Charles Petzold

Today is March 14, sometimes written 3.14 when celebrated as π Day. But I come to bury π, not to praise it.

The value we call π is commonly defined (for example, in Wikipedia) as “the ratio of a circle’s circumference to its diameter.” This certainly sounds like we’re talking about real objects and measurable quantities, but we’re not. If you measure a circle’s circumference and diameter, that ratio will be a rational number, and π is not a rational number. The values of 3.14 or 22/7 commonly associated with π are only approximations. The value of π is instead an endless succession of seemingly random digits.

A ratio of two numbers cannot be irrational unless at least one of those numbers is also irrational, and if a diameter or circumference of a circle is irrational, that dimension would not be measurable, and you can’t calculate the ratio. You can’t even indicate where π is on a number line using the classical geometric construction tools of straightedge and compass. (The √2, yes, but not π.)

When people define π as “the ratio of a circle’s circumference to its diameter,” they are speaking of a very, very, very, special circle: An ideal circle — a circle that exists only in the imagination. And if this ideal circle exists only in the imagination, that must mean that π as well exists only in the imagination. It has no existence in the real world.

Another way to approach π is algorithmically. Several mathematical algorithms generate the digits of π, and all you have to do is set one of these algorithms in motion. But if there’s anything I learned from writing The Annotated Turing, it’s that an algorithm is a process that occurs in time. An algorithm to calculate π needs a finite period of time for each step, and hence, it will never finish.

It is often said that π has an “infinite” number of digits as if that word “infinite” simplifies things. (Gosh, if π has an “infinite” number of digits, all we need do is run the algorithm “to infinity.”) But infinity is solely a mathematical concept. Like π, it is a product of the imagination rather than a representation of anything in the real world.

The word “infinity” is used much too promiscuously. I’m never in favor of banning words, but this one should only be used ironically or with great caution. Before the last syllable escapes your lips, alarms should being going off as if the Word Police were roaring down the street.

Infinity is not a quantity or a size or a measure of something. Infinity is a limitation. It indicates something that cannot be measured or cannot be finished.

One of the classical critiques of the concept of infinity is in Book III of Aristotle’s Physics. I quoted a couple sentences from Physics in Chapter 16 of The Annotated Turing, including this very wise observation:

Infinity turns out to be the opposite of what people say it is. It is not “that which has nothing beyond itself” that is infinite, but “that which always has something beyond itself.”

That’s from Book III, Chapter VI (near Bekker number 207a) from the translation by Robin Waterfield in the Oxford World’s Classics edition. It’s very similar to the Hardie & Gaye translation in W.D. Ross’s well-known edition of Aristotle:

The infinite turns out to be the contrary of what it is said to be. It is not what has nothing outside it that is infinite, but what always has something outside it.

But a couple weeks ago I picked up a more recent translation of Physics from the New Hackett Aristotle series translated by C.D.C. Reeve. In the whole chapter of Aristotle’s critique of infinity, Reeve avoids the word entirely, and instead favors the word “unlimited” for the Greek apeiron:

It turns out, then, that the unlimited is the contrary of what people say it is. For it is not what nothing is outside of, but rather what something is always outside of, that is unlimited.

This is certainly inspirational! Not the clunky prose, but the clever ploy: If the word “infinity” can be expunged from a translation of Book III of Aristotle’s Physics, surely it’s not needed for most other purposes!

It could be argued that the Greek apeiron and the English “infinity” (from the Latin infinitus) are quite similar. Both are negatives meaning “not limited or finite.” But through promiscuous usage, “infinity” has lost the sense of being a negative. Quite perversely, we regard “finite” as “not infinite,” and the useful word “finity” is archaic.

Which brings us to Georg Cantor. It’s commonly said that Cantor discovered “different sizes of infinity.” I’m sure I said something quite similar in The Annotated Turing, but I’ve come to reject that formulation.

A little background:

The natural numbers 1, 2, 3, 4, and so forth are said to be “countable” or “enumerable” because we can list them for however long we want. It is also possible to systematically list the positive and negative integers just by bouncing between them: 0, 1, –1, 2, –2, 3, –3, 4, –4, and so forth. Go on as long as you want.

Although the process is a little more complex, it’s also possible to enumerate rational numbers and even those numbers that are classified as algebraic because they are solutions of algebraic equations. These include the square root of 2, the 42nd root of 42, and many others. (This is all discussed in Chapter 2 of The Annotated Turing.)

But what’s not possible is to enumerate all the real numbers, which include the transcendental numbers such as π. Georg Cantor proved it quite definitively and in several different ways.

Nobody doubts the importance of Cantor’s contribution to set theory or his analysis of enumerable and non-enumerable sets. But is it really helpful to say that these represent “different sizes of infinity”? Is that the best we can do? More useful perhaps is this distinction: In one case, we can’t finish counting, and in the other case, we can’t even begin. That’s not a description of different sizes. It’s an acknowledgement of different limits to enumeration.

What mathematics does best is supply us with tools, sometimes to understand and quantify the real world, and sometimes to program computer graphics. In all circumstances that involve circles and angles and other mathematical entities, approximations to π are always suitable.

Nobody needs an exact value of π, and it’s a good thing too, because it doesn’t exist.