Mathematical Thinking in Computer Science

Why some arguments are convincing and some are not? What makes an argument convincing? How to establish your argument in such a way that there is no possible room for doubt left? How mathematical thinking can help with this? In this week we will start digging into these questions. We will see how a small remark or a simple observation can turn a seemingly non-trivial question into an obvious one. Through various examples we will observe a parallel between constructing a rigorous argument and mathematical reasoning.

How can we be certain that an object with certain requirements exist? One way to show this, is to go through all objects and check whether at least one of them meets the requirements. However, in many cases, the search space is enormous. A computer may help, but some reasoning that narrows the search space is important both for computer search and for "bare hands" work. In this module, we will learn various techniques for showing that an object exists and that an object is optimal among all other objects. As usual, we'll practice solving many interactive puzzles. We'll show also some computer programs that help us to construct an example.

We'll discover two powerful methods of defining objects, proving concepts, and implementing programs — recursion and induction. These two methods are heavily used, in particular, in algorithms — for analysing correctness and running time of algorithms as well as for implementing efficient solutions. You will see that induction is as simple as falling dominos, but allows to make convincing arguments for arbitrarily large and complex problems by decomposing them and moving step by step. You will learn how famous Gauss unexpectedly solved his teacher's problem intended to keep him busy the whole lesson in just two minutes, and in the end you will be able to prove his formula using induction. You will be able to generalize scary arithmetic exercises and then solve them easily using induction.

We have already invoked mathematical logic when we discussed how to make convincing arguments by giving examples. This week we will turn mathematical logic full on. We will discuss its basic operations and rules. We will see how logic can play a crucial and indispensable role in creating convincing arguments. We will discuss how to construct a negation to the statement, and you will see how to win an argument by showing your opponent is wrong with just one example called counterexample!. We will see tricky and seemingly counterintuitive, but yet (an unintentional pun) logical aspects of mathematical logic. We will see one of the oldest approaches to making convincing arguments: Reductio ad Absurdum.