 23, 24, 33f).

4. This self-contained existence is the same as consciousness/knowledge of self. (pp. 22, 23,32).

5. Freedom is the end of history, the self-contained Idea (God) seeking to know itself. It is men’s consciousness that they are free (pp. 24, 25, 34, 35).

2. Means to End:

a.This Freedom is internal, its means are external. (pp. 25, 26, 36).

b. Its Means are men's activities of will, evil, passions, private aims, selfish desires. (pp. 25-29, 36-41).

c. Specific individuals are “world-historical individuals” (pp. 38-43, 54 ff). Spirit is the "innermost soul of all individuals." (pp. 40, 56).

d. World-historical individuals satisfy both themselves and world history. Man's activities unconsciously achieve more than their in­tentions - namely the conscious purposes of the Spirit (pp. 43, 60). They are unhappy and expendable like hulls "falling off kernels" once the conscious purposes of the Geist are ful­filled. (pp. 41, 57, 43, 60).

e. The individual's morals are acted out within the dictates of the entire moral system of his culture, "But each individual is also the child of a people at a definite stage of its development" (pp. 37, 38, 52).

f. The content of the moral system arises from the Idea's dialectical movement. (pp. 38, 53).

g. Content not only has its source in Idea, but it changes along with the Idea. The new moral universal is "potentially present, not actually present." (pp. 38, 53).

h. Moral content changes with the Idea and is destroyed by world-historical individuals. (pp. 38, 39, 54). No absolute good or evil.

3. The State:

a. The State as realization of the Idea:

1. The State is the external realization of the Idea. (pp. 50, 1, 70-72).

2. The State is not necessarily political but the spirit of the people; their consciousness, their truth. It is the external moral life of the people. (pp. 52, 53, 73).

3. The State is larger than the will of the individual (pp. 49, 50, 69).

4. By obedience to the whole, Freedom is found. (pp. 49, 50, 69).

b. Law as Realization of Freedom:

5. The whole is the law even over the consensus of the majority (pp. 53-56, 75-78).

6. Freedom is not found already inherent in man by nature limited by the state, nor by self-renunciation in external familial structure, but in law/constitution (pp. 53-56, 75-78).

c. Legal Foundation of the State (Constitution):

7. 'The constitution is formulated on the basis of a people's spirit, art, philosophy, imagination, climate (pp. 59, 60, 83).

8. As the culture changes so does the Constitution. "A Constitution is not a matter of choice but depends on the stage of a people's spiritual development (pp. 60, 83), and "Constitutions give us no universally valid basis” (pp. 61, 85).

d. Religious Foundation of the State:

1. The idea of Freedom as the end of history and the means of its realization (subjective will and passion of ind­ividuals) are both united in the State. Freedom is found in the state while passion or will are subordinate to the consensus of the majority, law, and the Constitution of the State (pp. 62, 63, 86).

2. The Idea is expressed in the form of the State and the State's divisional "forms" of religion, art, philosophy and morality (pp. 66, 93).

3. Religion becomes morality. Morality and the State are unity of subjective individual and general will (seen before in Law and Constitution). Both religion and morality are forms of national attitudes (pp. 62, 63, 86).

4. Religion is the reasonable consciousness of the people. that seeks to unify the universal and particular. God is "transcendent" over the particular and "immanent" in the particulars (pp. 64, 88).

5. Religion is the basis of the State, this understanding of unity (pp. 65, 90).

6. From the nature of religion and the morality of the people arise the State and Constitution. (pp. 66, 67, 93) The Spirit of the people/State is thus represented by religion, art, and philosophy (pp. 66, 67,93).

7. National Spirits and their "definiteness" of art, religion, philosophy, science are changing forms of the Spirit in history (pp. 67, 94).

IV. CRITIQUE:

“Hegel’s Philosophy of History is important as a source of much evil, but (I think) of no good... The Absolute Idea is pure thought thinking about pure thought. That is all that God does throughout the ages –truly a professor’s God. Hegel goes on to say: ‘This unity is consequently the absolute and all truth, the Idea which thinks itself...’ There is no contrast of morals and politics, because States are not subject to ordinary moral laws. Such is Hegel’s doctrine of the State – a doctrine which, if accepted, justifies every internal tyranny and every external aggression that can possibly be imagined.” -Bertrand Russell

1. Religion and Providence:

a. The State, for Hegel the perfect embodiment of the Spirit or Divine Idea as it exists in history on earth. The State for Hegel is God. Moses had taught that society was not a law unto itself, but was accountable to the laws of God, a radical idea indeed in and age where the State and its gods were universally viewed as one. Hegel’s view in this respect is a turning back of the clock to ancient Near Eastern models rather than the historically discernable onward evolutionary advance of the zeitgeist he claimed.

b. Optimistic view of man’s perfectibility and “onward and upward” evolutionary thesis (compare 19^th century protestant liberalism’s emphasis on the progress and perfectibility of man) ran aground along lines foretold by Nietzsche, who predicted the 20^th century would be the bloodiest known to man. A century which saw two World Wars, massacres of populations exceeding the half-billion mark (Mao, Stalin), ecological crises, evolution of new technologies whose dark sides at least equal their bright sides (nuclear, biological, chemical, nanotechnology, cybertechnology, etc) all show man’s fall and the world of sin and death to be irrefutable presuppositions of our time. Hegel’s philosophy of perfectibility stands refuted on historical and empirical grounds. “Nor is there any reason, if reality is timeless [as Hegel, held], why the later parts of the process should embody higher categories than the earlier parts – unless one were to adopt the blasphemous supposition that the Universe was gradually learning Hegel’s philosophy” (B. Russell, A History of Western Philosophy, p. 735).

c. Providence as Hegel understands it is limited in its universality; its primary sphere is the advancement of humanity). It is exhibited merely in plants, animals, individuals and a few pivotal events, "When for example, an individual in great perplexity and need gets unexpected help." The Newtonian World Machine model left few chance events and data outside of its umbrella of regularities. Hegel attempted to save any talk of Providence by attaching it to Reason's work in history. Transcendence and a personal God with value for the individual, however, are lost (pp. 15, 20, 18, 24; cf. Kierkegaard’s existentialism was a reaction to Hegel; on his tombstone the inscription reads “That Individual”).

d. Only omniscience can devise a system encompassing all the world, as Hegel attempted to do. One cannot know the Geist is working through them. “The Idea is what Hegel happened to believe. The whole course of the universe is making it just such as Hegel thought it was” (Russell).

2. Deterministic Freedom?

a. History for Hegel is a determinism of thesis, antithesis, and synthesis (intellectual, logical, historical, and cultural determinism). It might seem paradoxical in such a context to speak of freedom, yet Hegel does. As Russell so poignantly observes, “Hegel uses freedom in a very peculiar sense. Freedom is the right to obey the police.” Hegel denies the reality of perfect or pure freedom; freedom such as we have it exists in a spectrum between pure despotism and anarchy. Judeo-Christian philosophy would agree with Hegel’s assertion that there is no freedom without law, but would repudiate the source of law which brings freedom being the state, or the pantheistic zeitgeist.

b. Marx derived his own historical determinism out of Hegel by only a few small but significant alterations. Hence Hegel’s nations becomes classes in Marx; Hegel’s Idea is replaced by methods of production. Hegel and Marx are otherwise virtually identical in most important respects.

c. Kierkegaardian vs. Hegel on freedom.

2. Relativity Thesis:

a. The dynamic (i. e. Dialectical) movement of the Geist in history permeates all categories of Hegel's philosophy (zeitgeist of the times). Its progression leads to new and changing national-spirits, whole cultures, laws, institutions, religions: moralities, etc. Nothing is permanent (cf. Heraclitus’ River). There is no universal moral standard/content except found in the Idea, in dialectic movement. Cultures and their moral bases represent vacillating moral Phoenixes. With so many categories of today’s society in transition (information, education, science and technology, morals, place of residence) it is all to easy to uncritically swallow the Hegelian thesis of universal historical relativity (Sociology of Knowledge Thesis). In our world of ubiquitous change and diversity some things do appear perennially (e.g. C. S. Lewis, The Abolition of Man; Mircea Eliade, The Myth of the Eternal Return: Or Cosmos and History). The fact that cultures do differ in many respects (cultural relativism thesis in anthropology; contra ethnocentrism) is a separate question from whether they ought to differ in every conceivable respect; in fact no one can accept anything and everything which may be done or conceived: relativism is always hypocritical whether in worlds real or ideal.

3. Hegel helps formulate the question, "How does a historian in a later epoch of the Idea's movement understand the events of earlier periods under his relativity thesis?" The his­torian transcends by entering into the "spirit" of the time investigated.

4. An Absolute goal of history is inconsistent with pantheistic philosophy. Indeed history itself becomes problematic in Hegelian philosophy, for there can be no phenomenological access to creation or consummation (cf. linear, teleological history presupposes origin/goal; cf. Eliade, Mircea, The Myth of the Eternal Return: Or Cosmos and History).

Emphasis on self-identity/ class identity comprise the application of "scientific method" to political structure; instrumental in the division of Europe. No nationalism before the 17^th century.