Dear all
Thanks for your contributions to this mini debat over the value of pi. I
found them all (and I mean all) interesting and thought provoking. I've
always been fascinated with mathematics and take a lot of pleasure from such
cerebral exercises.
I'd like to offer an incite into why it is that pi (and other numbers like
root 2) can be irrational even though they feature in real world
measurement.
Most of you have suggested (if I've understood your thoughts correctly) that
we can only approximate the "true" value of the circumference of a circle
because real world measurements are always limited in accuracy, i.e. they
have, to put it another way, a finite error bound that can never be reduced
to zero.
It is quite true of couse that all practical measurement can only go so far
in expressing the true value of what is being measured. The limitiations are
basically technological but could steadily improve with time with no
ultimate limit being known. (Leaving aside the principle of uncertainty in
physics)
So does this mean that we may start to predict the value of pi by direct
measurement? Well obviously we'll never get anything like the millions of
digit accuracy of pure calculation by fast number crunching computers.
No, I think this is missing a fundamental point - which is that the real
culpritts are not measuring devices at all. It's us, or rather, the human
mind!
When we solve real world problems or carry out tasks involving numbers we
implicitly go through a certain process. We translate the problem into
mathematics, do the calculations and then translate the result back into the
real world. (The interface between the two usually but not always involve
measurement).
This is what is known as mathematical modelling. However the key to this
riddle about pi lies hidden in our assumptions about the relationship of
mathematics to real world. Our mathematical models *approximate* the real
world. It is not the other way round.
When we do geometry we model the world with lines that have no thickness,
right angles that are exactly pi/2 radians, curves that are perfectly
smooth, and circles that are perfect circles. In essence they obey very
simple rules so that we can cope with them.
The physical world is vastly more complex than this. The ideal circle is a
mathematical abstraction, just as numbers are. We'll never get pi from
practical measurement because the objects needed to produce it don't exist!
They may come very close to it but a perfect circle of the idylic world of
human mathematics simply cannot be realised physically.
There's nothing mystical about this. It's just the way the world is. I
finish with a quote from a well know science fiction author - "the Universe
is not only stranger than we imagine but stranger than we can imagine"
Phil Hall
- [USMA:35913] Re: Value of Pi Philip S Hall
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