Math 300: History and Philosophy of Mathematics
History and Philosophy of Mathematics.
(Credit Hours:Lecture Hours:Lab Hours)
F, W, Sp
Historical development of important mathematical ideas and philosophies; implications for the mathematical curriculum.
Desired Learning Outcomes
The main purpose of this course is to learn mathematics through its history, to clarify and reinforce fundamental mathematical ideas, such as (but not limited to) not just who were the first to compute π to whatever degree of accuracy, but more importantly, why is π well-defined, i.e., why is the ratio circumference/diameter the same for all circles, or how the ancient civilizations figured out the exact formula for the area of circle without using integral calculus, as done in calculus courses.
Knowledge of the topics whose history is being discussed, e.g., basic algebra, arithmetic, geometry and calculus.
Minimal learning outcomes
The students should be able to demonstrate a historical perspective of mathematics. They should be able to exhibit with clarity both the ideas and their evolution over the period of their history. The students are expected to know the following periods of mathematical history:
- Mathematics of ancient civilizations, especially of Babylon and Egypt with special reference to:
- Arithmetic. Representation of (natural) numbers in Egyptian, Chinese, Mayan and Roman numerals. The evolution of our own (Indo-Arabic) number system. Ability to do arithmetic in different number systems and conversion from one number system to another.
- Geometry. Basic facts known at the time about triangles, circles, Pythagorean Theorem, Pythagorean triplets, Plimpton 322, Rhind Papyrus, proof of proportionality of sides of similar triangles, universality of π, and the formula for the area of a circle (without using integral calculus).
- Greek Mathematics. Evolution of Greek civilization with special reference to mathematics, its indebtedness to Babylonian and Egyptian math, life and contribution to math of Thales, Pythagoras, Zeno, Euclid, Apollonius, Eudoxus, Archimedes, Diophantus, Pappus and others. Discussion of Euclid's Elements, a sketch of his self-contained proof of Pythagorean Theorem.
- Mathematics of Arabia. Demise of the Greek math, Dark Ages in Europe, migration of Greek mathematicians to Arabia, translation of Greek texts into Arabic at the House of Wisdom at Baghdad. Arab mathematicians, in particular al-Khwarizmi, their contribution to mathematics, including transmission of Indian mathematics to Europe.
- Mathematics in China with special reference to the Chinese Remainder Theorem.
- Mathematics of Medieval India. Indian mathematicians, esp., Āryabhaṭa, Brahmagupta and Bhāskara, their solutions of the quadratic equation and the Pell equation, using the Pell equation to approximate square roots.
- Mathematics of Medieval Europe. Life and work of Fibonacci, especially his rabbit problem and congruent numbers (numbers that are areas of right triangles with all sides rational).
- Mathematics of Renaissance. Famous story behind Cardano's formula for solving cubic equations, discussion of the cube roots of unity, Cardano's formula and its proof.
- Emergence of Modern Mathematics. History of modern mathematical symbols (+, -, ×, etc.), introduction of letters for variables and constants by Viète and Descartes. The beginnings of modern number theory with Fermat. Invention of:
- Calculus, by Newton and Leibniz,
- Algebraic geometry, by Descartes and Fermat.
- Bernoullis and Euler. Work of Euler and his standardization of current mathematical symbols and terminology.
- French and German schools of Mathematics. Gauss, Riemann, Cantor, Dirichlet, Weierstrass, . . ., and Cauchy, Lagrange, Laplace, Fourier, . . . .
- Hilbert and his 23 Problems, and their impact on 20th century math.
Possible textbooks for this course include (but are not limited to):