If you were aboard the falling ship, you wouldn't notice anything unusual, though, because all the processes you'd use to observe clocks would also be slowed down. Time dilation is always something you observe in others, not yourself.

While energy and momentum are related, this comment is conflating the two in a way that might confuse someone just getting started with physics. It turns out that, properly framed, Newton's third law does kind of indirectly imply conservation of energy, but you need some work to get there.

In (Newtonian) physics, we think of things in terms of objects (which we assume are rigid, that is, they're like hard balls on a pool table and never squish or heat up or anything), forces, and position/motion.

Each object has, at any given moment, a position. That position is changing over time, and the change is called the object's velocity. And, in turn, the change in the object's velocity is called acceleration.

Changing the motion of a heavy object is harder than changing the motion of a light object. This is where we get the ideas of momentum (the velocity of an object times its mass) and of force (the acceleration of an object times its mass). Or, equivalently, you can think of force as the change in momentum over time, in the same way that acceleration is the change in velocity over time.

In the way I'm presenting it here, the definition of force gives you Newton's second law (the equation F = ma, that is, force equals mass times acceleration) for free: that's just what we mean by the word "force". It also gives you the first law: if force is zero, acceleration is zero, and therefore change in velocity is zero, too.

Newton's third law, on the other hand, says that all the forces in a system (possibly on more than one object) always add up to zero. In other words, if I apply a force +F to you, there is necessarily a force -F applied to me. If you're pushed forward, I'm pushed back. If you're pushed up, I'm pushed down. And so on.

But remember how we said earlier that one way to think of force is the change in momentum? Well, if all the forces in a system add up to zero, the total change in momentum must, therefore, also add up to zero. If you gain momentum upward, I must gain momentum downward; if you gain momentum to the right, I must also gain momentum to the left. So a modern way to describe Newton's third law is just that it describes conservation of momentum: you can never create or destroy momentum, only transfer it between objects.

> From low orbit the rotation is basically unnoticeable. The ground moves under you at 400-something m/s but you yourself are flying by at 7000+ so you're just trying to spot the ground move by under you slightly slower than expected.

Yes, all of which I already said in my own top-level comment. But that's not an issue of distance, really, it's an issue of Earth's mass - the same wouldn't be true of a proportional orbit around a tiny asteroid.

> there is nothing in the universe that says "there should be capsaicin".

There's nothing in the Universe that says "there should be capsaicin" specifically. But conditional on animals having particular receptors, there is something that says "plants with it can outcompete plants that don't".

Yes, evolution has (significant) random elements in terms of where you start on the fitness landscape and in terms of non-biological factors (e.g. "oh shit a meteor just hit the Earth) that can sometimes intervene. But evolution is tightly intertwined with game theory, and it isn't a coincidence that game-theoretic strategies show up all the time in evolutionary biology.

"Organisms, broadly speaking, will eat and reproduce" is just as iron-clad a law of our Universe as "objects will roll downhill" is. Maybe more so, since it's implied by abstract mathematical law and not even by the particular quirks of actual physics.

> There are, for all practical purposes, infinitely more traits that have never and will never be expressed than ones that we've ever seen.

This is true to some extent, but the frequency of convergent evolution shows us that some patterns really are just super useful. Wings have evolved independently in birds, insects, mammals, and even plants if you count the little fins on a maple leaf. That's not a coincidence.

"Just an accident" is maybe not giving credit to the forces involved here.

The plant itself isn't intelligent, but the process that makes it kind of is (in the sense that there is meaningful information encoded in which individuals reproduce or not). It isn't a coincidence that a chili pepper produces capsaicin any more than it's a coincidence that a water droplet takes a spherical shape: both are obeying mathematical laws, just not with any "intent" behind them.

> Imagine holding a ball at arm's length

This is a little misleading, because a tennis ball at arm's length would be a closer analogy to the view from geosynchronous orbit. A tennis ball has a diameter of about 2.5 inches, and the average human arm is ~25, so you're viewing from ~5 radii away. The Earth's radius is ~4,000 miles, so you'd be viewing from ~20,000 miles away (geosync orbit is 22k).

The view from the ISS is more like holding a ball (250 miles / 4000 miles) * 2.5 inches = 0.16 inches sorry, twice that, 0.32 (since one of those is a radius and one is a diameter) from your eye, at which point it would be brushing against your eyelashes.

But it turns out the rotation's still pretty slow even that close up.

Earth rotates fast in the sense that the linear speed of the rotation at the equator is fast. It does not rotate fast in the sense that it makes a full spin quickly (it doesn't; it makes a full spin in - from a distant perspective - 23 hours and 56 minutes).

If you were floating just above the equator and not co-rotating with the Earth, you would see the land below you flying by at an extremely fast pace (about 1.5x the speed of sound). But if you're in space, you're quite far from the surface of the Earth - at least a hundred miles or so - so that speed doesn't look too too fast.

In the very best case, where you're 100 miles above the Earth, not in orbit, and not co-rotating at all, you'd see the Earth move at a speed of about 0.2 degrees across your field of view per second. That isn't nothing, and you probably would notice it if you were paying attention, but it's easily lost in other motion. For comparison's sake, a finger at arm's length covers about 1 degree, so an object directly below you would take about 5 seconds to cross the width of your finger at arm's length. Or, put another way, it would move across your field of view at about the same rate as a person walking slowly on the other side of a football field from you.

Fundamentally, it looks slow because of parallax: faraway objects don't move across your field of view very quickly even when they're moving very fast.

In practice, though, you never get even that much, because:

• If you're in orbit anywhere near the Earth, you're actually moving faster than the Earth rotates (by a lot!), so you mostly see your movement over its surface. You're also higher than the 100 miles in our example: both active space stations (the ISS and China's Tiangong) orbit at about 250 miles up.
• If you're not in orbit, you probably just launched from the surface of the Earth, and you still have the horizontal momentum you started with, so you're still more-or-less co-rotating with the Earth.

> pushing a slug of cold saline into the coronary arteries

Is it weird that I kind of want to know what that would feel like?

Oh, I thought you were claiming an uncountable discrete subset of the reals (since your comparison was the rationals). Yes, obviously you can give any set the discrete topology (although the discussion of "between" suggests something more like an order metric?)

> Conversely, there are uncountable "discrete" ordered sets where nothing is between a number and its two neighbours.

That's not true, and it's (relatively) easy to prove.

Consider a set S that is a subset of the reals, with the property that for each s in S there exist two numbers u and l (for "upper" and "lower") such that u < s < l and there are no numbers x for which u < x < s or s < x < l. In other words, u is the "next biggest" number and l is the "next smallest" (this formalizes the idea you've stated informally).

For each s in S, consider the radius R = min(d(s,u), d(s, l)) (of course, R, u, and l all depend on s, but reddit markdown means I'm gonna skip the subscripts). This radius is basically just the "minimum spacing" around s. Such an R exists for each s, and is strictly positive. Since R is strictly positive, so is R/2. And since the rationals are dense in the reals, we can find two rational numbers a and b (again, also dependent on s) such that s - R/2 < a < s < b < s + R/2. In other words, we can find an interval of rational numbers (a,b) that does not overlap the corresponding interval for any other s in S.

Now, consider the function f: S -> (Q x Q) that takes each element s in S and maps it onto the ordered pair of the interval generated by the process in the previous paragraph. This function is clearly injective, since none of the intervals (a,b) overlap (so they certainly cannot be the same), but the set (Q x Q) is a Cartesian product of countable sets and therefore countable. Since we have an injection from S to a countable set, S is itself (at most) countable.

No, that's because the hard drive itself reserves some storage for the programs that manage it (or, if it's a new prebuilt computer, some space is taken by the operating system).

The net force on the wagon is not zero. If the child pulls with a force F (where we set our coordinates so that "forward" is positive), then the force on the wagon is +F and the corresponding force on the child is -F. If these were the only forces operating (say, if the child is on a surface with ~zero friction, as if they were on ice skates), the wagon would accelerate forward, and the child would accelerate backward.

But these are not the only forces involved. In particular, the child is pushing on the ground. The ground exerts a force back on the child - let's call it A for "anchoring" - in response. So the total forces are:

• The wagon experiences a force +F from the child pulling.
• The child experiences a force -F from pulling on the wagon and a force +A from pushing on the ground.
• The ground experiences a force -A from being pushed on by the child.

These forces do sum to zero, satisfying Newton's third law, but the forces on each object do not (assuming F and A are not zero, the child is the only one who could be experiencing zero net force here if A and F are exactly equal).