History of Mathematics

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, as well as 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. Two Historical Approaches to Geometry: Axiomatic, Algebraic, and Differential

1. The axiomatic approach (6 lectures)
Geometry in ancient civilizations. The elements of Euclid and the postulated V.

2. The algebraic approach (4 lectures)
Birth of analytical geometry. Related geometry and Euclidean. Projective geometry. Hilbert's Theorem

3. Non-Euclidean geometry. (4 lectures)



Parte III. Elements of History of the Mathematical Analysis.

1. Infinitesimal methods in the ancient Greece (4 lectures)

2. Medieval speculations (2 lectures)

3. The genesis of the Calculus (2 lectures)

4. The Calculus according to Newton and according to Leibniz (3 lectures)

5. Foundations of the Analysis in the eighteenth century (1 lecture)

6. Foundations and criticism in the nineteenth century (1 lecture)

7. Twentieth century and recent developments (1 lecture)

Part I

Basic bibliography:

A. Baker. Transcendental Number Theory. Cambridge University Press, 1975.
C. B. Boyer. Historia de la matemática. Alianza Universidad, 1986. (Available online)
L. W. Cohen, G. Ehrlich. The structure of the real number system, Van Nostrand, 1963.

Complementary bibliography:

R. G. Bartle. The elements of real analysis. John Wiley & Sons, 1964.
J. M. Ortega. Introducción al análisis matemático. Publ. UAB, 1993.


Part II

Basic bibliography:

F. Borceux. An Axiomatic Approach To Geometry. Springer, 2014. (Available online)
Euclides. Elementos, Clásicos do pensamento universal, n. 20, Universidade de Santiago de Compostela, 2013.
U. C. Merzbach, C. B. Boyer. A History of Mathematics, John Wiley & Sons 2011.


Complementary bibliography:

J. Gray. Worlds Out of Nothing. A Course in the History of Geometry in the 19th Century Springer, 2007.
M. J. Greenberg. Euclidean and non-euclidean geometries : development and history. Freeman and Co., 1980.
S. Kulczycki. Non-euclidean Geometry. Pergamon Press, 1961.
C. J. Scriba, P. Schreiber. 5000 Years of Geometry, Mathematics in History and Culture, Springer, 2015.
J. Stillwell. Mathematics and its History. Springer-Verlag, 1989.


Part III

Basic bibliography:

C. H. Edwards. The Historical Development of the Calculus, Springer-Verlag, 1979. (Available online)
A. Verdejo. Mujeres matemáticas: las grandes desconocidas. Universidade de Vigo. 2017.

Complementary bibliography:

U. Bottazzini. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Springer-Verlag, 1986.
C. B. Boyer. Historia de la matemática. Alianza Editorial, Madrid, 1999.
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.

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.

The assessment system will consist of continuous assessment and final exam.


The continuous assessment will be done through the completion of 3 works (one for each part) and 3 written tests (one for each part).


The final grade, both in the first and in the second opportunity, will not be lower than that of the final exam or that obtained by weighting the final exam with the continuous evaluation, giving the latter a weight of 30%. The grade of the continuous assessment will be maintained for the second opportunity.


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.

A student who does not take the final test in the first and second opportunity will be considered as not presented.

Both the continuous evaluation tests and the final exam will be the same in all the expository and interactive teaching groups of the subject.

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.