What Tangled Webs: The Hopf Fibration And Physics III

by Jochen Szangolies In the previous two installments of this series ([1], [2]), I have been engaged in the project of communicating a bit of the intuition behind the abstract notions of physics (and the necessary mathematics). My guiding principle in this attempt (essay in the literal sense) has been a famous quote of Hungarian…

Wheels Within Wheels: The Hopf Fibration And Physics II

by Jochen Szangolies In the last column, I have argued against the idea that understanding in mathematics and physics is transmitted via genius leaps of insight into obscure texts rife with definitions and abstract symbols. Rather, it is more like learning to cook: even if you have memorized the cookbook, your first soufflé might well…

Music Of The Spheres: The Hopf Fibration And Physics

by Jochen Szangolies Modern physics in its full mathematical splendor introduces an array of unfamiliar concepts that daunt the initiate, and often even bewilder the pro (or is that just me?). A part of it is just that it’s a complex topic, and its objects of study are far removed from everyday experience: a quark…

Nonstandard Humans: On Leaving Room For Error

by Jochen Szangolies Humans, Plato famously held, are featherless bipeds: uniquely singled out from other animals by walking permanently on two legs, which delineates us against fish, mammals, and reptiles, while not sporting the sort of plumage associated with birds. Of course, he knew little of the mighty T. Rex (although its featherlessness is subject…

The Large Language Turn: LLMs As A Philosophical Tool

by Jochen Szangolies There is a widespread feeling that the introduction of the transformer, the technology at the heart of Large Language Models (LLMs) like OpenAI’s various GPT-instances, Meta’s LLaMA or Google’s Gemini, will have a revolutionary impact on our lives not seen since the introduction of the World Wide Web. Transformers may change the…

Finding Your Self: Desire Paths In Identity Space

by Jochen Szangolies If you spend any time in a place with public parks, gardens, or simple green areas fenced in squarely by concrete walkways, you’ll be familiar with the sight of trampled paths cutting across the grass, tracing a muddy connection through that which the street would lead you around. Depending on your mood…

Russell’s Bane: Why LLMs Don’t Know What They’re Saying

by Jochen Szangolies Recently, the exponential growth of AI capabilities has been outpaced only by the exponential growth of breathless claims about their coming capabilities, with some arguing that performance on par with humans in every domain (artificial general intelligence or AGI) may only be seven months away, arriving by November of this year. My…

Superdeterminism: Quantum Mechanics Demystified Or The End Of Science?

by Jochen Szangolies The quantum world, according to the latest science, is, like, really weird. Cats that are both dead and alive, particles tunnel through impenetrable walls, Heisenberg can’t both tell you where he is and how fast he’s going, and spooky influences connect systems instantaneously across vast distances. So it seems that anything that…

The How Of Why: Not Quite A Review (Part II)

by Jochen Szangolies In the previous column, I took Philip Goff’s latest offering Why? The Purpose of the Universe as a jumping-off point to present some of my own rumination on life, the universe, and what it all means. While that prior installment was mainly concerned with looking outward, into the wider cosmos, here, I’ll…

The How Of Why: Not Quite A Review (Part I)

by Jochen Szangolies I’m inherently suspicious of overt declarations of having arrived at a certain position only through the strength of the arguments in its favor, even against one’s own prior commitments. If that were typically how things happen, then either there ought to be much more agreement than there is, or the vast majority…

Wrestling With Existence: Social Reality And Modalities Of Untruth

by Jochen Szangolies A proposition, like “it’s raining outside”, can either be true or false—it might be the case that it actually rains, or not. It is then seductive to think that somebody uttering such a proposition is doing nothing but making a factual claim, and in doing so, either tells the truth, or not.…

Münchhausen And The Quantum: Dragging Ourselves Out Of The Swamp

by Jochen Szangolies There seems no obvious link between tall war-tales, shared among a circle of German aristocrats in the 1760s, and quantum mechanics. The former would eventually come to form the basis of the exploits of Baron Münchhausen, the partly fictionalized avatar of Hieronymus Karl Friedrich, Freiherr von Münchhausen, famous for his extravagant narratives,…

Wigner’s Many Friends: Quantum Mechanics And Reality

by Jochen Szangolies Whenever I use words like ‘reality’, ‘truth’, or ‘existence’, I feel an almost irresistible urge to mark my vague sense of unease by liberal application of scare quotes. After all, what could such words even mean? They seem to denote concepts too vast and simultaneously slippery to be pinned down by a…

Gödel’s Proof and Einstein’s Dice: Undecidability in Mathematics and Physics – Part III

by Jochen Szangolies The simulation argument, most notably associated with the philosopher Nick Bostrom, asserts that given reasonable premises, the world we see around us is very likely not, in fact, the real world, but a simulation run on unfathomably powerful supercomputers. In a nutshell, the argument is that if humanity lives long enough to…

Gödel’s Proof and Einstein’s Dice: Undecidability in Mathematics and Physics – Part II

by Jochen Szangolies The previous column left us with the tantalizing possibility of connecting Gödelian undecidability to quantum mechanical indeterminacy. At this point, however, we need to step back a little. Gödel’s result inhabits the rarefied realm of mathematical logic, with its crisply stated axioms and crystalline, immutable truths. It is not at all clear…