Thursday, February 21, 2013
The Life of π
π (Pi) is odd, wouldn’t you agree? π is simply this: the circumference of a circle divided by its diameter. Hypothetically, at least, each of these has an exact value, but when you divide one by the other, there is no exact value: it can be calculated indefinitely, with an infinite number of decimal points, without any repeating pattern of numbers. Now the number 1/3, when expressed as a decimal also extends to infinity; but the digits repeat forever: 0.33333... π does not do that; π cannot, in fact, be expressed as a ratio of integers in this way. This has always seemed very weird to me, slightly mysterious. It seems to suggest some kind of open-ended-ness in the universe that we inhabit, especially as π plays such an important role in this universe. Not surprisingly, π (together with a few similar numbers) is also known as a transcendental number, that is (I suggest skipping the rest of this sentence, if you become queasy in the face of mathematics), a number that is not the root of any nonzero polynomial having rational coefficients.
It is perhaps worth pointing out that, although π plays in an important part in equations of cosmology, it is not a physical constant, in the way that, say, the speed of light or the strength of the various attractive forces in the universe are physical constants. It describes no property of reality. It in fact describes the relationship between two aspects of a pure concept: the perfect circle. No such thing exists, except in our minds. That we can conceive such things, and that there is such amazing coherence to mathematical concepts, is truly astonishing. Mathematics itself is transcendent, in the sense that it is beyond any mere physical manifestation of it. The number “3”, for example, “exists” independently of any group of three objects that we might encounter. And within that transcendent system called mathematics, π is further transcendent. Mathematics exists purely as a set of logically coherent concepts. Then, surprisingly, within this very system, we encounter “irrational” numbers such as π. Equally surprisingly, these concepts somehow “apply” to reality in some way.
Is it any wonder that human beings have always felt that there was something mysterious or even mystical about numbers? You do not have to believe that numbers have any magical or predictive capacity for this to be true. Once again, these magical and semi- or pseudo-religious interpretations obscure what is truly mysterious about numbers; but it is easy to see how one could slip into magical thinking. I emphasise here, as I have done time and time again in these pages, that the universe is wonderful and surprising in its own right, without having to ground that in any other being or reality. I stand in awe of, but do not worship, this universe into which we have come into being.