Thursday, May 17, 2018

Are the digits of Pi real?

 Martin Gardner
In an article published in Discover magazine in 1985, Martin Gardner wrote this:
As it happens, the thousandth decimal of pi is 9... The question: Was [this assertion] true before the 1949 calculation? To those of the realist school, the sentence expresses a timeless truth whether anyone knows it or not... [Others] prefer to think of mathematical objects as having no reality independent of the human mind.
This problem is quite old, as we have been discussing it for over two thousand years. The question about whether mathematical objects really exist or are a pure creation of our mind is a particular case of another problem, much more general, that debates whether ideas and concepts (like the dog species) really exist, or just this dog and that dog exist. This is the problem of universals, famous in the Middle Ages, which has not yet been solved to everyone’s satisfaction. In fact, at present, this debate is more virulent than ever.

The problem of universals can be given two different solutions:
• Realists say that universals really exist outside our mind. In turn, this solution is divided into two:
• Platonic Realism, so-called because Plato proposed the theory of ideas, according to which concepts really exist in a world apart from ours (the world of ideas), which in fact is more real than ours, as explained in The Republic by the image of the cavern. According to this theory, the value of p is something real that resides in the world of ideas.
• Moderate realism, a solution proposed by Aristotle, who asserts that every being is composed of a matter of its own, and a form that it shares with other beings of the same species. According to this theory, concepts (forms) exist in reality, but are inseparable from matter. Thus, the value of p would be part of the form of all circular objects or trajectories, and therefore it would exist outside us.
• Nominalists or anti-realists say that concepts have no real existence, because they are our own creation. In turn, this solution is divided into two:
• Conceptualism, defended by Pierre Abelard (the same Abelard connected to Eloise) and William of Ockham (or Occam), among others. This theory asserts that universals are concepts that do exist, although only in our mind, and differ from words (nouns), which are the names of concepts. According to this theory, the value of p exists in our mind, but only there, because in reality there are just objects, not circles, since the idea of a circle is an abstraction that does not exist.
• Strict Nominalism, defended by Roscelin of Compiègne, who asserts that universals are mere names, just words (hence the name of this philosophical current). According to this theory, p is nothing but a term whose properties only depend on the way we make use of it.
 Voyager spacecrafts golden record
Notice a curious consequence of this: if either of the nominalist theories were right, then when we send to the stars plates engraved with mathematical symbols (representing, for instance, the digits of p), in case it will be found in a remote future by some extraterrestrial civilization, we are doing a foolish thing, because mathematics, being a creation of the human mind, may not be understood by other intelligences different from ours. On the other hand, it makes sense to do it, if any of the realist theories were right, because in that case mathematics would exist outside our mind, would be a part of reality, and any extraterrestrial intelligence must share them with us.
The value of p is defined thus:
p is the quotient of the perimeter of a circle (usually called its circumference) divided by its diameter.
The English mathematician William Oughtred represented this quotient by the symbol p/d, the initials of the two terms: perimetron (perimetron, which means measure around) and diametron  (diametron, which means measure through). In the eighteenth century, the Swiss mathematician Leonhard Euler was one of the first to use the symbol p to represent the quotient, for he considered a circle with a diameter equal to 1.
The value of p is always the same, whatever the circle in question, but we can’t know its exact value: we can only obtain approximations. Today we know 10 trillion digits of p, so we have a very good approximation. About 4000 years ago, the Egyptians had a much less accurate approximation. This one:
16×16×(1/9)×(1/9) = (16/9)2 = 3,16049...
 Archimedesby Domenico Fetti
The Greeks were the first to come up with the strange idea that it is possible to deduce the properties of geometric figures simply by thinking, without measuring anything with rules and ropes. Applying this procedure, Archimedes obtained two very good approximations of p: the fractions 22/7 = 3,142857... (by excess) and 223/71 = 3,140845... (by defect). He got one even better by computing their average: 3123/994 = 3,141851... A few centuries later, Ptolemy proposed one even better: 377/120 = 3,141666... And towards the end of the fifth century, the Chinese Zu Chongzhi proposed the fraction 355/113 = 3,14159292..., which was also discovered in the West in the sixteenth century. The value of p with 20 exact digits is 3,14159265358979323846…
Are those really the digits of p, as the realists say, or is this just a pure mental elaboration, as the anti-realists say? We will continue talking about this in successive posts.

The same post in Spanish
Manuel Alfonseca

1 comment:

1. It is interesting that pi has an infinite number of digits despite having a fixed value of circumference/diameter.