The New York Times has recently introduced a new series of “Opinionator” articles on its Web site, articles that are all about mathematics. (One has to admit that this is not a typical subject for OpEd pieces.) The articles are written by Prof. Steven Strogatz, professor of applied mathematics at Cornell University. As he describes his objective:
I’ll be writing about the elements of mathematics, from pre-school to grad school, for anyone out there who’d like to have a second chance at the subject — but this time from an adult perspective. It’s not intended to be remedial. The goal is to give you a better feeling for what math is all about and why it’s so enthralling to those who get it.
Judging from his first two columns, Prof. Strogatz is an entertaining writer who is able to present some subtle ideas in a very accessible way. In his first column, “From Fish to Infinity”, he talks about what one might call the duality of numbers: they represent something very concrete, like six fish or six baseballs or six dollars, yet they have an abstract existence in which they follow their own rules.
Even though they exist in our minds, once we decide what we mean by them we have no say in how they behave. They obey certain laws and have certain properties, personalities, and ways of combining with one another, and there’s nothing we can do about it except watch and try to understand.
He also talks about an idea I always try to emphasize with students I tutor: that a number is something distinct from the name or symbol by which we refer to it. That is,
5, V, cinq, fünf, 8-3, 20/4
all refer to the same number.
In his second column, “Rock Groups”, Prof. Strogatz talks about the playful side of arithmetic, and uses the idea of arranging groups of rocks to illustrate, for example, why the sum of two odd numbers is always an even number. Perhaps more surprising, he shows how the sums of sequential odd numbers are perfect squares; that is:
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 =25
It is, perhaps, a way of thinking about the properties of numbers and arithmetic that is a bit different from what most people are used to; yet in some ways it is not that odd, at all.
Looking at numbers as groups of rocks may seem unusual, but actually it’s as old as math itself. The word “calculate” reflects that legacy — it comes from the Latin word “calculus,” meaning a pebble used for counting.
I think it’s quite possible that much of the early understanding of how numbers work came from little experiments like these.
In any case, I think it’s great to see a column like this one in a publication like the New York Times. I’m looking forward to reading Prof. Strogatz’s next article.
(Incidentally, the columns also furnish some great references to further sources of mathematical insight. One that is worth mentioning is the wonderful essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, by Prof. Eugene Wigner, Nobel laureate in physics.)