Alright, replies indicate I need to explain this in more detail.

Properly conceived, there is simply no difference between "simulated water" and "water". It's just water. But to understand that, one first needs to distinguish between two meanings of the word "computer". (1/11)

https://twitter.com/dearmadisonblue/status/2045559826168545530

The term "computer" can mean either an abstract system (i.e. a Turing machine), *or* a concrete instantiation of that system in physical hardware. Let's talk about the second case. A physical computer encodes abstract computational states (i.e. Turing machine states)... (2/11)

"The role of symmetry in physics" has always felt like a tautology to me.

The study of physics *begins* with the observation that certain aspects of reality remain regular across space, time, and structure. We could imagine (perhaps?) a completely orderless world. (1/8)

https://twitter.com/EricRWeinstein/status/2043012737949700310

One in which every particle does its own thing, obeying its own set of rules. Where what I experienced yesterday has no correlation with what I'll experience tomorrow. Where the behavior of an object in one room predicts nothing about its behavior in another. (2/8)

I think one of the conclusions we should draw from the tremendous success of LLMs is how much of human knowledge and society exists at very low levels of Kolmogorov complexity.

We are entering an era where the minimal representation of a human cultural artifact... (1/12) ...will be, generically, an LLM prompt. And those prompts will be, generically, orders of magnitude more compact than the artifacts themselves. The great success of coding agents, for instance, indicates that the source code of most software artifacts is orders of... (2/12)

New paper!

What if you could guarantee (using a mix of formal verification and PDE theory) that a neural network would *always* give you the correct answer, even when making inferences arbitrarily far away from the training data?

Introducing BEACONS. arXiv link below. (1/15)
Link: 
In physics, we often want to use neural networks to infer new solutions to systems of PDEs. But outside of the spatiotemporal ranges on which they're trained, even physics-informed neural networks (PINNs) struggle to extrapolate correctly. (2/15)arxiv.org/abs/2602.14853

I remember this moment very vividly. Apologies in advance for the arrogant and self-indulgent personal anecdote...

I originally went to university with the intention of becoming a pure mathematician, and for the first couple of years this dream seemed to be pretty safe. (1/10)

https://twitter.com/elocinationn/status/2010474719607427290

I generally came top (or near top) in exams, sat in on graduate-level courses, tackled some research problems, published some papers. I'd convinced myself that I could understand any mathematical structure if I just wrote down the rules and stared at them for a bit. (2/10)

Like @davidbessis and others, I think that Hinton is wrong. To explain why, let me tell you a brief story.

About a decade ago, in 2017, I developed an automated theorem-proving framework that was ultimately integrated into Mathematica (see: youtube.com/watch?v=mMaid2…) (1/15) x.com/vitrupo/status… The real reason for building it, beyond it just being a fun algorithmic puzzle, was because my then-collaborator @stephen_wolfram and I wanted to perform an empirical investigation: to systematically enumerate possible axiom systems, and see what theorems were true. (2/15)

What all of these explanations get "wrong" is that they try to explain what a tensor is. Yet tensors themselves are neither fundamental nor important.

The fundamental/important thing is the tensor *product*. A tensor is just an object assembled using tensor products... (1/9)

https://twitter.com/vikhyatk/status/1969151806501487096

The tensor product ⊗ is, conceptually, the most general (binary) operation that behaves "how a product should behave". In practice, this means that the order of brackets shouldn't matter, i.e. X⊗(Y⊗Z) should be the same as (X⊗Y)⊗Z, for any objects X, Y and Z... (2/9)

How many holes does a straw have?

Topologically, of course, it has 1: it's homeomorphic to a punctured disk. But intuitively it has 2: one at the top and one at the bottom. And this answer lies at the heart of the most rigorous axiomatization of quantum field theory. (1/20) In this intuitive picture, the two "holes" of the straw are 1-dimensional circles, and they're connected by a 2-dimensional cylinder (the straw itself). Mathematically, this relationship is called a "cobordism". Two n-dimensional manifolds are "cobordant" if they form... (2/20)

The desiccated "Theorem, Lemma, Proof, Corollary,..." presentational style is staggeringly counterproductive, if one's objective is actually communicating the underlying mathematical intuitions and thought processes behind a result. In reality, the process is more like... (1/4)

https://twitter.com/EliotJacobson/status/1942937846484177239

"First, I tried <standard method>, but it failed for <enlightening reason>, so I investigated whether I could exploit this fact to find <counterexample> with <property>, but all objects obtained through this technique ended up having <interesting property> in common.... (2/4)

What's a gravitational wave? Anything that distorts the shape of spacetime, but preserves its volume.

What's matter/energy/momentum? Anything that distorts the volume of spacetime, but preserves its shape.

A 🧵 on the Ricci decomposition theorem, as applied to gravity. (1/13) Classical gravity is a manifestation of the Riemann curvature of spacetime, which describes how your coordinate system distorts as you move from point to point. More precisely, the *connection* describes how the coordinate system distorts, and the Riemann curvature... (2/13)

Calling c the "speed of light" completely misses the point. Rather, c is the "spacetime exchange rate": how many units of space you can exchange for one unit of time.

In actuality, everything travels at the "speed of light", just not necessarily through space alone... (1/4) Rather, everything travels through both space *and* time, simultaneously, with a speed of c. If you're standing still, then all of your velocity is focused in the time direction (with none in the space directions), so you move through time with a speed of c. (2/4)

Sure, I’ll give it a go…

Consider a rotating disk. What does it mean to say that the disk has angular momentum? Well, imagine assigning a momentum vector to every point on the surface of the disk, and then slicing through the middle of the disk with a flat surface. (1/14)

https://twitter.com/ThePhysicsMemes/status/1927662467481469428

The "net flux” of momentum vectors through the surface is zero, since every momentum vector poking through the surface in one direction is counteracted by a momentum vector poking through in the opposite direction. In other words, the disk has no *linear* momentum. (2/14)

New paper alert!

Birkhoff's theorem tells us that the spacetime around a non-rotating black hole is indistinguishable from that around any other non-rotating compact object, like a neutron star.

But what if it's rotating? Turns out, the differences can be *huge*. (1/4)
Link:

Though the spacetime around an uncharged black hole depends on two parameters (mass and spin) by the no-hair theorem(s), objects like neutron stars have "hair" in the form of many other multipole moments: mass quadrupole, spin octupole, etc. (2/4)arxiv.org/abs/2505.05299

My recent "dunk" about encoding functions and the algorithmic/Kolmogorov complexity of the laws of physics may have seemed flippant, but it actually goes back to an old 17th century philosophical conundrum: the dichotomy between idealism and materialism.

Let me explain. (1/14)

https://twitter.com/getjonwithit/status/1911933388895756371

When attempting to model the world computationally, there are typically *three* computations that one needs to consider: the computation that the system itself (e.g. the universe) performs, the computation that the observer performs, and the "encoding function". (2/14)

New paper alert!

We developed the first automated theorem-proving framework for (hyperbolic) PDE solvers: now you can build *formally verified* physics simulations, with provable mathematical and physical correctness properties.

arXiv link and explanation in thread... (1/10)
Link:
Hyperbolic PDEs are the foundation of most simulations in hydrodynamics, electromagnetism, general relativity, etc. But solvers often become unstable, fail to preserve hyperbolicity, or introduce new extrema, unless one is very careful. (2/10)arxiv.org/abs/2503.13877

"General relativity doesn't admit black hole solutions. It only admits *wormhole* solutions."

I have previously made this statement and had people get confused by it. So let me try to clarify precisely what this means, using a neat analogy to real/complex analysis. (1/17)
When you first encounter the sine function, you probably see it defined in terms of triangles, which means that it really only makes sense for x between -pi and pi (or -180 and 180 degrees). But later on, you learn that the "true" sine function is much larger. (2/17)

Energy, momentum, pressure, stress, etc. are all just different ways of quantifying the same basic thing: how our perceptions of space and time get distorted over time.

And once you internalize this, it allows you to think about these concepts in a much more general way. (1/12) Suppose that the room you're in defines a 3-dimensional coordinate system (x, y, z). If the walls in the x-direction get pulled apart in the x-direction, and those in the y-direction get pulled apart in the y-direction at the same rate, etc., then the room is expanding. (2/12)

"Oh, I'm a *pure* mathematician, I don't write code/do calculations/etc.."
"Oh, I'm a *theoretical* physicist, I don't do experiments/analyze data/etc.."
Etc.

These kinds of statements are typically uttered with an air of intellectual smugness. But what are they really? (1/6) In actuality, this classification of certain tasks as "worthy" (e.g. proving theorems, developing models) and others as "beneath oneself" (e.g. doing calculations, writing code) is a signal of a fundamental absence of intellectual curiosity. (2/6)

To illustrate just how different neutron star and black hole metrics truly are, I simulated perfect fluid accretion onto a black hole vs. onto a neutron star of identical mass and spin.

Source code and simulation details in thread below. (1/4)

https://twitter.com/getjonwithit/status/1878094729809723568

To produce this, I simulated supersonic accretion of a perfect fluid, obeying an ideal gas equation of state, initially onto a standard black hole metric in Cartesian Kerr-Schild coordinates, and then onto a physically realistic rotating neutron star metric due to Pappas. (2/4)

There's something which almost every major neutron star simulation gets wrong, and it's related to a widespread confusion about the behavior of rotating objects in general relativity.

The confusion goes back to one of the most beautiful results in GR: Birkhoff's theorem. (1/14)
Here's one way to think about Birkhoff's theorem: suppose that you have a perfectly spherically-symmetric object that is uncharged and not rotating. Clearly, by measuring its gravitational field (i.e. the extent to which it curves spacetime), you can determine its mass. (2/14)

I'm often described (due to my work on computable physics) as being an advocate for the Simulation Hypothesis.
But I'm not.
In fact, I think the Hypothesis is nonsensical, and that the most famous argument in its favor actually proves the opposite of what it purports. (1/10) First, it is crucial to distinguish two things: a computation (which we can formalize as the operation of a Turing machine), and a simulation (which is a specific type of computation, whose output states are interpreted as approximate states of some *external* system). (2/10)