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 clearer about the ways they used words differently so that they could focus on substantive issues rather than language.

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 our verdict, not the other way around. But in mathematics (thank goodness for the consolations of math!), things are different. Continue reading# Author Archives: jamespropp

# 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.

# 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*) = 6*x* − *x*^{3} 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A pleasant dream has been replaced by your worst nightmare.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?” Continue reading

# 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. Continue reading

# 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 steps^{1} 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.

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.* Continue reading