The start of a new year seems like a good time to take a look at the big picture. A question from September raises a big topic: What is mathematics, in general? What does all math – arithmetic, algebra, geometry, trigonometry, calculus, and beyond – have in common?
What does it mean to do math?
Eh Doh Kue asked this:
I would like to know what all of mathematics is, or at least in general what mathematics is about. I know there are many areas of mathematics but is there something in common that all of these areas of mathematics have? I have heard one answer from Quora stating that abstraction is the essence. Is this true? I just want to know what it truly means to do math in general. I don’t think number crunching is really what mathematics is about. I do feel at this point that abstraction can be the essence of mathematics but I just wanted the opinion of other people passionate in math to see if abstraction is truly the essence of mathematics.
The reason I’m asking this is because I feel once I’ve found the essence of mathematics, I can use this essence as like a framework of Calculus, Trigonometry, etc.; I can think of everything in Calculus, Trigonometry, etc. in terms of this essence; I can see every or a lot of the areas of mathematics through the lens of this essence. That’s why I would like to know what this essence is, if it’s possible to answer this question in the first place.
I could have just answered, “yes, abstraction is essentially how I would describe it, too” (as we’ll see). But one word doesn’t make things very clear. Instead I gave a reference to a collection of answers to the question from Ask Dr. Math:
What is Mathematics?, a selection of answers from the Dr. Math archives
This is just a list of links, and you need to make a (free) account to read them; so I quoted from several of them, to give a taste of what various Math Doctors (including me) have said on the subject. I’ll do the same here, but at greater length.
1: Math is the study of abstractions
We’ll start with a question from 2001:
What is Math? What is math? Just how do you define it? Is it a study of patterns or whatever?
“Patterns” is not a bad thought, we’ll talk a little about that later. I answered:
Hi, Rob. If you asked a dozen mathematicians (or lexicographers, even) what math is, you would probably get a dozen answers. You might want to gather different ideas by looking it up in various dictionaries or encyclopedias, and reading the introductory chapters of several popular books on math, which may tell what the authors think math is all about. There is no one "correct" answer.
We’ll be doing a little of this, later!
My own favorite definition is that math is the study of abstractions. That is, we isolate one or a few features of some kind of object for study, and see what we can learn about the behavior of those features while ignoring everything else about them: features like number, shape, or direction. For example, when we work with numbers we are taking the concept of counting away from all other details about the things we are counting, such as color or name, and just thinking about how many there are. We learn to work with numbers as an abstract entity, so that we can add two numbers without having to think of them as representing two apples and three apples. When we finish our calculations with numbers, we can come back to the real world and know just how many apples we have.
When we use algebra to solve a “word problem”, we write an equation that represents all we know about the problem in terms of numbers only; then we solve that equation in the world of pure mathematics, and finally translate our answer back into the real world, adding the color and individuality back to the numbers.
It is often found that a concept that is first encountered in one part of our experience turns out to be useful in other areas as well. Having solved a problem in one context, we don't have to solve it again, because we solved it abstractly. For example, a common problem is to find the number of sides and diagonals of a polygon. It turns out that the same solution applies also to a question about the number of handshakes that occur if everyone in a room shakes hands with everyone else, and also to a problem about the number of different ways a student could choose two classes to take. They all look the same when you think of them abstractly. If I know the solution to one of these problems, I can transform a new problem into the known problem, and quickly find the answer. That kind of thinking is central to what math is.
We discussed these problems in Polygons and Handshakes. And this idea is why doing mathematics requires thinking abstractly, seeing past the “surface structure” of a problem (e.g. people in a room) to its “deep structure” (e.g. pairs of elements of a set), which is what the math deals with.
As you go more deeply into math, you find that we end up studying abstractions of abstractions, such as systems of objects that behave like numbers, but don't follow all the same rules. This turns out to be surprisingly useful; for example, rotations of an object in space, thought of as if they were a sort of number, can be handled very neatly, even though they work very differently from numbers in some respects.
This subject is sometimes called Abstract Algebra. We explore rotations and other transformations as objects that can be combined, in Slides, Turns, and Flips: How to Combine Them, and explicitly in Trouble with Transformations. We also touch on Abstract Algebra and is connection to rotations in What is Multiplication … Really?
As you can see, math goes far beyond arithmetic, or even algebra and geometry. All sorts of logical thinking fit this description. And math is a very creative field, involving exploration of the unknown, not just learning rules we are told to follow. Mathematicians invent abstract worlds, and discover all the surprises in them that are never noticed by those who don't look for the abstractions behind the reality.
This creative exploration takes us to the next idea …
2: Math is the derivation of theorems
Here is a similar general question from 2000:
What is Mathematics? Hi, Please tell me the definition of maths. If there is no single definition, is there a group of definitions?
Doctor Ian answered:
Hi Erum, Stripped to its barest essence, mathematics is the derivation of theorems from axioms. So what does that mean? It means that mathematics is a collection of extended, collaborative games of 'what if', played by mathematicians who make up sets of rules (axioms) and then explore the consequences (theorems) of following those rules. For example, you can start out with a few rules like: A point has only location. A line has direction and length. Two lines intersect at a point. and so on, and then you see where that takes you. That's what Euclid did, and ended up more or less inventing geometry. And that's what other mathematicians have done over the centuries, inventing arithmetic, and number theory, and calculus, and group theory, and so on.
In a sense, Euclid’s geometry was an attempt to see how much of geometry (much of which was already known) could be rigorously proved, starting from minimal assumptions. This idea is discussed in Who Moved My Postulate?
It's a little like what you do when you invent a board game like chess. You specify that there are such-and-such pieces, and they can move in such-and-such ways, and then you let people explore which board positions are possible or impossible to achieve.
The main difference is that in chess, you're trying to win, while in math, you're just trying to figure out what kinds of things can - and can't - happen. So a 'chessamatician', instead of playing complete games, might just sit and think about questions like this:
* If I place a knight (the piece that looks like a horse, and moves in an L-shaped jump) on any position, can it reach all other positions?
* What is the minimum number of moves that would be required to get from any position to any other position?
But they would also think about questions like this:
* What would happen if I changed the shape of the chessboard?
* What would happen if I allowed some pieces ('ghosts') to move through other pieces as if they weren't there?
* What would happen if I made the board three dimensional, or let pieces disappear for specified periods, or made them appear and disappear at regular intervals (for example, if a rook becomes invisible for three moves, then visible for three, then invisible again, and so on)?
* What would happen if I allowed more than two players, or let players take turns in parallel instead of in sequence?
Some such questions turn out to lead nowhere, while others just might lead to a fascinating new game! And that’s how new fields of math are sometimes invented; a little of that is described in Why Properties Matter: Beyond Addition and Multiplication; and you can find an example of “What if” thinking in What If: Inventing “Pseudo-Complex” Numbers.
We should add, however, that he is talking only about “pure chessamatics”! After studying how chess pieces move, you would then be in a position to study “applied chessamatics”, devising ways to use your new knowledge to win games – just as we can use our knowledge of how numbers work to do calculations, or use our knowledge of how shapes work to build things.
In other words, mathematicians are interested not only in what happens when you adopt a particular set of rules, but also in what happens when you change the rules. For example, mathematicians in Germany and Russia started with Euclid's geometry, but asked: "What if parallel lines _could_ intersect each other? How would that change things?" And they ended up inventing an entirely new branch of geometry, which turned out to be just what Einstein needed for his theory of general relativity.
We discuss this in Why Does Geometry Start With Unproved Assumptions?
Doctor Ian’s perspective is also found in an extended form at
3: Math is the study of structure and rigor
Next, a question from 1997, asking more particularly, what sort of thing mathematics is:
What is Math? Dear Dr. Math: Is math a science, an art, or some other anomaly?
Doctor Ceeks answered James, going beyond what was asked:
Hi, What a question! To me, mathematics is a discipline that seeks understanding of the patterns and structures of constructs of the human mind. Understanding has no end to its depth, and mathematics seeks the highest standards of understanding by demanding rigor in its foundations and in its development. Rigor is achieved by responsible attention to the principles of logic.
An important point here is that math works, not with things in the world, but with things we imagine – number, idealized shapes, and so on – abstractions. “Patterns” are what theorems are about, and “rigor” is what proof is about, so this combines the two ideas we’ve seen.
In a strict sense, mathematics differs from science, if we accept that science is the discipline that seeks understanding of the physical world by means of the scientific method. (The scientific method is the procedure by which hypotheses are proposed and subjected to experiments designed to expose weaknesses in the hypotheses.) The reason mathematics differs from this is because mathematics does not, in a pure sense, attempt to describe the physical world. Mathematical theorems are not tested against nature, but against logic.
Mathematics may or may not start with an idea from the real world (such as flat surfaces, or counting objects), but then imagines a perfect world defined only by a set of rules derived from that real thing, and tries to follow logic consistently. Or it may go several levels of abstraction away from reality, as when we imagine a new sort of “number” that follows only some of the rules of real numbers. An example of this is found in How Imaginary Numbers Became “Real”.
There may be much in common between mathematics and art, and since art is also difficult to define, it is difficult to discuss this relation. Certainly, there are definitions of art that would allow the inclusion of mathematics as an art. The creation of mathematics requires creativity, and I think most mathematicians would agree that some constructs are more beautiful than others, so that there is an aesthetic aspect to mathematics.
Not all students see the beauty in mathematics, which is partly because we don’t tend to teach about that beauty and creativity. That is a shame.
In any case, mathematics certainly is a human endeavor involving the exchange of ideas between human beings. It has been around for millennia, and shows no signs of abatement. These ideas are debatable and reflect my own personal opinion. Also, this response is really too short to explore all the aspects your question addresses.
One way to think of it is that mathematics (particularly as envisioned by the Greeks) is as the pursuit of truth. They wanted logical perfection, and in order to do that, it had to be about abstractions, facts about which could be assumed (by definition), rather than being guesses about the real world. With such a foundation, they could then write and prove theorems. Thus our three answers combine into a single picture: logical rigor requires making theorems, which requires abstraction!
4: Is math a science or an art?
We’ll end with a similar question from 2001, where we’ll dig a little deeper:
Science or Art? My question is, is math a science or an art? I understand how it can be used maybe for both. Math is used to form tessellations and they are a form of art. And science is all about numbers. But I do not understand how to support that math is a form of art or that it is scientifically figured out. It might even be both - but how do you support what you say?
I answered, supposing that the question is an assignment Erin was given. How would I write an essay on this topic?
Hi, Erin. I think you first have to find out what "science" and "art" mean in this question. The question is not, is math used in science or art, but is it AN art or A science. The terms you're dealing with are used in different ways, and your first need is to define them carefully. If you were asked this question by a human, ask what they mean by the terms; if it came from a book, ask another book (a dictionary) for details.
You may recognize a mathematician’s hand in this: We never talk about something without first defining it carefully!
So I looked up the words in a dictionary:
In my view of this question, the definition of "a science" from my dictionary that best fits our needs would be "a department of systematized knowledge as an object of study"; this doesn't necessarily have to be, as another definition says, "knowledge covering general truths or the operation of general laws esp. as obtained and tested through scientific method." That is, the mere fact that math is an orderly study of something makes it a science. It is different from natural sciences, however, in that we determine truth by proof rather than by observation and experiment. So it doesn't exactly follow the scientific method, but it's similar.
So math is not the same as what we usually call science, but it is parallel to it. Rather than studying physical things using numbers, it studies the numbers themselves.
On the other hand, the first definition my dictionary gives for "an art" is "skill in performance acquired by experience, study, or observation," and another is "a system of rules or methods of performing particular actions." The aesthetic aspect of "art" is only one of many definitions. Math (even apart from pictures) is often called beautiful by those who know it well; but that's not what I would mean in calling it an art. Rather, it is an art because it involves skill.
As an example, people talk about “the art of war”, which is not at all beautiful, but is a skill. Math is, in part, skill in solving problems, and in seeing logical connections.
The other thing you have to think about is, what is math? It's not just arithmetic, but a whole set of different fields in which we discover and prove theorems, use them to invent methods of calculation or construction, and ask new questions about completely imaginary objects. Students often miss the full extent of math, and get too small a view of it.
Here I’ve briefly summarized the other answers above: Math, which the question is about, is not just calculation, but exploration; it is not just using rote methods, but discovering them, solving unfamiliar problems and even inventing new things to ask questions about.
Putting all this together, I would say math is both an art (a system of skills used to do calculations, devise proofs, and so on) and a science (a system of knowledge about numbers, shapes, and other abstract entities, and a way of gaining that knowledge). That is, it's both a way of doing (an art) and a way of learning (a science). You'll want to make your own study of definitions, choose how you want to interpret the terms, and then learn what you can about what math is, in order to come up with your own opinion.
To really answer the question asked, you need to have a broad understanding of what math really is, beyond what most students see most of the time.
But not all math involves rigorous proof
After quoting from several of these answers to Eh Doh Kue, I added this:
I have to add, though, that doing mathematics (that is, being a “mathematician”) in the sense we’ve described is not the same as using mathematics. You can learn calculus, for example, without thinking fully in the way we describe, inventing and proving theorems for yourself; but understanding this bigger context can help you appreciate it more fully.
We don’t need to teach all students how to be creative mathematicians; some mostly just need the tools it provides. On the other hand, solid problem solving skills are very similar to the skills needed to prove theorems, and even to conjecture new theorems. And when you use any kind of math, it’s useful to be aware that the tools you are using fit into the framework we’ve examined here from various perspectives.
For examples of non-proof problem-solving that uses similar kinds of thinking, see Non-routine Algebra Problems.
So, what’s the bottom line? What view of math will help you as you pursue it in any of its forms? I would say you should understand that mathematics deals with simplifications of the real world, in which patterns are consistent and rules always hold, and also that the ideas of logic, rigor, and proof give it a kind of beauty that can keep you motivated as you study, with something new and unexpected often around the corner.