Posts Tagged ‘incompleteness theorems’
“I love to talk about nothing. It’s the only thing I know anything about.”*…
Try as they might, scientists can’t truly rid a space or an object of its energy. But as George Musser reports, what “zero-point energy” really means is up for interpretation…
Suppose you want to empty a box. Really, truly empty it. You remove all its visible contents, pump out any gases, and — applying some science-fiction technology — evacuate any unseeable material such as dark matter. According to quantum mechanics, what’s left inside?
It sounds like a trick question. And in quantum mechanics, you know to expect a trick answer. Not only is the box still filled with energy, but all your efforts to empty it have barely put a dent in the amount.
This unavoidable residue is known as ground-state energy, or zero-point energy. It comes in two basic forms: The one in the box is associated with fields, such as the electromagnetic field, and the other is associated with discrete objects, such as atoms and molecules. You may dampen a field’s vibrations, but you cannot eliminate every trace of its presence. And atoms and molecules retain energy even if they’re cooled arbitrarily close to absolute zero. In both cases, the underlying physics is the same.
Zero-point energy is characteristic of any material structure or object that is at least partly confined, such as an atom held by electric fields in a molecule. The situation is like that of a ball that has settled at the bottom of a valley. The total energy of the ball consists of its potential energy (related to position) plus its kinetic energy (related to motion). To zero out both components, you would have to give a precise value to both the object’s position and its velocity, something forbidden by the Heisenberg uncertainty principle.
What the existence of zero-point energy tells you at a deeper level depends ultimately on which interpretation of quantum mechanics you adopt. The only noncontentious thing you can say is that, if you situate a bunch of particles in their lowest energy state and measure their positions or velocities, you will observe a spread of values. Despite being drained of energy, the particles will look as if they’ve been jiggling. In some interpretations of quantum mechanics, they really have been. But in others, the appearance of motion is a misleading holdover from classical physics, and there is no intuitive way to picture what’s happening…
More on the development of our understanding of “zero-point energy” and on the questions that remain: “In Quantum Mechanics, Nothingness Is the Potential To Be Anything,” from @georgemusser.com in @quantamagazine.bsky.social.
For the most amusing of musings on nothing, see Percival Everett‘s Dr. No.
* Oscar Wilde
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As we noodle on nought, we might spare a thought for Kurt Gödel; he died on this date in 1978. A mathematician, logician, and author of Gödel’s proof. He is best known for his proof of Gödel’s Incompleteness Theorems (in 1931). He proved fundamental that in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular, the consistency of the axioms cannot be proved… thus ending a hundred years of attempts to establish axioms to put the whole of mathematics on an axiomatic basis. [See here for a consideration of what his finding might mean for moral philosophy…]
“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…
“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
Time travel…
Chino Otsuka, 1980 and 2009, Nagayama, Japan
In the photo series “Imagine Finding Me,” photographer Chino Otsuka revisits her childhood by digitally inserting herself in old photos of her as a child. Otsuka likens her double self-portraits to a kind of time travel:
“The digital process becomes a tool, almost like a time machine, as I’m embarking on the journey to where I once belonged and at the same time becoming a tourist in my own history…”
Chino Otsuka, 1975 and 2005, Spain
See (and read) more at Laughing Squid and at AGO.
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As we revisit Memory Lane, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978. Considered (with Aristotle and Frege) one of the most important logicians in history, Gödel published the work for which he is probably most widely remembered– his two incompleteness theorems— in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. He demonstrated that:
- If a system is consistent, it cannot be complete.
- The consistency of a systems axioms cannot be proven within the system.
Gödel’s theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Russell and Whitehead’s Principia Mathematica and Hilbert’s formalism, to find a set of axioms sufficient for all mathematics.
As the Anschluss swept Austria, Gödel fled to the U.S., landing at Princeton, where he joined Albert Einstein at the Institute for Advanced Studies. In 1951, as a 70th birthday present for Einstein, Gödel demonstrated the existence of paradoxical solutions to Einstein’s field equations in general relativity (they became known as the Gödel metric)– which allowed for “rotating universes” and time travel… and which caused Einstein to have doubts about his own theory.
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