## 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

## Course contents

SOME MATHEMATICAL TOPIC

Boolean Algebras

Boolean Rings

Boolean Algebras and sets

Intension and extension

Partial ordering

An axiomatic of Boolean Algebras

The Boolean Algebra 2

Anelli di Boole

Combinatorics

Combinatory of text-books

Combinatorial calculus from set-theoretic point

Combinatorial object with repetitions

THE BIRTH OF PROBABILITY

Before probability

Probability and moral systems

De Méré’s problem

Huygens’ contribution

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.

Montmort

De Moivre

Other trends in probability

Buffon

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

Laplace

Laplace’s precursors

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)

Logicism

The English culture at the beginning of 19th Century

Boole’ definition of probability

Other logicists

Criticism of the classical version of probability and other proposals

Frequentism in 19th Century

The subjectivists

A SHORT EXCURSUS INTO 20TH CENTURY PROBABILITY

Frequentism

Richard Von Mises

Reichenbach

Logicism

Wittgenstein

Keynes

The principle of indifference

Keynes’s axiomatic

Subjectivism

Ramsey

De Finetti

Axioms of probability calculus

Kolmogorov

Kolmogorov’s axioms

## Recommended readings

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

## Teaching methods

Teaching methods

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

Assessment will be made by a final oral, in which student must solve mathematical or interpretative problems.

## Assessment methods and criteria

Oral examination