Debatiary

I've noticed there are at least 5 quite different kinds of debate. What they're called isn't that important, so don't mind my names too much.

• Gladiatior
• Procedural
• Feminist
• Academic
• Collaborative

Televised debates are almost always gladiator debates. Representatives of factions duke it out, striving to show each other up and seize prime moments for one-liners. Two things set this debate style apart from others. First, it's done for the spectacle. Everything is pitched, above all, for the wider audience: pitched to persuade, pitched to win favor. Second, no one on stage can ever admit they're wrong or express much doubt. Not only would such an admission seem weak, but the speaker making it would be seen as failing to represent—or even as betraying—their ardent supporters, and they'd be punished.

The debates that go on in courtrooms and houses of deliberation are what I call procedural. They have clear rules that must be followed closely. These are similar to games, with distinct moves. They have a lot in common with gladiator debates, in that, for example, lawyers aren't primarily concerned with sharing the complete truth. Participants want to win for their clients or constituents, and they will use persuasion to do so. But procedural debates can also have an objective focus. A jury's purpose is to select the true story. A good lawmaker wants the law that works best for everyone.

Feminist debates are similar to what's seen in classroom discussions. Everyone has a perspective and this is politely respected. There is not a pressing need to establish fact versus illusion. That would be seen as rude. Another place this style is seen is in group therapy sessions. The emphasis is on sharing, respect, listening, and turn-taking.

Academic debates prioritize theories, interpretations, argumentation, and evidence. They have some of that in common with procedural debates and feminist debates. They are often competitive in the sense that participants will take clear positions and promote those. Ever human, academic debaters can still become offended when someone doesn't agree, which introduces an ironic element of interpersonal squabbling and feuding where, in theory, it ought to have disappeared.

Collaborative debates are what I consider true debates. They take all the best qualities of the others. Participants strive for truth, uncovering it by sharing and listening to and critiquing evidence. They take positions, but they freely change these, and they know that they can play Devil's Advocate and they won't offend their fellow participants. Everyone there knows that they are all there to advance their knowledge together. Winning means learning something new, helping to build a product, or even just imagining in a fresh way.

That last kind is rarely seen, except maybe between two or three close collaborators.

Stuck appearing

I have a recurring dream that there's a class I've been forgetting to go to all semester. However well I do in the others, this one is an F, probably an unavoidable F, come to think of it (how do I keep blotting it out so completely?), and I bet there's a test today or tomorrow. Fear rises. I don't remember how I'd find out when the next test is, or what's on it. Where did I put the syllabus? Don't I have one? It would be so embarrassing to go and see the professor—I probably won't. Sometimes, the dream includes the detail that I never finished high school, so I'm back in high school as an adult to get this out of the way, and I'm the only one like that, which makes me self-conscious. At first, I'm glad to finally get around to this. Then I realize I've somehow completely forgotten that class all quarter, and I'm going to fail it. So I'm back in high school remedially, but I'm failing it.

Considering that I spend so much time teaching high schoolers, and there actually were classes in college just like that, the only thing that's really surprising is that after I wake up, I still fully believe there's that panicky class I haven't been going to, and it takes a few minutes to convince myself it isn't true. On the other hand, I haven't finished my Master's project in 8 years and counting, so there's that.

This time, my mother was driving me to school. We stopped at a supermarket and spent a strange amount of time there, so I prepared mentally to arrive at lunch. She took a photo of the checkout conveyor and put it on Facebook (she's never used social media, and it didn't exist then). Back in the car whirring down the main road to school, I closed my eyes and imagined I was on a yellow bus; and I was, looking out the window. I tried to think about less typical things to photo and post (in the dream apparently my mom's conveyor belt snapshot was a total cliche). I fell asleep. When I woke up, my brother was to my left driving the car, his present age. "What are you doing here? How did you get here?" I asked. He sort of chuckled and said, "Hm. How DID I get here?"

Biosphere routines, feedbacks

I always feel slightly at odds with the term "climate change." Deep political stasis needed shattering with a slogan, and the phrase filled in. We had to unify and focus, pack up our smorgasbord of ecological concerns and send a single must-have item home. The old terrestrial alarms of the 19th and 20th centuries ended up putting decent people off. Numbing them. Too often our best information was seen as alarmism. It wasn't seen as fire alarms or canaries in coal mines, as notices we should at least hear and preferably investigate to see if we can help. As in all eras, some concerns were exaggerated, or turned out to be failed best guesses from partial facts, but so many weren't.

The stasis isn't unlike finding yourself saying "everything causes cancer!" to dismiss a new snippet on how you might get or avoid cancer. The rational action, of course, is to listen to the piece of information. It could save your life. (And it could save someone else's if you share.) If you believe that thinking about cancer will spoil your fun while you smoke barbecued lamb or spray paint a car door in an unventilated basement, I'll tell you what'll spoil your fun: dying unexpectedly. Talk about exaggerated concerns! It is untrue that everything causes cancer. It is true, however, that the dose makes the poison. So if you were rational, you wouldn't say "everything causes cancer!" You would listen and then try to glean what evidence there is and what dose of this factor could be threatening. Even too much water will poison you, but if a companion points out that the wild berries you're eating are poisonous, good luck with the reasoning that "everything is poisonous!"

"Everything's bad for the environment!" Are we on the same page about the difficulty we Earth aliens are having when we think about this? It's a lot easier to ruffle your flightless wings and stick your head in some mud. It's more difficult to think, "Wait a minute. Many things are bad for the environment. It isn't just one. But, hm, not everything is bad for the environment. And not everything is equally bad. Hm. So what are the worst things? And what are the best things? Where can we start today?"

It isn't actually that hard. It's just harder. And many people out there are setting the easy example of head-under-quicksand-it's-really-great-try-it. Humans love to imitate each other.

Climate change has helped focus our concerns. Most of us know it isn't just global warming from CO₂ that's looming on our radar. At the same time, I worry that "climate change" is both too vague and too specific. If someone doesn't feel as if a changing climate sounds all that bad—after all, we've lived through ice ages and hot spells, and plants will enjoy some extra CO₂—then "climate change" is just vague enough and just specific enough that this person may feel excused from considering any of the problems at all.

This leads to the question of whether we face one problem or many unconnected problems. And while there are many problems, I believe they are connected. These are dangers to the biosphere introduced by inefficiencies and inadequacies in the world's political and economic practices. In my opinion, we have plenty of information about how to upgrade those practices. The trouble is that it's happening so much more slowly than the speed of knowledge and understanding.

Sneak

Today I want to talk about Shakespeare. And not the stuff everyone always talks about—he was an actor, a playwright, a poet, likely gay, all actors at the time were men, he invented many words and phrases we still use, he was "the greatest," etc. I want to talk about three other things, one mentioned slightly less often, one mentioned rather less often than that, and one basically never mentioned. But I don't want to talk a lot. (Edit: too late.) I want to put the thoughts on the table for later.

One, mentioned slightly less often. Plays circa 1600 were still considered just about the lowest of low culture, and actors were looked down on or even despised. They were seen as riffraff. You know how some adults see young skateboarders in motley clothes as deficient human beings who should be cleared away? "NO SKATEBOARDING," don't you see the sign? Wait, why isn't there a sign! [Marches off to yell at some official.] In 1572, the British Parliament officially called actors "vagabonds and sturdy beggars." There were laws against hanging out in the streets while also having acting as a profession. Peculiarly like that sign against the skateboarders, right? There were guilds for other professions, the respectable ones, but unlike today, there was no Screen Actor's Guild protecting actors' human rights and wages.

Okay, I admit I'm exaggerating slightly. It had been rough, it had been like that, but things were changing, slowly, at least for several of the very best troupes. Shakespeare's troupe would perform for King James sometimes, and this association with royalty gave them some protection and extra funding. They would constantly perform for the public in open-air theaters, people of all classes, and not just in London, but also in the countryside. We have to remember the then-recent history of theater, which was still in evidence all around the country: actors are scum, plays are trash. Today Shakespeare is often seen as high culture, such high culture that we expect to feel dumb trying to decipher his torrent of metaphors, so much that if you merely claim that your movie or novel is based on Shakespeare, that immediately makes you seem legit.

This all very loosely and associatively reminds me of "Mad King Ludwig II" of Bavaria. Ludwig was called insane by his doctors—quite suspect doctors who happened to have been bribed by his enemies to find something wrong with him. What ensued from their examinations was 19th century fake news. Their critical evaluations of his psyche were splashed over the newspaper for everyone to read. In our era, that would already be seen as a terrible breach of privacy for a psychiatrist, but worse still was the diagnosis. Actually, Ludwig was just gay—hardly a mental illness, but sadly that isn't how people saw it back then. As I mention above, Shakespeare was probably gay or bisexual, though he married and had three children. How this might have interacted with his Catholicism, which was another taboo in early Church of England times, is a fascinating question, and who knows, might have something to do with the disappearance of all his books, manuscripts, diaries, etc.

Anyway, this young monarch of Bavaria was also popularly trashed for his building projects, which almost everyone considered a colossal waste of money. To be fair, the kingdom of Bavaria was not doing so well, and more money would certainly have been appreciated. But also to be fair, these were not state funds he was exhausting, but the family's funds. Now fast-forward a bit. His castles bring more money into Bavaria today, thanks to tourism, than any other industry there. And here's a commercial trifle many people care about: Neuschwanstein, his unfinished chef d'oevre on a mountaintop, gives us the image of a fairytale castle, the one lifted directly into Disney's logo. Our so-called mad king built the Disney castle before there was a Walt Disney alive to copy it. (If you're ever in that part of Bavaria and you haven't been up that mountain, I highly recommend the visit. You will not regret it.)

While King Ludwig may not have been a very good king—he wasn't even interested in being a king—his unparalleled vision probably mists into your mind when you think "fairytale castle." Today, we would likely consider Ludwig's extravagant construction projects not so much a waste of money, but rather a series of moon shot speculations that turned out unexpectedly well. Ludwig was not only insulted and smeared for turning "insane" enough to love men, interior decorating, and co-designing beautiful architecture. Tragically, he died young by assassination. It seems obvious that he was an artist born into the wrong profession: king.

Their biggest commonality is that the enduring value of their work was not recognized, and they had to contend with fairly brutal prejudices against their sexual orientation and even talents and religion (in Shakespeare's case).

Two, mentioned rather less often than that. Shakespeare and company were using a new technology. There has to be some reason we're virtually unaware of any plays after the ones by Sophocles in ancient Greece (most of us), and then suddenly we're aware of this Shakespeare bloke from 2000 years later. Does it strike you as odd, on reflection? What happened to all the other playwrights in that time? Were there any?

Yes, of course. Something new was happening that begins to explain this misalignment, and I mean something other than the printing press. Remember, there was no printing press in ancient Greece, and some of Shakespeare's plays were likely written out from memory by his fellow actors years after his death, so "lack of a printing press" doesn't explain the 2000 year gap at all.

Curious yet? Ok, what was changing was that for the first time in history, building technology was good enough that you could actually make a theater that was difficult to sneak into without paying. A few guards couldn't stop a crowd before. For the first time, you could charge everyone a price of admission, like at the movie theater. That's so standard for us today that we don't stop to wonder whether it began at some particular moment in time. It began in Shakespeare's day. Ticket prices were low—I've heard one penny—so basically anyone could attend, and yet barring people who didn't pay—to encourage payment—allowed at least the very best troupes to make a steady living. Few respected the actors or the art form, but if you were Shakespeare enough, you could be well-off from ticket sales. This here, my friendly readers, is a very early example of middle-class culture. It wasn't just kings paying for private entertainment or free religious/moral tales for everyone. Shakespeare was something like a Kickstarter baby. His generation of playwrights was more crowd-funded than any of its predecessors. (At least, this is what I read in a scholarly introduction to Antony and Cleopatra written by the editor, A.R. Braumuller. It's called "The Theatrical World," and it appears at the beginning of every book in the Pelican Shakespeare series.)

One other shift was that though the Catholic Church had long suppressed theater, which probably explains some of the big gap, England had split from the Catholic Church in 1534 with the foundation of the Church of England. I have no idea whether that was a factor, but it seems possible.

Three, one basically never mentioned. We give credit where it's due for words and phrases that come to us from Shakespeare. There are hundreds, thousands. But you know what's strange? When you compare Shakespeare to other top playwrights from his time, he actually invented (or recorded before anyone else) fewer words than they did. Not bad for a buncha riffraff, huh? It's just that his plays were so impactful that the words and phrases he did invent changed the whole language. Many consider his writing the beginning of modern English.

So—we especially remembered his inventions, even though there were fewer of them. Why?

There's a distant parallel in the works of the "Italians" Petrarch and Dante. I use quotation marks, because there was no Italian nation at the time, even no Italian language at the time. Those two poets spliced together the local languages, the bits they liked—kind of a "greatest hits" of ways people spoke around the peninsula—in their writings, synthesizing what eventually became modern Italian. They weren't the only good writers. But what they wrote stuck so much it defined a language, this time a language where no single language had existed before the publications. This all circa 1300, when, as I say, there was no such language as "Italian." In 1850, there still was no such language as "Italian," but Petrarch and Dante spoke up from 500 years before. Their works had gotten so much traction over the centuries that they were instrumental in standardizing a language for the new nation.

Incidentally, almost a total tangent. This "invention of a national language" is why Italian spelling is so consistent and intuitive that Italians don't even really use a word for "spelling," they just say "writing." The language "began" recently enough that the spellings haven't accumulated baggage from old pronunciation patterns. Like, "through" and "who" have baggage from ancient pronunciation patterns. Same with French words like "ancêtres," where the accent on ê denotes that the word used to be "ancestres," but the middle s is no longer pronounced. Also the final e is mostly silent, the s at the end is silent, and the n is not really pronounced anymore either, though it influences the a somewhat. None of that guff in Italian. To a native English speaker, spelling in Italian feels like cheating.

We had Chaucer and Shakespeare and then centuries of language evolution, and only then some modest attempts at standardizing and simplifying spelling, which had to honor the old and the new and countless works in between. With the new country of Italy, they just updated Petrarch and Dante and taught that to everyone. ("Just" is an atrocious disservice to how long this took and how much effort, given that only 2% of the peninsula spoke the language when the nation formed in 1861. But it sounds smoother, ha!)

Another similarly tongue-defining pair of works is the Tanakh/Old Testament (for Hebrew) and the King James Bible (for English, in fact contemporary with Shakespeare, and some believe he was on the writing team). Both have been widely read for so many generations that they've played a major role in defining and preserving the respective languages. We would have a language, but without a big culture of widely admired examples of how to express ourselves, the language would have drifted faster, and "English" words from 1600 would probably be totally incomprehensible by now. In a sense, the best writing almost conspires to keep itself remembered by sticking pins in the language. It says, "Slow down! Keep speaking my language my way so you can remember me!"

Think about this: if a video game console came out without any games, or with only completely bug-ridden, slapdash, unplayable games, then no one would bother to write any emulators for the console. No one would bootleg the games. No one would write games in the same style. Soon, no one would remember the console or the games. If there were any unique interaction idioms in those games, they'd be forgotten, and if you were to encounter them, they'd make no sense to you. This is something I like to investigate... I've toyed with many old classics whose interaction innovations simply didn't catch on, and though I find the awkwardness lovable, sometimes it's even more incomprehensible today than it was back then.

Do you remember VTech Socrates? You don't. It was my first and only game console as a kid. And it was decently educational, and the music settled so deep in my brain that when I finally heard it again recently on YouTube, for the first time since I was probably 8, it was as if I'd heard it a week ago. But there were almost no releases for Socrates, just the math/spelling/music skills cartridge that came with it and one other that could be bought. Yours truly should be the prime candidate for a person who would know what that other game was, yet I have no clue. The company went out of business quickly. You've never heard of it. Right? There is no VTech Socrates emulator. The games don't look like games from any other system.

Meanwhile so many people love Super Mario Bros (I'm not particularly a fan but I respect it), it's easy to get a Nintendo emulator. Because it's easy to get a Nintendo emulator, it's also easy to run any of the other games that came out for that system. Other games were very popular too, and keeping a clutch of these popular games alive means that less successful ones for the same console stay more hydrated and whole than they would have been otherwise. Because the "language" of that console, its style, its controllers, its interaction idioms, remains a prevalent influence, no one is going to forget how to run, play, or make any of those games very soon. That means someone who grows up playing only the newest games today will still be able to understand the old Super Mario Bros easily. The influence on the present keeps the past available. Looked at a certain way, it almost feels like a conspiracy. With religion, it feels even closer to conspiracy. But I'm just being colorful.

So yes, Shakespeare was a wordsmith and a gifted one, but his language is not more complicated than the language of his rivals (less complicated, actually), and he invented fewer words. But I suspect we remember his words (or the words of his troupe if you see the authorship as communal, and it was—plays were often written collaboratively in taverns) simply because the plays are better overall. Then again, I've never read or seen one of these rival plays by the likes of Christopher Marlowe and Ben Johnson, so I don't have more than a suspicion. Some of the rivals and rival works were more popular than Shakespeare at the time. Would I recognize the words they coined? I don't know.

For one argument that could back the "better overall" claim up, it's often said that Hamlet was the first "modern" character, the first character with so complex a personality, with realistic enough introspection and inner conflict that we don't see him as phony, abstract, stilted. The earliest high-water mark for lifelike character writing that's agreed on is Odysseus. But there's a big jump, both in time and in realism. Hamlet is kind of the Mona Lisa moment in character creation and development. All other characters before that just seem less real. Or so it's said in English departments! If that isn't a misconception, I can believe that this special incantation—the lifelike figment, the entirely believable ghost of a human machine—would be more memorable than inventing words or writing rococo sentences. Maybe "good characters" is the best explanation.

Certain questions

Most people judge a person's claims and opinions by that person's confidence. We don't always realize we are doing this. We may explain it to ourselves another way; it has the status of an unconscious bias.

For perspective, early experiments with electrical stimulation of living brain tissue during surgery began to reveal how we rationalize processes inside us that we don't understand. An electric current sent to one part of a person's brain would reliably make that person laugh. When asked why, the person would say the doctor's face was funny. Clearly that was not the actual reason.

It's the same with other, more naturally present brain processes. Since these experiments, researchers have found much evidence that the reasons we give for our opinions and actions are unreliable. We aren't being dishonest. (There can be a difference between honesty and truth.) We explain why we think we like this product or distrust this person, but often the evidence strongly suggests that our minds are fabricating an explanation so that we can feel whole and coherent. Often the actual reasons are much more basic.

This is why a little subtle dishonesty can be very persuasive. Our minds are already at a disadvantage in that our beliefs about why we choose this or that are often mistaken or inaccurate. Malicious actors can exploit this.

The phrase "con artist," after all, comes from "confidence man," someone who would use trickery to get taken into your confidence. Confidence is a key weapon in the confidence man's arsenal. Our country is presently run by a confidence man. His tactics aren't even a secret. Yet they still work on tens of millions of people who have heard that he's a habitual liar and a con artist. They believe that they're laughing because the doctor is making a funny face, not because of the brain surgery they know they're in, but their minds are being hacked.

For the reasons I've just outlined, we should be careful not only against knowingly accepting confidence as the reason we believe something—or someone—but also against unknowingly accepting confidence as our reason for believing.

I'll give you a better way.

First, listen not for what a person believes, when they speak, or how strongly, but for why. Why does the person say this? If they are biased or trying to trick you, there will be problems with their reasoning. Whether they know there are problems with their reasoning or not, they will try to distract you from talking about those problems openly and civilly. They will try to make this about you and them, for example, by guilt-tripping you or making you feel proud, instead of giving bulletproof reasoning or listening to questions with a good attitude, thinking about them, and being grateful to the people who bring them up.

It's like isolating two suspects at the station and questioning them separately until one says something inconsistent with the other's assertions. Here, though, you are not looking to make sure every thought coming from a person is consistent with every other. That worry is a red herring, one good liars know how to exploit. "A foolish consistency is the hobgoblin of little minds," remarked Ralph Waldo Emerson in his landmark essay, "Self-Reliance." Don't be afraid to disagree with what you've said before, or to observe two facts that seem contradictory, or even a hundred facts that seem contradictory. And let others do the same. It's smarter! It's kinder! It leads to discoveries! Eventually, even to agreement and consistency! See, you can find possible inconsistencies even in math proofs that are valid and known to work. Preoccupation with enforcing this unrealistic standard outside of math, or else with attaining an appearance of perfection outside of art, is following or planting a red herring. Let people be inconsistent; you are.

Try this instead. You're looking for how they support their beliefs and claims with evidence and logic. We may have a veil over our mind's inner workings and be deluded as to why we choose A over B, but logic either works or doesn't. Evidence is either accurate or inaccurate. There are shades of evidence from "none" to "extensive," yes, but the physical world does not change to match a wrong view; it stays the same. It's the view's responsibility to match the physical world and its potentials, not the world's responsibility to be how a confident person says. This is how we begin to extract ourselves from the problems I've outlined above. Don't make it personal, just look for why. What is the person's reasoning? What is the evidence?

Now comes the most important part.

How does the person handle being disagreed with?

To the trained eye, if you have no better source of information right now, this can sometimes tell you everything you need to know.

It's a window on how they reached their opinion in the first place.

If their first response is to undermine the person disagreeing by trying to make them look incompetent, stupid, or just plain bad, then that's a giant red flag. They are not relying on reason. They are making this personal. They are focusing on optics and its cousin side-taking rather than on facts and the tissue of reality and uncertainty that connects them.

A mostly accurate view can reflect reality quite well despite containing inconsistencies and paradoxes. The entire field of science is not invalidated every time a discovery resolves a point of confusion or rewrites part of a textbook. What picture in a few words can capture all the nuances and smooth out all the sharp edges? I know of no such picture, do you? A con artist will try to erase or deny these. An intellectually honest person, by contrast, will want to think about potential paradoxes and data gaps and talk about them and find out more.

It is not confidence to look for, but this kind of adventurousness. That is, if you really lack time, energy, or skill for a better, more thorough, more evidence-based approach.

A person who cannot admit uncertainty is less reliable than anyone else, when it comes to ultimate believability. No one is truly certain, 100%, about anything. Whoever claims to be is lying, often unknowingly. Knowledge in large part comes down to how we find, respond to, and manage uncertainty. If we decline to look for it, deny it when it's mentioned, and manage it by believing it doesn't exist, then we are con artists. Perhaps we are not malicious. Very likely we are not. Perhaps we do not even know we are con artists. Perhaps we are unknowingly conning ourselves for lack of training in a better way. Perhaps it would be more accurate and kinder to say we are unwittingly allowing our instincts and biases to con us. But if we are allowing our instincts and biases to con us, then we are opening ourselves up to attack by intentional con artists. And we are also magnets for all kinds of wrong ideas that people naively hold with great confidence, great, infectious, influential confidence.

Confidence is not bad. Do not take me the wrong way. Having an even keel of self-confidence and self-efficacy allows us to hear criticism and accept that we are imperfect and try again and improve. Healthy confidence is critical to living, learning, even listening. No one is always at exactly the same level of confidence, let alone entirely confident or entirely unconfident. It is a useful, an unavoidable and necessary, inner emotional signal, in its shades of presence and absence. But it isn't a reason to believe something. Confidence isn't logic. Confidence isn't evidence. And it isn't reliable.

Think of confidence as a quick sketch—a one-line drawing that may have artistic merit—rather than as the thing that's being drawn. You can admire that one-line drawing. You can feel it was made by a master. You can wish you had that vivacious boldness. You can feel inspired, comforted. You can imagine what the one-line drawing represents, even in great detail, if you want to make enough effort. But now do something for me, and more importantly for you and for people you know who listen to you. Put down the one-line drawing you've admired, and take a long, inquisitive, skeptical, deep look at the thing the sketch was a sketch of.

Forget the one-line drawing. What do you see?

Congratulations. Now you know how to do better.

Inner musical rhyme

It's funny, for some reason Western music mostly shuns the core Arabic-sounding (Gypsy/Jewish/Byzantine/Indian/etc) scale. All these different cultures around the world have found it. Occasionally classical or pop music deigns to find a chair for it, but we treat it as somehow pathological. Somehow we ought not to use it, for reasons I've never understood.

And I partly understand. It's one of the most difficult common scales to put in harmony. It just doesn't work with classic notions of chord and cadence. And that's probably related to the way traditions that prefer this scale tend not to be big on chords. Though I haven't done any sort of objective analysis, it seems that in monophonic, multitimbral traditions (one note at a time but multiple, possibly different instruments or singers sharing that note), it's often preferred. It makes for exciting melodies. If you want polyphonic harmony, in other words chords that move, you'll probably have to break out of the scale often.

But I think if you look around the world, there are 3 most common scales: major, pentatonic‡, and the Byzantine/Jewish/Arabian/etc one I'm lavishing with attention today. Cultures have discovered and rediscovered these scales ad infinitum because there are fundamental reasons the ear is drawn to them most.

‡ (The major pentatonic most of all for pentatonics, pentatonic meaning 5 notes in the scale, but I think there are reasons to emphasize the set of notes as one united scale even more than it is five modes, given that you basically can't play anything that sounds bad or off—try it! Sit at a piano and play the black keys, and tell me if you can really go wrong. It doesn't matter where your bass note is. Plonk on the black keys, and it'll sound good. This is the most universal core of music around the world. Major pentatonic, minor pentatonic, mixolydian position pentatonic, etc, whichever note in the pattern you start on, people around the world love it.)

What I've just realized about the Byzantine scale (in Western music it's called the Double Harmonic Major scale, but who cares, that makes it sound so specialized when it isn't at all) is that it's not only symmetric (I did know that before), the only perfectly symmetric scale in 7 notes (hm, I should check this claim for its conditions), but it's actually symmetric within its two halves. It's the most fractal-looking scale I've ever seen.

You can see what I mean... if 1 means the note is on, and 0 means it's off, here is the pattern of every regular Major key:

101011 101011

(If we're in C Major, this means there's a C in our scale, there isn't a C#, there is a D, there isn't a D#, there is an E, etc.)

You can see it's got a nice sort of symmetry, not a mirror, but repetition.

101011 <-> 101011

Here's Phrygian, ¶the¶ Flamenco scale:

110101 110101

¶ (Not exactly true. Flamenco music uses other scales also. But I think Phrygian most immediately harkens to what's different about Flamenco and Spanish music overall, for most people, in the lowered second note. It's a convenient and practical jumping-off point for Minor, Phrygian Dominant, Double Harmonic Major, and from there even the latter's popular mode Double Harmonic Minor. Also, I'm not concerned about whether we call Phrygian a scale or a mode, for sticklers out there, as we're comparing chromatic patterns with fixed beginnings and endings.)

It's the Major scale, flipped. Neato, right?

101011 101011 Major

110101 110101 Phrygian

Major sounds like it wants to rise and keep rising. Phrygian sounds like it wants to fall and stop. They're perfect opposites.

-

If you're unfamiliar with scales and modes, these 2 patterns live in a family of 7 connected musical patterns called the modes of the diatonic scale (diatonic meaning 7 notes). They are also called the Church modes, because they go back past Renaissance music to Gregorian chant. And even earlier than that: they go beyond the first musical notation, so we don't know when they started. (If familiar with modes, you can skip to the next section.) What I'm calling a "family" of modes (such a family is called a scale, and this 7-note/diatonic scale is the most standard scale around the world, at that), is kind of like a large piece of fabric, and the modes are like sections of it. Maybe we can pick on wallpaper—imagine some kind of tartan wallpaper in a big roll, if you like. This big piece of tissue, the common musical scale, is based on the idea of harmonics, which are whole-number multiples of a core frequency. It isn't critical here, but harmonics are essentially multipliers for sounds. Anyway to make a very long story short, this scale, this big "fabric" or "tartan wallpaper roll" is just the white keys on a piano.

In your mind, picture a rectangle of that striped wallpaper, and focus on the edges, which will look different depending on where you cut. If a scale is a big roll of wallpaper, a mode is a specific section cut from it, the stripes in the pattern situated wherever they are as a result of the cut at the beginning and the cut at the end of the section. Actually, the cut at the end occurs at the same point in the pattern as the cut at the beginning, the first time the pattern begins to repeat. That's always true with modes. And to go back and use my other metaphor, if the scale is a landscape, its 7 modes are like 7 big, overlapping photos of that landscape, or better still, 7 full panoramas with different center points. All this is most easily understood with your ears, by just playing the regular letter notes on a piano (white keys) or any other instrument: A, B, C, D, E, F, G. Hey, no sharps or flats yet! Now for the exciting bit. Starting on each letter and playing notes around it, you'll get a slightly different vibe. Actually, these vibes can feel very different from each other. There are 7 of those modes, one centered on each letter. They're like 7 windows looking out of the same castle on the same heathery, windswept valley, but focused on different parts of it with different emotions coming in through each window. There's the A window, the B window, the C window...

Major is the vibe you get when you start on C and play the white keys up or down to the next C. To be candid, just play anywhere, but always return to a C as the central station, the "home base" sound. Keep hitting C a lot, and also explore and play anything else, but just the white keys for now. So, whether it sounds lovely or excruciating, that's Major (C Major in particular, but I'll get to that). Phrygian, meanwhile, is the vibe you get from starting on E and playing the white keys up or down to the next E. And just like before, play all around, but only the white keys for now and keep returning to E as "home." Wherever you go every day, you start at home and end up at home, right? Travel the same way with music. If you don't return home, you'll feel lost or strange. Music uses that trick also, but most of the time, most of us return home at the end of the day, and the same is true in music. (Also, I know "only white keys" really sounds like some kind of racism, ha, and I'm sorry about that. There are pianos available that reverse the colors... though they tend to be more expensive because they look super rad and are in demand. Whichever palette your keyboard uses, interesting music tends to break out of this "only white keys" or "only black keys" thing.) Now, so far our keyboard strolling has demonstrated 2 of the 7 modes. The other 5 modes come from starting on F (Lydian), G (Mixolydian), A (Minor), B (Locrian), and D (Dorian).

There's much more, though, and this is just the tip of the iceberg of scale patterns. You can slide any pattern to start on any key of a piano, or on any frequency in the whole sound spectrum, including frequencies too high or low for a piano, or else drifting somewhere between the keys of a piano. For all starting points other than the mode's home letter on a piano, maintaining that mode's pattern or vibe will require black keys; or if we're between keys on the piano, we'll need more between-keys; if we're dealing with a musical staff and written notes, it'll require some sharps or flats. While C Major is all the white keys from C to C, F Major is almost all the letter notes from F to F, except not quite, because there will be no B key, but instead the B flat key to the left of it. And while E Phrygian is all the white keys from E to E, C Phrygian will have 4 black keys and only 3 white keys. The same goes for starting any of the modes on another letter. We break out of the white keys somewhere (or the black keys if the piano has reversed colors).

Note this is an artifact of piano design, not of sound itself. If you learn to sing and never look at a piano, there's no difference between E Phrygian and C Phrygian except where your voice is. The audio spectrum does not "think" or "feel" in terms of black keys or white keys, sharps or flats. Those are human constructs. On a guitar, sharp notes, flat notes, and natural notes look identical, because they are. A note is a frequency. Two notes are a pair of frequencies. Whether that pair sounds sharp, flat, or in tune depends on what you're used to hearing and what you're expecting, not actually on the frequencies themselves, not in any way that systematically matches the whole "sharps and flats" thing. Which isn't to say there's no reason for it, but sometimes we need to step back and look at ground truth.

One more point about naming. Often modes themselves are just called "scales," which is confusing. Basically all these patterns are scales, but "mode" is a special meaning of "scale" that refers to the (sibling) patterns you get from starting at different points in the same (parent) pattern. Let's pretend for a second, for illustration, that music is just numbers, not notes, not even sound. You could say the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2 ... are a scale, because, as you'll notice, it's a repeating pattern. If you started say on 4, that would create a different mode of the same scale: 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7 ... But since it's also a repeating pattern, it's also correct to call it a scale. It's just a scale that we're thinking of as a mode of another scale.

Minor mode (also called Aeolian):

A, B, C, D, E, F, G, A

Major mode (also called Ionian):

C, D, E, F, G, A, B, C

Phrygian mode:

E, F, G, A, B, C, D, E

And so on, for all 7 letters. The letters are shared, so it's all the same larger scale.

The spacing between the letters is not the same, so these modes have different sounds, feelings, atmospheres.

-

Back to the relationship between these two modes I was showing you, now that we've got a little background.

101011 101011 Major

110101 110101 Phrygian

Major is the second-brightest of the 7 diatonic modes (just below Lydian, the pattern starting on F). Phrygian is the second-darkest of them (just above Locrian, the pattern starting on B). You can see there's a kind of mirror of symmetry in between them, I hope. If so, that isn't your imagination. That isn't just a human construct. That's a real symmetry.

But here's the Byzantine scale:

110011 110011

It's symmetric, then it's symmetric inside each half. That's got to be related to why it's so goddamn compelling.

You can't get Byzantine by playing only white keys on a piano, unlike the others I've been talking about.

Also notice that you can get to it in two steps from the previous scale above it, Phrygian, and those two steps are actually the same step reflected in each half of the scale:

110101 110101

110011 110101

110011 110011

Traditional Spanish music just loves to navigate these three scales. They are called the Phrygian mode of the diatonic scale, the Phrygian dominant scale, and the Double Harmonic major scale, respectively, if we want to be "correct" (whatever that really means) and use the conventional names in English, down to which letters are capitalized. Personally, I call them Phrygian, Spanish Phrygian, and Byzantine. Works for me.

110101 110101 (Phrygian mode of the diatonic scale / Phrygian)

110011 110101 (Phrygian dominant / Spanish Phrygian)

110011 110011 (Double Harmonic major / Byzantine)

The last one is most recognizable as Middle Eastern or Gypsy or just downright foreign and exotic (though, believe me, there are much more foreign sounds out there).

Just in case you didn't believe anything eldritch was going on, let's visit the Hirajōshi scale, a family of pentatonic modes played on the koto, the zither-like national instrument of Japan. This classic Japanese sound is what you get by removing those two notes instead of moving them.

110001 110001 (Hirajōshi scale's corresponding mode, showing the alignment of patterns)

Detail: between the two halves in all the scales above, there's a note that isn't on. I'm using a tradition going back to ancient Greece of dividing scales into two "tetrachords," two groups of four notes, stacked one on the other. For most scales we use, there's one note missing between the two tetrachords. Hence the gap above. As the ancients would see it, we play a tetrachord, a little jump, and then another tetrachord. The jumped-over note is the same note an ambulance siren plays. It doesn't sound nice in many circumstances. (It's often left out in various ways, even though it's actually critical to the push and pull of melodic and harmonic development as for example in the Tristan chord, but that's another story.)

So the Byzantine scale actually looks like this:

1100110110011

And the first 1 and the last 1 are the same letter, the last being exactly one octave higher. For example, if we're in the key of E Byzantine, then the notes will be:

E F G# A B C D# E

By the way, the note skipped over in the middle of E Byzantine is an A#. It isn't part of the scale, but as I said, we shouldn't entirely ignore these notes, which are called tritones. A#, the tritone, is the line of symmetry for this octave. It's the harsh center. Blues loves tritones! And going back some centuries, baroque composers figured out they could use them to change keys seamlessly. (Remember C Major versus F Major? Or E Phrygian versus C Phrygian? That's changing key. It makes an immediately audible difference, and if you aren't careful, it can sound totally wrong and incompetent, an obvious mistake. Then again, are mistakes necessarily bad? Anyway, crafty use of these ignored, underdog, siren-ugly tritones makes this a moot point, because with them you can change to any other key and have it sound right and pretty amazing.) You can grimace at the tritone, you can pretend the tritone doesn't exist, but it's still right there in the middle, and it still has soul.

There's an Indian tradition called Carnatic, which is probably the most complex, nuanced scale system in popular use. It has about 22 notes available (give or take depending on the particular area or tuning approach) instead of 12. Singers work within that system with hundreds of different scales, literally, and they know how to work them accurately.

The first song that's usually learned in the Carnatic tradition uses the Byzantine scale. Of all the hundreds! That strongly suggests it's one of the most compelling scales to the ear. And that's obvious. It just is. But it's maybe the most "scientific" test a culture has done, given that Carnatic has such a high-resolution tuning system, and they could have picked a zillion other places to start.

I would say maybe it's the visual symmetry appealing to new Carnatic practitioners as a mnemonic device, but I doubt that applies when singing, very much. Maybe? Maybe the scale is only so popular around the world because it's somehow easier to remember. But I can also say I've always been very drawn to this scale, and I didn't know about the symmetry for years. And the symmetry within the symmetry, the fractal quality? I didn't know that at all, yet I suspect my ears did all along.

Exclusive selection

How are Geiger counters and what makes human brains unique related? There's a much more specific answer than "human brains came up with Geiger counters, and crow brains procrastinated for too long and didn't."

In my post-before-last I made passing mention of a board game called Nim and its role in the history of computing. Actually, Nim is more of a matchstick game than a board game, but that doesn't matter! Nim is normally played with 4 rows of matchsticks, arranged in counts of 1, 3, 5, and 7 (that's a total of 16 matchsticks). Other counts work. And you could use tallies in orange chalk, pennies, donkeys, or whatever. What matters is that we have several different groups of items. Each row is a group. Mathematically, each row is actually a set, in our case a set of distinct matchsticks sharing the same row. Synoptically, Nim is an ancient Chinese game about sets and how many items are in them. But it's more fun to pick up matchsticks than to think of mathematical sets, sizes of sets, etc. (Play it here. The player who picks up the last matchstick loses.)

Ok, but what about the history of computing? On September 24, 1940, Edward Condon, a 38-year-old physicist, patented a machine called the Nimatron that he'd recently presented at the 1939-1940 World's Fair in New York. The showcase at the fair had taken place on May 11, and he'd filed for a patent a couple weeks earlier, on April 26. This machine could play Nim better than most humans. Indeed, it won 9 out of every 10 games.

The idea of robots was not new to fairgoers. One of the attractions was a man dressed up in a metallic (bronze? tough to tell in black and white) suit, talking in staccato fashion about having a mind. (Correction: that "man dressed up" was actually the robot Elektro. It had me fooled!) Other attractions prompted audience discussions about the new, advanced mechanization in factories putting workers out of jobs. At least one person took the other side of the old fence and said what AI researchers like to say today: that the widespread fear of lost jobs had not actually happened, because mechanization changes the nature of our human roles and therefore of our jobs. The debate continues more than 200 years after it started. But all that aside, it's something quite different to walk up to a machine that promptly outsmarts you.

That's what happened for about 90,000 visitors at the fair, out of perhaps 100,000 who played the Nimatron. What's particularly interesting about the patent on this machine is that it's earlier than the first general computer, yet it uses a similar process. Somewhere in the back of Condon's mind, he was aware that more could be done with the gadgetry he was assembling. It could do more than just beat fairgoers at matchsticks (represented by lightbulbs).

A year later, in May, 1941, a machine developed by Konrad Zuse, the Z3, became the first general, digital computer, a contraption broadly equivalent to whatever you're reading this post on, just slower and with less memory. The principle had been worked out, the invention built. And it worked. (I'm simplifying here. Yes it worked, but if Condon didn't know the full capabilities of the process, neither did Zuse. As far as the latter was concerned, this was a very effective calculator. No one knew his Z3 was a computer in the proper sense until work in the 90s proved it. However, this began a lifelong career in computer science and engineering and manufacturing. He did knowingly build computers later. He was the first. He cracked it.) Let's return to Zuse's computer in a moment, though. After all, the Nimatron had in some ways beaten it to the punch.

One thing that made Nim particularly suited to such early development is that the winning strategy relies on binary numbers. It's actually quite easy, if you know how to count in binary.

1) Convert every row's matchstick count into the equivalent binary number.

2) Cross off pairs of identical digits. That is, with these binary numbers stacked up and alined on their right sides, just as in grade-school addition, go down each column and cross off any pairs of 1s that you find. For example, if there are three 1s in the rightmost column, cross off two of them, and bring the last down, writing 1 under the column. If there were five, you'd cross of two pairs, or four of them. You'd also write 1 underneath. If there are four, cross off all four. Write 0 underneath. Everything canceled this time. If there's one, don't cross off any. Write 1 underneath. This process will result in a new binary number made of the straggler 1s for columns that weren't completely canceled out. At most a single 1 will be left for each column. Like with grade-school addition, each column represents a specific place value, a specific location in the final answer. You get a new binary number, but the value of that number isn't important. What we care about is whether it's 0 or not. When you receive a board position that produces 0, that's bad.

3) For your next move, make the sum 0. If at all possible, that's what you want to do. And while you're at it, remove as many matchsticks as possible to do so. Making a move in Nim (I should have said this before, but there's no time like the present) means choosing a row and choosing a number of matchsticks to remove from it. You can remove any number of matchsticks, but only from one row, and you have to remove at least one matchstick. Do this so that the process in step 2 passes a 0 to your enemy to deal with, every turn (except the last, when you craftily leave them the final matchstick so they lose), and you will definitely win.

Step 2 actually amounts to performing the XOR (exclusive or) operation on the numbers, something very easy to accomplish with gizmotry, and a core operation in computing. It's also very easy for humans when we line the numbers up vertically on paper, as in normal addition. Step 2 may sound complicated, but I could teach you to do it in about one minute, even if the words above make no sense.

Following this strategy assures a win if you are the second player. If you are the first, it assures a win if your partner (opponent) makes a mistake. Otherwise, they have the advantage and can always win by following the strategy exactly. Summary: be the second player. Other summary: if you're the first player, you may be out of luck, but try to distract your opponent. Skeleton key: use the strategy.

What about Geiger counters? I was getting to that.

Ok, the Nimatron was born in a moment of inspiration. Edward Condon was working with Geiger counters, and he suddenly realized the internal mechanics could do something more than just counting. (I don't understand the details, as I've yet to research the claim "he realized that the same calibration circuits used in Geiger counters (although built with ordinary electromagnetic relays, not by valves), can be used to represent the numbers defining the state of a game.") Fast forward a bit, and he's built this clever party trick that's perplexing people at a New York City fair.

What just killed Edward was that he could have gone about a million steps further, without too much additional effort. The same principle and the same components are what got Konrad Zuse's contraption calculating, well, uh, in theory, anything that could ever be calculated. That's more than a game of Nim. No matter how much you like Nim, you have to admit that "anything that could ever be calculated" is a rather larger prize.

You may think I've now connected Geiger counters and what makes human minds special, right? But there's something I've kept up my sleeve, and if you've been reading the science news lately, you may have noticed.

Breaking news, then. Some neuroscientists have been examining layers two and three of the human cerebral cortex. These are the layers of cells that seem to be most unique to us. They're much thicker in humans than in our closest animal relatives. The cerebral cortex itself, of course, is also much bigger, but just blowing up the square meterage isn't necessarily going to make for qualitatively different experiences and powers. Something about layers two and three, though, stood out. What stands out on closer examination is that the cells in these layers can compute XOR.

This may not seem impressive. After all, I promised I could teach you to do an XOR on several binary numbers in one minute. And it seems to be the easiest thing a computer can do. Why is this impressive at all?

I know, it doesn't seem impressive, right?

Let's talk about neural networks for a minute. The earliest attempts to model human brain tissue resulted in what are now called, depending on context, finite-state automata, regular expressions, or regular grammars. To put it simply, when you sign up for a new account online and it gives you some rules for what your password should be like, the code that checks your password against those rules is a finite-state automaton. More to the point, the rules themselves are represented as a finite-state automaton. It's sort of like a mini, prototype brain that can only say whether your new password follows the rules given. You can actually think of the rules as the finite-state automaton (or regular expression, or regular grammar—the analogy with grammar, rules about letters, may possibly make more sense now). It's primitive, but that was our first semi-successful attempt to model brain tissue. We now use it all over the world trillions of times a second, if not more, as these conceptual rule widgets are at the heart of all coding languages in practical use. These are not neural networks, mind you. That was the next innovation.

Neural networks, to make a long story short, and I mean the artificial kind (ANNs, for artificial neural networks), use matrix math to try to match human pattern recognition and pattern creation. These networks of matrices can approximate any information process you want, using calculus behind the scenes to learn the right numbers to put in the matrices. This is much like, though not identical to, the way your brain can learn a vast array of different processes just by tweaking some connections with experience. And that is no coincidence. We were trying to mimic human learning, and, though it took a couple generations to get things moving at a good clip, today I think we can say we've succeeded far better than most people believed we would, even many who were working in the field.

"XOR, you promised XOR," I hear you saying. The most famous result early in research on artificial neural networks and their properties was that these networks, though they could learn many processes, could not learn XOR. This one little observation put a chill on the field for decades, because people who didn't entirely know what they were saying went around repeating the finding to each other. It became fashionable to make fun of ANNs for being overhyped, so overhyped they couldn't even do XOR. Ha ha. How silly those ANNs are.

ANNs can do XOR, actually. It just takes more layers of artificial neurons. While it is true that, using the basic matrix method we knew about decades ago, a single neuron (or even a single layer of neurons) cannot do XOR alone, just by looking at its inputs, it is also true that groups of neurons working together can do the same work as a single XOR operation. It's a group effort, if you will.

This may sound like a limitation of artificial neural networks, but actually all biological neurons examined until perhaps this last year—whenever the new finding occurred—had the same limitation as the simplified, mathematical, artificial neurons I've been talking about. There are no chimpanzee neurons that are known to be able to do XOR. Not without teaming up.

Yet the "most different" part of our brains has just been found to contain neurons that, completely on their own, can compute XOR.

So when I said I could teach you XOR in a minute, that's a funny statement. You have many cells that are doing XOR independently. I'd just be showing you how to do what a single one of your brain cells already knows how to do!

***

Maybe you'd use this capacity intuitively if you were playing Nim by the winning strategy. Instead of tallying things up, as in the 3 steps I give above, you could use your eyes.

As we scan the rows of matchsticks, we're looking for powers of 2. That's 1, 2, 4, 8, 16, 32, 64... but the only ones that show up for 16 matchsticks arranged into 4 rows are 1, 2, and 4. The setup of the game shows you: a row with 1 matchstick, a row with 1 matchstick and then 2 matchsticks, a row with 1 and 4 matchsticks, and a row with 1 and 2 and 4 matchsticks. That's the result of grouping into powers of two.

1
1 + 2
1 + 4
1 + 2 + 4

Which on the board looks like

|
| | |
| | | | |
| | | | | | |

Personally, I find it easier to think in big numbers first here, like this:

1
2 + 1
4 + 1
4 + 2 + 1

Mathematically it's identical either way, for the purposes of Nim (and addition, and XOR).

So you could see this:

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| |   |
| | | |   |
| |   | | | |   |

Or this:

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|   | |
|   | | | |
|   | |   | | | |

Or lining things up:

|
|   | |
|         | | | |
|   | |   | | | |

It's all equivalent.

You want there to be an even number of every group.

Wherever you see one of the clusters, look for another cluster just like it to cancel it out. If you don't find one, or if the other one has already been used to cancel another cluster, then the board is unbalanced. Every time you move, you want to make the board visually "balanced" in this way. That's it. That's basically the entire strategy. With only a few minutes of practice, you can "see" it. In this first board position, everything cancels out with another group.

I
I   | |
|         | | | |
|   | |   | | | |

|
|   | |
I         | | | |
I   | |   | | | |

|
|   I I
|         | | | |
|   I I   | | | |

|
|   | |
|         I I I I
|   | |   I I I I

Let's revisit XOR now. XOR in general means: "We're good as long as exactly one of the options passes. No more, no fewer. One."

Either you eat an apple, or you eat an orange, or you eat a grape.

Feel free to skip the next paragraph if this is crystal clear.

This rule is being followed if you eat an apple. It's being followed if you eat a grape. It's being followed if you eat an orange. It is not being followed if you eat nothing. It is not being followed if you eat an apple and a grape, or an apple and an orange, or a grape and an orange, or an apple and a grape and an orange. Just want to be complete about this XOR example! Many things are obvious until they aren't.

Apple XOR Orange XOR Grape. Pick a fruit and stop, and you're following my rule. Take exactly one.

The player who picks up the last matchstick loses, so perhaps you're thinking ahead and you see how that implies XOR. I'll give you a hint: the last. But it doesn't work how you'd expect. It works in the opposite way.

For our Nim strategy, the XOR we use means something like: "If there's exactly one uncanceled group for a given group size, then that's good for Satan." (I mean our opponent, sorry. Just trying not to mince words.) Basically, we want to break XOR every time it's our turn. We don't want to hand the matchsticks over in an XOR-abiding situation. We want to keep the rule broken for our opponent every time they see the board on their turn. There should be no lonely "exactly one" cluster. All the groups should be balanced, fully canceled out. It's a little counterintuitive, because at the end, on our opponent's last turn, the moment they lose, they'll be looking at nothing but an uncanceled group of 1 matchstick. You see, if you resist leaving the board in an unbalanced, XOR-abiding configuration, then because of the way the game works, this means you can put it in an XOR configuration in the final moment. On your last turn only, you make the board XOR-friendly. The player who picks up the last matchstick loses.

This was the first computerized game because the strategy's logic is so simple to apply with XOR. And the game is still kind of fun, because most people won't figure out the strategy on their own. (I didn't. In a few dozen games trying to figure out how the computer kept winning, I worked out that it was probably related to even versus odd numbers. That's as far as I got.)

Maybe detailing XOR for Nim hasn't helped anything. But I will claim that it quickly becomes possible to see the strategy. And after all, maybe that suggests individual neurons are handling the XOR. Is that necessarily true? No, I see no reason to believe that. But when you watch your own mind clasp these either/or calculations visually, and then looking for the odd group out becomes effortless, you have a more visceral appreciation of what those individual neurons just discovered are doing.