## Historical Projects in Discrete Mathematics and Computer Science

Citations: | 2 - 1 self |

### BibTeX

@MISC{Barnett_historicalprojects,

author = {Janet Barnett and Guram Bezhanishvili and Hing Leung and Jerry Lodder and David Pengelley and Desh Ranjan},

title = {Historical Projects in Discrete Mathematics and Computer Science},

year = {}

}

### OpenURL

### Abstract

A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development

### Citations

6050 | A mathematical theory of communication - Shannon - 1948 |

3836 |
J.D.: Introduction to automata theory, languages, and computation
- Hopcroft, Motwani, et al.
(Show Context)
Citation Context ... from Q × Σ into Q × D where 21 Technically (and, more accurately), we say that 2DFA can be exponentially more succinct in descriptional size than DFA for solving the same problems. 22 Two textbooks (=-=[35]-=-, [46]) cover 2DFA. One [35] follows the approach by Rabin and Scott, and the other [46] follows Shepherdson’s ideas. 23 In contrast, it is difficult to construct mechanically a DFA from a 2DFA based ... |

1157 |
M.: On computable numbers, with an application to the Entscheidungsproblem
- Turing
- 1936
(Show Context)
Citation Context ...cance, such as Blaise Pascal’s “Treatise on the Arithmetical Triangle” [53, vol. 30] from the 1650s or Alan Turing’s 1936 paper “On Computable Numbers with an Application to the Entscheidungsproblem” =-=[66]-=-. The projects are designed to introduce or provide supplementary material for topics in the curriculum, such as induction in a discrete mathematics course, or compilers and computability for a comput... |

837 |
Theory of recursive functions and effective computability
- Rogers
- 1967
(Show Context)
Citation Context ...ed about recursive functions via a treatment that emphasized partial functions from the outset to realize just how important Kleene’s contribution was. Thus Rogers’ excellent and influential treatise =-=[57]-=-, p. 12, contains an historical account which gives the impression that the subject had been formulated in terms of partial functions from the beginning. To summarize, in the mid-thirties there were s... |

686 |
Introduction to Metamathematics
- Kleene
- 1952
(Show Context)
Citation Context ...g computable functions to functions of multiple variables. 4.(c) In your own words explain Kleene’s definition of a Turing machine by reading carefully the following excerpt from section 67 of Kleene =-=[43]-=-. The machine is supplied with a linear tape, (potentially) infinite in both directions (say to the left and right). The tape is divided into squares. Each square is capable of being blank, or of havi... |

385 |
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme.” Monatshefte für Mathematik und Physik
- Gödel
- 1931
(Show Context)
Citation Context ... a general logical notion of computability in terms of specific types of functions, which then forms the basis of “Church’s thesis.” The project contains excerpts from the original work of Kurt Gödel =-=[23]-=- and Stephen Kleene [39] from the 1930s in addition to that of Turing. In computer science today a Turing machine is modeled by an automaton. A key result of John Shepherdson [62] from 1959 2states t... |

376 |
Representation of events in nerve nets and finite automata
- Kleene
- 1951
(Show Context)
Citation Context ...ction Hing Leung 18 In 1943, McCulloch and Pitts [51] published a pioneering work on a model for studying the behavior of the nervous systems. Following up on the ideas of McCulloch and Pitts, Kleene =-=[44]-=- wrote the first paper on finite automata, which proved a theorem that we now call Kleene’s theorem. A finite automaton can be considered as the simplest machine model in that the machine has a finite... |

263 |
An unsolvable problem of elementary number theory
- Church
(Show Context)
Citation Context ...s that Gödel’s rejection of λ-definability as a possible “definition” of effective calculability was the main reason behind Church’s announcement of his thesis in terms of general recursive functions =-=[8]-=-. Church made his announcement at a meeting of the American Mathematical Society in New York City on April 19, 1935. Below is an excerpt from his abstract: Following a suggestion of Herbrand, but modi... |

126 |
Automata and Computability
- Kozen
- 1997
(Show Context)
Citation Context ...Q × Σ into Q × D where 21 Technically (and, more accurately), we say that 2DFA can be exponentially more succinct in descriptional size than DFA for solving the same problems. 22 Two textbooks ([35], =-=[46]-=-) cover 2DFA. One [35] follows the approach by Rabin and Scott, and the other [46] follows Shepherdson’s ideas. 23 In contrast, it is difficult to construct mechanically a DFA from a 2DFA based on the... |

113 | The four-colour theorem - Robertson, Sanders, et al. - 1997 |

102 |
A symbolic analysis of relay and switching circuits
- Shannon
- 1940
(Show Context)
Citation Context ...m 1945 [69]. The second project begins with Claude Shannon’s analysis of the circuitry necessary to perform base two addition from his 1938 paper “A Symbolic Analysis of Relay and Switching Circuits” =-=[60]-=-. Taking a step backwards chronologically, the project examines addition and subtraction on a Chinese abacus, which, when used to its full potential, provides an excellent device for base sixteen (hex... |

100 | A note on the entscheidungsproblem - Church - 1936 |

98 | Introduction to Mathematical Logic - Church - 1956 |

83 |
On notation for ordinal numbers
- Kleene
- 1938
(Show Context)
Citation Context ...54) formulated yet another equivalent version of computability [54]. However, his work was less detailed than Turing’s. Lastly we mention that partial recursive functions were introduced by Kleene in =-=[41]-=-. In [43] he also generalized the notion of Turing computable functions to partial functions and showed that a partial function is Turing computable if, and only if, it is partial recursive. The impor... |

76 |
Grundzüge der theoretischen Logik
- Hilbert, Ackermann
- 1928
(Show Context)
Citation Context ...onsistent, a question which would have profound consequences for the foundations of mathematics. Continuing in this direction, in 1928 Hilbert proposed the decision problem (das Entscheidungsproblem) =-=[31, 32, 33]-=-, which asked whether there was a standard procedure that can be applied to decide whether a given mathematical statement is true. Both Alonzo Church (1903–1995) [9, 10] and Alan Turing (1912–1954) [6... |

72 |
Beiträge zur Begründung der transfiniten Mengenlehre II
- Cantor
(Show Context)
Citation Context ...s us to create infinitely many transfinite cardinal numbers. We will learn much of this by studying and working with the historical source [6], which is an English translation of two papers by Cantor =-=[4, 5]-=- that appeared in 1895 and 1897. More on Georg Cantor can be found in [15, 34, 48] and in the literature cited therein. We begin by reading Cantor’s definition of the cardinal number of a given set. N... |

67 |
General Recursive Functions of Natural Numbers’, Mathematische Annalen 112
- Kleene
- 1936
(Show Context)
Citation Context ...n of computability in terms of specific types of functions, which then forms the basis of “Church’s thesis.” The project contains excerpts from the original work of Kurt Gödel [23] and Stephen Kleene =-=[39]-=- from the 1930s in addition to that of Turing. In computer science today a Turing machine is modeled by an automaton. A key result of John Shepherdson [62] from 1959 2states that whatever can be comp... |

64 |
The Undecidable
- Davis
- 1965
(Show Context)
Citation Context ...showed that the three notions of Turing computable, general recursive, and λ-definable functions coincide. On page 72 of Gödel’s “postscriptum” to his 1934 lecture notes which he prepared in 1964 for =-=[13]-=-, Gödel states: Turing’s work gives an analysis of the concept of “mechanical procedure” (alias “algorithm” or “computation procedure” or “finite combinatorial procedure”). This concept is shown to be... |

52 | The Computer from Pascal to von Neumann - Goldstine - 1972 |

50 |
Lambda-definability and recursiveness
- Kleene
- 1936
(Show Context)
Citation Context ...equivalent to or weaker than recursiveness,” which indicates that, at the time, Church was not yet certain whether λ-definability was equivalent to general recursiveness. Kleene filled in this gap in =-=[40]-=- by showing that these two notions were indeed equivalent. Thus, in the full version of his paper [9], Church was already fully aware that the two notions of general recursiveness and λ-definability c... |

47 |
Graph Theory 1736-1936
- Biggs, Lloyd, et al.
- 1976
(Show Context)
Citation Context ...luded with three projects on graph theory for an upper-level mathematics or computer science course. The project Euler Circuits and the Königsberg Bridge Problem offers Leonhard Euler’s 1736 solution =-=[3]-=- to what today is phrased as finding a closed path in a graph that traverses each edge exactly once. The project Topological Connections from Graph Theory studies the idea of flow around a network (gr... |

47 |
Contributions to the founding of the theory of transfinite numbers
- Cantor
- 1915
(Show Context)
Citation Context ...covered in each. The project Are All Infinities Created Equal? touches on naive set theory, countability, and one-to-one correspondences between sets using excerpts from George Cantor’s original work =-=[6]-=- in the late 1890s. The project is well suited for a beginning undergraduate course in discrete mathematics. The idea of computability is raised in the project An Introduction to Turing Machines, base... |

45 |
Recursive predicates and quantifiers
- Kleene
- 1943
(Show Context)
Citation Context ... been speculating, and finally definitely proposed, that the λ-definable functions are all the effectively calculable functions—what he published in [9], and which I in [43] Chapter XII (or almost in =-=[42]-=-) called “Church’s thesis”. When Church proposed this thesis, I sat down to disprove it by diagonalizing out of the class of the λ-definable functions. But, quickly realizing that the diagonalization ... |

43 |
On undecidable propositions of formal mathematical systems
- Gödel
- 1965
(Show Context)
Citation Context ...el early in 1934, but, according to a November 29, 1935, letter from Church to Kleene, Gödel regarded it “as thoroughly unsatisfactory.”Instead, in his lectures during the spring of 1934 at Princeton =-=[24]-=-, Gödel generalized the notion of primitive recursive functions, which was introduced by him in his epoch-making paper on undecidable propositions [23]. 29 He did this by modifying a suggestion made b... |

42 |
The reduction of two-way automata to one-way automata
- Shepherdson
- 1959
(Show Context)
Citation Context ...l work of Kurt Gödel [23] and Stephen Kleene [39] from the 1930s in addition to that of Turing. In computer science today a Turing machine is modeled by an automaton. A key result of John Shepherdson =-=[62]-=- from 1959 2states that whatever can be computed via an automaton reading a tape in two directions (forwards and backwards) can be computed via an automaton reading a tape in just one direction. The ... |

40 |
Finitary Combinatory Processes – Formulation I
- Post
- 1936
(Show Context)
Citation Context ...te if we did not mention that around the same time and independently of Turing, but not of the work in Princeton, Emil Leon Post (1897–1954) formulated yet another equivalent version of computability =-=[54]-=-. However, his work was less detailed than Turing’s. Lastly we mention that partial recursive functions were introduced by Kleene in [41]. In [43] he also generalized the notion of Turing computable f... |

39 |
A Logical Calculus of the Ideas Immanent
- McCulloch, Pitts
- 1943
(Show Context)
Citation Context ...or using dynamic versus naïve recursive programming influences the effectiveness of computation. 528 Two-Way Deterministic Finite Automata 8.1 Introduction Hing Leung 18 In 1943, McCulloch and Pitts =-=[51]-=- published a pioneering work on a model for studying the behavior of the nervous systems. Following up on the ideas of McCulloch and Pitts, Kleene [44] wrote the first paper on finite automata, which ... |

29 | Science and Civilisation in - Needham - 2004 |

22 |
Journey Through Genius: The Great Theorems of Mathematics
- Dunham
- 1990
(Show Context)
Citation Context ...uch of this by studying and working with the historical source [6], which is an English translation of two papers by Cantor [4, 5] that appeared in 1895 and 1897. More on Georg Cantor can be found in =-=[15, 34, 48]-=- and in the literature cited therein. We begin by reading Cantor’s definition of the cardinal number of a given set. Note that in his writings Cantor uses “aggregate” instead of the more familiar “set... |

22 |
Ueber die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren
- Hierholzer, Wiener
(Show Context)
Citation Context ...f this direction is due to the German mathematician Carl Hierholzer (1840 - 1871); following Hierholzer’s premature death, this proof was prepared for publication by a colleague and appeared in 1873 [=-=[28]-=-]. Complete the proof sketch below by filling in the missing details. (Specific questions that you will need to address in your completed proof are indicated in italics.) Note: You may make use of the... |

21 | Why Gödel Didn't Have Church's Thesis - Davis - 1982 |

18 |
Computability and λ-definability
- Turing
- 1937
(Show Context)
Citation Context ...inability and general recursiveness just as he was ready to send off his manuscript, to which he then added an appendix outlining a proof of the equivalence of his computability to λ-definability. In =-=[67]-=- he gave a proof of the equivalence in detail. Thus, Turing introduced his notion of computability in 1936–1937 and, using some of the results of Kleene, showed that the three notions of Turing comput... |

17 | A Critical Exposition of the Philosophy of Leibniz - Russell - 1975 |

17 |
The Hellenistic Philosophers
- Long, Sedley
- 1999
(Show Context)
Citation Context ...project on the summation of numerical powers. The theme of propositional logic and truth tables is treated in a separate project by Jerry Lodder, and draws on sources from Chrysippus (280–206 b.c.e.) =-=[4]-=- to Ludwig Wittgenstein (1889–1951) [5]. A total of fifteen projects will be written: 1. Summation of Numerical Powers 2. Summation of Powers, Bernoulli Numbers, and the Euler-Maclaurin Summation Form... |

16 |
Mathematical Expeditions: Chronicles by the Explorers
- Laubenbacher, Pengelley
- 1999
(Show Context)
Citation Context ...uch of this by studying and working with the historical source [6], which is an English translation of two papers by Cantor [4, 5] that appeared in 1895 and 1897. More on Georg Cantor can be found in =-=[15, 34, 48]-=- and in the literature cited therein. We begin by reading Cantor’s definition of the cardinal number of a given set. Note that in his writings Cantor uses “aggregate” instead of the more familiar “set... |

15 |
Origins of recursive function theory
- Kleene
- 1981
(Show Context)
Citation Context ...e theory of λ-definable functions. Church proposed to identify the effectively calculable functions with the λ-definable functions. Here is Kleene’s description of these events, taken from page 59 of =-=[45]-=-: The concept of λ-definability existed full-fledged by the fall of 1933 and was circulating among the logicians at Princeton. Church had been speculating, and finally definitely proposed, that the λ-... |

14 |
Probleme der Grundlegung der Mathematik’, Mathematische Annalen 102
- Hilbert
- 1929
(Show Context)
Citation Context ...onsistent, a question which would have profound consequences for the foundations of mathematics. Continuing in this direction, in 1928 Hilbert proposed the decision problem (das Entscheidungsproblem) =-=[31, 32, 33]-=-, which asked whether there was a standard procedure that can be applied to decide whether a given mathematical statement is true. Both Alonzo Church (1903–1995) [9, 10] and Alan Turing (1912–1954) [6... |

12 | From ENIAC to UNIVAC, An Appraisal of the Eckert-Mauchly - Stern |

11 | A History of Vector Analysis: The Evolution of the Idea of a Vectonal System, Notre Dame - Crowe - 1967 |

10 |
Remarkable Mathematicians: From Euler to von Neumann
- James
- 2002
(Show Context)
Citation Context ...er of the twentieth century, having contributed significantly to the foundations of quantum mechanics, the development of the atomic bomb, and the logical structure of the electronic digital computer =-=[25, 36]-=-. Born in Budapest Hungary, the young von Neumann showed a gift for mathematics, received a doctorate in the subject from the University of Budapest and a degree in chemical engineering from the Eidge... |

8 |
Principles of mathematical logic
- Hilbert, Ackermann
- 1951
(Show Context)
Citation Context ...onsistent, a question which would have profound consequences for the foundations of mathematics. Continuing in this direction, in 1928 Hilbert proposed the decision problem (das Entscheidungsproblem) =-=[31, 32, 33]-=-, which asked whether there was a standard procedure that can be applied to decide whether a given mathematical statement is true. Both Alonzo Church (1903–1995) [9, 10] and Alan Turing (1912–1954) [6... |

8 |
An application of modular equations in analysis situs
- Veblen
- 1912
(Show Context)
Citation Context ...ctures of the American Mathematical Society in 1916. Although he remained interested in topology afterwards, he published little research in this area following the 1922 publication of Analysis Situs =-=[68]-=-. The extracts we examine are taken from [3, pp. 136 - 141]. Note: This project assumes the reader is familiar with basic notions of graph theory, including the definition of isomorphism and isomorphi... |

7 |
A History of Mathematics: An Introduction, Second Edition
- Katz
- 1998
(Show Context)
Citation Context ...The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development in ancient cultures =-=[38]-=-. By contrast, a course in finite mathematics is sometimes presented as a fast-paced news reel of facts and formulae, often memorized by the students, with the text offering only passing mention of th... |

6 | A history of Chinese Mathematics - Martzloff - 1997 |

5 |
Teaching Discrete Mathematics via Primary Historical Sources, http://www.math.nmsu.edu/hist_projects
- -Bezhanishvili, Leung, et al.
- 2003
(Show Context)
Citation Context ...ng covered. Be familiar with all details of a project before assignment. The source file for each project together with its bibliographic references can be downloaded and edited from the web resource =-=[2]-=-. The topics covered by the projects include set theory, mathematical induction, binary arithmetic, computability, graph theory, and the combinatorics of the Catalan numbers. They range in level from ... |

5 |
Commentarii Academiae Scientarium Imperialis Petropolitanque 7
- Euler, Novi
(Show Context)
Citation Context ... proof of his method. The method, if correct, leads to a formula for calculating the number of triangulations of an n-sided polygon which can be used to quickly calculate this number [17, p. 339-350] =-=[18]-=-. Later, Euler communicated this problem to the Hungarian mathematician Jan Andrej Segner (1704–1777). Segner, who spent most of his professional career in Germany (under the German name Johann Andrea... |

5 |
History of binary and other nondecimal numeration
- GLASER
- 1981
(Show Context)
Citation Context ...utstanding problem, inaugurated a mathematical technique, or offered a novel point of view on existing material. For example, although Leibniz was not the first to experiment with base two numeration =-=[22]-=-, his paper “An Explanation of Binary Arithmetic” presents the topic as a confluence of order, harmony, and ease of calculation. Here is a brief description of the projects appearing in this chapter, ... |

4 |
polygone convexe étant donné, de combien de manières peut-on le partager en triangles au moyen de diagonales?” Journal de Mathématiques Pures et Appliquées
- Lamé, “Un
(Show Context)
Citation Context ...the modern formula for the combination numbers or binomial coefficients. An upper-level project on combinatorics is Counting Triangulations of a Polygon, which presents Gabriel Lamé’s 1838 derivation =-=[47]-=- for the number of triangulations of a convex n-sided polygon in terms of a simple recursion relation. From this follows easily the modern formula for the “Catalan numbers” in terms of binomial coeffi... |

4 | Treatise on the Arithmetical Triangle
- -Pascal
- 1991
(Show Context)
Citation Context ...ried Leibniz, 1646–1716, [6]) 5. “Arithmetic Backwards from von Neumann to the Chinese Abacus” (Claude Shannon, 1916– 2001, [10]) 6. “Treatise on the Arithmetical Triangle” (Blaise Pascal, 1623–1662, =-=[9]-=-) 7. “Counting Triangulations of a Polygon” (Gabriel Lamé, 1795–1870, [8]) 8. “Two-Way Deterministic Finite Automata” (John Shepherdson [11]) 9. “Church’s Thesis” (Alonzo Church, 1903–1995, [5]) 10. “... |

4 | Excursions in Calculus: An Interplay - Young - 1992 |