Learning Discrete Mathematics and Computer Science
via Primary Historical Sources

Available projects

Sixteen projects developed before 2008 are posted at http://www.math.nmsu.edu/hist_projects/

Projects developed since 2008 are listed below. With each project there is a list of suggested courses where the project may be used and a list of topics covered in the project.

• Project 1: Sums of numerical powers in discrete mathematics: Archimedes sums squares in the sand

  Courses: Discrete mathematics (lower division), Calculus

  Topics: Summations, closed formulas, Riemann sums, areas, inequalities, telescoping sums, mathematical induction (optional), systems of linear equations (optional), recursive computation (optional)

• Project 2: Figurate numbers and sums of numerical powers: Fermat, Pascal, Bernoullii

  Courses: Combinatorics, Discrete mathematics (upper division)

  Topics: sums of powers, figurate numbers, binomial coefficients, combination numbers, Pascal's triangle, Bernoulli numbers, recursive definitions and computation, summations and inequalities, counting and geometry, mathematical induction, interplay between discrete sums and calculus.

• Project 3: Deduction through the Ages: A History of Truth Revised: 9/15/2009

  Courses: An introductory discrete mathematics course, a course in discrete mathematics for beginning computer science majors.

  Topics: Propositional logic, truth tables, the truth table of an "if-then" statement, elementary logical equivalences in propositional logic.

• Project 4a: Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce

  Courses: Discrete Mathematics (introductory)

  Topics: elementary set operations (union, intersection, relative difference, symmetric difference) and their (algebraic) properties, Venn diagrams, relation of set operations to standard language use, relation of elementary set operations to boolean algebra, inverse operations (optional), issues related to mathematical notation (optional), changing standards of rigor and proof (optional)

• Project 4b: Boolean Algebra as an Abstract Structure: Edward V. Huntington and Axiomatization

  Courses: Discrete Mathematics (introductory or intermediate), Model Theory, Abstract Algebra

  Topics: axioms for boolean algebra, boolean algebra properties (with proofs), two-valued model of boolean algebra, axiomatization, consistency, independence

• Project 4c: Applications of Boolean Algebra: Claude Shannon and Circuit Design

  Courses: Discrete Mathematics (introductory) for mathematics or computer science majors

  Topics: boolean algebra axioms and properties, application of boolean algebra to design and simplification of circuits, simplification of boolean expressions, disjunctive normal form, two-valued model of boolean algebra

• Project 5: Euclid's Algorithm for the Greatest Common Divisor (10/25/2009)

  Courses: Discrete Mathematics (lower division), Introduction to Computer Science

  Topics: Basic division properties of natural numbers, divisors and common divisors, greatest common divisor, basic reasoning, arguments and proofs, formal proof by induction (optional), basic algorithm design and formal description (pseudocode), reasoning about correctness of algorithms.

• Project 7: Striving for efficiency in Algorithms: Sorting

  Courses: Data Structures and Algorithms

  Topics: Sorting, quicksort, insertion sort, efficiency of algorithms, stack data structure.

• Project 8: Discovery of Huffman Codes

  Courses: Data Structures and Algorithms

  Topics: Huffman encoding, greedy algorithm, prefix-free code, optimal code, unit of information, amount of information, binary tree

• Project 11: Abstract awakenings in algebra: Early group theory in the works of Lagrange, Cauchy, and Cayley (10/9/2009)

  Courses: tba

  Topics: tba

• Project 12: Regular Languages and Finite Automata

  Courses: Automata and Formal Languages, Theory of Computation

  Topics: Regular Languages, finite automata, regular expressions, regular events, Kleene's theorem, Kleene's star operation

• Project 13: Henkin's Method and the Completeness Theorem

  Courses: Mathematical Logic (advanced)

  Topics: syntax of first-order logic, axioms and rules of inference, formal concept of proof, deduction theorem, satisfiability and validity, soundness, Henkin's method, completeness, Loewenheim-Skolem theorem, first-order logic with equality

• Project 14: Peano Arithmetic

  Courses: Mathematical Logic (advanced)

  Topics: Peano postulates, consistency and independence of Peano postulates, recursive definition of addition and multiplication, properties of addition and multiplication, first-order arithmetic

• Project 16: Undecidability of First-Order Logic

  Courses: Mathematical Logic (advanced)

  Topics: Different formulations of the decision problem, models, isomorphic models, intended model, one-way infinite line, two-way infinite line, upper half-plane, Turing machines, halting problem, undecidability of first-order satisfiability

• Project 17: An Introduction to Sets and Infinity via Richard Dedekind (5/11/2009)

  Courses: Discrete/Finite Mathematics (sophomore/junior level)

  Topics: Sets, Union, Intersection, Functions (composition, inverse, injective, and forward image), Set Equivalence, Infinity