# The Art of Insight in Science and Engineering: Mastering Complexity

Sanjoy Mahajan
Pages: 408
https://www.jstor.org/stable/j.ctt1287hhg

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Preface
(pp. xiii-xvi)
4. Values for backs of envelopes
(pp. xvii-xviii)
5. Part I Organizing complexity
• 1 Divide and conquer
(pp. 3-26)

As imperial rulers knew, you need not conquer all your enemies at once. Instead, conquer them one at a time. Break hard problems into manageable pieces. This process embodies our first reasoning tool: Divide and conquer!

To show how to use divide-and-conquer reasoning, we’ll apply it to increasingly complex problems that illustrate its essential features. So we start with an everyday estimate.

What is, roughly, the volume of a dollar bill?

Volumes are hard to estimate. However, we should still make a quick guess. Even an inaccurate guess will help us practice courage and, when we compare the guess with...

• 2 Abstraction
(pp. 27-54)

Divide-and-conquer reasoning, the tool introduced in Chapter 1, is powerful, but it is not enough by itself to organize the complexity of the world. Try, for example, to manage the millions of files on a computer—even my laptop says that it has almost 3 million files. Without any organization, with all the files in one monster directory or folder, you could never find information that you need. However, simply using divide and conquer by dividing the files into groups—the first 100 files by date, the second 100 files by date, and so on—does not disperse the chaos....

6. Part II Discarding complexity without losing information
• 3 Symmetry and conservation
(pp. 57-102)

The rain is pouring down and shelter is a few hundred yards away. Do you get less wet by running? On the one hand, running means less time for raindrops to hit you. On the other hand, running means that the raindrops come toward you more directly and therefore more rapidly. The resolution is not obvious—until you apply the new tool of this chapter: symmetry and conservation. (In Section 3.1.1, we’ll resolve this run-in-the-rain question.)

We use symmetry and conservation whenever we find a quantity that, despite the surrounding complexity, does not change. This conserved quantity is called an...

• 4 Proportional reasoning
(pp. 103-136)

When there is change, look for what does not change. That principle, introduced when we studied symmetry and conservation (Chapter 3), is also the basis for our next tool, proportional reasoning.

An everyday example of proportional reasoning often happens when cooking for a dinner party. When I prepare fish curry, which I normally cook for our family of four, I buy 250 grams of fish. But today another family of four will join us.

How much fish do I need?

I need 500 grams. As a general relation,

${\rm{new amount = old amount}} \times \frac{{{\rm{new number of diners}}}}{{{\rm{usual number of diners}}}}$. (4.1)

Another way to state this relation is that the amount...

• 5 Dimensions
(pp. 137-196)

In 1906, Los Angeles received 540 millimeters of precipitation (rain, snow, sleet, and hail).

Is this rainfall large or small?

On the one hand, 540 is a large number, so the rainfall is large. On the other hand, the rainfall is also 0.00054 kilometers, and 0.00054 is a tiny number, so the rainfall is small. These arguments contradict each other, so at least one must be wrong. Here, both are nonsense.

A valid argument comes from a meaningful comparison—for example, comparing 540 millimeters per year with worldwide average rainfall—which we estimated in Section 3.4.3 as 1 meter per...

7. Part III Discarding complexity with loss of information
• 6 Lumping
(pp. 199-234)

In 1982, thousands of students in the United States had to estimate 3.04×5.3, choosing 1.6, 16, 160, 1600, or “I don’t know.” Only 21 percent of 13-year-olds and 37 percent of 17-year-olds chose 16. As Carpenter and colleagues describe [7], the problem is not a lack of calculation skill. On questions testing exact multiplication (“multiply 2.07 by 9.3”), the 13-year-olds scored 57 percent, and the 17-year-olds scored 72 percent correct. The problem is a lack of understanding; if you earn roughly $5 per hour for roughly 3 hours, your net worth cannot grow by$1600. the students needed our next...

• 7 Probabilistic reasoning
(pp. 235-278)

Our previous tool, lumping, helps us simplify by discarding less important information. Our next tool, probabilistic reasoning, helps us when our information is already incomplete—when we’ve discarded even the chance or the wish to collect the missing information.

The essential concept in using probability to simplify the world is that probability is a degree of belief. Therefore, a probability is based on our knowledge, and it changes when our knowledge changes.

Here is an example from soon after I had moved to England. I was talking to a friend on the phone, of the old-fashioned variety with wires connecting...

• 8 Easy cases
(pp. 279-316)

A correct analysis works in all cases—including the simplest ones. This principle is the basis of our next tool for discarding complexity: the method of easy cases. We will meet the transferable ideas in an everyday example (Section 8.1.1). Then we will use them to simplify and understand complex phenomena, including black holes (Section 8.2.2.2), the temperature of the Sun (Section 8.3.2.3), and the diversity of water waves (Section 8.4.1).

Let’s start with an everyday example, so that we do not have to handle mathematical or physical complexity along with learning the new tool.

One August, upon becoming eligible...

• 9 Spring models
(pp. 317-356)

Our final tool for mastering complexity is making spring models. The essential characteristics of an ideal spring, the transferable abstractions, are that it produces a restoring force proportional to the displacement from equilibrium and stores an energy proportional to the displacement squared. These seemingly specific requirements are met far more widely than we might expect. Spring models thereby connect chemical bonds (Section 9.1), xylophone notes (Section 9.2.3), gravitational radiation (Section 9.3.3), and the colors of the sky and sunsets (Section 9.4).

A ubiquitous spring is the bond between the electron and proton in hydrogen—the bond that is our model...

8. Bon voyage: Long-lasting learning
(pp. 357-358)

The world is complex! But our nine reasoning tools help us master and enjoy the complexity. Spanning fields of knowledge, the tools connect disparate facts and ideas and promote long-lasting learning.

An analogy for the value of connected knowledge is an infinite two-dimensional lattice of dots: a percolation lattice [21]. Every dot marks a piece of knowledge—a fact or an idea. Now add bonds between neighboring pieces of knowledge, with a probability ρbondfor each bond. The following figures show examples of finite lattices starting at ρbond= 0.4. Marked in bold is the largest cluster—the largest connected...

9. Bibliography
(pp. 359-362)
10. Index
(pp. 363-389)
11. Back Matter
(pp. 390-390)