Mathematics without numbers
Name: tom karchesy
Date: Around 1995
Can you do mathematics without numbers?
It is terribly unfortunate that "mathematics" is only arithmetic for grade
schoolers (at least it was for me). Higher mathematics like geometry, set
theory, and trigonometry involve concepts other than just numbers.
Actually, I was reading recently that the ancient greeks may have decided
that numbers were not to be trusted after the catastrophe discovered by the
Pythagoreans (which they apparently tried to keep secret) that the square
root of 2 was not a rational number. Hence Euclid, and the development of
geometry as a replacement for arithmetic (arithmetic is all contained in the
geometry of line segments on a straight line). Grade schoolers really
should be informed that arithmetic and geometry are mathematical develop-
ments that are already over 2000 years old, and a lot has happened in
mathematics since then!
(Of course I guess it took a while for zero and place-holding numbers to
become accepted, but still . . .)
There are some numbers that are important as they represent fundamental
concepts. Examples are 0, 1, pi, e, and i. Amazingly, these are all neatly
related by the expression e^(i x pi) + 1 = 0.
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Update: June 2012