## The History and Concept of Mathematical Proof (2007)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Krantz07thehistory,

author = {Steven G. Krantz},

title = {The History and Concept of Mathematical Proof},

year = {2007}

}

### OpenURL

### Abstract

A mathematician is a master of critical thinking, of analysis, and of deductive reasoning. These skills travel well, and can be applied in a large variety of situations—and in many different disciplines. Today, mathematical skills are being put to good use in medicine, physics, law, commerce, Internet design, engineering, chemistry, biological science, social science, anthropology, genetics, warfare, cryptography, plastic surgery, security analysis, data manipulation, computer science, and in many other disciplines and endeavors as well. The unique feature that sets mathematics apart from other sciences, from philosophy, and indeed from all other forms of intellectual discourse, is the use of rigorous proof. It is the proof concept that makes the subject cohere, that gives it its timelessness, and that enables it to travel well. The purpose of this discussion is to describe proof, to put it in context, to give its history, and to explain its significance. There is no other scientific or analytical discipline that uses proof as readily and routinely as does mathematics. This is the device that makes theoretical mathematics special: the tightly knit chain of reasoning, following strict logical rules, that leads inexorably to a particular conclusion. It is proof that is our device for establishing the absolute and irrevocable truth of statements in our subject. This is the reason that we can depend on mathematics that was done by Euclid 2300 years ago as readily as we believe in the mathematics that is done today. No other discipline can make such an assertion.

### Citations

496 | A New Kind of Science - Wolfram - 2002 |

404 | Constructive Analysis
- Bishop, Bridges
- 1985
(Show Context)
Citation Context .... So he arranged to transfer to U. C. San Diego. At roughly the same time, Bishop became convinced that proofs by contradiction were fraught with peril. He wrote a remarkable and rather poignant book =-=[BIS]-=- which touts the philosophy of constructivism—similar 14 In fact, for the constructivists, the phrase “there exists” must take on a rigorous new meaning that exceeds the usual rules of formal logic. 2... |

301 |
The Theory of Groups
- Hall
- 1959
(Show Context)
Citation Context ...be as short as possible, and we want our collection of axioms or postulates to be as concise and elegant as possible. If you open up a classic text on group theory—such as Marshall Hall’s masterpiece =-=[HAL]-=-, you will find that there are just three axioms on the first page. The entire 434-page book is built on just those three axioms. 7 Or instead have a look at Walter Rudin’s classic Principles of Mathe... |

104 |
The unreasonable effectiveness of mathematics in the natural sciences
- Wigner
- 1960
(Show Context)
Citation Context ...trast, is here forever. What is marvelous is that, in spite of the appearance of some artificiality in the mathematical process, mathematics provides beautiful models for nature (see the lovely essay =-=[WIG]-=-, which discusses this point). Over and over again, and more with each passing year, mathematics has helped to explain how the world around us works. Just a few examples illustrate the point: • Isaac ... |

81 |
Axiomatic Set Theory
- Suppes
- 1960
(Show Context)
Citation Context ... look at Walter Rudin’s classic Principles of Mathematical Analysis [RUD]. There the subject of real variables is built on just twelve axioms. Or look at a foundational book on set theory like Suppes =-=[SUP]-=- or Hrbacek and Jech [HRJ]. There we see the entire subject built on eight axioms. 3 The Purpose of Proof The experimental sciences (physics, biology, chemistry, for example) tend to use laboratory ex... |

76 |
Grundzüge der theoretischen Logik
- Hilbert, Ackermann
- 1928
(Show Context)
Citation Context ...GRA] and [YAN] give a detailed historical accounting of the colorful history of the Hilbert problems. One of Hilbert’s overriding passions was logic, and he wrote an important treatise in the subject =-=[HIA]-=-. Since Hilbert had a universal and comprehensive knowledge of mathematics, he thought carefully about how the different parts of the subject fit together. And he worried about the axiomatization of t... |

72 | A History of Western Philosophy - Russell - 1945 |

36 | Grundlagen der Geometrie - Hilbert - 1899 |

32 |
Single axioms for groups and Abelian groups with various operations
- McCune
- 1993
(Show Context)
Citation Context ...ise. In mathe7 In fact there has recently been found a way to enunciate the premises of group theory using just one axiom, and not using the word “and”. References for this work are [KUN], [HIN], and =-=[MCC]-=-. 7smatics we set our definitions and axioms in place before we do anything else. In particular, before we endeavor to derive any results we must engage in a certain amount of preparatory work. Then w... |

22 |
Single axioms for groups
- Kunen
- 1992
(Show Context)
Citation Context ...tellectual enterprise. In mathe7 In fact there has recently been found a way to enunciate the premises of group theory using just one axiom, and not using the word “and”. References for this work are =-=[KUN]-=-, [HIN], and [MCC]. 7smatics we set our definitions and axioms in place before we do anything else. In particular, before we endeavor to derive any results we must engage in a certain amount of prepar... |

15 |
A case study of theorem proving by the Knuth-Bendix method discovering that x 3 =x implies ring commutativity
- Stickel
(Show Context)
Citation Context ...olean algebra, projective geometry and other classical parts of mathematics. Even some new theorems in Euclidean geometry have been found (see [CHO]). Results in algebra have been obtained by Stickel =-=[STI]-=-. New theorems have also been found in set theory, lattice theory, and ring theory. One could argue that the reason these results were never found by a human being is that no human being would have be... |

9 |
Proving Geometry Theorems with Rewrite Rules
- Chou, Schelter
- 1986
(Show Context)
Citation Context ...ers have been used effectively to find new theorems in Boolean algebra, projective geometry and other classical parts of mathematics. Even some new theorems in Euclidean geometry have been found (see =-=[CHO]-=-). Results in algebra have been obtained by Stickel [STI]. New theorems have also been found in set theory, lattice theory, and ring theory. One could argue that the reason these results were never fo... |

9 |
The Hilbert Challenge
- Gray
- 2000
(Show Context)
Citation Context ...roblem, and considerable praise and encomia were showered on anyone who did so. Today most of the Hilbert problems are solved, but there are a few particularly thorny ones that remain. The references =-=[GRA]-=- and [YAN] give a detailed historical accounting of the colorful history of the Hilbert problems. One of Hilbert’s overriding passions was logic, and he wrote an important treatise in the subject [HIA... |

4 |
Set Theory (3rd ed
- Jech
- 2003
(Show Context)
Citation Context ...assic Principles of Mathematical Analysis [RUD]. There the subject of real variables is built on just twelve axioms. Or look at a foundational book on set theory like Suppes [SUP] or Hrbacek and Jech =-=[HRJ]-=-. There we see the entire subject built on eight axioms. 3 The Purpose of Proof The experimental sciences (physics, biology, chemistry, for example) tend to use laboratory experiments or tests to chec... |

3 |
The Euler-Bernoulli beam equation with boundary energy dissipation
- Chen, Krantz, et al.
- 1987
(Show Context)
Citation Context ...ributions from the different groups—some numerical, some analytical, some graphical—reinforced each other, and the end result was a rich tapestry of scientific effort. The end product is published in =-=[CHE1]-=- and [CHE2]. This type of collaboration, while rather the exception today, is likely to become ever more common as the field of applied mathematics grows, and as the need for interdisciplinary investi... |

2 |
Analysis, designs, and behavior of dissipative joints for coupled beams
- Chen, Krantz, et al.
- 1989
(Show Context)
Citation Context ...rom the different groups—some numerical, some analytical, some graphical—reinforced each other, and the end result was a rich tapestry of scientific effort. The end product is published in [CHE1] and =-=[CHE2]-=-. This type of collaboration, while rather the exception today, is likely to become ever more common as the field of applied mathematics grows, and as the need for interdisciplinary investigation prol... |

1 |
Groups as gropoids with one law
- Higman, Neumann
(Show Context)
Citation Context ...ual enterprise. In mathe7 In fact there has recently been found a way to enunciate the premises of group theory using just one axiom, and not using the word “and”. References for this work are [KUN], =-=[HIN]-=-, and [MCC]. 7smatics we set our definitions and axioms in place before we do anything else. In particular, before we endeavor to derive any results we must engage in a certain amount of preparatory w... |

1 |
Principles of Real Analysis,3 rd ed
- Rudin
- 1976
(Show Context)
Citation Context ...hat there are just three axioms on the first page. The entire 434-page book is built on just those three axioms. 7 Or instead have a look at Walter Rudin’s classic Principles of Mathematical Analysis =-=[RUD]-=-. There the subject of real variables is built on just twelve axioms. Or look at a foundational book on set theory like Suppes [SUP] or Hrbacek and Jech [HRJ]. There we see the entire subject built on... |

1 |
The Honors
- Yandell
- 2002
(Show Context)
Citation Context ...d considerable praise and encomia were showered on anyone who did so. Today most of the Hilbert problems are solved, but there are a few particularly thorny ones that remain. The references [GRA] and =-=[YAN]-=- give a detailed historical accounting of the colorful history of the Hilbert problems. One of Hilbert’s overriding passions was logic, and he wrote an important treatise in the subject [HIA]. Since H... |