COMPLEMENTARY MATHEMATICS I
Learning outcomes of the course unit
Knowledge and understanding. Today mathematics education is a complex phenomenon calling for knowledge from different field, and not only from mathematics. History of mathematics, epistemology and philosophy of mathematics together with specific knowledge of didactical research give important contributions to education and to student’s culture. The course will contribute to the knowledge of relevant historic times of the mathematical thought, by a direct and/or mediate use of original texts. In this way it is possible to follow the evolution of concepts in times and under the influence of various cultural currents. The course will train student to an autonomous comprehension of an ancient text and to recognize the philosophic panorama or the didactic framework influencing the text. The aim is that student turns to able to read the advanced texts and documents produced by research of the field. It contributes to a wide outlook on mathematics.
Applying knowledge and understanding. Students will be required to find solution of problems from the original texts. Moreover they will become able to choose the suitable frameworks in which to insert the topics in order to devise and to conduct personal argumentation regarding the course subjects.
Making judgement. I will assess the students ability for identifying the historic and contents parameters. The final knowledge will help student on working in team and singly and on individuating the main features of the subjects.
Communication skills. Students will reach a firm knowledge of communication process for mathematics as a scientific subject and as branch of learning.
Learning skills. Students have to acquire flexibility in order to adapt themselves to the need of listeners whom their mathematical communication is addressed.
Course contents summary
Some mathematical tools for probability: Boole’s Algebras, Combinatoric.
The birth of probability: Before probability, First steps into probability, Ars Conjectandi of Jacob Bernoulli,
Probability in the 18th Century: Arbuthnot, Montmort, De Moivre.
Other trends in the probability: Buffon, From Leibniz to Bayes.
Probability in the 19th Century: Laplace, The logicism, The criticism to the classical interpretation of probability and the alternative options.
A short excursus into 20th Century probability: Frequentism following Von Mises and Reichenbach, Logicism in the meaning of Wittgenstein and Keynes, The 20th Century subjectivists Ramsey and De Finetti , The axiomatic of Kolmogorov
SOME MATHEMATICAL TOPIC
Boolean Algebras and sets
Intension and extension
An axiomatic of Boolean Algebras
The Boolean Algebra 2
Anelli di Boole
Combinatory of text-books
Combinatorial calculus from set-theoretic point
Combinatorial object with repetitions
THE BIRTH OF PROBABILITY
Probability and moral systems
De Méré’s problem
The years ’60 of 17th Century secolo
First steps into probability
The meanings and the synonims of the word probability.
The Jacob Bernoulli’ treatise and its relevance
Chapter I of Ars conjectandi
Chapter II of Ars conjectandi
Chapter III of Ars conjectandi
Chapter IV of Ars conjectandi
Chapter V of Ars conjectandi
PROBABILITY IN THE 18TH CENTURY
The divine Providence.
Other trends in probability
From Leibniz to Bayes
Probability as a relation
Digression on independent eventsi
Digression of conditional probability
Analysis of Essay
Bayes’ Theorem and its applications
PROBABILITY IN 19TH CENTURY
A short account on Laplace’s biography
Laplace and mathematics
Laplace and probability
The 1774 essay
Analytical theory (first part)
The Philosophical Essay
Analytical theory (second part)
The English culture at the beginning of 19th Century
Boole’ definition of probability
Criticism of the classical version of probability and other proposals
Frequentism in 19th Century
A SHORT EXCURSUS INTO 20TH CENTURY PROBABILITY
Richard Von Mises
The principle of indifference
Axioms of probability calculus
Hacking, I. (1975). The emergence of probability, Cambridge: Cambridge University Press
Todhunter, I. (1865). A History of the Mathematical Theory of Probability from the time of Pascal to that of Laplace, Cambridge & London: Macmillan and Co.
Costantini, D. (1970). Fondamenti del calcolo delle probabilità. Roma: Feltrinelli.
Marchini, C. (2011) Appunti delle lezioni di Matematiche Complementari 2010 – 2001, http://www.unipr.it/arpa/urdidmat/MC10_11
Lectures will be mainly in transmissive style, but with a steady dialogue with students which can be called to the blackboard for discussing problems, or for showing their understanding of and taking part to the course. Original texts will be projected and commented by students and teacher.
Home tasks will consist in problems or solutions of problems to be commented and analysed, for pointing out the peculiar features.
Assessment will be made by a final oral, in which student must solve mathematical or interpretative problems.
Assessment methods and criteria