Understanding Without Proof

Euler's famous equation   
relates the five most important numbers in mathematics:

• e, which tells you how quickly a quantity grows when its growth depends on how much has grown before;

• i, the square root of minus one;

• pi, the ratio of the circumference of a circle to its diameter;

• one, the amount of a single instance;

• zero, which is not there.

    [If you cannot see the equation above,
    here are two ways of writing it in plain text:

        e^(i pi) + 1 = 0
    and
         i pi
        e     + 1 = 0

    Euler, by the way, is pronounced `Oiler' in English.]

After proving this formula in a lecture, the mathematician Benjamin Peirce said

Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth.

(Quotation from E. Kasner and J. Newman, Mathematicas and the Imagination,
New York 1940)

Pierce was unable to express to others the meanings which underlie mathematics, although there is no doubt that he worked with them. He was able to prove Euler's equation to his own satisfaction, but his remark demonstrates `Proof Without Understanding'.

At least Pierce could prove the relationship. I was worse: when I first heard of Euler's equation I could neither prove it nor understand it.

Fortunately, understanding is now easier.

Recently, George Lakoff and Rafael E. Núñez ¹ argued that mathematics is based on `conceptual metaphors' that are

... a cognitive mechanism for allowing us to reason about one kind of thing as if it were another.

A `conceptual metaphor' is an

... inference-preserving cross-domain mapping ...

The authors argued that mathematics consists of metaphor piled on metaphor, blended and transformed, so people often do not realize the basis of it all.

Lakoff and Núñez provided evidence that infants can see the sizes of groups of up to four objects and recognize subtraction and addition prior to the development of language. They contend that arithmetic comes from an inference-preserving extension of this ability to larger numbers.

Moreover, they argue that there are actually four `grounding' metaphors (metaphors based on experiences many of us had as children); these are

• adding and taking away objects from a collection (playing with pebbles);

• construction of a larger whole from smaller objects (playing with blocks);

• measuring the width or height of something (by stretching our hands to the ends of the object or standing up to see how high it is);

• moving from one place to another (by crawling or walking).

These experiences provide us with four metaphors that work with arithmetic: four inference-preserving cross-domain mapping mechanisms that work consistently with each other and the world.

Measuring provides us with zero and moving provides us with negative numbers. By blending these metaphors, and insisting on consistency, we get zero and negative numbers for collections, too. And then by adding new metaphors based on existing arithmetic metaphors onto existing ones, we get the `empty set' and set theory....

To quote Daniel J. Solove ²,

Metaphors do not just distort reality but compose it.

These ideas change the salience of my understanding. No longer do I think of a metaphor as `merely' a figure of speech or as an aid to thinking. Instead, I have come to realize that much thought — and all abstract thought — is based on metaphors.

Consider Euler's equation:    .

The hard part is the first: the number e raised to a power with the product of the square root of minus one and pi. What is going on here?

Lakoff and Núñez argue that the expression makes sense if, but only if, we understand that mathematics consists of the metaphorical extension of familiar notions into unfamiliar areas.

First, it is straightforward to think of multiplying the number 2 with itself three times: two times two times two. The answer is eight.

The next step is to imagine multiplying two with itself some fractional amount, such as two and a half times. This is hard, since ordinary multiplication can only operate as an integral whole. However, we do know that two times two is four, and that two times two times two is eight. So if we were able to multiple two with itself 2.5 times, the result would be somewhere between four and eight. (It is approximately 5.66.) Some centuries ago, mathematicians figured out how to calculate such results. The procedures are not the same as those you follow to multiply two times two, but the idea is consistent with doing that.

It is easy to imagine multiplying a fractional number with itself: for example, 2.5 times 2.5, which yields 6.25. (Of course to understand 2.5, we need to understand fractions. Perhaps the base for that understanding comes from crawling part way to the cookie jar as a baby.)

The next step is a combination of the previous two: multiplying a fractional number with itself a fractional number of times. (Two and a half multiplied by itself two and a half times is a bit more than 9.88.)

The number e is a bit more than 2.71818. It is the number you find when you figure out a value that depends on its previous values. For example, air pressure on the surface of the earth depends on the air above the surface. That bit of air above the surface has a pressure, too, which depends on the air above it. The numbers of plants or animals in an ecology depend on the same number, with the additional constraints that the ecosystem can provide only so much food, and others will infect or eat them.

Both e and pi are fractions. You can multiply e by itself pi times — e raised to the pi power. The result is a little more than 23.14.

The square root of minus one is not a regular number like 2 or 3.15159; you cannot place it somewhere on the ancient `number line'; you cannot crawl to it. That is why, historically, it was called `imaginary'. It does not fit the ancient way of thinking about numbers. However, i does perfectly well when we think of it as a `lateral' number, as Gauss suggested. If you are crawling or walking up the ancient number line, you need to turn. Perhaps it is even better to think of i as suggesting a quarter turn.

pi, you will remember, is the ratio of the circumference of a circle to its diameter; this is another way of saying that it is the ratio of the circumference of a circle to twice its radius, since a radius is one half a diameter.

A rotation is a complete turn. By a consistent metaphor, you can think of this a crawling or walking around a circle. This means moving a distance that is twice times pi times the radius. If the radius is one, then a complete turn means going twice pi.

The distance for a half turn is one pi.

The number i is an indicator of a rotation, along a radius of one unit.

Originally, we insisted that e multiplied by itself some number of times be a result on the ancient, straight number line. But we can also talk about a situation in which we turn. Metaphorically, e can be extended to this notion, and extended consistently.

This is how i fits. It tells us to go out a unit and then crawl or walk around the circle for which that unit is the radius. The distance we are going to travel is pi.

And where do we end up? This is the beauty of the equation. We end up at the location of minus one on the ancient `number line'. When we add one to that number, the result is zero.

The key to understanding — not the key to mathematical proof, which is different — is that mathematics comes from consistently extending fundamental experience, such as crawling. Each extension is consistent with what went before, but a little different.

Mathematics is difficult because most people do not see the metaphors that give meaning. So all they learn is proof, which is boring when meaningless.

[ Brad DeLong inspired me to this notion by quoting Pierce. ]

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