Today@Dal

From the Scientific American website:

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.

Read the rest of this post at Scientific American.