Redefining Reality - The Intellectual Implications of Modern Science

th century, the Russian mathematician
Nikolay Lobachevsky and others realized that they had found
something incredibly deep and troubling. They had in their hands
a new geometry, a non-Euclidean geometry. But if there are two
geometries, which one is true?
If Euclid’s geometry had come under question, at least numbers
were thought to be well behaved; they obeyed certain undeniable
the part. This seems trivial and obvious, but in the second half of
the 19 th century, German mathematician Georg Cantor showed that
it is not the case.
without counting. Two sets are of equal size if there exists a way to
map the members of one onto the other so that each element of the
none left over.
yields the same number. That result seems strange, but it makes
However, Cantor’s work went further.
the real numbers, Cantor found that the set of real numbers has at
least one number that cannot be in the set of rational numbers. Thus,
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