# Ben Orlin's Guide to Understanding Math as Language

### It's more than just Xs and Ys and strange symbols...

As a former quant with six grandkids, my spidey-senses started tingling as soon as I heard about Ben Orlin’s mission to make math fun.

A native of St.Paul, Ben is a math educator and popularizer who is known for his “Math With Bad Drawing” blog and book series. Today’s conversation revolves around his excellent, original new book Math for English Majors: A Human Take on the Universal Language, which reframes math as a language, complete with nouns, verbs and grammar.

Like any mathematician worth his salt, Ben loves games (describing them as ‘puzzle engines’), and our fun conversation was packed full of game-like diversions and surprises. We discussed making sense of *sampling* through fantasy towns where 70% of inhabitants are lawyers (not a town I’d like to be in), threw in a bit of Lewis Carroll to discuss the assumptions built into *propositional logic* (sometimes it really is turtles all the way down) and pitied the Welsh kids learning how to count (it will make sense when you listen to the episode!).

We’ve summarized the episode’s key concepts, provided Apple/Spotify links, and shared the transcript below. You can also watch the episode on our YouTube.

As always, if you like what you hear/read, please leave a comment or drop us a review on your provider of choice. In the meantime, here’s a sizzle reel to whet your appetite:

# What Makes a Mathematician?

Ben’s worldview is very cool. Here are a few standout snippets:

**Mathematicians are Obsessives: **I’m sure you’ve heard of the term *Monk Mode*, when people hyper-focus on a single task and remove all distractions. There’s also *Founder Mode* — something that people on X are debating endlessly. But I’d like to give you an example of *Mathematician Mode. *Here’s Ben:

“But the research mathematicians that I know, they'reobsessive and obsessive about everything. They get excited about a mathematical idea and that'll be three years. Or they get excited about a card game and they become the best in the world at that card game. One very typical example, or well, extreme but typical is, I'm blanking on her name, but let's see a Swiss biker I think. I think maybe she was just working in Switzerland.

Anyway, but from the Tokyo Olympics—the ones that were delayed from 2020 to 2021—she studies partial differential equations.She was a mathematician who trained herself to be a fast biker and went and won the Olympic gold medal.And not just won it, but in the opening moments of the race, zipped off into the distance so that the people who'd been training together and knew the field and knew the world when they finished second thought that they'd won. Not realizing she'd finished minutes ahead, she'd just flown away.

And I don't know, there's something when a mathematician gets their hands on a problem, they just obsess until it is solved.”

**Mathematicians Can Be Very Flexible **…even in their understanding of reality. Pure math can be confusing to some, but it’s a powerful way to build on top of ideas. Here’s Ben:

Bertrand Russell has a line where he says, ‘Mathematics is the subject where we only study statements of the form A implies B, if this, then that. And we never really ask if A is true. It's all hypothetical, it's all conditional.’ Which is a very modern perspective on math.

I think [ancient] mathematicians viewed it as like, ‘No, no, we proceed from things that are absolutely certain, self-evident that we know are true.’

And that's not how modern mathematicians see it. They're like, ‘Well, we make assumptions.And an assumption is an assumption. It doesn't really matter if it's true. It's just a logical starting point.And then we reason from there.’

Which makes the field in its modern form so flexible and so powerful and often alienating for other people. It's like, ‘What do you mean? We don't know if any of it is true?’”

**Mathematicians Love Mysteries… **because good mathematicians are intensely curious, they want to get to the bottom of questions, big and small. Here’s Ben:

“There's so many open questions internal to math, still open questions about the behavior of the prime numbers.

I think the biggest mystery though, and the one that will never quite solve, it's not quite solvable, is the one that I think Einstein probably put most simply, which is,why is the world comprehensible?

Why do we live in a world that's patterned and lends itself to thought, and to solutions to problems? And not a world of formless chaos, which it's hard to imagine, but we can almost imagine, well, maybe just a world where nothing follows patterns and it's just madness everywhere.”

# Episode Links

# Transcript

**Jim O’Shaughnessy:**

All right. Here we go. Well, hello, everyone. It's Jim O'Shaughnessy with another episode of Infinite Loops. I have been very excited to chat with today's guest, who is Ben Orlin, for a lot of reasons. First off, he's going to school me on mathematics as he's a graduate, a summa cum laude graduate from Yale University. But I love the double major of math and psychology. That's very, very cool.

You are a math lover extraordinaire. You've written four books on the power of mathematical ideas. Your most recent one is coming out in September, and it's titled Math for English Majors: A Human Take on the Universal Language. Well, Ben, I was a economics major, so I'm probably even more fucked up in terms of that concern. But my first question is just a super easy one for you, Ben, just to get us started. Why is the answer to the question life, the universe and everything 42?

**Ben Orlin:**

Yeah. Well, thanks so much for having me on, Jim. And starting with the easiest question of all, the question that in Douglas Adams it takes supercomputers millennia to devise the answer to that question. Let's see, so why 42? I think they debate it in the Hitchhiker's books as to why 42. And they realize that they haven't figured out what the question of the life the universe and everything is. They know the answer is 42.

I think they debate whether it might be six times seven. Is that the question? Which should be fair? That's a great question. Because that's two times, three times seven, you got three prime factors. I don't know, that's a pretty cool question to ask, but maybe a little too trivial. And then they settle on how many roads must a man walk down before you can call him a man? And the answer being 42.

For me, why 42? It's a good question. I don't know, I think it might just be six times seven. **I think maybe we've been running around trying to find meaning in much more complex places when really, the secret to the universe is just, yeah, what is six times seven?**

**Jim O’Shaughnessy:**

And that's a great answer. And I was doing a little research before recording with you. And I found from Adams himself, he goes, "I made the number up," which I find really, really funny. Well, in your own words, you say a mathematician is hardly a reasonable person, more like a feral philosopher or a logician gone rogue. I love that. Which made me contemplate Pythagoras, Archimedes, Euclid.

And I, with the help of AI, started to say, "Okay, what common characteristics, personality type, psychological background would these people share?" And I thought the answer was pretty reasonable. It came back with, "Well, here's what they share. They are all obsessively curious. They are all intellectually stubborn, they are all eccentric. They all have a sense of wonder and awesome. And they all have the courage to challenge existing norms." And I thought that was a pretty good summation of what drives... I myself am only a layman. I love math, but unlike you, I'm not trained in it. I'm self-trained. Do you think that's right? Do you think those are drivers for great mathematical minds?

**Ben Orlin:**

Yeah, I think so. I mean, I think given another year or two and you're just going to be having this conversation with the AI. You won't need me because I think the first word when you mention those names, I was thinking, "Okay, what are the traits that unite mathematicians?" The one that always comes to mind for me is obsessiveness. There's something, and it's something that I actually don't share. I don't know. I see glittery ideas and I'm always drawn in 70 different directions. And my books really show that because I'll do chapters on everything.

But the research mathematicians that I know, they're obsessive and obsessive about everything. They get excited about a mathematical idea and that'll be three years. Or they get excited about a card game and they become the best in the world at that card game. One very typical example, or well, extreme but typical is, I'm blanking on her name, but let's see a Swiss biker I think. I think maybe she was just working in Switzerland.

Anyway, but from the Tokyo Olympics, the ones that were delayed from 2020 to 2021, and she studies partial differential equations. She was a mathematician who trained herself to be a fast biker and went and won the Olympic gold medal. And not just won it, but in the opening moments of the race, zipped off into the distance so that the people who'd been training together and knew the field and knew the world when they finished second thought that they'd won. Not realizing she'd finished minutes ahead, she'd just flown away.

And I don't know, there's something when a mathematician gets their hands on a problem, they just obsess until it is solved. Yeah, that's something that I think, yeah, across. You look across history and you look across different kinds of mathematics. And there's such variety in mathematicians as there is in any discipline. But yeah, obsessiveness is definitely something they share.

**Jim O’Shaughnessy:**

Yeah. And I would add the curiosity after the obsessive. I think that obsessively curious people have always fascinated me. And I find when I'm looking into various people's biographies or autobiographies, the number of mathematicians that show up is very high. And I was thinking about your book, math as a language, my own experience in that regard. I was not formally trained in math at all, but I became really interested when I was in my early 20s with quantum physics. I went through all of the layman books and Tao physics and all that kind of stuff.

And then I wanted to learn more. And I bought some books that had equations in them that literally, I closed the book. I'm like, "I am so fucked. I'm not going to be able to figure this out." But then I had the insight that you have, which is what if I just looked at math as a language as opposed to scary symbols and everything else? Let's start with the idea behind teaching math as a language, understanding math as a language and how that really opens the aperture of the student, be they a child or an adult, to being able to grok math better.

**Ben Orlin:**

Yeah, yeah, I think the language idea, it's funny because it wants unnatural for students and unnatural for teachers and experts in mathematics, but for very different reasons. For experts, they don't tend to think about math as a language because they already speak it fluently. And you and I, we don't think of ourselves as having like, "Oh, look at all the English words we're using in this conversation. Look at all the sentences we're constructing." It's like we're thinking about content not form.

And so, that's how math is for mathematicians. They make quibble about bad notational choices or they might say, "Actually, I would use this definition, not that definition," but they're not really thinking about the language. They're thinking about what's being conveyed through it. And for students it's unnatural because it's not like any language that they know. I mean, you can walk into math classroom trilingual, but none of the three languages you speak are very much like mathematics. It's so specialized and sculpted for these certain purposes and has such strange features.

And in particular, it's developed in this way where there are these little notational dances you do, you expand across the parentheses. And what is that about? English doesn't work that way. You don't have these little routines, these little notational subroutines where you move the symbols around the page and somehow that accomplishes something. There's just weird features to mathematical language. It's weird for students and it's weird for teachers, but I think it's the key thing that mathematics is both a set of ideas and a set of linguistic conventions for talking about those ideas. And you've got to learn them both in parallel. You got to move back and forth between them as you go.

That's, there's a lot of thinking I was doing, yeah, on working on this last book. But language kept rising to the surface for me as this is the way to unify that alienating student experience where you're like, "I don't know, I just do these 17 things in the right order and then I get 100%. And I don't really know what it means or what's going on." I'm like the person, the thought experiment, John Sorrell, the Chinese room where you're being handed slips of paper and you're copying things out of books and handing them back under the door. And it's like you're pantomiming some kind of linguistic process, but you have no idea what any of it means.

And mathematicians for whom that's totally alien because they're just talking. They're just communicating about ideas. And it's mastery of that language and its form that is the difference I think, that's the gap.

**Jim O’Shaughnessy:**

Another way that I used to think about math when I was trying to get younger people interested in it in my own family was as a musical score. Because math and music, to me at least, are so inherently linked, one defining the other. But let's go through what you say. You say that if you think of numbers as nouns and adjectives and you think of verbs as calculations and grammar as algebra, voila, you're going to be able to understand it a lot better. Teach me something using that construct.

**Ben Orlin:**

Yeah, sure. Yeah. Yeah. The pivot point I think is the equal sign and is the move from thinking about essentially arithmetic equations to algebraic equations. From the right perspective, this is barely a jump at all. It's just next stepping stone. From a student's perspective, it is not. And the difference I think is students tend to see equal sign as a command. Seven plus two equals bam. Now there's like you are a calculator, someone is pressing a button on your forehead. And telling you, "All right. Tell me seven plus two."

And once you've gotten used to the language of algebra, that's not how you read that. In fact, seven plus two, and I talk about this in the book, when you're first learning it, it feels like a verb. The plus is an action. I need to combine these two piles. But once you are an expert user of mathematics, it's not a verb. It's actually more like a preposition. It's like seven and two, seven with two. A conjunction or a preposition or something. Because you don't need to add the seven and two, you can just call it seven plus two. And nine is a synonym. It's a paraphrase of seven plus two, but it's not a necessary action. That seems like just, I don't know, a little linguistic pedantry. But it turns out to be crucial because when you run into X plus Y or any algebraic sum, there's nothing to do with X plus Y. That's why people find algebra so frustrating. They look at the equation, "Well, what am I supposed to do? 19 plus four, I know I'm supposed to write 23, I'm supposed to add them, but X plus Y, is it XY?" No, that's actually not XY. That's something else.

And you just need to let it sit there. You just need to not do anything with it, which is really hard when you've gone through a decade of math education already. And every single day it was like, "Do, do do. Follow this exercise, take these steps." And then suddenly you run into algebraic expressions and you're just supposed to let them sit there and read them as pieces of language.

Yeah, the equal sign I see as the pivot here, you need to stop seeing it as the call to action or call a drum roll preceding the answer. And you need to see it as the workhorse verb of mathematical language, which is just it's the to be verb, it's is equal to. Nine equals seven plus two is just the statement that those things are the same.

And so math, the language becomes this very weird repetitive discourse because almost every sentence is just, "Well, this is the same as that, and this is the same as that." And we see a page of equations, it's just a bunch of statements that things are the same as each other, but it turns out that's a really powerful way of thinking about the world.

**Jim O’Shaughnessy:**

Indeed. It also led me to think about Mr. Orwell's 1984, where one of the primary ways that they try to break down a person's ability to trust their own perceptions and logic is to get them to say, "Two plus two equals five." And I always found that to be more profound than many people think, because to me at least it's signaling a totalitarian government's attempt to basically negate and control base reality. If you can get people to believe two plus two equals five. I mean, am I wrong? Am I off base here?

**Ben Orlin:**

Yeah. No, no, no. No, yeah, that's something I read about a little bit in the book in my chapter on addition that that's the thing when Orwell is trying to reach for, what is the deepest base truth, what is the thing that if a totalitarian government can reach that they've got everything? Yeah, it's your basic perceptions of number and addition and equality that those are... It's funny, because they're both universal. Two plus two should equal four for everybody. But because they're universal, that makes them the deepest, most personal thing that you have. Your ability to perceive that, that's your deepest possession. And if someone can take that from you, then they've taken everything.

**Jim O’Shaughnessy:**

Yeah. And I guess in Newspeak, would be the party would find that double plus good. Right?

**Ben Orlin:**

Double plus good. That's right.

**Jim O’Shaughnessy:**

But it also made me start thinking about Lewis Carroll, who as you know is a logician, and Charles Dodgson, his real name. I got obsessed when I was young with this very brief, what the tortoise said to Achilles. Because essentially, what he is doing in the piece is he's got the tortoise challenging Achilles to justify a logical inference.

And so he says, "If A, the premise, and B, another premise, are true, then Z, the conclusion, must be true." And Achilles keeps adding prepositions and the tortoise keeps rejecting them. He keeps going, "Yeah, no, I don't believe you. I don't believe you." And then I think the point, and you can correct me because your knowledge of this is far greater than mine, but the point I took away from it is that logical reasoning, no matter how far you dig down, ultimately requires me and you to accept that and agree on the rules of logic.

How does that? And the reason I bring it up is because I was just trying to think of one of your main objections with what you're doing is to try to get people to get excited by math to show how really cool it is. And then you come up with this and Agrippa's trilemma and all of that kind of stuff. And I find that stuff fun, but-

**Ben Orlin:**

Yeah, me too. No, no. I love the story of Achilles and the tortoise. I think it was probably an intro philosophy class where I ran into that first. Yeah. And it's this there's strange set of features that math has, and people have been wrestling with it for ages. I mean, there's a reason Lewis Carroll in the 19th century put this in the mouth of figures from deep antiquity. Because these are the same questions, ever since we started recording things on paper, some of the same questions about the philosophy and the theory of math have been recurring.

That one, yeah, to draw it out a little more, the Achilles and the tortoise, I think it's riffing on the idea of them running the race. And so obviously Achilles is going to win the race when they run. And I think as Lewis Carroll tells it, the tortoise is like, "Wait, okay, yeah, we could run in just a minute. But just before we do that, I'm, I'm just puzzling out this logical dilemma that I'm having. If something is true, say A, the statement is true. And A implies B, so can we therefore conclude B?" And Achilles is like, "Yeah, yeah, yeah, let's go. Let's get moving, let's run."

And the tortoise is like, "Okay, that seems right. Oh, I want to conclude B. But isn't there kind of another rule you're using there? You're using this rule that if A is true and A implies B, then B is true. Isn't that that whole statement its own rule? And it's like, yeah, just the idea that you can apply a logical inference that you can move from premise to conclusion, I guess that is its own rule." Achilles grants that and says, "Okay, fine, there's one more rule. We write that down and then we're okay." But then the tortoise, yeah, just keeps saying, "Well, but we still can't quite conclude B, can we? Because isn't that whole thing you just wrote down, isn't that a rule?"

And so, I don't know enough mathematical logic to know how modern logicians would resolve that or refute that. But yeah, it gets to something about math, which is, I don't know. It's how do you get anywhere with math if we're just working with pure ideas that we're certain are true? And yeah, there's this question of how do you get anywhere? Bertrand Russell has a line where he says, "Mathematics is the subject where we only study statements of the form A implies B, if this, then that. And we never really ask if A is true. It's all hypothetical, it's all conditional." Which is a very modern perspective on math. That's not how, I don't know if Achilles, the mythical figure knew much math, but certainly that's not how Thales or Euclid would've seen math. They would've viewed it as proceeding even through the 17th century. I think mathematicians viewed it as like, "No, no, we proceed from things that are absolutely certain, self-evident that we know are true."

And that's not how modern mathematicians see it. They're like, "Well, we make assumptions. And an assumption is an assumption. It doesn't really matter if it's true. It's just a logical starting point. And then we reason from there." Which makes the field in its modern form so flexible and so powerful and often alienating for other people. It's like, "What do you mean? We don't know if any of it is true? You're just doing these little thought experiments. Give me something real. Give me something true." It's like, "That's not... Math doesn't do that kind of truth. It's a different kind of truth."

**Jim O’Shaughnessy:**

Which is one of the reasons I'm so fascinated by it because of its flexibility and its ability to be conducted that way. And one of the things that I've always thought is that and it's a question for you. I think people who understand abstractions get math much better than people who are much more concrete thinkers. And we had a pretty large societal change in my opinion, right around the beginning of the 20th century. Where if you look at the fellow Flynn who did the paper on, "Why are IQs going up?" And one of his conclusions was because, and he looked at not traditional Stanford-Binet IQ tests, but those types of tests of the late 1800s.

And he found that one of the big changes in people was the people from the late 19th century really thought in very concrete and specific terms. They would be baffled by the why is the bear white riddle. And then through a variety of things like innovations in physics, video games, a variety of things. He found that we were getting better and better at abstractions. As a teacher, how would you get somebody who's much more of a literal thinker to fall in love with math?

**Ben Orlin:**

Yeah. Math has a funny relationship between the concrete and the abstract. Every discipline has its concrete examples. In every discipline, as you move through schooling, you're moving from very concrete experience into this more abstract academic study. And in math, I don't know if that's accelerated or something or if we just go five layers more abstract is the nature of the subject. But yeah, it's such a big question. Often, what students need is experience of those concrete particulars that are going to generalize. Leaping straight into generalities just doesn't really get you anywhere. It means that you wind up pushing symbols around a page or concepts around your mind, which just amounts to the same thing. But it's all a game of chess. If you don't know what any of it stands for or how it cashes out, it's hard to find much meaning in it.

And so I can't remember who I'm quoting here, but there's someone, maybe one of the 17th, 18th century great mathematicians, who says that to study math is about finding the example that contains the seed of generality. It's about finding that concrete instance, which is going to grow into the beautiful full abstraction. And so yeah, you pick any topic and it's about for helping students meet a topic. It's about what's that specific example of a specific problem that's going to get you into that? That's why when we teach probability, dice are great, because six sides is enough. You can think about it, you can handle it, you can write out all the possibilities, but then you can start to see some surprising results and start to see where your intuition is going to lead you astray. And you can actually roll the dice or simulate it in a spreadsheet and see what is the truth here.

**Jim O’Shaughnessy:**

And that brings me to the other thing that really appeals to me about your work is the idea that bringing play and fun into it is probably a better way to teach, especially kids. I used to play a game with kids that I knew as Korean Dice, and it's played with five die, and your objective is to get the lowest score possible, and three is equal to zero. And so you have to roll the dice and you can take as many die away as you want. So if you've got five die in your hand and you roll all threes, you win. You get a zero. And you can take all of them, but you also must take one die away before you re-roll. So if you roll all sixes, you got to take one away. If you go above nine, you're bust. And if you want to continue to play the game, anyone who busts have to double the pot if there is no winner.

And I've found that that generated such interesting, A, fun. Everyone really enjoys it and has a really good time. We played with play money, Monopoly money. But it also really was a springboard for tons of questions about probabilities, about probability theory. Do you have any other methods you can share with us on how you could play a game and use that game to illustrate mathematical concepts and thinking?

**Ben Orlin:**

Yeah. Yeah, that's a great game, for one thing. I got to make a note after our conversation because my third book was a book of games including some dice games, but that's a really nice one. Yahtzee-style re-rolling, but I like the way that having to set aside one die creates a gradual pressure on you. It's like you're forced into some bad choices potentially. One thing I love to do with games, so I did this book on games and I went out of my way not to pick games that felt like classroom games. A couple of them come from teachers, but those are the ones that you play in the classroom and they don't really feel like classroom math. I didn't want to have-

**Jim O’Shaughnessy:**

And kids will never play outside of the classroom, right?

**Ben Orlin:**

Yeah, exactly. Yeah, you can do those polynomial flashcards. That's not really a game, that's a math exercise dressed up as a game. So yeah, so dice games like that are great. I think the way to extract the best mathematical conversations and thinking from that is to view those games as puzzle engines. They're just constantly, every move they're generating a new puzzle for you. And sometimes it's an easy puzzle. As you said, if you roll all threes and three is the lowest, then that's a very easy puzzle to think to do is set them all aside. And sometimes it generates a hard puzzle. So say you roll all twos, how many twos do you set aside? Two's better than average, is it? I'd have to think about that, so that's an already interesting question. So to me, when I do classroom visits or when I'm using them in my own classes that I teach, I like to find a good moment and hit pause. Be like, "Okay, here's the scenario. Here's what we're doing. What should we do?"

Just in any simple game like that, you'll suddenly get really rich concepts that you need to be able to evaluate the situation, because you'll need to consider, what are all the possible outcomes? And now you're starting to think about distributions. And you need to say, "Okay, well, which is the likeliest outcome?" And so you're thinking about probability and you might want to say, "Well, there's a five in six chance that something kind of bad happens, but a one in six chance you win the game right there." And so you need to start thinking about expected value too, and how do you take a weighted average of different outcomes? So one that I like, there's a simple game, a dice game too. You play with a 10-sided die, or you can generate it on Google Dice, where it's a five-by-five grid and you're rolling this 10-sided die and just writing the number somewhere in your grid. And you score points by having adjacent numbers be the same, so two sevens next to each other is worth 14, or three sevens in a row is worth 21. So it's really simple and it's so simple that you think, well, isn't everyone just going to have roughly the same grid? And I've done the game with a room of 50 people and the answer is no. People do not have the same grid. There's quintillions of ways to arrange the numbers, so even trying to accomplish the same objective with the same numbers, people will do it differently. And that's a great one for generating puzzles because you get down to the last two or three spaces left and it's like, "Oh, do I put this number here? That's next to a 10, so it'll score or whatever. I've got a one and I can put it next to the 10 where it'll score two points by being next to another one, but maybe I want to leave that space blank because if a 10 comes up, that's tons of points for me."

So sometimes you'll need to sacrifice a little bit, but also, do you want to maximize expected value or do you want to maximize your chance to win, which are different things. I'm sure you get similar dilemmas in finance where it's like, are you trying to maximize expected value, even if what that means is there's a one in a hundred chance of having a huge venture capital style payoff, or do you just want to maximize your median outcome or even minimize the bottom outcomes, avoid disaster, so look for an insurance policy? Anyway, these are things that run deep through game playing and through finance, and you can get there just with a simple dice game. There's enough structure there to cast off some interesting puzzles.

**Jim O’Shaughnessy:**

Yeah. And it also makes me wonder why people who are engaged in teaching math don't take that path more often. One of the things that I found specifically in finance was trying to get people to understand base rates. And there was a great experiment, I can't remember who did it, and I didn't put it in my notes. I'm just thinking of it as I'm listening to you now, but-

**Ben Orlin:**

I wonder if this is one of the Kahneman and Tversky ones.

**Jim O’Shaughnessy:**

It might be. Yeah, it might be.

**Ben Orlin:**

Go ahead.

**Jim O’Shaughnessy:**

Yeah, so the experiment is you take a random sampling of people, you tell them the following things. There is a town of 100,000 people. 70,000 of them are lawyers. 30,000 of them are engineers. When you randomly select just names and give them the names, the majority of people say, "Well, if my objective is to get the most right, I'm going to say they're all lawyers because I'll get 70%." And most people actually do that. But then they add meaningless descriptive information to the name, and it's like Jim is 64 years old and the hair which he has left is gray. Mary is 22 and likes to walk. When they add the meaningless descriptions, people largely begin to abandon the underlying base rate of 70,000 and 30,000. And then, the clincher, they add stereotypical information. Frank is 42 years old, wears glasses, likes mathematical puzzles, and doesn't like going out on Saturday night. And literally everyone, no matter how sophisticated they are, say, "I don't care if it was 99,999 lawyers and one engineer, he's an engineer."

**Ben Orlin:**

Yeah, no, I love that. Yeah, I love that line of experiments. Yeah. Yeah, definitely. It's something Kahneman writes about at some point. It wouldn't surprise me if those were some studies he did. Yeah, it's funny. There's so much there. I think statistics is a very unnatural process for us. You have to talk about the human operating system, and it's not built to look at spreadsheets. That's not what we're about. We can learn that because we're very malleable, but we're about details. We're about looking at the world and seeing all the richness of detail and about the social world and drawing quick social assumptions and calculations about people. So you tell me this person likes Sudoku and I'm like, "Lawyers don't like Sudoku. That's an engineer right there." It doesn't matter that the city is 99% lawyers, which is an interesting city where it's 99% lawyers. We may be headed there in some sense for the future.

But yeah, I ignore the base rate. Base rates. What are base rates? In ordinary everyday life, you never know the base rate. You just know what you see, and you got to go by what you see, so it's really hard. One of the things I think that math asks of us, and it's often very tough, is to disregard many forms of knowledge that we have. And it doesn't mean disregard them forever, but disregard them for the moment while you're doing the calculation, because there's a lot you need to throw out. Yeah, one that came off this, I was teaching an interest statistics course, and a student... Let's see, what was it? She had missed a few days, due to family emergency, and so I was trying to get her caught up on what we'd done in the meantime. It was just the basic idea that if you want to know something about a big population. Take a sample, find the average of your sample, and that's your best guess now for the average of the population, so you can't do better than that, especially if you've got a nice random sample.

But even that requires turning off some knowledge. So the example I started with, "Okay, say you're running a business, what business do you want to run? Do you want to run a boba tea business? Fun bubble tea." I was like, "Okay, so you got a new flavor. You don't know how much people are going to like it, so you go out and ask 50 people and their average score is seven out of 10, so what do you think it'll be for all of St. Paul, the whole city?" And she was like, "Well, probably a little higher than seven actually." And I was said, "Well, why higher than seven?" It was because she has lots of knowledge about boba tea and she was picturing a specific flavor and she pictured the people she knows.

So it's like, "Oh, I activated all this knowledge for you. Actually, you need to hold that in suspension. You need to set that aside." So the example that she likes better, that made more sense to her as to why you hold that aside is say, "Okay, there's a new species of animal that's just been discovered. I'm not going to tell you anything else about it, but my assistant is down there outside the building and has a hundred of these things, so let's bring one in and let's see what it weighs. Okay, it weighs 10 pounds. What do you think is the average weight?" And that one, she was like, "Okay, well I don't know anything about this animal." And she could have maybe been like, "Well, you don't get animals more than about 10 tons. You don't get animals smaller than tardigrades or something," but that's a pretty big range.

So she was willing to accept that. It was like, "Okay, I don't have any knowledge I can bring to bear. 10 pounds. That's the guess for the average." But it was funny, I had to construct an example for her that prompted her to ignore other information, because it's really unnatural. In math, it's such a purified form of reality. It's so narrow the information that makes it into a calculation, and that's what makes it powerful, but that's also can make it unnatural. It feels for people like, "Why would I forget all the things I know? I know a lot about the world. Do you want me to leave that on the side?" I was like, "Yeah, actually. While you're doing the math, you leave it on the side, and then you bring it back in."

**Jim O’Shaughnessy:**

That's a great way of doing it, and because I often think about why people have such a difficulty with understanding the law of large numbers versus the small sample. They don't intuitively understand that the small sample is going to have much wider deviations. And you ask them, "If hospitals in general give birth to five boys and five girls, where would you guess that a hospital that gave birth to nine boys on one night was located? In St. Paul or in Manhattan or New York City with all the boroughs?" And they always pick New York City with all the boroughs. Do you think that that's just an intuitive thing that messes up the way to think about it?

**Ben Orlin:**

Yeah, I was thinking about this a lot the most recent time I was teaching statistics, because I think the way we naturally tend to think about samples is almost exactly backwards. So I think you're totally right about sample size, and with coins, that's the way we mathematicians always try to explain it. You say, "Okay, think about coins. You flip 10 coins, you could get nine heads. That's not crazy. You wouldn't be slapping your face and be like, wake me up, I'm in a dream, if you flip nine out of 10 heads." If you flip 900 out of 1,000, you should be like, "You're in a simulation, you're dreaming, the coin is biased. There's no way that happens. That does not happen." That would be the weirdest event in the history of earth if you flip 900 out of 1,000 heads. And people can see that to varying degrees. To some extent, I think it's important to have just a gut feel for coins, for numbers. There's a few things. Building that intuition is not always so easy.

But the other thing that happens with sampling that's tough is the only thing you really need to worry about when it comes to random sampling, sample size is an issue, but it's bias in sampling. Ask any statistician. Statistics is all about handling the random error in sampling. And we've got great ways of putting constraints on that and knowing how much error might arise. But if your sample's biased, you're screwed. There's the famous story of the beginning of Gallup polling. I think this was the 1936 election when FDR was up for re-election, and Reader's Digest had a huge circulation and they polled their readership. And on the basis of 2 million in a country of about 100 million, they'd polled tons of the population. And they're like, "Okay, Landon is going to beat Roosevelt."

And then Gallup got a thousand people, tiny, tiny fraction, it was a few thousand, and said, "It's going to be FDR. He's going to win re-election." And it was the difference between a huge, huge biased sample, which doesn't really tell you that much, and a small random sample, which does. And I don't know. For students, I find, for people learning it, approaching samples from the outside, they get it backwards. Bias doesn't really occur to them. They don't worry about bias in a sample, but they're always worried about random error. You poll 50 people. It's like, "Oh, well, what if we just got 50 people who really, really boba tea or 50 people who hate boba tea?" It's like, "Well, that wouldn't happen randomly." 50, maybe. But actually, that's not really the thing you need to be worried about. The thing you need to be worried about is if you went out and asked your friends because your friends are not random. That's biased. But if you actually could just polled random people from St. Paul, you'd be fine. 50 is a great sample if it's actually random.

**Jim O’Shaughnessy:**

Yeah, and that's one of my little hobbyhorses or soapboxes that I like to be on. Self-selected samples. The challenge that I have is, A, getting people to understand what a self-selected sample is. And not to beat up on anyone but the book, The Millionaire Next Door. This book sold millions and millions of copies, and I read it and I'm like, "This is all utter bullshit." And I just started thinking about it and I'm like, "Well, wait a minute. What kind of millionaire will devote two days, or whatever the amount of time it was, to answering this very detailed questionnaire for a thousand dollars?" And the answer is that guy-

**Ben Orlin:**

Not a real millionaire.

**Jim O’Shaughnessy:**

Exactly. They have nothing in common with real millionaires in my opinion. It's pervasive through culture, though. Whenever I read something like, "Science says," or, "According to," I immediately... Okay, where are the planet axioms here? Is this a self-selected sample or is this a Reader's Digest sample? People don't intuitively go, "Okay, what type of person might subscribe to Reader's Digest?" That in and of itself is a biased group. And then your example of your friends, that's a super biased group. How do you disambiguate that? How do you get people to understand that?

**Ben Orlin:**

I wish I knew. To me, that's one of the great epistemological problems that we all face is you go around the world and you don't get a random experience of the world. You get your little slice. You get the people who are probably very similar to you in lots of ways. Yeah, no idea. I think about this a lot because just the questions that we all care about, but you don't get data on. How do people act when they're in conflict, when they're fighting with each other? Or what makes a good relationship? Or how do people approach death? You don't get a random sample of this. I don't know how the world approaches this. I know how my very particular demographic of the world approaches this. It's really hard to break out of. The first thing you can do, and I think math helps with this, is just to see that that's the problem, that you're seeing this tiny, tiny little slice that's very biased to be like you. The world around you is going to resemble you a little bit. The exercise I like is I'll ask a class, "Think about Wikipedia. What fraction of pages do you think have pictures? What percentage of pages? If you just pick a random Wikipedia page, what percentage are pictures?" I never go to a Wikipedia page without a picture. John Travolta. Yeah, they've got pictures of John Travolta. There's pictures everywhere. I've never seen anybody guess below 90%. Maybe one kid who didn't use Wikipedia very much guessed 80% or something, but everybody thinks it's going to be 95 plus. A lot of people think it's going to be effectively a hundred.

And then you say, "Okay, everybody in the room. Go find a couple of Wikipedia pages and report back." And so they'll go find the Wikipedia page for chicken. It's like, "Oh yeah, there's a picture of a chicken right there." And they come back and go, "Yeah, it's a hundred percent in the sample." And you say, "Okay, great." Now, the reason I use Wikipedia not Instagram or TikTok for some equivalent thing is social media doesn't give you a random button, but Wikipedia is beautiful human institution here in the 21st century. It's built to be accessible and transparent, and so they've got a random button. So, "Click random article, do it 10 times, and come back and tell me how many have pictures." And so a room of 20 kids, you get 200 examples, and it's about half.

Because what you realize is when you look at Wikipedia, you see, you go to the page for the Oppenheimer movie, and then you go to the page for the Barbie movie, it's like, "Yeah, they have pictures of both of those." But if you click a random article, you get the 2010 National Swimming Championships in Belarus, or you get a train station in Sri Lanka, or you get a midterm election in Ireland in 1997. You get stuff that you don't think of, that's not the pages you're thinking to go to. And you realize, "Oh, I'm just going to the big famous pages. I'm getting these very biased glimpses of reality." And if you can really randomly sample it, you see that so much is missing from your daily experience. We all see the big stuff. We all see the famous stuff. There's stuff that's visible and shows up all over and over again. And then there's the invisible dark matter of every population. We were talking earlier about concrete examples. I'm hopeful that Wikipedia as a real concrete example, where we know how many pages are on Wikipedia, we know how many we've looked at, can start to plant the seed if you have the conversation afterwards. And that's everything. It might seem like everybody on social media is a famous influencer, because you see famous influencers. They're famous. They're influencers. They show up a lot. Most people have very few followers on social media. That's just how most accounts are. You just don't see them because nobody's following them.

**Jim O’Shaughnessy:**

I love that example. And again, you also hit on my absolute favorite function of Wikipedia, which is the random button. Every day, every morning, I go to Wikipedia and hit the random button. And it's just so fun because lots of times it's, as you say, Irish elections in 1938 and O'Shaughnessy and O'Malley were having at it, and no pictures. But there are a significant number of times where it's something so cool that I'd never thought about, and then that ruins that day because I go down that rabbit hole all. But that is-

**Ben Orlin:**

Yeah, I love that. That's a great habit. I'll have to start doing that.

**Jim O’Shaughnessy:**

Yeah, it's really fun because you just never know. And it helps with a lot of things if you just are really, really curious about things. But also, you're talking about the epistemological challenges, and I'm also really interested in those. Language, for example. Do you think that people's native language influences how well they grok math?

**Ben Orlin:**

Yeah, it's a good question. There's a little bit of research on counting that I know of. I'm not an expert in this research area. But yeah, so Mandarin, which I don't speak, apparently has a more regular pattern for naming numbers than English does, particularly through the teens. We have this weird funky legacy of the Anglo-Saxon, whatever. So you get 10, then 11 and 12, and then the teens, and it's a little weird. It's not the same pattern as 21, 22, 23. But apparently in Mandarin it's more like the equivalent of ten one, ten two, ten three, ten four, and they get to two tens and one and two tens and two and two tens and three. And apparently, yeah, my daughter's five right now, kids at that age learn to count a little faster if Mandarin is their native language.

They compared English to Mandarin I think for bilingual speakers maybe, or I can't remember exactly where the study was done. And they also compared English to Welsh. I talk about this in the book because Welsh is so cool. I was hiking in Wales and came across this plaque that just lists the numbers, and they look pretty similar to English. You can see the parallels until you get to 15, which is, I don't know how to pronounce it, something like [foreign language]. And then 16 is 15 and one, like [foreign language], and then 17 is two and 15. And 19, to jump ahead, is four and 15. But 18 is not three and 15. 18 is two nines, which is a really cool name for 18. That's a great way to think about 18 mathematically. But also, yeah, kids who are learning to count in Welsh, they learn it a little slower.

So there's at least some evidence that the language you're speaking and how easy it is to grasp the naming system is going to affect the way you learn counting and the speed with which you learn how to count to a hundred. I suspect that after that, lots of cultural factors will swamp the specifics of the language. That would be my guess. But it's funny, the The Sapir-Whorf hypothesis is the language that people have for this.

The hypothesis that the language in which you speak and think about the world really shapes how you see the world. And it's funny because it's one that... My understanding is when psychologists really try to test it, the predictions don't tend to cash out. There tend to be faint effects, but it's actually not really the case that our thinking is totally constrained by our language as far as they can test. And yet philosophers and thinkers of all kinds keep coming back to the idea because it just feels... There's sort of an intuitive appeal to it. And I don't know, I'm one of those people who finds it intuitively appealing. It does seem like a lot of learning mathematics as an individual is finding the right language to pin down ideas.

And even research mathematics can be that. Mathematicians will often be lost in searching in a space, but then if you can just pin down the right definition, just saying what the thing is that you're looking at in just the right way. [*in Latin*], the philosopher calls them proof sharpens definitions. So it's a definition that's sharpened by your exploration, and now you're putting a name on it. Yeah, it seems to me that naming can be very powerful. Did I drift away from your question? I don't feel like I...

**Jim O’Shaughnessy:**

No, no. I'm just fascinated to listen to your response. I agree. And it actually makes me think of, let's go way, way back to ancient Sumeria, and the temple priests had very arduous tasks because the people would bring these offerings of grain and they had to account for it, and they're like, "Well, there goes our week." And I'm going to give them an honorable mention because of that, because we have the tablets for the discovery of zero, but it really belongs over in India, I think.

**Ben Orlin:**

Yeah, yeah. That's my understanding, yeah.

**Jim O’Shaughnessy:**

I started thinking about it though, and I wanted to get your read on it because zero is such a profoundly important, not just number, but concept. And it's not just mathematically important, it is also philosophically important. It is also important by the way you view the cosmos. It also gives birth to the idea of, hey, the void. So chat to me a bit about a concept like that. How would you teaching the concept of zero, it leads to limits and infinitesimals, and all of those things. Without zero, right, we're kind of screwed.

**Ben Orlin:**

Yeah, yeah. No, it's true. Yeah. So Charles Seife, I'm not sure how to pronounce his name, but has a book on zero, sort of a biography of zero. And yeah, it's funny, I feel when I look at zero, it's almost like the rabbit duck illusion where you can see it two different ways. Because from one angle, you're like, "Were they idiots in the past?" Yeah, if there's nothing there, there's nothing there. That's zero things. It just feels so obvious. And then from another angle, which is the one you were eloquently laying out, it is this incredible leap of intuition to imagine zero as a number, to think that anything could be gained from calling it a number. Because the Greeks even, in Greek mathematics, one wasn't always considered a number. One was the unit, it was like a thing, but they didn’t think the number should have more to it than that.

And even two was sometimes a little ambiguous as why that was a number. The number should kind of have a beginning and a middle and an end. So three is kind of your first number number. And they had a very concrete geometric way of conceiving of number. Euclid avoids multiplying more than three numbers together. He could do it computationally. He knew in his proof that there are infinite prime numbers. You could need to effectively multiply together arbitrarily many numbers, but he wouldn't do more than three on the page, because one number measures a length, and two numbers multiplied together gives you an area, and you multiply by a third number and now you have a volume, but four numbers multiplied together, what's that? That's nonsense. There's no fourth spatial dimension. So yeah, so I like trying to inhabit those other ways of seeing number because it makes you realize even in our, what feels like a very simple concept, there's lots of different perspectives one can have on it.

Okay, so I'm drifting away though. Circling back to zero. So what makes zero compelling and weird, I think it's not so hard to get the idea that you can label nothing as zero. What's weird is folding zero into your system of computation, right? What's weird is the idea that you can do arithmetic with zero. You can add zero and subtract zero and even multiply by zero. And that's weird, because it's sort of like it doesn't... It's not interesting arithmetic and zero times anything is zero. It's sort of this attractor point of multiplication. So it's kind of a leap to be like, okay, something's going to be gained by that. But it turns out it is. There's a very general long-term trend in thinking about number, spanning the last several centuries at least, but going back to the origins of zero, of moving away from a sort of natural way of thinking about number, which is as labels for quantities.

Like, okay, there's two cups on the table, that's two. There's three cups on the table, that's three. And to thinking about them instead as things you operate with, or sort of solutions to equations even. And so you kind of need to expand your notion of what a number is for zero to make sense. This is even more vivid with negative numbers where there's not negative three ducks in the pond. I don't know what that would mean. These little chasms of emptiness where a duck should be. Negative three doesn't do the things that we like numbers to do. And so there's a strict sense in which negative three isn't really a number at all, it's something else. But I guess there's sort of number definition one, and that's the one you kind of grasp first when you're learning math, the counting numbers. And then there's number definition two, which is sort of, we can extend that system of number, and suddenly it becomes less about labeling quantities in the world and more about patterns and letting our patterns propagate and extend.

Yeah, I've been thinking a lot about this recently, not with zero, but with negative times a negative being a positive, which you learn in, I don't know, I've taught to sixth graders. People hate it. People hate it, for good reason, because it doesn't make any sense. There's actually no concrete physical reason why a negative times a negative is a positive. It's just if you... Basically it's the distributive property. If you want to make the distributive property king of your multiplication and addition and how they relate, then you need negative times a negative to be a positive. But that's a very intellectually inside out reason for most people as they approach math.

And Zero has some of the same flavor to say, "Oh yeah, zero divided by something, that's still zero. Something divided by zero, no, no, don't do that. It's not that kind of number. Can't divide with it. Not the dividing kind of number." Yeah, so you need to embrace a version of math and a way of seeing math that lots of very, very smart people throughout history have not embraced. Archimedes didn't really think of negative numbers that way. And I'm not smarter than Archimedes, I'll tell you that much. But I live in a different culture that's given me a different inheritance, and so I can manipulate negative numbers in a way that he wouldn't have wanted to. He would've seen that as nonsense.

**Jim O’Shaughnessy:**

I love that way of looking at it too, because we often forget that we, as Newton is alleged to have said, which I think he actually did as a dig on Leibniz, where he said that if he's seen further, it's only because he stood on the shoulders of giants. And of course Leibniz was a dwarf.

**Ben Orlin:**

I wonder if... Was it Hook, maybe? I think Leibniz was...

**Jim O’Shaughnessy:**

Might've been Hook. That's right. That's right.

**Ben Orlin:**

I think it was Hook. Yeah, I think somebody says in a letter to Hook. And apparently it was a common phrase before that, but it's something about the phrase prior to that was a dwarf standing on the shoulder of giants, and when Newton applies it to himself, he just kind of leaves out the part where the guy on the shoulders is supposed to be shorter than other people.

**Jim O’Shaughnessy:**

Right. And what's funny to me though, getting back to kind of zero and negative numbers. So negative numbers make zero sense until you have zero. Then it's like, if you and I are going mountaineering, I can say, "Yeah, it's 300... We got to climb 300 feet. We certainly don't want to fall 300 feet." And that kind of makes it a practical like, holy shit, that negative 300, I'm dead. Don't fall 300 feet, right?

**Ben Orlin:**

Yeah, no, that's right. Yeah. Yeah, yeah. I was just going to say, I think what negatives let us do is they're the mathematical language of opposites. They let us unify opposite quantities and opposite directions and opposite ideas as if they all belong to the same continuum. You can think of, altitude is a really good one. And then debt of course, that's really the one that clinches, I think, the meaning of negative numbers. When I teach middle schoolers, that's what I haven't done in a few years now, but debt is the one that gets to them. It's like you owe four bucks. It's like, oh, that's real. Money is a sort of magnificent fiction we've created, but I owe somebody four bucks, that's a very vivid social reality. I owe four bucks and I owe that guy five bucks? Like, oh my gosh, I owe nine bucks. And then negatives take on a real meaning.

**Jim O’Shaughnessy:**

Yeah, I love that. My friend Annie Duke also says that you can help people understand math better if you challenge them to think of everything as a bet. In other words, like you're doing with the, oh, holy shit, I owe that guy three, I owe that guy seven, I owe... Oh, I don't want even talk to him, I owe him 20. And it becomes much more visceral. And it made me think about when I've had... So I've got six grandkids and I love math and I want them to get excited about it, and I'm showing a book here for people who are listening as opposed to watching. And it's about the golden ratio. My personal thing is that I think one reason people love things and non-math people with zero interest in math, they still know about Fibonacci, right? And if they do a little bit of homework, because the Fibonacci sequences are beautiful and they describe so much spiral galaxies, seashells, etc.

So the ubiquity of their explanatory power is really interesting. And I find that this is a book that has really pretty pictures. And it also ties in art with math. And I find that that's very helpful when you're able to say, "Okay, so let's look this." You don't even bring up math. And then they get it and they're like, "Huh? So one plus one equals two. Two plus one equals..." And they do the Fibonacci. And then you bring in five. What would you think about a ratio that shows up all over architecture and everything? Do you think there's a validity to that kind of way of teaching just to edify me, because if it's wrong, I'll do something else.

**Ben Orlin:**

No, sure, yeah. Yeah, definitely. People love beauty, it turns out. People don't necessarily love worksheets. In my teaching career, I've probably used more worksheets than I've used beauty, but maybe that isn't me doing it, right? Yeah, I feel like many of us have a very alienating experience of math in school. And I think it's easy, and sometimes I do it to blame teachers and including myself and the choices we make in presenting it, and you can blame other aspects of the educational system. What I tend to think of it as is it's an extraordinary project we're attempting, right? Mass education on the level in this society that we're attempting, it's huge. You can go back through the years and how many people were getting a four-year college degree a hundred years ago? 7% of the population, that may be only the male population.

Go back further, how many people are finishing high school? Or even up to World War II, how many people are finishing high school? Not that many. So the idea of this mass education where it's like, okay, a quarter of the population is going to learn calculus, that is a hugely ambitious undertaking. And maybe we could maybe target our ambition a little more intelligently sometimes. But okay, so what happens when you do that? You just have this system of mass schooling. It's a huge bureaucracy, and huge bureaucracies are not known for exciting people's sense of wonder and curiosity, it's not the thing they do best.

So what do you need as a teacher is you need to do that. You're the human in the room. And it's tough, because you have all these bureaucratic pressures on you. But one of the duties you have is you do have to awaken their sense of curiosity, and it is about awakening and/or igniting it, because it's there. The fuel is all there. People have a fascination with number and with pattern and with shape. And the fact that all those things turn out to be related, that numbers are a way of talking about patterns, and shapes can be a language of number, and number can be a language of shapes, that's cool. That's weird. That's a weird facet of this existence we happen to find ourselves in. And so any doorway into that I think is great.

A lot of math lessons inevitably in a good math education, they look pretty down to earth. You're working on a certain kind of question, you're wrestling with different sides of it, you're practicing a technique maybe. Not every lesson will look like that. A lot will. But the fairly small percentage of lessons that are explicitly about we're just having fun today, we're exciting some curiosity, those are really important lessons. Not every day of your life is equally impactful, and not every day of your math education is equally impactful. So those days, even if it's not 90% of days when you're seeing something vivid and electrifying and glimpsing the patterns of the cosmos, it doesn't have to be 95% of days. If you do that on 5% of days, that's pretty solid. That's better than we tend to do in math education.

**Jim O’Shaughnessy:**

Yeah, and then it's the linking them to the more practical aspects that I'm still struggling with. I use metal broads, chaotic diagrams because they're beautiful. I show Penrose tiles and kind of explain why they're important, but then linking them back, right?

**Ben Orlin:**

Yeah, no, that's right. I struggle with this too. Something I think a lot about with this fourth book that I was working on, because I'm a math teacher, but also a math popularizer, right? And what's the way you popularize math is you don't talk about school math, you talk about cooler stuff. You talk about metal broad set, you talk about fractals, you talk about, yeah, periodic tilings, Penrose tiles, the new aperiodic monotile that they've discovered. You talk about knots. There's lots of great stuff. There's great fodder. My first book, I talk about lottery tickets, I talk about the geometry of the death star, triangles in architecture. So there's great stuff out there. There's lots of beautiful math that actually people can get excited about. But factorized x^2 + 5x + 6, that's harder to get excited about.

And yet that core you learn in school, we don't always teach it, great. We could probably make some trimming here and there. There are there topics that, I don't know, conic sections I don't think really need necessarily to be taught in Algebra II. We can all make a little... Trim a little bit of fat here and there. But there's a lot of math, I'm sure as you know this from your quantitative work. There's a lot of math that just to access it, you do need that core of algebra. You need a good sense of proportion, which is a lot of what's taught in middle school math. You need a good command of the operations and what they mean in different settings, and that's elementary school math. Ratio and fractions, just a good grasp of that. And so that's a dilemma. I think it's beautiful that math has so many different sides, that there's the artistic beautiful side, and there's the really concrete, practical side. But how to weave them all together for students and get them excited about the parts that are maybe less naturally beautiful is tricky.

The thing I think about, the word that I come back to always is power or powerful. Though what students love in math is when they feel powerful, when they feel capable. And unfortunately given the way that our education system works, we have a room full of kids who are all the same age, all doing kind of the same work every day. And somebody probably finishes first, and it's usually the same kid every day who finishes first. The thing that makes people feel powerful in math is being better than others at it. Just as a descriptive matter, that tends to be how it goes, is that if you feel like you are good at math, then you'll probably like it. And if you feel like you are slow or bad at math, then you probably won't like it. And so what we need to do, and it's hard, but it's to give a different sense of the power of math, that the power of math is not your power to out compete someone.

It's your power to seize control over a weird question and pin it down and find an answer, an answer that's more elegant than you could have guessed or to a question that you never would've thought you could manage given the tools you have. Yeah, it's that sense of the power of math is I think the only way to really make it get through to all students. That's very abstract though. Like how do you get kids to get excited about quadratics? You need a good relationship with the kids so that they trust you and think like, okay, this is worth learning. This guy I trust says it, so okay, we're going to go along for the ride.

**Jim O’Shaughnessy:**

Yeah, one thing I always... On that note, one thing I try was to look historically at what empowered certain cultures and civilizations. And we're back to the Sumerians now, right? What empowered certain cultures to rise, become powerful, etc. And there's always math lurking.

**Ben Orlin:**

Yeah, yeah. Yeah, especially in agriculture, that you want to go to agriculture, you need to know the seasons. You want to know the seasons, it turns out there's this weird coded set of dots in the sky that are going to tell you how the seasons are changing and what's going on, and give you this notion of deep time that it's very hard to get here in the terrestrial realm. And so you get the Mayas, the civilizations in Indian subcontinent, right? Incredible astronomy with really sophisticated mathematical thinking, and all for that purpose, for command of the sky or being able to predict what's going to happen in the night sky.

**Jim O’Shaughnessy:**

Yeah, and I was fascinated by the connection between the Babylonians and the clay tablets and why Babylonia rose. But then I found a story that really interested me, and it just made me a little actually. And it was that the Babylonians were kind of first on the scene with counting and figuring out how to collect it and add and subtract and all that kind of stuff. And the rest of Sumeria just weren't having it. Like, whatever, our gods are more powerful than your gods. So what actually made Babylonia become one of the central city states of Sumeria was they were like, okay, we have a new God.

And they literally invented a new mythology where their God was the God that tore the known universe as we would call it, in half. He threw half up this way and created the sky, and he threw the other half down and created the flat earth. And he was the God who created everything. Ergo we're the shit. But the ultimate thing was probably the math. And I just wonder what metaphors would you find particularly attractive? Like you note power for kids. Are there others that get them excited that get them like, "Oh wow, maybe I should be interested in this."

**Ben Orlin:**

Yeah, yeah. There's not a period of history I know everything about, but the guy ripping the world in half, that's a pretty good... I feel like that's hard to imagine a seven-year-old being like, "Oh yeah, I don't care about it. Anybody ripping in the world in half, you don't have my interest right now." Yeah, I like that too actually. Somehow it's got a funny rhyme thousands of years later with Newton and the falling apple, which who knows if that happened, but he told the story a lot. Newton loved to tell that story of he sees the apple falling and suddenly he realizes like apple, moon, gravity. It's universal. It's one thing. The sky up there, this ancient Babylonian God apparently tore it in half and put half up there and half down here, and then it took Newton to put them back together. And it's actually universal gravitation unites them both.

For kids, kids love games. Games are great for kids. So dice games and number games and counting games. You mentioned drawing and art and shapes. There's a reason there's a lot of geometry in how kids like to approach math, just corners and edges and how many ways can you connect things. And puzzles and riddles too. One of the strange features of math is that what it does as a discipline is it takes these really fun, open-ended curiosity provoking riddles, and just beats them down with a mallet until they're totally flat and you can do a hundred of them and they're all kind of the same. And it's a very great service as a discipline. It's what makes, say, computers possible, because you can have routine algorithmic calculations. You do want to give people an experience though of the problem before it's been smashed with a mallet.

I mentioned quadratics earlier, this is older than the kind of age you're talking about, but you can frame problems in quadratics, factoring problems or solving a quadratic equation as an interesting one-off puzzle. You can pick one that has just the right features where it's like, oh, you can almost guess what those numbers are. If you can guess what the first one is, you can maybe figure out the second one. And so you can frame them as puzzles, and then eventually you're going to find a routine way to solve them all. But breaking off one and giving one problem space to breathe, I think that's often enticing for students. It's got to be one they can get a handle on. You can't give them something that they're totally clueless on. Just like throw a partial differential equation at me, much less at a ninth grader. "I don't know how to solve it." So you're just not going to get anywhere.

But, and this is the art of teaching, the science of teaching, is finding that problem where they can get a handle on it and they're going to be able to dive in and start exploring. I think that puzzles pull people in. You're right, there's millions of people who wake up every day and do Sudoku. They don't wake up every day and solve a system of linear equations, except for they do actually, because Sudoku is isomorphic to a system of linear equations, so they kind of are. So if you can find the puzzle in the math, I think that pulls people in. That's what games are too. Games are, like I said, they're puzzle engines. Every move is generating a new puzzle for you.

**Jim O’Shaughnessy:**

Yeah. If you were going to create a mathematical escape room, like you're in danger and you've got to solve this puzzle to get in that safe room, what would it look like?

**Ben Orlin:**

Oh my gosh, that sounds fun. I have to go through. I've got friends who are really good with manipulatives and have all the cool, yeah, different... My daughter loves to play with MAGNA-TILES, so it's got to be something fun you could build out of MAGNA-TILES, build a platonic solid out of MAGNA-TILES. One thing that comes to mind, I feel like good for an escape room is the 15 puzzle. There's 15 tiles in a four by four grid, so there's one empty space. You've probably seen them in gift shops. They'll have picture versions. But it was apparently, when this came out in, it was just about a hundred years before the Rubik's Cube actually, it was like 1880s. It was a huge national craze. People lost their minds over the 15 puzzle. The New York Times was doing fad piece articles on it. So a 15 puzzle where you have to rearrange, maybe for an escape room, you've only got an hour usually, so maybe an eight puzzle, a three by three instead of four by four to be a little more manageable.

Like I say, where you and you arrange it and if you get the picture, then you can press it down and open it up. I guess I'm assuming if I've got a decent budget to hire an architect or an engineer to build some cool math stuff for me. I don't know. Yeah. To some extent our reality is mathematical escape room, so there's lots of material there.

**Jim O’Shaughnessy:**

I love that line. What do you think maths' greatest mystery is?

**Ben Orlin:**

That's a good question. I don't know. Can I throw it back at you? Do you have a mystery that, a mathematical mystery or it has to be on me?

**Jim O’Shaughnessy:**

Nope.

**Ben Orlin:**

Okay. Yeah. Fair.

**Jim O’Shaughnessy:**

It's on you. You're the mathematician here.

**Ben Orlin:**

Yeah. Right. That's funny. There's so many open questions internal to math, still open questions about the behavior of the prime numbers. I think the biggest mystery though, and the one that will never quite solve, it's not quite solvable, is the one that I think Einstein probably put most simply, which is, why is the world comprehensible? Why do we live in a world that's patterned and lends itself to thought, and to solutions to problems? And not a world of formless chaos, which it's hard to imagine, but we can almost imagine, well, maybe just a world where nothing follows patterns and it's just madness everywhere. And you can maybe answer that with sort of an anthropic argument that in such a world we couldn't frame such a question. But I think that's, why does math work? I think is sort of the fundamental question about math.

Mathematicians love to talk about these cycles you get where, someone takes a, there's a concrete problem you're trying to solve, and so you go off, you try to find a solution, and then it leads to other weird fanciful math that has nothing to do with anything, and the mathematician spends a century just lost in a closet over there, exploring their own little ideas that have nothing to do with anything. And then they come out of the closet and the scientists are like, "Oh, good. That's just what we needed to solve protein folding." I was like, "Wait, they were working on protein folding back in the day? They had never seen a protein. They were just working in the closet." It's like, no, actually, something about the nature of logic and pattern means that we get cycles like that in mathematics. Yeah. I think that in the simplest form, why does math work or an even simpler form, why does thinking work as a way of understanding the world? That to me is probably the ongoing mystery of math.

**Jim O’Shaughnessy:**

Which of course brings us back to Charles Dodgson as Lewis Carroll, right? I wrote a term paper in college about Alice in Wonderland, essentially being his questioning, why the hell does all this stuff work? What if it didn't work? And I am fascinated with that story and him himself. I think he was kind of asking and getting the same type of question as the answer you just gave. Why? And if I remember the title of the paper correctly, it was something like the Neoclassical Era Through the Looking Glass or something like that. And when you do that deep dive, it's really interesting because when you put him in his set and setting, Victorian England, there were so many weird things about Victorian England.

They were the most prudish on the surface, but they had the greatest number of brothels in London than any other time. And so there were all of these discontinuities and these paradoxes. And he himself, if you read his stuff more as logic and philosophy, it's really a trip because he is really good at, which also leads me to Tesla. Tesla was obsessed by three, six, and nine. And I was trying to figure that out. Why was Tesla, the man, obsessed with it? And so I did a little research and wow, is that a rabbit hole?

**Ben Orlin:**

I don't know anything about Tesla. Yeah. Why three, six and nine?

**Jim O’Shaughnessy:**

One of his great quotes was to understand everything about the universe, think in terms of, what was it? One quote is three, six, and nine. And then so I feed that in and I start looking at books about it, and it gets you into numerical patterns, vortex mathematics, energy weapons. I had no idea that Tesla, Tesla was obsessed by scalar waves, and he concocted what he called a death ray. He didn't build it, thank God. But just this, seeing Tesla and reading his really, very few people have seen it because it's not broadly in publication, his very, very thin autobiography. I got obsessed with all this kind of stuff, and then a three, six, and nine, and suddenly you're reading about energy weapons and scalar waves, and all this kind of stuff. So, it's like he must've been onto something-

**Ben Orlin:**

Possibly both onto something and on something.

**Jim O’Shaughnessy:**

Exactly. But he fascinates me because, again, we were talking about Pythagoras and Euclid, and everyone else. He too was, his story, it's cinematic almost. This was a guy who was unarguably a genius, and he didn't care. He just didn't care. The only thing he cared about was learning more, figuring more stuff out. And that gets back to that insatiable curiosity. George Westinghouse totally screwed him. He had an agreement with George Westinghouse. He had arguably the keys to a fortune in his patents. And Westinghouse is like in this battle with Edison. And so Westinghouse comes and says to Tesla, "Dude, I really need you to not get your share of this." And Tesla's like, "Yeah. Whatever." Ripped up the contract. Died in poverty. And yet I am fascinated by guys like him because they all do come back to math in many instances. And so it leads me to another question I have specifically for you, because you're right about it. Obviously, the title of this podcast, Infinite Loops, and you have a very specific notion, which I agree with by the way, about how people think about the infinite. Help us there.

**Ben Orlin:**

Yeah. Right. Well, before we get into that, I love learning about the biographies of mathematicians, obviously. But I especially love when you have a thinker, who we don't think of as mathematical or a figure we don't think it was mathematical, but if they're efficiently curious, that curiosity will have led them at some point to mathematics, and how the role that math plays in their worldview and in their understanding of things is always fascinating. Lincoln being one of my favorite examples, loved Euclid. And so for him, like mathematical logic and argument was really central, I think, to how he thought about constitutional law, much less about math. Anyway, but going back to infinity. Yeah. Well, it's real hard to think about. It's very attractive-

**Jim O’Shaughnessy:**

No kidding.

**Ben Orlin:**

It's compelling, it's the thing. If you want to mention a mathematical topic and have an eight-year-old really dig in and just want to think about it and wrap their hands around, infinity is probably the one. I'm obviously not the originator of this point, but that infinity, we can only really think of it by negating what we know. That infinity is, that which does not end. I think David Foster-Wallace has a line like this. He wrote a book on mathematical concepts of infinity. And he talks about how the defining feature of our existence is it's finiteness, it's our mortality. And so there's something attractive and elusive, and impossible about imagining the infinite. And then what's wild about math is for the last 150 years or so, we've actually had a really good, well-developed theory of infinity.

Before that mathematicians mostly had to sidestep it. There was a lot of talk about the difference between a potential infinity and an actual infinity. Aristotle introduces that distinction, and to varying degrees, it's important in later thinkers. But then George Cantor comes along and he's like, "No. Okay. It's sets. We got it. Set theory. Bam." And then lays out this beautiful and insane theory of, and logically airtight, theory of different levels of infinity. The intuitions you have to give up in order to grasp the infinite are numerous. There's a lot of things that are true of finite numbers that cease to be true of infinite numbers. As someone with mathematical training, Cantor is my, that's what infinity is, I guess. And there are other ways of conceiving of it, but that's still, it's such a beautiful and weird one that he gave us. For me, that's still, when I think the infinite, rather than try to think unthinkable thoughts and get lost in the bottomless well of imagination, I tend to be like, "It's sets. It's power sets."

**Jim O’Shaughnessy:**

I love that. And the idea that, I still struggle to try to, another thing that I use a lot is, I love Bach. And so generally speaking, during a work day, like today, unless I'm doing an interview, Bach is play. And I think back, getting back to Douglas Adams again, he had a quote that went something along the lines, it's not a very nice quote because I like all of these musicians that he is referencing, but his quote was something like, "Beethoven's music teaches you what it's like to be Beethoven. Mozart's music teaches you what it's like to be a human. Bach's language teaches you what it's like to be the universe." Do you think, I know he meant it as a joke. He loved music. But I've always thought of Bach that way. I've always thought of Bach as the, since I was a kid, and accidentally, what five-year-old says, "Oh, this is great."

Well, I was lucky enough to have a father who loved music, and he had Bach on one day and he came upon me, and this is not my memory, this is him telling me about it. Apparently I was five, and he said he found me on a couch just like with my eyes closed, with this sublime little smile on my face. And he said, "Jim, what are you doing?" And apparently, my dad told me, I look at him and I said, "What is this magical music?" Do you think, I don't want to get really woo-woo here, but the music of the spheres, all of the ideas that really tie math into music, am I just being fanciful?

**Ben Orlin:**

No. Douglas Hofstadter would be the one ask. He wrote Gödel, Escher, Bach the book that unifies the great modern mathematician and the great artist, and then Bach-

**Jim O’Shaughnessy:**

One of my favorite books, actually, but what a slog. I'm not smart enough. And literally, I'll read a page, turn the page and then go, "I have no idea what was on the page that I just read," and go back.

**Ben Orlin:**

Yeah. Definitely a very ambitious book. Yeah. Leibniz says a quote that's something to the effect of, "Music is the soul calculating without knowing that it calculates."

**Jim O’Shaughnessy:**

Oh, I love that. That isn't one I'm familiar with.

**Ben Orlin:**

Yeah. I may be botching the translation, whatever, he didn't say it in English so I can translate it however I want. No, I think, I don't know. There's something deep in music. I guess, you mentioned I majored in math and psychology. And I came to psychology first, and then I was taking a little bit of math, but it was really sort of my last year in college I just loaded up on math classes so I could get the major and be qualified to go be a teacher of it. But psychology was sort of my first passion in college. Just like how do people think? What are people? How does this bizarre machinery operate? And music is there very much at the heart. I love Oliver Sacks text about Musicophilia.

**Jim O’Shaughnessy:**

Yeah. It's a great book.

**Ben Orlin:**

And there's something really deep about music in how we think. And math is the same. I think they may be braided together down there. My younger daughter is one now, almost one and a half, and she's just started getting into music and she's into it. It'll be like, I set the timer for my older daughter and the timer went off. I was just playing its little gentle beeping noises. And my 1-year-old just starts dancing. She's like, "Oh, wow." Or if I'm watching her while I'm listening to a podcast and then the podcast's outro music starts to play, again, she'll just start dancing. It's like there's something, she knows music is playing. I actually played her a pop song the other night and she went ballistic. She's like, "Oh my gosh. I didn't know music could be this good. I thought it was just timers and podcast outro music. I didn't know it could be like this."

I don't know. It's in the body and the brain, there's something really deep about music. And Bach, I don't know music theory well enough at all, I know zero to comment on what's woven into Bach, but the patterns are really rich. And even just how we hear, just to take sound waves and turn them into that experience we have is sort of a fascinatingly mathematical process. I've heard lots of mathematicians talk very fondly about the weird puzzles of tuning. Of how does an octave work? And you get stuck with irrational numbers because if you just want to have, go from a C to a C in the next octave up and you want to do it in eight steps, because that's an octave, actually you need an irrational number because you wind up with the 12th root of two or something like that being the way to do these half steps. So, can't speak to that, but that's a great Douglas Adams line though.

**Jim O’Shaughnessy:**

And that also gets back to Tesla because he said amplitude and frequency, they were everything. And then of course there were all these urban myths. You're too young, but Dan Rather, who used to do the evening news for CBS, had this bizarre instant in Manhattan, where these two guys came up to him and started accosting him, literally. Look at Wikipedia and type in, What is the Frequency Kenneth?

**Ben Orlin:**

I know the R.E.M album.

**Jim O’Shaughnessy:**

So, you know the R.E.M album, but it's a wild story. Just check it out. Check out, What is the Frequency, Kenneth? Because-

**Ben Orlin:**

Cool. Yeah.

**Jim O’Shaughnessy:**

Have a couple hours because it's a really deep rabbit hole, way than I understood because I knew the song too, but I remember the incident when it happened. Everyone thought Rather was crazy. But it turns out, no, no, no, these were two real guys and then, we don't have enough time, and it's not a math subject really, but what happens to these two guys-

**Ben Orlin:**

Everything's a math subject in the end.

**Jim O’Shaughnessy:**

That's true. What happens to these two guys though is really interesting. I'm getting the hook from my nanny here. I would go on forever and ever, and that's why her job is so important, Ben. Well, first off, the book comes out in September. It's going to be available everywhere?

**Ben Orlin:**

Yeah. It should be anywhere that you buy books. Yeah. It should be around. Math for English Majors and subtitle is, A Human Take on the Universal Language.

**Jim O’Shaughnessy:**

Yep. And we will have links to any preference you have to where people might want to buy it. And we'll link to your other books as well, because they're all fun. I love your stuff on drawing, for example. A mathematician who loves drawing, who cannot draw. I can't draw either. But our final question on the show is the following. We're going to wave a magic wand and we're going to make you the emperor of the world. You can't kill anyone. You can't put anyone in a reeducation camp. You can't insist that they endlessly do math or flashcards for math. We're going to give you a magical microphone and you can say two things into it. And if you're familiar with the movie Inception, you're going to be able to incept the entire population of the world and everywhere, whenever their morning is, everyone's going to wake up and say, "You know what? Unlike all the other times I got two great ideas and never acted on them, I am going to act on these two ideas starting today." What you going to incept in the world, Ben?

**Ben Orlin:**

Yeah. Well, first I should have thought about this more deeply. I heard you ask this question a bunch of times. I'm always curious, do people ever try to go for the reeducation camp or the mass murder? You were very careful to give that caveat. Has that come up before? Where somebody said, "Well, what I would do is…”

**Jim O’Shaughnessy:**

Actually, you are the first person to ask me that question. And when I was trying to think of what should I ask people as the final question, that would be fun and that would get people thinking, I started talking to a bunch of friends and one particular friend said, he shall remain nameless.

**Ben Orlin:**

Okay. Yeah. A quiet ruthlessness there. It's funny, your guests always seem so genial and friendly. None of them seem like the reeducation camp types, but I guess you never know until it happens.

**Jim O’Shaughnessy:**

Well, it's my little specific piece, like we were talking about earlier. Do not take this sample as being legitimate in any way, other than to make me add those two exclusions, so that nobody has them.

**Ben Orlin:**

Yeah. I think I'll go back to the beginning of our conversation, and I think my two will be, first, be curious, ask questions, know more. And the nice thing about these also is these will work, if I could just remember these myself and wake up and really drive through on these. Because the world is so complex. There's so much one could be curious about. It can almost be like there's just so much noise you just shut down. You're like, "I don't know. There's too many questions. I have no idea how any of this works, so I'm just going to give up." But if you can just carve off piece and just be curious, and keep asking, and keep asking, and like, "Okay. Well, I didn't get an answer there, but I'm going to come back tomorrow and ask more questions." I don't know, that's what humans are best at when we can get the ball rolling.

So I'll say, number one, be curious. And number two, be obsessive. Don't be a stalker or something, but be obsessive about what you're making, what you're building. Because the best moments of my creative work, the moments when I feel like I'm making the best things that I can make is when I vanish into them. And it's not all the time. It sort of can be all the time. It's like when I'm really deep into a book, and I need to have all those parts moving around in my brain. But obsessiveness, yeah, just gives you, I don't know, enlists these powers that humans have that we're not always using. And in some ways, those two point different directions. To be curious as to sort of be open to lots of different possibilities, to hit the random article on Wikipedia. And then to be obsessive is to go five miles deep once you find a good random article that you're excited about. So, those would be my two, I think. Be curious and then once your curiosity catches on something, be obsessive. Let yourself obsess.

**Jim O’Shaughnessy:**

I absolutely love those, were very simpatico. When I was doing the first version of What Works on Wall Street, I didn't really realize that I was behaving ridiculously obsessively. My wife looked at me and she's like, "You didn't realize?" And of course this is in the nineties when computer monitors were this big, and the towers. And she goes, "Jim, when we used to take the kids to Nantucket, I had to plan the packing just really carefully because you brought three computers with you so that you could be continually running the tests on the CompuSet data." And I went, "And you find that obsessive?"

**Ben Orlin:**

It's only three.

**Jim O’Shaughnessy:**

Yeah. That was my point.

**Ben Orlin:**

It's fewer than there are people on the trip. It's fine.

**Jim O’Shaughnessy:**

Exactly. That's right. Well, Ben, this has been really delightful and fun. We'll have to, and as many as often happens to me, I had a bunch of other stuff here, but maybe we will have a chat again on another episode because this is just an infinitely interesting topic to me. Let's be really clear. I make no claims to being a mathematician or-

**Ben Orlin:**

I'm sure you're underselling a lot of your quantitative skills. I think the things are, it's a vast set of things I don't know. That's what makes the conversation fun.

**Jim O’Shaughnessy:**

That's exactly right. Ben, thank you so much for joining me and …

**Ben Orlin:**

Yeah. Thanks so much, Jim. I really enjoyed the conversation.

**Jim O’Shaughnessy:**

As did I. Cheers, Ben.

**Ben Orlin:**

Thanks.

This was a fascinating conversation. Thank you very much for it!

Have you come across W. Brian Arthur’s math-language-based critique of economics? He argues (I think) that economics is bad at exploring and explaining change because historically, it’s modelled using algebraic rather than algorithmic mathematics. He suggests this is partly why economists conceptualise in terms of “nouns”, rather than “verbs”; which connects to why they’re bad at describing non-equilibrium states. It’s an interesting idea about how ‘language’ is not neutral: it both shapes and reflects how we think about the world. Link to the paper below.

You also mentioned money and debt. Have you noticed parallels between hierarchies of language and money? Eg English as today’s ‘common language’ and the USD as today’s ‘common currency’, which resolves problems of cross-border value exchange? Perhaps it speaks to thinking of both money and language as communication technologies, with different ‘forms’ being more or less appropriate and useful in different contexts?

Thanks again!

https://sites.santafe.edu/~wbarthur/Papers/Nouns_Verbs_JEBO.pdf