216 | John Allen Paulos on Numbers, Narratives, and Numeracy

0:00:00.0 Sean Carroll: 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 go to school for a... 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. 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.

0:01:12.1 SC: And why is that? And in the essay, tries to argue that it's basically because when you're at home and you're trying to learn math, you're gonna learn it from your parents and chances are good that your parents hate math [chuckle] because a lot of people hate math. It's considered okay to hate math. To say, "Oh, I'm not so good at 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 Paulos, famously, he's the author of Innumeracy: Mathematical Illiteracy and It's 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 illiteracy, 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.

0:02:18.2 SC: In the podcast, we're gonna 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 bit about the substance of math, because it's not just that we need to teach math and we need to get better at it, but what kinds of math should we teach? And I think that there's a thing that is said over and over again in my circles, that we actually don't spend nearly enough time teaching probability and statistics to people of all sorts of backgrounds. Whether you're gonna be a professional scientist or whether you're just gonna 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 bit 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.

0:03:27.6 SC: So I think this is 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/seanmcarroll, 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 gonna be able to give out three college scholarships this year of $10,000 each. So if you go to bold.org/scholarships/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 gonna be and what great things they're gonna do. And with that, let's go.

[music]

0:04:38.2 SC: John Allen Paulos. Welcome to the Mindscape Podcast.

0:04:40.5 John Allen Paulos: It's a pleasure to be here, Sean.

0:04:42.9 SC: So you've, in some sense, maybe tell me if this is fair or not, but in some sense, you've devoted your life to fighting against innumeracy, or maybe you wanna say to promoting numeracy. You wrote a wonderful book with a title Innumeracy. How do you yourself define numeracy or innumeracy?

0:05:03.1 JP: 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 it's... One of the points I like to make over again is the dubiousness of precision. And so people are very precise and the precision is unwarranted. So that's my defense for not giving a more precise definition of numeracy.

0:05:46.3 SC: 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. Some people have bigger vocabularies than others but you read or you don't. Whereas when it comes to math, there's 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 that there's enormously more math that I don't know than what I know. [laughter]

0:06:14.8 JP: I think that to some extent that's still a case with with literacy. I mean, people who can read the newspaper, you give them a book by I don't know, Heidegger or even a normal person reading a book by Heidegger, he would be baffled. So there is kind of a continuum although it's a little bit more binary with literacy than it's with numeracy.

0:06:39.4 SC: And I think that... But there are different kinds of mathematical knowledge, and 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. Like what I do, I'm no better at calculating a tip or doing my taxes than anybody else's. I don't know about you.

0:07:00.7 JP: Yeah, no, it's the same.

[laughter]

0:07:03.7 JP: Sorry, what?

0:07:05.9 SC: So how do you think about that difference? How do you think about math as opposed to just multiplying and dividing?

0:07:14.0 JP: Well, 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 typist. Why don't you write a novel or converse?" And so that's why one of the obstacles in teaching math, people think they know it already. I know how to multiply, I know how to plug numbers into formulas. Is there something more? But there's a lot more. Patterns, logic, structure of various sorts and it maps onto everyday events often enough that it's worth knowing even if you're not gonna 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 or statistics or people who are in algebraic topology don't necessarily understand much about Banach spaces. But that's the case in a lot of things. I mean, my son's a lawyer and 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.

0:08:39.5 SC: Right. Yeah. It's specialized. Let me just... Let's get some examples on the table so people have an idea of what we're talking about in the book. Like 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 7/15."

[laughter]

0:09:03.0 JP: Right. That's... Well, lemme back up. If the probability a woman has two sons, if the given... Well probability two sons given she's got at least one son, that's 1/3, because you can't have girl girls. So probability two boys, given at least one boy is 1/3. The probability of two boys given at least one boy born in the summer is 7/15, which is close to 1/2. So it's... The more precise you are about... You can set up the sample space and convince yourself, but the more precise you are about the boy she does have, if they... Has two boys, at least one of whom is born on July 4th then the probability is almost 1/2. So the more... I mean, it's hard without writing out the sample space, but it's typical of problems... In 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 naturally in terms of probability and many people, "Oh, everything has a reason." And and the fact that propositions, like you just mentioned are counterintuitive, that most famous example is a birthday puzzle. 23 people sufficient for... If you have 23 people in the room probability is one half that at least two have the same birthday.

0:10:51.3 SC: Well, that's why I wanted to discuss this example in particular, because even the number 7/15 is maybe not... I don't really care what that number is. This is not gonna come up very often, but it's an example of conditional probability of probability of one thing and how it changes, given some extra information. Is that something that people struggle with, especially in your experience?

0:11:16.3 JP: Yeah. Conditional probability, revising one's probability estimates is something we do in everyday life. The only thing is if you study probability or base rule in particular, you can refine it. So everything we do in probability in a sense, are refinements of notions that develop naturally. The notion of mean, median and mode or central tendency, there are all kinds of words in English or other natural languages average so so, whatever, 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 alder, national languages too, far out, disparate, very different, far out is interesting 'cause it's far out on a distribution. And same thing with probability, as Mo here commented, people generally have been...

0:12:19.2 JP: Speaking pros all their life, and they've generally been speaking probability all their life. It's just that they speak it 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. Any... Yeah, it's common sense to say, well, it's very likely, more likely that she picked the two-headed coin. But, but only Bayes' theorem says that that probability rises from one half to eight ninths. So as I said probability statistics are refinements or distillations of kind of hazy, nebulous everyday notions.

0:13:04.0 SC: Well, the classic example of this, that royaled, I guess it wasn't the internet it seems 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? 'Cause maybe there's still people out there who never heard of it.

0:13:19.2 JP: Yeah, there are actually, in my... At least in my experience, the Monty 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 let's say 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... Says to the guest, Are you sure you don't wanna switch from one to two? Let's say he opens door three. You sure you don't wanna switch from one to two? You chose door one, I've opened door three. You wanna switch your bet to two. And often people say, No, it doesn't make any difference. There's two doors open. Chances are 50/50 one half [0:14:23.0] ____, I might as well stay here.

0:14:25.9 SC: Yeah.

0:14:26.8 JP: But he should 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. So were he to switch, he would raise his probability of winning the car from one third to two-thirds. But many people including, there's a story that may be apocryphal, but the great Paul Erdős a mathematician was supposedly baffled by this. I find it hard to believe.

[laughter]

0:15:03.0 SC: I find it hard to believe myself too. Yeah. But, I mean, maybe I don't, because the way that you stated the problem, everything made super duper perfect sense. I do think that sometimes when people state problems like this, they kind of want people to get the wrong answer just to explain the trickiness of it. So they, kind of hide that fact that Monty Hall will only ever open the door where the car is not there. Right? And that does change the problem.

0:15:34.5 JP: Yeah it does, does change the problem. I mean, it's one thing I tell my students in probability and in math in general, being really clear about what 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 that you're gonna get 500 heads [laughter]

0:16:08.2 SC: That's right.

0:16:09.6 JP: But with regards to Monty Hall, there's... What I try to do is show the relevance of 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 there called Taunty Hall.

[laughter]

0:16:28.9 JP: And [laughter], and there's a guest. And the same thing. The only thing is if he picks the door, that 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 he 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 wanna switch to 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 than the other two doors. But he wants to limit his exposure, so he should stick there. So it's the kind of dual problem, he should stick with what he picked. And the relevance of that, let's say to COVID is you wanna limit the number of people you come in contact with. I mean, there are versions of this with a hundred doors or whatever.

0:17:24.8 SC: Right.

0:17:27.8 JP: But in any case.

0:17:28.5 SC: 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?

0:17:41.7 JP: I think they are, I mean maybe fear is, induces one to think more clearly than does greed. I'm not sure, although greed does a good job too.

[laughter]

0:17:53.5 SC: 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.

0:18:20.5 JP: Yeah. I think that's true. I mean you don't know. Primitive man didn't need, probability to realize if there's a rustling in a bush to run even though he'd be wrong a lot of the time.

0:18:35.0 SC: 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.

0:18:42.5 JP: Yeah.

0:18:43.0 SC: 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 it can go from high entropy to low entropy.

0:19:00.8 JP: Right.

0:19:01.2 SC: And the answer is those two numbers have to be exactly the same because the underlying... This is symmetry of reversibility of the underlying system, and Loschmidt 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.

0:19:31.8 JP: 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 to be 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."

0:20:02.8 SC: Oh, Yeah. Right. People don't wanna believe that.

0:20:04.8 JP: Yeah. Okay. I mean, so QAnon conspiracy theorists, the election deniers, once they publicly state the nonsense they believe, it's very hard to get them back.

0:20:16.5 SC: Well, you've talked in this book and elsewhere a lot about pseudoscience and conspiracy theories and and so forth. What is the connection to numeracy or innumeracy there?

0:20:27.7 JP: Well, I think, it's just clear thinking there. In that case, you've gotta know some facts. You've gotta know a modicum of arithmetic probability. And if you don't, you're more easily fooled. Like 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 is 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.

0:21:18.2 SC: Right.

0:21:18.7 JP: The relevant conditional probability is the 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 that don't know the difference between if A, then B and, if B then A, which is basically what conditional probability is.

0:21:47.2 SC: And this kind of feeds in with, in the case of the conspiracy theories, some kind of wishful thinking 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.

0:22:07.7 JP: That is definitely the case. In fact, cognitive foibles in general are a big part of what leads people astray. And we're all vulnerable to the anchoring effect and availability, error confirmation vice. One that is... Doesn't get as much publicity is the conjunction fallacy. You have somebody let's say he is a senator, US senator, and he's everybody... He's very moderate, he's intelligent, rectitude is the word that comes to mind for everybody that 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?

0:23:01.2 SC: [chuckle] Right.

0:23:02.8 JP: [chuckle] And most people will say, "Well," or at least a lot of people probably the latter given what she said about him, 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, factoids, little details are available. So it's easy to cobble together a superficially plausible story. And there's this trade-off between probability and plausibility. You add more details, which you can glean from the internet that your story becomes in a way more plausible the same way this senator thing work, but less probable. And if you're gullible to begin with, and you have all these seemingly, precise details, you can fool people.

0:24:01.3 SC: 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?

0:24:19.6 JP: I guess, yeah, it is possible to distinguish. It is a matter of clear thinking, but yeah, there are other components. In fact, one thing I tried to do in my writing is set up 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 mathematics by the way, then there's 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, [0:24:56.9] ____ for different reasons in humor or jokes is for the [0:25:03.5] ____ in math is [0:25:05.3] ____ something. But there is a kind of continuum. And 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 so in essence, puzzles are very mathematical and whether they're mechanical puzzles like Rubik's Cube and auto group theory and that, or verbal puzzles like you have Monty hall. So, there are similarities, but there are... So it is kind of a continuum there.

0:25:41.5 JP: There's not a chasm between, here's numbers and here's narratives. There's a connection. There's differences. One is that, in mathematics or science, the logic is extensional. If you have a proof or something every time you have a three, you could put in square root of nine or cube root of 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, "Oh, the happiest day of my life was the 80th anniversary or 110th anniversary of Millard Fillmore's death." Even if that was her wedding.

0:26:15.6 SC: That was the day, Yeah.

[laughter]

0:26:17.0 JP: That was the day. But, are you crazy? That the happiest anniversary is 110th anniversary? So you can't do that. Then also just the whole kind of psychological mindset that you do reading a story, science fiction or whatever, just for the enjoyment, you suspend disbelief. Okay, let's go with it. But in math or science or statistics, you do just the opposite. You suspend belief so you're not bamboozled. You wanna really prove it. So there are lots of differences, but still they're both human endeavors and they're not the totally distinct things storytelling and theorem proving or number crunching.

0:27:00.5 SC: 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 theorist, who made a quote that I will never stop quoting, which is, "Never brings statistics to a story fight."

[laughter]

0:27:26.1 SC: People love their stories, and I guess maybe, I don't know how intentional it is, or it just seem 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.

0:27:42.4 JP: Exactly. I like to use jokes, puzzles, anecdotes, a 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 gonna be judged. But you get the same idea across without the formalism. There's a limit to it, you can't, but you can get a good deal of mathematics across with stories. In fact, one of my books, I think, therefore I laughed I... It was based 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.

0:28:31.0 SC: Very loosely.

0:28:31.8 JP: You get the joke, you get the relevant philosophical point. And, so you can see it in the book, here's a collection of stories, jokes, anecdotes that get points across.

0:28:44.7 SC: Well, I'll confess, I have not actually read your books that discuss math and humor, but I... 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? 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?

0:29:15.8 JP: Not too much. There are a couple of journals of humor that kind of touch on it, but 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 differential.

[laughter]

0:29:36.4 JP: So, yeah. And so I... Even in this book, I bring up some philosophical issues about Eugene Wigner, unreasonable effectiveness of Mathematics, narratives. There's a collection of metaphors, justified true belief is not equally knowledge. And, so...

0:29:56.8 SC: I can tell that you had some philosophical background as well as the usual mathematical background.

0:30:02.3 JP: Yeah. As an undergraduate, I skipped around. I majored for a while philosophy and in English and physics. I kept coming back to math, but, I always wanted to write. So in a sense, I do both.

0:30:14.5 SC: Got the right... Got the exactly the right thing. So the... 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 one has to be numerate?

0:30:38.9 JP: Well, a facility with arithmetic, first of all. And, of course, probability and logic and notions in the philosophy of science. Just everyday notions, like what's the placebo, what's the double blind study? Which, most people are innocent of. In fact, I just wrote a piece for 3 Quarks Daily about, why... Quizzes for congressional aspirants... I mean at any time you apply for a job, especially a high tech job, you are interviewed and they give you some problems. "Can you program this, some python or whatever?" But yet you can run for Congress, can run for president and there's no such test, and perhaps there should be, although you'd have to drag people into it or shame them into it, but you wouldn't have Herschel Walker saying, why are we making clean air, we just send it to China and they send us dirty air and evolution is a hoax, and it's kind of embarrassing.

0:31:51.5 SC: I do wanna... I wanna note, your mention of 3 Quarks Daily, that's one of our favorite websites here at The Mindscape podcast, so I'm glad that you put in a plug for that.

0:32:00.8 JP: Oh yeah, no, I enjoy it as well. In fact, I write a kind of semi regular column for them.

0:32:07.1 SC: 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 shouldn't be allowed to vote if you can't solve a quadratic equation. And it seems a bit extreme to me. I mean, I understand the...

0:32:43.0 JP: Yeah, no, I agree.

0:32:45.1 SC: The motivation, but should politicians be allowed to run for office if they can't solve a quadratic equation?

0:32:52.4 JP: Yeah, I think that's a bad example, but being able to... Having some feel for... Scaling. Like, we can scale up things. Like, every time I go to a movie, I'm always amazed that you can get a soda that's eight inches high or 10 inches high, and the 10 inch high one is 50 cents more than the eight inch high one, even though it's probably... 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 the size of cities scale up and with the fractional exponent Geoffrey 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, as well as being able to estimate things. It's a very rough estimate, you could tell somebody, the Empire State building is two miles high, and you don't have to know the exact height, happens to be 1776 feet, but two miles high, do you really realize that?

0:34:09.8 JP: So some appreciation for normal estimates. I mean, if you ask people... 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 I can see people saying, Wow, I better be careful. But Friday, Saturday, Sunday is three sevenths of the week, [chuckle] which is 43%. So, same thing, a four-day holiday weekend or 400 people are gonna die, 35,000 or more a year do die on the highway. So it's a normal four days, it's not... So some feeling for relevant magnitudes, for scaling, for estimating, for sequencing. Some things you have to do in a certain order, there are lots of the puzzles, so like that. And they'd also involve quadratic equations, which I don't think that would be a reasonable requirement.

0:35:12.1 SC: No, it's just memorizing a formula. Yeah, I do get that. But it's interesting 'cause all the examples you're giving are a little bit different than what we actually teach people in high school or even in college. We take Geometry and Trigonometry, maybe 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?

0:35:47.4 JP: Well, it's hard to generalize 'cause 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 and everyday life. I think scaling, as I said, estimation and so on. And just the ability to kind of think outside the box, one thing I talked about it in an argument, pro or anti-abortionists, that it's kind of fanciful, but it is interesting that people who are absolute, super absolutist with regard to banning abortion, I think that it would 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.

0:36:51.3 JP: And there's a story I like that I prefaced, the abortion story with about George Bernard Shaw supposedly... And again, the story might be apocryphal, he's 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. Then he said, "Would you sleep with me for 10 pounds?" And she says, "No, who do you think I am?" And he says, Well, we already established that, now we're just haggling a bunch of details. But it's a stupid story, sexist story, but it's relevant to this argument I'm gonna make. Imagine that because...

0:37:33.0 JP: 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 the 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 absolutists, 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. 'Cause that's 10 amounts to abortion. And so they'd have to be... 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, sorry, George Bernard Shaw story is once you get them off of this totally absolutist position, then the rest is haggling about the details. You can go for 15 weeks or 20 weeks or whatever. So in any case.

0:39:01.9 SC: 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. I myself, when it comes to science, often complain that we teach science as a list of facts, right?

0:39:22.0 JP: Right.

0:39:22.2 SC: 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 new stories about science in the media. And I guess probably the same thing is true with math also, that there's a different kind of math that is equally good that would be way more relevant to people's reasoning in everyday circumstances.

0:39:43.9 JP: 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. But, no, but I agree, but I think it should be part of a general... I mean, general knowledge is important as well. And it's... And teaching weariness, skepticism, suspending belief and so on is important. And it's connected to other 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 did 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.

0:40:53.1 JP: And nobody said, how did this come into being immediately? 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... How did [0:41:01.9] ____ life lights come into being so immediately? How did this... I come into being? And instead of saying, Well, to use the broken word, it evolved, the same way cities and economy evolved, but they are most accepting of capitalism, which... And least accepting of the analogous process when it comes to life. And so...

0:41:30.5 SC: Yeah, I mean this, 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.

0:41:46.2 JP: Yeah.

0:41:46.7 SC: I actually, I don't know whether or not that makes sense. Is that something that can be taught in your experience?

0:41:54.0 JP: To some extent. I mean, there... It does often devolve into something kind of silly, but I mean, I think getting people to know a lot, [laughter] not just facts, but friends and connections between disparate fields like economics and evolution is, I mean, if you know something, not a lot necessarily, I mean, it helps if you know a lot, but if you know something and are taught to bring things together to try to relate, to have a kind of more holistic attitude towards knowledge, I think that's a worthwhile endeavor. But you're right. I mean, it is hard. I'm not sure how you go about teaching critical thinking because people always are [0:42:42.9] ____ drawn me on to some fact the key, the key formula, the key thing.

0:42:48.2 SC: Yeah.

0:42:48.8 JP: And often there is no clear thing.

0:42:51.8 SC: Well, and often it's a sort of against their interests, right? People don't wanna 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.

0:43:04.5 JP: Yeah, exactly. [laughter], Well, I mean, I talk a lot about logic in the book and paradoxes and their relevance in everyday life. I mean, even in the stock market, it... The efficient market hypothesis says that, information about a stock is immediately available to everybody, but it... Most markets aren't all that efficient. But it's... You can have a... Create a kind of relevant example, a relevant, something that's relevant to the wire's [0:43:40.9] ____ paradox, the efficient market hypothesis is true to the effect that most people think it's not true [laughter] 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 word? What's the... What's the worth of doing that? That's every information, every bit of information is already priced in. Or what do I wanna do that for. So... Well, a lot of these paradoxes are relevant to broader themes. And again, to your point, that it'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 or read who's counting.

0:44:37.1 SC: Yeah, exactly. Read who's counting. Speaking of which, this is a perfect segue 'cause 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 in my new book, The Biggest Ideas in the Universe, I try to teach people the basics of classical physics. 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, "I just don't understand equations. I will never be able to understand it." And I say, "Look, you understand two plus two equals four. That's an equation. 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 are you willing to do a little bit more work than you usually do?" What do you think of that angle? Is that a plausible story?

0:45:40.0 JP: I think so. Stephen Hawking allegedly said that if you put an equation in your book, you cut the readership by a half. So if you put a hundred equations in a book...

0:45:56.1 SC: Very, very tiny.

[laughter]

0:45:57.2 JP: But I think it's false. But yeah, you just have to... Have to do it carefully. You have to embed the equation in a discussion in which it makes sense. You can't just boldly say, "Here's the formulas." But I think that's a good idea if you... 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.

0:46:30.8 SC: Well, we try. But I'm wondering about the level of abstraction. 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, not E equals MC squared but [0:46:44.7] ____. That is... There's some journey that the learner has to go on to really wrap their heads around it. And I'm wondering if it is really something that 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 gonna do in high school?

0:47:14.5 JP: I think you could reach a lot of people. I wouldn't say everybody. But you could reach many more people than our reach now because if... It was done right.

0:47:26.8 SC: Yeah, give them the opportunity.

0:47:30.1 JP: Yeah, give them the opportunity. Require in your book.

[laughter]

0:47:33.5 SC: Exactly. There you go. I've been fishing for that one. But yes. But...

0:47:38.7 JP: Actually... What? Sorry.

0:47:40.0 SC: Go ahead.

0:47:41.0 JP: No, I was just gonna say, two and two equals four is always put forth as a kind of standard simple fact. But everything depends on context. It's not always the case. If you take two cups of popcorn then 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. 'Cause before they had mastered the vector analysis, when they saw a bear to the 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. And math like adding integers and simple vector analysis, this silliness is clear, is 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 gonna challenge you. You can say the most abstruse sounding nonsense.

0:49:00.1 SC: Well, this is the famous anxiety people have when they take math classes about word problems, right? You can memorise 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, right?

0:49:17.6 JP: Yes, yeah. Exactly. That goes back to my narrative numbers continuum.

0:49:22.2 SC: Yeah.

0:49:22.9 JP: That is by far more important and...

0:49:28.2 SC: You mentioned already the phrase the unreasonable effectiveness of mathematics. And of course...

0:49:34.2 JP: The what?

0:49:35.6 SC: The unreasonable effectiveness of mathematics and physics, which is a famous phrase from Eugene Wigner.

0:49:42.2 JP: Yeah, right.

0:49:42.6 SC: 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.

0:49:53.6 JP: Right. Right. I'm very skeptical of it. 'Cause we learn about numbers by playing with little pebbles and putting them 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 off them. And also we learn about physics by walking through the world. So mathematics is kind of an idealisation and abstraction of everyday things that we do. So I don't think it's all that unreasonable. If you abstract, if you idealise what you do when you're playing with pebbles and twigs and moving around, it's not surprising you get a mathematics that's gonna be effective. It grew out of things that worked.

0:50:42.6 SC: Then there are some. That's a great... That's another great segue because I've been recently thinking in part, 'cause I had a podcast interview with Justin Clark Doane, 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 [laughter] 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?

0:51:22.0 JP: 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 in philosophy and that's still kind of mathematics that I'm... At least initially was most interested in. And I talk about Gödel's theorem, but an unstandard proof of it using ideas from complexity theory, Greg Chaitin, and... Yeah. So I think a little bit of some logic it doesn't have to be, you know, go too far, but people don't know the difference being affirming a consequence and all these Latin terms.

0:52:08.3 SC: Sure.

0:52:08.3 JP: And I think they should, they don't have... They probably just focus on memorizing the terms instead of understanding them. But.

0:52:17.7 SC: Can you say more about proving Gödel's theorem using complexity theory [chuckle]

0:52:22.3 JP: Yeah. A sequence is 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, 0 1, 0 1, 0 1, you can generate that by just saying "zero one repeated."

0:52:43.6 SC: Right.

0:52:44.0 JP: A thousand times. But you can never generate something more complex than the generating algorithm. And so it uses that and Berry's 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 gonna be things or statements that you're not gonna be able to prove because of the limited complexity of any logical system. A nice example that's kind of silly, but I like, Berry's 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. That by pushing, you can't reach it.

0:53:31.8 SC: Right.

0:53:33.2 JP: So it's, yeah, I sketch it a little bit more than that, but I like the Chaitin's proof of Gödel's Theorem better because it's connected to more basic stuff about the notion of complexity. Complexity is something that's in the world and is an important topic in computer science and from it follows Gödel incompleteness in without going through and working at Gödel numberings and so on.

0:54:00.2 SC: No, that's fascinating. I didn't really know about it. So just to make sure I get it right, Gödel's Theorem is saying that if you have a system that you assume is consistent, which you can't prove, but if you assume it's consistent, there's roughly speaking, going to always be statements that are true, but unprovable in that system.

0:54:17.7 JP: Yeah. That it would be neither provable nor disprovable.

0:54:21.4 SC: Right, Right.

0:54:22.6 JP: It's just undecidable.

0:54:23.8 SC: Undecidable. Exactly. And so what you're saying is that that kind of follows from a counting argument that you... 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.

0:54:45.5 JP: Right. Beyond our or the system's complexity horizon.

0:54:51.5 SC: 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.

0:55:06.8 JP: Rewarding. But, I'm not sure it'd be useful to too many people on the street. They'd have to have a kind of theoretical vent, I guess. But Computer Sciences is... A big part of it is about classifying sets, high code integral complexity and getting algorithms that are harder and harder to break that, not necessarily, and quantum algorithms, like the just regular ones involving prime numbers and simple facts about the number theory.

0:55:39.3 SC: Right.

0:55:39.8 JP: But, so in essence sense, they're important, there's... GH Hardy mathematician once wrote a book called "The Mathematician's Apology", in which case he said, he only pure math and only number theory." Pure number theory is the only thing that's worth our reverence. And you got applied mathematics he acted like it was pornography or something. But actually somebody once wrote a review of his book, a one sentence review of GH Hardy's apology. And he said, the world sickens from such cloistered clowning.

[laughter]

0:56:20.3 JP: But the funny thing is that number theory which he thought was so pure, you couldn't carry on a modern economy without being able to transfer trillions of dollars over oceans and around the world instantaneously. So this so even number theory was... It was very applied. You can't tell what's applied. And same thing about relativity and Makowski, Reman and so on, they weren't talking about physics but they constructed the tools.

0:56:51.0 SC: It's also fascinating to me how people get worked up about these mathematical issues. I was just reading a little bit about Georg Cantor and his proof of the different kinds of infinity and...

0:57:04.0 JP: Yeah.

0:57:04.6 SC: How Leopold... I guess, Leopold Kronecker...

0:57:08.0 JP: Yeah. Right.

0:57:09.0 SC: Really gave him a hard time, like really just tried to ruin his career 'cause he had proven that there were different kinds of infinity.

[chuckle]

0:57:14.0 JP: Yeah. Right. He used to... There's only the integers and everything else is...

0:57:19.9 SC: Yeah.

0:57:19.9 JP: It's made up.

0:57:23.1 SC: 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?

0:57:51.4 JP: That's a good question. I don't know. I mean, it's a very beautiful subject, trans finite arithmetic and trans finite set theory in general. But, I don't know, if... But wait, one thing yet about Chroniker, he thought only the integers exist, but there's lots of connections between the two. But one that I find and discussed in the book about is the number E 2.718 and so on, which is, it plays a big role in math, finance, everything else, all kinds of the instances. But this one I like, some people look at their computer phone and randomly pick a real number between zero and a 1000. Okay? You have to pick rational if you're doing it with a computer, but still it's close enough. So pick a real number between zero and a 1000, then keep on doing that until the sum of the real numbers you picked is over 1000.

0:58:46.4 SC: Okay.

0:58:48.0 JP: So I might... I may pick 502, 308, 607. The third number would be over. So if you have a whole... 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 1000 is E.

[chuckle]

0:59:11.0 JP: Which seems weird. Really?

0:59:12.2 SC: That seems weird. Yeah.

0:59:14.3 JP: Yeah. So that's weird. I mean, there... So Chroniker was... Even his beloved, just integers, positive integers give us rise to the number of E which is irrational, transcendental and so on.

0:59:26.8 SC: It is... Yeah. I get that it's hard to get around the appearance of these numbers. One of my favorite blog posts I ever wrote was on Pi Day, you know, Pi Day March 14th.

0:59:38.0 JP: Yeah.

0:59:39.0 SC: 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...

0:59:46.0 JP: Yeah.

0:59:46.7 SC: General relativity, the right hand side is eight pi G T mu nu. And so pi appears there in the equation for gravity.

0:59:56.0 JP: Yeah.

0:59:57.0 SC: And why is that? And, it's a very interesting story having to do with the fact that spheres have pi in them when you calculate the area of the sphere.

1:00:06.0 JP: Yes.

1:00:06.8 SC: 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.

[chuckle]

1:00:18.5 JP: Yeah, No, I think you can, I mean, actually, you look at the Ramanujan, the famous Indian mathematician who died early. He has... 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? And...

1:00:36.0 SC: Yeah.

1:00:37.0 JP: He did. And nobody, I mean, that, G H Hardy writes about him. He said he was the only... I'm paraphrasing it, but the only person he ever loved, platonically is, well, for him it would be, was Ramanujan. I mean, he just fell in love with the amazing kind of, resonance that Ramanujan had with the mathematical universe.

1:01:02.9 SC: Yeah. Ramanujan is a great example, almost as a counterexample. 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? Yeah.

1:01:16.0 JP: Right. Yeah.

1:01:16.8 SC: 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, seem to just see things out there in the world of the natural numbers and the... And continued fractions and things like that, that were pretty amazing.

1:01:36.7 JP: Yeah. It is. And in... And to this day, incomprehensible, like, how did he come to that?

1:01:43.0 SC: Yeah.

[chuckle]

1:01:44.0 JP: I often... He didn't necessarily believe in proofs. I mean, Hardy had a... He came from India and Hardy too give him some conventional mathematics that would reprove things, and he often just intuited it in some sense. And, so a lot of his amazing results are just statements and people... Mathematicians work feverishly and say, Oh, yeah, I proved it. This is why it's said like that.

[chuckle]

1:02:08.8 SC: 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?

1:02:20.5 JP: I think so. Yeah. I... It's hard to make predictions, but, yeah, I think so. And not just mathematics, I mean, everyday life, everyday humans, sometimes I fear.

[chuckle]

1:02:35.4 SC: And I do wanna 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 it 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.

1:02:56.6 JP: Now, Ramsey theory is, it's just the idea that with a big enough set and a big enough number of connections among the elements, you're gonna have... Necessarily have some bit of order. The order's gonna gonna be there. If you have six points, and you connect them with lines six nodes, you connect them with lines, there's necessarily gonna be, some of them with blue lines, some of them with red lines. There's necessarily gonna be a triangle where all three are the same and is... Results of that sort. And with bigger sets, you need a much bigger set to have, you know, more order, but then it's impossible to not have any bit of order.

1:03:41.6 JP: In fact, I mean, in general that's an idea I've always liked that the impossibility of total disorder 'cause if you have total disorder on a higher level in a sense, those order, I mean that's statistical mechanics, I mean this order. And then at the higher level you'd get that bit of definition of temperature which is more macro. And there are other elements like Kaufman, you connect the light bulbs in some random way on and off then 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 a game of life. I mean, you get these random things become computers and Wolframs Artwork and Conway's Life game and so on. So an order arises no matter what, which I find an intriguing kind of result in a kind of generalized sort of way.

1:04:51.5 SC: 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...

1:05:04.0 JP: Right. Yeah. Right. Yeah.

1:05:04.8 SC: 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?

1:05:17.4 JP: 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 takes longer in some cases but... Which is kind of a neat result in a way.

1:05:31.5 SC: 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...

1:05:39.5 JP: Yeah.

1:05:41.6 SC: 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... Or 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?

1:06:27.4 JP: 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, it's one of the reasons for Trump. I mean resentment drives a lot of his base. But I don't buy just because somebody feels that you know, presenting mathematics, presenting physics, presenting history, I mean looking at something seriously, trying to understand it relating it to other things, I... That's not elitist, that's human.

1:07:15.7 SC: 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.

1:07:34.6 JP: Oh, yeah, yeah. Right. I mean, if you are... Very dull look on your face and you hit your students on the knuckles with a ruler.

[chuckle]

1:07:45.0 JP: And tell 'em to go sit in the corner with a dunce cap, yeah, you're not... That's not a way to get them to appreciate or love mathematics, physics, history.

1:07:54.8 SC: I mean...

1:07:55.4 JP: 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 map yourself onto them in some way. I mean if that person who's enlightening you to use that term is kind of repellent, boy, you're not gonna wanna learn much from that person.

1:08:18.8 SC: 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.

1:08:28.6 JP: Right.

1:08:29.9 SC: 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?

1:08:39.6 JP: I think there is, but... And science is something that people understand. I mean, yeah, they don't understand the details, but they know the moon's up there and the universe is expanding and light goes this fast. And you know, and even just the speculative theories, the multiverse and so on, that's something that engages people's imagination. And so there's gonna be more stories about advances in physics or science or biology as well. Whereas math, I mean you get a new result, a new consequence, sort of action, a choice or Banach's theory, you can take a part of a sphere and put it together and make it twice as large.

1:09:26.1 SC: Right.

1:09:26.6 JP: I mean, it strikes people as just, you know, that's just hocus pocus. That's, you know, mathematicians, you know, whatever. So I mean that... But as far as outreach in general, I mean I think STEM is more widely understood as what is 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 innumerical for lack of a better term, but there is no better term.

[chuckle]

1:10:00.8 SC: There's no better term and there's no better person who's done more to combat it than you have. So John Allen Paulos, thanks very much for being on the Mindscape podcast.

1:10:07.9 JP: Thanks very much, Sean. I truly enjoyed it.

[music]