The classroom felt deserted that day, as always. The room was large enough to seat a hundred students, but there were only twenty or so there now. Most of them were in the back row so that they could slip out after attendance had been taken or work on some project of their own during the lecture.
Very few undergraduates wanted to be mathematicians. In fact, Ishigami was probably the only one in his entire class. And this course, with its lectures on the historical background of applied physics, was not a popular one.
Even Ishigami wasn’t all that interested in the lectures, but he sat in the second chair from the left edge in the front row. He always sat there, or in the closest available position, in every room, at every lecture. He avoided sitting in the middle because he thought it would help him maintain objectivity. Even the most brilliant professor could sometimes err and say something inaccurate, after all.
It was usually lonely at the front of the classroom, but on this particular day someone was sitting in the seat directly behind him. Ishigami wasn’t paying his visitor any attention. He had important things to do before the lecturer arrived. He took out his notebook and began scribbling formulas.
“Ah, an adherent of Erdős, I see,” said a voice from behind.
At first, Ishigami didn’t realize the comment was directed at him. But after a moment the words sank in and his attention lifted from his work—not because he wanted to start a conversation, but out of excitement at hearing someone other than himself mention the name “Erdős.” He looked around.
It was a fellow student, a young man with shoulder-length hair, cheek propped up on one hand, his shirt hanging open at the neck. Ishigami had seen him around. He was a physics major, but beyond that, Ishigami knew nothing about him.
Surely he can’t be the one who spoke, Ishigami was thinking, when the long-haired student, still propping up one cheek, remarked, “I’m afraid you’re going to hit your limits working with just a pencil and paper—of course, you’re welcome to try. Might get something out of it.”
Ishigami was surprised that his voice was the same one he’d heard a moment earlier. “You know what I’m doing?”
“Sorry—I just happened to glance over your shoulder. I didn’t mean to pry,” the other replied, pointing at Ishigami’s desk.
Ishigami’s eyes went back to his notebook. He had written out some formulas, but it was only a part of the whole, the beginnings of a solution. If this guy knew what he was doing just from this, then he must have worked on the problem himself.
“You’ve worked on this, too?” Ishigami asked.
The long-haired student let his hand fall down to the desktop. He grinned and shrugged. “Nah. I try to avoid doing anything unnecessary. I’m in physics, you know. We just use the theorems you mathematicians come up with. I’ll leave working out the proofs to you.”
“But you do understand what it—what this—means?”
Ishigami asked, gesturing at his notebook page.
“Yes, because it’s already been proven. No harm in knowing what has a proof and what doesn’t,” the student explained, steadily meeting Ishigami’s gaze. “The four-color problem? Solved. You can color any map with only four colors.”
“Not any map.”
“Oh, that’s right. There were conditions. It had to be a map on a plane or a sphere, like a map of the world.”
It was one of the most famous problems in mathematics, first put into print in a paper in 1879 by one Arthur Cayley, who had asked the question: are four colors sufficient to color the contiguous countries on any map, such that no two adjacent countries are ever colored the same? All one had to do was prove that four colors were sufficient, or present a map where such separation was impossible—a process which had taken nearly one hundred years. The final proof had come from two mathematicians at the University of Illinois, Kenneth Appel and Wolfgang Haken. They had used a computer to confirm that all maps were only variations on roughly 150 basic maps, all of which could be colored with four colors.
That was in 1976.
“I don’t consider that a very convincing proof,” Ishigami stated.
“Of course you don’t. That’s why you’re trying to solve it there with your paper and pencil.”
“The way they proved it would take too long for humans to do with their hands. That’s why they used a computer. But that makes it impossible to determine, beyond a doubt, whether their proof is correct. It’s not real mathematics if you have to use a computer to verify it.”
“Like I said, a true adherent of Erdős,” the long-haired student observed with a chuckle.
Paul Erdős was a Hungarian-born mathematician famous for traveling the world and engaging in joint research with other mathematicians wherever he went. He believed that the best theorems were those with clear, naturally elegant proofs. Though he’d acknowledged that Appel and Haken’s work on the four-color problem was probably correct, he had disparaged their proof for its lack of beauty.