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Continuous and Discrete

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As far back as the year 2000, a bookstore on Charing Cross Road in central London bore a sign that said “Any Amount of Books.” These days one often hears people conflate not only “amount” and “number” but also “less” and “fewer,” as in “There were less students in class today.” Alas, the confusion is even more common in North America than in England.

Is it just a simple conversational error that only the grammatically fastidious find grating, or is there something more to it? The truth is that mathematicians recognize the gravity of the error as well. In fact, far from being a mere linguistic slip, this error does a profound disservice to concepts that are at the very foundation of modern technology.

The fundamental distinction that is glossed over in that usage is the one between the continuous and the discrete. Now “continuous” is a word that is ubiquitous in day-to-day conversation, and its meaning is well-understood, at least in the sense that the common-sense understanding is consistent with its technical or mathematical meaning. (To understand the full ramifications of continuity, one has to dig deeper.) Simply put, if someone says, for example, that she has worked continuously for twenty years in a particular office, she means that there were no breaks or gaps in her service at that office during that twenty-year period.

On the other hand, “discrete” is not a word that occurs often in common parlance, although people seem to understand it well enough. It is difficult to define it precisely – one has to start with the notions of a set and a one-to-one correspondence between sets and go through the basic ideas put forward by the great nineteenth century mathematician Georg Cantor. (There are many books where they are discussed, but a beautiful and perspicuous description of them can be found in the book “Satan, Cantor, and Infinity” by Raymond Smullyan. As one might guess from the title, the book is accessible to anyone with a junior high school mathematics background. It is a delectable read.) The meaning of “discrete” becomes clear, however, when one uses it in an example: one has one child, two or more children, or none at all. One instinctively understands that it is absurd to talk about 1.2 or 3.5 children. The same thing applies to apples or oranges in a basket.

So, without going into a detailed construction of real numbers, an ordinary person understands that some things, such as children, books, or cars can only be counted, whereas certain other things, such as water, milk, or the weight of a person have to be measured. Discrete objects are counted, while continuous ones are measured.

Lest one should dismiss these thoughts as the idle ruminations of a disgruntled fusspot, let us observe that the difference between continuity and discreteness is the basis for the profound and spectacular developments in science and technology that define the 21st century as well as the second half of the 20th. One often hears that ours is the digital age. What does it mean? It means, for example, that music recorded in the old days was analog, meaning that the signals were continuous.

In contrast, when music is digitized, the signals are sampled at distinct points in time. Yet if the number of sampling points is large enough, and the duration between successive sampling points very close to, but distinct from, zero, then our ears cannot distinguish between the continuous and discrete signals. In other words, it is beyond our powers of resolution. And this sampling at discrete time or space intervals is at the heart of digital technology, the hallmark of our times. Thus when we confound the continuous and the discrete and speak of the “amount” of people, for example, we are in effect saying that digital and analog technologies are the same. Of course, in mathematics itself, there are entirely different sets of ideas and techniques for dealing with continuous as opposed to discrete problems. Any mathematician worth her salt will tell you that they are very different ways of mathematical thinking. (The two points of view meet, however, when one considers asymptotics, i.e. what happens in the long run. This is rather like two parallel lines meeting in the far distance, at what mathematicians call the point at infinity.)

As the great Henry Fowler, author of “A Dictionary of Modern English Usage,” said, the ultimate arbiter of correctness of a word or a phrase is usage. So it behooves those of us who care about the words we use and their meanings to raise alarm bells about the lumping of “amount” and “number”, or “less” and “fewer” as synonyms. Otherwise we will be stuck with them forever and have nobody else to blame. In that spirit, one only hopes that, in typical English fashion, that sign outside the bookstore in London has spurred many an enraged stickler-for-precision into action.

Acknowledgments: It is a pleasure to thank John Rennie and Keith Johnson for helpful comments and suggestions and Aileen Penner for the illustration.

Note: Continuity is a property of functions. For sets, the corresponding property is connectedness. However, in the interests of keeping the discussion simple and easy to understand, this was not mentioned in the article.

The views expressed are those of the author and are not necessarily those of Scientific American.

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