The Anthropic Principle – Toolbelt of Knowledge

Toolbelt of Knowledge: Concepts
Algorithms
Equivalence
Emergence
Math
The Anthropic Principle
Substrate-Independence
Significance

Today’s topic is something I’ve long struggled to wrap my head around. In fact, I’ve gotten it wrong in previous blog posts, and only recently have I come to really understand it. This topic is called the Anthropic Principle, and it’s a method for inferring knowledge about our surroundings by observing that we are there.

Suppose you wake up in a locked room. There is a note on the ground, telling you that you are in an experiment. There are a thousand rooms just like this one. Nine hundred ninety-nine of them have their doors painted red, and one has its door painted blue. You look up, and notice your door is blue.

Before the experiment began, the researchers flipped a coin. If it landed heads, a thousand people would be drugged and each put into one of the rooms. Nine hundred ninety nine would find themselves in rooms with red doors, and one in a room with a blue door. If the landed tails, only one person would be drugged and wake up in the room with the blue door.

The note says you must guess whether the coin landed heads or tails. If you get the answer right, you get a hundred dollars. Which do you bet, heads or tails?

On the surface, it seems like the odds are equal, 1:1. After all, a coin can either land heads or tails, and each possibility has one person behind a blue door. However, we’re ignoring part of the story, so let’s look at the probabilities of each possibility.

If the coin landed heads, you would only have a probability of 1/1000 of landing in the room with the blue door, and a 999/1000 chance of landing with a red door. If the coin landed tails, you would be in the room with a blue door, and have 0 chance of a red door.

Let’s put these together. The chances of you getting a red door, written as heads:tails, is .999:0. The chance of you landing at a blue door is .001:1. This means, if you wake up in a room with a blue door, it is a thousand times more likely the coin landed tails than it landed heads!

Wait a minute, you say. If the coin landed heads, someone would have to be with the blue door. How do you know that’s not you? The answer is, we don’t have absolute certainty. However, the fact remains we have a thousand to one odds against it, so the reasonable bet by far is to choose tails.

Now that we’ve gone through that example, let’s take a step back and remember the big picture of what we did. We were given a little bit of information, and based on the fact that we were there, deduced more information. That is the Anthropic Principle.

There are many applications of the Anthropic Principle, and we will talk about some of them in future posts. For now, let’s apply it to the situations most people turn to first: life on Earth and in the universe.

The history of science has shown us that we are not nearly as significant in the grand scheme of the universe as we might like to think. The sun and planets do not go around the Earth, the Earth goes around the sun. Neither the sun nor the Earth is a special feature of the cosmos; there are trillions of stars in each galaxy, and trillions of galaxies in the universe. A significant fraction of the stars are sun-like, and a significant fraction of stars have Earth-like planets. There is nothing special about the sun or the Earth.

This idea, that we aren’t special in the universe, is called the Mediocrity Principle (also known as the Copernican Principle). Because the Mediocrity Principle applies to so many things we know about, it is easy to assume it applies to things we don’t know about too. Perhaps, for instance, life like us exists all over the universe.

But in the realm of the unknown, the Anthropic Principle steps in and says, “not so fast.” We don’t live on a random planet, we live on a planet where the conditions were right for life to emerge, and continued to be right for life to evolve, until intelligent life appeared. Regardless of whether life is common or rare in the universe, this observation would be the same; we are in a place where the conditions are right and have been right for intelligent life. Thus, even if it turns out Earth is the only planet in the entire universe where intelligent life exists, we should not be surprised.

Just like there are people who want us to occupy a special place in the universe, there are people who want everything about us to be commonplace. Both of these beliefs are fallacious. We know our planet and our star are not special, but we don’t have nearly enough evidence to determine how common life is. Given only the information we have, there is roughly the same probability that there are a billion trillion civilizations in the universe as there is that we are the only one.

On a larger scale, the Anthropic Principle can be applied to the universe as a whole. The laws of physics, as we currently understand them, have a bunch of parameters that don’t seem to follow any pattern. However, it seems as though if any of them were slightly different, life could not exist at all.

Let’s apply the Anthropic Principle to this question. Assume we don’t know whether life exists or not. If the physical constants must be what they are, and every other combination is impossible, we would bet against life existing without hesitation. After all, the number of permutations that allow life are vastly dwarfed by the number of permutations where life is impossible. So if there is only one combination, it would be vastly more likely to be one where life is impossible.

However, if the physical constants could have been different, we have another story. It would mean they could be tuned to allow for life to exist, making a universe that allows life vastly more likely.

The fact that we are here, that we observe ourselves to exist, is evidence via the Anthropic Principle that the physical parameters of the universe could have been different. And we know this without even knowing how they were set!

So then, what are the possible explanations for why the parameters of the universe allow life to exist? It could be that there are many universes, each with their own physical constants, most of which don’t have life. It could be that the universe was created for life, either intentionally or unconsciously. See the Multiverses post for more discussion on these. And finally, it may be that if something is unobservable, it is the same as not existing. If this is true, then even if there is a multiverse, all universes have life, because if a universe cannot be observed, then it doesn’t exist. We’ll talk more about that in the next installment of the Consciousness series.

If the Anthropic Principle still confuses you, that’s okay. I’ve been thinking about science and philosophy for years, and I’ve only come to understand it a few months ago. Still, it’s worth spending the time to understand, because it opens up new paths by which to explore deep and interesting questions.

Consciousness – Physicalism

Consciousness:
The Hard Problem
Dualism
Physicalism
Idealism
Identifying Consciousness

Vast Minds

Identity, Self, and Other

The Question of Meaning

Recommended Pre-Reading:
Representational Realism
Existence and Natures
Knowledge of Reality
The Hard Problem

In the philosophical zombie thought experiment, there are two possible routes to follow. Last time, we explored what happened if we assumed p-zombies are possible, and something could exist that could act like, talk like, and be physically identical to a human, yet have no consciousness. It would follow that consciousness is something independent from, yet able to interact with, physical reality. Today, we are going to explore the other route, what we get if we assume something physically identical to a living human must be conscious, and it is impossible for it not to be. This is called physicalism.

For physicalism, we must suppose that something about the physical world is equivalent to consciousness. Right away, this seems weird. The way we conceptualize consciousness is completely different from the way we conceptualize physical systems. Consider light. Light doesn’t have any color, it’s just a wave in the universe-spanning electromagnetic field. In order to become color, it has to enter your eye, interact with the cells on your retina, be transformed into electrical signals, which make electrical patterns in your brain.

The physicalist must say these electrical patterns are color and shape and texture. The collection of neurons in your brain firing and buzzing together not only correlates with your consciousness, it is your consciousness.

This seems crazy. After all, if you think about a color, and you think about a bunch of cells shooting electrical pulses to one another, the two thoughts are nothing alike! In fact, they seem so different from each other that it feels like they must be fundamentally different things.

However, let’s remember that in analytical thinking, we might discover truths that go against our intuition. Last year, in the Nature of Reality series, we did a discussion on how we can know truth if our conceptions of reality are just representations. We concluded that if the math/logic of our conceptual representation is isomorphic with (the same as) the math/logic of reality itself, then our conceptual representation is true.

The models we create of physical systems, the images we draw and imagine and look at, are just representations. The picture in your mind of cells sending electrical pulses between each other is not reality-as-it-is, it is a conceptual representation. Consciousness, on the other hand, is not a representation. It is the direct experience of reality.

Let’s be clear. The computer you see in front of you is not a direct experience of reality. It is an image created in your brain. The direct perception of reality is the fact that you perceive an image, regardless of its contents.

This is important, so let’s go over it again. Consciousness is like the chalk marks on a chalkboard. Many things can be written on a chalkboard. They can be erased, and replaced by something else. You can write words or equations, or draw pictures. The things we write on chalkboards convey meaning to those watching. Yet all of it, regardless of what it means, comes from the chalk marks.

Your mind lets you experience many things. You see sights. Hear sounds. Feel feelings. These things are parts of consciousness. You find all kinds of meaning in the sights, sounds, feelings, and other senses. You can learn about the universe, keep a relationship going with another person, experience the imaginary world of a story. Yet all these things come from sights, sounds, and feelings. All these things come from consciousness. All the things we do with the contents of our consciousness are models, representations, not true reality. But consciousness itself, the sensations of sights, sounds, feelings, and other senses, is a little piece of direct reality.

When we apply this to the paradox of neural patterns and conscious experience, we find it is not so contradictory after all. The mental model of neurons and electrical pulses is just a representation, whereas consciousness is reality-as-it-is. If they don’t seem like the same thing to our intuitions, that is completely fine. It’s the logic and math that matters. If the math/logic of the model of neurons and electrical pulses is the same as the math/logic of consciousness, then those neural patterns are a valid model of the real system, which is consciousness.

If this is true, what does it mean metaphysically? To be conscious as we know it, a brain needs billions of neurons working together. A single neuron is about as conscious as a rock. The phenomenon of large numbers of things coming together and demonstrating new, holistic behavior is called emergence. The reverse of emergence, the process of examining systems in terms of their constituent parts, is called reductionism.

by Chlodulfa on Deviantart

If the conscious part of a brain can be reduced to neurons, what is consciousness reduced to? If consciousness and the conscious part of the brain are equivalent, then the same logic that reduces the conscious part of the brain must also reduce consciousness.

Does this mean what we said a moment ago about neurons and rocks is incorrect? Does it mean a neuron, despite being just one cell, is a tiny bit conscious? What about the things the neuron is made of? Molecules, atoms, and sub-atomic particles? If we say the fundamental physical level of reality has no consciousness, but brains do, then somewhere along the train of emergence there must be a step where the smallest possible building blocks of consciousness come into existence.

This idea, that somewhere along the hierarchy of emergent complexity the smallest units of consciousness appear from nothing smaller, seems non-scientific. Stuff doesn’t just pop into existence without any reason. One way to resolve this paradox is to say the fundamental level of physics and the fundamental level of consciousness are the same. Consciousness doesn’t magically appear at some level, rather it breaks up and keeps breaking up into smaller and smaller pieces the further down the physical reductionist ladder we go. When we get down to sub-atomic particles, we find that electrons, quarks, and the like all have a tiny bit of the stuff that comes together and makes consciousness. This would mean everything in the universe is a little bit conscious, a theory called panpsychism.

As the philosopher David Chalmers describes it, panpsychism contains the hypothesis that, just like electrons and other sub-atomic particles have properties like mass and electric charge, they have another property: a tiniest possible amount of consciousness. This wouldn’t be consciousness as we know it, with colors and concepts and perceptions, but a tiny bit that builds up and joins together with the other bits of consciousness from the other particles it interacts with, and in large enough numbers and the right organization, become the consciousness we recognize in ourselves.

Panpsychism isn’t the only physicalist option, though. If we can find a different ladder of reductionism to follow, we might end up at a place other than sub-atomic particles. As it turns out, matter isn’t the only dimension the universe can be reduced down. There is also information.

At the fundamental level of information, we find bits. Bits are things that can be one of two possibilities: on or off, yes or no, true or false, 1 or 0. It may be that any amount or type of information can be reduced to bits, or if not, to some other indivisible basis. So maybe consciousness can be reduced to bits.

By Christiaan Colen on Flickr

To me, this makes more sense than panpsychism. Consciousness seems to be correlated with information, not matter. Intuition would say a giant brain with the same number and configuration of neurons as mine should be exactly as conscious as I am, despite being made of a lot more matter. And if we could simulate a brain using electrons rather than cells, it seems to me like it would be just as conscious too. Of course, intuition can often lead us astray in science, especially when we’re talking about consciousness, so experiments would have to be done before we say anything confidently.

There is a scientific theory that looks into the possibility that consciousness comes from information, Integrated Information Theory. It looks at networks of information where each bit is connected to other bits, and changes based on the signals it gets. The brain is such a system, each neuron functioning as a bit, either firing an electrical impulse or not. According to Integrated Information Theory, consciousness is formed according to each state of the brain related to all other possible states it could be in.

Integrated Information Theory is a first step toward a scientific understanding of consciousness, but it is almost certainly not the whole answer. Just as Newton’s theory of gravity gave way to Einstein’s theories of relativity, Integrated Information Theory will probably give way to something much deeper and more insightful. Some avenues to pursue would be the time lag between signals and patterns passed along by neurons, and the fact that much of the brain seems not to contribute to consciousness, despite being very active.

Information is not something constrained to one universe or another. It is a mathematical construct, which means it transcends material reality. There is something deeply wondrous about the idea that consciousness comes from timeless, spaceless truths, made real by physical systems acting out their natures. Not from a religious or spiritualistic narrative, but from analytical thought and a scientific worldview.

Why Faster-Than-Light Travel Allows Time Travel

In our everyday experience, things can always move faster. Give a car a little more gas, and it will speed up from 100 miles per hour to 110 miles per hour. Give a rocket a little more thrust, and it will speed up from 1000 miles per hour to 1050. But weird stuff starts to happen when things get close to the speed of light, and to understand it, we have to talk about space and time.

Our brains automatically think of space and time as absolutes. A yard in a straight line is a yard, no matter who is measuring or calculating it. The present is a special moment in time, and it exists right now all across the universe. A minute is a minute, and it is the same for everyone everywhere. It’s intuitive and obvious. And none of it is true.

To understand why, we have to look at a little theory called Special Relativity. It is one of Einstein’s most famous insights, and one of the reasons he is known worldwide as the face of genius, because he questioned our natural understanding of space and time and found a deeper truth.

The simplest place to start is to draw a coordinate plane, with position going horizontally and time going vertically. As an object moves through time, it moves upward on this graph. We ignore the other two dimensions of space, both because a 4-dimensional diagram is hard to draw and to look at, and because they aren’t necessary for the concepts we’re interested in.

If something is staying still according to its coordinates, its path goes straight up. If something is moving in these coordinates, its path goes up at an angle. If it is accelerating, its path goes up along a curved path. Light travels at 45-degree angles.

Now that we’ve set up our coordinates, we define a term called a reference frame, a set of coordinates where zero is set to a specific location and speed. It might be tailored to an object, or just a point in space. From the origin (0,0), we draw four lines at 45-degree angles. These lines below the x-axis are the paths light from the past takes to reach the object at point 0, and the lines above the x-axis are the paths light takes coming from the object at point 0. These are called light cones.

Let’s look at everyday relativity we all know well. Suppose you’re standing still, your friend is driving by at 50 miles per hour, and a truck is driving in the same direction at 100 miles per hour. If we switch into your friend’s reference frame, they are the ones sitting still in their car, the truck is moving forward at 50 miles per hour, and you are moving backward at 50 miles per hour. In the reference frame of the truck, your friend is moving backward at 50 miles per hour, and you are moving backward at 100 miles per hour.

Now you might think, “What’s the big deal? Just speed up until you catch up to light, and then you’ll be going faster than it.” Well here’s where the craziness comes in. You see, light always travels at 45-degree angles on the space-time diagram in every reference frame, no matter how fast you or any other person or object is moving. If you are standing still on the Earth, and your friend takes a bullet train past you at 1000 miles per hour, the speed of light for you is the same in all directions as the speed of light for your friend: c. Your friend does not calculate light in front of them traveling any slower than the light behind them. They calculate both light beams traveling at the same speed, c.

The light cone must always be at 45 degrees, not skewed as it is in the middle diagram.

What does this mean? Well the math is complicated, but in order to get light to travel at c in all reference frames, space and time get messed up. In your reference frame, the time axis points straight forward in time, not any direction in space. Your friend’s time axis points in the direction through space-time as if they are not moving. On our graph, this means your time axis points straight up, but your friend’s time axis points along the path they are going to take though space-time at their current speed. Because the speed of light stays constant, this needs to be counterbalanced by your friend’s space axis changing too. In the transformation between your reference frame and your friend’s reference frame, space and time get rotated toward each other, and vise versa.

This is why nothing can go faster than the speed of light. No matter how much you speed up, light will always be traveling at c ahead of you, and you can never catch up to it no matter how much you accelerate. In the reference frame of someone standing still and watching you, you go closer and closer to the speed of light, but never reach it. This causes your time to slow down, your mass to increase, and your shape to flatten. You of course don’t notice any of this stuff, because it’s not happening in your reference frame.

Make a note of the fact that time passes slower for someone traveling close to the speed of light. This is going to be important later on.

If we look carefully at the way the axes change when we transform between coordinate systems, we’ll see that this means a universal “now” doesn’t exist! To show this, let’s take the x-axis, which is the slice of space-time where t=0, and what we think of as “now” across the universe. But your friend’s x-axis point in a different direction through space-time, meaning their “now” slice is different from yours! Mind blown, right? To drive the point home, let’s suppose there is a firecracker set to go off at a certain time 1000 miles away. According to your “now” slice, the firecracker is about to go off. But according to your friend’s “now” slice, the firecracker has already gone off!

There is no universal “now.” Each point in space-time is its own “now,” both in time and space. “Now” for you is only now for you, everyone and everything else has their own “now.”

There is nothing special about the t-axis in our diagrams. You can move in one direction, and then you can stop and turn around and go the other direction. In the same way, there is nothing special about the x-axis. If something were somehow able to travel faster than light, there would be nothing stopping it from going in a slightly future direction, then turning around and going in a slightly past direction. The fundamental boundaries are the light cones, not the t- or x-axes.

This means, if you were somehow able to make a ship go faster than light, all of the space-time between your future and past light cones would be open to you. This means you could set off going slightly backward in time, then turn around and go slightly backward in time the other direction, and get back to where you started before you began. It doesn’t matter what the method is, if you can travel faster than light, you can go back in time.

If you could go faster than light, all of the space-time outside of your light-cones would be available to you, but you would be locked out of your past light cone, unless you take a roundabout path. However, a past light cone is the same kind of barrier as a future light cone, so if you have the technology to cross the future light cone, that same technology can probably let you cross the past light cone, and you won’t have to go faster than light to time-travel; you’ll be able to do it while staying in the same place.

Warp drive, hyperspace, whatever your method is, the math doesn’t lie; according to Special Relativity, if you can travel faster than light, you can also go backward in time. But what about taking shortcuts? What if you don’t have to travel the vast distance between stars, but can get there in a single step? What if we could use a wormhole?

A wormhole is a theoretical object that comes out of Einstein’s other theory, General Relativity. Wormholes are interesting enough that we might give them their own discussion, but all we need to know today is that a wormhole is a shortcut between points in space-time. You can think of it like a doorway, but instead of leading to another room, it leads to a different planet.

You might think a wormhole is something that picks you up and teleports you away. This is a misconception. As you walk through the wormhole, nothing is happening to you that doesn’t happen when you take a walk down the street. The two ends of the wormhole might be light years apart the normal way, but on a path through the wormhole, they are only separated by a few feet. This isn’t just a metaphor, it’s literally true.

Wormholes, it turns out, also allow for time travel. To demonstrate this, let’s start with a simple setup, a wormhole where the two mouths are five feet away from each other and synchronized in time. If you look through the wormhole, you can see your own back five feet in front of you.

We put one of the wormhole mouths on a spaceship, and fly it around near the speed of light. Remember from earlier, if something is moving near the speed of light, time is slowed down for it. This means time is passing slower for the mouth of the wormhole on the ship than it is for the mouth on Earth—but only on the path through space from Earth to the ship. On the path through the wormhole, time is passing at the same rate on both sides. You could step through, have tea for half an hour with the astronauts, and when you step back, half an hour would have passed on Earth.

Think about this. On a path through space, time is passing slower on the ship than it is on Earth. But on a path through the wormhole, time is passing at the same rate on both sides. What is going on here? It’s a paradox! Two different ways of calculating the same problem give us two different answers. Which is right? Is time flowing at different rates, or isn’t it?

The answer is, both calculations are correct. Time is flowing at different rates on a path outside the wormhole, but it is flowing at the same rate on a path through the wormhole. This means the wormhole is not only connecting two points in space, but also two points in time.

Suppose the astronauts decide to return to Earth. When they land, less time has passed for them than on Earth. Mission control says, “Hey, you’ve finally arrived.” An astronaut says, “What do you mean, ‘finally’?” The astronaut looks through the wormhole, and on the other end, the same mission control member says, “You’ve landed? But we still see you flying around up there!” Outside the wormhole, the mission control member chuckles and says, “I remember having this conversation a week ago.”

By putting one mouth of a wormhole on a spaceship and flying it around near the speed of light, and then landing, the team has created a gateway through time. In our example, the difference between wormhole ends is one week; step through the end that traveled on the ship, and you’ll find yourself a week in the past. Step through the end that stayed home, and you’ll find yourself a week in the future.

What about quantum entanglement? Can’t we send messages instantaneously by measuring one particle and instantly affecting another one light years away? Wouldn’t this achieve faster-than-light communication without time travel? The answer is no, because in order to send information by quantum entanglement, the two parties must compare notes via traditional channels. Also, it’s incorrect to say whose measurement affected whose, because in some reference frames person A measured their particle first, and in other reference frames person B measured their particle first. When measuring entangled particles, there is no causation, only correlation.

In science fiction, we see faster-than-light travel all the time, but time travel usually takes some special magic sauce. This isn’t because of science, but because easy time travel would ruin the plot. Another reminder of the difference between narratives and reality. We may try hard to come up with a loophole that doesn’t allow time travel, but the fact that there is no universal “now,” and all spacelike trajectories are open to a faster-than-light traveler, nails the box shut. I am all for imagination, of course, but when it comes to reality, despite how strange it may be, we should allow ourselves to follow the evidence where it leads.

Economics: Inequality

Economics:
The Purpose of the Economy
A Problem-Solving Mindset
Production and Distribution
Motivations and Incentives
Inequality

We keep hearing these days that a very small percentage of people in the world own a large percentage of the wealth, and those numbers are getting more extreme. Usually, this is meant to shock us, assuming we will automatically see it as an indication that our economy is unfair. But does it really mean that?

The first time I heard about economic inequality, it was in a bar graph in a college textbook. It was relatively flat for most of the space, shooting upward at the rich end. The final bar, the one percent, went all the way to the top of the figure, wrapped around to the bottom, and stretched to the top again. And then again. Five times.

I was surprised, but I didn’t feel it was necessarily unjust. As long as things were getting better for everybody, I thought, what did it matter if the people at the top had a million, a billion, or a trillion dollars?

We might naively think we can see how the average person is doing by dividing the GDP over all of the people, the GDP per capita. But that would only be valid if the GDP were proportionally spread across all of the people in the country, and that is not the case. To get a better picture of what is happening with the average person, we need to look at the median income. Median income takes the person directly in the middle, with an equal number of people in the country who are richer and poorer.

US GDP, adjusted for inflation, since 1993.

Median income in the US since 2000.
The red line is raw dollars, and the blue line is adjusted for inflation.

In the United States, adjusting for inflation, the GDP has been on a fairly steady upward trend, but the median income has remained about even. The amount of wealth in the country is increasing, but it is not getting distributed to the average family. This means all that extra wealth is going to people who are already in the upper brackets. This is disheartening for many people, and makes them lose motivation and fear for their financial security.

Despite this, it is still possible that the average person’s condition is improving. This is because as technology and manufacturing improves, stuff gets cheaper. With the same amount of money, people can afford more, better stuff. On the other hand, some things are getting more expensive, like healthcare and college, so that argument is somewhat flaky. We would also hope that people accrue money over time, and the median is preserved by older people passing away and younger people coming into the workforce, but that doesn't always work out.

Inequality feeds on itself. The more money you have to start with, the better education, tools, facilities, resources, and services you can afford, which allow you to live healthier, longer lives, and make even more money. The richer you are, the more influence you can have on politics and the media, turning them in your favor. Add to this the fact that there is wealth inequality between races and genders, and we have a recipe that readily triggers a lot of people’s sense of injustice.

If we see inequality as a problem, how can we go about fixing it? First, we have to ask what we are aiming for. It’s very hard to find a reasonable goal where we can say we have solved the problem. So perhaps the answer isn’t to try to make a paradise, but just to strive to make things better than they are now.

Since the problem is inequality of wealth, an obvious solution is redistribution. One way to do this is through philanthropy. It may surprise you, but there are a lot of rich people who see inequality as a problem, and donate their money to help. Another way is through government taxation and distribution programs at the city, state, or federal level, depending on the specific problem in question.

The goal wouldn’t be just to give outside assistance, but to create strong, prosperous communities. This can be done by supporting local businesses, supplementing underpaid jobs, and empowering individuals through a basic income. And there are surely many other options I haven’t thought of.

When we talk about economic inequality, we mean more than just the difference in wealth between the rich and the middle class and its rate of  change. It also matters whether the average people are getting better or worse off, and how likely that trend is to continue in the future. As are all things with economics, it’s a complicated subject, and we shouldn’t settle for answers as simple as “it’s a disgraceful injustice,” or, “it’s nothing to worry about.” It’s a real issue, not just philosophical, and that means we have to look at the real consequences, both intended and collateral, of our solutions.

Imaginary Numbers and Beyond

Everyone knows what a number is. It is an amount of something. We have the natural numbers, 1, 2, 3, … We add zero and negative numbers to get the integers. Between the integers, we have rational numbers, which can be written as fractions or ratios. And we have the real numbers, which include numbers like π and √ 2  that cannot be written as fractions. It would seem like that’s all the numbers, because if we try to fit anything else in, it counts as a real number.

But let’s do something weird. If we multiply a number by itself, we get its square. 4² = 4*4 = 16. We can do the opposite, what is called the square root. √ 16  = 4. But when we play around with this, we notice something. A negative times a negative equals a positive, so -4*-4 = 16. Therefore √ 16  = 4 and √ 16  = -4. It might seem weird that there are two answers, but that’s all right. If we find a square root in a calculation in physics, it just means there are two right answers. For instance when you calculate the moment in time when a cannonball will be a certain height after it is fired, you find it’s at that height twice, once as it goes up and once as it comes down.

But consider the following operation: √ -1  . What is the answer? It’s not 1, because 1*1 = 1. It’s not -1, because -1*-1 = 1. So does it just not have an answer? No, it does. Everything in math has an answer. If the answer can’t be found in what we already know, we have discovered something new. We define √ -1  as i, and see what happens from there. In official math, the set of all real numbers *i is called the imaginary numbers, and the set of all combinations of imaginary numbers plus real numbers (for example, 8 + 4i) is called the complex numbers.

Let’s play around with i. i*i = -1, so i*-i = 1 logically, 4i*i = -4, and the same holds true when you substitute 4 with any other number. But again, what happens when we get to √ i  ? It’s not i or -i. It’s not anything that doesn’t include i. Do we have to postulate a new type of number? Something like j = √ i  ? Miraculously, we don’t. √ i  can be written as a number that doesn’t involve j or any other letter besides i. That number is:

√ i  = 1/√ 2  *(1 + i)

Hold on. That looks weird. Let’s do the calculation to make sure it actually is the answer. We start by squaring it:

(1/√ 2  *(1 + i))²

By association, this is equal to

1/√ 2  ²*(1 + i)²

The left part is easy.

1/2*(1 + i)²

Next, we need to know the rules for how to square groups of numbers that are added together. (1 + i)² does not equal 1^{2 +} i². It equals, not forgetting the 1/2

1/2*(1² + 1*i + i*1 + i²)

We simplify this to

1/2*(1 + 2i + i²)

i² = -1, so we have

1/2*(1 + 2i - 1)

The 1 and the -1 cancel each other out, so

1/2*2i

1/2*2 = 1. So when we simplify it completely, we are left with

i

And there it is! We have just proven √ i  = 1/√ 2  *(1 + i). No new dimensions of numbers are required.

Are imaginary numbers just a math thing, or do they have applications to the real world? One significance of imaginary numbers is that they represent things that don't exist. For instance, you can calculate the moment in time when a cannonball in flight will be higher than its highest point, and you get an imaginary number. On the other hand, sometimes complex numbers are shortcuts we can take to make math easier. For instance, in the famous Schrodinger equation in quantum physics, momentum is represented by an imaginary number. We could represent it by another dimension of real numbers and put in more sines and cosines, but imaginary numbers make it a whole lot easier.

But we can do other things besides take the square root. Complex numbers obey a rule called commutativity, which means 4*5 = 5*4, and the same is true with any other pair of numbers. But what if it weren’t? In math, it’s perfectly okay to ask questions like that. For this one in particular, we get a new, 4-dimensional set of numbers called the quaternions. Their units, and their basic operations are

1, i, j, k

i² = -1,     j² = -1,     k² = -1

i*j = k,     j*k = i,     k*i = j

j*i = -k,     k*j = -i,     i*k = -j

i*j*k = -1,     k*j*i = 1

Notice the differences between the 3rd and 4th rows. If we switch the order of multiplication, we get a minus sign. This is weird, and you may wonder if it even makes sense, or if the mathematicians who dreamed it up were smoking something. To assuage your fears, there is a more intuitive way to understand it, and that is to use matrices. Although the quaternion i can be thought of as the same as the complex i, it can also be written as a matrix, as can j and k, and even 1.

To multiply matrices, you take the first row in the first matrix and the first column in the second matrix, multiply each pair of numbers in the order they appear in the row and column, add the results, and put the answer into a new matrix in the place where the row and column cross. If that was as confusing to read as it was for me to write, here is a single step as an example. Suppose we want to calculate i*j in matrix form. Specifically, we want to know what the top right element will be. To do that, we choose the first row of i and the last column of j

The top right element of the result will be

0*0 + -1*1 + 0*0 + 0*0

Or

-1

Do that with every other combination of i rows and j columns, and you’ll find that you end up with k, just like we expected. And the same is true with every other product combination. I won’t prove it here, because that would be a lot of work and no one would read it, but you can work it out for yourself if you like, or you can take my word for it. In any case, hopefully you are convinced that the quaternions make sense, and aren’t just random gibberish spouted by people who want to be seen as smart.

Quaternions aren’t just a quirk of math. They are useful in modeling 3D rotations, and they are used in all kinds of simulations, movie special effects, and video games. This goes to show that math is everywhere, and even the weird, out-there mathematics can have a practical use.

Are quaternions the end, or are there larger sets of numbers still? There are. By a process called the Cayley-Dickson construction, which I know nothing about, you can get an infinite amount of them, increasing in size by powers of 2. Beyond the quaternions there are the 8-dimensional octonions, and then the 16-dimensional sedenions, and on and on. What are these number groups useful for? Heck if I know. But it sure is fun to know they exist!

Hopefully, this discussion has shown you a glimpse into the wild depths of mathematics. If not, then at least you will be able to wow your friends with imaginary numbers, and brag that you know what the square root of i is.