Marion Deutsche Cohen
Portrait of the Mathematician as a Young Child
Sometimes at night, insomnia or not
and forgetting sheep, staying abstract
I would count. Once I decided to count to a million.
It took about a week.
At first I was good.
Clearly, slowly, each four-digit number completely annunciated..
But after ten thousand there were too many digits.
I got tired of thinking which digits they were.
So I decided, instead of counting to a million
I’d count to ten-thousand one hundred times.
True, I’d have to keep track of two countings
but there’d be fewer digits
and less thinking
about which digits they were.
Treatise on the Smaller Positive Integers
(1) Properties of One
Some of them are new-age. Some are downright religious, like “Hear O Israel, the Lord our God, the Lord is One.” But there are also the properties I thought about in high school and first-year college, the properties I listed in that treatise. One is mystical but it’s also mysterious.
“One is the loneliest number,” says the song. But to be somebody’s one, or to have a one, that wasn’t lonely. There being no others was the beginning of warmth and security. I once wrote in a poem, “So tired am I that I want there to be one of everything.”
There’s a painting called “One”. It’s a scribble-scrabble of lines, curves, and funnels.
There’s the word “unique”, no such word as “two-nique”. There is also “universe” and the pronoun “one” used like the generic singular “you”.
n points on a line divide the line into n+1 segments. But in order for a thing to be in n stages, it must undergo n-1 changes. And in order to add up n numbers, we do the adding n-1 times.
There is one-to-one correspondence. There could also be two-to-two correspondence but nobody talks about that.
One times any number is that number. A thing fits into itself one time.
When we define something, we define one of that thing.
There are more advanced mathematical properties of one, like the simplest continued fraction, it has all ones. But that would come later, after grad school, after teaching. For now, what I wrote down were the properties advanced mathematicians call trivial. They mean that in a bad way but I still don’t.
(2) Portrait of the Mathematician as a First-Year Undergrad
I was in love with Schumann’s Piano Quintet Opus 44, I was in love with Jeff, I was in love with all the points and lines that could exist, and I was in love with two.
A straight angle is two right angles, a revolution is two straight angles.
A book has two covers, a page has two sides.
The lamp goes on and off, the door open and closed.
The pencil two ends, one to write, the other to un-write.
Translation is between two languages.
The moon has two faces. In the playground two children per seesaw.
An angle can be bisected or quartered, any reciprocal of a power of two.
DeMorgan’s Laws true for any number of sets, but most easily proven for two, and proving for one wouldn’t say much.
And our very words, “both” vs. “all”, “singular” vs. “plural”, “between” vs. “among”. In German “zweifelt” means “doubtful”.
A sentence has two parts, subject and predicate.
People have two eyes, two hands, but only one nose, only one mouth. We have one of what’s in the middle, two of what’s on the sides.
A closed line that crosses itself in n points divides the plane into n+2 spaces.
And the formula for the angle-sum of a polygon with n-sides, (n-2) times 180 degrees.
But what might have been most important, “It takes two to tangle.” Yes, two is the number in a truelove. Why does it have to be two? The answer boiled down to the number one, the one and only, along with the person who has the one and only, that has to make two. I wanted there to be a definite enough reason why two is the first and best non-lonely number, a logical reason to be as monogamous as I was and am. I also wanted there to be a reason why, back in 1961, the number of genders also happened to be two, was that a coincidence?
And so, that year, I kept on thinking about two. The first counting number after one. The number of possibilities for anything, any property, either a thing satisfies that property or it doesn’t. There had been, and would continue to be, plenty of other integers, very large integers, and fractions, functions, and topological spaces. I was in love with all of those things but I still needed more two, I took time from my calculus homework to think and write about two, I had finished exploring one but I wasn’t finished with two.
(3) A Triangle Is a Three-Pointed Star.
Three points determine a plane.
We can see and touch only three dimensions.
Associativity has to do with three elements. So does distributivity. So does transitivity.
In order to prove generalized associativity, we need only prove regular associativity. Some things boil down to three, not two.
From a vertex of an n-gon can be drawn, to some other vertex, n-3 lines that are not sides of the n-gon.
Fermat’s Last Theorem involves three terms.
There are past, present, future. There are mother, father, child.
Sneezes happen in threes.
“My hat, it has three corners. / Three corners has my hat.” That song wouldn’t be as haunting with two corners.
I wasn’t really finished with three but I was ready for four.
(4) There Is Not a Four-Pointed Star.
But four is the common value of two plus two, two times two, two-to-the-two, and two with two in all the higher processes.
Four is the first non-prime.
Every integer is the sum of four squares.
And a crossing polygon must have at least four sides.
Oh, nowadays I could google to find all the special properties of four. But in 1961 there was no google and if there had been, I wouldn’t have wanted to use it.
There are four corners. But there could just as well be five. There are four directions. But there could just as well be five. Yes, four was getting slightly less fundamental. I was beginning to be finished with four. And I didn’t move on to five. Four was enough.
Oh, I did believe that every integer, even a large integer, had simple properties but I didn’t feel like exploring them. I loved five-pointed stars, and five is the first number to make stars. There’s also the 3-4-5 right triangle, and the 5-12-13. And five Platonic solids. And I once wrote in a poem, “Five is the first actual number.” But I didn’t feel like exploring five, five was even less fundamental than four. I would go on to learn about the Five-Lemma in category theory, and how five is the first degree of a general polynomial equation which can’t be solved with an algebraic formula. But that would be learning, not exploring. Five was still a small integer but maybe not quite small enough.
Having Fun Mimicking Ramanujan
Hello there, zero. Hello there, one.
You’re both prime, aren’t you?
And you’re the sum of your digits, the product of your factors.
It’s the fun of the mathematician who’s not a savant.
It’s the revolt of the theory creator
the mathprof who’s not a problem solver.
Goodbye, three-thousand five-hundred seventeen.
It was nice knowing you
if, indeed I am capable of knowing you.
And I kept on quipping
though no longer aloud.
Addicted to the Beginning
“God created the integers. The rest is the work of man.” -- L. Kronecker
Another mathematician said “a lot of work was done by forgetting what he believed.”
God hadn’t yet created the fractions, nor the irrationals, certainly not transcendentals. ButHe was about to. It had taken him a while to be ready. To be almost-finished with the integers.
No, he wasn’t quite finished with the integers. Just like I wasn’t quite finished with straight lines. Or arithmetic. And now I’m thinking about Kronecker’s delta. Like God, Kronecker wasn’t finished with the integers. He wasn’t even finished with zero and one.
Today as he counts we sit with him.
We massage, we sway, we make eye contact.
We’re his labor coach.
We count the contractions, or we count something.
Together and gently we take each separate
Each as it comes
Until he says he’s had enough, it was just false labor.
Another time he’ll have real labor.
We’ll sit with him again, breathe and sway.
We’ll count to the end, or far enough.
We’ll be his doula, we’ll help with the babies
We’ll be his blessingway, we’ll pass them around
We’ll hold and admire each brand new one.
Then we’ll hand them back
To him, the new mother.
Silent and respectful, we’ll watch as he bonds
As he bonds with each
Each unique and precious one
And each two, each three, each twelve, each fourteen
And whatever comes after twenty-nine.
In the bedroom, opposite our bed, hangs that mobile ($3 at a flea market). It’s not sophisticated like Calder but it’s cute. 20 cute discs, sort of shiny but paper, 10 on top, 10 on bottom. some an inch, some an inch and a half, and some black, some grey. In the morning, when I’m awake but not yet moving, I count the discs, first the bottom ones, then the top. I count, that is, the ones I can see. If a disc is eclipsed by another disc, or if it’s at an angle which makes it two rather than three dimensional in my line of sight, or if a grey one is in front of some shadow, I won’t see it so I won’t count 20.
I’ll count LESS than 20 and that’s explainable. But if I should ever count MORE than 20, what then, what of that?
I think of dreams where, although we no longer have cats, I keep finding them all over the house. And I think of that math fiction story where, at the end, someone sees three jellybeans on the table, eats one, and then still sees three jellybeans. Jon my husband the philosophy Ph.D. says one of the questions in the philosophy of math is, why does math conform to the physical world? The “unreasonable effectiveness” of math. 3 minus 1 is 2, but why should that mean there should be two jellybeans left out of three if we eat one? That story is called “The Secret Number”, it’s an integer strictly between 3 and 4 and it’s evil.
I think of Alice in Wonderland, where she discovers things that weren’t there before. Could my cute little non-Calder mobile sprout extra discs? Where would those discs come from? They don’t have to come from anywhere, they can just suddenly BE there. They can be evil. And WHICH would be the extra evil ones? I wouldn’t know which ones to get rid of. I wouldn’t know whether getting rid of them would do any good.
Two is Worse than One
In the small room we notice, on the floor towards the left, a large roach. Then, in the sink, we notice a large roach. “How did it move so fast?” I ask, but Jon says “No, there are two of them.”
That was a horror. Two is worse than one, more worse than you’d think, infinitely worse b/c…. well, in math it takes only two linearly independent vectors to generate, via linear combos, an infinity of them. So yeah a whole plane of roaches. Whole floor, whole ceiling, or whole wall.
Three would be even worse.
List of Things There Are Ten of
10 fingers, 10 toes
and I have a pamphlet titled “10 reasons to get rid of the death penalty”.
True, only 5 stages of grief.
But each stage
like each of the 5 fingers
occurs at least twice.
Part (2) of "Treatise on the Smaller Positive Integers" appeared in The American Mathematical Monthly.