The course aims at introducing the students to the epistemological study of mathematical processes, by means of a detialed analysis of some significant examples stemming from various mathematical disciplines.
Expected learning outcomes
At the end of the course, students will have acquired some basic knowledge about the construction of mathematical theories, and about their epistemological meaning.
1 The bad influence of Mathematics on Philosophy. G.C. Rota opinion. Brauwer's and Ramsey examples indicating philosofical problems. 2. Viceversa: the influence of Philosophy on Mathematics 3. The axiomatic method; does it produce knowledge? Three examples from graph theory, topology and elementary geometry. Hilbert formalism. 4. Rapresentation theorems. The lack of points and Stone representation theorem. Its philosophical and onthological meaning. 5. From the particular case to the universal law. Induction. 6. The axiom of choice, the Zoen Lemma, The good order axiom. Logical equivalence but neither operational nor descriptive equivalence. Zorn Tychonoff and Weierstrass, Stone weierstrass. Transfinite induction and Zorn lemma in the mathematical practice. 7. Lakatos' normative.Examples on Euler-Poincare and the magic numbers of Davies and Hersh. 8. The end of intuition: from patologies to the general case. Weierstrass function. The measure with the concept of category according to Baire. There daoes not exist a unique meaning of "large". The point of view of G. Bachelard. 9. Generalisation via extension and analogy. The Ekeland minimum principle and its Bourbaki version due to Brezis. The weak maximum principle at infinity. Ahlfors parabolicity criterium and the open form of the weak maximum principle at infinity. Cantor-Schroeder theorem and the Tarski analog in the theory of Boolean algebras. 10. Infinity: potential and actual. Axiomatic definition of the real numbers. Archimedes postulate. A non-Arhimedean field.
Prerequisites for admission
A basic mathematical knowledge.
Classroom lessons 42 hours
Lecture notes and bibliograohy suggested in class.
Assessment methods and Criteria
The final examination consists of an oral exam .
In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them. -T Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.