ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Mathematics, Statistics, Mathematical Analysis and Optimisation
Areas: Algebra, Geometry and Topology, Mathematical Analysis
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable
The process of development of concepts and theories over time is part of the study of any discipline. In this subject it is intended to address some of the most important facts in the history of mathematics and its influence today and to know the work of some of the most prominent mathematicians. And to use historical reflection to approach the different conceptions that exist today about the nature of mathematical knowledge.
Part I. Need, existence and uniqueness of real numbers
1. The incommensurables in Greek mathematics
Pythagoras and numerical mysticism. Figurative numbers. The Pythagorean pentagram. Golden reason. Hippasus of Metapontum and the discovery of incommensurables. The paradoxes of Zenón. The geometric algebra. Eudoxus of Cnidus and the comparison of reasons between incommensurable magnitudes.
2. Existence and uniqueness of real numbers
Ordered fields. The axiom of the supreme. Unicity of ordered fields verifying the axiom of the supreme. Construction of real numbers by Cauchy sequences of rational numbers. Properties. Existence of an ordered field verifying the axiom of the supreme.
3. The trascendence of "pi"
Polynomials in several variables. Symmetric polynomials. The elementary symmetric polynomials generate the algebra of the symmetric polynomials. Transcendence of "pi".
Part II: From the Euclidean geometry to the structure of the universe
1. The Elements, of Euclid, and the V Postulate
2. The long way to the non Euclidean geometries
3. The geometry of Lobachevski and Bolyai
4. The Program of Erlangen
5. The hyperbolic geometry
6. Geometry and reality
Parte III. Elements of history of the Mathematical Analysis.
1. Infinitesimal Methods in the ancient Greece.
2. Medieval speculations.
3. The genesis of the Calculus.
4. The Calculus according to Newton and according to Leibnitz.
5. Foundations of the Analysis in the eighteenth century.
6. Foundations and criticism in the nineteenth century.
7. Twentieth century and recent developments.
Parte I
Baker, A. Transcendental Number Theory. Cambridge University Press, 1975.
Boyer, C.B. Historia de la matemática. Alianza Universidad, 1986.
Collette, J.P. Historia de la matemática. Siglo XXI de España editores, 1985.
Joaquín M. Ortega, Introducción al análisis matemático, Publ. UAB, 1993.
R. Courant, H. Robbins, ¿Qué es la matemática?, Aguilar, 1971.
Parte II
J. W. Anderson. Hyperbolic Geometry. Springer, 1999
R. Bonola. Geometría no euclidiana. Editorial Calpe, Madrid, 1923
H. M. S. Coxeter, Fundamentos de Geometría. Ed. Limusa-Wiley S.A., México D.F., 1971.
J. Dieudonné. Abregé d'Histoire des Mathématiques, 1700-1900. Herman, Paris, 1986.
Euclides, Elementos, Editorial Gredos. Madrid, 1991.
M. J. Greenberg. Euclidean and non-euclidean geometries : development and history. Freeman and Co., 1980.
S. Hawking. Dios creó los números: los descubrimientos matemáticos que cambiaron la historia, Editorial Crítica, 2006
S. Kulczycki. Non-euclidean Geometry.Pergamon Press, 1961
H. P. Manning. Introductory Non-Euclidean Geometry. Dover, 1963 (Re-edición dun texto de 1901)
L. A. Santaló. Geometrías no euclidianas. Eudeba, Buenos Aires, 1966.
A. Seidenberg. Elementos de geometría proyectiva, Compañía Editorial Continental, México, D.F. 1965
J. Stillwell. Mathematics and its History. Springer-Verlag, 1989
J. Stillwell, Sources of hyperbolic geometry. History of Mathematics 10, Amer. Math. Soc., Providence, RI; London Math. Soc., London, 1996
Part III
A. D. Aleksandrov, A . Kolmogorov, M. A. Laurentiev y otros, La matemática: su contenido, métodos y significado, Tomos 1, 2 y 3. Alianza Editorial, Madrid, 1973-1974.
U. Bottazzini, The Higher Calculus: A History of Real and Complex Analysis fromEuler to Weierstrass. Springer-Verlag.
C. B. Boyer, Historia de la matemática. Alianza Editorial, Madrid, 2003.
C. H. Edwards, The historical development of the Calculus, Springer, 1979.
K. Ríbnikov, Historia de las Matemáticas. Mir.
G. F. Simmons, Ecuaciones diferenciales con aplicaciones y notas históricas, McGraw Hill, 1993.
Contribute to achieve the general, specific and transversal competences included in the Report of the Degree in Mathematics of the USC and, in particular, the following ones:
Written and oral communication of knowledge, methods and general ideas related to the history of mathematics (CG4).
Use of bibliographic resources on the topics of the subject, including Internet access. Use of these resources in different languages, especially English (CT1, CT5).
Knowing how to present hypotheses and draw conclusions using well-reasoned arguments, being able to identify logical flaws and fallacies in arguments (CG2, CE4).
Specific competences of the subject:
To know some of the most important facts in the history of mathematics, and how to characterize different periods, framed in their historical context, recognising its relationship with the Mathematics studied in the degree. To appreciate the differences in formalization, abstraction and rigor in different historical periods. To be able to analyze the different types of mathematical proofs and the problem of the existence of mathematical objects in each historical period. To place the most relevant mathematicians and their contributions in their time. To handle bibliographic references of the history of mathematics.
SCENARIO 1 (adapted normality)
The curriculum of the degree contemplates for this subject three types of sessions: lectures, in which the professor will develop the program; the interactive lessons, in which an active participation of the students will be looked for, by means of the accomplishment of works, the discussion and elaboration of conclusions, ...; and tutorial sessions, which aim to monitor learning. Its format will be adapted to the progress of the course at the time of its completion.
Tutorials will be in person or via email.
SCENARIO 2 (distancing)
Teaching will be partially virtual, according to the distribution organized by the Faculty of Mathematics. The virtual classroom of the course will be used, in which students will be provided with complete notes of the contents of the subject.
Tutorials will be via email or Microsoft Teams.
SCENARIO 3 (closure of faculty)
Teaching will be totally remote. The virtual classroom of the course will be used, in which students will be provided with complete notes of the contents of the subject. Periodic synchronous sessions will also be planned through Microsoft Teams (as many hours per week as expected in the case of in-person attendance) in which the contents of the subject will be worked on and the students' doubts will be solved.
Tutorials will be via email or Microsoft Teams.
The assessment system will consist of continuous assessment and final exam.
SCENARIO 1 (adapted normality)
Continuous assessment will be done through student participation in the classroom and the completion of assignments.
The final grade will not be lower than the final exam or the one obtained by weighting the final exam with the continuous assessment, giving the latter a weight of 30%.
SCENARIOS 2 (distancing) and 3 (closure of facilities)
Continuous assessment will be done by means of deliveries by the students of certain tasks or works proposed by the professor. These deliveries will be made through the virtual classroom.
For the final exam, the indications established at the moment by the Faculty will be followed, being carried out in person if possible and in a non-face-to-face way otherwise.
The final grade will not be lower than the final exam or the one obtained by weighting the final exam with the continuous assessment, giving the latter a weight of 50%.
In those cases of fraudulent behavior regarding assessments the precepts gathered in the “Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións” will be applied.
ON-SITE WORK AT CLASSROOM:
Lectures: 42 hours
Interactive lessons: 14 hours
Tutorials: 2 hours
Total hours on-site work at classroom: 58
PERSONAL WORK OF THE STUDENT: 92 hours
Total hours of work: 150 hours
Active and regular participation in scheduled activities. Search on the bibliographic references to expand and improve the knowledge of the topics of the program. Never hesitate to ask what is not well understood, or any questions that the development of the program raises.
CONTINGENCY PLAN
Adaptation of the methodology to Scenarios 2 and 3:
Scenario 2: Written material will be provided (accessible through the virtual campus) so that students can follow the subject online.
Scenario 3: Written material will be provided (accessible through the virtual campus) so that students can follow the subject online. Regular synchronous sessions will be organised using Microsoft Teams.
Adaptation of the assessment system to Scenarios 2 and 3:
Same procedure as in Scenario 1 with the exception that all the works will be proposed and delivered through the Virtual Campus. In this case, the final grade will not be lower than the final exam nor that obtained by weighing the final exam with continuous assessment, giving the latter a weight of 50% (instead of the 30% established in Scenario 1).
Antonio Garcia Rodicio
- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813144
- a.rodicio [at] usc.es
- Category
- Professor: University Professor
Juan Francisco Torres Lopera
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813137
- juanfrancisco.torres [at] usc.es
- Category
- Professor: University Lecturer
Lucia Lopez Somoza
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- lucia.lopez.somoza [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
María Ferreiro Subrido
- Department
- Mathematics
- Area
- Geometry and Topology
- maria.ferreiro.subrido [at] rai.usc.es
- Category
- Ministry Pre-doctoral Contract
| Monday | |||
|---|---|---|---|
| 16:00-17:00 | Grupo /CLE_01 | Spanish | Classroom 09 |
| Tuesday | |||
| 16:00-17:00 | Grupo /CLE_01 | Spanish | Classroom 09 |
| Wednesday | |||
| 17:00-18:00 | Grupo /CLE_01 | Spanish | Classroom 08 |
| Thursday | |||
| 16:00-17:00 | Grupo /CLIL_02 | Galician, Spanish | Classroom 02 |
| 17:00-18:00 | Grupo /CLIL_01 | Spanish, Galician | Ramón María Aller Ulloa Main Hall |
| 05.19.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 02 |
| 05.19.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 03 |
| 05.19.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 07.14.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 02 |