Today is March 14, 2015, or 3/14/15, or Pi Day to celebrants of the mysterious number π: 3.1415…
Yes, I know — it’s hokey, and who the hell cares anyway, besides engineers and math teachers and other geeks? It’s worth thinking about because π is one of the most useful numbers ever discovered — or invented, according to your point of view — and its usages and importance have evolved right along with the rest of mankind’s knowledge.
Π is defined as the ratio of the circumference of a circle to its diameter; if you have a circle with a diameter of exactly 1.0-foot, the distance around the circle is approximately 3.1415-feet. (I say approximately because π never resolves to an exact value; it is what mathematicians call an irrational number.) Since scientists and engineers model nearly everything as perfect instances of abstractions which don’t actually exist in nature — circle, line, triangle, sphere, catenary, and so forth — π makes an appearance in all sorts of counter-intuitive places, everywhere from chemistry to economics to cryptography.
Here is where it gets fun: If those geometric figures don’t actually exist in nature, if they are in fact human inventions created to simplify how we think about things, how is it that they are so useful for predicting so many real, physical events with great exactitude? Why do they remain useful for understanding and predicting matters that the first users of π could never have conceived?
Did humankind invent mathematics whole, or is mathematics something which is ‘out there’ and which humankind has discovered? What can it actually mean to say that the number π even exists?
The physicist Mario Livio says we started out by inventing numbers — two hands led inevitably to the concept of the integer 2, for example — but that, once invented, future mathematical work was constrained by the rules of the invention. Humankind invented two, say, but then discovered thirty-seven according to the rules created for integers. This sounds reasonable — at first — but if π is merely a tool, how can it answer so many different needs without modification? Why do entirely new branches of mathematics having no apparent relation to geometry continue to find π so useful?
“The fundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never seen any reason to modify.”
Bertrand Russell
This is no small matter; such questions had put mathematics in a deep philosophical crisis at the start of the last century: What is math?
Bertrand Russell argued that mathematics is a branch of logic, and co-authored with Alfred North Whitehead the 3-volume Principia Mathematica to prove it. Whether or not he succeeded remains to this day a subject of dispute, but whenever an exasperated math teacher exclaims “It’s just logic,” she is quoting Russell.
Remember the Pythagorean theorem? That the square of the hypotenuse of a right-triangle is equal to the sum of the square of its sides? That finding was disbelieved for centuries because the ancients had no concept of any number other than integers; real and irrational number were unknown to them. How, then, could the length of the hypotenuse of a right-triangle having legs equal to one be equal to the square-root of two? What could it mean to even speak of the square-root of two?
When different types of numbers were discovered, or invented, and π, insinuate themselves into even religious disputes.
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.
1 Kings 7:23
This is quite clear: the ratio of a circle’s circumference to its diameter is 3; inerrantists lose. “N-o-o-o,” they will answer. “Those measurements are inherently inexact and, besides, what about the thickness of the bowl? What if the diameter was measured to the outside the bowl, that is, and the circumference was measured around the inside of the bowl?” Increasing the diameter, the denominator, and decreasing the circumference, the numerator, pushes the ratio downward toward 3. Explain that the author had to mean 3, however the measurement was made, because only integers were known, or invented, at the time Kings was written, and they’ll exult that the verse then proves the divine inspiration of the text, because what else could explain that the ancients got it right?
Have I ever mentioned that the average apologist is a hopeless idiot?
So, Happy Pi Day. If you figure it all out, I’d be grateful if you let me know what you came up with.