General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

2.2 Elementary Discrete Math
2.2.1 Mathematical Foundations: Natural Numbers
We have seen in the last section that we will use mathematical models for objects and data structures
throughout Computer Science. As a consequence, we will need to learn some math before
we can proceed. But we will study mathematics for another reason: it gives us the opportunity
to study rigorous reasoning about abstract objects, which is needed to understand the “science”
part of Computer Science.
Note that the mathematics we will be studying in this course is probably different from the
mathematics you already know; calculus and linear algebra are relatively useless for modeling
computations. We will learn a branch of math. called “discrete mathematics”, it forms the
foundation of computer science, and we will introduce it with an eye towards computation.
Let’s start with the math!
Discrete Math for the moment
Kenneth H. Rosen Discrete Mathematics and Its Applications, McGraw-Hill, 1990 [Ros90].
Harry R. Lewis and Christos H. Papadimitriou, Elements of the Theory of Computation,
Prentice Hall, 1998 [LP98].
Paul R. Halmos, Naive Set Theory, Springer Verlag, 1974 [Hal74].
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The roots of computer science are old, much older than one might expect. The very concept of
computation is deeply linked with what makes mankind special. We are the only animal that
manipulates abstract concepts and has come up with universal ways to form complex theories and
to apply them to our environments. As humans are social animals, we do not only form these
theories in our own minds, but we also found ways to communicate them to our fellow humans.
The most fundamental abstract theory that mankind shares is the use of numbers. This theory
of numbers is detached from the real world in the sense that we can apply the use of numbers to
arbitrary objects, even unknown ones. Suppose you are stranded on an lonely island where you
see a strange kind of fruit for the first time. Nevertheless, you can immediately count these fruits.
Also, nothing prevents you from doing arithmetics with some fantasy objects in your mind. The
question in the following sections will be: what are the principles that allow us to form and apply
numbers in these general ways? To answer this question, we will try to find general ways to specify
and manipulate arbitrary objects. Roughly speaking, this is what computation is all about.
Something very basic:
Numbers are symbolic representations of numeric quantities.
There are many ways to represent numbers (more on this later)
let’s take the simplest one (about 8,000 to 10,000 years old)
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