[FOM] Real numbers

[Lucas Wiman]

 >Harvey:
 >If pi were missing, then the circles of unit radius would not have an 
area.
[Dean;]
 >That's a beautiful answer, and correct.  I see no difficulty in there 
being
 >objects to which infinitely many things stand in a given relation.  For
 >example, regarding the number pi as the sum of every x such that x is a
 >value of the decimal expansion of pi.
  >3  + 1/10 + 4/100 + 1/1000 .
 >What's wrong with the statement that pi is the sum total of every 
number in
 >this series?

Here is how real numbers are generally defined (or a somewhat analogous 
treatment):  if q1, q2, q3, q4,... is a series of rational numbers, it 
is called a Cauchy sequence if the following condition holds:

for every m > 0, there exists an N such that for all n, p>N, |qn-
qp|<1/m.  (|x| denotes the absolute value of x).

Intuitively think of these as sequences whose elements get closer 
together "at infinity."  That is, given any quantity, no matter how 
close to zero, you can find a point in the sequence beyond which the 
elements are closer together than this.

Real numbers are Cauchy sequences of rational numbers.  Two real numbers 
a1,a2,... and b1,b2,... are equal (in a non-philosophical sense) if 
a1-b1,a2-b2,... gets arbitrarily close to zero.  So yes, pi is the 
sequence of partial sums you define:

3, 3.1, 3.14, 3.141, ...

It is also:

4, 4--4/3, 4-4/3+4/5, 4-4/3+4/5-4/7, ...

I think Vaughan got closer to the truth than Harvey did.  The essential 
problem with leaving pi out is not that it can't be approximated 
arbitrarily well, but rather that it leaves a hole in the real numbers.  
There are many sequences of reals which doesn't converge to any real if 
pi is left out.  Sequential completeness is the fundamental idea of a 
continuum as determined by decades of research in analysis by some of 
the greatest mathematicians of all time in the nineteenth century 
(Riemann, Balzano, Weirstrauss, Cantor and Dedekind).  Actually, the 
generally called-for condition is the least upper bound condition, which 
is a bit different.  I'm not going to go into it, but suffice it to say, 
I think you could greatly benefit from an elementary analysis book.  I 
recommend Rudin, Principles of Mathematical Analysis.  Rosenlicht's 
Introduction to Analysis isn't as good, but it's a good deal cheaper.  
Any theory of the real numbers which leaves out the ability to do 
analysis (in some form or other) is deficient and can be scrapped on 
general principle.

You would probably also greatly enjoy Weyl's The Continuum, which has 
been nicely reprinted by Dover.  It is a logical / philosophical 
analysis of the continuum with a philosophy you might enjoy.  It is 
quite short, and very cheap.

- Lucas Wiman