“Jewish Mathematics”?

Quick math-personality quiz: What is seven-and-one-fourth minus three-fourths, expressed as a mixed number (a whole number plus a proper fraction)?

What matters isn’t what answer you get but how you arrive at it; your thought-process will reveal what kind of thinker you are. So please stop reading now and continue once you’ve found the answer.

Got the answer? Here are two common ways of getting it:

You could convert 7 1/4 into 29/4, subtract 3/4 from that to get 26/4, and reduce that fraction to get 13/2, or 6 1/2.

Or, you could reason that, because increasing each of two numbers by 1/4 doesn’t change the difference between them (or to put it in daily-life terms, the height-difference between two barefoot people doesn’t change if they both put on 1/4-inch shoes), 7 1/4 minus 3/4 equals (7 1/4 + 1/4) minus (3/4 + 1/4), which equals 7 1/2 minus 1, or 6 1/2. Alternatively, you could reason that 7 1/4 minus 3/4 equals (7 1/4 − 1/4) minus (3/4 − 1/4), which is 7 minus 1/2, or 6 1/2; same idea, same answer. Pictorially:

The red lines are all the same length, and the length of each red line equals the difference between the two numbers on the number line associated with its endpoints.

Did you solve the problem the second way, nudging the two numbers upward or downward? Congratulations: you’re thinking like a German. But if you solved the problem the first way, converting the mixed fractions into improper fractions, then I have bad news: you’re thinking like a Jew.¹

That doesn’t mean you’re actually Jewish; it’s possible that some of your math teachers were. You might not have known that they were Jewish at the time; they might have had wholesome Aryan looks and deceptively Christian names. And you may have been too young to realize that they were infecting you with Jewish mathematics.

Nine Years of Blogging About Math

[This month’s post is a transcript of a talk I gave on March 13, 2024, as part of a UMass Lowell “Conversation Starter” event on the topic of scientific literacy and communication. I was addressing other members of the Kennedy College of Sciences, so I treated mathematics as a subfield of the sciences, though I feel that math is an art as well as a science.]

Quick: What 19th century mathematician is this?

If a journalist had buttonholed the nineteenth-century pure mathematician shown above and asked him what his revolutionary ideas about geometry were good for, and if the mathematician had answered “Someday people will use my ideas to figure out exactly where they are, not by looking up at the stars but by looking down at little boxes they carry around in their pockets,” the mathematician’s colleagues would have been justifiably outraged by such unscientific sensationalism.

No points awarded for recognizing this guy.

There was no way Bernhard Riemann or anyone else in 1854 could have known, guessed, or dreamed that his new kind of geometry would give someone named Albert Einstein the mathematical language he needed in order to express his theory of general relativity, which now is a key part of the geopositioning technology embedded in your smartphone, a century and a half later.

Plus and Times Set Free

This essay chronicles the liberation of plus and times from narrow notions about what sorts of things can be added and multiplied — and, relatedly, what sorts of things numbers should be. With the advent of abstract notions of fields and rings in the late 19th and early 20th centuries, algebraists took addition and multiplication far beyond the borders of Number and in so doing showed that the border was artificial and an obstacle to progress.

One way I’ll bring a 20th century perspective into focus is to show how an abstract point of view can unify many of the disparate number systems (and not-quite-number systems) we’ve looked at so far, using the simple but powerful idea of modding out, freed from number-centric bias.

As a warm-up to all that, I want to tell you about an under-publicized paradox of mathematical history: The French mathematician Augustin-Louis Cauchy (1789–1857), the person who did more than anyone else to bring complex numbers into calculus, didn’t believe in that most famous of complex numbers, sqrt(−1). (See my essay “Twisty numbers for a screwy universe” if you need a brief reminder about the twisty ways of imaginary and complex numbers.) Cauchy couldn’t have been clearer; he wrote “We completely repudiate the symbol sqrt(−1), abandoning it without regret because we do not know what this alleged symbolism signifies nor what meaning to give to it.”

And yet, even as he rejected sqrt(−1), Cauchy accepted the symbol i that Euler had introduced for sqrt(−1). In fact, Cauchy didn’t just accept i; he glorified it. So what exactly did Cauchy think complex numbers were, or could be?

What “a 96 percent chance” doesn’t mean

When people find out I’m a mathematician, they assume I’m into numbers. I find this assumption frustrating (and a little sad) since there’s so much more to math than numbers, but the truth is that I am into numbers — so into them that I’m writing a book about them.

It’s also the case (though less than it used to be) that when people find out I’m a mathematician, they assume I’m into NUMB3RS. And I’m not.

Vectors from Leibniz to Einstein

And how naive to have imagined that the series ended at this point, in only three dimensions!

           − 2001, A Space Odyssey, by Arthur C. Clarke

The German philosopher-scientist Gottfried Leibniz dreamed of a universal language and a method of calculation to go with it, so that, if disputes arose on any subject, a disputant could exclaim “Calculemus!” (“Let us calculate!”) and the method would yield the answer.

It was an audacious dream and Leibniz knew it, but, hoping to bring one small corner of his dream-language to life, he attempted to invent a “geometry of situation” that would blend algebra and geometry. He adopted 􏰀♉︎ (the zodiacal symbol for taurus) to signify the relation of congruence, and he tried to reduce all the traditional notions of geometry to properties of ♉︎.¹ Leibniz aimed to devise a kind of “Solid Geometry 2.0” that would allow him and others to reason about geometric situations by first translating the situations into statements involving ♉︎􏰀 and then making algebra-style deductions from those statements using suitable axioms.

Leibniz wrote to his friend the Dutch mathematician Christian Huygens in 1679 that he envisioned a language that would describe not just geometry but the actions of machines. “I believe that by this method one could treat mechanics almost like geometry, and one could even test the qualities of materials.” Leibniz failed to convince Huygens and others to help him develop his ideas, but with hindsight we can see glimmerings of the idea of vectors in what he wrote. If only he’d based his system on translational congruence – the relation that holds when one figure can be obtained from another using only sliding, not rotation – he would’ve gotten closer to the modern concept of vectors and everything that the concept led to, including (most recently) chatbots.

Numbers Far Afield

“To learn all that is learnable;
to deliver all collected data
to the Creator on the third planet.
That is the programming.”

(Vejur, née Voyager 6, in Star Trek: The Motion Picture)

Imagine a vast growing sphere centered on the Earth with a radius of fifteen billion miles, the distance a beam of light travels in a day. This is the anthroposphere: the patch of the universe into which humanity and its artifacts have spread. At its periphery – indeed, defining its periphery – is a one-ton device hurtling away from us at forty-thousand miles per hour while sending radio signals toward Earth with a broadcast power of twenty watts. That’s only a tiny fraction of the wattage used by commercial radio stations on Earth, yet somehow Voyager 1, with its puny transmitter, is still executing its mission of gathering data about our solar system and relaying that data back to far-off Earth. And part of the technology that makes this feat possible is an algebraic construct that nobody even dreamed of two centuries ago: the arithmetic of finite fields.

Marvelous Arithmetics of Distance

No reckoning allowed
save the marvelous arithmetics
of distance

         (from Smelling the Wind by Audre Lorde)

Suppose a child comes up to you and says “I know 1 is odd and 2 is even, but I think 4 is more even than 2, and 1/2 is more odd than 1.” You might be tempted to reply “There’s no such thing as ‘more’ or ‘less’ odd; a number either is odd or it isn’t. And fractions aren’t odd or even; they’re just fractions.” But if you did, you’d be missing an opportunity for some serious and far-reaching fun.

Every adult who teaches kids about math should be aware that in advanced mathematics the “school rules” don’t always apply, and a kid who says something that appears to ignore the rules of the road might be intuiting some of this off-road mathematics. In the particular example I gave, if you refrain from correcting the child but instead say “Okay, I’ll play” and help the child explore the consequences of their nonstandard point of view, the two of you might end up re-inventing p-adic numbers!

If you’ve gotten this far in your life without meeting any p-adic numbers, you might think you don’t need these spooky creatures. But beware. Mathematician Paul Garrett writes: “The p-adic integers and related objects are already right under our noses, if only we can see them. … It is unwise to ignore them.”

When Five Isn’t Prime

How far would you go to save a theorem? Would you invent a new kind of number? That’s what the mid-19th century German mathematician Ernst Eduard Kummer did, and while he was partly driven by the hope of proving Fermat’s Last Theorem, that wasn’t actually the theorem he was trying to save.

Math for Your Ear

I didn’t have time to compose an essay this month¹, so I can’t offer any of my own writing for you to read (unless this 500-word trifle counts). But I feel I should offer you something, so I decided I’d tell you what I’ve been reading lately in a pop-math vein. Or rather, what I’ve been listening to, since these days I mostly listen to audiobooks.

If your tastes resemble mine, the enjoyment you’ll derive from the audiobooks I recommend will exceed the pleasure you would have taken from the essay that I didn’t write!

My five recommendations are:

The Triumphs of Sisyphus

To err is human; we all make mistakes. But some mistakes have worse consequences than others. According to Greek myth, King Sisyphus of Ephyra made the especially big mistake of cheating Hades, the God of Death. Twice. Hades said “Fool me once, shame on you; fool me twice, you have to roll a huge boulder up a hill only to have it roll back down again, forever and ever.” I like to imagine Sisyphus counting his steps, repeatedly reaching the same top number just before the heart-breaking moment when the boulder returns to the bottom and Sisyphus is back at 1 again. Ironically, this kind of cyclical counting has played a big role in helping us non-mythical people catch our own mistakes in calculations and, more recently, in helping us correct errors in messages we send and receive across vast distances.

We use Sisyphean counting to keep track of time: the hours go from 1 up to 12 and then back to 1 again. When we waltz, we keep track of our movements with “one, two, three, one, two three …” (until we stop consciously counting and just enjoy moving to music in accordance with Leibniz’s observation “Music is the pleasure the human mind experiences from counting without being aware that it is counting”). But the simplest kind of cyclical counting people do (usually unconsciously) goes “one, two, one, two, …” with every odd number in the counting process replaced by “one” and every even number replaced by “two”; Leibniz would probably say it’s part of the pleasure of taking a walk.