# Is Math Art? Part I

Is math art?

For many, math brings to mind pencils and paper, obscure symbols, and dusty chalkboards. Even if you like math, you probably don’t think of it as being in the same category as painting or poetry.

Why ask if math is art?

Two written pieces prompted me to ask this question.

The first is a reflection by a mathematician looking back on his choice of career. The other is a critique of modern mathematics education.

A Mathematician’s Apology

Mathematician G. H. Hardy published “A Mathematician’s Apology” in 1940. In it, Hardy writes, “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”

The idea of math as patterns is something I’ve written about before. Hardy then articulates a further idea: “The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way.”

This description touches on artistry in math. But the idea that math’s patterns “must be beautiful” may be puzzling. What does Hardy mean by that?

A more recent essay, by an elementary school math teacher, takes an even stronger stance.

Lockhart’s Lament

Fifteen years ago, research mathematician Paul Lockhart decided to devote himself to elementary school teaching. In 2002 he wrote “A Mathematician’s Lament” as a critique of the current state of K-12 math education. The essay is brilliantly written: funny, sharp, and thought-provoking. (While I don’t endorse every one of Lockhart’s views, I highly recommend the piece.) Hardy spoke of patterns and beauty; Lockhart was much more direct:

“Mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.”

Lockhart imagines a world where all arts are squeezed into the educational format that is now used to teach mathematics. In that world, art is stripped of meaning, reduced to something children hate and fear — because of the way it is taught. Lockhart exposes what he sees as a tragic distortion of the art of math:

“Everyone knows that something is wrong. The politicians say, ‘we need higher standards.’ The schools say, ‘we need more money and equipment.’ Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, ‘math class is stupid and boring,’ and they are right.”

Mathematics, Lockhart says, has been distorted beyond recognition. “The fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.”

Finally, “Mathematics is the purest of the arts, as well as the most misunderstood.”

Why?

Hardy and Lockhart both viewed mathematics as art. And, from what I can tell, they are not alone. Many of the mathematicians I have personally encountered have expressed similar thoughts.

Some mathematicians feel that they are artists. Why? What quality of mathematics pushes its practitioners to describe it with words like “beautiful,” “elegant,” and “poetic”?

I’ll try to answer that in the next post.

# On Math and Meditation

I find it almost impossible to explain the experience of being involved in math.

To describe something, a person uses words. If I have seen a pineapple before, and you have seen a pineapple before, and we both know the word “pineapple,” we can use the word “pineapple” to communicate. But if you’ve never seen such a fruit, I have to be creative to get you to understand. I might describe its stringy chewiness, bright yellow color, and piquant aroma; I might draw parallels to your other experiences, perhaps compare it to other foods. That way, we could communicate, even though you don’t have a previous concept of “pineapple.”

So here I am, every week in this blog, trying to explain math ideas in creative ways — such that, even if you’ve never savored a good piece of it, you can appreciate math’s unique flavor. This particular post will try to explain, in the best words I can find, what the experience of math is like for me.

What it’s not

• Anything like what you did in middle school.
• Adding, subtracting, multiplying, or other arithmetic operations. Or, manipulating algebraic symbols, remembering formulas, or drawing accurate pictures. In my experience, these are tools — useful, but not the heart of the experience. They are there to facilitate the more interesting ideas that come later.
• Getting the “right answer.” That framework is pretty much irrelevant.

What it is (sort of)

Thinking about math involves multiple steps, much the way that baking a cake involves measuring, mixing, and applying heat.

Math starts with a translation step. When reading mathematics, you have to peel the words and symbols off the page, and turn them into a concept in your head.

If I read, for example, about an “open sphere S in R^4,” I first translate that symbolic language to semi-English words:  “the interior of a four-dimensional sphere.” Then I mentally review what “interior,” “sphere,” and “four dimensions” mean. (I’m not going to explain what these are here — I’m just bringing them to show that I translate symbolic statements into ideas I can understand.)

This translation process gives me a kind of mental picture of the sphere and its properties. I then hold that picture in mind and move on to the next phrase. A slightly sped-up version of this happens when I am sitting in class: when my professor refers to a difficult concept, I’ll pull up associations and mental pictures to make sense of what he or she is saying.

All of this translation can be difficult sometimes. It’s not to figure out anything deep — it’s just for seeing what exactly is being referred to. But when I encounter this type of challenge, even though it happens before I encounter the actual math, I’ll keep working at it, because I know that the concepts will be interesting once I get ahold of them.

Once the concept is in my head, there is … well … there’s the math.

Math requires you to take a concept and stretch it, turn it over, invert it, or break it into pieces. It involves taking a logical argument and deciding whether it really makes sense, creatively employing tools to solve a problem, and often inventing tools on your own. More than almost anything else I know, it really works the mind.

The world of the concrete, of grass, clouds, chairs, and pencils, falls away. In its place, I consider an abstract scene: the grand, sweeping architecture of calculus; the straight, clean spaces of linear algebra; the chopped-up landscape of discrete mathematics.

The experience is completely absorbing in a way that I find almost meditative. My mind has no space for other thoughts. Everything else must quiet. With intense focus and hard thinking, I can often see the elegance of a mathematical pattern: something beautiful, outside of the self, that isn’t like anything else I know.

I can’t explain what math looks like from the inside, if you’ve never seen it, any more than you can explain the flavor of a pineapple to someone who has never tasted an acidic fruit. But I can say that pineapples are tingly, bright, sweet and little sour; and I can say that understanding math abstractly is like coming upon a mountain or a waterfall after a long hike. And that if the moment of understanding is like coming upon a waterfall, then, the experience of thinking is like the exhilaration of maneuvering through a dense growth of forest.

For me, climbing that path is almost as thrilling as reaching the destination.

# Math Backwards: Part II

This post contains a little more technical stuff, with less explanation, than many of the posts on this blog. If there is something you don’t understand, feel free to ask about it below. Or skim it – you won’t miss out on the main idea.

In the last post, we looked at this mystery number:



We left off with a question: what is the value of this sum, written as a “regular” number?

To keep things from getting too messy, I’m going to make up a new name for the mystery number. I’m going to call it “M”: that’s a nice short name, and it reminds me of the word “mystery.”

The letter M here is an example of notation. I am choosing to represent the mystery number as M, instead of as a big fraction. This type of neatness is generally useful. The more abstract an idea, the more important it is to have a good way to put it on paper.

Unfamiliar notation can make written math look intimidating, at least until you learn what each symbol stands for. But good notation makes things clearer. An eloquent discussion of notation can be found in a classic book on problem solving, George Pólya’s How To Solve It.

I have two ways to approach it, though there are certainly more out there that I didn’t think of. My first method involves some algebra.

Method Number One

Take a look at this number again.



Notice that if we peel the outermost “1+1/” off this number, we end up with another copy of M, albeit in a slightly smaller font. Because of the fractal nature of M, the bottom half of the fraction is the same as the entire number. That gives us two different ways to write the mystery number: as M, or as 1 plus 1 over M. We can write that as an equation:

$1+frac{1}{M} = M$.

By writing M in two different ways, we learned something about it. This equation means that if you take 1 plus 1 over the mystery number, you get the mystery number again.

Now, if we can find a “regular” number that satisfies this equation, we know it must be the mystery number.

Well, if $1+frac{1}{M}$ and $M$ are equal, that means they both represent the same number. They are two different ways of writing the same thing. So, if I subtract 1 from each of those terms, I’ll get a new number, but both sides will still be equal. That’s the idea behind a lot of algebraic manipulation: when two terms are equal, they represent the same number, so if you do the same thing to both of them, they will still be equal.

$1+frac{1}{M}-1= M-1$.

Now, $1+frac{1}{M}-1$ is the same as $frac{1}{M}$. So,

$frac{1}{M} = M-1$.

Since these two terms are equal, we can multiply each by M and again get two equal terms. So,

$M times frac{1}{M} = M times (M-1)$.

Multiply this out, and the M’s on the left cancel. On the right side, we multiply everything in the parenthesis by M. So,

$1=M^2-M$.

We still don’t know which number makes this true, but the equation is starting to look workable. Now, subtract 1 from both sides again.

$0 = M^2-M-1$.

This is a special type of equation called a quadratic equation, and it has a 0 on one side. Using something called the quadratic formula, which I will not explain in depth right now (though it does have good reasons behind it if you want to learn about it!), we find

$M=frac{1+sqrt{5}}{2}$.

Hmm, that’s interesting.

Before we talk about why that’s interesting, let’s take a different perspective on this number.

Method Number Two

Let’s look at a number that’s similar to the one we’re interested in: $1+frac{1}{1+frac{1}{1}}$. It almost looks like a mini-version of M, but it’s not quite the same, because it stops repeating itself pretty quickly. There is no “…” in this number.

First, let’s look at the fraction in the second term of the denominator: $frac{1}{1}$. This is equal to 1.

Next, the denominator as a whole: $1+frac{1}{1}$ is equal to $frac{2}{1}$, or 2.

Finally, let’s look at the bigger fraction: $1+frac{1}{1+frac{1}{1}}$ is equal to $frac{3}{2}$.

What if we added another layer, another “1+1/”, to the outside of this fraction? We’d have $1+frac{1}{1+frac{1}{1+frac{1}{1}}}$. We know what most of this fraction looks like already. The whole thing is equal to $1+frac{1}{1+frac{3}{2}}$, which is the same as $frac{5}{3}$.

So far, by working outwards, we’ve got a little sequence: there’s $frac{1}{1}$, then $frac{2}{1}$, then $frac{3}{2}$, then $frac{5}{3}$. If you add another layer to the outside of the fraction, you get $frac{8}{5}$. Add another, and you get $frac{13}{8}$.

You might notice a pattern here. These fractions involve a sequence of positive integers. Each fraction is made up of some integer in the sequence, divided by the previous integer in the sequence. And the integers are: 1, 1, 2, 3, 5, 8, 13…

It’s the start of the Fibonacci sequence.

That’s weird.

Maybe this pattern will stop after a little while – there have certainly been stranger coincidences in mathematics. Or maybe the pattern continues forever. But there’s no way to directly check – no matter how many fractions we evaluate, there is always room to add another “1+1/” on the outside of the fraction.

But I think the Fibonacci pattern continues forever.

Actually, I don’t just think it. I know it. And I can prove it.

Here is my proof.

To make things easier to write, I’m going to call the first Fibonacci number $F_1$, the second Fibonacci number $F_2$, and so on. I’m also going to start with the fact that for some number of layers, the fraction is equal to a Fibonacci number divided by the Fibonacci number before it. I’m going to call that number of layers $n$. So when there are $n$ layers, the fraction is equal to $frac{F_n}{F_{n-1}}$.

We just calculated this for a bunch of values of $n$, and we know it’s true for all of those. But for right now, we’re just assuming that $n$ is some specific number of layers for which the fraction is $frac{F_n}{F_{n-1}}$. We’re not specifying which number $n$ actually is.

Now for some more algebra. By adding another layer to $frac{F_n}{F_{n-1}}$, we get

$1+frac{1}{frac{F_n}{F_{n-1}}}$.

This is equal to

$1+frac{F_{n-1}}{F_{n}}$

Which is equal to

$frac{F_n+F_{n-1}}{F_{n}}$.

So far so good. Now there’s one important thing we know about Fibonacci numbers: each is the sum of the previous two Fibonacci numbers. In other words, $F_{n-1}+F_n=F_{n+1}$. That tells us that the fraction above is equal to

$frac{F_{n+1}}{F_{n}}$.

Let’s review what we just did here. We showed that if a fraction with $n$ layers is equal to $frac{F_n}{F_{n-1}}$, then a fraction with $n+1$ layers is equal to $F_{n-1}+F_n=F_{n+1}$. That’s the same formula as $frac{F_n}{F_{n-1}}$ for $n$ layers, but with $n+1$ in place of $n$.

We already know that when $n=2$, this works: when there are two layers, the fraction is equal to $frac{F_2}{F_{1}}$, or $frac{1}{1}$. Now, since the fraction works for $n$, it works for $n+1=3$ as well. And since it works when $n=3$, it must work for $n=4$. And since it works for $n=4$, it must work for $n=5$… and so on. This proves that for any positive integer $n geq 2$, this fraction with $n$ layers is equal to $frac{F_n}{F_{n-1}}$. The pattern continues forever.

As $n$ gets bigger and bigger, the value of $frac{F_n}{F_{n-1}}$ gets closer to a particular number. That number is known as the Golden Ratio.

The number M is what you would get if you added infinitely many layers to this fraction. You’d get the number that $frac{F_n}{F_{n-1}}$ approaches as $n$ gets bigger and bigger.

In other words, M is the Golden Ratio. As we showed with Method One, M is $frac{1+sqrt{5}}{2}$.

I don’t know about you, but I find this connection pretty amazing.

Well, there you have it. We took a strange-looking math idea, played with it, and ended up finding connections to some totally different ideas. Hopefully it was a different experience from that of the drills and rules you might associate with math.

There are a lot of follow-up questions I could ask on this topic.

But I’ll let you ask the questions yourself.

# Count to 1023 On Your Fingers: The Video

In the last post, we talked about base systems. It turns out that’s all you need to know to be able to count to 1023 on your fingers.

Here is how to do it.

Thank you to Nik for filming!

# Count to 1023 On Your Fingers

If you have “31” potatoes, how many do you have?

Really. It’s not a trick question. So give it a try — what do you think? How many potatoes do you have when you have “31”?

When I wrote 31, you likely assumed that I meant “three groups of ten, plus one more.” You understood that 31 means that quantity. It was like seeing the written string of letters “pie” and thinking of a round, fruit-filled pastry dessert.

But you may also know that the letters “pie” don’t always refer to a dessert. If I were writing in Spanish, “pie” would refer to someone’s foot.

The string of letters P-I-E has two meanings. Each makes sense in its own context.

The same thing is true of 31. In most situations, 31 means “three groups of ten, plus one more.” That’s because nowadays, we use what is called the base ten system.

We all relate to base ten as “normal.” But it is important to understand what is going on when we go from 9 to 10.

To understand what base ten really does, imagine that I’m a potato farmer, with, say, 1436 potatoes to ship to my friends outside of Idaho. I have special crates to put my potatoes in. Each potato crate holds exactly ten potatoes: no more, no fewer.

I use my potatoes to fill as many crates as possible. Any single potatoes left over, after all the crates are full, are put on the side. Those single potatoes are represented by the rightmost digit (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9). So, with 1436 potatoes, and crates that hold ten each, I will have 6 potatoes left over.

You probably know how this pattern continues. The second-to-rightmost digit works like the rightmost digit, but on a bigger scale. I pack all the ten-potato crates into big sacks. Each sack holds ten crates – that’s ten crates, not ten potatoes. I fill as many sacks as possible, and then count how many crates are left over. 1436 potatoes leaves me with 3 crates not in sacks along with the 6 potatoes not in crates.

And of course, the third digit to the right works the same way. Not only am I shipping my friends some of my mountain-grown potatoes, I actually have an order from a distributor on the East Coast. I load the sacks into trucks, and each truck holds ten sacks. The third-to-rightmost digit represents the number of sacks left when I have filled as many trucks as possible. If there are 1436 potatoes, that means that once the trucks are filled, there are 4 sacks, 3 crates and 6 potatoes left.

You might have noticed a pattern here. Each crate holds ten potatoes. Each sack holds ten groups of ten potatoes: that’s ten times ten potatoes, or one hundred potatoes. Each truck holds ten times ten times ten potatoes: one thousand potatoes. Every step up in size – that is, every digit further to the left – holds ten times as many potatoes as the last one. If, say, there were a barge that held ten trucks, it would hold ten times ten times ten times ten potatoes: ten thousand potatoes.

This is why we can also write 1436 in a way similar to something you may remember from grade school: “1 group of one thousand potatoes, 4 groups of one hundred potatoes, 3 groups of ten potatoes, and 6 single potatoes.” It is the same as writing, “1 group of (ten times ten times ten) potatoes, 4 groups of (ten times ten) potatoes, 3 groups of ten potatoes, and 6 single potatoes.” Likely none of this is revolutionary: you probably realized it already, on some level.

But with this understanding now explicit, other base systems suddenly become interesting. To jump to another base, there is only one thing you need to do. Change “ten” to another number.

Imagine you go to a new Base Sixteen potato farm, and are shocked to see larger crates! Base sixteen represents how potatoes would be sorted at that farm. Instead of crates that hold ten potatoes, sacks that hold ten crates, and so on, the crates hold sixteen potatoes, the sacks hold sixteen crates, and so on. To the people who store those potatoes, 1436 means “1 group of (sixteen times sixteen times sixteen) potatoes, 4 groups of (sixteen times sixteen) potatoes, 3 groups of sixteen potatoes, and 6 single potatoes.” That’s a lot more potatoes than the “1436” from my farm.

In base sixteen, the set of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is not enough. Eleven, for example, can’t be written in base sixteen as a combination of those digits. Usually, people use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F to represent zero through fifteen in base sixteen. So, 6A means “6 groups of sixteen, plus eleven more.” Base sixteen is also called hexadecimal. It is used a lot in computer science, including as a way to represent colors in HTML. (I’m a fan of #8904B1 and #00FFBF.) This very website, like many others, uses hexadecimal notation to specify the font color and background color.

You may be wondering how all of this helps you count to one thousand and twenty-three on your fingers.

Believe it or not, it IS possible to count that high. All you need is the notion of base and a little creativity.

I’ll show you how in the next post.

# Turtles All the Way Down

The point of this post is not to teach or explain anything directly. Instead of walking through the steps of an idea, I am going to tell you three stories. The stories are meant to get you thinking. In my opinion, all three of them point to a single topic.

By design, there are no straightforward answers given here… not even the name of the topic. (Hint: It’s not “turtles.”)

A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: “What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.” The scientist gave a superior smile before replying, “What is the tortoise standing on?” “You’re very clever, young man, very clever,” said the old lady. “But it’s turtles all the way down!”

Steven Hawking, A Brief History of Time

The old lady in this story has a strange scenario in mind. The Earth, she says, is sitting on a turtle. That turtle is standing on a second turtle. The second turtle is standing on a third. And so on: turtles all the way down.

Of course, giant space turtles are impossible in a practical sense. No one really takes that part of the argument seriously. But the old lady’s reasoning holds a deeper logical flaw.You probably got a sense of it when you read her words. There is something not quite right, though what exactly is wrong may not be immediately obvious.

The issue lurks somewhere the old lady’s final statement. Even if there are turtles holding up the Earth, they cannot go “all the way down.” Her scenario provokes discomfiting questions: How far is down? How many turtles are there? How can there be no bottom turtle?

The Footrace

Hawking’s tale is reminiscent of another famous story: the footrace described in one of Zeno’s Paradoxes.

A race has been called between a tortoise and the Greek mythological figure Achilles. Achilles is known for being swift, and the tortoise is slow. Because of this, the tortoise gets a head start.

One imagines the turtle ten yards ahead of the starting point. The race begins. Achilles quickly makes it to the ten-yard line. But by that time, the tortoise has moved further forwards, perhaps to the twelve-yard line.

Achilles can reach the twelve-yard line, but by then, the tortoise has moved still further. Though Achilles gets closer and closer to the tortoise, he can never pass him: whenever he reaches the tortoise’s most recent location, the tortoise will have already moved ahead.

Of course, this is not how things play out in real life. If this paradox governed real motion, all sorts of commonplace things would be impossible. But they are possible, and they happen all the time. A slow runner can lose a race to someone faster. A car in the left lane of a highway can pass a car on the right.

Zeno’s fictional race conflicts with our experience. And yet, when presented this way, his logic is not easy to refute.

The Endless Notebook

“What the Tortoise Said to Achilles” is another paradoxical piece. This one is written by Alice in Wonderland author Lewis Carroll. It is a story about the same characters mentioned in Zeno’s Paradox, and it can be read here.

The Tortoise strikes up a conversation with Achilles, as the two approach the finish line of their race. They start with an innocuously simple question (which I cannot explain better than Lewis Carroll did!) But soon the Tortoise’s dizzying list of inquiries fills up Achilles’s notebook. His ideas never seem to stop.

And in fact they never will. Just as it was with the old lady and Zeno, there is something wrong with the Tortoise’s logic, but it is hard to put one’s finger on exactly what.

There’s an unsettling catch in each of these three arguments. In Hawking’s story, the turtles go “all the way down.” In the footrace of Zeno’s paradox, Achilles gets “closer and closer” to his opponent. And in Carroll’s story, the Tortoise always writes down yet another hypothetical step.

These scenarios are obviously not naturally possible. And yet they hold an appeal. They seem to make sense. And each seems to make sense because it rests on a subtle logical assumption. Somewhere deep in each argument lies a premise that seems true, but isn’t. But what?

Do each of these stories have something in common, besides turtles and tortoises? Are two alike, and one different? And what is the real problem with each argument?

Feel free to comment below.

# The Integer Hat Shop

Say you’re going shopping for some new hats to add to your wardrobe. You walk into the hat shop and pick up a nice baseball cap. But you don’t just want one—you want three. Or maybe you want five. Point is, you will walk out of the store with a specific number of hats. That number won’t be 1.5, or the square root of 7: it will be a positive integer.

Why the word “integer?” Why use such a term?

Math is full of not-quite-English words that make many people cringe.

But the words aren’t meant to confuse people. No one decided to use the word “integer” just to make it difficult for an eighth grader to do his homework (even though it can seem that way). The words are not used make math seem complicated or exclusive.

They’re shortcuts—used so that a concept doesn’t need to be explained every time it’s invoked. Every language, including English, uses shortcuts in this way. “Cookie” is a shortcut that refers to a round, bite-size, crumbly baked dessert. “Quack” is a shortcut that refers to the sound that ducks make. Some ideas don’t require shortcuts, like the sound that zebras make. But we would make a word for it if we referred to it a lot.

Math isn’t (usually) talking about cookies, zebras or ducks. It needs its own shortcuts – that is, its own special words.

Take the word integer. Integer is a shortcut term.

You can think of integers as the units used to count whole objects. Those objects can’t be broken into pieces. For example, 3 is an integer. You can buy 3 baseball caps: that’s one baseball cap, plus one baseball cap, plus one baseball cap. But you can’t buy 1.5 baseball caps: that would be one baseball cap, plus a half of a baseball cap. That doesn’t make sense. If you have to rip up a baseball cap to get the number of hats you want, that number is not an integer. So 1.5 is not an integer.

What about the “positive” part? To understand positive integers, look no further than their rivals, the negative integers.

Imagine a baseball cap made out of some fantastical antimatter. It looks kind of like a regular baseball cap, but it is different in one important way. When it comes in contact with a regular baseball cap, both vanish! They cancel each other out.

Regular hat.

Anti-hat.

In this way, the negative number -2 is the “antimatter” of the positive number 2. When you add them together, both vanish. You are left with 0.

You can add up hats and anti-hats, just like you can add positive and negative integers. This kind of concrete idea can really help when you’re doing arithmetic. Let’s say you add 4 anti-hats to 5 regular hats. Then 4 anti-hats will vanish right along with 4 of the regular hats. That leaves 1 regular hat in your hands. It is a concrete way to understand the equation (-4) + 5 = 1.

What about 6 regular hats added to 10 anti-hats? All cancel out except for 4 anti-hats. That is like saying 6 + (-10) = -4. Positive integers correspond to baseball caps, and negative integers correspond to anti-caps.

Zero is another interesting concept. It doesn’t fit neatly into either the “positive” or “negative” category. If you have zero hats and zero anti-hats, do you have a positive number of hats? Do you have a negative number? You don’t really have either. Zero gets its own category: still an integer, but neither positive nor negative.

Zero hats.

I can’t speak for all people, but for me, this kind of analogy is the best way to understand a new math concept.

It is unsatisfying to learn a technique, memorize it, and repeat it without understanding. But with a picture in your mind, your approach changes immediately. I can mentally pick up a hat and move it around, smash together hats and anti-hats, and toss around an ephemeral zero. Once I’ve done that enough, I can skip the hats and do the same things with integers.

This is the process I have gone through for almost every math concept I have ever learned. To understand integers, I thought of solid objects like hats. To gain an intuition about angles, I thought about toothpicks, connected, at the corners, by balls of clay. To learn about derivatives, I thought about curvy shapes, speeds, and snapshots.

Math isn’t supposed to be scary. It comes from real, basic ideas: ideas that make sense to the human mind, like numbers, shapes, and speeds. When these ideas are singled out for abstract discussion, they need their own special vocabulary to go with them. “Integer” means something more specific than a word like “amount.” So it often pays to use it.

Math ideas are precise, so to express them, we need precise terminology. That’s it. That’s how math language develops and why we use it.

So next time you hear a math term that seems intimidating, keep in mind: it’s probably a lot simpler than you think.