You know what’s often missing from math class? Yes, candy bars, but even more important than that: coherence.

Math class shouldn’t be a mishmash pile of facts, thrown together haphazardly, like an academic version of *The White Album*. It should be a perfectly interlocking tower of truths, climbing upwards with singular purpose—an academic *Sgt. Pepper* or *Abbey Road*.

A good class isn’t a greatest hits record. It’s a concept album.

In that spirit, I’ve been taking each topic in the secondary math curriculum—algebra, geometry, calculus, etc.—and trying to boil it down to its one-word essence. Here are the rules of the game:

*You must choose a single word to complete the sentence, “[Branch of math] is the mathematics of _____.”*

For example, you might say, “Topology is the mathematics of dinosaurs,” or “Category theory is the mathematics of abstraction,” or “Combinatorics is the mathematics of sadness.” (To be clear, only one of those is remotely accurate; you have my sympathy, combinatorists.)

*You must pick a word that laymen would understand.*

No fair saying “Algebraic geometry is the mathematics of varieties.” What the heck is a “variety”? Like, varieties of breakfast cereal? You shouldn’t need training in a subject just to understand its definition.

*Your word should encompass as much of the subject as possible.*

It might not be a perfect fit. It’s okay if you don’t nail every detail. (Even *Sgt. Pepper* has lots of songs that don’t really fit the theme.) But you want to capture the spirit and scope of the subject.

So you could say, “Algebra is the study of equations,” but that’s weak sauce. Yes, 90% of algebraic work involves creating and manipulating equations. But it totally misses the essence. First, inequalities and expressions can be just as “algebraic” as equations; and second, lots of equations (like 2 + 2 = 4) aren’t remotely algebraic.

That’s the game. If you want to spend some time thinking of your own before being poisoned by my answers, then read no further.

Now, here’s what I’ve got:* *

This one’s a slam dunk. Calculus is already the most beautifully unified course in the high school curriculum, so the one-word treatment fits well here.

The first half of calculus deals with *derivatives*, which tell us how a quantity is changing at a specific moment. “That car is moving 80 miles per hour.” “The city is growing by 80,000 people per year.” “The clown is getting 10% angrier every time you insult his shoes.” It’s all about change. The second half of calculus deals with *integrals*, which allow us to aggregate lots of little, incremental changes to see their cumulative effect.

If you haven’t already cued up David Bowie’s “Changes” on YouTube, then what are you waiting for?

I don’t mean John-and-Yoko relationships, or even John-and-Paul relationships. I’m talking about relationships between quantities.

Pretty much all of elementary algebra boils down to the question, “How do *y* and *x* relate?” You explore different *types* of relationships: linear, quadratic, exponential. And you represent those relationships different *ways*: graphs, tables, equations.** **

“Shape” is a pretty good answer, too. Certainly, you spend most of geometry class studying shapes: triangles, rhombuses, spheres… But I like to think that geometry begins with more elemental ideas. Points. Lines. Planes. Right off the bat, you reckon with abstract concepts like “dimension.” That stuff’s more fundamental than the idea of a shape.

Look around, wave your hands in front of your face, and think about the weirdness of this three-dimensional universe that we occupy. Geometry is the mathematics of *that*. Yes, shapes are important in geometry, but its essence is deeper—and freakier.

My first thoughts were “triangles” and “waves.” But those two are pretty darn different, and that gap points to trouble. How are we going to reconcile such disparate halves? Is this class going to be our version of *Let It Be*—so promising in its conception, but a hot mess in its result?

Fear not! The coherence is there. Fundamentally, trig is about two different ways of understanding position.

First, you can count east/west steps (the x-coordinate) and north/south steps (the y-coordinate).

Or second, you can give a direction (the angle θ) and a distance (the magnitude *r*).

This captures basically all of trig. The SOHCAHTOA stuff, for example, is all about manipulating and integrating these two notions of position. Circle trig, meanwhile, is about translating between them: the sine and cosine functions swallow your direction angle, and then spit up the *y* and *x* coordinates.

To clinch the matter, what is the most powerful historical application of trigonometry? Navigation—which is to say, finding your position on earth.

Sometimes we know exactly what’s going to happen.

For everything else, there’s probability.** **

Maybe I’m overthinking this one. Maybe I should just say “data.” (And maybe the Beatles shouldn’t have released the insufferable song “Wild Honey Pie,” but there’s no going back now.)

But you know what? There’s something vacuous about picking “data.” It’s too obvious. Too technical. It rubs me the wrong way.

So let me make my case for “populations.” Notice how “data” always comes in the plural? Statistics is, fundamentally, about understanding groups. Think of mean, median, mode, range: these are all ways of summarizing a population, of boiling a diverse group down to a few illustrative numbers. Or think about percentile and z-score: these are ways of specifying an individual’s place in relation to the population. And finally, think about hypothesis testing: there, we’re using samples to draw inferences and conclusions about whole populations.

When mathematicians call a statement “true,” they mean something very specific. It’s different from what scientists, artists, plumbers, and politicians. (To be honest, I’m not sure *what* the politicians mean.) They mean that the statement in question follows irresistibly from others that we have already laid out.

So what is logic? It’s the study of how mathematicians use that little word: “true.” As you might expect, it underpins every other branch of mathematics, and it’s a worthwhile field of inquiry in its own right, too.

So, to summarize:

Agreed?

Topology is the mathematics of nearness?

If you change the “?” to a “!”, this could be a Hallmark Card to the mathematician in your life that you love on St. Valentine’s Day.

I’ve previously said topology is the mathematics of spaces, which I think is a nice counterpoint to the geometry answer. It’s the abstract study of spaces and functions between them, paring down to the bare bones necessary to say anything about them at all.

It’s not one word, but I was talking about topology to someone and trying to explain that it wasn’t just some weird kind of geometry and came up with “Topology is what you know about something if you don’t know how big it is.” The more I think about that the more I like it.

Can someone please explain to me whether calculus is the mathematics of anything useful, or just another way for my math professor to frustrate two hundred college kids at once?

Calculus is the study of change – very useful to Engineers. From an academic stand point, it sets the foundations for more interesting and advanced Maths. Rather than ask why frustrate kids, the curriculum should be changed to support science and technical reasoning from day one of schooling, so things like calculus can be put into a real world context with inquiring minds ‘investigating’ this new topic, not learning by rote to pass exams only.

I’m a statistician. We don’t make the headlines much, but behind many scientific studies – including many in climate science, medicine, population studies and the social sciences, there is a statistician or (someone filling that role) using statistical tools to analyze the data. Calculus forms the backbone of almost everything I do. Without calculus, statistics would not be possible. Other fields have a similar relationship with calculus, including computer science, economics, and physics, to name a few.

Consider pi.

It seems reasonable that there would be a constant that relates the diameter of a circle to the length of its circumference; it’s less obvious, but still reasonable, that it’s a linear relationship. That constant happens to be π.

It seems reasonable that there would be a constant that relates the radius of a circle to its area. Again, it’s less obvious, but still reasonable, that it’s a quadratic relationship.

What I find really awesome about this, though, is that the same constant relates the radius to its length and its area (namely, π). And calculus makes it really easy to prove, since the derivative of πr² is 2πr dr.

There are lots of useful applications for calculus, but this is one cool fact. 😉

a “calculus” is a collection of math “tools”.

“Calculus”… i.e., “the calculus of differentials

and integrals”? well, it’s useful for scientists and

philosophers and other suchlike intellectually-

-curious beings.

also for institutions hoping to separate sheep

from goats in some seemingly-honest way.

alas.

i always knew i was gonna be a math major.

except during freshman year when i took

the standard first-year calc course. that

very nearly did me in.

then i got “linear algebra” and things started

making sense again (as i always kind of knew

they eventually would).

life gets better. your milage may vary.

I like your one word description of category theory! A group of interested students is gathering up to form a category theory study group over the summer to delve further into the subject. I encourage anyone interested in the subject to join us at Category Theory Summer Study Group 2015. No significant prerequisites required.

Eugenia Chang said that category theory is the mathematics of mathematics. Five years ago I would have agreed with that, but now it seems to have a competitor: Homotopy Type Theory is another mathematics of mathematics. It is not as well developed yet as category theory but it shows great promise. A hundred years ago Zermelo-Fraenkel set theory tried to be the mathematics of mathematics but it failed miserably.

>Zermelo-Fraenkel set theory tried to be the mathematics of mathematics but it failed miserably.

Nice one, Charles!

>Homotopy Type Theory is another mathematics of mathematics

Note that we can use the ‘generalized the’, since we are category theorists, so both of them can be the mathematics of mathematics 🙂

one poster’s miserable failure is another’s wild success.

look again. look closer.

Logarithms are the mathematics of growth.

mhh maybe

Statistics is the mathematics of guessing.

What would you say to that one?

Beautifully and succinctly put – as usual. I am reading this at 6:18 am, and while that’s way too early to be reading blog posts on the way to work, it’s a great way to start my day and get my brain in gear. Thanks for sharing your talent!

Thanks for reading!

Algebra is the mathematics of rearranging. It goes beyond simply identifying relationships to learning how to restate the same mathematical relationship in multiple ways: If e = mc^2 then m = e/c^2 and c = √(e/m).

Algebra is the mathematics of substitution. I am hampered by the one-word requirement. I would like to say algebra is the mathematics of substitution in formulas.

Hmm… “rearranging” strikes me as just a way of restating and paraphrasing relationships. In your example, Paul, we’re really just manipulating the relationship established by Einstein.

Similarly, I’d argue “substitution” is just a way of exploiting pre-established relationships. E.g., if we know E = mc^2, and we plug in m = 0.0001 gram, we’re just working with a specific instance of a relationship.

“Rearranging” puts an emphasis on the rebalancing aspect of the word al-jabr. “Relationships” strikes me as static; “rearranging” is active.

Mmm, that’s well said. I like that a lot.

It’s good you started with tautology, b/c I think that’s where my brain wants to go. Geometry/measurement, logic/reasoning, trigonometry/triangles. Probably because of your one word rule.

Yeow Meng Chee suggested the very good:

Category Theory is the mathematics of nonsense.

I’ll add one more:

Topology is the mathematics of shape.

I think “shape” fits topology better than geometry. The key concept in geometry is the concept of “angle” (therefore lengths, sizes, etc. come to matter). It’s topology that concerns itself with the shape of an object in a more intrinsic sense- a torus versus a sphere, never mind the radii, never mind if the hole is a bit off-centre or the surface is a bit bent…

Therefore I would suggest:

Geometry is the mathematics of angles.

I think geometry is the mathematics of size. Angles are an important tool for determing size, but so are other geometric ideas.

“Size” isn’t bad, but would seem a little over-inclusive (set theory deals with the size of sets; arithmetic deals with the sizes of numbers; neither is terribly geometric). Maybe “measurement”?

Measurement might be better left for analysis, in which measure (and related weaker notions like Jordan content is fundamental). In this sense, combinatorial or discrete analysis may be seen to be a subarea of analysis that specializes in discrete measures, which not surprisingly parallels the partition of probability into discrete probability and continuous (measure theoretical) probability.

Geometry includes the word measurement in its name (Earth measurement), and the metric is the important distinction between traditional geometry and topology (finite projective geometry might disagree here), the intention of geometry has usually been to convey spatial information. For finite geometry, there is some distinction between the geometries and other incidence structures or 1-dimensional topological spaces.

We matched on the first three!

For Trig, I said “triangles,” which I knew was weak. I’m not sold on “position,” though.

For Probability, I said “chance,” but I like “uncertainty” better.

For Statistics, I said “data,” again knowing it was weak. “Populations” seems just as weak to me, though. But maybe that’s because I don’t care for statistics!

For Logic, I said “thinking.” I do like “truth,” though. It’s interesting–the former makes logic an internal thing, while the latter makes it external. If that makes any sense.

I’m just glad to know that someone feels the same way about The White Album that I always did!

Yeah, my thinking has evolved – I still love a lot of stuff on there, but it just doesn’t hang together! Too many tracks to skip.

How about this: logic is the mathematics of implication. “Truth” is a little distracting to me, because it suggests that we’re concerned about what is true — we’re not. We just want to know what *else* is true if A, B, and C are true. Are A, B, and C actually true? That question falls outside the scope of logic — “above my pay grade,” as they say.

Yeah, I think if you’re standing inside the field of logic, then “truth” sounds all wrong (“I don’t know what’s true, I only know what follows from accepted propositions”).

But if you stand outside the field, then “truth” seems more apt. Logic, after all, is the study of the processes mathematicians use to establish truths.

Statistics is the mathematics of estimating. I think that captures it better. if we know the truth about the population, we don’t need statistics. The key thing is that we need to estimate things because the truth is unknown

Fantastic. I felt like I was back in my freshman Calculus course . . . my prof had a similar personal philosophy of math: simplify the big picture, get them to understand why, and then worry about the details.

Game theory is the mathematics of decisions.

Cryptography is the mathematics of secrets.

Really good and interesting

I like ‘estimating’ for statistics, but I think there should be something better. I think the essential characteristic is being able to know something (not necessarily with certainty) about the world that you haven’t seen from a sample of it that you can see. Filling in the missing bits and drawing conclusions from an obstructed view. (If you can measure something across the entire population, you don’t need statistics.) But I’m not sure how to say that in one word.

Then again, I’m a statistician, and therefore biased. Or at least picky. The way it’s often taught, statistics is the mathematics of p-values. And that’s tragic.

Yeah, possibilities seem to include:

-Estimation

-Inference

-Extrapolation

-Guesswork…

Or, if you want to go with popular perceptions:

-p-values?

-deception in the social sciences?

I like “geometry is the mathematics of lines”. What about points? Well, two points determine a line and two lines determine a point, with the bonus that you can see the point.

geometry “is the mathematics of” *space*, not “lines”.

spac*es*, i’d’ve said. but i didn’t want to get all felix

klein about it and pretend that geometry is group theory.

“two lines determine a point”? what, “infinity” comes in somehow?

euclid never did this.

I agree, blw, statistics is definitely NOT the mathematics of p-values! Sometimes we’re interested in populations (and in many situations where non-statisticians take notice), but other times, the only “population” of interest is the population of errors around a specific measurement. Also, we’re often unable/unwilling to say anything about the population after data is collected – but we’re perhaps willing to say something about the mean and the variation of mean.

perhaps inference is what we are looking for? “Statistics is the mathematics of inference” I think I like estimating better – inference almost seems too broad.

Extrapolation?

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I agree with you on calculus being about change (though I said “changes” so as not to confuse it with numismatics). I think you’ve nailed probability, statistics, and logic as well (though the logicians might argue that “truth” is not a valid predicate).

However, I’d say *topology* is the mathematics of space. Geometry on the other hand really is concerned with shapes.

And for trigonometry, the only correct answer is “triangles”. That’s what the word means, and pretty much every important trig identity has an elegant proof-by-picture of triangles that makes it almost self-evident when drawn.

Elementary algebra is arithmetic with letters. We don’t have a one-word synonym for that, so it stays three words. Arithmetic with letters.

Actual algebra is something more like the mathematics of patterns. Perhaps “relationships” is better here (though it might be best for differential equations). Too bad “isomorphisms” isn’t lay-accessible.

I also said:

• Number theory is the mathematics of integers. (Or primes.)

• Set theory is the mathematics of containment.

• Analysis is the mathematics of rigor. (Though I’m not satisfied with this one.)

Does anyone have a good, non-technical, one to three word summary of linear algebra?

Yeah, I can go either way on topology/geometry. Topology as space and geometry as shape is probably a better way to go.

As for trig, obviously “triangles” is perfectly defensible, but it’s not my favorite, for several reasons:

1. It’s overinclusive, because there’s lots of Euclidean facts about triangles that don’t feel like “trigonometry” to me (e.g., congruence theorems, or the decomposition of all polygons into triangles in order to prove things about them).

2. The right triangle gives birth to the unit circle, and the unit circle gives birth to the sine wave. The triangle is the origin of the lineage, but its descendants strike me as equally central to the subject. An analogy: I wouldn’t say that the study of polygons is just “the study of triangles,” even though everything we prove about n-sided polygons boils down to triangles in the end. Obviously trigonometry USES a lot of triangles, but I see them as the tool for a higher purpose, not the purpose itself. (Construction is the science of building, not the science of bricks.)

3. So what is the purpose of trig? Well, look at its historical and practical uses: astronomy (the position of stars), navigation (the position of your ship on earth), vectors (the position of points in space), complex numbers (specifically, their positions in the Argand plane)… all about position.

As for algebra, with “relationships” I was straining for something that could capture both elementary school algebra (where “arithmetic with unknowns” is pretty good) and modern algebra (where something like “the structure of number systems” is more apt). But “relationships” is probably too broad; a Twitter pal pointed out that geometry deals with all sorts of spatial relationships, none of them terribly algebraic.

Also, I love “containment” for set theory.

Sure, geometry also talks about triangles, but that’s because trigonometry is essentially a detailed, in-depth look at one specific part of geometry. Namely the part dealing with triangles.

Meanwhile, lots of other fields talk about position, without any such relation to trig. For instance, calculus talks about rates of change of position; elementary algebra talks about loci of positions satisfying equations; and measure theory talks about convoluted collections of positions.

One thing that does set trig apart is its consideration of periodic behavior. Waves, orbits, any sort of repetition really. Maybe trigonometry should be considered the mathematics of cycles?

Although at the end of the day, I suppose an argument could be made that trigonometry primarily deals with how circles interact with triangles and lines. So in that sense, one might just conclude that trigonometry is the mathematics of deathly hallows. Maybe run that one by your students and see if it helps?

In “Unknown Quantity”, John Derbyshire offers this definition of “trigonometry” without comment: “the study of numerical relationships between the arc lengths and chord lengths of a circle” (p. 87).

I reflected on that here: http://curiouscheetah.com/BlogMath/trigonometry-as-the-study-of-circles/

“Relationships” is a good choice for algebra. Most of the time the relationship is some type of equivalence (equations, isomorphisms, etc), but sometimes it includes partial orders and other relations. Don’t be put off by examples of relationships in geometry and other areas because those examples merely underscore the algebra that is inherent in those subjects, as well as the reasons for unifying and overlapping the subjects.

I am in love with this idea. The only thing I’m mad about is that I didn’t think of it myself. I work with students struggling with mathematical ideas and when it comes down to it I often have to present them complicated and subject specific terms to which they can make no connection to if they are too embedded. Needless to say I do a lot of boiling down in my everyday work and I just love this one word way of doing it! Thank you!

Actually, there’s one way you can significantly improve upon this: give the one word answer to Mathematics is the study of ______. Our subject as a whole has suffered from the “billions of puzzle pieces” problem that has led people to believe (eroneously) that math is the study of numbers, shapes, etc. Just like Biology is the study of life and Economics is the study of scarcity, Mathematics needs a single word answer:

Mathematics is the study of structure.

Now, in all of your above examples, you can replace “mathematics” with “structure”. Algebra studies the structure of relationships. Calculus studies the structure of change. Probability studies the structure of uncertainty. Statistics studies the structure of populations. Analysis studies the structure of measure. Logic studies the structure of arguments (truth comes in many forms, and the flavor we’re dealing with is a specific type). Not only does this open the mind to new mathematics (graph theory is mathematics because it deals with configurations, incidence, adjacency, … structure), but it also clarifies and advertises the utility of mathematics: everything with structure (geology, molecules, social networks, architecture, music, etc) can invite applications of mathematics.

By the way, excellent topic; I have actually been saying some of the same characterizations to students for a few years now, and have even used them in statements of teaching philosophy.

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“calculus” is the study of *continuous* change.

let’s get with the program here.

Personally, “calculus is the mathematics of approximation” works best for me.

What about set theory: the mathematics of sets, obviously, but can we go deeper than that?

Also, one-word definition of number theory: the mathematics of patterns?