It's Pi Day, so I thought I would have some fun with numbers, while also revisiting a very old argument about the nature of numbers I've written about before.
The mathematical constant Pi, the ratio of any circle's circumference to its diameter, has fascinated people for millennia. The fact something infinite (& 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 you then throw in Euler's Identity, and how Pi connects to other fundamental concepts of math & science, you start to understand why 3.141593... 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.
The sound of meditation for some people is full of deep breaths or gentle humming. For Marc Umile, it's "3.14159265358979..."... "There are many things that could not be built without implementing the constant pi," Umile said. "The great engineering marvels like the arch or suspension bridges we cross over, the tunnels spanning within mountains or even under the water that we drive through. ... Without it, everything would be incomplete or in danger of collapse."
Designing any structure with cylindrical components involves pi, as the formula for area is pi multiplied by the square of the radius... Mathematicians know that pi is irrational -- it cannot be represented as one number divided by another -- and transcendental, meaning it is not algebraic. That means, theoretically, that its digits will continue on indefinitely without ending in repetition -- in other words, the digits won't suddenly continue infinitely as 5s after 3 trillion digits (Pi's digits were calculated out to a record 2.7 trillion places in December by French computer scientist Fabrice Bellard).
That also means, mathematicians theorize, that any string of numbers you can imagine is somewhere in pi -- for instance, look for your birthday. Coincidentally, "360," the number of degrees in a circle, occurs at digits 358 to 360.
On the other hand, the true "randomness" of pi's digits has never been proven, which is frustrating, said David Bailey, a technologist at Lawrence Berkeley National Laboratory who is still working on this question. "For all we know, just out beyond where we calculated, there are no more 5s," he said.
Wikipedia has a rough history of the computation of Pi.
The earliest evidenced conscious use of an accurate approximation for the length of a circumference with respect to its radius is of 3+1/7 in the designs of the Old Kingdom pyramids in Egypt. The Great Pyramid at Giza, constructed c.2550-2500 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits; the ratio 1760/280 ≈ 2π. Egyptologists such as Professors Flinders Petrie and I.E.S Edwards have shown that these circular proportions were deliberately chosen for symbolic reasons by the Old Kingdom scribes and architect. The same apotropaic proportions were used earlier at the Pyramid of Meidum c.2600 BC. This application is archaeologically evidenced, whereas textual evidence does not survive from this early period.
That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to Ancient Egyptian, Babylonian, Indian and Greek geometers. The earliest known textually evidenced approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value. The Indian text Shatapatha Brahmana gives π as 339/108 ≈ 3.139.
Archimedes (287–212 BC) was the first to estimate π rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygons and calculating the outer and inner polygons' respective perimeters... The average of these values is about 3.14185... Around AD 265, the Wei Kingdom mathematician Liu Hui provided a simple and rigorous iterative algorithm to calculate π to any degree of accuracy. He himself carried through the calculation to a 3072-gon and obtained an approximate value for π of 3.1416. Later, Liu Hui invented a quick method of calculating π and obtained an approximate value of 3.14 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.
Around 480, the Chinese mathematician Zu Chongzhi demonstrated that π ≈ 355/113, and showed that 3.1415926 < π <3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value would remain the most accurate approximation of π available for the next 900 years. </p>
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. I once had a professor who jokingly called it proof of "spooky order" in the universe.
Euler's identity is considered by many to be remarkable for its mathematical beauty. 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. 0 on the other hand is a very useful, multipurpose constant which can be used in countless algebraic capacities].
- 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 nonconstant polynomials and lead to deeper insight into many operators, such as integration.
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. And so far no one has "invented" a way to make 2+2=5 & still have it make any kind of sense.
On the other hand, what are "2" & "4"? Mathematics has sometimes been described as the language of science & 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 & a very old debate about the nature of mathematics. Discovery or invention?
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."
A pretty simple way to resolve this would be to say it's both a discovery & an invention, but that would be too easy & rob people of thousands of years of arguing.