Foundations of Mathematics (2017), UCSCIMAT14

Teacher: F.Beukers, teaching assisant: Wesley Strik

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 it's proof is both simple and very clever.
Another feature of the course is the introduction to the mysteries and paradoxes of the concept 'infinity'. 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.

Aim: After completing this course students are able to:

The classes will be a mix of lecture sessions and exercise sessions. Exercises 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 graded.

As our guide we use the book by
G.Chartrand, A.D.Polimeni, P.Zhang,
Mathematical Proofs, A transition to advanced mathematics.
Pearson, third edition 2013.

Classes take place on Wednesdays 13:45 - 15:30 and Fridays 11:00-12:45 in Newton D and they will be in the form of a combined lecture and exercise class.
Homework is assigned weekly and must be made and handed in by each student individually. Homework will always be graded; these grades will determine 20 percent of the overall grade for the course.
Exams: There is a midterm and final exam, each counts towards 40 percent of the overall grade.

Material covered so far: