Gödel’s Theorem

Godel and Einstein

Kurt Gödel and Albert Einstein

Kurt Gödel[1] was a Platonist,[2] logician and mathematician who developed the intention of making a profound and lasting impact on philosophical mathematics. His next task was to think of something! Amazingly, at the age of twenty five, he achieved his goal, publishing his incompleteness theorem.

A good friend of Albert Einstein’s, Einstein once said that late in life when his own work was not amounting to much, the only reason he bothered going to his office at the Institute for Advanced Study at Princeton was for the pleasure of walking home with Gödel.

John von Neumann wrote: “Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement.”[3] Continue reading

The Halting Problem – there is, definitively, more to thinking than computation

Alan Turing

Alan Turing

Kurt Gödel’s Incompleteness Theorem was inspired by David Hilbert’s question “Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?” Hilbert played the same role regarding Alan Turing’s proof of the halting problem. Hilbert had asked: “Is there some mechanical procedure [an algorithm] for answering all mathematical problems, belonging to some broad, but well-defined class?”[1] In German this is called Entscheidungsproblem – the decision problem.[2]

Turing found that he could answer this question by framing it in terms of a Turing machine[3] – could there be a program that could determine whether any other arbitrary computer program and input would eventually stop or just loop forever? This was called the halting problem.

“Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist.”[4]

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The Kali Yuga: René Guénon’s Traditionalist Critique of Quantitative Modernity

Guenon 02 Portrait Right-Facing (Larger)

Rene Guenon (1886 – 1951)

The modern anti-modern critique of modernity is by no means a recent phenomenon; it begins rather with the responders to Jean-Jacques Rousseau and his Jacobin followers in the late Eighteenth Century.  It is sufficient in this regard to mention the names of Edmund Burke (1729 – 1797) and Joseph de Maistre (1753 – 1821) and of their successors, S. T. Coleridge (1772 – 1834) and François-René de Chateaubriand (1768 – 1848), to suggest the range and richness of immediately post-revolutionary conservative-reactionary discourse.  In the Twentieth Century, José Ortega y Gassett (1883 – 1955), Oswald Spengler (1886 – 1936), and T. S. Eliot (1888 – 1965), among others, continued in the line established by French réactionisme.  In Ortega’s case and in Spengler’s this continuation entailed incorporating the iconoclastic skepticism of Friedrich Nietzsche into the discourse, with numerous qualifications.  In Eliot’s case, it meant rejecting Nietzsche’s atheism and taking up from Chateaubriand and Coleridge the apology for Christian revelation and for a theological, as opposed to a secular, view of existence.  René Guénon (1886 – 1951) belongs by his dates with the generation of Ortega, Spengler, and Eliot; like Eliot, Guénon is a theist, but despite his favorable treatment of Catholicism he is less identifiable as a Christian than Eliot.  Guénon, who late in life converted to a Sufi-like sect of Islam, sees Catholicism as the vessel of Tradition in the West, but elsewhere Tradition has other forms that are valid in their own contexts.  Spengler’s Decline of the West undoubtedly made an impression on Guénon, much as it did on Guenon’s younger contemporary Julius Evola (1898 – 1974).  Guénon and Evola knew one another and mutually influenced one other.  Both Guénon and Evola together exemplify a branch of modern critical anti-modernism affiliated much more than casually with the Twentieth Century occult revival.

Guénon at one time, in the 1920s, edited the chief French-language occult periodical, La Gnose or “Gnosis.” Yet Guénon, a fierce un-masker of religious mountebanks, can hardly be accused of employing mystic obscurantism to push a doctrinaire agenda.  Guénon’s interest in occult topics, even more than Evola’s, strikes one as rigorous and objective.  As for Guénon’s awareness of ideological deformations of reality, it ran to the acute.  The driving force of deformation, in Guénon’s analysis as in Evola’s, is the stultifying massiveness of modern society, with its conformism on an unprecedented scale, and its receptivity to oratorical manipulation.

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Nicolas Berdyaev on the Despiritualization of the West

Berdyaev 01 Portrait Face Forward

Nicolas Berdyaev (1874 – 1948)

My long-term ongoing project involves reading backwards into the critique of modernity, resurrecting from the archive writers who fifty, seventy-five, or even one hundred years ago, intuited prophetically where such trends as democracy, utilitarianism, and the technocratic conception of science were taking mankind – and who foresaw accurately just how deformed morally and socially Western civilization was likely to become.  The writers in question, with a few exceptions, are today largely forgotten or are remembered under a false image or for spurious reasons.  The names of Karen Blixen, Gustave Le Bon, Jorge Luis Borges, Julius Evola, René Guénon, Hermann Keyserling, Peter Ouspensky, Oswald Spengler, T. Lothrop Stoddard, and Sigrid Undset, among others, have appeared in a series of articles, most of them at The Brussels Journal.  I wish, however, to devote the present occasion to a renewed discussion of the Russian writer-philosopher Nicolas Berdyaev (1874 – 1948), whom the encyclopedias of ideas classify variously, not to say confusingly, as a Christian Existentialist, a Russian Nietzschean, a Neo-Platonist, a follower of Vladimir Solovyev, or an out-and-out mystic and subjectivist.  Berdyaev is perhaps a bit of each of these, while being also much more than any of them.  Academic philosophers have either never heard of Berdyaev or, knowing of him at second hand, perhaps from an encyclopedia article, and being unable to fit him into any Positivist or Postmodern framework, dismiss him summarily.

One might fairly assert that Berdyaev did himself little good publicity-wise by cultivating a style of presentation which, while often resolving its thought-processes in a brilliant, aphoristic utterance, nevertheless takes its time, looks at phenomena from every aspect, analyzes every proposition to its last comma and period, and tends to assert its findings bluntly rather than to argue them politely in the proper syllogistic manner.  In Berdyaev’s defense, a sensitive reader might justifiably interpret his leisurely examination of the modern agony as a deliberate and quite appropriate response to the upheavals that harried him from the time of the 1905 Revolution to the German occupation of France during World War II.  If the Twentieth Century insisted on being precipitate and eruptive in everything, without regard to the lethal mayhem it wreaked, then, by God, Berdyaev, regarding his agenda, would take his sweet time.  Not for him the constant mobilized agitation, the sloganeering hysteria, the goose-stepping and dive-bombing spasms of modernity in full self-apocalypse.  That is another characteristic of Berdyaev – he is all at once leisurely in style and apocalyptic in content.  Berdyaev was quite as apocalyptic in his expository prose as his idol Fyodor Dostoevsky was in his ethical narrative, and being a voice of revelation he expressed himself, again like Dostoevsky, in profoundly religious and indelibly Christian terms.  Berdyaev follows Dostoevsky and anticipates Alexander Solzhenitsyn in his conviction that no society can murder God, as Western secular society has gleefully done, and then go its insouciant way, without consequence.

The titles of Berdyaev’s numerous books, especially when taken in chronological order, tell a story all by themselves: The Meaning of the Creative Act (1916), The Meaning of History (1923), The End of Our Time (1924), Christianity and Class War (1931), The Destiny of Man (1931), The Fate of Man in the Modern World (1934), Christianity and Anti-Semitism (1938), Slavery and Freedom (1939), Spirit and Reality (1946), and The Beginning and the End (1947), among many others.  There is also a posthumous Truth and Revelation (1954).  I call attention to the earliest of the listed titles, The Meaning of the Creative Act.  Berdyaev began his career as a philosophical writer (he never completed his doctorate) with an ambitious study of aesthetics, his theory of which locates the purest manifestation of the highest value of his worldview, freedom, in the labor that generates the work of art and beyond that in all the highest effects of the artwork in its context.  At the end of Berdyaev’s life, he wrote the essays that constitute Truth and Revelation, one of his several ventures into the philosophical-theological sub-genre of theodicy, in which he invokes a “creative response to the appeal of God.”  Whereas in the Catholic and even more so in the Lutheran and Calvinist variants of Christianity there is, according to Berdyaev, a strong “sociomorphic” or “legalistic” distortion of Christian doctrine; in Russian Orthodox commentary, by contrast, “the coming of the Christ has been understood not as a reparation for sin, nor as the offering of a ransom, but as the continuation of the creation of the world and the appearance of the New Adam.”  In Berdyaev’s view, “What God expects from man is not servile submission, not obedience, not the fear of condemnation, but free creative acts.”  Berdyaev adds in an aside that, “I wrote on this subject some while ago in The Meaning of Creativeness,” that is, The Meaning of the Creative Act.  Thus Berdyaev’s work exhibits a remarkable closure, returning at the end to its beginnings, linking as it were its omega with its alpha.

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Moloch is But a Vassal of Our True Enemy

Back in 2010, I commented to a post at VFR:

Nominalism is satanic, I’m telling you. It’s a device to destroy man. Convicted nominalism has to end in suicide, whether cultural or personal. If there are no transcendent values, but rather only and merely our own personal, private preferences, then our personal private preferences are false to facts. This is a little tricky to see, until we draw the analogy to the schizophrenic. The schizophrenic’s impression that there are black helicopters pursuing him are peculiar to him. The black helicopters are not really there. So we understand that his impressions are illusions. But nominalism says that the values we apprehend in things and people and activities, like the black helicopters, are not objectively real. And this means that our feelings of value are—just like the schizophrenic’s black helicopters — hallucinations. They are false. Nominalism says that there is in reality no value out there to be had.

But to say that there is no value really to be found in the world is nihilism. And the consistent nihilist, who has the courage of his convictions, cannot believe that his own life, or anyone else’s life, or the life of his nation, are worth a hill of beans. So he cannot find any way to defend them—none at all. And this will result in death, one way or another, even if only through the sheer lassitude of utter ennui.

I thought at the time I sent that comment to Lawrence, God rest his soul, that in characterizing a school of epistemology as satanic I was perhaps engaging in a bit of rhetorical hyperbole. Firing for effect, as it were.

But then, the other night, I was reading An Exorcist Explains the Demonic: the Antics of Satan and His Army of Fallen Angels, by Father Gabriele Amorth, SSP. Father Amorth was for many years the exorcist of the Diocese of Rome. I read the following passage from his explanation of Satanism (beginning on page 30):

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The Form of Forms is Itself Formless

That which has no form cannot be conceived – and vice versa. It’s easy to see that this is so when we try to think of what a square circle is like, or a four-sided triangle.

But, let’s talk about God.

To put the same thing another way: that than which no greater can be conceived by any mind cannot be conceived by any mind. If that than which no greater can be conceived could be conceived by any mind, then that mind would insofarforth understand how its conception of that thing could be surpassed, and would realize that in conceiving of it he had not yet quite conceived of that than which no greater can be conceived. He would realize that he had not been thinking of the unsurpassable. He would, i.e., realize that he had been thinking, not of God, but rather of something like Gaunilo’s Island, than which always some greater island might be conceived.

So, here’s the shocking consequence of these considerations: Not even God can conceive himself.

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The Autophagy of Falsehood

It is easy to see that thoroughgoing skepticism devours itself. If we can’t know the truth, then we can’t know that we can’t know the truth.

Postmodernism, likewise, obviously. If all texts are tendentious, then the texts that propagate postmodernism are tendentious.

But here’s a question: do all false propositions devour themselves? Are they autophagous? I.e., is it the case that if any false proposition were true, it would under the force of some necessity or other – logical, causal, historical, etc. – be false, or meaningless?

I think it may be.

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Incarnation & Transubstantiation are Formally Analogous

Transubstantiation stymies us in the same way, and for the same reason, as the Incarnation. In both cases, God takes embodiment in a finite creaturely vessel. The Logos takes the form of man and of bread (and likewise of Church, and Word – but tace re them for the nonce). These forms remain what they were. Jesus the man is still a man – Good News for us, since only qua man could he make strictly human reparation to God for the sins of Man, thus healing the cosmic wounds particularly inflicted by men – and the bread is still just as bready as ever – again, good, or we could not eat him, and so partake his Body and its sacrificial redemption of all our predicaments. The human nature is not driven out of the man by the divine nature, and the breadiness of the bread is not driven out by the divinity of it. On the contrary, they are each perfected. When God becomes man, a man – and, so, Man in general – becomes the God, so that men (can) become gods. Likewise, when God becomes bread, the bread becomes the supersubstantial Bread of Heaven: it becomes the God, who is the manna that feeds the angels, and the other members of God’s Body. Us.

We are what we eat, deo gracias.

In both cases, the soma remains soma; and, so, as soma, divine participant and influence in this world – a solid, as heavy as any stone, and so therefore scandalous to any who would pass by.

The true question is this: why should either Incarnation or Transubstantiation so scandalize us? Is it not only, merely, that these Incorporations of the divine into his creation are difficult for us to comprehend?

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Plato’s Cave

Plato’s Cave

Plato’s allegory of the cave appears in Book VII of Plato’s most famous and longest dialog, The Republic. Plato’s dialogs frequently star Plato’s teacher Socrates as a character. The dialogs involved discussions and philosophical arguments between various characters, some of whom were based on real people. Plato particularly disliked the sophists who were professional rhetoricians and who seemed to care more about money and social success than truth. In fact, Plato accused them of teaching their students how to make the worse argument appear better – enabling their students to convict the innocent and set free the guilty.

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Foundationalism: in praise of vagueness

Foundationalism: in praise of vagueness

Thoughts inspired by teaching epistemology for the first time and listening to the podcasts of Jordan Peterson

Epistemology became a major topic for analytic philosophers because they trace their intellectual origins to Descartes and the British empiricists. Descartes dismantles the foundations of his beliefs and then tries to rebuild them on certain grounds. Having used the method of doubt to tear everything down, including even mathematics, he finds irrefutable evidence of the existence of his own mind and then tries to prove that the “external world” exists.

Descartes

René Descartes

The British empiricists take their inspiration from Descartes, accept his distinction between mind and body and plump for body as the truly real. Following Galileo and Montaigne’s distinction between primary and secondary qualities, they attempt to dispose of mind-related secondary qualities as merely “subjective” and thus nonexistent.

Strangely enough, analytic philosophers complacently flip between claiming that physical reality is the paradigm of the real and also thinking that the existence of the external world is questionable and in need of proof. It is the latter that drives the analytic philosopher’s interest in epistemology. Continue reading