(October 25, 2018) An important concept in category theory, indeed in mathematics itself, is deciding which attributes to ignore when you conduct mathematics. Context matters. When you’re signing a contract to buy a car, you may not care what color the contract is printed on. But when you’re sending out your wedding invitations, you might care about the paper color a lot. So in our first Math Circle on category theory, we played a lot of Odd One Out.
ODD ONE OUT
The students found it easy to conclude that the grey duck was the odd one out when all the others were yellow and every other attribute was identical. But what to do when multiple attributes change? After vigorous debate over some pre-determined groups of objects,* the students created their own odd-one-out challenges for each other. They used markers, cubes, pentominoes, playing cards, shape tiles, and rocks.
“Even though this one is the odd one out because of color, you’re wrong. A different one is the odd one out.”
“You’re right that this one is the odd one out, but your reason is wrong.”
Again and again, students stated versions of the above two comments. Serious mathematical discoveries were going on here:
- Changing assumptions changes the answer.
- There can be multiple paths from beginning to end.
- You have to state your reasoning or your conclusion won’t stand up to scrutiny.
Soon, the students changed the wording of their replies:
“You’re right that this one is the odd one out because of color, but that’s not what I had in mind. I was thinking that a different one is the odd one out.”
“You’re right that I was thinking that this one is the odd one out, but your reason is not the reason that I was thinking.”
Do you see the significance of the change in wording?
The students were essentially creating physical proofs, except that here, the conclusion/proposition wasn’t stated at the beginning as it is in a mathematical proof. You could say that the students created the building blocks of mathematical proofs. To paraphrase mathematician Eugenia Cheng,** proofs are like storytelling with a beginning, middle, and end. In the beginning you state your assumptions and definitions. In the middle you state your reasoning. In the end, you state your conclusion (i.e. “ta-da!”). In our group, the beginning, middle, and end of the proof emerged through asking questions, positing conjectures, rejecting or accepting conjectures, and then finally a statement of the author’s reasoning.
The students proved that “this one is the odd one out because it is the only one that”
- is a non-primary color.
- has the white cubes totally enclosed in a boundary.
- doesn’t start with the letter B.
- has no symmetry.
- has no other objects in the group that share a shape.
- has a rough texture.
- has writing of a slightly thicker font.
- etc., etc., etc.
No one set up a group where the obvious attribute (color, size, etc.) was the exception. The exceptional attributes were hidden. Just like in an interesting math problem! I expected this activity to take 5-10 minutes, but it took much longer. I had set an alarm to give us a three-minute break after 50 minutes. When that went off, A said, “Wow, Math Circle goes fast!” It felt like we had just gotten started. We could have done this activity all day. Even when we moved on to other activities, the students took odd-one-out interludes to continue challenging each other.
Another aim of category theory is abstraction – seeking the underlying structure of things in a way that allows you to see a similar structure in seemingly very different things. By “things” I mean mathematical things. By “mathematical things,” I mean things that have a logical structure. I have seen Cheng extend “things” to non-mathematical things. She applies category theory to real life. I hope to do this later in the course, but for now we’re sticking to logical things.
We explored Cheng’s juxtaposition of the symmetries of equilateral triangles to permutations of the numbers 1,2, and 3.*** It may have seemed like this was an activity about properties of triangles, types of symmetries, and one way to calculate a permutation (listing it out). The students did conclude that an equilateral triangle has 6 symmetries and a list of three digits has 6 permutations. But the big question was this: is that just a coincidence? Or (dramatic music here) could they really be the same problem?
One thing I love about doing “high-level” math with younger students (here, ages 10-12) is that they are not blinded by too many preconceived notions about math. (Of course, they’re blinded by some. I see five-year-old students convinced that any problem that’s not about number theory is not mathematics and that there’s only one right answer no matter what and that every question has an answer and that the teacher knows it and that there’s only one way to get the answer and and and…. Okay, I’ll get off my soapbox now.)
“Of course, it’s the same problem,” S right away announced. The others quickly agreed with her. We talked briefly about how the underlying structure of things can be the same despite the obvious differences. This idea blew my mind when I first encountered it at an age much older than these students. I suspect that at least some older students might think it’s just a coincidence.
I put “high-level” in quotes because category theory is generally not taught to students before graduate school or possibly third-fourth year college. Cheng is leading a movement to make it accessible to middle- and high-school students. Math Circles in general are hoping to make deep mathematical study accessible to younger and younger students.
In our last few minutes, we played an informal round of Would You Rather:
- Would you rather have some golden eggs, or a goose that lays golden eggs?
- Would you rather have a goose that lays golden eggs, or a machine that makes geese that lay golden eggs?
- Would you rather have a machine that makes these geese, or a machine that makes machines that make these machines?
This discussion was rich, going off in many philosophical and mathematical directions. Cheng gives this example**** to make a point about abstraction: “in order to build a machine to do something rather than doing it yourself, you have to understand that thing at a different level.” You have to analyze every step and every implication of those steps. This is some of the thinking we do when our goal is abstraction. We were out of time at this point, so never really did get to a discussion about this perspective. We’ll start there next week.
All of this in 75 minutes – phew! Fortunately, we have five more weeks.
*Here are the puzzles on paper that the students debated:
** How to Bake π , Eugenia Cheng, pp66-68
*** Cheng, p17
**** Cheng, p31