Why math?
. develops good thinking,
both raw muscle (it is demanding, like physical exercise),
and specific habits of mind (esp. analytic, rigorous, etc.)
[programming is also useful for this]
. aesthetic: beautiful / part of culture
(like Bach, like Dante)
. scientific: basic structure of world
(more deeply: shows up everywhere!
Overlap of science with life outside its domain is *mediated* by math;
if X in sociology is like Y in chemistry, it's not b/c there are
chemicals in society,
but b/c they share the same *abstract* (viz, *mathematical*) pattern)
You learn the basic patterns, and perhaps to recognize said patterns
(this is more pronounced in science).
[That is, there are *common* patterns.
It's not surprising that various objects have their own fields of
study;
what's powerful and surprising is when some fields have overlap.
EG, chemistry is very powerful b/c it covers the study of all manner of
physical things
(there is a common theory underlying water, fire, air, wood, metal,
earth, etc., etc.).
Different sciences use math differently,
but there are big commonalities (esp. calc / lin alg / stats, hence
their centrality;
more subtly, PDEs):
there is a "math for chemists" and "math for physicists" and "math for
economists" and "math for engineers" (yes, exactly those classes) -
which is not surprising, as the fields are different
...but it's the same core math!]
[the sciences have their own content: physics is *not* simply math, but
it uses math,
and the theory *is* math (the *semantics* and *validity* of the theory
is science).]
[distinguish math and science clearly;
note also that scientific cum quantitative perspectives are not nec.
useful in a particular area,
but they are surprisingly useful: biology -> medicine, ]
The unreasonable effectiveness of mathematics
...but NB: also very *ineffective* in other areas
(see Chaitin)
Beauty:
Bertrand Russell:
Mathematics, rightly viewed, possesses not only truth, but supreme
beauty — a beauty cold and austere, like that of sculpture,
without appeal to any part of our weaker nature, without the
gorgeous trappings of painting or music, yet sublimely pure, and
capable of a stern perfection such as only the greatest art can
show. The true spirit of delight, the exaltation, the sense of
being more than Man, which is the touchstone of the highest
excellence, is to be found in mathematics as surely as poetry.
(The Study of Mathematics, in Mysticism and Logic, and Other
Essays, ch. 4, London: Longmans, Green, 1918.)
Paul Erdős expressed his views on the ineffability of mathematics
when he said "Why are numbers beautiful? It's like asking why is
Beethoven's Ninth Symphony beautiful. If you don't see why,
someone can't tell you. I know numbers are beautiful. If they
aren't beautiful, nothing is."