Euler, Erdős

This is just a collection of my thoughts on debates/discussions and how being a mathematician has shaped them. It's mainly sparked by a group chat I had with friends over cultural appropriation and, even though this is a math blog, I think it is an important example of how versatile math ideas are. Most importantly, I will hone in on a pet peeve:

Look at the dates on those tweets! I'm going to go full-on meta and use analogies to explain why I hate analogies. Hopefully, by the end of this post you will appreciate how not quite so ironic this is. Here we go:

Some people ask me why I love math so much; they wonder if math is just about calculating bigger numbers. I used to think the same way too until I realized math is more about logic than anything else I can think of. Math, to me, is all about making assumptions and finding out what the consequences of said assumptions are. The types of assumptions we make range from straight forward, e.g. \( C \) is a circle with radius of \( 1 \), to somewhat abstract, e.g. \( E/F \) is a Galois extension, to very obscure, e.g. \( F(-) \) is a functor. The obvious thread that permeates all areas of math is the use of logic (proofs) to make conclusions (theorems). What I find very interesting and would like to discuss here is how mathematicians critique theorems. Suppose I make the following claim and give a proof for it too:

Claim 1. All unit circles have a perimeter of \( 2 \)

Proof: Suppose you have some unit circle: \( (x-a)^2 + (y-b)^2 = 1 \). Let \( \alpha(t) = \left< a + \cos \theta, b + \sin \theta \right> \) be its paremetrization, then the perimeter is given by \[ \begin{aligned} \int_0^{2\pi} |\alpha'(t)| \, dt &= \int_0^{2\pi} 1 \, dt \\ &= 2 \end{aligned} \]\(~\Box\)

If you think my claim is bogus, which it is, then here is what you should NOT do. You shouldn't say:

Mathematicians deal with counterexamples all the time. But if you are giving me a counter-example then it is your job to first prove that your counterexample satisfies my hypotheses. In this particular case, the hypothesis is the shape is a unit circle. Before telling me that your counterexample does not have a perimeter of \( 2 \), tell me why you think it is a unit circle. The advantage of this is it weeds out most baseless counterexamples and we're able to focus on the important ones. Then the first counterexample is immediately seen for what it is, a square. Second counterexample is not a unit circle, since the radius is \( 2 \) not \( 1 \). The third has the equation \( x^{\frac{28}{15}} + y^{\frac{28}{15}} = 1 \) which is not quite the unit circle \( x^2 + y^2 = 1 \) but very close.

So the shapes are not direct counterexamples but you are onto something. To change my mind, you could argue that

• the unit circle can be circumscribed in the square and it is odd that there is such a huge difference in perimeter: \( 2 \) vs. \( 8 \).
• when you double the radius of a circle, then you double the perimeter. But the larger circle has perimeter \( 4\pi \), not \( 4 \).
• the shape is very close to a unit circle and so their perimeters should also be very close. My shape has perimeter a lot more than \( 2 \) and in fact very close to \( 2\pi \).

Another perfectly reasonable response to my claim is critiquing the logic. There are a lot of deep ideas that I actually did not mention explicitly. These are moments where my logic possibly fails.

You said the unit circle is \( (x-a)^2 + (y-b)^2 = 1 \) : yes, that comes from the definition of a circle and Pythagoras' theorem. I can explain it or give you a reference to it.

You gave the parametrization for the circle. What is that? That's some trigonometry. I can explain it or give you a reference.

Where does that integral come from? It's the arclength formula. I can explain it or give a reference.

Where does that \( 1 \) in the integral come from? It's the magnitude of a derivative. I can explain it or give a reference...

Where does the final \( 2 \) come from? It's the fundamental theorem of calculus (FTC); it was proven by Newton & Leibniz several hundred years ago. I can... Wait a minute, I made a mistake. By FTC, the final answer should be \( 2\pi \) not \( 2 \). Thanks!

I now fix the claim:

Claim 2. All unit circles have a perimeter of \( 2\pi \)

At this point, you will mull over the proof for a while, read the references I gave, and then hopefully accept my claim as true. Sometimes, you will find a flaw in my proof or come up with a solid counterexample in which case I will retract my claim and attempt to fix it. Overall, this is what a conversation about a new theorem in math might look like.

Now I'm going to try and apply it to how we debate in general. Specifically, how we use analogies to try show that someone's argument is ridiculous. In this case, analogies are acting sort of like counterexamples. Let's say we are talking about cultural appropriation and I make the following claim:

Claim 3. Iggy is an example of cultural appropriation

Proof: She is white but she raps in AAVE. I do not think that is okay.\(~\Box\)

A few things to note:

• Whereas in math, we have clear-cut definitions, in real life, such definitions don't exist. I have not defined what cultural appropriation is because the topic is too broad. My “proof” is not a justification of why my claim is true but more of an explanation of what I mean.
• Conversations in real life are rarely in the claim-proof format but I think it is good practice to use the format in your interpretation before you start critiquing. What I probably said was, “Iggy is appropriating black culture cause she is white but she raps in AAVE.”
• This is not about whether my claim is justified; it's about the proper way to respond to it with a counterexample.

You might think I'm being ridiculous but here is how you should NOT respond:

“Is Justin Timberlake appropriating as well?”

“Is French Montana appropriating as well?”

“Is Drake appropriating as well?”

So if you think my statement was bogus and you think Justin Timberlake (JT) is a good counterexample to it, then that's fine. But it is your job to first tell me why he is relevant. What does JT have to do with Iggy? Until you properly explain why you're bringing him up, there isn't much I can do to respond. First identify the premises in my claim and briefly explain why they apply to him as well. I do not know how JT can be made relevant but at least the second question can be rephrased as:

“French Montana is white and raps in AAVE. Are you not okay with him too?”

This is now a valid analogy: you have pointed out what you think my premises are: 1) Iggy is white. 2) Iggy raps in AAVE. Then you found an example of another artist who satisfies the same premises and for whom I will probably make a different conclusion. Now I'm forced to fix my reasoning:

She has an australian accent when she talks normally but she uses AAVE in all her raps. I am not okay with how she tries so hard to sound like that. On the other hand, French Montana raps the way he speaks and I don't mind it.

“Drake who is from Canada occasionally uses Jamaican Patois. Why is no one accusing him of cultural appropriation?”

Iggy built her whole career on another people's dialect and Drake did not. His occasional use of Jamaican Patois is okay with me. However, I will say people still make fun of him for it.

As you can see, my original claim was not very clear and it should have raised some concerns. It was your use of examples and your explanation of why you think the examples are relevant that helped me clarify my standing. In the process, we were able to focus on what my premises were. This is how you keep a healthy discussion moving forward.

Now back to my meta-analogy. The issue at hand was: Don't use analogies! I could have given a logical reason on why I don't like analogies but then such abstractness may not have been properly appreciated. Instead, I gave an example with math theorems and an analogy in the context of a regular discussion; I also took care to point out some of the places where the two examples are actually different. Hopefully, you now understand what my tweets meant:

Claim 4. Analogies/Examples are only useful if you first put in the effort to explain why they are relevant.

Proof: This post.\(~\Box\)

If you disagree, show me the flaw(s) in my reasoning and/or give me one or more relevant counterexamples. Don't just say, “Jesus used parables all the time.”