The Development of the Idea of Mathematical Proof: A 5-Year Case Study

1996 ◽  
Vol 27 (2) ◽  
pp. 194 ◽  
Author(s):  
Carolyn A. Maher ◽  
Amy M. Martino
Keyword(s):  
Mathematical Proof

Trusting Computers Through Trusting Humans

2008 ◽  
pp. 3460-3474
Author(s):  
Alison Adam ◽  
Paul Spedding
Keyword(s):  
Quality Assurance ◽  
Software Engineering ◽  
Computer System ◽  
Software Quality ◽  
Software Verification ◽  
Mathematical Proof ◽  
Quality Standards ◽  
Software Quality Assurance ◽  
The Social

This article considers the question of how we may trust automatically generated program code. The code walkthroughs and inspections of software engineering mimic the ways that mathematicians go about assuring themselves that a mathematical proof is true. Mathematicians have difficulty accepting a computer generated proof because they cannot go through the social processes of trusting its construction. Similarly, those involved in accepting a proof of a computer system or computer generated code cannot go through their traditional processes of trust. The process of software verification is bound up in software quality assurance procedures, which are themselves subject to commercial pressures. Quality standards, including military standards, have procedures for human trust designed into them. An action research case study of an avionics system within a military aircraft company illustrates these points, where the software quality assurance (SQA) procedures were incommensurable with the use of automatically generated code.

Trusting Computers Through Trusting Humans

2011 ◽  
pp. 384-398
Author(s):  
Alison Adam ◽  
Paul Spedding
Keyword(s):  
Quality Assurance ◽  
Software Engineering ◽  
Computer System ◽  
Software Quality ◽  
Software Verification ◽  
Mathematical Proof ◽  
Quality Standards ◽  
Software Quality Assurance ◽  
The Social

This chapter considers the question of how we may trust automatically generated program code. The code walkthroughs and inspections of software engineering mimic the ways that mathematicians go about assuring themselves that a mathematical proof is true. Mathematicians have difficulty accepting a computer generated proof because they cannot go through the social processes of trusting its construction. Similarly, those involved in accepting a proof of a computer system or computer generated code cannot go through their traditional processes of trust. The process of software verification is bound up in software quality assurance procedures, which are themselves subject to commercial pressures. Quality standards, including military standards, have procedures for human trust designed into them. An action research case study of an avionics system within a military aircraft company illustrates these points, where the software quality assurance (SQA) procedures were incommensurable with the use of automatically generated code.

scholarly journals PENGGUNAAN BAHAN AJAR STRUKTUR ALJABAR UNTUK MENINGKATKAN KEMAMPUAN PEMBUKTIAN MATEMATIS PADA MAHASISWA IKIP PGRI PONTIANAK

2017 ◽  
Vol 8 (2) ◽  
pp. 60
Author(s):  
Jamilah Jamilah ◽  
Syarifah Fadillah
Keyword(s):  
Mathematics Education ◽  
Experimental Method ◽  
Study Design ◽  
Algebraic Structure ◽  
Teaching And Learning ◽  
Mathematical Proof ◽  
Case Study Design ◽  
Class A

This study aims to know the effect of teaching and learning using Algebraic Stucture materials based on mathematical proof to improve mathematical proof ability. The type of this study is experimental method with One Shot Case Study design. Population in this study is students of 6th semester in Departement of Mathematics Education and sample is students in class A and B which are consist of 83 students. The results show that teaching and learning using Algebraic Structure materials based on mathematical proof can improve mathematical proof ability with 16Z0bs=9,74>1,645=Z0,05"> . The total score of mathematical proof ability is 44,34 from 100 or 44,34 % (good). In each aspect, teaching and learning using Algebraic Stucture materials based on mathematical Proof can improve the ability to construct evidence includes the use of methods of proof and the use of fact, the concept and principles of mathematics by 26,99 from 50 or 53,98 % (good) and 17,14 from 50 or 34,70 % (good) in the aspect of the ability to read a proof to determine truth or mistake by seeing the correspondence between the system of axioms, premises, or theorems existing. Keywords : algebraic structure material, mathematical proof ability

Trusting Computers Through Trusting Humans

2009 ◽  
pp. 2435-2449
Author(s):  
Alison Adam ◽  
Paul Spedding
Keyword(s):  
Quality Assurance ◽  
Action Research ◽  
Software Engineering ◽  
Computer System ◽  
Software Quality ◽  
Mathematical Proof ◽  
Quality Standards ◽  
Software Quality Assurance ◽  
The Social

This article considers the question of how we may trust automatically generated program code. The code walkthroughs and inspections of software engineering mimic the ways that mathematicians go about assuring themselves that a mathematical proof is true. Mathematicians have difficulty accepting a computer generated proof because they cannot go through the social processes of trusting its construction. Similarly, those involved in accepting a proof of a computer system or computer generated code cannot go through their traditional processes of trust. The process of software verifi- cation is bound up in software quality assurance procedures, which are themselves subject to commercial pressures. Quality standards, including military standards, have procedures for human trust designed into them. An action research case study of an avionics system within a military aircraft company illustrates these points, where the software quality assurance (SQA) procedures were incommensurable with the use of automatically generated code.

The Development of the Idea of Mathematical Proof: A 5-Year Case Study

1996 ◽  
Vol 27 (2) ◽  
pp. 194-214 ◽  
Author(s):  
Carolyn A. Maher ◽  
Amy M. Martino
Keyword(s):  
Small Group ◽  
Mathematical Thinking ◽  
Mathematical Proof ◽  
Individual Interview ◽  
Longitudinal Case Study ◽  
Whole Class ◽  
Mathematical Justification ◽  
Insight Into

This longitudinal case study presents a sequence of episodes that document the mathematical thinking of one child, Stephanie, over a 5-year period. Her development of the idea of mathematical justification spans grades 1–5. Stephanie worked on several combinatorics tasks in small-group, whole-class, and individual interview settings. The documented events indicate Stephanie's progress in classifying, organizing, and reorganizing data. The study provides some significant insight into the process by which Stephanie learned to make proofs, within a setting that encouraged the development of her ideas.

NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF

2019 ◽  
pp. 1-35
Author(s):  
WILFRIED SIEG ◽  
PATRICK WALSH
Keyword(s):  
Proof Theory ◽  
Mathematical Proof ◽  
Formal Proofs ◽  
Bernstein Theorem ◽  
Search System ◽  
Inference Mechanism ◽  
Mathematical Proofs ◽  
Computational Work ◽  
Definitional Extension

Abstract Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/. We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation. Hilbert 1918

Trusting Computers Through Trusting Humans

2008 ◽  
pp. 1152-1166
Author(s):  
Alison Adam ◽  
Paul Spedding
Keyword(s):  
Quality Assurance ◽  
Software Engineering ◽  
Computer System ◽  
Software Quality ◽  
Software Verification ◽  
Mathematical Proof ◽  
Quality Standards ◽  
Software Quality Assurance ◽  
The Social

This article considers the question of how we may trust automatically generated program code. The code walkthroughs and inspections of software engineering mimic the ways that mathematicians go about assuring themselves that a mathematical proof is true. Mathematicians have difficulty accepting a computer generated proof because they cannot go through the social processes of trusting its construction. Similarly, those involved in accepting a proof of a computer system or computer generated code cannot go through their traditional processes of trust. The process of software verification is bound up in software quality assurance procedures, which are themselves subject to commercial pressures. Quality standards, including military standards, have procedures for human trust designed into them. An action research case study of an avionics system within a military aircraft company illustrates these points, where the software quality assurance (SQA) procedures were incommensurable with the use of automatically generated code.

Trusting Computers Through Trusting Humans

2009 ◽  
pp. 2760-2774
Author(s):  
Alison Adam ◽  
Paul Spedding
Keyword(s):  
Quality Assurance ◽  
Software Engineering ◽  
Computer System ◽  
Software Quality ◽  
Software Verification ◽  
Mathematical Proof ◽  
Quality Standards ◽  
Software Quality Assurance ◽  
The Social

This article considers the question of how we may trust automatically generated program code. The code walkthroughs and inspections of software engineering mimic the ways that mathematicians go about assuring themselves that a mathematical proof is true. Mathematicians have difficulty accepting a computer generated proof because they cannot go through the social processes of trusting its construction. Similarly, those involved in accepting a proof of a computer system or computer generated code cannot go through their traditional processes of trust. The process of software verification is bound up in software quality assurance procedures, which are themselves subject to commercial pressures. Quality standards, including military standards, have procedures for human trust designed into them. An action research case study of an avionics system within a military aircraft company illustrates these points, where the software quality assurance (SQA) procedures were incommensurable with the use of automatically generated code.

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