On the role of mathematics:
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... [U]nderpinning every technological advance is the often
unrecognised use of mathematics.  ...  To take one example: the whole
of solid state electronics (and hence much of the fabric of modern
society) could not have been envisaged, let alone developed without
mathematics.  It was mathematical analysis which predicted from basic
equations of physics the "quantum mechanical tunnel effect".  This is
such a subtle and counterintuitive effect that it is unlikely ever to
have been stumbled upon by accident.  However, without it we would
have no miniaturized electronic devices, no personal computers, no
pacemakers, no global telecommunications network, no mobile phones,
no CD players, no .......... !

(Excerpt from a speech given by Professor Brailey Sims at the
Merewether High School (NSW Australia) End of Year Assembly, 25
November 1998.)