Properly Trained Common Sense

Toolbelt of Knowledge: Practices
Skepticism
Listening
Deconstruction
Rationality
Mindfulness
Steel Manning
Common Sense

On this blog, we have not treated common sense too kindly. “Common sense” is what we use to mean “you just know,” without having to go through all of the tedium of proving something. In general, it is a catch-all term for mental shortcuts that get us to answers quickly without getting bogged down in confusion. Because of this, common sense can be incredibly idiotic. However, there are also times when it is wise and practical. Today, we’re going to look at how to use common sense well, so that you come to conclusions that have high probabilities of being correct and useful.

The first type of common sense is a reality check. After you have gone through a process of deduction and calculation, does the answer you get make sense? For instance, suppose you are doing a homework problem to calculate the speed of sound, and the answer you get is 30 miles per hour. Do you turn it in? You may not know what the real answer is, but you do know you don’t hear a sonic boom every time a car speeds up nearby! Common sense says you better check your calculations.

Not considering the consequences of the choices we make or put off making can be considered a breach of common sense. If you’re making rice and you fill the pot with grains, it will overflow as it absorbs the water. If you have a pain in your wisdom tooth, waiting to see if it goes away on its own is not worth the risk that it’s infected. When you vote for politicians and leaders, you might want to avoid the ones who like to beat people down. Actions and consequences. Common sense.

Our next use of common sense is to accept true statements that are extremely difficult or impossible to prove. There is a form of argument called the syllogism, which is two premises and a conclusion. An example would be, “All men are human, Frederick Douglass was a man, therefore Frederick Douglass was human.” In the abstract, the syllogism looks like this:

A: If B, then C.
B: B.
C: Therefore, C.

It is obvious that if someone accepts both A and B, they must conclude C to be true as well.

But think about the statement in bold. It is a premise in itself, a hidden premise of the argument. So let’s bring it out of its hiding place, and add it to the syllogism as a premise Z.

Z: If A and B, then C.
A: If B, then C.
B: B.
C. Therefore, C.

There we go. Now we know the whole truth: if someone accepts A, B, and the hidden premise Z, they must conclude C to be true as well.

Oh no! In bringing out the hidden premise to prove the syllogism, we have discovered yet another premise! X: If Z, A, and B, then C. It becomes clear that there is a pattern: for every hidden premise we find, there is yet another premise hidden behind it. Rather than two premises and a conclusion, the syllogism in its true form has an infinite number of premises!

This is an interesting puzzle for the philosophy of logic. But for our everyday problem solving, the two-premise syllogism is good enough, and it is fine to act as if it is absolutely proven to be true. This is our next use of common sense: to take the improvable foundations of logic as if they are proven to be true.

Even when an argument is logically sound, common sense can sometimes veto one or more of its premises. For instance, we can go to the classic example of a faulty syllogism, “All men have beards. Socrates was a man. Therefore, Socrates had a beard.” Our common sense says wait a minute, only some men have beards, not all of them! Despite the argument being formally valid, the first premise is false, meaning the conclusion is invalid, and we can’t know if Socrates had a beard without more information.

However, we must remember, just because something is common sense doesn’t necessarily mean it’s true. At first glance, quantum physics seems to go against common sense, with photons and electrons behaving sometimes as particles and sometimes as waves, and seeming to teleport from place to place. But there is an enormous amount of theoretical and experimental evidence pointing to quantum physics being true, and we have a large amount of technology, such as the laser, that would not function otherwise. Therefore, despite quantum physics going against common sense, we have every reason to believe it is true.

This illustrates the problem with common sense: sometimes it is wrong. If we hear an idea that goes against our common sense, it is important to hear the arguments supporting it, and to give those arguments a good mull over, with the attitude that we might allow ourselves to be convinced to let go of our common sense belief.

Perhaps the worst danger of invoking common sense is to avoid looking at a question in its full complexity, and make out anyone who disagrees with us as fools. We all know of people who have defended their religious or political beliefs by saying, “it’s common sense,” brushing us off, and sending the message that because we don’t agree with their “common sense” view, our thoughts on the matter aren’t worth hearing. That’s not common sense, it’s stubbornness, and we must keep ourselves accountable not to fall into that kind of behavior.

Like rationality, common sense is not something we automatically have. In order not to cause more problems than it solves, common sense must be trained, and a great way to do that is to practice all of the other skills in the Toolbelt of Knowledge.

Why I Write about Things I Don't Believe

Toolbelt of Knowledge: Practices
Skepticism
Listening
Deconstruction
Rationality
Mindfulness
Steel Manning
Common Sense

On this blog, I write about all kinds of philosophical ideas. One of the major goals is to help me think about all of these interesting topics, and to figure out what I believe. But I also write about things I don’t believe, like Dualism, Idealism, and the creation solutions to the fine-tuning problem. Why do I do this? There are a few reasons. Firstly, taking the time to think through ideas I don’t believe helps me understand the ideas I do believe. Secondly, when I write fantasy, I want my alternative philosophy to be coherent. And thirdly, thinking about all kinds of philosophical ideas is fun, whether I think they correspond to the real world or not.

We will begin by discussing something called the Steel Man technique. A good competitor always wants their opponent to be at their best, because only then can the competition truly reveal who is stronger, faster, more cunning, or whatever. Although I don’t like to think of intellectual discussions as competitions, the same principle applies: the only way to know whose ideas are truer is if the very best arguments are made for all sides.

image attribution

There is a fallacy often committed during debates or arguments, when one person sets up a weak caricature of the other’s position and knocks down that caricature, acting as if this means they have knocked down their opponent’s argument. This is called the Straw Man fallacy. Most open-minded individuals will usually be able to spot one, and the person who makes the fallacy will lose credibility.

But if someone does the opposite, and builds up their opponent’s position to be at its very best, such that their opponent would agree wholeheartedly at how it is put, they have made a Steel Man. Knocking down a straw man is a cheap and dirty trick. Knocking down a steel man is a well-earned and noble victory.

In a less competitive setting, such as when two people are having a friendly conversation, a steel man can keep things positive by letting the other person know you understand where they’re coming from. It is also helpful to you, whether you are with someone or thinking on your own, because understanding what you don’t believe helps you better understand what you do believe.

When world-building fantasy stories, I strive for philosophical coherence. We all know of science fiction and fantasy stories where the authors throw in whatever they feel like without worrying about how much sense it makes. Sci-fi realism, on the other hand, is when a writer takes scientific ideas and language seriously, and uses them appropriately to craft a story that might plausibly take place in the future. Similarly, medieval realism is an idea applied to fantasy such that the geography, economics, politics, armor and horses and all of that stuff, make sense. Philosophical coherence is like sci-fi and medieval realism, but with philosophy. For instance, in The Mentor, the Hero, and the Trickster, there is a magical world the main characters can go to called the Unconscious Realms, which are built according to my conception of what a world based on Berkelian Idealsim world would be like.

When I write about things like idealism and dualism, I am building up my skill in philosophical coherence. In my fantasy books, philosophical coherence will help me give them the depth and richness I strive for.

In order to further grow my skills of philosophical coherence and steel manning, I am going to start a new series called “Best Arguments Against,” where I present the arguments against my beliefs that I find the most compelling. We’ll talk about God and quantum physics and all kinds of awesome stuff. So make sure to check back regularly, because it’s going to be a fun time.

NaNo Results 2019

November is over, and I happily stand on top of 50,000 words of extreme future science fiction! The immortal Maki Tanaka First Spring has been wandering along on his million-year journey across the Milky Way. The story is not finished yet, with perhaps another 10,000 words left to go in the first draft, which I plan to finish this month.

After the first draft is finished, my wonderful friends in my new writing group and I will swap stories and critique each other. We will trade advice on what is going well, what needs work, and what to focus on. If any of you are reading this, thanks for being absolutely amazing!

The next step in my book will be to develop the characters, so they are more than props for the main character’s journey. In November’s last days, I didn’t even give the supporting characters names, going instead for placeholders like Leaderman, Otherguy, and Cap2. So they will be the first priority in preparation for draft 2.

After that, I will revisit the structure of the story, taking into account the advice from the alpha readings. Some scenes need to be scrapped, others fleshed out. When I write a first draft, I have to include the transitions between scenes, such as car rides and time spent in waiting rooms, so that I know how the characters are feeling when the interesting parts happen. These transition scenes have very little importance, and can be reduced to a few short sentences, or cut completely.

As happens in NaNoWriMo, I was rushed every day to get my word quota on the page. Because of this, the story jumps from scene to scene, and characters were made up on the fly. Some of the sections feel more like heavy outlines than immersive stories, and not one of the characters other than Maki himself is interesting. Draft 2 will probably be at least 100,000 words, putting it in the normal range of adult science fiction. After that, it may require further structural drafts, and then a prose draft, where I edit each paragraph and sentence to be the best it can be in terms of wordcraft. And after that, it will be time for query letters and publisher submission!

When will it be published? Probably not for a long time. I would be surprised if it takes less than a year. But I can guarantee that sometime in the hopefully not too distant future, you will see the name Christian Horst on the spine of a book in the sci-fi section of a bookstore.

The Doomsday Argument

Series on the Anthropic Principle:
The Anthropic Principle
The Doomsday Argument

65 million years ago, dinosaurs roamed the earth. Then, all of a sudden, they were gone. Something happened, we think it was a meteor strike, that made the earth uninhabitable for them. 251 million years ago, a major volcanic eruption and the ensuing global climate change killed most of the life on the planet. In total, there were 5 major extinctions in Earth’s history.

Human stories are full of tales of the end of the world. In Norse mythology, there is Ragnarok. In the Bible, Armageddon. In modern times, we have the Terminator, Galactus, disaster movies. We have fears of climate change, asteroid strikes, nuclear war, and uncontrollable artificial intelligence. It’s clear we have a question deep within our psychology: is humanity about to end?

The astrophysicist Brandon Carter thought so 35 years ago. The argument went like this: assume human population continues to grow exponentially until some cataclysmic event reduces our numbers to near or total extinction. If the population doubles every 50 years, then by the Anthropic Principle, you had a 50% chance of living in the final 50 years of human civilization, and a 50% chance of dying sometime in the entirety of human history before then.

If this were the entire human population over all time, probability says we would expect to find ourselves in the spike on the right side.

This seems crazy, but it isn’t immediately obvious why. When I argued against it way back in the day, I was completely wrong. The first thing I said was that because you and I are you and I, we are not randomly selected from human history. This is nonsense. If we choose randomly from all humans who ever have or ever will live, you and I are among the possible choices. It’s a perfectly valid framework for asking questions, and that’s what the Anthropic Principle does.

The other bad argument I had against it was this: “Pick a point in history between the dawn of agriculture and the end of time, and the Doomsday Argument will give the same result: humanity is about to end.” This is true, but it is not a counter to the Doomsday Argument. The fallacy was that I switched from a random sampling of humans to a random sampling of time. And because population increases over time, a random sampling over time gives the earlier times undue weight.

So then what is the counter-argument? There are a few. The first one that jumps out to me is the fact that it is really hard to think of a scenario that will make humanity go totally extinct. Think of your favorite existential disaster: catastrophic global warming, nuclear annihilation, super-virus, supervolcano, autonomous weapons that decide they want to destroy all humans. If any of these were to wipe out the vast majority of the human species, it would be a terrible, tragic event unparalleled by anything in human history. However, we only need a small number to survive, and they will be able to repopulate the planet and restore civilization.

A big enough asteroid strike could make the Earth uninhabitable, but we have the technology to see asteroids that big years before they would hit us, sometimes decades, plenty of time to nudge them off-course. There are no supernova or gamma ray burst progenitors close enough to harm us with their explosions, nor alien civilizations close enough to invade. And the sun won’t get hot enough to turn Earth into another Venus for another few hundred million years.

What could cause humanity to go extinct? There is only one realistic example that I can think of: superintelligent AI that wants to exterminate humanity. How likely is this? Well, that deserves a discussion of its own. We can rest assured that there are many extremely intelligent people thinking about this topic, and working hard to foresee the possible risks and dangers of developing artificial intelligence. Of course, there is always a possibility that we will miss something, but the more we think about it and work on it, the more likely we are to spot the dangerous paths and go around them.

The other thing that could cause humanity to go extinct lies in the unknown unknowns. It may be that some technology will be invented that is extremely easy to make, and can wipe out humanity. This is sometimes called a “black ball technology.” If such a potential technology exists, then anyone with the right equipment, materials, and recipe would be able to destroy humanity, and given that there are billions of people on the Earth, we would be in serious trouble.

However, the very thing that makes us vulnerable to black ball technology also guards us against it: technological progress and expansion. Once we are able to leave our home planet and start new civilizations on other planets and moons and giant artificial space habitats, then we will be able to survive even something that destroys all life on Earth.

A second problem with the Doomsday Argument is that it assumes humanity will continue to grow exponentially, and then be cut down to a level so low it cannot recover. We already know this model is incorrect, because the rate of human population growth is slowing. If we apply the anthropic principle to another function, say, a linear increase, we find that we have a slightly more than 50% chance of living within the final third of human history. That final third could have a range of anywhere between 100,000 years and 150,000 years. And 50% means there is an equal probability of living outside of that time. So with linear growth, the Doomsday Argument doesn’t tell us much of anything at all!

Since the area under the curve is about equal in each section, we would be about equally likely to find ourselves in either of them.

The final problem with the Doomsday Argument is that it assumes the growth of civilization will stop with some catastrophic event. There are plenty of other models of human population that work perfectly fine with the anthropic principle. For instance, if it looks more like a bell curve, we would be more likely to find ourselves near the top. If it levels off, we would be equally likely to find ourselves anywhere along the level period. If it looks bumpy and wavy, we are more likely to find ourselves near one of the peaks. It’s very difficult to say what the human population curve will look like in the future, because every model carries with it plenty of assumptions.

There is one sense in which the Doomsday Argument seems to get things right. When we look out into the universe, we see a vision of a possible future where life, humanity’s descendants, and perhaps alien species, thrive and live around every star and in the spaces between. The number of people in such a multitude of societies astronomically outnumbers the number of humans who have ever lived. Therefore, regardless of what the population curve looks like, the odds of living on such a civilization’s planet of origin before it spreads out into the galaxy is astronomically small. It’s like winning the lottery, and the reward being another lottery ticket, which wins again, and is rewarded with a third lottery ticket, which wins yet again.

Yet here we are. And we have two options. One, we can look at the staggeringly improbable odds that we would find ourselves at this time in such a universe, and hang our heads in despair, declaring that these odds mean such a civilization is doomed never to happen. The other option is to accept that the future has not happened yet; it depends on what we do now. So since we have already won the cosmic lottery, let’s do our part to build the machine that generates our winning tickets. The world hasn’t ended yet, so as Samuel L. Jackson says, let us act as though it intends to spin on.

Why are Some Things Impossible?

Impossible. We’ve all heard this word. Sometimes it is used as an excuse to give up. Sometimes as a reason to pursue other things with our time. Sometimes in the context of science and technology. When we want to encourage imagination, we tell our kids (and sometimes adults) that nothing is impossible.

If something is impossible, it can’t be done. Seems straightforward enough. But what makes the difference between whether something is possible or impossible? In asking this question, we find there is not one single answer, but several tiers, each nested within the one before.

Logically impossible

At the most bedrock level, we have logical impossibilities. These are things that cannot exist or cannot be done because they contradict themselves. These include mathematical contradictions, as well as things that contradict their own definitions.

For example, suppose among the men in a certain town, there is a barber who shaves those, and only those, who do not shave themselves. Does the barber shave himself? If so, he is not the barber. Does he not shave himself? If so, then he shaves himself. Therefore, a barber who fits this description cannot exist.

Other things that are logically impossible: A square circle in cartesian coordinates. A solution to a 2x2 sudoku puzzle with a 1 in the top left and a 2 in the bottom right. A legal crime. A dry ocean.

Physically impossible

We don’t live in a universe where everything logically consistent is possible. Everything that exists has a nature, described by laws of physics, and these natures render a whole host of things impossible.

I should pause to mention the distinction between the true nature of reality and our current theoretical understanding of it. Science is very much a work in progress, so there are many things we don’t know yet, and many we think we know but are wrong about. Nevertheless, there are things that are logically possible, but impossible within the true laws of physics, whatever they turn out to be at perfect resolution.

With that in mind, we can talk about what would be impossible if our current understanding were correct, and let it be a proof of concept. Some things that are impossible in our universe, but logically acceptable, are perpetual motion, going faster than light or backward in time, getting out of a black hole, and reducing the total entropy of a closed system.

Technologically impossible

Even within the laws of physics, many things are not possible to us yet, because we do not have the technology to do them. This tier is vast and rich, and in considering what is inside it, we can imagine mind-bogglingly bizarre futures.

In the near term, we have things like mind-computer interfaces, autonomous cars, and quantum computers. A little further out, we might expect to invent nuclear fusion power plants, space elevators, superconducting power lines, weather control, conscious artificial intelligence, biological immortality, revival of extinct species, human colonies all over the solar system, and so much more.

These may seem like science fiction magic to untrained ears, but the difference between them and magic is that people can envision a path toward inventing them that makes sense in the context of modern science.

Economically impossible

Even when we know how to do things, we still need the will and the resources for them. As humanity finds new ways to harness energy and resources, especially in space, more and more things become possible. These include things like space ships with spin gravity, particle accelerators that dwarf CERN, interferometer telescopes the size of the solar system, and buildings tens of thousands of meters high.

But when we’re talking about the merely economically impossible, we don’t have to limit ourselves to such small scales. We could build an orbital ring around the Earth’s equator, orbiting just outside the atmosphere, with launch platforms for shuttles and rockets. We could mine asteroids and use their materials to build giant ships in space. We could envelop the sun entirely in solar energy collectors, a Dyson sphere. We could build giant reflectors to direct all of the sun’s light in one direction, effectively a rocket thruster that could move the solar system. All under known science, and with technology that has already been invented.

Cognitively impossible

Here we come to the last, most easily surpassed level of impossibility. These are things which are impossible only because people believe they are. As soon as people put their minds and efforts toward it, it becomes possible. Many of the great inventions of history came about because someone or some group of people decided they were going to do what everyone else assumed to be impossible, from the Wright brothers inventing the airplane, to the USSR sending a man to space, to the US putting boot prints on the moon.

Today, the biggest example of someone challenging the cognitively impossible is Elon Musk. From SpaceX building bigger and better rockets, to the Boring Company digging tunnels under Los Angeles to solve traffic congestion, to Neuralink connecting people’s brains with computers. He, those who work for him, and many other entrepreneurs, bring the cognitively impossible into existence.

Other things that seem cognitively impossible: World peace. Meeting the basic needs of everyone on Earth. Tolerance and good will across political and religious divides. Governments that work for everyone. A business culture that cares about the poor, the workers, and the environment. Cultures where there are no social minorities or majorities. Cities floating on the ocean. Powering civilization and the global economy with sustainable resources.

All of these things are possible and doable given the technology and economic infrastructure we have now. Why don’t we? Well, there are a million reasons, all of which are unique to their particular challenge. But many of them can be overcome if enough people or the right few see through the curtain of the way things are, and aim their sights on achieving what many brush off as impossible. Not everything is possible. But the number of things that are is vast.

NaNoWriMo 2019

It’s November again! That means I, along with hundreds of thousands of other writers, will be jamming out a 50,000-word novel in 30 days.

I’ve done NaNoWriMo three times before, and each time, I upped the ante. In 2016, my goal was just to finish a story, which ended up less than 30,000 words. In 2017, I made myself write no less than 1,000 words every day. In 2018, I aimed for and reached 50,000 with a junk novel, using the sense of fun to motivate myself to meet the daily goal. And now in 2019, I’m going all in. This year, not only am I aiming for 50,000 words by the end of the month, I am writing the first draft of a story that I want to continue to work on afterward, and aim to get published.

That’s right, I’m about to start my first book that I’m aiming to publish. Goodbye fanfictions and practice stories, the real game is beginning!

This book’s working title is Earthbound: A Galactic Odyssey. It’s a hard science fiction story taking place a million years in the future. Inspired by Isaac Arthur’s video, “Hitchhiking the Galaxy,” humanity’s first gardener ship has seeded colonies to the edge of the Milky Way, and is ready to head off for the Large Magellanic Cloud. One of its citizens, an augmented, immortal human, decides he would really like to see Earth again. The story follows him as he travels back across the galaxy at slower-than-light speeds, revisiting some of the civilizations that have grown out of the colonies he helped plant. There will be space ships, AI, technology, speculative societies, and all kinds of futuristic stuff, almost all grounded in known science and realistic speculation.

Since it is intended to be a full-on adult sci-fi novel, chances are high it will end up much longer than 50,000 words, and take longer than one month to finish a full draft.

The writer’s group I was in last year didn’t work out, but I found a new one, which looks promising. Hopefully, this group can stick together, and we can give each other the feedback and energy we need in order to succeed as authors. If any of you from NaNo and Beyond are reading this, hi! Here’s to realizing our dreams!

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.