Intro. [Recording date: June 9, 2025.]
Russ Roberts: Right now is June ninth, 2025, and my visitor is mathematician and writer Paulina Rowinska. Her guide, which is our subject for at present, is Mapmatics: A Mathematician’s Information to Navigating the World. Paulina, welcome to EconTalk.
Paulina Rowinska: Hi there, comfortable to be right here.
Russ Roberts: Now, I ought to begin by saying that I occur to be a map lover. I at all times have liked maps. I really like totally different sorts of maps. I am not a collector, however once I had extra room in my life bodily, I’d save a number of maps that I simply occurred to like as a result of I like the thought of getting them. However I wish to let listeners know that this isn’t going to be a dialogue about simply the type of map that involves thoughts whenever you hear that phrase. There’s a bit about that, and we’ll discuss it. However, what you imply, Paulina, by a map, and what the remainder of us imply by a map, shouldn’t be the identical. So it is a very wide-ranging guide throughout science, enterprise, on a regular basis life, drugs, and it is actually a few map as a illustration of actuality and never what we usually consider as, say, a factor that you just fold up that you just used to maintain in your automotive that can assist you discover out the place to go. Is {that a} truthful abstract?
Paulina Rowinska: Sure. There’s positively extra to maps than I assumed earlier than writing the guide. So, there are very various kinds of maps, and the way I take into consideration the map, and the way additionally many mathematicians take into consideration the map, is illustration of actuality. So, type of the mannequin of actuality. And, it may be the map–although I do not know the way many individuals nonetheless have the paper, folded maps. I believe that is much less and fewer in style method to make use of maps. Totally on our telephones, which can be a sort of a map. However then, now we have, like, tube maps. I used to stay in London, and with out the tube map I’d have been misplaced. So, that is additionally a map, although it would not appear to be one.
Russ Roberts: Yeah. We’ll discuss that. And, for these of you not from London, a tube is a subway in London.
Russ Roberts: I truly wish to begin with the old style type of map, both the sort you might need had on the wall whenever you had been in school–and you write one thing very stunning. You say, “Every time we open a map, we maintain in our arms the outcomes of centuries of mathematical analysis.” And, in fact, you do not simply imply the type of maps that we maintain in our arms, however the maps that we maintain in our heads. It is actually a stunning method to consider it.
However, I wish to begin with the roadmap or a map on the wall. I do not suppose most individuals suppose that has something to do with arithmetic. They suppose: ‘Properly, yeah, we take an image possibly from up above after which we type of crop it.’ And, in fact, that’s not the historical past of map making. Even shut. So, discuss, as an introduction, among the ways in which arithmetic had for use to attract the maps that started many, many, many centuries in the past; after which in fashionable occasions even.
Paulina Rowinska: Yeah. So, mapping is difficult. Why? As a result of the earth shouldn’t be flat, and maps are flat basically. And, it is a essential downside.
So, in arithmetic, there is a very attention-grabbing theorem that reveals that once we attempt to take one thing just like the earth–so a sphere–onto a flat piece of paper, there shall be issues. One thing shall be not working, one thing shall be distorted, be it distances or angles, and we can not get round it. So, that is the fantastic thing about arithmetic. The concept says so. We simply cannot get round it.
And, we nonetheless are making maps, and they’re nonetheless working. However, by understanding that one thing must be distorted, we may be extra cautious once we make maps and once we have a look at maps.
So, for instance, most of us are conversant in Mercator Projection. So that is, I believe, nonetheless the most well-liked world map. And, properly, what’s attribute about it’s the nearer to the poles we get, the larger the issues look–the greater the continents look. So, now we have this large Greenland, after which now we have, like, tiny Africa. And, I believe most of us are conscious that this possibly shouldn’t be precisely how it’s on this planet, actually. However, truthfully, I personally, although I do know this, each time I have a look at it, I do not understand how enormous the distinction is. It actually skews our actuality.
And that is due to the arithmetic. That is due to the best way we mission the globe onto a chunk of paper.
So, I like to consider projections. So, that type of translating the 3-D, three-dimensional globe onto a two-dimensional map. I wish to suppose: Think about you could have a clear globe with simply the continents, or no matter you wish to translate, drawn on them, and you set a lightweight bulb within the globe. And you’ve got a chunk of paper, which is able to turn out to be your map. And, the shades, the shadows of the continents, shall be mirrored on this piece of paper.
However the query is, the place do you set this piece of paper? Do you wrap it someway across the map, like a cylinder around–
Russ Roberts: Across the globe. Yeah.
Paulina Rowinska: Or do you set it simply subsequent to it? Do you set it additional away? And, this actually adjustments dramatically what the map appears to be like like. And, there’s actually stunning arithmetic behind it. And, we are able to write down equations or think about mathematically what this map will appear to be and what shall be distorted, what shall be preserved. And, once more, what I wish to stress is I am not saying that one among these methods is healthier than the opposite.
I do know individuals hate Mercator map as a result of they declare, like, racist. As a result of what it reveals is principally the international locations which can be within the north–they’re largely richer and nonetheless extra powerful–unfortunately‚ they’re proven as greater. And, the thought is that this map reinforces it. And, it would possibly if we do not perceive that it is the arithmetic behind it. As a result of this map, Mercator’s map, it is also very helpful. It is made for navigation. It is made for navigation. So, if we use it for navigation–well, nonetheless the maps we use for navigation in on a regular basis life–so, like, Google Maps–are primarily based on one thing just like this map.
So, once more, it isn’t that a few of these projections are higher than others. It is simply we have to perceive what they distort and what they maintain, and suppose: Why are we making the map? What can we need to get out of this map? And, select the projection accordingly.
Russ Roberts: In case you’re planning a visit throughout Greenland, you would possibly wish to use a special projection when it comes to how a lot meals you’re taking or what number of adjustments of clothes, relying on what technique of transport you employ.
However I believe that is a extremely stunning picture, this picture of the globe casting a shadow.
So, for those who wrap the piece of paper like a tube, like a cylinder across the equator, and you consider tracing the shadow that comes from the sunshine bulb contained in the globe, properly, across the equator you are going to be tracing principally the shapes of the international locations. However, as you progress away from the equator, you’ll be able to see very clearly that the shadow goes to be distorted, similar to one’s personal shadow in direction of sundown begins to elongate. It is a unbelievable picture.
How huge is Greenland, truly? Do you could have a comparability that will assist us perceive that?
Paulina Rowinska: Yeah. So, Greenland continues to be very huge. It is the most important island on this planet. Okay. Do not get me unsuitable, it is nonetheless large. However, whereas it appears to be like like the dimensions of Africa on Mercator’s map–I do not know the quantity. I wrote it within the guide, I believe it is like 14 occasions smaller. You most likely know higher.
Russ Roberts: No, I do not. It is a homework project for the listeners.
Paulina Rowinska: However, the distinction is simply actually enormous.
Russ Roberts: Yeah. That is very cool.
Russ Roberts: And, in fact, I do not suppose you wrote about this, however I did not take into consideration this till we simply began speaking. While you construct a globe, it’s important to print the shapes of the international locations and continents and oceans on a sphere. It is a sphere–even although it isn’t a sphere, however the globe is a sphere; it is an approximation. That is not really easy both, is it? Or did they do it regionally in small chunks in order that it isn’t distorted?
Paulina Rowinska: Yeah. So, truthfully, I am not precisely positive the way it’s performed now. However, truly, Mercator started–so the one who created Mercator’s map, he began like that. So, he began from the alternative downside. He was making globes, and the way they had been making it, they had been making these wedges, printing–printing, drawing these wedges and pasting them onto a globe. So, it is type of the alternative downside.
And I believe this impressed him to suppose in the wrong way.
And, once more, that is all math, because–it’s one among my favourite theorems of arithmetic that we can not actually translate simply between dimensions, between totally different shapes with totally different curvatures. That is the phrase that we should always most likely be utilizing. And you recognize if we–if it nonetheless sounds too summary, making maps–like, truthfully, have you ever ever tried to place a Band-Assist in your knee? It simply would not stick. Have you ever ever seen a lollipop? It isn’t–like, the wrapping has all these wrinkles, and that is the very same purpose. So, when you see it, you see it all over the place.
Russ Roberts: I wish to discuss coastlines. That is one thing I do know a bit bit about earlier than I learn your guide, however you could have a number of wealthy description of it. This isn’t, I believe, intuitive to most individuals. Neither is the map downside, by the best way. I believe for those who ask individuals about projections versus a globe, a three-dimension all the way down to two-dimension, they’ve heard of it, however they have not actually thought of it being a giant downside. They simply suppose, ‘Yeah, yeah, it is an approximation.’ The concept that it is a mathematical reality shouldn’t be intuitive to non-mathematicians.
And, equally, what I will say subsequent is admittedly unintuitive: and that’s that the shoreline of England is so long as you need it to be, versus a quantity. I imply, absolutely the shoreline of England, the perimeter of, for example, the United Kingdom–more precisely than England–the perimeter of the UK, it is a quantity, no matter it’s. You simply exit, and also you might need to take a bicycle wheel with a stick and also you’d roll round or no matter, otherwise you’d take a ruler. Clearly, we all know the way to measure that, proper? And but that is not a well-defined downside. And that’s stunning, I believe, whenever you first hear it. And, at first, I believe you suppose, oh, that is simply clearly a trick. However, it isn’t a trick.
So, I assumed–you use a good looking instance, and I believe it is the identical point–about measuring a desk with a set stick of a sure dimension. That helped me see it–in a technique of seeing it. So, use no matter you need. Discuss: why is it onerous to give you an correct measure of a rustic’s shoreline?
Paulina Rowinska: So, simply, to start with, you mentioned it isn’t intuitive for non-mathematicians. It is also not intuitive for mathematicians.
So, I believe that is the factor about arithmetic. Generally the outcomes are usually not intuitive. And, as I mentioned, I typically am, like, ‘What do you imply we can not measure a shoreline?’ It simply feels unsuitable. And so, the purpose we’ve–yes. After we measure a desk, once we measure a straight small line, we all know the way to do it. Even when we do not have a ruler, if now we have a stick that we all know the size of, we are able to simply put it one after the opposite and approximate–it is between 5 sticks and 6 sticks–and we all know kind of what it’s. And, for a lot of shapes we are able to do this. And, that is type of how we do it with coastlines. Clearly, no person takes nowadays, like, a stick and goes round England, however the concept is analogous.
The issue is that if the stick could be very lengthy, we’re lacking lots of the irregularities. There are some–especially the gorgeous components of the shoreline, with all of the cliffs and all of the inlets and bays and all these peninsulas–we are lacking them. So, we take a shorter stick; and we’re capturing far more, however we’re nonetheless lacking some. And, the shorter the stick is, the extra we’re capturing. And, it is the identical factor as–no. Relying on what scale of the map we’re using–if we attempt to measure the coast of the UK utilizing only a world map, there are usually not many particulars there. It isn’t going to be very lengthy. However, if we have a look at a really detailed map, large-scale map of the UK, now we have far more particulars. So, we’re capturing them, and the quantity grows and grows and grows. So, for a traditional, common form, it ought to converge to the precise quantity. There may be one. Right here, there’s none.
Now, does it imply that every one the evaluating totally different international locations and their borders or coastlines is meaningless? No, it isn’t. As a result of if we agree on the best way and the dimensions of the map we’re utilizing, we are able to make significant comparisons.
However sadly, totally different businesses, totally different international locations, they use totally different strategies. There isn’t any settlement. After which we are able to simply declare no matter we would like. And, I am positive you’ve got seen some individuals will say that the Nile is the longest. Some individuals say that the Amazon River is the longest. And, clearly, there are different points. Generally we do not know the place the river actually begins, and so forth. However, one of many points is it actually relies upon how we measure it.
Russ Roberts: So, I wish to say one thing, once more for listeners who’re listening to this for the primary time: A shoreline in a real-world nation is, by definition, not a straight line. A shoreline. There are different borders that might be both a straight line or near it–we’ll discuss these in a minute. However, a shoreline from space–from outer house, from a satellite–has a sure form. And, for those who take that image from outer house, clearly, it’s important to management for the truth that you are far-off. However for those who’re strolling on the seashore, the ins and outs on the granular–to use a nasty phrase, sandy degree; however by granular, I imply small–there’s as a lot element as you possibly can probably think about. Which is to say that in some sense, for those who suppose a crude shoreline that stretches for, let’s simply say, a mile, and it is fairly straight. So for those who simply pretended that it’s straight, you get some measure. However, for those who needed to do higher than that and be extra correct and also you tried to observe every little tiny flip and bend, you begin to understand that there is an immense quantity of potential house in that bending.
I assume a technique to assist listeners give it some thought is: inside your physique, your gut ought to be–I do not know, what–a foot and a half lengthy? How lengthy might it’s? However, in fact, the thinner it’s and the extra it is turned backwards and forwards and backwards and forwards on itself, and for those who begin to say that the turns may be near infinitely small–which can be true for those who had been down in your knees with a magnifying glass attempting to determine the coast–you could make it just about so long as you need. You may squeeze a number of distance in there. Is that what is going on on?
Paulina Rowinska: Sure. And, what I like to consider is typically whenever you go mountain climbing and somebody asks you–like, you are attempting to estimate a distance–in a straight line? Straight-line distance is often very brief. It’s like, ‘Oh, in half an hour we’ll be there.’ However then, it takes you six hours as a result of it’s important to go round. Or for those who’ve ever pushed in a spot when there are a lot of mountains and the street actually goes backwards and forwards, you didn’t anticipate that typically for those who use a smaller-scale map. It is, like, ‘Okay, we’re virtually there.’ However, truly, it’s important to travel. And, that is precisely what’s occurring.
Russ Roberts: And also you gave the instance of shared borders of nations are measured–they have totally different lengths, if the borders are usually not straight.
Paulina Rowinska: Yeah. So, the one who truly appeared it the primary time, Richardson, he appeared at–I believe it was Portugal and Spain–the border. And he is, like, ‘Wait a minute. Why is not the border the identical size? It is the identical line.’ And it is simply the international locations measured it otherwise.
And, I believe we’re getting higher at it. It is a lot more durable now to search out these excessive variations as a result of simply international locations are agreeing a bit extra, and it is often like one huge company. However, I nonetheless managed to search out some significant variations.
And, simply to make it clear, once we discuss variations, it isn’t that the distinction is, like, ‘Oh, one nation is, like, 100 kilometers extra. It is double the quantity, or triple the quantity.
Russ Roberts: Yeah. The quantity you report[?] is: Portugal and Spain share a border, and Portugal says the size of that border is–I do not know if that is, when this was performed. However, it was 1,214 kilometers, and Spain thought it was 987. That is a massive–it’s not prefer it’s virtually a complete p.c. No, it is extra like 25%, as an approximation of what sort of error. It is a horrible error. And it isn’t an error: it is only a totally different methodology, I assume, for calculating the space.
Paulina Rowinska: Yeah. It isn’t an error. Each numbers are right in a roundabout way.
Russ Roberts: Which is loopy.
The final instance I wish to take of this, which is entertaining primarily for the footnote. There’s a number of good footnotes on this guide. I believe it is my favourite.
So, for those who have a look at Colorado, the state in America, it is a rectangle. It is obtained 4 sides, and so they’re all straight traces, and it appears to be like like they’re 90-degree angles, roughly, for the corners. And then you definately begin to suppose, properly, wait a minute, it is not likely 90 levels precisely, as a result of it is curved, and the curvature of the earth is totally different up by the northern border than by the southern border. Okay, okay, okay: That is going to make it a bit difficult. However, it is extra difficult. You declare that Colorado has 697 sides. That is greater than 4. How is that potential?
Paulina Rowinska: It’s potential as a result of the road drawn on the map was straight. However then, you could have people–and keep in mind, we’re speaking a couple of centuries in the past, when the expertise was very totally different. And, you had individuals truly going on the market within the land and marking the border. And, to start with, individuals make errors: it is unimaginable to do it in a straight line. However then, I do not know, you could have a bit pond that it’s important to go round, or you could have a bit forest, no matter that is. And, these are very small variations. You actually should have a large-scale map to see them, so it’s important to zoom in. So, they did a extremely good job, truly. Like, it is roughly a rectangle, minus the curvature. However, it’s attention-grabbing that one thing we assume–these are straight traces, that is it. In the actual world, they don’t seem to be precisely straight traces. And this, we’re speaking about place that did not have many rivers or something like that. So, think about in a jungle, it is unimaginable to attract a straight line.
Russ Roberts: Really, I usually use Google Maps on my telephone, however I went and used Apple Maps. Apple Maps truly provides you a globe whenever you ask for a rustic. It would not simply provide you with a projection on a flat floor. It tries to present a 3-D illustration on a flat floor. I do not even wish to begin with what’s unsuitable with that, or some distortions, but it surely appears to be like actual. It appears to be like prefer it’s properly performed. And, for those who begin to look intently and enlarge it, pinch it out together with your fingers, you see that, oh my gosh, down there within the south, it is not straight. In case you look intently, it’s beginning to go down a bit. What occurred there? And, the reply is: It isn’t straight.
So, it has 697 sides. And, that is the footnote that I liked. There is a title for that. Is there actually? Did you make this up, Paulina? Is that this true?
Paulina Rowinska: No. No. I put a supply in there. This isn’t my proper [?]. I believe I noticed it in–I believe it was Atlas Obscura that reported on that for the primary time, so credit score goes to them. However, I used to be like, ‘I have to put this on this guide.’ Yeah. And, there’s a title, and do not ask me to pronounce this title proper now, however there is a title for this form, which is totally ineffective.
Russ Roberts: No. It isn’t. It is nice for events or EconTalk trivia contests. So, I will pronounce it. I am positive I will pronounce it incorrectly, virtually by definition, proper? It is a hexahectaenneacontakaiheptagon.
Paulina Rowinska: Properly performed.
Russ Roberts: That could be 698 sides. However, in line with the–if I pronounced it accurately, it is 697.
Russ Roberts: Let’s discuss subway maps. Totally fascinating. What did they used to appear to be in London, and who had this perception of the way to make them much less correct and extra helpful?
Paulina Rowinska: Sure. So, when the transportation began in London–public transportation in London or in any city–you have a couple of traces, a couple of stations. It is quite simple. So, you’ll be able to simply put it on–you take a geographical map of London, and you set these stations there, and everyone seems to be comfortable. However, the extra difficult the system becomes–now London has an enormous system. You may go from one facet to the opposite; you’ll be able to spend hours on the Tube. After which, it grew to become so difficult as a result of the traces are usually not straight. They simply go round and switch and curve. And other people had been afraid to make use of public transport as a result of they had been, like, ‘Oh my gosh, I will get misplaced. I do not know the way to do this.’
And, Henry Beck, who was not a mapmaker–he was, properly, draftsman for some engineering company–but in his free time, he was, like, ‘What if we did with the Tube map one thing that engineers do with electrical circuits?’ So, a map that forgets concerning the geography however solely focuses on the stations and the connections between them. As a result of, in the long run of the day, that is what we use public transportation maps for. We aren’t those driving the bus or the Tube. We simply have to know which station–where to take the road, which line to take, the place to get off, and that is the vital bit.
So at present, the map–obviously, this isn’t the map that Henry Beck drew, but it surely’s an identical concept. And, he was not the primary to give you this concept. He simply grew to become probably the most well-known. And, there are different cities the place the maps appear to be that. And, now now we have this iconic Tube Map with straight traces. So, they’re all vertical, horizontal, or diagonal. The distances between stations–they all look, or most of them look, they’re equal–which shouldn’t be true in any respect.
And, I’ve made this error of assuming that, yeah, yeah, this is similar distance; it is only a brief stroll. No. Some tube stations are simply, like, three-minute stroll away, and a few are, like, 40-minute stroll away. However they appear the identical. As a result of, in any other case it simply turns into so difficult that no person would ever use this map–because why would you? And, many, many, many cities all around the world–I stay in Boston space now, and it is the identical factor right here. The map shouldn’t be geographical, as a result of why would it not be? It simply serves a special objective.
So, we name these maps topological. And, topology is a really bizarre space of arithmetic, I’d say. It solely appears to be like at connections inside shapes, but it surely would not actually care concerning the actual form. So, a circle and a sq., for topologists, is similar factor, since you simply can squeeze it and simply elongate it. So, so long as you do not make any holes in it, you are fantastic. So, that is why we name it a topological map, as a result of that is what it’s. It cares about connections and order of the stations fairly than the precise shapes.
Russ Roberts: And what’s attention-grabbing, in fact, is which you could’t superimpose the subway map or the tube map on the geographical map. It actually would not match, which is on objective, as a result of if it did match, you’d want both a a lot greater map–I do not suppose you mentioned–you obtained to put in writing the stations down. So, it isn’t simply it curves round. It will get difficult as a result of there are stations that have–five totally different traces undergo them, and the concept that you’ll print that on an precise map, properly, it looks like a extremely good concept, besides you’ll be able to’t learn it as a result of they will should be so small.
It is simply an interesting instance of the ability of approximation. And, one of many themes of your guide is that maps are simplifications. You must depart issues out or they don’t seem to be helpful, as a result of a map that had no simplifications would not be a map. It might be the actual world and be too huge to fold to place in your pocket. It is a totally different math result–right?–the variety of occasions you’ll be able to fold a chunk of paper.
Paulina Rowinska: Sure. And, that is proper, that is what maps are. And once more, I maintain coming again to this on this book–why did I even write this guide? Why it is vital to know, possibly not all the main points of arithmetic, however the concept that each map maker makes a map for some objective. So, we have to know what this map is helpful for, what it isn’t helpful for, and what’s there–and I believe that is an important bit–what’s lacking? And once more, I am not saying that if one thing is lacking within the map, it is a dangerous factor. Typically, it is an excellent factor, as a result of we simply have to have what is required and what’s essential. After which, the map is helpful. However we want to pay attention to that.
Russ Roberts: So, I wish to discuss a bit bit about supply of packages. And so, it is an previous problem–mathematical downside. It goes again to–is it Euler [Leonhard Euler]? It is Euler, proper? It isn’t Gauss [Carl (Friedrich) Gauss]. It is Euler. Gauss is on this guide. Gauss and Euler–two of my favourite individuals. I am not a mathematician, however I’ve a number of love for each of them.
And, there is a basic downside: You have to go to a bunch of various locations, both delivering issues or making gross sales calls. And, what are the alternatives of how one can make that journey? Clearly, you do not wish to spend an infinite period of time. You wish to economize on time.
You’ve this nice quantity, which is–I believe the average–not the typical, however some drivers for UPS [United Parcel Service] make 135 stops in the midst of a day. Properly, what number of methods might you make 135 stops? Properly, the reply is rather a lot. You say greater than the celebs within the universe, cells within the our bodies of all human beings, bushes on the Earth put collectively.
That is a number of routes.
Some are the worst, some are one of the best. However you do not want one of the best. That is one other instance of the type of simplification we’re speaking about. You need a route that is, quote, “ok.” That is shut to the best–if potential. Is probably not, however that is not horribly lengthy. So, how does math play a task in that?
Paulina Rowinska: Yeah. So, this is likely one of the extraordinarily tough issues, and one of many issues the place the variety of methods to do one thing grows very, very quick with the variety of stops we’re including.
So: with 5 stops–if I am a supply driver and I’ve to make 5 stops–I’ve 120 methods to do this. However then, we develop very, in a short time to numbers that we can not even think about. And, keep in mind that daily the stops are totally different. So, at present I order a package deal. Tomorrow you order a package deal. And so they want to determine each morning the way to do it quick.
And, for those who’ve ever seen a supply driver, they’re actually quick. They drive with open doorways, and they also soar out, throw the package deal, and are available again in. As a result of that is what they’re paid for. That is what UPS, or DHL [Dalsey, Hillblom, and Lynn], or Amazon, or whichever drivers–companies–they satisfaction themselves in that they’re quick.
So, you want one thing quick. That, each morning you get a route. You already know precisely the place to go, which strategy to go.
And, as a result of there are such a lot of methods to do this, probably the most {powerful} laptop on this planet won’t be able to compute, each morning, all these routes and spit out the shortest one or the quickest one. As a result of this might take, like, centuries most likely. Which does not actually make a lot sense.
Now, we additionally know[?] the driving force to do–first, he goes to the furthest, or she goes to the furthest place, after which comes again right here, and then–it would not make sense. Proper? Like, if there are two stops subsequent to one another, they most likely must be served very shut in time collectively. So, you want a route that is ok.
And for that, we’re utilizing totally different algorithms.
And, truly, they’re all secret nowadays. So, as a result of that is what the corporate’s success depends upon, actually. Like, the quicker they go, the extra clients they’ll serve.
That, you recognize, it is all about gasoline effectivity and the time. Like, they’ve hundreds upon hundreds of supply drivers, and so they should pay all of them, and so they most likely ought to pay extra to them.
So, you recognize, it is about effectivity.
So, we’re creating algorithms–so, type of methods of determining the quickest, probably the most optimum route.
Now once more: we can not actually discover the quickest. However now we have superb algorithms that guarantee that the route is no less than pretty much as good as X. So, it could be, like, 10% worse than the probably quickest one. And, it is ok. So long as we are able to get a quick reply each morning.
So, yeah. That is, once more, arithmetic at its greatest. Additionally, real-life arithmetic. As a result of in theoretical arithmetic, we love the reply. We love one of the best reply.
However, in on a regular basis life, we’re utilizing alternative ways, totally different algorithms, to determine the good-enough solutions. And, there are such a lot of algorithms to do this, and so they give totally different solutions, however most of them give some reply that is just–okay.
Russ Roberts: You give an instance within the guide. A crude algorithm or crude heuristic can be: decide a cease, after which go to the closest cease to it. That is not a nasty beginning place for a heuristic or a rule of thumb. Presumably, FedEx, DHL, and UPS use some enchancment on that. However, within the instance, I believe you gave, drivers develop a sure instinct. And, if I keep in mind accurately, a number of these firms use some mixture of math and driver expertise or know-how to assist design that route. Is that right?
Paulina Rowinska: Sure. So, once more, I used to be attempting to piece collectively some items of knowledge, I felt, as a result of it is all prime secret. However, sure, UPS had–one day, they determined: We’re going to make every part automated. We’re going to optimize that. They paid some huge cash, invested some huge cash, employed mathematicians or laptop scientists to determine this out. And, this was a giant failure. It simply did not work.
After which, the drivers had been, like, ‘Wait, why are you, I do not know, making us drive at 8:00 A.M. or 9:00 A.M. within the morning subsequent to a college when there is no strategy to drive? It would not make any sense.’ So, they type of know their space higher. They know which turns are extra harmful or take extra time, and so they simply know what works, what would not. However, additionally, we wish to make it a bit extra optimized. So, combining these two, the human experience–the human data, and the human intuition–and the mathematical insights, and it seems that this offers one of the best outcomes.
Russ Roberts: So, some listeners might keep in mind the interview–I believe it was with Vernon Smith–where we had been speaking about Federal Categorical, and Federal Categorical utilizing the–it’s referred to as a hub-and-spoke system. However, on the time, there was just one hub, and it was Memphis, Tennessee. And, each package deal that FedEx picked up was taken to Memphis; after which it was flown from Memphis that evening to its vacation spot.
And, everybody would at all times say to me once I would discuss this, ‘Clearly, that is inefficient. Why would you make each package deal go to Memphis? In case you’re flying a package deal from Oakland, California, to Berkeley, California, silly to make it go all the best way to Memphis.’ And then you definately begin excited about the fact–and it is a good method to consider it–we thought of it otherwise once we talked about it earlier than. However, okay, so let’s simply use one aircraft. So let’s go round to all of the locations that persons are sending packages, and we’ll simply decide up all of the packages, after which we’ll take the aircraft, and we’ll drive it to all of the locations the place the packages are going, and we’ll ship them. Not solely would it not take centuries to determine what one of the best route was that evening, however even that evening it will take a really, very, very very long time. And so, you want a number of planes.
And so, what Fred Smith understood is: for those who ship and decide up in 40 cities, you should utilize 40 planes. It begins in a metropolis: it picks up all of the packages; they’re all delivered to that airport by vehicles. You fly it to Memphis; they’re all distributed to the planes going again to the place they got here from. And, you want 40 planes, which is a tiny, tiny quantity in comparison with for those who’re attempting to attenuate the space {that a} package deal traveled. So, it is the identical downside. It is precisely the identical set of mathematical points, I believe. Inform me if I am proper.
Paulina Rowinska: Yeah. No. I do not suppose even the main points matter a lot. Once more, it is about how–also, what are we minimizing? We’re not minimizing solely distance, proper? We’re minimizing the time, the effectivity, the fee. So, each firm must resolve what is the precedence and what’s extra vital. That is exactly–also the purpose I am attempting to make–is that arithmetic, it is one factor on paper and one factor if we wish to apply it. And, it is okay if we take arithmetic and make a little bit of an artwork out of it. We take some outcomes and depart different outcomes so that they swimsuit our software.
Russ Roberts: I am curious if in scripting this guide, this concept of simplification and leaving issues out whenever you make a map, you mentioned you begin to see sure issues all over the place. Did you begin to see that in your personal life? Did you discover the need of leaving out particulars in any respect? Or is there anything you realized in writing a guide about maps?
Paulina Rowinska: To begin with, once I was writing the guide, I needed to make so many choices about what to depart out and what to maintain in. And, truthfully, it is more durable than I assumed it will be, as a result of to do this you actually, actually need to know the issue. It is the simplest to present all the main points and maintain the opposite ones.
However, I positively like taking a look at, I do not know, infographics and any knowledge visualizations. Probably the most {powerful} are those which have the precise quantity of knowledge that the reader wants. And, I keep in mind listening–I believe it was one other podcast about–you know on the planes now we have these leaflets that inform us what to do in case of an emergency. And, I did not understand how a lot effort goes into creating them. As a result of, you need individuals to learn, however no person will research a complete guide of all the main points. And, I actually–like now, taking a look at totally different planes, how otherwise they appear, and to see what they put in, what they pass over, and what’s an important factor. As a result of, once more, we’re people, and there is solely as a lot data we are able to maintain and course of. So, the bottom line is the way to make us act.
One other instance I keep in mind: 20 years in the past at this level, I visited Germany, and they’re actually good with recycling. However there have been, like, I swear, 20 totally different trash bins. Brown glass, inexperienced glass, white glass. I used to be so confused. I got here from a rustic that wasn’t actually recycling at that time, so I used to be simply standing in entrance of those trash bins and attempting to determine the place trash, like, my tea bag. And, I am fairly positive that since then Germany determined to really cut back the variety of these totally different bins. And to really enhance effectivity, as a result of, if you’re so confused, you are simply going to place every part into combined recycling or combined bag, which will not get recycled in the long run. And, I do not know the precise particulars of how the recycling works there, however I believe it is the identical concept: We must always serve people and never attempt to be the artist and attempt to be, like, the most effective.
Russ Roberts: Yeah. It is an important instance of the significance of imperfection and the right is the enemy of the great. I’ve the identical factor right here. You may be within the airport and there is three bins–there’s not 20, there’s three.
And, there’s one other situation which you did not point out, but it surely’s apparent, is that not everyone speaks the identical language. So, you are going to use some type of visible illustration. And there is no tea bag on any of these bins.
So, if it is a tea bag, you are going to should make a classification resolution, and it is typically not apparent. And, you end up throwing one thing away in one of many bins or placing it in one of many bins. You haven’t any concept whether or not you’ve got performed the best factor, the unsuitable factor. Ought to you could have emptied the liquid?
So, yeah, nobody desires to learn a 400-page book–no one will learn a 400-page book–about the way to throw away trash, the way to recycle or throw away one thing on the airport. So, you have to use some crude, imperfect system.
And, it is actually metaphor for all times, truly. I will not say something extra about it. You may add something to that you really want, and listeners can fill in these blanks. However, do you wish to say anything on that query of imperfect however helpful?
Paulina Rowinska: One factor I have been learning–so this yr, I have been finding out science journalism in grad faculty. And, writing the guide, I had time. I actually might take my time to undergo every part and make it–it’s not perfect–but as good as I might make it.
Now, we had been studying to put in writing science information. And it is a complete totally different story, as a result of you could have information, and in three hours or in at some point most, it’s important to have a narrative out. And, if I began to essentially look into each single element and one of the simplest ways to cowl the story–and that is what I used to be doing at first, do not get me unsuitable, I missed some deadlines.
So, that is one other instance. And each approaches are most likely proper, but it surely relies upon. If we simply wish to talk information, we higher be quick. If we wish to write a good looking characteristic story with attention-grabbing characters, we are able to take our time. However, yeah. I believe ok is sweet sufficient. And, that is what maths and maps educate us as properly.
Russ Roberts: This jogs my memory of a number of the debates that economists get into and others get into about rationality. I remember–I am positive I talked about it within the episode I did–I believe I did a couple–with Gerd Gigerenzer, the German psychologist. And he talked about this good system for allocating sufferers to the emergency room. There have been 15 variables. The docs needed to give a rating in any respect 15 variables after which use a pocket calculator to determine whether or not to ship anyone to the ICU [Intensive Care Unit]. And, not solely was that system outperformed by the three-step heuristic that he advised or the docs had been utilizing, however they weren’t even going to use that.
So, to name that probably the most efficient–it wasn’t. However even when it had been, the truth that individuals would not use it means it isn’t probably the most environment friendly.
So, it forces you to begin considering otherwise about effectivity.
Russ Roberts: I do not suppose you mentioned a lot on this guide about AI [artificial intelligence]. When did the guide come out initially?
Paulina Rowinska: So, properly, it got here out 2024–exactly a yr in the past, just about.
Russ Roberts: The Darkish Ages. Medieval occasions. Earlier than BAI [Before AI–Before Artificial Intelligence].
Paulina Rowinska: Sure. Yeah. It is insane how a lot progress has been made within the final two or three years. I used to be writing the guide, and with out ChatGPT [Generative Pre-trained Transformer]. I do not know the way I did that.
Russ Roberts: It is a miracle.
Paulina Rowinska: Yeah. It’s actually loopy to see how the world has modified. And many individuals truly write in feedback and opinions: There’s nothing about AI. And I am like, ‘Properly–‘ And, I am not saying there was nothing occurring. In reality, in my very final chapter of out–
Russ Roberts: There’s a bit. There may be–
Paulina Rowinska: Yeah. I wrote about, for instance, self-driving cars–
Russ Roberts: We’ll come to that. Yeah–
Paulina Rowinska: Yeah. However, you recognize, I imply: One half was the progress is just–I believe 5 years in the past, individuals knew that AI is a factor, however no person was speaking about it. Now it is like a Subject Quantity One. And I am not speaking about in science, however simply you go on the road and folks discuss AI.
And likewise, there are such a lot of issues I disregarded as a result of it is only one guide, and I simply wrote–like, possibly there shall be a sequel that shall be AI. Who is aware of?
Russ Roberts: Positive. It isn’t a criticism, however I’ve to make this dangerous joke: However, the guide was written in B.C.–Earlier than Claude [Claude, an AI chatbot written by Anthropic]. Sorry about that.
So, I am not criticizing you for not having something about AI within the guide. However, what I am enthusiastic about is what ideas you might need on AI when it comes to each map-making and arithmetic.
There are individuals which can be saying now that arithmetic is–it’s not going to be a subject as a result of AI can do every part, can show every part. It will show each conjecture quicker than human beings. I am skeptical about a few of that. However, it is going to have an effect on arithmetic, and it is definitely going to have an effect on utilized mathematics–which is what we’re speaking about. We’re speaking about a mixture of principle and utilized. So, what are your ideas on that?
Paulina Rowinska: Properly, to start with, every part is in progress. So, at any time when I hear somebody making, like, a particular assertion about it, I am, like, ‘All proper. I imply, I assume you recognize higher. However I do not know.’
My view is that I do not suppose arithmetic will die out as a subject, however it should change. And it is altering already. So, type of in the identical method that calculators modified the sphere. Proper?
And like, so, I believe–when you suppose, for instance, about proving a giant theorem, typically it is a collection of proving mini-theorems inside it. So, it is like many steps to get to this huge chunk.
And, I believe AI can do a few of these steps, or will be capable of do a few of these steps. They’re type of mechanical. They contain some creativity, however they don’t seem to be, like, do not require the brightest minds.
Now, what I believe AI will not be capable of do–at least for an extended time–is to, to start with, determine what the steps must be; and second, how all of them match collectively. And, I believe that is the place people set in, to essentially suppose out of the field. To consider the way to show it, the way to determine all of it out.
And, I believe, proper now I consider AI as a really superior calculator that may do far more than a calculator, however type of to do the issues that often take extra time than not. And, I could be unsuitable. And, it is getting increasingly more superior. So, I believe it could possibly show extra superior issues. It could possibly suppose in a extra superior method.
However, I nonetheless suppose for no less than some time–and an extended time–we’ll want the mathematician to essentially take into consideration the way to method an issue. Which issues are related, as properly? And, what inquiries to ask.
So, I believe nonetheless individuals shall be in management for some time.
However, once more, I could be very unsuitable. I additionally do not work actively in mathematical analysis proper now, so I could be lacking one thing. However that is my feeling. If I needed to guess my cash, I’d say AI could be very useful. Will probably be very useful. Will probably be increasingly more concerned. And, I believe it’d turn out to be a bit like a calculator in some unspecified time in the future. Simply that the mathematician will not spend time computing an integral, however will spend time considering the way to use this integral.
Russ Roberts: That is nice. I used to be excited about how persons are speaking about it in comparison with, say, 5 years in the past. And I believe we’re simply on the fringe of that. There’s so many issues recently, I’ve discovered it to be–it’s helpful, but it surely’s greater than helpful. It is type of enjoyable, and it is type of thrilling that it may be performed. And, it is kind of marvelous within the old style sense of the phrase ‘marvelous’–it’s a marvel.
And, I am looking for an analogy, the best way individuals discuss it. I believe it is a horrible analogy, however I will run it by you and see what you suppose. It is such as you’re in a rustic and so they haven’t got pizza, after which anyone opens a pizza place and, ‘Oh my gosh, I’ve by no means tasted something’–like, you consider pizza: sushi is considerably on this class. It is like, it isn’t a number of issues which can be actually prefer it.
And, with regards to your nation or your metropolis or your city the primary time, it is like everybody’s speaking about it. ‘Have you ever tried that pizza?’ And, quickly there’s pizza outlets all over the place. And, in fact, a few of them are usually not superb, and a few of them are higher than others, and a few are unbelievable, and folks wish to attempt all of them. I discover myself attempting all–I take advantage of Claude and I take advantage of ChatGPT, which I now name Claudette, as a result of when I’ve a woman–Claudette–talking to me on ChatGPT, it is so thrilling. Or possibly it is like a brand new pet. You bought a canine and a cat, after which anyone says, ‘Properly, what about an ocelot?’ ‘Oh, wow. See, that man had an ocelot? That was so cool.’ I do not know. It is one thing. Individuals wish to discuss it.
Paulina Rowinska: I believe that is part of it. However, I additionally suppose that there’s a bit–for instance, this picture technology, all social media persons are producing photos. Additionally, ‘Please do not do this as a result of this takes a lot vitality. That is the worst factor for the planet.’ Simply saying that. And, it is thrilling. Like, I’m typically, like, ‘Wow. It could possibly do that, it could possibly do this.’ And, it is enjoyable and thrilling. However, I believe–so that is just like the pizza half.
However, I believe there’s also–it’s actually turning into a giant a part of our lives. In all probability examine it a bit extra possibly to social media, one thing that at first was like, ‘Oh, what is that this Fb factor?’ And, now it simply grew to become, like, a factor. If you’re not on social media, you do not exist. Information occurs on social media and every part occurs there. So, it grew to become actually only a huge a part of our lives. And, I believe that is what’s occurring with AI.
However, I additionally suppose there’s–one factor is, like, an everyday person: it is all enjoyable, and it is simply enjoyable and video games. However then, you could have the analysis half and the usage of AI in drugs, for instance. That is actually saving lives–or endangering lives, relying the way it’s used. So, I believe it is greater. I believe it is greater than every part else.
Russ Roberts: Do you suppose it is greater than pizza? I do not know. I do not know, Paulina.
However, I believe that you just’re proper. I believe the analogy can be for those who felt pizza was so scrumptious that you just needed to have it no less than as soon as a day–cold within the morning for breakfast, or lunch, or dinner–so possibly twice a day. After which, individuals would say, ‘Properly, need a pizza?’ ‘No, I already had it for lunch.’ ‘Oh, okay.’ ‘Did you could have pizza for lunch?’ ‘No.’ ‘Oh, nice. We’ll have it for dinner.’ And, you are right–it’s all over the place. Yeah. And, it is just the start. In all probability.
There is a temptation, like in every part, to imagine that every one developments proceed without end, in order that this explicit exponential leap in its capabilities will simply get much more dazzling. And, that is probably not true. However, it could be. Such as you say, I would not guess towards it. And I would not guess on it.
Paulina Rowinska: Yeah. And I believe in some unspecified time in the future it’s going to turn out to be not even that thrilling. It should turn out to be a brand new factor and folks will transfer on to one thing else. However, I believe it’s going to keep. I simply do not see how we are able to return.
Russ Roberts: Yeah. You could be proper.
Russ Roberts: I wish to shut with–you talked about driverless vehicles. I believe everybody’s had an expertise, for those who’ve ever used Google Maps or Waze to navigate in your automotive: Abruptly you are on the unsuitable street. You’ve got taken an exit ramp; there have been a pair selections; you picked the unsuitable one. It thinks you are on the best one for some time, and also you suppose you are okay; after which swiftly, to your horror, you understand you are on this different one which’s veering off at a horrible angle. And, you level out that for driverless vehicles, the extent of accuracy–I imply, it is wonderful how correct Google Maps is, utilizing it in your automotive. Or Waze. However with driverless vehicles, the accuracy must be a lot, rather more correct.
How are they doing that? And, clarify why that is vital. It is type of apparent, however possibly individuals have not thought of it.
Paulina Rowinska: Yeah. So, it is rather vital as a result of the automotive has to–so think about whenever you’re driving a automotive, it’s important to take into consideration so many issues. You must take into consideration the street, the place you are going. You must take into consideration wanting. Additionally navigating all this equipment–brakes and every part. You must suppose: Is there a toddler going to leap out on the street and I will should brake quick? What are the situations? Is it raining? Do I’ve to decelerate? Oh, there’s an ambulance coming, so I’ve to cease or change lane.
All these items. It is some of the excessive multitasking we do as human beings, I believe.
And, you could have a machine that has to do all that. Now, it is very useful if the machine no less than is aware of the street very properly and in addition is aware of what to anticipate so it could possibly give attention to the surprising. Proper? So, if it is aware of the place precisely the street is, what it appears to be like like, the place the site visitors lights are, and so forth., and so forth., it could possibly give attention to the kid leaping out doubtlessly. So, that is why it is so essential for self-driving vehicles to have very, very correct maps. By correct, I imply it actually must know the place the curb is on the street.
Now, only a caveat–I used to be writing it two, three years in the past. Might need modified drastically since then as a result of the expertise retains altering. And I do know that vehicles with AI are getting extra and more–they’re extra in a position to regulate to novelty. So they could need–I consider that sooner or later, they will be like human drivers: that they will be okay with going to a spot they’ve by no means been earlier than. However for now, self-driving vehicles are type of largely working in particular areas, particular cities, once they know the town very properly.
So, what occurs you probably have vehicles which can be actually going street by street and mapping and taking photos on their cameras and all these things to create this map? And, there’s additionally this type of downside that you’ve, for instance, a self-driving automotive that is updating the map whereas it is utilizing the map to drive additional. So it is type of like a loop. And it is, you recognize, an attention-grabbing, advanced downside to unravel.
So, yeah: I imply, the self-driving vehicles are getting higher and higher. They used to get fully distracted by–I do not know, in the event that they’re driving by–there’s asphalt after which there’s grass–they are simply, like, ‘Whoa. What’s that?’ Now, I do not suppose they’re making these errors anymore.
So, they’re getting higher than human drivers. As a result of, persons are not that nice at driving, truthfully. And, although many people declare we’re one of the best drivers on this planet. However, yeah.
So, I believe that is the significance of those maps–to type of assist the automotive no less than know what to anticipate. Once more, I am not an professional there, so if I mentioned one thing very unsuitable, please depart a remark. Yeah.
Russ Roberts: They’re going to let you recognize.
However I am curious: I assume it is a satellite tv for pc downside to determine the place the curb is and within the automotive itself. And, I assume they’ve got–as a footnote to what you mentioned, by the best way, I took what I believe was a driverless automotive. There wasn’t a driver in it. There’s some dispute about what sort of overrides firms can do. However, once I was in San Francisco the final time, I took a driverless taxi plenty of occasions, and I liked it. The primary 30 seconds was a bit bit horrifying, however then you definately in a short time understand it is a a lot safer experience, no less than in its current–again, if it is actual, I do not know–but in its present manifestation, than most–many–human drivers that I’ve had prior to now. So, it was actually fairly a pleasant expertise.
However, have you learnt how they’re bettering the accuracy of the visible illustration that the vehicles are utilizing?
Paulina Rowinska: So, I am undecided. I believe, what I imagine–it’s a mixture of higher satellite tv for pc photos. And, so the identical method that, for example, common Google Maps are bettering.
And likewise, the extra vehicles there are, the extra knowledge is coming to enhance. So, you recognize, you probably have, like, a whole lot and hundreds of vehicles going the identical street, they are often, like, ‘Oh, this map shouldn’t be precisely correct.’ Or, ‘That is the spot the place typically dangerous issues occur.’ I do not know.
So, once more, I am not precisely positive. And I’d think about that each firm has barely alternative ways, however I believe it is simply with extra knowledge that is coming in.
And sure, as you mentioned: self-driving vehicles truly really feel a bit safer. And, I believe it’s true additionally as a result of they’re much extra danger averse. So, in the event that they’re not sure, they will cease. They’re going to decelerate. Whereas persons are: ‘I am speeding, so you recognize what? I’m not precisely positive if I can flip right here, however I am going to do this anyway.’
So, I believe that is the type of the advantage of them. They’re–they haven’t got emotions. They do not have feelings, they do not get offended, they do not get drained. And, I believe that is the large good thing about them.
Russ Roberts: Yeah. I agree.
We did not get to every part in your guide. There are lots of, many different attention-grabbing chapters on Columbus, earthquakes, mapping the ocean flooring. However, let’s shut, truly, with the chapter that you just want you’d had. You may have been twice as lengthy. I discussed we did not get to cholera and John Snow and the deal with of the water pump, however: What’s your favourite? Do you could have a favourite? Was there a chapter there that you just fell in love with, you would like you needed to do it twice as lengthy, however your editor mentioned, ‘Nyeah, you’ll be able to’t do that, Paulina. Sorry.’?
Paulina Rowinska: I imply, there have been a couple of. One is I’d like to have extra sorts of projections and type of actually have a look at the bizarre ones as properly. As a result of there are such a lot of bizarre projections which can be used for one very explicit factor. And, I really like that. I believe that is stunning.
And, one other chapter is the one that you just truly talked about about mapping epidemics or mapping crime–kind of maps that actually save lives.
And, this was a humorous chapter, as a result of that is the chapter that my editor, who was at all times, like, ‘No, there’s an excessive amount of maps.’ Like, ‘Make it a bit less complicated.’ And this one, she was like, ‘Wait, why did not you set any maps?’ And, I used to be like, ‘No, that is all maps.’ And he or she was like, ‘Wait, however that is simply, like, frequent sense.’ And this, I discovered–the neatest thing I’ve heard about this guide.
So, I talked about how we are able to discover issues, or individuals, or the supply of an epidemic by type of utilizing one thing that appears like frequent sense, but it surely’s truly deep arithmetic.
And I liked reporting on this chapter as a result of I had no concept about all these items about geographic profiling–so how we are able to slim down the seek for serial criminals.
And I had pretty interviews, and I believe I’d love to put in writing extra, as a result of that is the chapter once we actually discuss life-or-death conditions, and I believe this actually brings house the message that maps and maths collectively are actually vital.
Russ Roberts: My visitor at present has been Paulina Rowinska. Her guide is Mapmatics. Paulina, thanks for being a part of EconTalk.
Paulina Rowinska: Thanks.