Pi is 3
“Pi
is 3” ..... or is it 3.14159 ? ...... or something else? It started as a Christmas Day debate amongst
teenage cousins, all of us reasonably intelligent high school graduates about
to enter university and embark on careers as varied as the arts, teaching and
(in my case) engineering. My cousin's
argument was simply that, in ancient times “Pi is 3” was a true statement because
it worked. He argued that monumental works were constructed on the basis of
this belief (not strictly correct), thus proving its validity. Perhaps the
subtlety of his argument was lost in the heat of the midsummer day discussions
.... but it seems to me that the relevant question was not about mathematical
accuracy, but rather on the nature of truth. After 50 more years of reflection on life, the
cosmos and the nature of truth, it seems to me that much debate on this issue,
and on many other important issues, is clouded by semantics, the inadequacy of
language and the misunderstanding of both the written and spoken word.
But
first let me explore the statement “Pi is 3” – in particular, the definitions
of the terms "Pi", "is" and "3".
·
"Pi"
is defined as "a mathematical constant, the ratio of its circumference to
its diameter". The inference of "constant" is that it is
immutable (unchanging) and thus independent of the method of its determination.
·
"is"
means "be", that is "identical to". It does not mean
"approximates" or "approaches".
·
"3"
is a rational number, without an associated tolerance. In other words,
"3" has an absolute value of 3.0000000 to an infinite number of decimal
places. It is not “3 plus a little bit”.
There
are many methods of approximating pi, providing various degrees of accuracy and
speed of convergence, but none yielding a precise value. For example, pi = 3.14159265358979323846264338327950288419716939937510
is much more precise than pi = 3, but it is still not exact.[1]
While
empirical observations demonstrate an apparent consistency in the relationship
between the circumferences of circles and their diameters, it requires a leap
of faith to claim that this relationship is immutable (an unchanging constant).
While most reasonable people accept such a hypothesis (based on the enormous
number of times that has been shown to work in practice), that there is an immutable
relationship (towards which the various calculated values approach) remains in
the realm of faith.
At
this point I must declare that I am but a humble engineer, and certainly no
mathematician. But, as I stated at the beginning, this discussion is about the
nature of truth, not about the accuracy of mathematics.
Many
children believe that Santa Claus will bring them presents at Christmas time if
they are well behaved. Parents willingly promote this myth, while
simultaneously recognising that their children will grow out of such childhood
fantasies in time. However, the myth of Santa Claus is underpinned by historical
fact – the story of Saint Nicholas[2],
Bishop of Myra (in modern Turkey) during the late third and early fourth
centuries. This was a turbulent period, with Nicholas enduring Diocletian’s
persecution and participating in Constantine’s 325 AD Council of Nicaea. But
Saint Nicholas was a pious man, particularly revered for his generosity and care
for children. Thus was secured Nicholas’s association with the Santa Claus
myth.
The
Santa Claus myth approximates a behavioural ideal, while the story of Saint
Nicholas gives us a closer approximation to that ideal. However both fall short
of the absolute ideal. 3 approximates pi, 3.1416 gives a better approximation
of pi, but both fall short of revealing the real pi.
And
so it is with much of our understanding …… the earth was flat, then it was
round and now it is an oblate spheroid. The smallest particles of matter used
to be atoms, then they were protons and electrons, and more recently they are quarks.
Even our understanding of mathematics
has evolved …. from our simple primary school concepts through geometry,
algebra and calculus to imaginary numbers, differential calculus, set theory, mathematical
logic, topology, game theory and many more.
Our
behaviour is grounded in our beliefs, and they in turn on our perception of truth.
While truth is absolute, the information available to us only approximates that
truth, and we must be prepared to re-examine our beliefs when more reliable
information becomes available.
We
now know that pi is an irrational number, the ratio of a circle's circumference to its diameter …… not much use to
us mere mortals. So an approximation to the truth, 3.14159, is close enough for
most of us most of the time.
Truth
is immutable, and my cousin was wrong. “Pi is not 3”…. and never was.
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