There's two common uses of this term - in philosophy of science and in mathematics (arguably it was also a philosophical issue when first described). u/o0oo0oo0oo0ooo's answer (as well as the Stanford philosophy page) are good resources on the philosophy of science one. Edit: It seems like that answer was erroneously removed - if you were actually asking about that, let me know and I can summarize it as well.

In mathematics, incommensurability is best understood through the classical Greek conception of numbers around Archimedes' time. The fundamental entities were the whole numbers 1, 2, 3, ... . There was no decimal notation, but it was obvious that numbers could measure lengths, and that lengths between the whole numbers existed. They developed a system equivalent to fractions, but they thought of them slightly different. If some interval had length 3/2, that meant that you could break it up into 3 intervals, each of which was the result of breaking up some reference interval into 2 parts. In general, fractions were not their own numbers, but could only refer to some kind of relationship between the numerator and denominator, which were numbers. Measuring a length using fractions actually meant you were comparing the lengths of various intervals, and those intervals are called commensurable - literally, co-measurable, or having a common measure.

This was ruined with the discovery of the 45-45-90 right triangle. If this triangle had legs with length 1, then its hypotenuse has length √2. As you may have learned, √2 is an irrational number, which means it cannot be written as the ratio of two whole numbers. This fundamentally broke the previous conception of the relationship between lengths, fractions, and numbers - we had found two lengths that were incommensurable, not able to be measured relative to each other (using the only kind of number that existed at the time, whole numbers).

They had discovered that their notion of length was broader than their numerical notion of measure. This was just a feature of classical geometry (some lengths were commensurable, some were not), but nowadays we don't make that distinction as much because people are comfortable using real numbers and Cartesian geometry, where irrational lengths are just another number. Also, nowadays instead of the above, we say that commensurable means expressible as a rational number, and incommensurable means irrational. This definition is identical, though it loses some of the historical context.

There's a significant incentive for programmers early in their career, which is as a resume builder. College students are often advised to put school or independent projects on github to use during interviews.

Charged particles are coupled to the electromagnetic field. These means that they can change the shape of the field, and if the field is nonzero in their vicinity, they will experience some kind of force.

For electric charges, this picture is intuitive. Two electrons generate an outward-pointing electric field. If they're next to each other, each electron's field pushes the other one away, and they move apart.

Whenever a charged particle has some kind of motion associated with it, it generates loops of magnetic field around it. Similarly, whenever a charged particle has some kind of motion and is in a magnetic field, it will experience a force perpendicular to the direction of the magnetic field. The geometry is a lot more complex, so you can get things pushing on each other at odd angles, but in many cases the directionality of the loops and the forces cancel out, and you get a normal attractive force.

In everyday circumstances, most magnetic fields are associated specifically with electrons. There are three common kinds of motions of electrons. Their intrinsic spin, which generates a dipole field focused on the electron. Their orbit around atomic nuclei, which also generates a dipole field but focused on the center of the atom. And their motion through a wire (or through space, like in a thunderbolt), which generates circulating magnetic fields around it.

Most magnetism you see is some kind of interaction of these three types. For example, a bar magnetic picking up a paperclip. The inside of the bar magnet has a bunch of electrons' intrinsic magnetic fields lined up. The electrons in the paperclip feel this, experience a rotational force to line up their magnetic fields with the bar magnets' field, and then once they're lined up experience an attractive force toward the bar magnet. An electromagnet also generates a magnetic field, but using the bulk motion of electrons through the wire, which also allows it to pick stuff up.

>Even at locally finite & bounded speed. >

How would that work? It seems like in the actual transition from a point into a space, speed wouldn't even be definable. Is there an explicit construction of such a contraction?

We know that on the surface of the Earth, there is a maximum distance you can get from any other object. We could have confirmed this empirically by placing transmitters at points around the world and seeing how far away they were, but we instead we proved it conclusively by showing that the Earth is a sphere, and given the way that distances work, it doesn't make sense for something to be more than 20,000 km away along Earth's surface. If you try to, the object will seem to appear closer behind you, though from the outside we know it just moved across the point opposite you on the Earth's surface.

There's an analogous fact about spacetime. Minkowski space, which seems to be what we live in, does not geometrically allow for something to accelerate above the speed of light. It's not simply something we haven't observed, it's more like trying to get more than 20,000 km away on Earth's surface - the way velocities work in this geometry make that physical action kind of nonsensical. We 'proved' things can't go faster than light by experimentally confirming various effects that we would expect in a Minkowski space like length contraction and time dilation.

It may be that things can go faster than light, but if so it will require new physics, and will mean we don't really live in a Minkowski space, but something that usually behaves similarly but something is different. This is very probably the case, since we know that general relativity is an incomplete theory, but so far no extensions allow for faster-than-light travel either.

Wireless internet connections use light in the radio frequency ranges.

If you want to send someone a signal with visible light, you can turn the light on and off to communicate in Morse code. This isn't as time-efficient as using a whole display, but it allows you a lot of flexibility - the receiver can stand anywhere, and the sender can turn the light on and off without worrying whether it is oriented correctly. Computers do something similar. The encoding system is much more complicated than Morse code, but the basic idea of changing how the light is emitted over time to transmit data is the same.

These radio frequency ranges have wavelengths between millimeters and meters (they go longer, but the longer wavelengths usually aren't' used for data channels). For these long wavelengths, many physical barriers are translucent, the way cloudy glass is for visible light. The transition isn't perfect, but we intentionally use an encoding that can handle a little bit of noise so you can still receive those signals.

> So it shouldn't matter at what speed the spacecraft is traveling since the thing it's doing work against (propellant) is always traveling at the same speed as the spacecraft?

That is exactly right.

From the spaceship, when I expel some exhaust, I gain some constant amount of energy. So, it makes sense to me that me acceleration is steady.

From the ground, for a spaceship to have steady acceleration, it must be gaining more and more energy per second. The faster I am going, the more energy I need to gain a little bit more speed. This extra energy comes from the fact that the fuel is moving along with the spaceship - from the ground, the fact that it's moving means it has extra kinetic energy to expend on propelling the spaceship. If you do the algebra, you'll find that the extra quadratic terms on the spaceship's energy and the fuel's energy exactly cancel out.

So the actual effect (constant acceleration of the spaceship) looks the same, but the accounting of which object has how much kinetic energy looks completely different.

I also had this problem. It seems unintuitive that somehow going from 50 to 60 mph adds a different amount of oomph to an object than going from 60 to 70 mph.

I'm going to base my answer on this physics stackexchange answer. I'll rephrase it to make the mathematical steps more clear, but I recommend reading that answer as well if you know high school algebra well.

Hopefully you have the intuition that energy is some property that objects have which can cause internal transformations in matter. Whenever an object collides and as a result deforms or heats up, that's kinetic energy acting on the object.

So let's look at a situation that makes it easy to account for all the kinetic energy - a clay ball hits a wall. It will splatter, and the size of this deformation can be used to calculate the energy. If two clay balls of equal size hit a wall and splatter to the same extent, they had the same amount of energy. Let's say the balls were moving with speed v, and had an amount of energy E.

Our first step is to change this experiment. Instead of throwing the balls at the wall, we throw them at each other. They splatter in the air instead of on the wall, coming to a complete stop. They were both still moving with speed v, but in opposite directions. When we do this experiment, we find that the amount of deformation each ball undergoes in this collision is the same as when they hit a wall. This makes sense - each ball has energy E, so the entire collision has energy 2E, but it's spread out through twice as much mass. Each ball undergoes an E's worth of explosion, the same as if they each hit a wall.

Our second step is to change this experiment once again. Throw the balls a each other, but now observe them from a car moving forward at speed v. The ball we're following looks like it's hanging in the air, while the other ball is moving toward it at speed 2v. Now to see what happens when they collide, you'll need to keep track carefully. To a person standing on the ground, the two balls stopped moving completely, and fell straight down. But since we're in a car moving forward at speed v, the collided balls are now moving backward at speed v. From the car, it looks like the 2v ball came in, collided, and knocked the combined system backwards.

Now the crucial step - other than this change in relative speeds, the collision looks identical. Us being in the car doesn't change the events. The splatter of each ball is the same size. So the moving ball went from 2v to v, and yet it delivered 2E's worth of energy. Going from 0 to v grants an object E worth of energy, and yet going from v to 2v grants at least 2E (in reality more, since the two-ball system is still moving, so it didn't lose all its kinetic energy to the splatter). We got here not by assuming any kind of mechanics, but by assuming that the laws of physics work the same when we're moving as when we're stationary - if we do all our math from a moving car, we should still observe the same things happening, even if we observe them happening at different speeds.

To get the exact quadratic relationship, we have to account for the fact that the two balls are now moving backwards with velocity v, which means there is an extra 2E's worth of energy stored in their motions. That's a total of 4E. If velocity v has energy E, while velocity 2v has energy 4E, that means doubling the velocity quadruples the energy, which is only possible if E(v) ~ v^(2). Then E(2v) ~ (2v)^2 ~ 4 v^2.

This argument suggests that this relationship does not come from the inherent properties of objects, the way electromagnetic energy does. That energy is somehow stored in the bonds between nuclei and electrons. Kinetic energy is some kind of property of the geometry of space and time, so just thinking about the symmetries of motion can tell us what the relationship between motion and energy must be.

> From my (limited) understanding, entangled particles don’t remain entangled if they are far apart or if something else “touches” one of them, which is not at all how it is explained in the article.

Entanglement is not an all-or-nothing deal. Entanglement is correlation + superposition. If two particles are in a 'large' superposition (they are simultaneously in multiple states which are very distinct from each other), and their states are highly correlated with each other, they are very entangled. This is what typical spin entangled states look like. One particle is 50% spin up and 50% spin down, the other is 50% up and down, and they are maximally correlated - they will always be spinning in opposite directions. When you observe one particle, you precisely know the state of the other.

However, it's also possible to have very weak entanglement. For example if one particle is in a location superposition, and it interacts with another particle from a distance, their trajectories are now entangled. But this entanglement can be arbitrarily weak. If you observe the location of one particle, you gain a tiny amount of information of the other one - if you thought the location would be clustered around some target, you know that cluster might have shifted a little bit. But it doesn't particularly help you narrow down the location.

There's no strict line between weak entanglement and no entanglement. Extremely weak entanglement still exists in macroscopic interactions. Our classical world arises from continuous weak entanglement of the environment correlating everything's state, so we don't see any disconnected parts of the world existing in large superpositions.

That's what the skeptics are saying. The tardigrade is entangled with the qubit in a very weak sense. If you had a godlike view that could measure the exact quantum state of every particle in the tardigrade's body, you would notice a very slight correlation between them and the state of the qubit. The tardigrades are not in a clean set of two possible states that exactly correspond to the qubit, they are in the same multitude of microstates that continually interact with surrounding objects and some probabilities of those microstates have shifted around just a little bit.

There have been experiments that do actually entangle macroscopic objects, e.g. large mechanical oscillators. The 'Results' section is the most informative - not only does there have to be interaction with some microscopic entangled states, but the states of the oscillators should also be highly correlated, and the statistical properties have to demonstrate that they are in fact in a non-classical superposition. The tardigrade experiment did not do this clear work.

This high level description isn't fundamentally different to how you would actually entangle macroscopic objects. We know how to create microscopic superposition states, and we know that entanglement spreads by interaction. That's the point of the Schroedinger's cat experiment - the decaying atom interacts with the cat through the detector+poison vial, and this interaction entangles them, putting both in a very distinct superposition of vastly different states (decayed and not decayed, and dead and not dead).

The question here is whether the tardigrades were meaningfully entangled with the qubit states by this specific interaction (acting as a dielectric on capacitors within the quantum system). The skeptics say that the interaction of the tardigrades with the actual qubit states is so weak, there is effectively no correlation between the two.