Functions like sine and cosine have some pretty nice properties: they're always differentiable and well-behaved (i.e. don't ever blow up to infinity). Apparently.

Yet it turns out that such functions are actually impossible, at least over the complex numbers. (1/9) As soon as we start extending sine and cosine to the complex plane, we see that they start blowing up - they don't remain nicely bounded like they do over the reals. We could try imposing bounds on them, but that would quickly break their differentiability. (2/9)

"Duality" is a deep concept in mathematics, but an intuitive way to think about it is in terms of tables.
When we lay out data in a table, we are familiar with the idea that each row represents a different "entity", and each column represents a different "property". (1/10) If we flip it over (swapping rows for columns), now we have a new data layout in which each "entity" is really a property, and each "property" is really an entity. It's clearly the same data, just represented differently; if we flip again, we get back to where we started. (2/10).

Gödel's first incompleteness theorem is commonly proved by means of a diagonal argument. But, in retrospect, we can see that what Gödel was really doing was proving that Peano arithmetic is Turing-complete, and then applying an argument from computational irreducibility... (1/15)

First, the standard proof. In Peano arithmetic (integers with multiplication), the fundamental theorem of arithmetic guarantees that all integers greater than 1 admit a unique prime factorization. So a sequence of integers can be encoded uniquely as a single integer... (2/15)

Moths are attracted to lights because of the same mathematics that underlies twistor theory and compactification in theoretical physics: projective geometry.

It all starts from a simple observation: translations are just rotations whose center is located "at infinity". (1/11) So if we take an ordinary space (like the 2D plane) and we adjoin a hypothetical point "at infinity", then the distinction between translations and rotations disappears: as the center of rotation moves further and further away from the object being rotated,... (2/11)

As this is pretty much the only statement of my personal philosophical outlook on metaphysics/ontology that I've ever made on here, I should probably provide a little further clarification.
It starts from a central idea from philosophy of science: theory-ladenness. (1/11)

https://twitter.com/getjonwithit/status/1813766177862353132

As argued by Hanson, Kuhn, etc., raw sense data is filtered through many layers of perception and analysis before it may be said to constitute "an observation". So making a truly "bare metal" observation of "reality" (uninfluenced by theoretical models) is impossible. (2/11)

At first glance, the idea that distributions of stars in the outermost regions of the universe directly affect physics here on Earth seems pretty absurd.
But, as it turns out, reconciling the concept of rotation with general relativity requires some pretty absurd things. (1/16) Relativity is based on the idea that physics is the same in all "inertial" frames, i.e. all frames of reference that are not accelerating. If I move every object in the universe one meter to the left, then I haven't changed the laws of physics: only the coordinate system. (2/16)

"Quantum mechanics is just thermodynamics in imaginary time."

It sounds like a hand-wavy, quasi-philosophical statement. But with a brief dive into the relationship(s) between hyperbolic and parabolic PDEs, it becomes possible to formalize it mathematically. (1/16)

(2nd-order) PDEs may be classified broadly as hyperbolic, elliptic or parabolic, based upon their relationship with time and causality. Hyperbolic PDEs have a notion of time/causality: a perturbation somewhere in the domain propagates out, wave-like, with a finite speed. (2/16)

In physics, one often thinks of space and time as being fundamental, pre-existing concepts, and proceeds to define everything else (energy, momentum, forces, etc.) in terms of them. But it doesn't need to be so - symplectic geometry shows us how to go the other way. (1/16) Suppose you have two functions, f and g, and you want to quantify "How much does f change as a result of a flow generated by g?"
Well, that's exactly what the Poisson bracket {f, g} measures. [We'll come on to precisely what a "flow generated by g" means in a moment…] (2/16)

Fibrations/bundles/sections/etc. are slightly opaque-sounding terms for an otherwise very intuitive idea: that you can parameterize a collection of spaces in terms of points from a different space, and then assemble new spaces out of points taken from that collection... (1/15)

Take the simple example of a curve on a 2D plane. That curve is a 1D space, and at every point along it there is a line (i.e. a different 1D space) that is tangent to it. So we can say that the collection of tangent lines is parameterized by the points on the curve. (2/15)

"The boundary of a boundary is always empty."
A huge amount of (classical) physics, including much of general relativity and electromagnetism, can be deduced directly from this simple mathematical fact.
Yet, on the surface, it doesn't seem to have much to do with physics. (1/10) Some spaces (like spheres) don't have boundaries. But, when the boundary exists, it's always one dimension lower (codimension-1). A disc is a 2-dimensional space, but its boundary is a 1-dimensional circle.
But what's the boundary of a circle? Well, it doesn't have one. (2/10)

A few people have asked what "fully covariant computation" means, with regards to my last post. I'm currently writing up a big paper about this, but since that will take a while to finish, let me try explaining the basic idea.
Consider, for a moment, general relativity... (1/9) Often, when we think of solving the Einstein equations, we think of defining initial data on a spacelike hypersurface (a Cauchy surface, or "instantaneous snapshot") and then evolving it forwards in time.
But general covariance means that this is not the only way to do it. (2/9)

Here's a quick story about comonads, and how they can be used to unify various concepts in functional programming, category theory, rewriting systems and multicomputation. It starts with a practical problem: how to implement better compositional rewriting in Mathematica. (1/11) First, a brief recap of some basic functional programming/category theory. A “functor” is just a Map in Mathematica: it allows one to apply a function to every element of a data structure (e.g. a list), whilst still preserving the "shape" of that data structure. (2/11)

Shock waves, solitons and spacetime singularities: what do sonic booms and water ripples have to do with cosmic censorship and the determinism (or otherwise) of general relativity?
Brace for a brief adventure in hyperbolic partial differential equations (PDEs). (1/20)

PDEs may be broadly classified into three types: elliptic, which are time-independent and where information propagation is instantaneous; parabolic, which are time-dependent and diffusive; and hyperbolic, which are time-dependent and non-diffusive (i.e. wavelike). (2/20)

[This is part 2 of my series on tensor calculus, differential geometry, etc.]
These blobs are called "tensors", and the "rank" of a tensor represents how many legs it has. Here's a rank-4 tensor with two contravariant indices (inputs) and two covariant indices (outputs). (1/11)

https://twitter.com/getjonwithit/status/1724541884087885882

We can build higher-rank tensors out of lower-rank ones by stacking the blobs on top of each other; for instance, here we have assembled a rank-4 tensor out of two rank-2 tensors. As an equation: "M_{j}^{i} N_{l}^{k} = T_{j l}^{i k}". This is called a "tensor product". (2/11)

Lots of pop-sci talks about general relativity and quantum mechanics being “incompatible”, but what does that really mean? And how do quantum field theory, curved spacetime and (effective) quantum gravity fit into that picture? It's a surprisingly interesting story... (1/17)

One of the most important features of quantum mechanics is that it is *linear*: when we add two solutions to the Schrödinger equation together, the result is always another solution to the Schrödinger equation. Physically, this corresponds to the superposition principle. (2/17)

[This thread is hopefully(!) the first in a series on tensor calculus, differential geometry and the foundations of general relativity...]
This is a blob. This blob has no inputs and no outputs, so we shall refer to it as a "scalar", and we denote it by an "S". (1/10) This is a one-legged blob. This blob has one input (denoted "i") and no outputs, so we shall refer to it as a "vector", and we denote it by "V^i" with a superscript. The superscripted (input) index "i" is referred to as a "contravariant" index. (2/10)

What actually *is* curvature? It's a surprisingly hard question, and one which wasn't satisfactorily answered until the early 20th century, thanks to the work of Tullio Levi-Civita, Gregorio Ricci-Curbastro and other (largely Italian) differential geometers. (1/9) The (deep) answer?
Curvature is a measure of how much a space fails to be parameterized by a single coordinate system.
The room you're in right now is not (appreciably) curved, meaning that you can describe every point within it uniquely with a set of (x, y, z) coordinates. (2/9)

The Rabin-Scott theorem is one of the (philosophically) deepest mathematical results I know. When properly understood, I claim that it can't help but alter your view of reality in a fairly foundational way. Yet its typical textbook presentation obscures much of this depth. (1/8) Suppose you have a non-deterministic computer (formally, a non-deterministic finite automaton) which can perform a whole tree of possible actions from a given input state, and I have a deterministic one, which can only follow a single path. Clearly yours is better, right? (2/8)

What might spinning black holes be telling us about (the futility of) time travel?
The metric tensor in general relativity has three positive eigenvalues and one negative eigenvalue; geometrically, this means that three coordinate directions yield positive distances... (1/10) ...and one yields negative distances. We interpret the positive directions as space coordinates and the negative direction as time; we have freedom of motion in the positive/space directions, but are "locked" to progress monotonically in the negative/time direction. (2/10)

Everything we know about fundamental physics may be summarized by the statement:
"Nature doesn't care about coordinate systems."
Indeed, rather remarkably, all of our most foundational theories of physics appear to have (essentially) no content *apart* from this statement. (1/9) If you start from a smooth, 4-dimensional Lorentzian manifold (spacetime) and want to start "doing physics" on it, it is helpful to define local space and time coordinates at every point. But there is much freedom in how to do this, with many permissible coordinate choices. (2/9)

Indeed, and it's rather beautiful (and leads to some potentially interesting predictions)! First, imagine a multiway system in which a black hole forms on some branches of history but not on others. Because the BH is a relatively "ordered" state... (1/7)

https://twitter.com/mikeysee/status/1717866770424647804

...compared to the "pure" background hypergraph, the branches on which the BH didn't form eventually become entropically dominant over the ones in which it did. This has the effect of reducing the overall amplitude associated with the presence of the BH... (2/7)