Embodied Mathematics

1. The Dove Metaphor

`One of the main points that I would like to argue is that mathematics is metaphorical in nature. What does this mean? I’ll discuss what a metaphor is later, but for now just understand a metaphor as reasoning about one thing as if it were another thing. For example, consider this metaphor about problem solving. Visualize a dove flying through the air. In order for the dove to fly the air must be thick enough for the dove’s wings to be able to push on it and propel the dove. A dove couldn’t fly in space because there would be nothing to push against. There has to be some friction between the dove’s wings and the air that it’s flying through. On the other hand, if there’s too much friction, the dove can’t move at all. A dove couldn’t fly through corn syrup, because it wouldn’t be able to move at all. So, the dove needs an environment where there’s a balance between friction and freedom. Compare this to the skills that are needed to solve a problem in math. When approaching a math problem, one must have some structure. Some of the rules we just have to learn. For example, when we’re solving a triangle we just have to know that the squares of the two sides add up to the square of the hypotenuse and that the sum of the interior angles is 180 degrees. But, in addition to knowing these rules, we also must be flexible in how we apply them.

For example, think about problems in which it’s not totally obvious how to go from one step to the next. The rules don’t tell you how they’re supposed to be applied, and since every problem is unique it takes some clever thinking to figure out problems. Now think about what we’ve just done, we’ve taken a fairly difficult thing to understand, how to go about solving problems, and conceptualized it in terms of something that’s not so difficult to understand, how a bird flies.

2. Metaphors in Everyday Life

In this section I would like to talk about metaphors and how they structure the way that we think. But before we can discuss what a metaphor is, let’s consider some examples of metaphors:

“This class makes my blood boil!”

“I was just blowing off steam”

“She erupted”

“She exploded”

“He blew his top”

What do statements like these really mean? Has a class ever literally made your blood boil? I don't think that you could live to tell about it if this actually happened to you. Have you ever actually blown off steam? Certainly not. And obviously people don’t erupt, explode, or blow their tops in the real world. But if people don’t actually do these things, then how do we make sense of statements like these?

Obviously, these statements don’t mean what they literally say, but they still have some meaning. All of these statements are about people who are angry. Classes don’t literally make people’s blood boil, although sometimes it feels like they, they just make you frustrated and sometimes upset. In the same way people don’t actually blow off steam, but rather they do things that they know will make them less angry, like going for a run or counting slowly to ten.

Now the question becomes, how is it that we understand that these people are angry? The answer that I would like to propose is that we understand anger metaphorically as heat. This metaphor is common in the way we talk, we call someone who is easily provoked a “hot head,” or say that they have a “fiery temper.” If you reflect on the way you usually speak with this in mind, you’ll find that metaphors are everywhere.

Another common metaphor that we use in everyday life is “Time is Money.” We say things like “reading this is a waste of time,” “that you for giving me your time,” “Spend your time wisely,” and “Bush is buying time in Iraq” (National Public Radio, September 11, 2007). At this point, one might put her foot down and say “no, time literally is money, you can literally waste time, we do spend time, and we do ‘buy time.’” Some of these metaphors are so deeply engrained in the way that we think that we find it impossible to think about time without thinking about it in terms of money. But I argue that this is a fact about our culture, and doesn’t necessarily have anything to do with what time is. There was time before there was money, and people talked about time before money.

It’s easier to show how ridiculous some of these statements are when we consider them literally than others. If you’re still not convinced, consider this. All the money in the world could never buy you back time. You can’t pay any amount of money and get the time you spent in school today back (although you might really like to).

Examples of metaphors are everywhere. We talk about our feelings in terms of space by saying things like “I’m feeling down today,” “what you said to me earlier really picked me up,” or “I’m feeling really run down.” In all of these examples we understand up to be good and down to be bad. Another popular metaphor is “Love is a Journey.” We say things like “They’re relationship isn’t going anywhere,” “they’re in a dead-end relationship,” “we’ll have to go our separate ways,” and “it’s been a long, bumpy road.” Look for examples in your own conversations of times when someone is using a figurative expression to explain something.

Now that it’s apparent that we use metaphors in our daily speech, the question becomes why do we use metaphors and not just say what we literally think? The answer that I would like to propose is that it’s really hard to say what we literally think about some things. In some cases it’s easy, I can tell you about where I parked my car or what time I woke up this morning without using metaphor. Things that are concrete, or things that we can see, hear, touch, act are easy to talk about with simple, literal language. But other things that you can’t sense, like emotions, feelings, and time are abstract and difficult to talk about literally. So us humans came up with a simple solution, we talk about abstract things as if they were concrete things.

In the examples that we’ve already considered we spoke anger (an emotion), as if it were heat, love as if it were a journey, mood as if it were a location in space, and time as if it were money. In the case of “Anger is Heat,” we used imagery usually associated with heat. When you express your anger by saying “I’m going to blow my top” other people instantly know what you’re talking about and they can imagine what you’re feeling.

This is what we mean when we say “metaphorical.” A metaphor is just figurative speech that explains an abstract thing, like emotion, in terms of a concrete thing like heat. The literal meaning of metaphor goes back to its Greek roots. “Meta” means “beyond, transcending or next to” and “phor” means to carry. So a metaphor is a phrase that carries meaning beyond what the words are literally saying. Incidentally, the Greeks use the word “metaphor” as both a rhetorical device, as we’ve been talking about it, but also to refer to the baggage claim at an airport. In this case “metaphors” are literally carrying your luggage to you. Metaphors are extremely common in our daily speech, they structure our thoughts, and as I would like to discuss now, also extremely common in mathematics.

3. Metaphors in Math

Now that you have experience looking for metaphors in daily speech, can you think of any metaphors in mathematics? What concrete things do we use to think about numbers? Oftentimes we think about arithmetic as motion along a path. We say things like:

“How close are these two numbers?’

“3 is far away from 234,567”

“4.7 is near 6”

“Count up to 20, without skipping any numbers”

“Count backwards from 20”

“Count to 100, starting at 20”

We can do a similar analysis of statements like these as we did for other metaphorical statements. Obviously, numbers can’t be close to one another in the same way that San Diego is close to Tijuana or La Jolla High School is close to Wind and Sea Beach. The opposite is just as true, numbers can’t be far away from each other in the same physical sense as we are far away from China. When you step back and think about it, it’s kind of strange to say that we are counting backwards from 20. Who said that to count “20, 19,18,…” is to count back and to count “1,2,3…” is to count forward? Yet we all know that to count “1,2,3…” is to count up and to count “3,2,1…” is to count down. To make this point more clear, let’s see if we can think of a different metaphor that we use to talk about numbers.

We also think of numbers as objects in a collection. When you were learning to add and subtract, people probably asked you questions like: “If you add four oranges to six oranges, how many oranges do you have?” or “If you take two apples from five apples, how many apples do you have left?” In these questions numbers aren’t thought of as points on a line that can have distances from each other, but as the size of a collection.

Now I hope its apparent that the way that we think of numbers can be flexible. Once again, when you really stop and think about it, numbers are kind of weird things. If I were to ask you to show me the number three, you would probably do something like hold up three fingers or draw the symbol “3” on a piece of paper. But if you really think about it, both of these ways of demonstrating are inadequate for what you’re trying to show. In the first case you’re showing me three objects, but not the number three itself. In the second case you’re showing me a symbol that we use to represent three. Hopefully now you can see that the only way that we have to reason about abstract things like numbers is to reason about them in terms of concrete things.

Now I would like to consider a problem that uses both metaphorical ways of thinking about numbers. Try this problem in algebra.

John is five years older than Mary is. Four years from now, he will be twice her age. What is the present age of each?

What do we do to solve this problem? Well, we start by assigning a variable for Mary’s age, x. Since the problem says that John is five years older than Mary, we represent John’s present age as x + 5. Which of the metaphorical schemas that we discussed before are we using in this set of representations? It seems like we’re considering age to be something like a timeline and we consider John’s age to be five points to the right of Mary’s age. Next we translate the second sentence given to us in the word problem into a mathematical sentence. But for this we must use the understanding that age is something that is quantifiable.

Examples of metaphors in mathematics abound just like examples of metaphors in daily speech. In formal mathematics we talk about a function as being a dependence between two quantities, or a mapping from one set of values to another. However, these definitions aren’t very useful to people who aren’t already familiar with what a function is. So instead of relying on definitions, we will use the “a function is a factory” metaphor. This makes sense of statements like:

“The function f(x) = 2x +3 takes an input of three and produces an output of nine”

“Two plus two makes four”

“For each input a function produces a specific output

Just like in the previous cases, functions don’t actually produce or make anything. They also don’t take anything, but we can still make sense of these statements because we can link what a mathematical function is doing with our experience of a factory. We’re all aware of what a factory does, it takes in raw materials (like wood and epoxy), does some set of operations (like shapes, smoothes, and epoxies) and produces some output (like a surfboard).

I hope that this part of the discussion has made it clear to you that metaphor is an important part of how we reason and how we reason about math. Now I would like to discuss a tool called conceptual blending.

4. Conceptual Blending

As we have seen, metaphor is a powerful tool we have for thinking about mathematics. Another powerful tool that is related to metaphor is called a conceptual blend. To introduce what a conceptual blend is, I would like to pose a riddle:

Imagine a surfer who starts walking down the path toward the beach at dawn and arrives at the beach at dusk. She surfs and camps for three days and at dawn on the third day starts walking up the path, arriving at the top at noon. Make no assumptions about her starting or stopping or pace during the trips.

Riddle: Is there a place on the path that the surfer occupies at the same hour of the day on the two separate journeys?

Use this space to show how you reasoned about the problem and state your answer.

Yes! There must be a spot on the path that she occupied at the same hour of the day on the two separate journeys. This answer makes perfect sense if you just imagine that she made the two separate journeys on the same day.

Narrator: {Jenna walking down the path} What makes this riddle difficult? We can easily imagine a surfer going down a hill, some of you might have even walked down this very path to Black’s Beach. We can also imagine a surfer going back up the path on a different journey. {Jenna going up the path} To come up with the answer to the riddle we must imagine that the surfer is going up and down the hill on the same day. Of course it’s impossible for one person to both go up and down the hill at the same time, but we that doesn’t mean that we can’t imagine it. All we have to do is blend the two separate journeys together. Most of the elements will stay the same; we are talking about the same surfer, the same path, and the same duration but we’ve gotten rid of other features like the absolute date. Let’s say that in real life she made the trip down on a Friday and made the return trip on a Sunday, we don’t imagine our blend to be either a Friday or a Sunday, we just omit that aspect. Once we form this picture in our minds the answer is obvious, of course she crosses herself. {Jenna crosses Jenna 2} We don’t know where on the path she would meet herself or at what time of day, but we do know that it would have to be somewhere, at some time. {Simple animation two mental spaces, then blended} As you can see, by blending these two pictures together, we can see things that we couldn’t see when the two were separate.

Often we are doing the same thing in mathematics. Under closer examination, many seemingly new mathematical concepts turn out to be blends of two familiar concepts. A striking example of this is the Cartesian co-ordinate space. Cartesian co-ordinates are the axis that we use to graph equations. The story behind how Descartes came up with this representation is pretty interesting. He was serving in the French military and was idly lying in his bunk and watching a fly. He realized that at any point in time, the fly’s position could be described by its distance from the orthogonal boards that support the bunk above his.

This insight is interesting with respect to the blend because what he was able to do was blend ideas from Algebra and Geometry. Algebra and Geometry had both been around for thousands of years, but for the most part they were separate fields. What Descartes was able to do with the idea of the co-ordinate space, was to find a link between functions and the figures that represent them in space. He was able to see two distinct things as one.