Greg Hodes, a fellow interlocutor of our group, recently expressed doubts that Godel had proven his “incompleteness theorem”. Even if that were true, Godel had surely made the last and best effort to do so with help from some later day tweaks by a few other mathematicians. It seems intuitive that a system cannot be self-referential, but for mathematicians a rigorous proof was required. Even without a proof, however, the problem with systems in general illustrated the need for a new way of thinking. The maturation of dialectical materialism in the 20th century grew out of the antecedent problems in philosophy that Godel hoped to transcend with his work in mathematics. The remnants of the philosophical extremes that preceded Godel’s efforts had persisted well past the 19th century and continue to generate confusion today. On one side were the metaphysical Platonists, which included Godel who expected the opposite outcome from his work, and on the other, the empirical Positivists.
While philosophical discussion is usually esoteric, it is often concurrent with concepts in folk-wisdom. One favorite paradox told by atheists that reflects the problem Godel was tackling involved the question of whether god could create a stone she couldn’t lift. In mathematics, the paradox was posed as a question of whether the set of all sets was itself a set, or alternatively, in physics, what the difference is between inside and outside, if a point can be in both places, at a black hole boundary for instance..
In his magnum opus,K. Godel tried to lay the controversy to rest in 1931. While limited to the foundation of arithmetic, the theorem had implications, as if often the case, beyond its intended bounds. Godel’s incompleteness theorem replaces the word logic of the Principia Mathematica with index numbers which he manipulates using Cantor’s set theory in accordance with apodictic Aristotelean logic [uses the law of the excluded middle]. Aristotle contrasted apodictic with its opposite, dialectic, just like a binary “proof” could be opposed to reason. Kant further contrasted certainty [where something either is or is not true] not only with possibility but also with probability. By the mid-20th century, complex analysis, using numbers freed from their imprisonment on the Euclidian plane, was responsible for most advancement in engineering and technology . At least, Godel’s imposition of rigor compelled philosophers to search for an improved mode of thinking.
Godel’s result surely disappointed those who had hoped that a systematic approach to reality was possible. Many philosophers since have developed methods to replace rigid systems that were modeled to resemble Plato’s perfect universe. But theologians, whose livelihoods depend on a metaphysical foundation have generally not done so. Amidst the developing turmoil of contemporary life, it seems like a worthy goal to encourage new ways of thinking , by first eliminating the one that Godel proved problematic if not impossible.
In Austria, Godel’s native country, German [ Hegelian] metaphysics which attempted to explain reality with no prospect of demonstrable proof, had given rise to its opposite, the empiricism of the WienerKreis. Known as positivism, with Kant as their savior, the movement followed the lead of one Ernst Mach. General relativity owes its origin to Mach, but in other matters, his radical empiricism was rejected as an impediment to science by Einstein as well as the Phenomenologists because of the extreme relativity it entailed. Einstein’s key to relativity was the derivation of reality with respect to time. Change requires time, but in a strict metaphysical paradigm time does not exist. The issue of time persisted among Godel’s contemporaries, especially in Heidegger’s ontology which he passed on to Sartre.
What Godel, a confessed Platonist, set out to do was save idealism from the trashbin of history. Ironically, as a member of the WienerKreis, whose group motto was “Nothing Metaphysical”, he was able to sew a thread of absolutism into their radical empiricism. What he really proved was that the only thing that is permanent is “change”. Much esoteric discussion has taken place on this subject, but in an attempt to avoid the infinite regress the theorem entails, different terms are often used or forbidden to make it work.
As an example, the emerging debate among atheists is of particular interest. A prominent atheist, Sam Harris, has recently drawn attention to the concept of free will. As the possible basis of American Law, free will would do well with the backing of a good theorem like Godel’s. Harris concedes that the concept of free will may be a necessary myth, but that history really unfolds deterministically. For the atheist Sartre, on the other hand, people are forced to choose their own trajectory and doing anything less is to relinquish their “authenticity”. Sartre’s theory of consciousness deserves some further attention if clearing the muddled minds of a potential agency is important.
The words that Sartre uses to explain consciousness accept the validity of Godel’s theorem. In order to see a system, one must be outside it. If stasis is the main characteristic of metaphysics, then ex-stasis would characterize an exterior perspective. Using the Greek “ekstasis”, he describes consciousness as culminating in a choice that halts the perpetual transcendence characteristic of the Godel paradox. Turing, as well, defined humanness as an isomorphic algorithm able to halt that very same transcendence because it is self-correcting. Advantage: free will.
Religious adepts are attracted to the ecstasy of transcendence and equate it oftentimes to the work of the soul, a noun which they think must exist as the origin of an action. Whether induced by drugs or meditation, transcendence is an exhilarating high but no substitute for a sustainable solution to the challenges of existence. For Sartre it is the consciousness of consciousness that makes us human. A famous quote by Frantz Fanon illustrates the point from a political perspective. He said, “Human beings who are not considered as such by their masters should not be bound by principles that apply to humanity.” The anti-colonialist movement thus ignored bourgeois rules of engagement to the consternation of international capital.
If what “IS” determines what is known because the constructs of thought as represented most purely in mathematics are not sufficient to encompass reality, then what method should we adopt. Exactly when needed most, the notion that existence precedes essence was actively suppressed in the interests of corralling the masses into a megachurch of mythology. Portraying materialism as “vulgar” materialism, the mythmeisters were bolstered in their stance by a relentless anti-communist campaign that rejected all things Russian. The straw schools of Platonism and Positivism would not have persisted as long as they did if dialectical-materialism had been taken seriously. Sartre managed to make a version of it palatable to the anti-Stalinists, but the original concepts were in no need of exegesis.
A quote from Engels illustrates this point: “In nature there are no irreconcilable contradictions, no forcibly fixed boundary lines and distinctions, and if contradictions and distinctions are met with in nature, it is because we alone have introduced their rigidity and absoluteness into nature”. Even clearer were the positive words of Lenin in Materialism and Empirio-criticism: “Nature, like history, is subject to the dialectical law of motion.” Even though the equally unsatisfactory options of Platonism [and other metaphysical philosophies] versus positivism still lingered in the air, Lenin’s seminal work in 1909 would have by 1931 obviated their needless opposition, but the masterpiece suffered from its association with totalitarian communism and the onslaught of reactionary propaganda.
In his work, Lenin attacked Mach and the positivists. His immediate interest was to reverse the influence positivism was having on his political rivals, but the enduring conclusions of his book gave courage and clear thinking to a century of progressive science. He explains what materialism is not: “Materialism to a Humean must appear to be metaphysics, dogma, a transgression of the bounds of experience.” Hume was not anti-intellectual yet here he was accompanied by those whose doctrine rejected the claims of reason. Lenin emphasized that dialectical materialism is not mechanical materialism. A mind interacting with matter is the endless dialectical nature of science. Addressing the crisis, then current in physics, he blames “the breakdown of the old laws and basic principles and the rejection of an objective reality existing outside the mind [which had led to] the replacement of materialism by idealism and agnosticism”.
All of the foregoing was interesting history by 1970 as the standard theory of quantum mechanics took shape. By then a quark had meaning beyond its “absurdness” intended in the writing of James Joyce, even beyond the mathematical interpretation assembled by Gell-Mann , because it explains what goes on in a particle accelerator. What better proof that a certain way of thinking is appropriate in the navigation of reality.
Finally, it should be noted that the fideists share a fundamental strategy with the atheists. Within the range of their consciousness, fideists are not illogical. They contemplate the one true god, the god of all gods. To do otherwise would expose them as inauthentic, an emotionally distressing condition from which they seek to escape. The absolute which they profess imbues them with faith. Their faith is the foundation of their action, not unlike the faith aroused in a child whose dialectical engagement with gravity eventually leads to an important Human action: walking.
Consciousness of who we are is the beginning of our adaptation to the circumstances of our existence. Facilitating that consciousness is a worthy goal of the intellectual. Godel’s theorem was important, but really, Lenin beat him to the Truth.
A chronology of consciousness: 1909, Lenin; 1910, Russel and Whitehead; 1927-31, Godel; Turing, 1936; Sartre in Being and Nothingness, 1943.
Raymond Smullyan’s popular explanation of Godel’s proof:
Examples of statements that a printing machine might print out.
P*x [means the machine will print x]
NP*x [means the machine will never print x]
PR*x [means the machine will print xx]
NPR*x [means the machine will never print xx]
NPR*FOO [means machine will never print FOOFOO]
NP*FOOFOO [means machine will never print FOOFOO]
NPR*NPR* [means machine will never print NPR*NPR*]
And here is the apodicity of the argument: [see footnote #3]
Either the machine prints NPR*NPR*
Or it never prints NPR*NPR*
If the machine prints NPR*NPR* it has printed a FALSE statement
If the machine never prints NPR*NPR*,
Then NPR*NPR* is a TRUE statement that the machine never prints.
Conclusion: either the machine sometimes prints false statements OR there are true statements that it never prints.
Any machine that prints only true statements must fail to print some true statements.
OR any machine that prints every possible true statement must print some false statements also.
Photo credit: AK Rockefeller