Posts Tagged ‘Gödel’
“This incompleteness is all we have”*…
The impulse to “systemitize” morality is as old as philosophy. Many now hope that AI will discover and organize moral truths. But Elad Uzan suggests that Kurt Gödel’s work on incompleteness demonstrates that deciding what is right will always be our burden…
Imagine a world in which artificial intelligence is entrusted with the highest moral responsibilities: sentencing criminals, allocating medical resources, and even mediating conflicts between nations. This might seem like the pinnacle of human progress: an entity unburdened by emotion, prejudice or inconsistency, making ethical decisions with impeccable precision. Unlike human judges or policymakers, a machine would not be swayed by personal interests or lapses in reasoning. It does not lie. It does not accept bribes or pleas. It does not weep over hard decisions.
Yet beneath this vision of an idealised moral arbiter lies a fundamental question: can a machine understand morality as humans do, or is it confined to a simulacrum of ethical reasoning? AI might replicate human decisions without improving on them, carrying forward the same biases, blind spots and cultural distortions from human moral judgment. In trying to emulate us, it might only reproduce our limitations, not transcend them. But there is a deeper concern. Moral judgment draws on intuition, historical awareness and context – qualities that resist formalisation. Ethics may be so embedded in lived experience that any attempt to encode it into formal structures risks flattening its most essential features. If so, AI would not merely reflect human shortcomings; it would strip morality of the very depth that makes ethical reflection possible in the first place.
Still, many have tried to formalise ethics, by treating certain moral claims not as conclusions, but as starting points. A classic example comes from utilitarianism, which often takes as a foundational axiom the principle that one should act to maximise overall wellbeing. From this, more specific principles can be derived, for example, that it is right to benefit the greatest number, or that actions should be judged by their consequences for total happiness. As computational resources increase, AI becomes increasingly well-suited to the task of starting from fixed ethical assumptions and reasoning through their implications in complex situations.
But what, exactly, does it mean to formalise something like ethics? The question is easier to grasp by looking at fields in which formal systems have long played a central role. Physics, for instance, has relied on formalisation for centuries. There is no single physical theory that explains everything. Instead, we have many physical theories, each designed to describe specific aspects of the Universe: from the behaviour of quarks and electrons to the motion of galaxies. These theories often diverge. Aristotelian physics, for instance, explained falling objects in terms of natural motion toward Earth’s centre; Newtonian mechanics replaced this with a universal force of gravity. These explanations are not just different; they are incompatible. Yet both share a common structure: they begin with basic postulates – assumptions about motion, force or mass – and derive increasingly complex consequences. Isaac Newton’s laws of motion and James Clerk Maxwell’s equations are classic examples: compact, elegant formulations from which wide-ranging predictions about the physical world can be deduced.
Ethical theories have a similar structure. Like physical theories, they attempt to describe a domain – in this case, the moral landscape. They aim to answer questions about which actions are right or wrong, and why. These theories also diverge and, even when they recommend similar actions, such as giving to charity, they justify them in different ways. Ethical theories also often begin with a small set of foundational principles or claims, from which they reason about more complex moral problems. A consequentialist begins with the idea that actions should maximise wellbeing; a deontologist starts from the idea that actions must respect duties or rights. These basic commitments function similarly to their counterparts in physics: they define the structure of moral reasoning within each ethical theory.
Just as AI is used in physics to operate within existing theories – for example, to optimise experimental designs or predict the behaviour of complex systems – it can also be used in ethics to extend moral reasoning within a given framework. In physics, AI typically operates within established models rather than proposing new physical laws or conceptual frameworks. It may calculate how multiple forces interact and predict their combined effect on a physical system. Similarly, in ethics, AI does not generate new moral principles but applies existing ones to novel and often intricate situations. It may weigh competing values – fairness, harm minimisation, justice – and assess their combined implications for what action is morally best. The result is not a new moral system, but a deepened application of an existing one, shaped by the same kind of formal reasoning that underlies scientific modelling. But is there an inherent limit to what AI can know about morality? Could there be true ethical propositions that no machine, no matter how advanced, can ever prove?
These questions echo a fundamental discovery in mathematical logic, probably the most fundamental insight ever to be proven: Kurt Gödel’s incompleteness theorems. They show that any logical system powerful enough to describe arithmetic is either inconsistent or incomplete. In this essay, I argue that this limitation, though mathematical in origin, has deep consequences for ethics, and for how we design AI systems to reason morally…
Eminently worth reading in full: “The incompleteness of ethics,” from @aeon.co.
And as if that were not enough, consider the cultural challenge implicit in this chart:
More background at “Cultural Bias in LLMs” (and here and here).
* Charles Bukowski
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As we own up to it, we might recall that it was on this date in 1942 that actress Hedy Lamarr and musician George Antheil received a patent (#2,292,387) for a frequency-hopping radio communication system which later became the basis for modern technologies like Bluetooth, wireless telephones, and Wi-Fi.
Hedy Lamarr made it big in acting before ever moving to the United States. Her role in the Czech film Ecstasy got international attention in 1933 for containing scandalous, intimate scenes that were unheard of in the movie industry up until then.
Backlash from her early acting career was the least of her worries, however, as tensions began to rise in Europe. Lamarr, born Hedwig Eva Maria Kiesler, grew up in a Catholic household in Austria, but both of her parents had a Jewish heritage. In addition, she was married to Friedrich Mandl, a rich ammunition manufacturer with connections to both Fascist Italy and Nazi Germany.
Her time with Friedrich Mandl was bittersweet. While the romance quickly died and Mandl became very possessive of his young wife, Lamarr was often taken to meetings on scientific innovations in the military world. These meetings are said to have been the spark that led to her becoming an inventor. As tensions in both her household and in the world around her became overwhelming, she fled Europe and found her way to the United States through a job offer from Hollywood’s MGM Studios.
Lamarr became one of the most sought-after leading women in Hollywood and starred in popular movies like the 1939 film Algiers, but once the United States began helping the Allies and preparing to possibly enter the war, Lamarr almost left Hollywood forever. Her eyes were no longer fixed on the bright lights of the film set but on the flashes of bombs and gunfire. Lamarr wanted to join the Inventors’ Council in Washington, DC, where she thought she would be of better service to the war effort.
Lamarr’s path to inventing the cornerstone of Wi-Fi began when she heard about the Navy’s difficulties with radio-controlled torpedoes. She recruited George Antheil, a composer she met through MGM Studios, in order to create what was known as a Secret Communication System.
The idea behind the invention was to create a system that constantly changed frequencies, making it difficult for the Axis powers to decode the radio messages. The invention would help the Navy make their torpedo systems become more stealthy and make it less likely for the torpedoes to be rendered useless by enemies.
Lamarr was the brains behind the invention, with her background knowledge in ammunition, and Antheil was the artist that brought it to life, using the piano for inspiration. In 1942, under her then-married name, Hedy Kiesler Markey, she filed for a patent for the Secret Communication System, patent case file 2,292,387, and proposed it to the Navy.
The first part of Lamarr and Antheil’s Secret Communication System story did not see a happy Hollywood ending. The Navy refused to accept the new technology during World War II. Not only did the invention come from a civilian, but it was complex and ahead of its time.
As the invention sat unused, Lamarr continued on in Hollywood and found other ways to help with the war effort, such as working with the USO. It wasn’t until Lamarr’s Hollywood career came to an end that her invention started gaining notice.
Around the time Lamarr filmed her last scene with the 1958 film The Female Animal, her patented invention caught the attention of other innovators in technology. The Secret Communication System saw use in the 1950s during the development of CDMA network technology in the private sector, while the Navy officially adopted the technology in the 1960s around the time of the Cuban Missile Crisis. The methods described in the patent assisted greatly in the development of Bluetooth and Wi-Fi.
Despite the world finally embracing the methods of the patent as early as the mid-to-late 1950s, the Lamarr-Antheil duo were not recognized and awarded for their invention until the late 1990s and early 2000s. They both received the Electronic Frontier Foundation Pioneer Award and the Bulbie Gnass Spirit of Achievement Bronze Award, and in 2014 they were inducted into the National Inventors Hall of Fame…
“If a ‘religion’ is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one”*…
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.
In the 89 years since Gödel’s discovery, mathematicians have stumbled upon just the kinds of unanswerable questions his theorems foretold. For example, Gödel himself helped establish that the continuum hypothesis, which concerns the sizes of infinity, is undecidable, as is the halting problem, which asks whether a computer program fed with a random input will run forever or eventually halt. Undecidable questions have even arisen in physics, suggesting that Gödelian incompleteness afflicts not just math, but — in some ill-understood way — reality…
A (relatively) simple explanation of the incompleteness theorem– which destroyed the search for a mathematical theory of everything: “How Gödel’s Proof Works.”
* John D. Barrow, The Artful Universe
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As we noodle on the unknowable, we might spare a thought for Vilfredo Federico Damaso Pareto; he died on this date in 1923. An engineer, sociologist, economist, political scientist, and philosopher, he made several important contributions to economics, sociology, and mathematics.
He introduced the concept of Pareto efficiency and helped develop the field of microeconomics. He was also the first to discover that income follows a Pareto distribution, which is a power law probability distribution. The Pareto principle, named after him, generalized on his observations on wealth distribution to suggest that, in most systems/settings, 80% of the effects come from 20% of the causes– the “80-20 rule.” He was also responsible for popularizing the use of the term “elite” in social analysis.
As Benoit Mandelbrot and Richard L. Hudson observed, “His legacy as an economist was profound. Partly because of him, the field evolved from a branch of moral philosophy as practised by Adam Smith into a data intensive field of scientific research and mathematical equations.”
The future leader of Italian fascism Benito Mussolini, in 1904, when he was a young student, attended some of Pareto’s lectures at the University of Lausanne. It has been argued that Mussolini’s move away from socialism towards a form of “elitism” may be attributed to Pareto’s ideas.
Mandelbrot summarized Pareto’s notions as follows:
At the bottom of the Wealth curve, he wrote, Men and Women starve and children die young. In the broad middle of the curve all is turmoil and motion: people rising and falling, climbing by talent or luck and falling by alcoholism, tuberculosis and other kinds of unfitness. At the very top sit the elite of the elite, who control wealth and power for a time – until they are unseated through revolution or upheaval by a new aristocratic class. There is no progress in human history. Democracy is a fraud. Human nature is primitive, emotional, unyielding. The smarter, abler, stronger, and shrewder take the lion’s share. The weak starve, lest society become degenerate: One can, Pareto wrote, ‘compare the social body to the human body, which will promptly perish if prevented from eliminating toxins.’ Inflammatory stuff – and it burned Pareto’s reputation… [source]
“Neoliberalization has meant, in short, the financialization of everything”*…
Investing and deal-making occupy an outsized role in popular depictions of “business” like HBO’s Succession and Showtime’s Billions. They also occupy an outsized share of our elite: Over the last five years, the nation’s top business schools have sent nearly thirty percent of their graduating classes into finance.
But the buying and selling of companies, the mergers and divestments, the hedging and leveraging, are not themselves valuable activity. They invent, create, build, and provide nothing. Their claim to value is purely derivative—by improving the allocation of capital and configuration of assets, they are supposed to make everyone operating in the real economy more productive. The practitioners are rewarded richly for their effort.
Does this work, or are the efforts largely wasted? One might default to the assumption that an industry attracting so much talent and generating so much profit must be creating enormous value. But the elaborate financial engineering of the 2000s, which attempted an alchemy-like conversion of high-risk loans into rock-solid assets, and then placed highly leveraged bets against their performance, led to the collapse of some established Wall Street institutions, massive bailouts for others, and a global economic meltdown. Mergers and acquisitions, meanwhile, appear largely to be exercises in wheel-spinning: “M&A is a mug’s game,” explains Roger Martin in the Harvard Business Review, “in which typically 70%–90% of acquisitions are abysmal failures.”…
Hedge funds and venture capital funds appear to badly underperform simple public market indexes, while buyout funds have performed roughly at par over the past decade. Of course, some funds deliver outsized returns in a given timeframe; even a random distribution has a right tail. And there are managers whose strong and consistent track records suggest the creation of real value.
In other words, most fund managers are generating the results that one might expect from an elaborate game of chance—placing bets in the market with odds similar to a coin flip. With enough people playing, some will always find themselves on winning streaks and claim the Midas touch, at least until the coin’s next flip. Except under these rules of “heads I win, tails you lose,” they collect their fees regardless…
In the U.S., finance, insurance and real estate (FIRE) sector now accounts for 20 percent of GDP– compared with only 10 percent in 1947. The thorough and thoughtful analysis– and critique– of the frothier components of that sector excerpted above is noteworthy, beyond its quality, for it’s origin; it is an early product of a new conservative think tank, American Compass.
Read it in full: “Coin-Flip Capitalism: A Primer.”
Pair with “What Kind of Country Do We Want?“, a resonant essay from the amazing Marilynne Robinson.
(image above: source)
* “Neoliberalization has meant, in short, the financialization of everything. There was unquestionably a power shift away from production to the world of finance… Neoliberalization has not been very effective in revitalizing global capital accumulation, but it has succeeded remarkably well in restoring, or in some instances (as in Russia and China) creating, the power of an economic elite. The theoretical utopianism of neoliberal argument has, I conclude, primarily worked as a system of justification and legitimation for whatever needed to be done to achieve this goal.” — David Harvey, A Brief History of Neoliberalism
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As we look beyond price to value, we might recall that it was on this date in 1936 that Alan Turing submitted his paper, “On Computable Numbers” for publication; its full title was “On Computable Numbers, with an Application to the Entscheidungsproblem.” In answer to Hibert’s and Ackermann’s 1928 challenge, Turing demonstrated that some purely mathematical yes-no questions can never be answered by computation; more technically, that some decision problems are “undecidable” in the sense that there is no single algorithm that infallibly gives a correct “yes” or “no” answer to each instance of the problem. In Turing’s own words: “…what I shall prove is quite different from the well-known results of Gödel … I shall now show that there is no general method which tells whether a given formula U is provable in K.”
Turing followed this proof with two others, both of which rely on the first. And all rely on his development of type-writer-like “computing machines” that obey a simple set of rules and his subsequent development of a “universal computing machine”– the “Turing Machine,” a key inspiration (to von Neumann and others) for the development of the digital computer.
“Nothing happens until something moves”*…
What determines our fate? To the Stoic Greek philosophers, fate is the external product of divine will, ‘the thread of your destiny’. To transcendentalists such as Henry David Thoreau, it is an inward matter of self-determination, of ‘what a man thinks of himself’. To modern cosmologists, fate is something else entirely: a sweeping, impersonal physical process that can be boiled down into a single, momentous number known as the Hubble Constant.
The Hubble Constant can be defined simply as the rate at which the Universe is expanding, a measure of how quickly the space between galaxies is stretching apart. The slightest interpretation exposes a web of complexity encased within that seeming simplicity, however. Extrapolating the expansion process backward implies that all the galaxies we can observe originated together at some point in the past – emerging from a Big Bang – and that the Universe has a finite age. Extrapolating forward presents two starkly opposed futures, either an endless era of expansion and dissipation or an eventual turnabout that will wipe out the current order and begin the process anew.
That’s a lot of emotional and intellectual weight resting on one small number…
How scientists pinned a single number on all of existence: “Fate of the Universe.”
[Readers might remember that the Big Bang wasn’t always an accepted paradigm— and that on-going research continues to surface challenges.]
* Albert Einstein
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As we center ourselves, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978. A logician, mathematician, and philosopher, he is considered (along with Aristotle, Alfred Tarski— whose birthday this also is– and Gottlob Frege) to be one of the most important logicians in history. Gödel had an immense impact upon scientific and philosophical thinking in the 20th century. He is, perhaps, best remembered for his Incompleteness Theorems, which led to (among other important results) Alan Turing’s insights into computational theory.
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. — John von Neumann
“The laws of nature are but the mathematical thoughts of God”*…
2,300 years ago, Euclid of Alexandria sat with a reed pen–a humble, sliced stalk of grass–and wrote down the foundational laws that we’ve come to call geometry. Now his beautiful work is available for the first time as an interactive website.
Euclid’s Elements was first published in 300 B.C. as a compilation of the foundational geometrical proofs established by the ancient Greek. It became the world’s oldest, continuously used mathematical textbook. Then in 1847, mathematician Oliver Byrne rereleased the text with a new, watershed use of graphics. While Euclid’s version had basic sketches, Byrne reimagined the proofs in a modernist, graphic language based upon the three primary colors to keep it all straight. Byrne’s use of color made his book expensive to reproduce and therefore scarce, but Byrne’s edition has been recognized as an important piece of data visualization history all the same…
Explore elemental beauty at “A masterpiece of ancient data viz, reinvented as a gorgeous website.”
* Euclid, Elements
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As we appreciate the angles, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978. A logician, mathematician, and philosopher, he is considered (along with Aristotle, Alfred Tarski— whose birthday this also is– and Gottlob Frege) to be one of the most important logicians in history. Gödel had an immense impact upon scientific and philosophical thinking in the 20th century. He is, perhaps, best remembered for his Incompleteness Theorems, which led to (among other important results) Alan Turing’s insights into computational theory.
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. — John von Neumann
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