The name of an
excellent book on
mathematical philosophy, copyright 1953 by
Lillian R. Lieber, PhD. The book is 359 pages long, and was published by Rinehart & Company, Inc., of
New York. Its
Library of Congress catalog card number is 53-5355.
Infinity is written in prose, but with line breaks that make it look like free verse. The author explains at the beginning that it is not meant to be read as verse, but some sections are so eloquent that it is difficult not to. The book also features funny, abstract, modernistic illustrations by Dr. Lieber's husband, Hugh G. Lieber, in addition to the requisite mathematical illustrations.
The book has the easiest to read explanations and readable proofs of complex mathematical concepts I've yet found. What Infinity lacks in formulae and strict proof, it makes up in clear, intuitive explanations, history, and references. It also serves as a treatise on the nature of science (observation), and its relationship with art (intuition), and pure mathematics (reason). Some of the topics it covers are: Potential Infinity; Hyperbolic, Elliptic, and Parabolic (Euclidean) Geometries; Cantor's proof of enumerability and non-enumerability of sets; Dedekind Cuts; the Differential and Integral Calculus; etc. You get the idea, it's mostly all to do with Cantor's set theory.
Here is the first part of chapter 14 (with some abridgments), to give you an idea of what the book is like:
So far then you realize that
man's yearning for
the infinite
has not been fulfilled in the
physical world.
Even the
entire physical universe
is not infinite,
so far as we know.
Even the total number of electrons
in the entire physical universe
is not infinite
The old-fashioned idea that
the earth was flat
and extends to infinity
turned out to be false,
for as we now know
the earth is a sphere
and thus is not infinite,
or, as we say,
it is unbounded but finite.
And similarly
our three-dimensional universe
has also turned out to be
unbounded but finite.
Thus, wherever we look in the
physical universe
we have not found infinity.