It's Pi Day, so I thought it might be interesting to think about and have some fun with numbers, while also revisiting a very old argument about the nature of numbers.
The mathematical constant pi is the ratio of any circle's circumference to its diameter, and has fascinated people for millennia. The fact something infinite, and whose numbers don't repeat over the infinite, is derived from something as simple as a circle, is in many respects a "beautiful" yet complex feature of the universe. And did I mention we still don't understand everything about it?
If one then throws in things like Euler's Identity, and pi’s connections to other fundamental concepts of math and science, one starts to understand why it has fascinated people for a very long time. But what are pi, 0, 1, or for that matter, numbers in general? Here's a nice little mind-bender that's been around for a while: Is mathematics a human invention, or a discovery made by humanity?
It seems like a simple question, but not quite.
Earlier today, the latest trailer for Marvels Avengers: Endgame was released and has been breaking the internet with over 20 million views in just the last 14 hours.
The Marvel characters, as well as the elements of their universe, are brought to life through the use of computer generated imagery rendered through billions upon billions of polygons modelled to interact with each other in order to create a fantasy. And that fantasy, as well as all others which use similar methods, owe themselves to the math created out of endeavours such as Archimedes searching for a value of pi so many years ago.
The concept and numerical value of pi, or "3.14159265358979...", is at the very core of the everyday life around us. Every engineering marvel which involves a cylindrical component (e.g., bridges, roads, tunnels, etc.) and the formula for area, which is pi multiplied by the square of the radius, or technical innovation which involves continuous change, have been borne out through humanity knowing and understanding what pi is.
Among the qualities of pi:
- Irrational: It cannot be represented as one number divided by another (i.e., a ratio). One property of the irrational numbers is that when trying to express them as decimal fractions, one gets an infinite and non-repeating sequence.
- Transcendental: It is not algebraic. That means, theoretically, that its digits will continue on indefinitely without ending in repetition. More technically, there does not exist a non-zero polynomial equation with rational coefficients of which pi is a root.
- All strings of numbers exist within pi: Your significant other’s birthday, my mother’s bank account number, the password to your iPhone, and every assortment of numbers to ever exist is somewhere within pi.
- On the other hand: Pi is not known to be random, or at least its randomness has never been proven. Therefore, after a certain point, for all we know there might not be any 7s or any 2s.
From Steven Strogatz at The New York Times:
To get a feeling for this world-changing idea, imagine measuring the distance around a circular track near your house. To obtain an estimate, you could walk one lap and then consult a pedometer app on your phone to see how far you traveled. A pedometer computes the distance straightforwardly: It estimates the length of your stride based on your height (which you typed into the app), and it counts how many steps you’ve taken. Then it multiplies stride length times the number of steps to calculate how far you walked.
Archimedes used a similar method to estimate the circumference of a circle, and so to estimate pi. Again, imagine walking around a circular track … By taking enough steps, and making them small enough, you could approximate the length of the track as accurately as you wanted. For example, paths with six, 12 and 24 steps would do an increasingly good job of hugging the circle … A man obsessed, he went from 6 steps to 12, then 24, 48 and ultimately 96 steps, using standard geometry to work out the ever-shrinking lengths of the steps to migraine-inducing precision. By using a 96-sided polygon inside the circle, and also a 96-sided polygon outside the circle, he ultimately proved that pi is greater than 3 + 10/71 and less than 3 + 10/70.
Take a moment to savor the result visually:
3 + 10/71 < π < 3 + 10/70.
The unknown value of pi is being trapped in a numerical vise, squeezed between two numbers that look almost identical, except the first has a denominator of 71 and the last has a denominator of 70. By considering polygons with even more sides, later mathematicians tightened the vise even further. Around 1,600 years ago, the Chinese geometer Zu Chongzhi pondered polygons having an incredible 24,576 sides to squeeze pi out to eight digits:
3.1415926 < π < 3.1415927.
By allowing the number of sides in the polygons to increase indefinitely, all the way out to infinity, we can generate as many digits of pi as we like, at least in principle. In taming infinity, Archimedes paved the way for the invention of calculus 2,000 years later. And calculus, in turn, helped make the world modern. Archimedes’s mathematical strategy is used in computer-generated movies, approximating Shrek’s smooth belly and trumpet-like ears with millions of tiny polygons. The smooth glide of an Ella Fitzgerald song is digitally represented in streaming audio by an enormous number of bits.
In every field of human endeavor, from reconstructive facial surgery to the simulation of air flowing past a jet’s wing, billions of tiny, discrete elements stand in for an inherently smooth and analog reality. It all began with the computation of pi. Pi represents a mathematical limit: an aspiration toward the perfect curve, steady progress toward the unreachable star. It exists, clear as night, with no end in sight.
But what are numbers?
Let's start with a "simple" truth ... at least in base 10.
Now, on one level, it's a fundamental truth of existence. It should be true within this universe, whether humanity is here to know it or not. It should be as true today as it was 10,000 years ago. And it should be as true here on Earth as it would be if you were floating somewhere in the Eagle Nebula, 7,000 light years away. Mathematical truths are also apparent in nature. For example, a Fibonacci sequence, first described by Indian mathematicians possibly as early as the 6th century BC and popularized in the West by Leonardo of Pisa (a.k.a. Fibonacci) in 1202 to describe rabbit populations, is a sequence of numbers which is the sum of the previous two numbers. The sequence begins with 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and continues infinitely.
Fibonacci numbers have various connections to observed systems in nature and accurately describes progressions which have applications to economics and computer science. The arrangement of leaves on stems, the branching of trees, the number of petals on a flower, and the spirals of seed heads and pinecones follow the pattern of Fibonacci’s sequence. The generational lineage of honeybee drones follow a Fibonacci sequence (i.e., each drone has one parent, two grandparents, three great-grandparents, five great-great-grandparents, and so on.) Fibonacci numbers approximate the “golden spiral,” a logarithmic spiral whose growth factor is φ, the golden ratio. The golden ratio appears in architecture, art, and music, with many arguing the resulting proportions tend to be pleasing to most human tastes.
And, if arguably examples of mathematical proofs are discovered in the life around us, attempting to invent a math which goes against those observations doesn’t really work. So far no one has "invented" a way to make 2 + 2 = 5 and still have it make any kind of sense.
On the other hand, what are "2" and "4"? Mathematics has sometimes been described as the language of science and as such could be described as a tool. Mathematical concepts are not observable. A pulsar, Earth's atmosphere, a certain species of frog, all exist in nature as observable things to study within an empirical framework.
The number "1" only exists as a human construct on a piece of paper, classroom board, or computer screen. It's a symbolic representation of an idea used to express other ideas, just like +, -, $, %, and even the period I'm going to end this sentence with right now.
And if that wasn't enough, all of this leads to Plato and a very old debate about the nature of mathematics: Did humans invent mathematics to help describe nature, or are we just discovering something that is intrinsic to nature itself?
From Julie Rehmeyer at Science News:
Those who espouse discovery note that mathematical statements are true or false regardless of personal beliefs, suggesting that they have some external reality. But this leads to some odd notions. Where, exactly, do these mathematical truths exist? Can a mathematical truth really exist before anyone has ever imagined it? ... Plato is the standard-bearer for the believers in discovery. The Platonic notion is that mathematics is the imperturbable structure that underlies the very architecture of the universe. By following the internal logic of mathematics, a mathematician discovers timeless truths independent of human observation and free of the transient nature of physical reality. "The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on," says Ulf Persson of Chalmers University of Technology in Sweden, a self-described Platonist.
However, Plato has his detractors.
If the mathematical ideas are out there, waiting to be found, then somehow a purely abstract notion has to have existence even when no human being has ever conceived of it. Because of this, [Barry] Mazur, [a mathematician at Harvard University] describes the Platonic view as "a full-fledged theistic position." It doesn’t require a God in any traditional sense, but it does require "structures of pure idea and pure being," he says. Defending such a position requires "abandoning the arsenal of rationality and relying on the resources of the prophets."
Indeed, Brian Davies, a mathematician at King's College London, writes that Platonism "has more in common with mystical religions than with modern science." And modern science, he believes, provides evidence to show that the Platonic view is just plain wrong. He titled his article "Let Platonism Die."
And then there is Euler's Identity:
Derived in the 18th century by Swiss mathematician and physicist Leonhard Euler, the equation "connects" multiple fields of mathematics and considered one of the most beautiful statements in mathematics. I once had a professor who jokingly called it a “family portrait” proof of "spooky order" in the universe.
Three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants.
- The numbers 0 and 1, which serve double purposes as both standard integers and constants. The number 1 serves as a numeral under addition and subtraction, but as a constant under multiplication, addition, or exponentiation. This latter idea can be understood by realizing that any number or variable can be defined as itself multiplied by 1.
- The number π, which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis (π ≈ 3.14159).
- The number e, the base of natural logarithms, which also occurs widely in mathematical analysis (e ≈ 2.71828).
- The number i, imaginary unit of the complex numbers, which contain the roots of all non-constant polynomials.