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.

Hofstadter was too artistic, the Nagel and Newman book was too long, Wikipedia is too thorough or too simple. This presentation is just right; short enough to comprehend while including the important details.

Gödel’s Proof relies on the liar paradox, "this statement is false." Gregory Chaitin produced a similar result using the Berry paradox, demonstrated by the fifty-seven letter expression "the smallest positive integer not definable in under sixty letters."

Hagen von Eitzen expanded the predicate, giving two expressions of the Gödel sentence at the bottom of the page.

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.

If all this talk of substitution and self-reference remind you of programming, recursion, or lambda calculus, have a look at Incompleteness Ex Machina:

In this essay we'll prove Gödel's incompleteness theorems twice. First, we'll prove them the good old-fashioned way. Then we'll repeat the feat in the setting of computation. In the process we'll discover that Gödel's work, rightly viewed, needs to be split into two parts: the transport of computation into the arena of arithmetic on the one hand and the actual incompleteness theorems on the other. After we're done there will be cake.

As someone who is quite interested in formal methods (briefly: the application of mathematics and proofs to "real-world" systems), understanding Gödel's and Tarski's incompleteness theorems was an existential moment for me. They underpin a lot of my current ideas about what "factual truth" is and the limitations of mathematics and modeling techniques. For those very curious, this has applications to physics models and simulations as well: Fundamental Limits of Cyber-Physical Systems Modeling touches on some undecidable issues when constructing even very simple models of systems.

Haven't heard this take before:

What is tantalising, and perhaps unique, about his argument for an afterlife is the fact that it actually depends on the inevitable irrationality of human life in an otherwise reason-imbued world. It is precisely the ubiquity of human suffering and our inevitable failures that gave Gödel his certainty that this world cannot be the end of us.

I think Modernism is Aristotelian.

I understand exactly where you are coming from and I don't think Gödel thought of himself as a post modern (I think he was a royalist politically)

The effect of his proof was by my definition the definition of post-modernism. The history of mathematics has a pre-Gödel Hilbert or modern period and a Post-Gödel or postmodern period perfect. :)

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I don't think that Gödel should be included in the nothing means anything crowd. and I do agree that mathematics is or should be immune to the stupidities of the worst manifestations of post-modernism is other fields.

Abstract:

In this essay we'll prove Gödel's incompleteness theorems twice. First, we'll prove them the good old-fashioned way. Then we'll repeat the feat in the setting of computation. In the process we'll discover that Gödel's work, rightly viewed, needs to be split into two parts: the transport of computation into the arena of arithmetic on the one hand and the actual incompleteness theorems on the other. After we're done there will be cake.

Featuring: the ghost of ~sub-inconsistency~

This kind of logic always makes my head spin a bit, but these results are surprisingly important for understanding some of the concerns driving (post-)modern mathematics. Thinking about them in terms of programs definitely makes the results feel a bit more natural!

This is not so much an academic paper as it is a blog post written in LaTeX. If you already know about Gödel, it's a pretty good read; if not, but you know a bit about programming, you'd probably be best served by skimming through section 1, reading sections 2-6, then looking back through 1 if you're still curious about anything there.

Évariste Galois: Galois essentially founded modern abstract algebra. Galois died in a duel at age 20. No one knows what motivated the duel, but the night before the duel he was so sure of his own death he wrote many letters to his friends containing all the mathematics he had discovered. How monumental his contributions were was not realized until decades after his death.

Georg Cantor: Cantor made major contributions to set theory, especially with regards to infinite sets. His papers about infinity were very controversial. Some mathematicians believed infinity belonged in the realm of philosophy. Some theologians believed that only God was infinite, and thus Cantor’s argument was a foray into pantheism. This extreme criticism led Cantor into great depression. Henri Poincare, (one of my favorite mathematicians), quipped “later generations will regard [Cantor’s set theory] as a disease from which one has recovered”. He lived the final years of his life extremely disturbed.

Felix Hausdorff: Hausdorff contributed to set theory and topology (my favorite subject!). Although he attempted to get a position as a professor in the United States, Hausdorff was unable to escape Germany as the Nazis rose to power. He and his wife committed suicide to avoid concentration camps.

C. P. Ramanujam: Ramanujam made major modern advances in algebraic geometry, one of the most difficult fields of current mathematical research. Despite being one of the most prolific, brilliant mathematicians of the mid-20th century, he constantly felt that he was not contributing enough. He tortured himself with the exceedingly high standards he placed for himself and for all of mathematics. Eventually schizophrenia overtook his life, and he committed suicide.

Pythagoras: Pythagoras discovered the theorem which bears his name (though some historians believe the Egyptians may have discovered it first). According to legend, Pythagoras thought beans were horrid things because they symbolically represented testicles. Thus when a mob chased him to execute him, and he was chased to a bean field, he stopped. Because of his refusal to enter the bean field, he was murdered.

Kurt Gödel: Gödel researched logic, set theory, and foundations of mathematics. He is best known for his incompleteness theorems, which had a huge impact on mathematics, and especially on the philosophy of mathematics. During the last years of his life, Gödel became obsessed with the fear that someone would poison his food – so much so that he starved himself to death.

Disregarding what's already in this thread (Vonnegut!): Ken Kesey: One Flew Over The Cuckoo's Nest. The Grandfather of LSD culture, but also an incredible author. The visceral descriptions and unreliable narrator make for a very interesting read to say the least. Then there's the colorful cast of characters... Joseph Heller Catch-22. A dark comedy with a Kafkaesque bent. Reflects upon a lot of society in the modern age. Albert Camus: The Stranger. The definitive piece of absurdist fiction. Also very short and exciting. Jean-paul Sartre: No Exit. OK, yes I like French existentialists. No Exit (or In Camera, The Others, or a couple other translations iirc) is the source of the oft-misunderstood quote "HELL IS OTHER PEOPLE". A short play with characters who find themselves spending an afterlife together. Thomas Pynchon: V and Gravity's Rainbow. Haven't read the rest of his work, but these are brilliant, wide ranging, complex woven narratives featuring masses of interconnected characters and events over years. Taking on one of these novels is a journey itself with their dense prose (and occasional lyrical interludes), confusing subplots and gargantuan length. But it's a journey full of rewards as well. and on a slightly different note Douglass Hofstadter: Gödel, Escher, Bach. A description of elements of number theory and Gödel's Incompleteness Theorum and application to computing, art, consciousness, music, puzzles, and more. Also poetic interludes.

An absolutely fascinating article about language and how it could be tied to consciousness and human evolution (cultural and otherwise).

I've been reading Hofstadter's Gödel, Escher & Bach lately (yeah yeah, I know, stereotypical much) and this article has a lot of interesting themes that tie into it. Self-as-illusion and self-as-self-reference seem to be really high on my "(positive) shit that keeps me up at night" list

Time for your weekly scheduled Aeon article.

This one's about ant colonies. Here's a nice snippet:

The system that ant colonies use to organise their work is a distributed process. Like division of labour, distributed processes can take different forms. A distributed process is not the opposite of division of labour – but it’s different in important ways. Primarily, in a distributed process, there is never central control, while in division of labour there might be. A leader can tell one citizen to make candles and another to make shoes. In a distributed process this would happen through local interactions, for example with people who want to buy candles or shoes – creating demand that is filled by an entrepreneur who then meets the demand.

At least in the short term, a system organised by a distributed process and one organised by division of labour could look the same: the same individuals could do the same task over and over. An ant might do the same task day after day. It might go out to forage, come back to the nest, engage again in the interactions that stimulate it to forage, and spend the night among other ants that recently returned from foraging. The next morning, it is again in a situation in which it is likely to forage, and this could continue day after day. However, in different conditions, the ant might do another task, and so its role is not fixed.

Distributed processes and division of labour can both be effective, but they don’t function in the same way. For division of labour, specialisation can lead to better work. By contrast, in a distributed process, the fact that individuals are interchangeable makes the whole system more robust and more resilient. If the individual who performs a task gets lost or becomes unfit to do it, another can step in. The individuals don’t have to be all alike, but the differences among them are not large enough to affect the viability of the system.

I've been fascinated with consciousness and the concept of "I" for a while now (and coincidentally started reading Gödel, Escher & Bach last week), and this ties in pretty nicely to that. Could ant or bee colonies be thought of as being sentient in some way? There's actually a ton of really interesting research into bees regarding this; basically each bee can be seen as a neuron, and bee colonies can solve problems and "think ahead" even though each individual bee isn't very clever. I'd assume this would apply to ants as well.