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

Alan Turing

Alan Turing

Kurt Gödel’s Incompleteness Theorem[1] 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?”[2] In German this is called Entscheidungsproblem – the decision problem.[3]

Turing found that he could answer this question by framing it in terms of a Turing machine[4] – 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.”[5]

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Apparitions of the Gods

Gods 01 Muses by Andreas Mantegna (1496)

The Muses (1496) by Andreas Mantegna (1431 – 1503)

“The dove – the rood – the loaf – the wine.”

Men know the gods because they have seen or intuited them, but not all men have seen or intuited the gods, and some men are incapable of seeing or intuiting them.  The gods, moreover, sometimes disguise themselves so as to test men, or they appear in and as omens and auguries, which the dull of mind and the wicked of heart invariably either miss entirely through their mental obtuseness or, through self-serving prejudice, blatantly misread.

I. The gods appear in and as their attributes, which again only those who have vested themselves in the proper lore and the requisite discipline can correctly interpret. Who would see the gods must enjoy a gift of pre-attunement, even before he bows under the discipline and engraves the lore in his heart that will let him see them. Such a man is called a poet.  The ancient Boeotian teller of the gods, Hesiod, whom scholars assign to the late Eighth Century or early Seventh Century BC, bears a name that means simply “The” (he or hos) “Poet” (aiodos), suggesting that the Boeotians, or at least those of them in the vicinity of Mt. Helicon, recognized his special talent and accorded him the status owing thereto.  That status may claim itself paramount because the community must communicate with the gods, just as the gods must communicate with the community, and an efficient go-between nicely serves the requirement both ways.  One misthinking modern school argues through Hesiod’s name that any particular poet is a non-existence, as though no one could write a poem, as though poems constituted themselves, authorless, and as though therefore no one really ever saw Hesiod’s gods or heard them speak.  This thesis of a literary fantasy amounts, however, merely to another kind of noetic obtuseness.  Someone wrote Hesiod’s poems, obviously, and if Hesiod were the invention of that someone then that someone nevertheless would have seen Hesiod’s gods – through his invention, as it were, and taking Hesiod’s name, but equally in a vision such that the seeing must guarantee its own authenticity and such that He remains The Poet.

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