#incompleteness theorem

[adistraughtjester]

Gödel’s incompleteness theorems could be used for so many world building concepts

[noosphe-re]

In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history. Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths. But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency. His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.

Natalie Wolchover, How Gödel’s Proof Works, Quanta Magazine, July 14, 2020

[maxwellshimbo]

RIP Kurt Gödel you would have loved "What if kinkshaming was my kink?"

[lthmath]

Celebrating Kurt Godel's birthday!

He was born on 28th April 1906. Considered along with Aristotle, Alfred Tarski and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century.

Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

"Kurt Gödel was pals with Einstein, dubbed "the most important logician since Aristotle" by one of the greatest mathematicians of the 20th century, and now he's a magnetic finger puppet. Mike drop!"

[maxwellshimbo]

Neither provably true nor provably false but a secret third thing (unprovable within the current system of axioms and theorems)

[maxwellshimbo]

Me, reading about Gödel's incompleteness theorem again:

[maxwellshimbo]

me, reading about the proof to Gödel's incompleteness theorem: So this is basically the same as "What if kink shaming was my kink?"

[bluewatsons]

Siobhan Roberts, Waiting for Gödel, The New Yorker (June 29, 2016)

The mathematician Kurt Gödel’s incompleteness theorem ranks in scientific folklore with Einstein’s relativity and Heisenberg’s uncertainty.

In June of 1975, the Office of the White House Press Secretary announced President Gerald R. Ford’s picks for the National Medal of Science. One went to the Austrian-born mathematician and logician Kurt Gödel. Nicknamed Mr. Why by his parents, Gödel was known to a subset of his constituents as, simply, God. He received fan mail from all over the world, archiving it into files of “autograph requests,” “inquiries from students and amateurs,” “letters of appreciation,” and “crank correspondence.” A self-described “dunce fool of Mathematics” in West Bengal wrote seeking Gödel’s “Guruship,” and a svelte math teacher in California confessed that she’d taken the liberty of enlarging a photo of Gödel to make a poster for her classroom. (She’d also taken the liberty of enclosing a snapshot of herself.) Ultimately, Gödel came to be compared not only to his friend Albert Einstein but also to Franz Kafka. Such was the nature of his contribution—only a handful of theorems, but all of them monumental and fantastical.

Gödel’s masterpiece was his incompleteness theorem, which ranks in scientific folklore with Einstein’s relativity and Heisenberg’s uncertainty. Promulgated in Vienna in the early nineteen-thirties, the notion of incompleteness threw mathematics into a hall of mirrors, where it reflected upon itself to alluring, if disorienting, effect: the theorem proved, using mathematics, that mathematics could not prove all of mathematics. Of course, it has a proper and technically precise formulation, but the late logician Verena Huber-Dyson paraphrased it for me as follows: “There is more to truth than can be caught by proof.” Or, as the British novelist Zia Haider Rahman put it in his award-winning début, “In the Light of What We Know,” “Within any given system, there are claims which are true but which cannot be proven to be true.”

So although Gödel published his results eighty-five years ago, the theorem endures in the popular imagination. This month, a group of latter-day Gödelians convened on Thursday nights for a crash course in incompleteness, on the roster at the Brooklyn Institute for Social Research. One syllabus text promised “Gödel Without (Too Many) Tears,” while supplementary reading unpacked his work “in Words of One Syllable.” According to “Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse,” the chameleon interpretations emerge in discussions of math, philosophy, computer science, and artificial intelligence, as one might expect, but also in ruminations on physics, evolution, religion, atheism, poetry, hip-hop, dating, politics, and even the Constitution. (Famously, Gödel informed the judge at his U.S. citizenship hearing about an inconsistency that he had discovered in the Constitution, which would allow a dictator to rise in America.)

The students in the incompleteness class, held at the Brooklyn Commons, included a computer scientist obsessed with recursion (that is, self-referential things, like Russian nesting dolls or Escher’s drawing of a hand drawing a hand); a public-health nutritionist with a fondness for her “Philo_sloth_ical” T-shirt; a philosopher in the tradition of American pragmatism; an ad man versed in the classics; and a private-school teacher who’d spent a lonely, life-changing winter reading Douglas Hofstadter’s “Gödel, Escher, Bach,” which won the Pulitzer in 1980. Even Hofstadter, a professor of cognitive science at Indiana University, still has Gödel on the brain. He delivered two talks on the logician recently, billing him as “The True Inventor of Programming Languages and Data Structures.” When I told Hofstadter about the course—not everyone’s ideal Thursday night out on the town—he said, “It ought to be quite fun.”

In total, eight people enrolled for the class, a fitting number, since “8” rotated by ninety degrees is “∞,” and infinity is, in a sense, where the trouble began with this trippy episode in the history of mathematics. In the nineteenth century, the German mathematician Georg Cantor launched an investigation into various sizes of infinities, thereby inventing set theory, which became an overarching paradigm for mathematics. “A set is a Many that allows itself to be thought of as a One,” Cantor said. But, from nearby realms of logic, paradoxes emerged, such as the Russell set, the set of all sets that do not contain themselves. In the early twentieth century, Bertrand Russell and Alfred North Whitehead, with “Principia Mathematica,” and David Hilbert, with “Hilbert’s Program,” attempted to construct a solid foundation for mathematics, creating a formal system based on axioms and rules. Gödel, with his incompleteness theorem (two theorems, actually), put an end to these dreams. He proved, to borrow from the course outline:

For any consistent axiomatic formal system that can express facts about basic arithmetic:

There are true statements that are unprovable within the system.

The system’s consistency cannot be proven within the system.

Puzzled? Russell admitted the same. He wondered, “Are we to think that 2+2 is not 4, but 4.001?”

A mathematician is said to be a machine for turning coffee into theorems, and at that Gödel excelled, although he said that the coffee in Vienna was wretched. For Peter O’Hearn, an engineering manager at Facebook and professor at University College London, the incompleteness “wow moment” was fuelled by visits to the brewpub during graduate school. O’Hearn is the co-recipient of this year’s Gödel Prize—he and a colleague, Stephen Brookes, invented concurrent separation logic, a revolutionary proof system for computer software. “Gödel’s theorem has a major impact on what all computer scientists do,” he told me. “It puts a fundamental limit on questions we can answer with computers. It tells us to go for approximation—more approximate solutions, which find many right answers, but not all right answers. That’s a positive, because it constrains me from trying to do stupid things, trying to do impossible things.”

At the Brooklyn Commons, a keg of Turmeric Sunrise kombucha was the popular drink during class breaks, which people spent pondering the pronunciation of “Gödel.” (“Goy-del?” “Good-ul?” “Girdle?”) I asked my classmates whether they had heard of the sci-fi writer Rudy Rucker’s book “Spacetime Donuts,” which includes a minor character named Professor G. Kurtowski. The answer was no, but as the American pragmatist noted, “Kurtowski is significantly easier to pronounce than ‘Guuuudel.’ ” In Gödel’s archives, there is a full file of letters from Rucker. “I was looking for a guru,” Rucker told me. He first visited with the great man in 1972, as a graduate student, and thereupon Gödel joined his other mentors—Frank Zappa, Robert Crumb, William Burroughs, and Thomas Pynchon. Pynchon, in his novel “Gravity’s Rainbow,” paraphrased incompleteness by mapping it onto Murphy’s Law, “that brash Irish proletarian restatement of Gödel’s Theorem—when everything has been taken care of, when nothing can go wrong, or even surprise us … something will.” “Yeah,” our instructor said. “Metaphorically, yeah, I can see it.”

As the course progressed, the Gödelians remained divided in their enlightenment. An early verdict, from a lawyer studying for the bar, was that Gödel, with his loopy logic, seemed to be cheating. Later, the private-school teacher raised his hand and declared that the last bit of chalk-and-talk instruction made perfect sense. From my distractible perspective across the table, though, his semi-eureka experience looked like the misinterpretation of a sugar high, since he’d just eaten seven Oreo Thins from the pile of communal snacks. And, anyway, it would all come to nothing if not incompleteness.