Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas - 216 | John Allen Paulos on Numbers, Narratives, and Numeracy

Episode Date: October 31, 2022

People have a complicated relationship to mathematics. We all use it in our everyday lives, from calculating a tip at a restaurant to estimating the probability of some future event. But many people find the subject intimidating, if not off-putting. John Allen Paulos has long been working to make mathematics more approachable and encourage people to become more numerate. We talk about how people think about math, what kinds of math they should know, and the role of stories and narrative to make math come alive.  Support Mindscape on Patreon. John Allen Paulos received his Ph.D. in mathematics from the University of Wisconsin, Madison. He is currently a professor of mathematics at Temple University. He s a bestselling author, and frequent contributor to publications such as ABCNews.com, the Guardian, and Scientific American. Among his awards are the Science Communication award from the American Association for the Advancement of Science and the Mathematics Communication Award from the Joint Policy Board of Mathematics. His new book is Who's Counting? Uniting Numbers and Narratives with Stories from Pop Culture, Puzzles, Politics, and More. Web page Wikipedia Amazon author page Twitter

Hello, everyone. Welcome to the Mindscape Podcast. I'm your host, Sean Carroll. So as I'm recording this, I recently was reading an article by Clive Thompson on the effect that the COVID pandemic has had on education. And the fact, of course, there was a whole thing where students couldn't go to school for a while and people had to learn remotely. But what about not university students, but lower levels, high school and secondary school students? Well, it turns out, the effect of the pandemic was bad on education here in the United States for sure. And that's maybe not surprising. You know, a lot of the resources, a lot of the usual ways that we did things just weren't available. Even if we could have found ways that were just as good, it takes time to do that. And so certain students are really hurt by that. But interestingly, the decline in scores, which is sort of across the board, is much higher in math than it is in reading. So for whatever reason, there was a much bigger deleterious effect of homeschooling on kids trying to learn math than kids trying to learn reading. And why is that?

And the essay tries to argue that it's basically because when you're at home and you're trying to learn math, you're going to learn it from your parents. And chances are good that your parents hate math because a lot of people hate math. It's considered okay to hate math, to say, oh, I'm not so good at math. math. I don't really understand that stuff. It's too hard in a way that it's not considered okay to hate reading or history or other forms of knowledge. And that's a weird thing and it's probably a bad thing. And one of the people in the world who's done the most to combat this feeling is today's guest, John Allen Palos. Famously, he's the author of enumeracy, mathematical literacy and his consequences. That's back in 1988. But he's a mathematician at Temple University,

and he's kept up this fight to get people to overcome their innumeracy, their mathematical literacy, and to learn more and to be more comfortable learning math, including in a brand new book called Who's Counting, Uniting Numbers and Narratives with stories from pop culture, puzzles, politics, and more. In the podcast, we're going to talk a little bit about math, but largely in some sense we're talking about the relationship between ordinary people and math. What is the math that people should know? How should we teach it to them? Why don't they want to learn it? Why is it so frightening? Is it their fault? Is it our fault for how we teach it? How could we do better? But then we also do talk a little about the substance of math because it's not just that we need to

teach math and need to get better at it, but what kinds of math should we teach? And I think that there's a thing that it said over and over again in my circles that we actually don't spend nearly time teaching, probability and statistics. To people of all sorts of backgrounds, whether you're going to be a professional scientist or whether you're just going to be doing anything else in the world, there's a knowledge of probability that is very, very helpful that somehow we do not successfully convey to people in the educational system. So we talk a little about conditional probabilities and how you can get an intuition for things like that and why math works at all and how it can be conveyed using jokes and humor, which makes everything a little bit better. So I think this is

kind of an important topic. I'm not sure that we solved it once and for all, but paying attention to it is one step in the right direction. Occasional reminder that here at Mindscape, we have a Patreon feed. Go to patreon.com slash Sean M. Carroll, and you can get both ad-free episodes and the ability to contribute to the Ask Me Anything questions that we have. And yet another occasional reminder that we have the Mindscape Big Picture Scholarship. It's been going very, very well. I think we're going to be able to give out three college scholarships this year of $10,000 each. So if you go to bold.org slash scholarships slash mindscape, you can either contribute to it if you're at that stage of your life or apply for it if you're at that stage of your life. So I'm looking forward to seeing what the winners are going to be and what great things they're going to do. And with that, let's go. John Allen Polos, welcome to the Midescape podcast. That's a pleasure to be here, Sean. So you've, in some sense, maybe tell me this is fair or not, but in some sense you've devoted

your life to fighting against innumeracy, or maybe you want to say to promoting numeracy. You wrote a wonderful book with the title innumeracy. How do yourself define numeracy or innumeracy? I think innumeracy is just the ability to deal reasonably well with. numbers, probabilities, logic. So there's no hard and fast precise definition, but, and then numeracy, of course, is an inability to deal with basic numbers, basic probability, basic logic. And again, one of the points I like to make over again is the dubiousness of precision. And so, you know, people are very precise.

precise and the precision is unwarranted. So that's my defense for not giving a more precise definition of numeracy. No, that's completely fair, I think. I guess the reason why I'm wondering is because I'm thinking about literacy, which is the obvious comparison. And in some sense, literacy is more of a yes-no question. You know, some people have bigger vocabularies than others, but do you read or you don't? Whereas when it comes to math, there just seem to be a continuum. Like, I know a lot of math. I've learned a lot. I use it all the time. I still feel there's enormously more math than I don't know than what I know. Well, I think that to some extent, that's still a case with literacy. I mean, people who can, you know, read the newspaper, you give them a book by, you know, I don't know, Heidegger, or even a normal

person reading a book by Heidegger would be baffled. So there is kind of a continuum, although it's a little bit more binary with literacy than it is with numeracy. And I think that, but there are different kinds of mathematical knowledge. I think that maybe some people who are not into the world of math, maybe don't appreciate what mathematicians do for a living versus calculating. You know, like what I do, I'm no better at calculating a tip or doing my taxes than anybody else is. I don't know about you. Yeah, no, it's the same.

Sorry, what? So how do you think about that difference? How do you think about math as opposed to just multiplying and dividing? Well, I mean, one point I've made in a couple of my books is that math is much greater than computation. That's most people's kind of myopic view of it. But math is to computation as literature is to typing. Nobody says you're a great type as well. why don't you write a novel or converse?

And so, you know, that's why one of the obstacles to teaching math, people think they know it already. I know how to multiply. I know how to plug formulas into numbers into formulas. Is there something more? But there's a lot more. I mean, patterns, logic, structure of various sorts. And it, you know, it maps on to everyday events often.

enough that it's worth knowing, even if you're not going to be a mathematician. Even among mathematicians, I find that people who are analysts often don't understand some, what I think is an obvious point in probability of statistics or people who are in algebraic topology don't necessarily understand much about Bonnach spaces. But that's the case in a lot of things. I mean, my son's a lawyer. and you know whenever my wife or I have a question we know not to ask him because he's a very good lawyer he makes a lot of money but he only knows what he knows right yeah it's specialized let me let's get

some examples on the table so people have an idea of what we're talking about in the book like right right at the beginning you say the following thing if at least one of a woman's two children is a boy born in summer. The probability she has two boys is seven 15th. Right. Let me back up. Probability woman has two sons if the given, well, probability two sons given she's got at least one son, that's one third because you can't have girl, girl.

So probability two boys given at least one boy is one third. The probability of two boys, given at least one boy born in the summer, is 7.15s, which is close to a half. So the more precise you are about, I mean, you can set up the sample space and convince yourself, but the more precise you are about the boy she does have, if it has two boys, at least one of whom is born on July 4th, then the probability is almost a half. So the more, I mean, it's hard without writing out the sample space, but it's typical of many problems in probability. They're counterintuitive. That's one reason people have problems with probability.

Their probabilistic vocabulary is limited to one in a million maybe or 50-50 or sure thing. And on top of that, people don't think natural. in terms of probability and many people, oh, everything has a reason and the fact that propositions like you just mentioned are counterintuitive. The most famous example is the birthday puzzle, 23 people sufficient for, if you have 23 people in the room, the probability is one half that at least two have the same birthday. Well, that's why I wanted to discuss this example in particular because the number 715th is maybe not, you know, I don't really care what that number is. This is not going to come up very often. But it's an example of

conditional probability, of, you know, probability of one thing and how it changes given some extra information. I mean, is that something that people struggle with especially in your experience? Yeah, conditional probability. Revising one's probability estimates. It's something we do in everyday life, the only thing is if you study probability or Bayes rule, in particular, you can refine it. So everything we do in probability, in a sense, are refinements of notions that develop naturally. I mean, the notion of mean, median, and mold, or central tendency, there are all kinds of words in English or other natural languages, average, so-so, whatever, a complex of words that essentially mean 50-50 or the mean average. The same thing with variants. There's all

kinds of words in English that Albuy national languages too far out disparate, very different, far out as interesting because it's far out on a distribution. And the same thing with probability. I mean, as Moliere commented, people generally have been, you know, speaking pros all their life. and they're generally been speaking probability all their life. It's just that they speak with a very bad accent. I mean, a simple example in Bayes theorem, if you have two coins and one of them is two-headed and one of them is fair, you don't know which is which.

You pick one up and flip it three times. You get three heads in a row. It's common sense to say, well, it's very likely, more likely that you pick the two-headed coin. But only Bayes theorem says that that probability rises from one. half to eight-nights. So as I said, probability statistics are

refinements or distillations of kind of hazy, nebulous everyday notions. Well, the classic example of this that Royaled, I guess it wasn't the internet, it seemed before the internet was around, but the Monty Hall problem got people

very worked up, right? Why don't you remind us of what the Monty Hall problem is? Because maybe there's still people out there who've never heard of it. Yeah, there are, actually. at least in my experience. The money hall problem has to do with a TV show a number of years ago, let's make a deal, in which there's a host and a guest and three doors, behind one of which is a car.

And the host asks the guest to choose the door behind which he hopes is a new car, and if he's correct, he gets a new car. So that's a, the guest picks door one. then the host, who knows where the car is, always opens one of the other doors where he's certain that there's no car. So he always opens a door behind which there's nothing. And then he asks, he says to the guest, are you sure you don't want to switch from one to two? They say he opens door three. You sure you don't want to switch from one to two.

You chose door one. I've opened door three. You want to switch your bet to two. often people say, no, it doesn't make any difference. There's two doors open. Chances are 50-50, one-half that might as well stay here. But he shouldn't switch because the probability he was right the first time is one-third. The probability the car is behind one of the other two doors is two-thirds.

But since the host opens door three, that two-thirds probability is now focused on the unopened door. him. So worried his switch, he would raise his probability of winning the car from one-third to two-thirds. But many people, including them, there's a story that may be apocryphal, but the great Paul Erdoche, mathematician, was supposedly baffled by this. I find that hard to believe myself, too. Yeah. But, but, I mean, maybe I don't, because the way that you stated the problem, everything made super-duper perfect sense. I do think. I do think. think that sometimes when people state problems like this, they kind of want people to get the wrong answer just to, you know, explain the trickiness of it. So they kind of hide that fact that

Monty Hall will only ever open a door where the car is not there, right? And that does change the problem. Yeah, it does change the problem. I mean, it's one thing I tell my students in a to, in probability, math in general, being really clear about what you're going to, it does change the problem. you're saying is very important. In fact, yesterday I asked my class, I said, if you flip a coin a thousand times, what's the most likely number of heads you get? And two, is that a likely number of heads to get. And of course, 500 is the most likely number of heads. And no, it's not very likely as you're going to get 500 hits. But with the guy to Monty Hall, what I try to do is, you know, show the relevance of puzzles, the puzzles. The puzzles. I do talk about to everyday life. So I imagine a dual game, which instead of Monty Hall, there's this psychopath called Taunty Hall.

And there's a guest, and the same thing. The only thing is, if he picks the door, the crucial door, there's not a car there. There's a little gun that shoots out a toxic mist of gas and sickens the person choosing that door. So again, the person picks a door, let's say, picks door one, Taunty Hall always opens the door behind which he knows there's nothing. Let's say he opens three, and then he asks the guest,

do you want to switch the door two? And now he should say no, because for the same reason, he was right one-third of the time, two-thirds of the probability of the other two doors, but he wants to limit his exposure so he should stick there. So it's a kind of dual problem. You should stick with what he picked. And, you know, the relevance of that, let's say, to COVID, is you want to limit the number of people you come in contact with.

I mean, there are versions of this with 100 doors or whatever. But in any case. And did you, do I remember in the book you claim, I might be completely misremembering this, but people are better at getting that version of the problem right than the original Monty Hall problem? I think they are. I mean, maybe fear is, and Dusis is one to think more clearly than does greed.

I'm not sure. Although greed does a good job, too. Greed does a pretty good job, but I think it's actually, maybe it's just not about thinking clearly, right? You know, I pretty firmly believe that the human brain was not evolved and optimized to do math problems, but there's sort of situations we find ourselves in that are analogous to math problems that we are that our brain is pretty good at and maybe being scared of something is closer to that instinctual correctness than looking out for a reward. Yeah, I think that's true. I mean, you don't know. A primitive man didn't need probability to realize if there's a rustling in a borsh to run.

Even if it's a lot of the time. It's interesting because I've been thinking for various reasons, both teaching and research, about the arrow of time and the second law of thermodynamics recently, and reading back in the history of it, or even hearing what modern physics professors tell their students, that there's this issue that came up in the 1870s about the number of ways a system can go from low entropy to high entropy versus the number can go from high entropy to low entropy. And the answer is those two numbers have to be exactly the same, because the underlying symmetry of reversibility of the underlying system, and Lohschmidt famously made this objection to Boltzmann.

But there's plenty of textbooks you can read that will tell you, no, there's more ways to go from low entropy to high than high to low, and they're making the same mistake about conditional probabilities, I think. They're conditionalizing on starting in low entropy, which is a big cheat. Yeah, no, that's an interesting point. Yeah, that's true. Actually, there's one kind of political instance of thermodynamics, for lack of a better term. And that's Baldini's principle that it's much easier, sometimes called the bullshit principle,

much easier to produce bullshit than it is refuted. And Mark Twain had a similar comment. He said it's much easier to con people than it is to convince them they've been conned. Oh, yeah, right. People don't want to believe that. Yeah, I mean, so Q and on, conspiracy theorist, election deniers, I mean, once they publicly state the nonsense, they believe, it's very hard to get them back. Well, you've talked in this book and elsewhere a lot about pseudoscience and conspiracy theories and so forth. What is the connection to numeracy or innumeracy there?

Well, I think it's just clear thinking there in that case. I mean, you've got to know some facts. You've got to know a modicum of arithmetic probability. And if you don't, you're more easily fooled. I mean, the prosecutor's paradox is relevant, again, to conditional probability. There's some crime and there's a lot of evidence and they arrest somebody. If that person's innocent, the probability of the evidence, He was around the murder.

He did have big shoes. He did talk to people around there. So the probability of that evidence, given he's innocent, is low. But that's not the relevant conditional probability. The relevant conditional probabilities are probability he's innocent, given the evidence. And the defense attorney will bring in all the other people who have these characteristics. And the probability of innocence given evidence is much higher often than the probability. of evidence given innocence, but it's very easy if people don't understand about conditional probability

or just conditional statements. I don't know the difference between if A then B and if B then A, which is basically what conditional probability is. And this kind of feeds in with, in the case of the conspiracy theories, some kind of wishful thinking, right, or some kind of search for clarity in a simple system that covers everything. And so maybe this is an example where human psychology and bad math work hand in hand to lead people to wrong conclusions? Well, that's definitely the case.

In fact, cognitive boyables in general are a big part of what leads people astray. And we're all vulnerable to them. Anchoring effect, availability error, confirmation bias. One that doesn't get as much publicity is the conjunction fallacy. You have somebody that's say he's a senator, U.S. senator, and he's everybody, you know, he's very moderate. He's intelligent, you know, rectitude is the word that comes in mind for everybody thinks of him. He lives modestly with his wife and his daughter, who's unfortunately very sick. But in any case, given that background, what's more likely that this senator took a bribe from a lobbyist

or that he took a bribe from a lobbyist and used the money to pay for his daughter's expensive operation? And most people say, well, or at least a lot of people, probably the latter, given what you said about them. But it's more likely that he took money from a lobbyist period because it's always easier to satisfy one condition than two or more conditions. And what that has to do with the Internet and fake news is given the Internet, all kinds of odd facts, factoid's little details are available. it's easy to kind of cobble together a superficially plausible story. And there's this tradeoff between probability and plausibility. You add more details, which you can glean from the Internet, your story becomes in a way more plausible, the same way this senator thinks it worked,

but less probable. And, you know, if you're gullible to begin with and you have all these seemingly precise details, You can fool people. And a lot of this, what we're talking about is not maybe what I would necessarily think of as a math in the traditional math curriculum, so much is just logic and clear thinking. Do you distinguish between these two things, or is it all one set of sensible thoughts to you? I guess it is possible to distinguish. I mean, it is a matter of clear thinking. But, yeah, there are other components.

In fact, I mean, one thing I've tried to do in my writing is, kind of set up a links between stories and statistics or narratives and numbers, whatever. Even in my first book, Math and Humor, I talked about the similarities between jokes and and mathematics, by the, both math and jokes depend on logic, although the logic might be perverted, the patterns might be different. But you use some of the same tools, we do you out absurdum, for different reasons. And humor and jokes is for the absurdum, and math has disprove something. But there is a kind of continuum. And in the continuum between math and jokes is puzzles. They're more substantial than jokes, but they do share with math also this aha moment. And

And so in essence, puzzles are very mathematical, and whether they're mechanical puzzles like Rubik's Cube and a lot of group theory in that or verbal puzzles like you got Monty Hall. So, you know, there are similarities, but I mean, there are, you know, so it is kind of a continuum. There's not a chasm between here's numbers and here's narratives. There's a connection. I mean, there's differences. One is that in mathematics or science, the logic is.

is extensional. If you have a proof or something, every time you have a three, you could put in square of nine or cube were at 27. It doesn't make a difference. But you can't just, a woman can't just say, or a man can't just say, ah, the happiest day of my life was the 80th anniversary of, a 110th anniversary of Miller Fillmore's death. Even if that was her wedding day. That was the day, yeah. What was the day? But, you know, are you crazy that the happy anniversary is a 110th anniversary? So you can't do that. And also just the whole kind of psychological mindset.

And when you do reading a story, science fiction or whatever, just for the enjoyment, you suspend this belief. Okay, let's go with it. But in math or science or statistics, you do just the opposite. You suspend beliefs. So you're not bamboozle. You want to really prove it. So there are lots of differences, but still, they're both human endeavors, and they're not totally distinct things, storytelling and theorem-proving or number crunching.

Well, you have the word narrative right there in the title of the new book, so it's clear that you recognize that. And I guess it is interesting how human beings, we use math, but we love stories. I recently did a podcast with Peter Dodds, who is a statistician complex system theorists, who made a quote that I will never stop quoting, which is never bring statistics to a story fight. People love their stories. And I guess maybe I don't know how intentional it is or it just seemed to be the best way to do it, but what you're doing in your books is telling people math and helping them learn math through the device of fun little stories.

Exactly. I like to use jokes, puzzles, anecdotes, little vignettes. And you can often get the math across without rousing people's kind of too many people's innate, if not fear, discomfort with numbers or fear they're going to be judged. But you get the same idea across without the formalism. I mean, there's a limit to it. But you can get a good deal of mathematics across with. With stories. In fact, one of my books, I think therefore I laugh. I was inspired by a quote by Wittgenstein who said you could write a good and serious book in philosophy that consisted entirely of jokes. Where joke is interpreted very loosely. If you get the joke, you get the relevant philosophical point. And so the conceding the book is, you know, here's a collection of stories, jokes, anecdotes that, different points across. Well, I'll confess, I have not actually read your books that discuss math and humor. But now that you brought up the idea, I'm fascinated by the concept of a continuum between

math and jokes. And it does make sense that there's this aha moment, right? I mean, a joke is somehow confounding our expectations somehow, and a math puzzle is somehow resisting immediate analysis. Has there been, I don't know, academic work? on the relationship between these two things? Not too much. There are a couple of journals of humor

that kind of touch on it. But I mean, I think that's one of the appeal of counterintuitive results in math. They're kind of like jokes. Like, what? You can't do that? It's continuous, but not differentiable. So, yeah.

And so I, even in this book, I bring up some philosophical. issues about Eugene Vickner in the unreasonable effectiveness of mathematics narratives is a collection of metaphors justified true belief is not equally knowledge

I can tell that you had some philosophical background as well as the usual mathematical background yeah as an undergraduate I skipped around I majored for a while philosophy and English and physics I kept coming back to math but

always wanted to write. So in a sense, I do both. Got exactly the right thing. So a lot of the examples we've been talking about here are either logic or probability. There's a lot more to math than that, obviously. If you were put in charge, if you were the emperor of math education now, what is it that you would say people should learn? What is the minimum basic knowledge of math that one has to be numerate? well facility with arithmetic first of all

and of course probability and and the logic and notions in the philosophy of science

I mean just everyday notions like what's a placebo what's a double-blind study which you know most people are innocent of

in fact I just wrote a piece for three parts daily about what quizzes for congressional aspirants. I mean, anytime you apply for a job, especially a high-tech job, you are interviewed, and they give you some problems. Can you program this in Python or whatever?

But yet you can run for, you know, Congress, you can run for president, and, you know, there's no such test. And perhaps there should be, although you'd have to drag women people into it or should shamed them into it, but you wouldn't have Persia Walker saying, you know, why are we making clean air? We just send it to China and they send us the dirty air and

evolution's a hoax. It's kind of embarrassing. I do want to, I want to note your mention of Three Quarks Daily. That's one of our favorite websites here at the Weinstein podcast. So I'm glad that you put in a plug for that. Oh, yeah, no, I enjoy it as well.

In fact, I write a kind of semi-regular column for them. But let's take seriously this somewhat provocative idea that we should have standards for politicians or leaders or something like that. And we should have standards, but it's very hard to get it right. I mean, for various reasons in my Ask Me Anything episodes of the podcast, I recently revealed my love as a high school kid for reading Robert Heinlein novels. and Heinlein once said that, you know, you shouldn't be allowed to vote if you can't solve a quadratic equation.

And that seems a bit extreme to me. Like, I mean, I understand the motivation, but should politicians be allowed to run for office if they can't solve a quadratic equation? Yeah, I think that's a bad example. But being able to having some feel for like scaling, like you can't scale up things. Like every time I go to the movie, I'm always amazing. you can get a, you know, a soda that's 8 inches high or 10 inches high, and the 10-inch high one is 50 cents more than 8-inch high one, even though it's probably, you know, the volume is much greater,

or ordering a pizza. So, I mean, scale, people don't realize that things scale up squares or cubes, geometric things, but even, you know, size of cities scale up with a fractional exponent, Jopry West, talked about that. I talked about that. So it's a kind of basic notion that I think some people should have some idea about it, as well as being able to estimate things.

It's a very rough estimate. You tell somebody on the prior to state building is two miles high. You don't have to know that exact height happens to be 1776 feet, but two miles high, do you really realize that? So some appreciation for normal estimates. I mean, even if you ask people, you know, there's a study that's done that 42% of heart attacks occur on Friday, Saturday, and Sunday. And people attribute this to heavy drinking and partying.

And, you know, I can see people saying, wow, I better be careful. But Friday, Saturday, and Sunday is three-sevenths of a week, which is 43%. So, I mean, the same thing, a four-day holiday weekend, 400 people are going to die. 35,000 or more a year do die on the highway. So it's normal four days. So there's some feeling for relevant magnitudes, for, for escaping. for estimating, for sequencing, some things you have to do in a certain order. There are lots of puzzles like that.

And those don't involve quadratic equations, which I don't think that would be a reasonable requirement. No, it's just memorizing a formula. Yeah, I do get that. But it's interesting because all the examples you're giving are a little bit different than what we actually teach people in high school, right, or even in college. We take geometry and trigonometry, maybe. be calculus, and I love all these subjects dearly. But as far as I know, at least when I was in

high school, we didn't take probability, statistics, scaling estimates, anything like that. Are you implicitly suggesting a radical revision of the secondary school math curriculum? Well, I mean, it's hard to generalize, because some schools, I think, do, and the best schools do. But I don't know about a radical revision, but certainly an addition of such topics that are most relevant to politics, popular culture, everyday life. I think, you know, scaling, as I said, estimation and so on. And just an ability to kind of think outside the box. One thing I talked about an argument of pro- or anti-abortionist that is, you know, is kind of fancy-abortion.

but it is interesting. The people who are absolute, super absolutist with regard to banning abortion, I think it could be useful to kind of get them to admit that in certain cases, I'm not talking about rape or incest, but in certain regular cases, they should have abortion should be there. And there's a story I like that I prefaced abortion story with about George Bernard Shaw, supposedly, again, the story might be apocryphal, sitting next to a woman at this posh dinner party. And he said, would you sleep with me for a million pounds?

And she says, yes, I will. And she laughs and giggles. And then he said, would you sleep with me for 10 pounds? And she says, no, what do you think I am? And he says, well, we already established that. Now we're just haggling a bunch of details. But, you know, it's a stupid story, sexist story. but it's relevant to this argument I'm going to make.

Imagine that because of some cosmic catastrophe or toxins in the environment or whatever, that women who became pregnant became pregnant with 10 to 20 fetuses. That's one assumption. And two, imagine that advances in technology and birth procedures. and neonatal technology enabled doctors to save some or all of the fetuses if they intervene in the first three months. So if that's the case,

what would people who are absolutist opponents of abortion do if people got, the woman got pregnant with 10 or 20 fetuses? They can't just say, well, we'll take some of them and let the other ones go because that's 10 amount to abortion. And so they'd have to maintain their position, they'd have to accept a 10 to 20-fold increase in birth rates, which I think they wouldn't do. And so, again, just to get away from the absolutist position and the relevance of that to the, so George Bernard Shaw story, is once you get them off of this totally absolutist position, then the rest is haggling about the details. You go for 15 weeks or 20 weeks or whatever.

So in any case. I think, yeah, the concept that the kinds of math that we teach people in high school or whatever, I really think that's something we should take much more seriously. You know, I myself, when it comes to science, often complain that we teach science as a list of facts, right? A list of true things rather than as the process, the empirical, hypothesis testing process, which is much more central to it. And people come out not really understanding what to do with news stories about science in the media. And I guess probably the same thing is true with math also. There's a different kind of math that is equally good that

would be way more relevant to people's reasoning in everyday circumstances. Yeah, I think that's true, except in the case of physics or science in general, those stories do make it into popular press, whereas there's very rare that any breakthrough in, or big result in mathematics will be written about. No, but I agree. But I think it should be part of a general, I mean, general knowledgeism is important as well. And it's, you know, teaching, weariness, skepticism, suspending belief and so on is important. And it's connected to a lot of things.

I mean, it's interesting. I write about people who are most pro-free enterprise have no problem accepting the complexity of an economy. And they don't say, wow, how do they come into being all of a sudden? But it does. You can go to any store in the country, any convenience store, and you can get a Snickers bar or half a gallon of milk. You can get any kind of clothes or shoes anywhere you want. And nobody said, how did this come into beans? But yet they make this some people make the intelligent design people so called make the same make the comparable argument

They look at it came up how did life come into into being so immediately how did this I come into being And instead of saying well to use the The verboten word it evolved the same way cities cities and economy evolved but they are most accepting of capitalism, and which and least accepting of the analogous process when it comes to life. Yeah, I mean, I guess this is a common theme of what you're saying, is it's not just clear thinking, but a consistency of thinking across domains. And people always talk about we should teach critical thinking or something like that. I actually don't know whether or not that makes

sense. Is that something that can be taught in your experience? to some extent. I mean, it does often devolve into something kind of silly. But, I mean, I think getting people to know a lot, not just facts, but trends and connections between disparate fields like economics and evolution is, I mean, if you know, if you know something, not a lot necessarily, I mean, it helps if you know a lot, but you know something. and are taught to bring things together, to try to relate,

have a kind of more holistic attitude towards knowledge. I think that's a worthwhile endeavor, but you're right. I mean, it's hard. I'm not sure how you go about teaching critical thinking because people always are going on to some fact, the key formula, the key thing. Yeah.

And often there is no clear thing. Well, and often it's sort of against their interests, right? People don't want to reach certain conclusions, and the human brain is really, really good at reaching the conclusions we want to reach, not the ones that the data are forcing us to go toward. Yeah, exactly. I mean, I talk a lot about logic in the book and paradoxes and their relevance to everyday life.

I mean, even in the stock market, the efficient market hypothesis, says that information about a stock is immediately available to everybody. But most markets aren't all that efficient, but you can create a kind of relevant example, a relevant, something that's relevant to the Liars paradox. Efficient market hypothesis is true to the effect that most people think it's not true. Because if most people think it's not true, then they'll say, oh, there's a way I can make more money, and they'll do all kinds of

contortions, research that, and by doing that, they'll make it efficient. And if they already think it's efficient, then what's the worth, what's the worth of doing that? That's every information, every bit of information is already priced in, or what do I want to do that for? So a lot of these paradoxes are relevant to, you know, broader themes. And again, to your point, that's generally not taught in a math class, it's hard or any class. So you kind of have to come upon it yourself or whatever. I'll read

who's counting. Yeah, exactly. Read who's counting. Speaking of which, this is a perfect segue because I wanted to shift gears a little bit. You have the new book out. I have a new book that recently came out that also, well, I wanted to contrast it a little bit because my, you In my new book, the biggest ideas in the universe, I try to teach people the basics the basics, the difference being from other books that I do the math, that I show them all the equations. There's over 100 equations in the book, all the way up to Einstein's equation. And so one of the angles I try out when I come across a skeptic is who says, you know, I just don't understand equations.

I will never be able to understand it. And I say, look, you understand 2 plus 2 equals 4. that's an equation right and it's a matter of degree not of kind to understand Einstein's equation it's just a little bit more complicated there's no such thing as people who can't understand equations it's just you know are you willing to do a little bit more work than you usually do so what do you think of that of that angle is that a plausible story i think so i i think uh you know if stephen hawking allegedly uh uh said that uh if you put an equation in your book you cut the readership by a half.

So if you put 100 equations... Very, very tiny. But I think that's false. I mean, but, yeah, you just have to have to do it carefully. I mean, you have to embed the equation and discussion in which it makes sense. You can't just ball and say, here's the formulas. But I mean, I think that's, you know, that's, you know, that's, you know, that's, you know, So I think that's a good idea.

I'm sure you did. I've read some of your stuff. You do write very well, and it is embedded in a context that gives it meaning. Well, we try, but I'm wondering about the level of abstraction, right? I mean, now that I've done this experiment, I'm curious to see whether people enjoy it or not. But when you talk about Einstein's equation for general relativity, you know, not E equals MC squared, but RMU minus 1FRG-G-M-New, That is, there's some journey that the learner has to go on to really wrap their heads around it. And I'm just, I'm wondering if it is really something that, you know, some people just aren't going to do or aren't willing to do?

Or is it, could we teach everybody Einstein's equation and those other kinds of higher math that we're certainly not going to do in high school? I think you could reach a lot of people. I wouldn't say everybody, but you could reach many more people than I reach now because if it was done right. Yeah, given the opportunity. Yeah, given the opportunity, requiring your book. Exactly. There you go. Fishing for that one, but yes.

But, I mean... Actually, sorry. Go ahead. No, I was just going to say, I mean, two and two equals four is always put forth as a kind of standard. Simple fact, but everything depends on context. I mean, it's not always the case. If you take two cups of popcorn and add two cups of water to it, you get three cups of soggy popcorn, not four. So any bit of mathematics can be misapplied.

And there's a story of the bear hunters who became extinct shortly after they mastered vector analysis. Before they had mastered a vector analysis, when they saw a bear to the number. northwest, they shot it. But now that they know vector analysis, they see a bear to the northwest, they shoot one arrow to the north and one arrow to the west, and the bear gets away. So in math, like adding integers and simple vector analysis, this silliness is clear as apparent. But if you get into more complex mathematics, you can say something equally stupid. But it goes by. It's very easy to intimidate people if you're a mathematician or a physicist because people aren't going to challenge you.

You can say the most abstruse sounding nonsense. Well, this is the famous anxiety people have when they take math classes about word problems, right? Like you can memorize how to manipulate the equations and maybe get the right answer. But if you need to translate from words into equations, that's harder. but in some sense that's by far the more important skill. Yeah, exactly. That goes back to my narrative numbers continuum. Yeah.

That is by far more important. You mentioned already the phrase, the unreasonable effectiveness of mathematics. And, of course, the unreasonable effectiveness of mathematics and physics, which is a famous phrase from Eugene Vigner. which, by the way, I'm a little skeptical that it's true, that it's unreasonably effective. I kind of think that no matter what physics turned out to be, we would find math for it after the fact. Right. I'm very skeptical of it because we learn about numbers by playing with little pebbles and putting it together.

You take this one and that one, that's addition. Learn about geometry by looking at little twigs and extending them and making little triangles of them. And also we learn about physics by our working through the walking through the world. So mathematics is kind of an idealization and abstraction of everyday things that we do. So I don't think it's all that unreasonable. If you abstract, if you idealize what you do when you're playing with pebbles and twigs and moving around, it's not surprising you get a mathematics that's going to be effective. It grew out of things that worked.

But then there are some, that's another great side way because I've been recently thinking, in part because I had a podcast interview with Justin Clark Dohn, who is a philosopher of mathematics. So I've been thinking about the foundations of mathematics and mathematical logic. And it is the part of math that I understand the least. I really, really struggle with like geometry and topology I can do. But when you get into proving relationships between models and axiom systems and things like that, I just really, really struggle with it. But am I correct that that's part of that was part of your mathematical research? Yeah, yeah.

My degree was in PhD in mathematics, but I was most interested in my thesis and papers were in logic, in model theory and non-standard logics. And, yeah, I was interested, as I said, as an undergraduate philosophy, and that's still kind of mathematics that I'm, you know, at least initially, was most interested in. And I talk about Gertl's theorem, but an on-standard proof of it using ideas from complexity theory, Greg Schaedon. And, yeah, so I think a little bit of some logic doesn't have to be, you know, go too far, but. people don't know the difference being, you know, affirming a consequent in all these Latin terms. Sure. And I think they should. They don't have.

They probably just focus on memorizing the terms instead of understanding them. Can you say more about proving girdle theorem using complexity theory? Yeah. A sequence is as random if the shortest program that generates it is about as long as the, sequence itself. And it's not random if you can generate, I mean, 0101, 01, you can generate that by just saying 01 repeated a thousand times. But you can never generate something more complex than the generating algorithm. And so it uses that and varies paradox to show that, you can't speak loosely, you can't generate 10 pounds of theorems from five pounds of axioms.

There's always going to be theorems or statements that you're not going to be able to prove because of the limited complexity of any logical system. A nice example, that's kind of silly, but I like. A very paradox says you're in an elevator, you're very short, the building is very tall, press the first button that you can't reach. by that mission, you can't reach it. So it's, you know, I sketched a little bit more than that. But I like the shaping proof of girl's theorem better because it's connected to a more basic stuff about the notion of complexity.

Complexity is something that's in the world. And it's an important topic in computer science. And from it follows girl incompleteness in without going through and working at girl numberings and so on. No, that's fascinating. I didn't really know about it. So just to make sure I get it right, I mean, Gertl's theorem is saying that if you have a system that you assume is consistent, which you can't prove, right? But if you assume it's consistent, there's, roughly speaking, going to always be statements that are true but unprovable in that system. Yeah, that would be neither provable nor disprovable.

Right, right. It's just undecidable. And undecidable, exactly. And so what you're saying is that that kind of follows from a counterfeitable, counting argument, that you can just imagine that there's, I don't quite see the argument, but I get that it could be there, that given a finite axiomatic system, you can only reach so many provable statements, and there's a lot more out there that are neither provable nor disprovable.

Right. Or beyond our R or the systems, complexity of horizon. And is that kind of higher level abstract mathematics also useful to people on the street, beyond conditional probabilities and things like that? Are these sort of wilder realms of mathematics also rewarding? Rewarding, but I'm not sure to be useful to too many people on the street. They'd have to have a kind of theoretical bent, I guess. But computer science is a big part of it is about classifying sets

to have coding for their complexity and getting algorithms that are harder and harder to work. break, not necessarily quantum algorithms, but the disregular ones involving fine numbers and simple facts about the number theory.

So in that sense, I mean, they're important. I mean, there's G.H. Hardy, mathematician, I once wrote a book called The Mathematician's Apology, in which case he said he only pure number theory. Pure number

theory is the only thing that's that's worth our reverence. And he got, you know, applied mathematics. He acted like his pornography or something. But actually somebody once wrote a review of his book, a one-sentence review of G.H. Hardy's apology. And he said, the world sickens from such cloistered clowning. But the funny thing is that number theory,

which she thought was so pure. I mean, you couldn't carry on a modern economy without being able to transfer trillions of dollars over oceans and around the world instantaneously. So even number theory was very applied. You can't tell what's applied. And the same thing about relativity, Mankowski, Riemann and so on. They weren't talking about physics, but they constructed the tools. It's also fascinating to me how people get worked up about these mathematical issues. I was just reading a little bit about Geyorg Contour and his proof of the different kinds of infinity

and how I guess Leopold chronicer really gave him a hard time, like really just tried to ruin his career because he had proven that there were different kinds of infinity. Yeah, right. He, you know, there's only the integers and everything else is made up. I kind of, I've been wondering, again, for research level, reasons. Do we really need infinity or do we really need a continuous infinity or could we just imagine that reality just works on either a finite or at least accountable set of things? And we're sort of kind of just amusing ourselves but not really making productive understanding of the

universe by thinking about all these more difficult levels of infinity. That's a good question. I don't know. I mean, it's a very beautiful subject, transfinite arithmetic. and transfinite set theory in general. But I don't know. But one thing about chronicer, only the integers exist, but there's lots of connections between the two. But one that I find in discussing the book about is the number E,

2.718 and so on, which plays a big role in math, finance, everything else, all kinds of instances. But this one I like, you tell people, at their computer or phone and randomly pick a real number between zero and a thousand. Okay, you have to pick rational if you're doing it with a computer, but still, it's ghosting up.

So pick a real number between zero and a thousand. Then keep on doing that until the sum of the real numbers you've picked is over 1,000. Okay. So I need to pick 502, 308, 607, the third number would be over. So if you have a whole, you know, many, many people do this or you do it yourself many, many times, the average number of numbers you have to choose before you get a sum over a thousand is E. Which seems, really? That seems weird. Yeah.

So that's weird. I mean, so Chronicle was, even his beloved just integers, positive integers, gives rise to the number E, which is irrational. France and General and so on. It is, yeah, I get that it's hard to get around the appearance of these numbers. You know, one of my favorite blog post I ever wrote was on Pi Day, you know, Pi Day, March 14th, which is also Einstein's birthday. And so I wrote about the fact that in Einstein's equation that we were just talking about, the one for general relativity, the right-hand side is 8 Pi-G-T-Mu-New.

And so pie appears there in the equation for gravity. And why is that? And, you know, it's a very interesting story having to do with the fact that spheres have pie in them when you calculate the area of a sphere. And Isaac Newton gave us a law for forces rather than for fields. So all these numbers are there. And yeah, maybe that's a good reason not to try to discretize the world too much. Yeah, no, I think you can. I mean, actually, you look at Remenugin, the famous.

as Indian mathematician who died early. He came up with all sorts of crazy identities that involve Pi and E and infinite sums. And you say, how did he ever come to that? Yeah. He did. I mean, G.H. Hardy writes about him. He said it's the only, I'm paraphrasing it,

but the only person he ever loved, platonically, well, for him it would be, was Ramanujan. I mean, he just fell in love with the, you know, the amazing kind of resonance that Ramanujan had with the mathematical universe. Yeah, Ramanujan is a great example, almost as a counter example,

but as a version of this idea that I like to mention that the human brain is not meant to do math. We have to train ourselves, right? You know, we're meant to make rough-and-ready things, rough-and-ready estimates, but not so much more precise calculations. But not all brains are equal at it.

And he is an example of someone who really did in a way that no one understands, seemed to just see things out there in the world of the natural numbers and continued fractions and things like that that were pretty amazing. It is. And to this day, incomprehensible. Like, how did he come to that? And often he didn't necessarily believe in proofs. I mean, Hardy had a, he came from India,

Hardy gave him some conventional mathematics that would be proof things. And he often just intuited it in some sense. And so a lot of his amazing results are just statements. And then people, mathematicians work fever. So he said, oh, yeah, I proved it. It makes you wonder, do you have feelings about the future of artificial intelligence as mathematical proof generator? Could we get a lot more proofs for theorems,

once the AIs really get good at it? I think so, yeah. It's hard to make predictions, but yeah, I think so. And not just mathematics. I mean, everyday life, everyday humans sometimes I fear. And I do want to give you a chance to, there was one little piece of advanced math that appeared in the book that I thought was very amusing.

And I would like to understand better myself. So, you know, it's getting late in the podcast. We can indulge ourselves a little bit. which is Ramsey theory. You use it as an example of how complexity appears in unexpected places. So I'll give you a chance to explain that a little bit. Now, Ramsey theory is just the idea that with a big enough set and a big enough number of connections among the elements,

we're going to necessarily have some bit of order. The order is going to be there. If you have six points, you connect them with lines, six nodes, you connect them with lines. there's necessarily going to be, some of them with blue lines, some of them with red lines, there's necessarily going to be a triangle where all three are the same. And results of that sort. And with bigger sets, you need a much bigger set to have more order,

but it's impossible to not have any bit of order. In fact, I mean, in general, that's an idea I've always liked, that the impossibility of total disorder. Because if you have total disorder, on a higher level, in a sense, those order. I mean, that's statistical mechanics. I mean, disorder, and then at the higher level, you get a definition of temperature, which is more macro. And there are other elements like Kauffman, you connect light bulbs in some random way,

on and off, and if you have some rule, if two of the three inputs are on, it'll go on or off. And no matter how kind of random you make the rules, after a while you get some pattern that keeps appearing. And I mean, it's like the game of life. I mean, you get these random things become computers and Wolfram's work and Conway's life, life game and so on. So order arises no matter what, which I find an intriguing kind of result in a kind of generalized sort of way. Yeah, I'll confess I don't completely understand it myself. I would love to understand this better because I do know there are these examples which are very provocative, like you just said, of orderliness emerging.

Emergent is the word that I would like to use out of the underlying lack of order. But I don't understand how robust it is. Is it inevitable? Does it always happen? Or are we cherry-picking examples where it happens? And I'm just not really sure. I think, you know, again, I'm just talking off the top of my head here. But I think it always happens.

It just maybe it takes longer in some cases, which is kind of a neat result. Oh, it's very, very important if it's true. Yeah, that's why I don't know. It would be nice to have a, I don't know. There's probably other people who do know much more than I do about that. But okay, so good. We've indulged ourselves a little bit of less practical mathematical speculation. But to bring it back to close things up, you know, you've done an enormous amount for spreading the word of mathematics, as it were, to a broad set of people.

And I'm sure in various ways large and small you've gotten pushback about, is this elitist? Is it paternalistic? you know, are you just getting annoying people for letting them, for making them not just get on with their lives and thinking about all this abstract stuff? So how do we get people excited and interested and educated about math in a way that doesn't come across as elitist and paternalistic? I don't know. I don't think elitism is part of it.

I don't see why here's some interesting stuff. It's relevant to some stuff you might be interested in. Here's how it works. Why that's viewed as elitist. I mean, of course, it is by a lot of people. I mean, one of the reasons for Trump. I mean, resentment drives a lot of his base. But I don't buy it.

I mean, just because somebody feels that, you know, presenting mathematics, presenting physics, presenting history, history, I mean, looking at something seriously, trying to understand it, relating it to other things. That's not elitist. That's human. Well, and maybe not to present this as a leading answer, but I think that being human, being warm, being engaging, being likable is probably goes a longer way toward making the math palatable

than we want to admit. It's not all just about the math. We're still human beings at the end of the day. Oh, yeah, yeah. I mean, if you are a very dower look on your face and hit your student on the knuckles with a ruler and tell them to go sit in the corner with a dunce cap, yeah, that's not a way to get them to appreciate or love mathematics, physics, history. But, yeah, you're right. I mean, I think it's easier to learn from a professor or from anybody if you, in a sense, like them or can kind of.

kind of map yourself onto them in some way. I mean, if that person who's enlightening you to use that term is kind of repellent, you're not going to want to learn much from that person. You did mention in passing that big stories in physics get more play in the news or in science than big stories in math do. Do you think that's changing? Do you think that there's more and better math outreach and, public engagement today than there was 50 years ago?

I think there is, but science is something that people understand. I mean, they don't understand the details, but they know the moon's up there, and the universe is expanding, and light goes as fast, and, you know, and even just speculative theories, multiverse and so on, that's something that engages people's imagination, And so there can be more stories about advances in physics or science or biology as well. Whereas math, I mean, you get a new result and a new consequence of the Xima choice or Bonox here. You can take a part of a sphere and put it together and make it twice as large.

I mean, it strikes people as just, you know, that's just focus focus. That's, you know, mathematicians, you know, whatever. So, I mean, but as far as outreach in general, I mean, I think STEM is more widely understood as why it's important. And again, it's hard to generalize about pedagogy, but more places do a good job, even though there's still a vast number of people who are enumerate, but like a little better term. But there is no better term. There's no better term. and there's no better person who's done more to combat it than you have. So John Allen Polos, thanks very much for being on the Mindscape podcast.

Thanks very much, Sean. I truly enjoyed it.