Proof in math

There are been three recent pieces about mathematical proof that have caught my attention in the last couple of months:

(1) Most recently Keith Devlin’s piece here:

(2) Evelyn Lamb’s coverage of Leslie Lamport’s talk at the Heidelberg Laureate Forum (including, importantly, a link to a paper in which Lamport takes a critical look at a proof in Michael Spivak’s Calculus textbook)

A Computer Scientist Tells Mathematicians How To Write Proofs

(3) Numberphile’s “All Triangles are Equilatleral” video featuring Carlo Séquin whose Art and Math collection is always a pleasure to look through:

These three pieces have kept me thinking about proof in math for a while now. I was sick over the weekend and spent a little time browsing through the Museum of Math’s public lectures and found this nice one from Steven Strogatz where, by happy coincidence, the topic of proof in mathematics also comes up:

In this lecture Strogatz discuses a proof of the fact that the area of a circle is equal to \pi r^2. He also wrote an article about the same proof in the New York Times:

Steven Strogatz discusses the proof that the area of a circle is \pi r^2

The combination of the recent writing on proofs in mathematics and watching Strogatz’s lecture gave me an idea for a fun way to talk about proofs in mathematics with my kids. Not in any formal way, but (1) just to show them some easy “proofs,” and (2) to show them that it is ok to question something even if it is supposedly a mathematical proof.

I’ve used this format for a talk previously after learning about Jordan Ellenberg’s concept of “algebraic intimidation” (and you’ll see in our first video from today how much that talk still bothers my younger son!)

Jordan Ellenberg’s “Algebraic Intimidation” and the series 1 + 2 + 3 + 4 + . . . = -1/12

The two topics for today were (1) the proof that Strogatz used to show that the area of a circle is \pi r^2 and (2) the proof that \pi = 4. The proofs are actually quite similar and it turns out that one of them ends up with an incorrect result (though I won’t say which one!).

At the end of each proof I asked my kids what they thought. Funny enough, my younger son does not believe either of them because he is uncomfortable with the use of infinity in the proofs. My older son believes the first one but not the second one.

Here are those two talks – the first is about the area of a circle:

and the second is about the value of \pi:

So, a fun morning with the boys talking through a few “proofs.” I really like the lessons in both Devlin’s piece and in Strogatz’s lecture about using proofs to tell stories and to illuminate. Lamb’s piece about Lamport reminds us that details are important, though, and the two pretty similar proofs I went through with the boys this morning serve as a reminder that the details can actually be pretty subtle (but fun, of course).