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Jonathan Gorard Profile picture
@getjonwithit
Jul 10 • 4 tweets • 1 min read • Read on X
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)
"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)
...So I tried relaxing <axiom> to see whether <related property> could be removed, and this led me to realize that <intermediate lemma> is actually crucial to the structure of <related object>..." Etc. You occasionally get these insights from (very good) mathematical talks. (3/4)
Yet, when it comes time to write the paper, it's all just "Proposition, Lemma, Proof, Corollary". Place all the burden on the reader to reconstruct the actual thought process. IMO Bourbaki et al. did incalculable damage to the art of effective mathematical communication. (4/4)

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More from @getjonwithit

Jonathan Gorard Profile picture
Jun 2
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)
...describes how the connection distorts. So the Riemann curvature is effectively a second derivative of your coordinate system. The Ricci decomposition theorem then says that the Riemann curvature can be decomposed into two pieces: a "trace" part and a "trace-free" part. (3/13)
Read 13 tweets
Jonathan Gorard Profile picture
May 31
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) Image
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)
If you start moving, then now a little bit of your velocity vector points in one of the space directions, so a little bit less must point in the time direction. So you move through time slightly slower than c, such that your overall speed through space *and* time remains c. (3/4)
Read 4 tweets
Jonathan Gorard Profile picture
May 29
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) Image
Image
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)
But the "total flux" of momentum vectors (i.e. the total amount of momentum intersecting the surface, irrespective of direction) is clearly non-zero, because the disk is rotating. This discrepancy between the total flux vs. the net flux is what we call "angular momentum". (3/14)
Read 14 tweets
Jonathan Gorard Profile picture
May 9
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) Image
Image
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
Most calculations and simulations of neutron stars assume that such moments don't matter (i.e. that the geometry is well-described by the Kerr metric of a spinning black hole). We show that this isn't true, for physically realistic neutron stars spinning at moderate speeds. (3/4)
Read 4 tweets
Jonathan Gorard Profile picture
Apr 16
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)
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)
The encoding function is the computation that maps between states of the system and states of the observer. You might think that the system's computation is the only one that matters, but science is full of examples where the state of "the observer" matters too. (3/14)
Read 14 tweets
Jonathan Gorard Profile picture
Mar 19
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) Image
Image
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
Worse, a solver may converge to a solution that is mathematically correct but physically invalid (e.g. because it decreases total entropy, or fails to conserve mass/energy/momentum). Such problems get more pronounced in the presence of discontinuities, like shock waves. (3/10)
Read 10 tweets

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