Burn Math Class: And Reinvent Mathematics for Yourself (2016)

4. On Circles and Giving Up

4.2. Impostor Syndrome

Author: Wait, impostor syndrome? How is that a similar problem?

Mathematics: GIVE ME A MINUTE. I’LL EXPLAIN.

Author: Sorry, continue. What’s the problem?

(Mathematics clears its throat.)

Mathematics: SOMETIMES PEOPLE RECOGNIZE ME. I MEAN, I LOVE PEOPLE, BUT THE ENCOUNTERS CAN BE RATHER. . . AWKWARD. THEY ASSUME I KNOW THINGS I DON’T. THEY’LL COME TO ME WITH THESE QUESTIONS AND THEY’RE SHOCKED WHEN I DON’T HAVE AN ANSWER FOR THEM. I DON’T KNOW WHO OR WHAT THEY THINK I AM. I WISH I COULD SOMEHOW CONVEY TO THEM (THAT)²’S NOT ME. . . ANYWAYS, THAT’S A DIFFERENT PROBLEM. THE NEED TO BE UNDERSTOOD. . . OR THE DESIRE. . . WHATEVER. I GAVE UP ON THAT A LONG TIME AGO.

(Hey! Giving up is the theme of the chap—

Author: Shut up, Narrator. Not now.

Mathematics: STILL, THAT ASIDE, IT WOULD BE NICE TO FEEL LESS LIKE AN IMPOSTOR. . .TO KNOW SOME OF THESE THINGS EVERYONE EXPECTS ME TO KNOW, YOU KNOW? IT WOULD MAKE THOSE EVERYDAY INTERACTIONS MUCH LESS UNCOMFORTABLE. PLUS, MAYBE IF I TAKE THE EFFORT TO KNOW THESE THINGS I’M SUPPOSED TO KNOW, SOME PEOPLE MIGHT RECIPROCATE THE EFFORT. TRY TO UNDERSTAND ME BACK. WHO KNOWS? THERE MUST BE ONE, OR, MAYBE EVEN A FEW PEOPLE OUT THERE WHO WOULD TRY. . . BUT I CAN’T JUST SIT AROUND BEING MYSELF AND HOPING TO BE UNDERSTOOD. . . FIRST INEED TO PRACTICE BEING WHAT THEY EXPECT ME TO BE. SO I CAN BE BOTH. OR EITHER. OR NEITHER. WHATEVER I NEED TO. WHATEVER HELPS.

Reader: Wow, that escalated quickly. . .

Mathematics: SO ANYWAYS, BECAUSE OF ALL THAT, I THOUGHT I’D TRY TO INVENT SOME OF THE BASICS FOR MYSELF. I MEAN, I’M LITERALLY MATHEMATICS. I SHOULD BE ABLE TO DO THIS, RIGHT?

Author: Seems reasonable.

Reader: So, any luck?

Mathematics: NO. STUCK IMMEDIATELY. COULDN’T EVEN INVENT THE MOST BASIC THINGS.

Reader: What did you start with?

Mathematics: SOMETHING I GET ASKED ABOUT A LOT. ABOUT CIRCLES. I DON’T KNOW WHY EVERYONE THINKS I CARE SO MUCH ABOUT THESE THINGS. THEY’RE JUST SHAPES. THERE ARE SO MANY MORE INTERESTING THINGS OUT THERE! BUT STILL. . . THAT KIND OF THINKING IS HOW I ENDED UP WITH THIS IMPOSTOR SYNDROME IN THE FIRST PLACE — NOT KNOWING THE THINGS PEOPLE EXPECT ME TO. SO, BACK TO THE MUNDANE. . . IT WAS A QUESTION ABOUT THE DISTANCE AROUND A CIRCLE. A DUMB, CHILDISH QUESTION. THAT’S WHY YOUR QUESTION REMINDED ME OF IT. JUST A CIRCLE. THEY’RE SURPRISINGLY DIFFICULT TO DEAL WITH.

Author: Is it the curviness?

Mathematics: THAT’S IT EXACTLY. HERE’S THE PROBLEM I HAD. LET’S CALL THE DISTANCE ACROSS A CIRCLE d. THE d STANDS FOR dAMMIT IF I CAN’T EVEN DO THIS I SHOULD JUST GO LIVE UNDER A BRIDGE. OR dISTANCE. I HAVEN’T DECIDED YET. ANYWAYS, I WAS TRYING TO FIGURE OUT HOW MANY d’S IT TAKES TO WALK ALL THE WAY AROUND THE CIRCLE. I COULD TELL THAT THE DISTANCE AROUND IT WAS MORE THAN 2d, BECAUSE THE DISTANCE AROUND THE TOP HALF IS CLEARLY BIGGER THAN d. THAT WAS OBVIOUS. THE ONLY OTHER PROGRESS I MADE WAS TO FIGURE OUT THAT IT HAD TO BE LESS THAN FOURd’S, BECAUSE IF I IMAGINED A SQUARE AROUND THE CIRCLE, THEN THE DISTANCE AROUND THE SQUARE WAS 4d, AND THAT WAS CLEARLY LARGER.

Reader: Sounds familiar.

Author: Yeah, that’s almost exactly what we did when we were stuck on our problem.

Mathematics: MY BEST GUESS WAS THAT THE NUMBER OF d’S IS SOMETHING LIKE 3, BUT I COULDN’T FOR THE LIFE OF ME FIGURE OUT WHAT THE EXACT NUMBER WAS, SO EVENTUALLY I JUST GAVE UP. I HAD BEEN WORKING ON THE PROBLEM FOR A LONG TIME, AND I DIDN’T WANT ANYONE TO KNOW THAT IHADN’T FIGURED OUT THE ANSWER. CIRCULAR REASONING IS EMBARRASSING ENOUGH, BUT CIRCULAR REASONING WHILE REASONING ABOUT CIRCLES ON SUCH AN EMBARRASSINGLY SIMPLE QUESTION ISN’T THE KIND OF THING I WANTED TO MAKE PUBLIC.

Author: No need to be embarrassed.

Mathematics: PLENTY OF NEED. BEING WHO I AM, IT WOULD HAVE BEEN ON THE FRONT PAGE OF EVERY TABLOID IN THE VOID. SO LATE ONE NIGHT I SNUCK DOWN TO THE VOID’S LOCAL ABBREVIATION STATION — DISGUISED AS ACCOUNTANCY SO AS NOT TO ATTRACT ANY ATTENTION — AND I PICKED OUT A SYMBOL TO HIDE MY IGNORANCE.

Reader: That’s basically what we did too.

Mathematics: IT’S STRANGE THAT NONE OF US CAN UNDERSTAND SOMETHING SO SIMPLE.

Author: It sure doesn’t seem simple.

Mathematics: MAYBE IT ISN’T. EITHER WAY, I PICKED OUT A “GIVE-UP SYMBOL” AND USED IT TO WRITE THIS:

Author: I see you also used the Molière trick. What’s with the music symbol?

Mathematics: WELL, I WAS GOING TO USE THE “NUMBER” SYMBOL # TO REMIND ME IT WAS A NUMBER, BUT I’M NOT TERRIBLY FOND OF NUMBERS, AND I DON’T LIKE BEING REMINDED OF THEM TOO MUCH. THIS LOOKS SIMILAR ENOUGH. SEEMED LIKE A GOOD COMPROMISE.

Author: Sounds reasonable. Wait, your best guess was that was something near 3?

Mathematics: YEAH. JUST A GUESS, THOUGH. I DOUBT IT’S EXACTLY 3.

Reader: Our best guess for our area number # was something like 3, too.

Mathematics: INTERESTING. . . THE PROBLEMS DO SEEM EERILY SIMILAR. I WONDER IF THE TWO NUMBERS ARE THE SAME.

Author: No way. What are the chances of that?

Mathematics: WHO KNOWS? MAYBE WE COULD CONVINCE OURSELVES THAT THEY HAD TO BE THE SAME NUMBER.

Author: Haven’t you been paying attention? We don’t know either of these things. How could we possibly figure out if they were the same?

Mathematics: WE WOULD NEED A WAY TO RELATE AREAS AND DISTANCES. RELATE SOMETHING TWO-DIMENSIONAL TO SOMETHING ONE-DIMENSIONAL. I DON’T KNOW HOW T—

Reader: Well. . . one-dimensional things sort of look two-dimensional when we draw them. . . Like how a line almost looks like a long thin rectangle.

Author: Yeah, but it’s not.

Reader: No no. I know. But work with me here. Say I draw a “line” of length ℓ on a piece of paper. It’s not really a line, right? I mean, it’s not actually one-dimensional. If we zoom in on it, it would just be a really thin rectangle with length ℓ and some really tiny width dw. So we all know that thin rectangles look almost like lines. Maybe we could do something similar for circles and invent a way to relate areas and lengths. If we get lucky and everything works out nicely, we might be able to see if the two numbers were the same.

Author: This sounds interesting. . .