A Perfect Circle?
My approach to knowledge is that it is impossible to know anything for certain. Which means it's okay with me if it's good enough.
Imperfect Knowledge
To know something fully you have to possess all the information about that thing. It would not be enough to be able to record every particle that makes it up, you would also have to record all the properties of those particles: their types, masses, locations, momentary velocities, charges, spins, etc. The space required to write down all that information would necessarily be larger than the thing itself.
A human brain has a finite number of neurons, which make a finite – huge, but finite – number of synaptic connections, and so has a finite amount of "space" to store information. And because it has a finite amount of space, it can never have full knowledge of itself. There isn't enough room to store all the information about all of its own neurons and synapses inside those same neurons and synapses. So it certainly cannot have full knowledge about the rest of the person it's inside. Or the world around it. Or the universe.
I accept this limitation, because it allows me to relinquish the pursuit of perfect knowledge. I am freed, instead, to use approximations, to say that some particular knowledge is "good enough." But good enough for what?
Well, I'm a person, I live my life at "person" scale – things that matter to me from moment to moment are usually somewhere between micrometres and thousands of kilometres, between micrograms and tonnes, between milliseconds and decades. Much bigger than electrons, much smaller than galaxies. So if I'm faced with a problem in that range, I need to be able to use my imperfect knowledge to predict or generate an outcome that is also within that range.
Say you throw a ball, I don't need to know exactly how many protons and neutrons and electrons make up the ball, or their individual locations or kinetic energy or anything like that. I can approximate, and say it's "about 5¼ ounces" and travelling "about 45 miles per hour", and work out where to put my hand to catch it.
That approximation is called a "model", since it's – literally – a simple model that represents an incomprehensibly complex thing. Models are really important for letting my finite and very limited brain try to interact with the almost inifinite complexity of the world around me in any meaningful way.
And that approximation – considering a couple of gajillion particles as a single object with a single mass and a single velocity – works well enough for us to play catch, but it also works well enough for us to use it as the basis to make predictions about other events. What if we throw the ball harder? What if we throw a heavy rock instead of a ball? Based on what we observe we can write a formula, then we can plug in numbers like the weight of an object and the speed it's thrown, and use that to calculate where that object will land. Then if we do it a bunch of times, with a bunch of objects and a bunch of speeds, we can build confidence in the formula, and confidence that our measurements are in fact "good enough."
The formula is also called a "model", since it's a simple model of an even more incomprehensibly complex set of interactions and events. No computer today would be able to track all the particles involved, or calculate the squajillions of interactions between them, but I could work out a simple formula in my head (or with pen and paper, at least). It may be imperfect, but "good enough" is so much more useful than "couldn't work it out in time" ...or at all.
However, even if we test it a hundred times, and get it right all hundred of those, that isn't proof – what we've done is build confidence. It can be very high confidence, supreme confidence even. We can be practically certain that we can use our formula to predict where any object of any mass at any speed would end up, but we have to remember that there are limits.
What about the wind? If a ball is light enough, or the wind is strong enough, we'll have to add new variables to our formula, and take new measurements each time. But say we do that, and we test the new formula a thousand times under various conditions, and it gets it right all thousand of those – that's still not proof. What other variables have we missed? Does the temperature matter? Or the elevation above sea level? And what about if the ball were the size of a galaxy? Or a neutron? Can we be sure that the formula still works reliably? Or that we're even able to take accurate measurements at those scales?
We also have to remember that the ball isn't just a ball – it's a bajillion particles with their own masses and velocities and behaviours. We aren't measuring and feeding all of them individually into our formula. And if we were, how do we know there aren't other interactions between the particles that we're ignoring? Like, what makes them stay together in roughly the same shape? Our formula doesn't say anything about that. It's still a good formula, we're really really confident that we can use it to make useful predictions, but it's not perfect.
Making Predictions
The process of making a formula is itself a whole thing. Usually it starts with observing something in the real world, a physical event, that we want to be able to predict or influence in a predictable way. A ball we want to catch maybe, or an apple falling from a tree.
Then comes the hard part: coming up with a narrative explanation, a story, that fits our observations. "A body in motion or at rest will remain so unless acted on by an outside force." "Two objects are attracted to each other with a force that is proportional to their combined mass and inversely proportional to the square of their distance." That sort of thing.
And then, the fun part: turning that narrative into a formula. Working out how to take measurements that we can plug into that formula. Using the calculations to make predictions about real world events. And using the outcomes to either build confidence in the formula, or to come up with refinements and improvements, or to scrap it altogether and start over.
There is an entire field dedicated to the process of coming up with formulae and then coming up with specific ways of testing them – identifying variables that we can control under particular circumstances, performing the actions, comparing the results with our predictions, and documenting the whole thing rigorously so other people can also set up their own tests and verify it for themselves. This field is called "science", and the process is the "scientific method."
Eventually, after various iterations and trials and revisions we might come up with a formula that we're really confident about. We might get to learn the limits of where it's applicable and where it doesn't work so well. And we might have shared it with other people who have also independently run their own tests and fed back into the process too. It's still not perfect – there's no proof, the narrative and its formula remains a theory – but it's a really good one, and we know where and how to use it.
A Big One
In the late 1600s a guy in England came up with some of these narratives and formulae; in fact I paraphrased two of them earlier in this piece. The second one, about objects being attracted to each other, was called his "law of universal gravitation." It might have been called a "law", but today we often refer to it as a "theory" – in the sense that it, like everything else I'm talking about, isn't perfect. It's one of those "good enough" formulae – one that we have, collectively as a society, been using and verifying and building confidence in, for over four hundred years now. We've found some limitations, the extreme situations where the (relatively) simple formula doesn't lead to reliable predictions – but for the vast majority of cases it's shown itself to be incredibly reliable, and we as a society have used it to do some pretty incredible things.
While I'm talking about that English guy – his name was Isaac Newton, by the way – I should probably point out something that he himself said about this "law", and which has held true ever since. Neither the formula, nor the narrative it represents, ever explained why it happens, it only helped describe how. We can predict where a thrown ball will land, but we still can't explain why it falls. And that's fine, the formula is still useful – as long as we can make the predictions.
There are implications from these models, too. If a ball and the Earth are both attracted to each other, then why doesn't the Earth move up? Partly we can make sense of this by incorporating the first narrative/model I mentioned earlier – that things at rest want to stay at rest. (This is one of Netwon's "laws of motion", incidentally.) Further, their obstinance is proportional to their mass. In other words, if you want to make a very big world move enough that you can detect it, you need to apply a very big force. A small ball, however, doesn't take much force at all before it starts moving enough that we can see it and measure it. So the ball drops obviously, but if the world moves, it's so little that we couldn't see it anyway.
Another implication comes from the fact that it is, after all, a model. The Earth isn't a single object, it's – as far as we can estimate – about 130,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (that's 1.3×1050) atoms, each of which is made up of a couple of dozen electrons and protons and neutrons, and those protons and neutrons are made up of smaller particles (quarks and such) ... Each of those particles feels the attractive force from every other particle.
Aside: In fact, they feel it from every particle in the entire universe; however, as the distance increases the force lessens by the distance squared. So a when particle on the moon might be 30 times as far away as a particle on the far side of the Earth, the gravitational force would be 1/900 as strong. A particle on the sun might be about 11500 times as far away, so the attraction would 1/130,000,000 as much. At a certain distance you can say the force is close enough to zero that, for most intents and purposes, it can be ignored.
To reduce all of those particles to a single object, to make the maths reasonable, it turns out it's good enough to take the approximate average of all of their locations, and declare that point as the "centre of mass." Anything that is attracted to all of those particles is essentially attracted to that location. Another way to look at it would be to calculate all the attractive forces to all those particles (as vectors – strength and direction) and take their average. That singular "average force" points towards the centre of mass, and gives a single value for how strongly an object is attracted to that point.
That means that the ball is (or more accurately, all the particles in the ball are) pulled towards the Earth's centre of mass. And the particles in the Earth are pulled towards the ball, too. Those closest to the ball's centre of mass might even fly upwards – at half the distance, the attraction is four times as strong; a hundred times closer, ten thousand times stronger – and because they are much lighter than the ball, they might move much, much faster than the ball. But they aren't strongly attached to the rest of the world. The Earth is kind of flexible, a little bit gooey – especially at the surface – and constantly in motion. So part of the Earth might probably be pulled up towards the ball, but for any Earth particles further away from the ball a good fart nearby would probably affect their movement more than gravitational attraction to the ball.
What is up? (Baby, don't hurt me)
Most of the time, for most of the history of humanity, the overwhelming force that affects our day to day existence is gravitational attraction to the Earth. It's the force we fight against when we stand up, the one that pulls us down if we lose balance and fall. So our bodies contain a pretty good gravity detector, which we use to keep our balance and not fall over. Except that it's not specifically a gravity detector, it's not even really a force detector, it's actually an accelerometer. It works for us, and we can explain it using yet another of Newton's formulae: Force = mass × acceleration
Neither the mass of the Earth nor the mass of a person change very much from moment to moment, so in terms of that formula we can say the mass is constant. That means it's not a variable, so it can be written out of the formula. I.e. when mass is constant, Force = acceleration
And since gravity is the dominant force we feel most of the time, our formula can be reduced down to: gravity = acceleration
We even have a number for it: 32 ft/s2
So most of the time, an accelerometer works as a pretty reliable gravity detector.
And because our accelerometer/gravity detector is built into us, it's natural that we have a word for what we sense with it. In English that word is "down." When we're standing on the surface of the Earth, the average force of all the particles of the Earth pulling on us registers in our accelerometer and we can sense which way is down. And, like the ball from before, down is towards the centre of mass.
That's cool, because it doesn't matter where you are, "down" means "towards the middle." Every person, every ball, every molecule of dirt or water or air, everything is pulled towards the centre of mass. And as the particles and objects move, the centre of mass also moves a little, but there's always a centre of mass. Always an average.
When everything gets pulled towards a single point there's a natural tendency for them to form into a ball. Particles closer to the point block those further out, and the further ones move around trying to find a spot closer to the point; and layer by layer, piece by piece, they settle into the low spots, filling it out and forming into a sphere. There are other factors, of course – as with everything I've written so far, this is a "good enough" approximation – for example there are other forces, like the forces that make some atoms cling to each other to form crystals. The particles in a crystal don't flow so freely; but if there are enough particles in total (for example, the number of particles in the Earth) it doesn't matter much if they form into crystals or rocks, there's still plenty of fluidity to flow around and make a roughly spherical shape.
Fun fact: if you shrunk the Earth down to the size of a pool ball it would be a more perfect sphere than an actual pool ball, and would be almost as smooth (it'd probably have a texture like fine grit sandpaper.) [citation provided] Mountains are huge, but the Earth is so much bigger it's hard to comprehend.
We find the limits of using an accelerometer as a gravity detector when other forces get involved, though. For example, if you get spun around by the arms fast enough you experience a "centrifugal force" (which is a whole other "good enough" that I'm not going to go into) which can be even stronger than the gravitational attraction towards the Earth. By which I mean: when you get spun around, your legs fly out to the side instead of hanging straight down. Your sense of "down" shifts, to line up with the average force/acceleration you are experiencing.
It also breaks when you jump off something high. The accelerometer works by resisting the forces; if you're free to accelerate along with them (i.e. fall) it registers as close to zero, and you get a weird floating feeling.
As a society we know this. We have learned to use the word "down" to mean the direction you would feel if you were standing still on stable ground. This gives us a reliable frame of reference when communicating with other people. And we have other related words, like "up" – which means the opposite direction to "down." Your individual sense of "up" might change sometimes, but we have come to understand that the sky is up, and the soil is down. And most of the time that's definitely "good enough."
However if you want to do something more extreme, like travel really far, or go really fast, or fly into space, you need a more precise way to talk about the shape of the world and the directions of the forces you'll be experiencing. "The sky is up" doesn't cut it if you consider that there's sky around all of the Earth – in fact, most of the sky is "down" compared to your own local sense of which way up is, since more of the atmosphere is below the horizon and hovering around the far side of the planet than there is above you. So we can instead say "down is towards the average gravitional force you are experiencing" and "up is the opposite direction"; but even that stops being useful if, say, you're far enough out in space that the distances from all the other particles in the universe means the gravitational force you're experiencing gets close enough to zero to not matter. Or if you're constantly falling, and all your accelerometers depend on you not falling for them to work properly.
Fortunately, for most of us, this will never be a problem. So our understanding of up and down is usually good enough.
Footnote: 150 g, 70 km/h, 9.8 m/s2 ... I had a particular audience in mind when writing
