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.

Mathematical Flimflam

[Audio version available at http://mathenchant.org/050.mp3.]

I’m a pure mathematician with no background in applied mathematics. But lately I’ve been striving to make a name for myself in the less-crowded field of mis-applied mathematics, and bogus science more broadly.

Now you may be asking yourself, is bogus science really less crowded a field than good science? After all, if Sturgeon’s law (“Ninety percent of everything is crap”) applies to science, then we can expect crappy science to predominate over the good kind. But bogosity transcends mere crappiness. For something to be bogus, I think there must be an attempt to deceive. Or at least, there must be the appearance of an attempt to deceive. Sometimes the appearance is itself a sham, and that’s the kind of second-order bogosity I enjoy practicing, when I try my hardest to act like someone who genuinely believes (and wants others to believe) a nonsensical theory.

My forum is the Festival of Bad Ad Hoc Hypotheses (BAHFest), held periodically in various locations around the world (San Francisco, Seattle, Cambridge, Sydney, and London). It’s a celebration of well-argued and thoroughly researched but completely incorrect scientific theories. BAHFest is dedicated to the proposition that no matter how absurd a premise is, you can find a way to abuse the tools of science to support your cause and make people laugh in the process. (Or make nerds laugh, anyway.)

BAHFest was the brainchild of Zach Weinersmith whose Infantapulting Hypothesis got the game going.

Carnival of Mathematics #170

I’m hosting issue number 170 because I have a thing for the number’s largest prime factor, but it turns out there’s a reason for a Martin Gardner fan like me to appreciate the number itself: 170 is the number of steps¹ needed to solve a classic mechanical puzzle called The Brain invented by computer scientist Marvin H. Allison, Jr., described by Martin Gardner in his Scientific American essay “The Binary Gray Code”, and still available from Amazon.

The Brain, aka The Brain Puzzle, aka The Brain Puzzler.

Here’s what Gardner says about The Brain:

It consists of a tower of eight transparent plastic disks that rotate horizontally around their centers. The disks are slotted, with eight upright rods going through the slots. The rods can be moved to two positions, in or out, and the task is to rotate the disks to positions that permit all the rods to be moved out. The Gray code supplies a solution in 170 moves.

A Mathematician in the Jury Box, or, “But how should we define ‘intoxicated’?”

Back in the 1990s, when I was serving on a jury in a one-day trial, my mathematical temperament got me in hot water with my fellow jurors; fortunately, my outside-the-classroom mathematical training got me out of it. But that doesn’t come in until the end of the story.

The case featured a couple of surprising twists — which is in itself surprising, since even a single twist is unusual in a one-day trial. It had seemed at first like a very straightforward drunk-driving charge. The defendant went to a party, drank some alcohol, left the party feeling unwell, got into his car, drove off, and blacked out, though with enough advance warning of his impending unconsciousness that he was able to pull over to the side of the road and turn on his hazard lights before passing out. A police officer found him slumped over the wheel of his car. The officer smelled his breath and it smelled of alcohol. The District Attorney presented these facts confidently, as if this was going to be an open-and-shut case. But then, in the kind of surprise you see only on television, the defense attorney asserted (with medical records to support his assertion) that in fact the defendant was diabetic, that someone with diabetes can go into hypoglycemic shock if they ingest a little bit of alcohol on an empty stomach, that the breath of someone in hypoglycemic shock is often nearly indistinguishable from the breath of someone who is drunk, and that the amount of alcohol that the defendant had drunk at the party was, according to witnesses, well under the amount that would cause blood alcohol concentration to reach .08% (the legal definition of “too much”).