Post by Terry JamesI am trying to understand what math is, what numbers are? What relations do
they have to the natural world? Are they abstractions of the mind? If so
what distinguishes them from other abstractions? Would it be possible to
discover phenomena theoretically beyond mathematical description? If not
then is it possible, at least in theory, to conceive such phenomena? What
is the relationship between math and logic? Is all logic quantifiable? Is
all math necessarily logical? What is the relationship between math and
music? Are there books that specifically deal with the above questions?
You might be interested to look up the history of the development and
publication of Principia Mathematica (not to be confused with a similarly
named but completely different work by Isaac newton) in the early part of
the 20th century, work by Bertrand Russell and Alfred North Whitehead. They
were trying very hard to codify and contain the notion of "number" and, in a
way, mathematics in general. In particular, Bertrand Russell felt that he
had sealed up the concept in a perfect little package that was utterly air
tight. Then, along came Kurt Godel, who found that there were some flaws in
the notion. It's a fascinating story. Douglas Hofstadter refers to Russell,
Godel, and the Principia in a number of his works.
For a look into the some of the great minds of mathematics, and some
thoughts on what math is, a good book is "God Created the Integers: The
Mathematical Breakthroughs That Changed History", edited by Stephen Hawking.
The title of the books is a quote more aimed at condemning man's need to
make things complicated than anything else. :-)
Mathematics is a mighty broad subject, and almost anything can be described
using mathematics. Math is purely invention, in the sense that it doesn't
refer directly to anything in the real world. That is to say, "2+2" is an
abstract concept, without any meaning except what it's assigned. Fortunately
for us, there are so many places where mathematics maps onto the real world
(so, I can group two apples with two oranges and get four fruits, and my
mathematical expression was able to predict that).
When you wonder about logic, you may need to study that concept carefully
and separately... some people use "logical" and "illogical" to describe
actions that are based on the subjective decisions and impulses of the
person acting. You want to avoid that trap, or mixing math and logic will be
very confusing. Most people, when talking about logic and math will make
reference to "formal systems", which is another fascinating study (and much
the subject of the Principia, above).
Elements of music can be described mathematically in many ways, as evidenced
by the fact that we can write a score or digitize a sound. I don't believe
that anyone has ever come up with a comprehensive mathematical model to
describe the emotions felt when listening to a piece of music, or created a
predictive model for whether a given person will enjoy a musical performance
(although I'm sure there are many, many record company executives who would
love to get their hands on something like that...)
--
Jack