1.3.2 Dichotomies

You're driving along the highway, impatient to reach your destination. You look at the dashboard clock, and you look at the speedometer. You have 28 minutes to go, your destination is still 25 miles away, and you have to park the car. Are you going fast enough?

For now, we ignore the risk of a speeding ticket, and we focus on the two familiar devices just mentioned: clock and speedometer. Together, they represent a function: Given a time on the clock, there is a simultaneous speedometer reading that gives the speed of the car at that instant of time. For convenience, we assume that you glance at the speedometer each time a number changes on the clock (but ignore speed fluctuations between clock readings). Then there is just one speedometer reading for each time. In mathematical language, speed is a function of time.

Many of the functions we will study in this course will be functions of time. In the speedometer-clock example, the time variable is discrete; i.e., given a time value, there is a next value (except at the end of the domain) and there is a previous value (except at the beginning of the domain). For the digital clock on your dashboard, time moves in steps of one minute — or perhaps one second, if your clock is that precise.

You may prefer to think of time as moving continuously, i.e., with no identifiable next or previous time after or before any given instant. Some automobile dashboard clocks are analog clocks; that is, they have hands rather than digits. Furthermore, the speedometer always has a reading, whether or not the digital clock has just recorded a new time, so we may think of speed as a function of continuous time. Since the speedometer is not likely to be a perfectly accurate measuring device, there is a slightly different, highly theoretical function that represents the real speed of the car as a function of continuous time.

Consider the function determined by the U.S. Census data every ten years since 1790. On one hand, we know that the census is not a perfectly accurate measurement. On the other hand, we can imagine that there is some exact U.S. population at any instant of continuous time. Unlike a real speed, which may or may not change continuously, the exact, real, highly theoretical, and unobservable population cannot change continuously. Why not?

These dichotomies:

• discrete versus continuous
• real versus observable or measurable
• exact versus approximate

will be frequently recurring themes throughout our study of the interaction between mathematics and the world around us.

All three of these dichotomies interact with and overlap a fourth:

• calculational versus conceptual functions

The functions we observe, whether as tabulated data, graphical relationships, or measurements, can never be known at more than a handful of inputs, and the calculations we do with these functions can never be exact.

On the other hand, we can often model these functions by mathematical formulas, about which we know (or at least can hope to know) everything. (You did this yourself in Activity 1 of Section 1.2 when you fit a linear formula to per capita cigarette consumption data.) You can easily be seduced into believing that these formulas are the most important objects of study, that they represent exactness or truth, and that everything else is, at best, approximate. In fact, the true relationships of science, social science, and engineering are very seldom described exactly by formulas. Nevertheless, formulas that approximate these realities are often convenient models because we can study and manipulate the formulas to gain insights that might remain obscured by messy reality.

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Contents for Chapter 1