What is infinity, exactly?

For over 2,000 years, humanity insisted that infinity was only potential (roughly, this means you can always add "one more," but you never actually arrive). Aristotle thought so; Gauss just said so. Then Cantor showed up with a sort of hearsay we're still fighting about to this day.

He treated infinite collections as completed objects in and of themselves. Then he proved there are strictly more of them than anyone imagined. Kronecker called Cantor a "corrupter of youth." Poincaré called Cantor's work a "disease." Cantor died in a sanatorium.

Turns out, Cantor was right all along. Here's what's going on. (1/18) The distinction is sharper than it sounds. A potential infinity is a process: you keep counting 1, 2, 3, ... and you never stop. At no point do you hold "the completed collection of all natural numbers" in your hand. An actual infinity says: that collection exists, right now, as a single object. You can examine its properties. You can compare it to other collections. You can ask difficult questions about its "size." You get the idea. Cantor's great heresy was insisting on the latter, and then doing some math with it. (2/18)

Something you may not have heard of is Malament-Hogarth spacetimes. Besides sounding like a holographic Yu-Gi-Oh card, these spacetimes allow us to compute what we thought was uncomputable. To understand this, we have to talk about what we mean by "computable." (1/16) The Church-Turing thesis is often misread as a "theorem," but it's more like a claim that "effectively computable" just means "computable by a Turing machine." To me, it's more like a definition. It's also often upgraded to a STRONGER assertion about physics: that no physical process can compute anything a Turing machine can't. This version (the physical Church-Turing thesis) is also NOT a theorem. It's a HYPOTHESIS about the laws of nature. (2/16)

What if the universe isn’t actually made of points, waves, fields, particles, or even Lagrangian submanifolds, but instead… natural transformations? The Yoneda Lemma, a theorem that identifies elements u ∈ F(A) with transformations Φᵤ: hₐ → F, makes this view mathematically concrete. The proof of the Yoneda Lemma takes perhaps three lines (hence why it’s a “lemma” despite its weight). 👇🧵 (1/22) Its implications, however, reach algebraic geometry, representation theory, and even parts of theoretical physics. If you truly want to actually understand Tannaka duality, Isbell conjugation, or Grothendieck’s schemes, then you don’t get terribly far without Yoneda. But how can chasing idₐ be this impactful? The “Yoneda perspective,” that objects are their relations, is forced on you by the math. (2/22)

You've probably been seeing viral videos saying particles take "all possible paths." It's based on an extreme misunderstanding of path integrals. Let's disambiguate what the path integral is about and why Feynman's tool isn't a literal map of reality. Firstly, we need to stop saying electrons "go through both slits." That's not what quantum mechanics (even textbook QM) says. 👇🧵 (1/20) It's a hangover from misinterpreting wave functions in 3D space when they really live in something else called "configuration space." It confuses a calculational trick with physical ontology. Time for some rigor. \int D\phi, e^{iS[\phi]/\hbar People love to say that particles explore every possible path simultaneously—even going back in time or to the moon. Firstly, what is this word “possible”? Possible isn't a physics word. Do you mean to say every continuous path in R^4? Every once differentiable path in R^3? What is it? (2/20)

“All the towering materialism which dominates the modern mind rests ultimately upon one assumption; a false assumption. It is supposed that if a thing goes on repeating itself it is probably dead; a piece of clockwork. 👇🧵 (1/15) People feel that if the universe was personal it would vary; if the sun were alive it would dance. This is a fallacy even in relation to known fact. (2/15)

Think you know what energy is? You probably don’t. That’s okay. Einstein probably didn’t either, at least not in the context of his own masterpiece, General Relativity. Forget the pop-sci soundbites you hear from people like NDT. Energy is NOT simply “mass in motion” or “mass because E=mc²” or even the neatly “conserved currency of our universe.” These definitions (to the degree they’re definitions) don’t hold up in dynamically curved spacetime. 👇🧵 (1/20) Most likely, your GR instructor glossed over energy, perhaps mumbled something about “pseudo-tensors” under their breath, then quickly changed the subject. Why the rush? Why the evasion on such a supposedly fundamental concept? The full, honest treatment is extremely messy, deeply controversial, and fundamentally unresolved even after a century. Einstein himself wrestled with it, and the compromises he made are still debated today. Let’s talk about that mess. (2/20)

Philosophers who use Gödel's incompleteness theorem to make claims about "fundamental limits of human knowledge" have made a category error. It's about axiomatization, not epistemology. 1/ Firstly, epistemology is just fancy technical jargon for "what and how we can know." So, knowledge. Questions in the field of epistemology are questions that deal with the nature, sources, and limits of knowledge. The "nature" of this knowledge even includes defining what knowledge is (see Gettier's problem), but this is beside the point. 2/

“Entropy is geometry. And geometry is entropy.” This is a new finding by Gabriele Carcassi, and I'll explain the reasoning below, along with the math. Don't worry, I'll hold your hand (metaphorically, of course, unless you're into that). (1/19) Firstly, note that entropy is NOT a measure of disorder… It's a way you count states. From this, we'll see that geometry *is* entropy, and entropy *is* geometry. Let's start with classical mechanics... (2/19)

The Free Energy Principle: a 'theory of everything' that includes not only brains, societies, but perhaps even the universe itself… It's known for being considerably convoluted but the principles underlying it are actually straight forward. If the FEP is right, then is your entire reality a 'controlled hallucination.'? What does that even mean? And how does this relate to entropy? 1/13 Here's the core idea, stripped of the (intimidating) math. Imagine you're in a completely dark room. You can't *see* anything, but you still have expectations. You *predict* where the walls are, where the furniture might be. Now you see that your brain doesn't merely receive but *infers*. This is the difference between passivity and activity. 2/13

Godel's incompleteness theorem (all consistent formal systems aren't "complete" (provided it models arithmetic)) and Turing's theorem (you can't always determine if a program halts) are what you've likely heard of already. There are various other no-go results in philosophy / math, like Cantor's theorem, Rice's, Lob's, Tarski's undefinabilty as well… What most people don't know about is that there's just *one* theorem that underlies all of these: Lawvere's fixed point theorem. 1/13 When a function maps elements from one set to another, Lawvere showed that if you have a "nice" function (technically, a "fixed point operator") that can map elements from a set of functions to another set, you'll *always* find a fixed point (an element that maps to itself). Importantly, we don't assume the existence of this operator. We *derive* it. That's the power of this theorem. 2/13

“There is no wave function...” This claim by Jacob Barandes sounds outlandish, but allow me to justify it with a blend of intuition regarding physics and rigor regarding math. We'll dispel some quantum woo myths along the way. (1/13) Most people think of quantum mechanics as being about wave functions. What if Ψ isn't' fundamental? What if it's just a mathematical convenience? What happens to the associated devices like Hilbert spaces and state vectors? To Jacob, sure, they're useful, but they're not "real."

(I understand that you should technically be dealing with the endomorphisms of Hilbert spaces rather than direct members of them, but this is something unnecessary to get into currently.) (2/13)

Is the universe countably infinite or uncountably infinite?... I used to think language describes reality uniquely. However, Putnam showed that the Löwenheim-Skolem theorem says otherwise. Specifically, the concept of “infinity” has different meanings in different models. It's quite abstract but let me explain. (1/10) First, what's a “model”? In a formal theory, it's a set of axioms and rules (actually it's what “satisfies” those). Examples would be set theory or arithmetic. A “model” for that theory is like a mathematical world (technically called by model-theorists a “Universe,” interestingly enough) where all those axioms and rules are true. Some say it's like an interpretation, but I'd say it's more like what's being “referred to.” So, if you have a theory that says, “There exists an infinite set,” a model for that theory would be a mathematical structure that contains an infinite set. Super simple so far. (2/10)

"Everything is a Lagrangian submanifold..." Forget particles. Forget waves. Alan Weinstein quipped that the universe is built from something entirely different: Lagrangian submanifolds. What are those, and why should you care? To grasp Lagrangian submanifolds, you first need to know about phase space… (1/10) Usually people say phase space (incorrectly) when they actually mean “state space” so lets' define this phase space. It's an abstract space (not spacetime) where each point represents a particle's state is given by its position (q) and momentum (p). So, phase space is a space of (q, p) pairs. You can extend this to N particles, of course. (2/10)

The canonical quantization you're taught -- replace {f,g} with [f,g]/iℏ -- is, well, ill-defined. You can always add a classical 0 that becomes non-zero quantumly (thanks to non-commutation). But there's another way: geometric quantization. (1/7) It all starts w/ symplectic geometry. Think of a classical phase space, but forget coordinates for the moment. What you require is something called a symplectic form -- a closed, non-degenerate 2-form, ω... (2/7)