Foundations of Mathematics

This course introduces the students to academic mathematics. The big difference with high-school mathematics is its emphasis on proof. The student learns about logic and various forms of proof, such as the direct method, proof by contradiction and proof by complete induction.
These concepts will be applied to various fields of mathematics, such as set theory and number theory. Along the way, the student becomes acquainted with the language and notations of mathematics.
The course highlights the main attraction mathematics has for its practitioners: the joy of solving a puzzle. Every proof contains a sparkle of ingenuity, and there is great intellectual satisfaction in discovering the essential step in a proof, or admiring the brilliance of someone who found it before you. A typical problem is for instance the question whether the square root of 2 is a fraction. The answer came as a great shock to the ancient Greeks and its proof is both simple and very clever.
Another feature of the course is an introduction to the mysteries and paradoxes of the concept ʽinfinityʼ. Are there more real numbers than integers? (Yes.) Is the set of fractions larger than the set of integers? (No.)

Finally, there is a big emphasis on writing proofs. A proof should be logical, clear and do precisely what it should: convince a reader of the truth of some mathematical statement. Writing good proofs is a difficult art, which requires practice and the highest intellectual precision.

There will be lecture sessions and exercise sessions. These last sessions form the most important part of the course, as the course is focused on students making their own mathematical discoveries and writing them down correctly. Students are required to regularly hand in exercises, which will be returned and can be handed in again after corrections. Because a mathematical proof is never ʽapproximatelyʼ right, it may take a few cycles of corrections/editing before such a hand-in is acceptable.