At first, Milo climbs the stairs quickly. But he slows down and, finally, he sits—he realizes these stairs go on forever, just like the line. Someone tells Milo he wouldn’t like Infinity anyway, since there, “they can never manage to make ends meet.” Milo looks up to see the left half of a child. When Milo says he’s never seen half a child before, the boy says he’s actually .58 of a child. He explains that his family is average, with a mother, father, and 2.58 kids. He’s the .58 kid. The boy says it’s great—all average families have 2.58 children, so he has friends. And he’s the only person small enough to drive 3/10s of a car, since average families also own 1.3 cars.
Milo is initially excited to get to the top of the stairs—but he has to confront that it’s impossible to reach Infinity. Finding this half-child introduces Milo to averages, which are much more literal in the Lands Beyond than in the real world. Again, this is absurd—average families don’t actually have half children. But it allows the novel to illustrate how averages work by making them real.
Milo insists that averages are just imaginary, but the child disagrees. He points out that if Milo had no money but was with other people who had money, using averages, Milo would have money. The child offers more examples, but they just confuse Milo. Milo insists that none of this is possible. The child, though, says that the nice thing about math (and other subjects) is that impossible things can be real, just like Infinity. You can know it’s there and even if you can’t reach it, it’s still worth looking for. Milo bids the boy goodbye and heads back downstairs. He realizes he’ll have to learn a lot more to rescue the princesses.
The child essentially suggests to Milo that math (or anything else, for that matter) doesn’t have to make perfect sense, or be totally logical, to be fun. Indeed, it’s possible to read this passage as justification for the book as a whole. The novel is fantastical and presents a number of things that aren’t real, but it’s still entertaining, worth reading, and worth picking out its lessons. And this encounter—and the book—shows Milo just how much he has to learn. His education will never be over, even once the novel ends.
When Milo reenters the workshop, he laments that everything in Digitopolis is too hard for him. The Mathemagician comforts Milo and notes that the easiest thing to do is to be wrong—but it’s not worth it. Attempting to understand everything he’s seen, Milo asks why things that are correct aren’t actually right. The Mathemagician sobs that it’s been like this since Rhyme and Reason were banished, which he shouts is all Azaz’s fault. The Mathemagician refuses to discuss the matter with his brother, since Azaz didn’t answer his last friendly letter. Milo accepts the letter, which is entirely numbers. Milo doesn’t get it and suggests Azaz didn’t either, but the Mathemagician bellows that numbers are the same no matter where you are in the world.
While Milo is sad that everything seems too difficult for him, he’s no longer rejecting learning outright just because things are hard—rather, he wants to learn more. And again, it’s a sign of how much Milo has changed that he can now frame a question to the Mathemagician about why things are so hard in the Lands Beyond. The Mathemagician encourages Milo to keep trying to learn. However, the Mathemagician shows Milo that despite being able to dispense some good advice, he’s also very set in his ways—to the point that he can’t, and won’t, communicate with his own brother.
Tock asks for the Mathemagician’s permission to rescue Rhyme and Reason. When he hears that Azaz has already agreed to this, the Mathemagician refuses—he’ll never agree with his brother on anything. Milo thinks carefully and, using logic, is able to show the Mathemagician that he and Azaz are actually in agreement, since they agree to disagree with whatever the other agrees with. The Mathemagician cries out, but he gives his permission. Then, he warns Milo that the demons will know about him—and once they appear, it’ll be too late. He also has a grave problem to tell Milo about, though he can’t talk about it until after Milo returns.
Milo has learned a lot about logic if he’s able to beat even the Mathemagician at his own game. Things might be hard in Digitopolis and the Lands Beyond more broadly, but Milo is still learning how to navigate the land. And despite the Mathemagician throwing a fuss, he seems to agree very easily to help Milo get past the demons. So, he may not be as dedicated to fighting with his brother and keeping the princesses locked up as he initially made it seem.
The Mathemagician transports Milo, Tock, and the Humbug to the edge of Digitopolis. Ahead is a narrow path leading to the mountains. Milo laments that he won’t be able to get the car up the road, but the Mathemagician warns Milo that in Ignorance, he’ll have to take things step by step anyway. The Dodecahedron appears out of nowhere with Milo’s gifts. The Mathemagician gives Milo his own magic staff, and then he and the Dodecahedron bid the travelers goodbye.
The Mathemagician saying that Milo will have to take things step by step in Ignorance is humorous, because it means two different things here. Milo has to walk and literally step through Ignorance, and he also has to take things slowly, carefully, and logically. Giving Milo a magic staff means that Milo has learned what he needs to about math—and now he can go on to work problems for himself.
As Milo climbs the difficult path, the light fades. It’s not dark like the night, it’s “more like a mixture of lurking shadows and evil intentions.” As a heavy mist descends, Milo grabs Tock’s tail and wonders if they should wait until morning. Just then, a dirty bird quips that “They’ll be mourning for you soon enough.” When Milo says they need a place to spend the night, the bird says it’s not Milo’s to spend. As Milo tries to explain what he wants, the bird continually interjects to call Milo rude and tell him to use other words that make more sense. Tock asks the bird not to interrupt, but the bird says it’s his job—he’s the Everpresent Wordsnatcher.
The Everpresent Wordsnatcher continues the novel’s habit of playing with homophones, words that sound the same but have different meanings and sometimes different spellings. By using them, the Wordsnatcher is able to trip Milo make up and make it impossible to say—or learn—anything useful or intelligible. This shows that having a firm grasp of language doesn’t make a person good or helpful—language, just like anything else, can be abused and manipulated.
Milo asks the Everpresent Wordsnatcher if everyone in Ignorance is like him. The bird says folks here are worse, but he’s from a place called Context. The Humbug suggests the bird go back home, but the bird shudders—he spends “almost all [his] time out of it.” The bird then says he’s not a demon and flies away. When Milo shouts at him to wait, the bird replies, “thirty-four pounds.” The Humbug, pretending to be brave, suggests they find the demons. Tock starts to tremble.
The Everpresent Wordsnatcher’s one-liner about spending time out of Context is funny, especially since the bird seems to fit right in in Ignorance. He may be from Context and may call Context home, but within the context of Ignorance, he makes perfect sense. The Humbug may have absorbed some of the same lessons Milo has, given that he’s trying to be brave here.
Milo, Tock, and the Humbug continue to climb until they come across an elegant gentleman leaning against a dead tree. The man’s expression is totally blank—he has no face at all. But the man greets the travelers, and Milo is shocked to meet someone so nice. The man requests Milo to help him with some small jobs. He asks Milo to move a huge pile of sand from one place to the other using tweezers. He asks Tock to empty a well with only an eyedropper, and he assigns the Humbug to dig a hole with a needle. The travelers work on their tasks for hours.
At first, Milo is drawn in by this pleasant, mysterious man. He allows himself to assume that this is a person he can trust—which, from the looks of things, seems like a mistake. But this just goes to show that it’s okay to make mistakes, and that this is something everyone does—even someone as wise as Tock.