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# The Phantom Tollbooth: Chapter 15. This Way to Infinity Summary & Analysis

Summary
Analysis
Eight miners rush in with a huge, bubbling cauldron. It smells delicious. The Mathemagician offers Milo, Tock, and the Humbug something to eat. They all quickly finish their bowls. The Mathemagician keeps serving them again and again—and Milo feels hungrier with every bowl. When the miners take the cauldron away again, the Humbug is 23 times hungrier than he was before; he had 23 bowls of soup. The Dodecahedron says this is the kingdom’s specialty: subtraction stew. Here, the more you eat, the hungrier you get. Milo asks how anyone gets enough, and the Mathemagician says that here, people eat when they’re full and eat until they’re hungry. It’s an economical system that ensures that people always have “more than enough,” even when they have nothing.
Just as in Dictionopolis, Milo has to adjust to a totally new way of moving through the world. Because he’s not familiar with math concepts, it’s much harder for him to make this transition. It’s also worth noting that logic is a crucial part of math—this is why, in Digitopolis, people tend to rely on logic more than they did in Dictionopolis. But again, because there’s no rhyme and reason guiding Digitopolis, the subtraction stew and eating habits in Digitopolis are pretty nonsensical—food only works this way in theory, not in practice.
Themes
The Dodecahedron walks Milo through how this works with a math exercise and then assures the ravenous Humbug that he’ll be full in time for dinner later. Milo muses that he only eats when hungry, which the Mathemagician deems “curious.” Suddenly, the mine vanishes and he, Milo, Tock, and the Humbug are standing in the Mathemagician’s circular workshop. Everything is labeled with various measurements. When Milo asks if the Mathemagician always travels like that, the Mathemagician says he normally takes the shortest distance—unless he needs to be multiple places at once, in which case he multiplies. He writes 7 x 1 = 7 and suddenly, there are seven Mathemagicians.
In his workshop, the Mathemagician shows Milo that with the right baseline knowledge, it’s possible to look at the world only in mathematical terms. Where Milo might see that something across the room is a few steps, or a few seconds, away, the Mathemagician knows exactly how far away things are. And while it’s of course fantastical and absurd that the Mathemagician can multiply himself to be in several places at once, it also shows readers clearly what multiplication is and how it works.
Themes
Milo gasps, but the Mathemagician says it’s easy with “a magic staff.” The Humbug snaps that it’s only a big pencil—but the Mathemagician says that still, you can do lots of things if you know how to use it. The Mathemagician then shows Milo how to make things disappear by giving him a complex problem that equals zero—when Milo solves it, the written numbers disappear. Next, the Mathemagician shows Milo the biggest number they have (a three that’s twice the Mathemagician’s height) and the longest number (an eight that’s as wide as the three was high).
The Mathemagician’s magic staff might just be a big pencil, but he makes the case that a pencil can tell a person all sorts of things. If one knows how to do math, it’s possible to make things disappear, multiply, and divide, all on paper. So, Milo is finally seeing that there’s a purpose to learning math. Math problems aren’t just useless and boring—he can actually do things with them.
Themes
Related Quotes
Tock comes to Milo’s rescue and says Milo would actually like to see “the number of greatest possible magnitude.” The Mathemagician coaches Milo through adding one to a massive number, again and again—Milo can never stop, he says, because the biggest number is always bigger than the number you currently have. It exists, he says, the same place where the smallest number is. He asks Milo to divide 1/1,000,000,000 in half again and again until Milo asks if it ever stops. When Milo asks where you could keep something so tiny, the Mathemagician says it stays in a box so tiny it’s invisible, tucked within furniture just as tiny. But he’ll show Milo where to find it.
Infinity is more of a concept than an actual number, which the Mathemagician shows here through this exercise. But this doesn’t mean infinity isn’t still fun to think about. Moreover, it’s a mark of how much Milo has changed that he’s curious about infinity at all—such a thing wouldn’t have interested him before coming to the Lands Beyond. He's gaining the confidence to ask questions about things he doesn’t understand, and the novel presents this as a positive change.
Themes
The Mathemagician leads Milo to a window, where a line stretches to the ground and keeps going forever. If he keeps going forever and turns left, he’ll find the land of Infinity. Milo says he doesn’t have that much time, so the Mathemagician leads him to a flight of stairs. Milo bounds up the stairs, asking Tock and the Humbug to wait for him.
This is another matter of perspective: Milo realizes the line goes on forever, and he’s never going to reach Infinity. But because Milo probably believes that stairs stop eventually, he goes into this thinking Infinity is something he can actually reach.
Themes