Episode Transcript

Marcus DU SAUTOY: This is a little fairy tale that we all get told as mathematicians.

That’s Marcus du Sautoy, who is a mathematician, at Oxford University in England. As for this fairy tale:

DU SAUTOY: I don’t know whether it’s true or not, but who cares?

The story takes us back to Germany in the late 18th century, and a schoolboy named Carl Friedrich Gauss.

DU SAUTOY: The young Gauss, sitting — 8, 9 years old in his class, the teacher wants to get a little bit of rest, decides to set them a problem that it will take them ages to actually do.

Young Carl Friedrich Gauss would become one of the most remarkable mathematicians in history. There are more than 100 theorems, formulas, models, and other math terms named after him. But that was later. At the time of our story, he’s just a very bright young student. And the teacher gives the class this problem to solve. As Du Sautoy said, it’s not a particularly interesting problem. The teacher says:

DU SAUTOY: “You’ve got to add up the numbers from one to 100.” And most of the class set off and they go one plus two, that’s three, plus three, six —.

Go ahead, try it for yourself. Add up the numbers from one to 100. I’ll give you a minute.

Are you done? If not, then you are quite a bit slower than young Gauss.

DU SAUTOY: Carl Friedrich Gauss immediately writes down a number on his chalkboard, slams it down on the desk and says, “There it is.” The teacher thinks he’s being impudent but looks down and sees — “But that’s the correct answer, how did you get that so quickly?”

You may be thinking, well, Gauss is plainly a math prodigy, so he just added up the numbers in his head really fast. That’s not what he did. According to the story, here’s what Gauss told his teacher:

DU SAUTOY: He said, “Look, the rest of the class, they’re all starting at the beginning and just plodding on through this journey. I combined the beginning and the end of the journey. So one plus 100 is 101, two plus 99 is also 101, three plus 98, 101. So you’ve got 50 pairs of numbers adding up to 101. So that’s 50 times 101, which is 5,050. If the teacher came back and tried to give him an even greater challenge, “Okay, what about the numbers from one to a million,” that’d be taking the rest of the class ages. This is the power of a shortcut — doesn’t matter what number, the way of thinking will get you to the answer.

“The power of a shortcut.” What sort of power are we talking about here? And is it a power we really want? We’ve all been taught, since we were kids, that the right way to do something is to be thorough, diligent, tireless. You’ve probably heard the saying, “There are no shortcuts in life.” What if that saying is way wrong? Marcus du Sautoy believes it is. He has just written a book called Thinking Better: The Art of the Shortcut in Math and Life.

DU SAUTOY: Finding these shortcuts is often hard work. That’s like building a tunnel underneath a mountain. The first dig requires a lot of work and energy and time. But once the tunnel is created, the rest of us can then pour through, once we’ve learned the trick.

You don’t have to be lazy to love shortcuts. But it doesn’t hurt. Today on the show: the latest installment of The Freakonomics Radio Book Club, in which we interview an author and hear key excerpts from their book. Like this:

DU SAUTOY: A shortcut is not a fast way to finish your journey, but rather a stepping-stone to beginning a new one. It’s a pathway cleared, a tunnel dug, a bridge constructed to allow others to quickly reach the frontiers of knowledge so they can make their own journeys into the darkness. Equipped with the tools that Gauss and his fellow mathematicians through the ages have honed, stretch out your arms for the next great conquest.

Today, Marcus Du Sautoy will teach us how shortcuts can be applied to practically anything — music, psychotherapy, even politics.

DU SAUTOY: This is one of the extraordinary powers of mathematics.

Math itself is a shortcut. Language is a shortcut. This episode is a shortcut.

*      *      *

Marcus du Sautoy isn’t just a mathematician. He also writes plays, and he plays music, and he preaches the virtues of math on television and elsewhere in the U.K. At Oxford, he’s the Simonyi professor for the public understanding of science. It’s a position first held by the evolutionary biologist Richard Dawkins. Du Sautoy took over in 2008.

DU SAUTOY: Science is having such a big impact on society. It’s a bit like a superpower, so it needs its ambassadors to try and create bridges between those who might fear the science. So that’s a slightly more outward-looking role.

Stephen DUBNER: You describe the job of a mathematician as “discovering the ways of thinking smarter.” I’m curious how successful you think mathematicians have been in that realm.

DU SAUTOY: Incredibly successful. What a mathematician is very good at is looking at many examples of something happening and realizing there’s a commonality to them, and that even though the numbers might be very different, what’s making them tick is the same.

Even if you know nothing about math, you know that math underlies nearly everything. Medicine, engineering, computing — even music and art and probably love. As we learned from young Carl Friedrich Gauss, math is a language brimming with symmetry — and with patterns. One plus 100 equals the same as two plus 99, and so on. Being so pattern-based, it also lends itself to shortcuts. Think about how we can draw conclusions from a massive set of data while only having to touch a few pieces of that data.

DU SAUTOY: This used to be an advert in the U.K. for a particular type of cat food. And it said “Eight out of 10 cats prefer this particular type of cat food.” And we had a cat and I never remember anybody coming round and asking our cat what cat food it likes.

Back when that ad was being run, there were about 7 million cats in the U.K. So how many of those 7 million cats — or, more likely, how many of the humans who lived with cats — how many had to be surveyed for the company to make this claim?

DU SAUTOY: It’s an incredibly small number. It’s in the order of 250 cats will give you confidence level of saying, “Well, 19 out of 20 times, that will get me 5 percent away from the true value.” I was staggered when I first learned that at university, but once I’d learned this shortcut, that actually you can get away with quite a small sample set to give you this huge insight onto 7 million cats, that’s an amazing shortcut. But it needs to be used very carefully, because one of the problems is that you can be biased in the way you’re picking your sample.

DUBNER: So assuming that the cats were randomly selected, can you explain how that massive statistical reduction is possible, to survey just 250 cats that will actually represent the 7 million?

DU SAUTOY: Maybe I’ll change the problem a little bit, because it might illustrate how you can get access to a lot of things with a small number. So, here’s another problem: how many people do you need in a room for there to be more than likely to have two people with the same birthday? Now, to start with, you might say, “Well, hold on. That must be a huge number.” What you don’t appreciate is, “Oh I know, actually, I’m asking about all the ways of pairing people up.” So the number of different pairs that I can create with 23 people actually covers over half the possible birthdays in the year.

DUBNER: So I only need to have 23 people in a room and there’s a 50 percent chance that at least two of them will have the same birthday, is that right?

DU SAUTOY: That’s correct. Which is, I think, very counterintuitive.

DUBNER: So what are the odds that if there are only two people in the room, they have the same birthday?

DU SAUTOY: There’s a one-in-365 chance. And that’s incredibly small.

DUBNER: So, Marcus, my birthday is August 26th. What’s yours?

DU SAUTOY: My birthday is also August 26th. Now are you, are you — so yeah, that’s an amazing —.

DUBNER: You don’t sound as excited as I am.

DU SAUTOY: No, I actually, I really am, because that’s really unlikely. Is that really true?

DUBNER: It is really true. I mean, I’m cheating a little bit because I read your book in which you divulge that your birthday is August 26th, but mine really is August 26th. I’m not lying.

DU SAUTOY: We should become a double act and go do this probability problem, and we’ll always be guaranteed to have two in the room.

DUBNER: So, Marcus you’ve written this book in praise essentially of shortcuts, at least the mathematical variety, but others as well. Now, as you know well, for many people, the very word “shortcut” is a pejorative. But you are making the argument that they are or can be at least a strong positive. So let’s just start there, explain the philosophy behind that argument.

DU SAUTOY: I wanted to make very clear that this wasn’t about cutting corners. I think that when you hear “shortcut,” that pejorative side is about how you’re not doing it properly. But no, these are shortcuts which get you to your goal, but get you there without having to do a lot of laborious, boring, hard work. I talk in the book about Aristotle’s idea of two sorts of different work — praxis, work that you love doing for its own sake, so you don’t want a shortcut for that. And poiesis, which is work to attain a goal. And in general, that’s the one you want to get to quickly.

DUBNER: So one thing that I found so interesting is, you make the argument that mathematics is essentially a language or science of shortcuts to some degree, but then you persuade me, at least, that many other things are systems of shortcuts, including language itself. Which, the minute you say it, makes total sense to me, but I’d never thought of it that way. So can you talk about what you mean by that? One example you use in the book is the word “chair.”

DU SAUTOY: Yeah, absolutely. Because you could have had a new word for every single different type of chair there is. I’m sitting at the moment on a swivel chair on wheels, but it could be a little stool, which we also would categorize, perhaps as a chair. Actually Wittgenstein talks already about this power of language, to not have to come up with a new word for absolutely everything, that would be incredibly inefficient. I experienced this when I was doing my doctorate. I had this very complex structure that I kept on talking about. And I was getting bogged down. I just couldn’t see the wood for the trees. And my Ph.D. supervisor said “Give it a name.” I didn’t understand that I was allowed to do that but once he gave me permission, I realized, “Oh my gosh, now I don’t have to get bogged down in all the words describing this and it’s, it’s much easier to go forward.”

DUBNER: What was the name?

DU SAUTOY: I actually called it “a friendly ghost.” It was a sort of ghost-like structure hiding behind the things I was interested in. And it was very helpful, very powerful. In academia, people don’t really like funky names, and I sent this off to a French journal, and they wrote back and said, “We love your mathematics, but we can’t let you call this thing a friendly ghost.” The French are very controlling of language, perhaps more than the English. But I stuck to my guns and it’s now in the literature.

DUBNER: There have been Nobel and many other prizes won by people like Danny Kahneman for explaining how mental shortcuts or heuristics lead to poor decision-making, and poor outcomes. So is your argument, to some degree, a corrective?

DU SAUTOY: Yes! In fact, that’s really why I called the book Thinking Better. Because I was sort of bouncing off that book of Kahneman’s, Thinking, Fast And Slow. He explains in that book, thinking fast, our intuitive first go at solving a problem, often leads us to the wrong conclusion. And that’s generally relying on heuristics, the idea of what’s happening locally around me actually is what’s happening globally. Very often that’s not true. So he gets lots of examples where fast thinking leads you astray, and he says, “No, no, we’ve got this system two, this slow thinking, which is much more analytical.” So that slow thinking is what Kahneman says, “Yeah, that’s the way we need to get solutions.” Now my thesis is that, “Hey, that doesn’t have to be slow.” I’m saying maths is full of ways that you can think carefully, but it doesn’t have to be slow.

Here’s another passage from Du Sautoy’s Thinking Better, one of my absolute favorites, about how shortcuts abound even where people do not.

DU SAUTOY: It is interesting to note that humans weren’t the first to exploit the power of mathematics to assess the best way to tackle a challenge. Nature has been using mathematical shortcuts to solve problems long before we arrived. Many of the laws of physics are based on Nature always finding a shortcut. Nature is lazy, like humans, and wants to find the lowest-energy solutions. It’s extremely good at sniffing out shortcuts. Invariably it has a mathematical rationale to it. And often the discoveries of shortcuts by humans materialize out of our observations of how Nature solves a problem.

One example of a natural shortcut Du Sautoy writes about is the honeybee.

DU SAUTOY: When it’s making its hive, it chooses to make these hexagonal cells. And you ask, “Well, why didn’t it do it with squares?” It turns out that the amount of wax you need to use to make cells, which are hexagonal, is smaller than the amount of wax to make, say, squares or triangles. And wax is very expensive for a bee to make.

How did the honeybee discover this shortcut?

DU SAUTOY: It didn’t do the sort of calculus that we’ve now used to show that that’s the smartest way. It used a kind of evolutionary process. The bees that made the hexagons were the ones that survived better. Take light. Light is a great example of nature finding the shortcut, because if you think about light passing through, when you’re in a swimming pool, and you see your legs slightly foreshortened, or you put a ruler in, you see it bending, that’s because light finds the fastest way out of the medium that it’s slowest in, the water, and spends more time in the air. You might say, “Well, that’s extraordinary. So how does it know?”And that is actually exploiting quantum physics to try out all of the possibilities. And then it collapses into the shortcut solution, the fastest way out to get to the destination.

DUBNER: Can you talk for a moment about the efficiency of shortcuts used by single-celled slime molds, and how they seem to beat humans?

DU SAUTOY: Yeah, this is extraordinary, the way that certain natural things seem to be kind of mathematicians. This slime mold, which is tasked with trying to find its way through a maze. One example is they laid out stations where the Tokyo Underground went. And they were interested, what’s the best way to connect up all of these train stations? And to represent the train stations, they put this little bit of food, which slime mold likes. And sure enough, after a while it had settled on a wonderful kind of network through all of the stations in Tokyo. And it had replicated the most efficient way to connect up these stations. Same with ant colonies. Terribly clever at eventually finding these very efficient paths to food and things.

DUBNER: But Marcus, what’s the lesson we should take away from the slime-mold story? Because one could imagine a conclusion whereby humans are stupider than slime mold, which I think we would agree on most dimensions is probably not true. Or maybe there’s a conclusion whereby humans have a hard time staying out of our own way.

DU SAUTOY: Well, first of all, I think that it’s clever to tap into understanding when nature has solved a problem before you and take advantage of that. And in a lot of design cases, we’re understanding this. The skin of a shark is incredibly well made. We don’t have to reinvent the wheel. Let’s understand how that’s worked.

Shark skin, we should say, is a rough surface that’s patterned with grooves and ridges that make it hard for microorganisms to attach themselves. It’s being used as a model to create similar surfaces in hospitals and elsewhere to cut down on microbial infections.

DU SAUTOY: But there is also a little word of caution you need to use when looking at the slime mold. Because there are cases where you can set out where the stations are, and it won’t come up with the most efficient way to do it. What happens is that the slime mold sort of gets stuck in what we call a local minimum. I think that’s where we humans are sometimes much better at looking at the overall picture and saying, “Oh no, you’ve got stuck.” And that’s the amazing thing with machine learning and A.I., because there’s a similar principle at work there. But very often the A.I. works by moving around a landscape of possibility, and sometimes it can get stuck in these local minimums that it thinks is the most efficient solution. And one of our skills is actually forcing it to sometimes take a step out of where things are getting worse to actually explore the terrain, to see whether there might be even more efficient places to settle on.

DUBNER: Many of the problems we try to solve today as a society are inherently difficult to solve — that’s why they still exist. So, one thing I’ve noticed is that when clever people do solve difficult problems, they often redefine the problem they’re trying to solve — what I mean is, they focus on the root cause of the problem, which many others have perhaps overlooked, since it’s easier to fix the symptoms of that root cause. And solving the symptoms doesn’t necessarily even address the root cause. But if you can redefine the problem and address the root cause, at least you may have some options. With that in mind, I’m curious, do you see, Marcus, in the world that there are categories of problems for which there are viable shortcuts and those for which there are simply not?

DU SAUTOY: Yes. And I would say this is one of the extraordinary powers of mathematics, to look in on itself and understand its limitations. So we have this extraordinary case of problems that we don’t know yet that they don’t have shortcuts, but we believe that by their very nature, the problem cannot avoid having to go through all the hard work of trying all the different possibilities. And an example of that is something that people may have heard of, the traveling salesman problem. You’ve got a network of cities you’ve got to visit, lots of roads joining those cities, and you’ve got to plot the path which is the shortest distance such you visit all of the cities. Now you might say that sounds perfect for a mathematical algorithm that just looks at the network and then spits out very quickly the shortest path. But we actually think this is a problem where there’s no way you can avoid just having to try all the different paths that are possible.

DUBNER: And if someone happens to hear this and is able to solve it, they should know there’s a sizable sum of money attached to solving this problem, correct?

DU SAUTOY: Exactly.

DUBNER: I guess it makes me think there are three categories of people, at the very least. Those who successfully create shortcuts through a lot of effort. Those who may use a lot of effort, but fail to create shortcuts. And then the rest of us who just let people like you, mathematicians, work really hard to come up with shortcuts that we can then exploit. So is there any hope that we, the laity, can actually come up with our own shortcuts? Or do we really need to leave that to professionals like you?

DU SAUTOY: No, I think there is an opportunity for people to come up with shortcuts. You need to be immersed in this world of different ways of thinking, trying to put the beginning and the end of the problem together or lifting yourself out of the landscape, looking from above to see what’s going on. My book is trying to move people from this sort of category two person to category three that says, “Okay, but somebody has done all of this work. I’m just going to sit back and take advantage of it.” But I’m hoping that a little bit of the category three, we can push over to category one.

DUBNER: What share of category three do you think you could realistically get over to category one? Half of 1 percent?

DU SAUTOY: I was going to say actually 10 percent.

DUBNER: Oh, you’re so optimistic, I love that.

DU SAUTOY: I am an optimist and I think once people start to see the power of thinking in this mathematical way, often it doesn’t require you knowing a lot of technical mathematics. It just requires you taking a step backwards, understanding that there are some clever techniques.

*      *      *

The Oxford mathematician Marcus du Sautoy is a bit of a Renaissance man.

DU SAUTOY: I always really enjoyed things like music, theater, the arts.

But mathematics is plainly his true love.

DU SAUTOY: Yeah, I kind of fell in love with mathematics and decided I wanted to be a mathematician when I was 12 or 13.

I asked Du Sautoy what the world might look like if the mathematicians were in charge.

DU SAUTOY: I do think it would probably be a much better place. We do have the tools to just look at problems and to not get stuck in the local mess of things. Would there be war, for example, if mathematicians were running the show? I don’t think so. Because I think that it comes down to understanding that there’s more to gain if we collaborate rather than if we compete with each other.

It’s worth pointing out that Napoleon was one leader who was quite inclined toward mathematics. He is often credited with having discovered a geometric theorem related to equilateral triangles. But the math-loving Napoleon also loved to wage war, so Marcus du Sautoy’s vision of a world without war is perhaps imperfect. Still, it is tempting to think of a system in which more of our leaders were at least conversant with math. Why? Because mathematics is an exceedingly logical language. It’s also a powerful tool for solving problems, or at least untangling problems into solvable components. In the U.S. and elsewhere, a great many of our politicians are trained not in math but in law. Is there logic involved in the study of law? Sure, of a sort. Is legal thinking concerned with problem-solving? Sometimes — although you may have to squint to see it. So if our leadership were more mathematically inclined, what might be the upsides?

DU SAUTOY: Politicians by their nature are having to do several steps down the line to see the impact of things. And often I feel there’s a real inability to understand just very basic logic to see what the impact of certain policy changes would be. One of my beliefs is we should make every politician do a critical-thinking course before they’re let loose on running countries.

DUBNER: So I couldn’t agree with you more that politics — and policymaking, even — are areas where that sort of logical thinking is underused or underappreciated, and I understand why — the incentives sometimes conspire against it. That said, I would argue that the median politician has more real-world leverage than the median mathematician. So do you think you’re losing by winning with math?

DU SAUTOY: I think that you’ve put your finger on something really important here, which is, yeah, sometimes the fact that a mathematician lives in this far-too abstract world can mean that they perhaps are not engaging with the social implications of their thought process. I’m not saying that mathematicians or scientists should be running the world. There are decisions that are less scientific in nature. For example, making decisions about legalizing drugs. The evidence is really very strong from a data perspective to legalize drugs. However, I understand that there are other political issues that need to be taken into account in making that policy decision. I think we’ve seen that very clearly with the pandemic that it’s a team effort, and politicians, certainly in the U.K., have been talking a lot about, “We’re following what the scientists do.” But that’s not really true. They certainly are taking into account the scientific insights. But ultimately, this is not just about the science; there’s an economic realm to put this in, there’s a social realm to put this in. And that is what the top politicians are tasked with, is taking all of these components, and making a policy decision.

DUBNER: When I think about the work that you do in mathematics and the many people who’ve come before you for centuries, creating a language of shortcuts, it’s basically resulted in the fact that I don’t personally need to know any math at all to do quite well in life, and to use all the math that you and your predecessors have come up with. And therefore there’s almost no incentive for me to engage in your dark art of mathematics. So do you see it as a case where shortcutting has gone too far, where it takes a realm like mathematics and removes it from the arsenal of too many people?

DU SAUTOY: We’ve been too successful and done ourselves out of being part of society? Well, I think first of all, it’s actually really useful to know the power of mathematics as a tool. For example, one of the shortcuts I talk about is calculus. Now I’m not expecting everyone to suddenly do a first course in calculus because that’s quite technical, but—.

DUBNER: Yeah, and don’t worry, we’re not going to, no matter how much you might expect.

DU SAUTOY: Oh shucks, okay. But knowing what it can do, that if you’re running a business, that calculus can really hone in on that sweet spot between supply and demand, price, that is really useful. So, we need some mathematicians in our team who’ve got these tools. Otherwise, you’re going to be sunk just trying things out and making a lot of mistakes. The engineer is nervous of shortcuts because, yeah, they don’t want to cut corners, they need that building to stand, the bridge to cross the river. But again, that’s what calculus does. It sort of allows you to experiment beforehand, find the low-energy solution to creating your building, and then you build it.

DUBNER: Given all the technology and the shortcuts that we’ve been afforded by being born in this era, how well do you think we’re using that to our advantage to move on to the next useful thing? I think of the famous quote from Peter Thiel, “We wanted flying cars. Instead, we got 140 characters.” So that’s a little bit reductive to say that all of technology, all of mathematical shortcuts, have been reduced to just Twitter alone. But you see his point. Do you think that our rate of progress using these shortcuts is sufficient or are we kind of coasting?

DU SAUTOY: I’ve not got such a negative view of the progress that we’re making. We’re in a very interesting stage where artificial intelligence may be taking a lot of work away from us. And obviously there’s a lot of fear about that. But I think this is a potentially very positive moment where, again, we can get it doing the work we’re really not interested in doing, so we free ourselves up for the work that we do want to do. I type in a search term into Google and magically, there’s the website. I could have wasted hours and hours trying to find that website. I mean, I think medicine is one of the most extraordinary. If you look back 100 years, the progress we’ve made — and you could call these some sort of shortcuts, the ability to build up an immune system due to a vaccine, that’s an incredible way of tricking the body into a shortcut. So you’re ready for when the nasty virus comes. So I think we’re actually doing very well in applying our science. I think we’re in a really funny time where in some sense, people are understanding that science is having a massive impact on their lives, the technology they enjoy, the fact that they can access things so extraordinarily quickly on the internet, the vaccines that we’ve been able to produce in Oxford and beyond. Yet there is a kind of strange sense that, “Are we going backwards?”

The fear of new technologies is not a novel concept — in fact, it’s a feature of just about every new technology ever invented. Change can be unsettling, especially when the change affects our livelihoods, as is the case now with machine learning, artificial intelligence, and so on. We wrestled with this idea in a recent episode, called “How to Stop Worrying and Love the Robot Apocalypse,” episode No. 461. Still, it is a natural fear, that we humans could shortcut ourselves right into irrelevancy. Marcus du Sautoy, however, is not concerned; as he noted, he’s more of an optimist. Here he is again, reading from his new book Thinking Better:

DU SAUTOY: Slackers, take note. I think laziness is our saving grace, what will protect us against the onslaught of the machine. When a computer is faced with a problem, we know what it will say: “Well, I’ve got these computational tools, so I can just bash my way through the problem.” But I often look at a problem and think, “This is just getting too complicated. Let me try to step back and figure out a shortcut.” Because a computer doesn’t get tired and it’s not going to be lazy, maybe it will miss things that our laziness takes us to. Because we don’t have the ability to plow right through problems the way a computer would, we’re forced to find clever ways to handle them.

DUBNER: So, reading your book, it struck me that the greatest shortcut ever invented is maybe the internet because it operates along a couple of really important dimensions: speed and reach. So tell me whether you agree or disagree, and what I really want to get to is what you see as the upsides and downsides of this massive shortcut we’ve all been handed.

DU SAUTOY: I think for most people, it is being used as a shortcut, and perhaps this is a good example of one of the shortcuts I talk about in the book, which is the wisdom of the crowds. We’re seeing this being used very powerfully in science at the moment with citizen-science projects. For example, in Oxford, we have this wonderful project called Galaxy Zoo, which is looking at all of the images that telescopes have taken of the stars out there. We don’t have enough Ph.D.’s and postdocs to be able to look at all of these images. So very cleverly, they created this citizen science project, where even an amateur can classify galaxies as spiral or elliptical. And I would say that the most successful examples of the internet working, things like Wikipedia, are taking advantage of the fact that we’ve got multiple people building this extraordinary space.

DUBNER: Let me ask you about a very practical and policy-related category of shortcut. And that’s the one practiced by firms like Uber and Airbnb, who build businesses around models that are kind of illegal or quasi-legal at best. And there’s this old saying: “It’s easier to ask forgiveness than to get permission.” So what’s your assessment of the shortcuts used by firms like that?

DU SAUTOY: It was in some sense a shortcut that I recognize in mathematics as well, because sometimes the breakthroughs happen from breaking the rules, and you can get very stuck inside a particular system, a way of thinking, and in order to find the new path, the secret passage, sometimes you have to break the rules. And the one I rather like is when we came up with a new number, the square root of minus-one an imaginary number. At first sight, you say, well, there isn’t a number which when you square, it equals minus one. And for centuries, people just wouldn’t admit this number into the canon of mathematics. But we broke the rule and said, “Why don’t we try?” Let’s just try something, see if it works, and then maybe you can change the system.

DUBNER: What were the long-term benefits, then, of using imaginary numbers?

DU SAUTOY: One of the most extraordinary is that when we were trying to land planes using radar, it turned out the computers just weren’t fast enough to do the calculations if we just used normal numbers. But when we then exploited the power of imaginary numbers — which seemed sort of illegal because, you know, where are these numbers? — actually it allowed us to do the calculations much more quickly. And so we understood where the planes were, we were able to land them, whilst if you didn’t use imaginary numbers, the planes would have crashed.

Math doesn’t make moral judgments; people do. Some people may find that a firm like Uber creates so much value for consumers that it’s perfectly fine that they skirted the law to do so. Others may feel that laws aren’t meant to be broken, and that the slope becomes very slippery when you start weighing the value of a law against the utility of breaking it. And let’s not forget: there are hard problems where even breaking a law won’t help. Here’s one last excerpt from Thinking Better:

DU SAUTOY: Sometimes it’s just as important to know when there are no shortcuts to the problem you’re trying to crack. Knowing that the long way around is the only way to your destination will prevent you from wasting time in the hope of finding the shortcut. And if you are going to do all the work, then it’s worth knowing that you are not wasting your time.

Most problems we try to solve likely lie somewhere in the middle, where a shortcut may help but the attainment of said shortcut is itself a lot of work.

DU SAUTOY: I talked to a cellist, an international cellist that I know. I’m learning the cello at the moment, and I was desperate for some shortcuts to be able to play the Bach Suites, which is my ultimate goal. Now, she said there are some shortcuts, but on the other hand, to physically change your body to be able to play these pieces, that requires hours and hours of practice to get your muscle memory going. Yet she did identify shortcuts that she uses, which is seeing patterns on the page, which she knows how to play already because often they are scales or arpeggios, which is why we get a musician to practice their scales and arpeggios, because those will be shortcuts when you suddenly see this in a Bach passage. It’s like reading. You don’t have to read every letter. I see a word. So there are shortcuts there, even for somebody like a musician.

DUBNER: So your book includes a brief section on whether shortcuts can be useful in psychotherapy. And we should say that your wife is a psychologist. And the focus of what you write is on cognitive behavioral therapy, which many, many, many in the field of psychology have hailed as one of the few true breakthroughs in psychology, as both pragmatic and short-term. So you seemed to become very excited about the possibility of shortcuts in this realm, but then you seemed to become a little bit disillusioned with the realization that the human psyche is complex and dynamic enough to maybe reject that type of shortcutting that might work in other realms. So I’m curious where you land on that?

DU SAUTOY: Yes, I think that depends on the problem you’re facing. This is a message that applies to many of the shortcuts. So I talked to Susie Orbach, who’s a psychologist, and she had this nice way of describing some of the problems that people are facing, that — you know, it’s hard to learn a language. It’s even harder to unlearn a language. My first language is English, imagine trying to unlearn English. And so that’s the point, some people when they’re coming to therapy have got such entrenched ways of thinking from experiences they’ve had in childhood that you’re fundamentally having to change the way the brain works. So there are some cases where you cannot shortcut the time on the couch that you are going to need to reset the brain. However, there are some modalities where being aware of what your thought process is, is already enough to short-circuit the algorithm which was always sending you into depression. You’re sort of stuck inside the system of the way you’re thinking. What C.B.T. often helps you to do is to take a step up and look at the way that thought process is happening and understand the trigger which always sends you down into depression.

DUBNER: Let me ask you one last question. I wonder if you can help me come up with some shortcuts for how we make Freakonomics Radio. Because I love it, but it’s very labor-intensive. So it starts with an idea. And then there’s research. The producers will do a lot of research to find the appropriate guests and stories. Then the producers will produce preps for the interviews, and I’ll read those preps and I’ll do my own research. We do the interviews, we transcribe them. Then we take all that, we draft a script, we edit and rewrite and rewrite the script. We record it, we re-record, we fact-check, we score it with music. And even though there are, I’m sure, thousands of shortcuts built into this process — we use computers for everything, and that’s wonderful — the labor intensivity doesn’t seem to change that much. And it’s not like the quality is necessarily improving, either. A current episode isn’t ten times better than an episode we made five years ago with less technology. So I’m wondering if we are broken or if you can help somehow with a wonderful shortcut.

DU SAUTOY: I don’t think I can, and I think you should celebrate the fact that the quality of your program is because of it needing all of that hard work. And as you say, there are already shortcuts that you’re using as part of the process. You’re not reinventing the wheel every time. And I think you wouldn’t do this work if you didn’t enjoy actually the satisfaction of climbing that mountain each week, each month.

DUBNER: I am so satisfied by that answer, in part because you make me feel like the effort that I enjoy — and I do enjoy the effort — is the route to where I want to get. And I guess that’s what it’s really all about, right? You have to assess every goal and every possible route and ask yourself, what am I optimizing for here? I’m not optimizing for the 30-second microwave meal, at least not every day.

DU SAUTOY: Exactly. When you’re going on holiday, I don’t want to shortcut the holiday because it’s about spending time. The point is, I don’t want you to use shortcuts for everything and spoil something you enjoy doing. I’m dedicating my life to mathematics, the art of the shortcut, but I actually enjoy spending the time and having that aha moment when I find a little tunnel, which gets me through to the other side. And if it was all too easy, I don’t think I would enjoy it.

That was Marcus du Sautoy; his new book is Thinking Better: The Art of the Shortcut in Math and Life. Hope you enjoyed this episode of the Freakonomics Radio Book Club.

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Freakonomics Radio is produced by Stitcher and Dubner Productions. This episode was produced by Mary Diduch with help from Zack Lapinski and Jeremy Johnston. Our staff also includes Alison CraiglowGreg Rippin, Ryan Kelley, Jasmin Klinger, Eleanor Osborne, Emma Tyrrell, Lyric Bowditch, and Jacob Clemente. Our theme song is “Mr. Fortune,” by the Hitchhikers; the rest of the music this week was composed by Luis Guerra. You can follow Freakonomics Radio on Apple PodcastsSpotifyStitcher, or wherever you get your podcasts.