Air from Archimedes

When I was ten, I read with astonishment that with each breath, I was inhaling molecules that were breathed by the mathematician Archimedes over two thousand years ago.

This sort of invocation of chemistry as a magic history-spanning bridge can be traced back to James Jeans, the English scientist and mathematician, who in his 1940 book “The Kinetic Theory of Gases” wrote: “If we assume that the last breath of, say, Julius Caesar has by now become thoroughly scattered through the atmosphere, then the chances are that each of us inhales one molecule of it with every breath we take.” The science writer Sam Kean recently wrote an entire book, “Caesar’s Last Breath”, that takes this proposition as its starting point.

In between Jeans and Kean, other writers making the same point have replaced Caesar by Archimedes or Jesus or da Vinci. I prefer Archimedes, because he was the first of the ancient Greek mathematicians to come to grips with really big numbers and to connect the macroscopic and microscopic realms; in “The Sand Reckoner” he calculated how many grains of sand would fill the universe as the Greeks understood it.

As I write this essay in April 2020, human society has been violently tipped on its side, and the eight billion or so people who share this planet have come to realize how small the world has become epidemiologically. We’ve also become fearfully conscious of the contents of the air we bring into our bodies. Perhaps now is a good time to take a deep and hopefully healthy breath and think a bit about how the content of our lungs connects us to people far away in space and time, situated in a past that, even at a remove of a few months, feels very distant.

The Square Root of Pi

Bella: You gotta give me some answers.

Edward: “Yes”; “No”; “To get to the other side”; “1.77245…”

Bella (interrupting): I don’t want to know what the square root of pi is.

Edward (surprised): You knew that?

— Twilight

March 14 (month 3, day 14 of each year) is the day math-nerds celebrate the number π (3.14…), and you might be one of them. But if you’re getting tired of your π served plain, why not spice things up by combining the world’s favorite nerdy number with the world’s favorite nerdy operation?

Chess with the Devil

“As far as God goes, I am a nonbeliever. Still am. But when it comes to a devil — well, that’s something else.”

— The Exorcist (William Peter Blatty)

Sometimes a key advance is embodied in an insight that in retrospect looks simple and even obvious, and when someone shares it with us our elation is mixed with a kind of bewildered embarrassment, as seen in T. H. Huxley’s reaction to learning about Darwin’s theory of evolution through natural selection: “How extremely stupid not to have thought of that.”

This phenomenon often arises as one learns math. Mathematician Piper H writes: “The weird thing about math is you can be struggling to climb this mountain, then conquer the mountain, and look out from the top only to find you’re only a few feet from where you started.” In the same vein, mathematician David Epstein has said that learning mathematics is like climbing a cliff that’s sheer vertical in front of you and horizontal behind you. And mathematician Jules Hedges writes: “Math is like flattening mountains. It’s really hard work, and after you’ve finished you show someone and they say “What mountain?””

Drawings by Ben Orlin. Check out his webpage: https://mathwithbaddrawings.com/

What Proof Is Best?

“You don’t have to believe in God, but you should believe in The Book.”

— Paul Erdős

Creating gods in our own image is a human tendency mathematicians aren’t immune to. The famed 20th century mathematician Paul “Uncle Paul” Erdős, although a nonbeliever, liked to imagine a deity who possessed a special Book containing the best proof of every mathematical truth. If you found a proof whose elegance pleased Erdős, he’d exclaim “That’s one from The Book!”

I’m a fan of Erdős, but today I’ll argue that the belief that every theorem has a best proof is misguided.¹

The Null Salad

“If you have arugula, basil, celery, dandelion, and endive leaves, how many different tossed salads can you make?” That question, or something like it, was asked in a Math Bowl that I participated in back in high school, during my halcyon days as a mathlete.¹ Actually, “halcyon days” are supposed to be calm days, and quiz-show-style math-smackdowns aren’t known for being calm.  It was certainly an un-halcyon moment when my Math Bowl teammates were urgently saying we should buzz in with the answer 32 to that salad question, and I was saying we needed to figure out whether the judges would think that a bowl containing no ingredients at all was a valid salad.  While we were debating the issue, the other team buzzed in with the answer 32, only to be told “That is incorrect.” Our team immediately buzzed in with the answer 31, which seemed likely to be the answer the judges were looking for.

We got the points, but I liked the other team’s answer better. The idea of an empty salad might seem like a purely mathematical fancy, but half a dozen years later I saw a restaurant menu that offered the null salad, or rather “Nowt, served with a subtle hint of sod all” (for the unbeatable price of 0 pounds and 0 pence).² 

Cartoon by Ben Orlin. Follow him on Twitter @benorlin! Read his blog at https://mathwithbaddrawings.com! Buy his books from your local independent bookseller!

Let Us Define Our Terms

It was a truism of mid-twentieth century popular intellectual culture that many disagreements were “merely semantic” and could be resolved if only people would agree on the meanings of the words they used, or at least were more clear about the different ways they used words so that they could focus on substantive issues rather than language.

Cartoon by Jules Feiffer. Permission pending.

It’s not hard to see that this idea has serious limitations. For instance, even though many legal issues surrounding abortion hinge on different definitions of the word “life”, when it comes to the moral side of the debate, definitions don’t change anyone’s mind. Usually we each choose the definition that matches an outcome we’ve decided on, not the other way around. But in mathematics (thank goodness for the consolations of math!), things are different.

Guess Again: The Ehrenfeucht-Mycielski Sequence

The nice friendly way to play Twenty Questions is to select in your mind a secret something (a person, place, or thing) and to give honest answers to a bunch of true/false questions about it. A less nice way to play is to keep changing what you have in mind so that you can answer “No” to every question. That’s not a good way to keep friends, but something very much like it is a good way to generate a quasi-random sequence of bits.

Cartoon courtesy of Ben Orlin. Order his new book “Change is the Only Constant” now!

Sphere-Packing

Is there a way to pack more than 4 disks of diameter 1 into a 2-by-2 square?

Obviously not. But is there a way to pack more than 4000 disks of diameter 1 into a 2-by-2000 rectangle?

Again, obviously not — except that there is a way! (See my essay “Believe It, Then Don’t” for details.) So packing problems can be tricky.

Calculus is Deeply Irrational

You’re lying on a beautiful beach when you feel a tap on your shoulder, and suddenly you’re not at the beach at all — you’re in a classroom. The student who woke you looks apologetic, and from the front of the room the teacher is staring at you expectantly. He points at the blackboard on which he has written the function f(x) = 6x − x³ next to its graph. “I said: how can we find the maximum value achieved by this function on the interval from 0 to 2?”

A pleasant dream has been replaced by your worst nightmare. But into your still-sleep-fogged conscious mind rises a catchphrase, your only chance for salvation. “Um… Take the derivative and set it equal to zero?”

My Favorite Theorem

2019 is a great year to learn some calculus. Not only are there the videos of Robert Ghrist and Grant Sanderson, but there’s a wonderful new book out by Steven Strogatz. Strogatz has spent the last thirty years growing into the kind of writer who could produce the book about calculus that the world needs, and now he’s produced it. In a few months Ben Orlin will be coming out with a book of his own, and the chapters I’ve seen make me wish I had the power to magically forget calculus (temporarily), so I could have the experience of encountering the subject for the first time through Orlin’s delightful combination of lively prose and cutely inept drawings. And as if that weren’t enough, this year we also have David Bressoud‘s clarion call for teachers to improve the pedagogy of calculus by putting its standard topics back into something like the order in which they were discovered. Calculus is having a gala year.

The celebration is long overdue.¹ Calculus is one of the triumphs of the human spirit, and a demonstration of what perfect straight things (and perfect curvy things) can be made from the crooked timber of humanity. It’s given us a way of seeing order amidst the variety and confusion of reality, hand-in-hand with the expectation that when things happen, they happen for a reason, and that when surprising things happen, it’s time to look for new forces or additional variables.

One of my favorite theorems is a calculus theorem, but it’s not a theorem anyone talks about very much. It may seem mundane (if you’re mathematically sophisticated) or silly (if you’re not). It’s seldom stated, and when it is stated, it’s a lowly lemma, a plank we walk across on the way to our true destination. But it’s a crucial property that holds the real number line together and makes calculus predictive of what happens in the world (as long as we stay away from chaotic and/or quantum mechanical systems). It’s called the Constant Value Theorem, and it can be stated as a succinct motto: “Whatever doesn’t change is constant.” (This is not to be confused with the motto “Change is the only constant”, which happens to be the title of Orlin’s book.) I’ll tell you four things about this theorem that I find surprising and beautiful.