A Beautiful Mind Incorrect Maths?

Date: 04/22/2002 at 00:33:45
From: Kerry Cue
Subject: A Beautiful Mind incorrect Maths?

Dear Dr Maths,

I am a journalist working for the Herald Sun, Melbourne, Australia. 
This is Australia's largest circulation newspaper. But I was a maths 
teacher for 10 years.

Problem: In all publicity shots for "A Beautiful Mind" there is a 
picture of Russell Crowe looking through a window covered in Maths. 
Usually on his forehead is the statement

   0 less than pi less than one

You will find this statement (or Domain definition) on the official 
Web page, http://www.abeautifulmind.com/main.html .

I have published in the paper that this is wrong. A reader disagrees.
Can this statement ever be true?

Kerry Cue

Date: 04/22/2002 at 11:02:54
From: Doctor Ian
Subject: Re: A Beautiful Mind incorrect Maths?

Hi Kerry,

Note that sometimes 'pi' is the ratio of the circumference of a circle 
to its diameter; and sometimes 'pi' is just a variable that stands for 
something else, usually something beginning with 'p': perimeter, 
probability, price, population, and so on. 

(Why would you do this? Mathematicians often use Greek letters for 
variables; also, if you have more than one quantity whose name begins 
with 'p', you can only use 'p' to represent one of them. Then you have 
to look around for other alternatives: the Greek pi, the Hebrew peh, 
the Russian equivalent, and so on.)

Without knowing the context in which the equation 

  0 < pi < 1
    -    -

appears, there is no way to say whether it's correct or not, for the 
same reason that you can't say whether

  0 < x < 1
    -   -
  
is correct or not, without knowing a lot more about the context in 
which the equation appears. 

(Note that just above the equation is another equation, in which pi is 
used as a subscript on another variable, which suggests that he's 
using pi as a variable, and not in its 'usual' sense.)

For what it's worth, even when pi _does_ represent the ratio of the 
circumference of a particular circle to its diameter, that ratio can 
depend on the shape of the space in which the circle appears. The 
following item from the Dr. Math archives

   Why is Pi a Constant?
   http://mathforum.org/library/drmath/view/57828.html 

shows how that ratio for a particular circle can be equal to 2. If you 
move the circle farther towards the pole opposite the pole from which 
the radius is being measured, the circumference continues to shrink 
while the radius continues to grow. Thus, you can make the ratio as 
close to zero as you desire.  

So there are at least two ways in which the 'domain definition' 

  0 < pi < 1
    -    -

could be true: (1) It defines the range over which some variable whose 
name is 'pi' is defined; or (2) it defines a range of latitudes on a 
sphere for which the ratio of circumference to diameter for a circle 
centered at the north pole is between 0 and 1. 

But there are certainly lots of other ways in which it could be true 
as well. 

The moral of the story is that it's very difficult to write an 
equation that can't be true under _some_ circumstances - even if those 
circumstances make the equation useless. The same is true in English.  
A sentence like 

  President Bush is taller than President Lincoln.

is 'obviously' false... unless it's uttered, for example, by someone 
who is collecting figurines of American Presidents, and noting a fact 
that happens to be true of two objects in his collection.  

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/22/2002 at 19:54:14
From: Kerry Cue
Subject: A Beautiful Mind incorrect Maths?

Dear Dr Maths,

Many thanks for doing the hard work overnight while I slept soundly 
in my bed in Australia.

The answer that pi can be used as a variable solves the problem. As 
if there weren't enough letters in the Greek alphabet. Why couldn't 
they keep pi sacred? Is nothing sacred anymore?

I disagree about the circle and pole analogy. The radius of any line 
of latitude should be measured from the core of the earth.

I hope you had a Happy Pi Day on 3.14.

Thanks again.

Kerry Cue

Date: 04/22/2002 at 20:08:26
From: Doctor Ian
Subject: Re: A Beautiful Mind incorrect Maths?

Hi Kerry,

I'm glad I was able to help.  

The point of the circle and pole analogy is that if you're a 
two-dimensional being on the surface of a sphere, there _is_ no 'core' 
from which to measure anything. There is only the surface; so it is 
only possible to measure along the surface.

You look at it and think it's 'wrong' because you're living in three 
dimensions; but the people in the analogy are not. The trick is to 
imagine what might happen if you started trying to measure circles in 
a similarly curved space, in which you can only sense three dimensions 
out of four.  

If the idea still seems 'wrong' to you, may I suggest that you read 
the delightful little book _Flatland_?  You can find it online at

  http://www.math.brown.edu/~banchoff/gc/Flatland/ 

It's partly math, and partly political satire. It will definitely 
stretch your mind a little. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/