In 2016, Peter Trueb calculated 22.4 trillion digits of π. Ahead of π Day on 14 March, he repeats on the nature of π and its role in math, science and philosophy.π as we identify it and love it is 3.141592. It’s only an approximation, but it’s exact enough for most computations and additional precise than any value used in Europe before 1500 AD. Only the beginning of calculus improved the knowledge of π to more than 100 decimals at the start of the eighteenth century. The computation was performed with the y-cruncher code1, which implements the wonderfully fast converging Chernovtsy formula.
But the number π is more than its decimal digit. In hexadecimal notation, it reads 3.243F6… or 11.00100… in binary notation, which force seem extraneous at first. Suddenly still, the representations of π in base 16 and base 2 are peculiar. The reason for this lies in the BBP formula, named after David Bailey, Peter Borwein and Simon Plouffe who published it in 1973. This formula yields an algorithm for the calculation of the nth hexadecimal digit of π without the need to analyze any of the earlier digits. The y-cruncher code I used adopts this method to cross-check the hexadecimal digits of π, thus I could be sure that no mistake happened during the 105-day-long computation.
Is there any motive to compute π to trillions of digits? Johann Heinrich Lambert has recognized π to be irrational and Ferdinand von Lindemann has shown it to be transcendental — but proof of its imaginary regularity is still absent. Mathematicians explain a number to be normal if all possible substrings of equal length happen with the same asymptotic frequency in its figures of any base. Thus, if π is normal then all the digits 0–9 should seem with a probability of 10% and all substrings of length 2 should have a frequency of 1%. As presently as I had the hexadecimal and decimal digits presented, I calculated the frequency of all substrings up to length 3 in these illustrations. Unfortunately, I couldn’t find any hint that π is not normal.
Visualize two friends discussing a reprint on population leanings. One of them wonders about π appearing in a statistical formula because he doesn’t see any joining between the circumference of a circle and the calculated population. Eugene Wigner uses this story to show his own bewilderment about the irrational effectiveness of mathematics in natural sciences. Why do lunar objects behave giving to the non-trivial mathematical concept of a second derivative as expressed in Newton’s laws of motion? How can these equations predict the movement of planets with an accuracy well than one part per million despite the free-fall results of Galileo x¨=g≈π2ms−2 having a rather crude experimental basis and a very different experimental context?
Wigner’s reflections have a direct connection to a millennium-old philosophical dialogue about the ontological nature of mathematical objects and numbers such as π. The two main views are known as mathematical Platonism and nominalism. The former consider numbers to be abstract items, which exist independent of human language or opinions. According to a mathematical Platonist, there exists, for example, an object 2, which exemplifies the number of balls being released by Galileo from the leaning tower of Pisa. By contrast, mathematical nominalism denies the presence of mathematical items — in this view, my calculation of π would be considered as a creation rather than a discovery of a pre-existing number. One of the most stringent arguments for mathematical Platonism is the indispensable use of mathematical objects in science6. With this reasoning, we should be ontologically devoted to all and only those entities that are essential to our best understanding of the world around us. As our most positive physical theories rely on numbers such as π, π exists.