calculus book - Mathematics and Computer Science

Chapter 1
The Language of Mathematics
Die Mathematiker sind eine Art Franzosen; redet man zu
ihnen, so übersetzen sie es in ihre Sprache, und alsbald ist
es etwas ganz anderes.
Mathematicians are like Frenchmen. Whatever you say to
them, they will translate into their own language, whereupon
it becomes something completely different. —Goethe
1.1 The Nature of Mathematics
Mathematics is unique among human intellectual endeavors; it is not
art, philosophy, or science, but it shares certain features with each. As a
language, mathematics is unparalleled in its ability to express features
of the physical world with astounding accuracy. At the same time,
mathematics has no intrinsic connection to the real world; the objects
of mathematics are concepts, and do not have physical existence in
the same way that stars, molecules, or people do. Conversely, stars,
molecules, and people are not mathematical objects, though they do
possess attributes (height, mass, or temperature, for example) that can
be modeled by mathematical concepts.
It is remarkable that the language of mathematics can be used to
describe and predict natural phenomena, but this fact is not itself part
of mathematics. Mathematical concepts seem to exist independently,
in a (metaphorical) Platonic universe apart from the physical world,
waiting to be discovered rather than created. Mathematics is guided to
an extent by aesthetics, and though there are no graduate-level courses
on what constitutes beautiful mathematics, mathematicians agree to a
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