Proof Technology in Mathematics Research and Teaching

Abstract

The bedrock of mathematical activity is undoubtedly the concept of proof. Considering the abstract and conceptual nature of mathematical proofs, the act of proving is sometimes "thought to be the aspect of mathematical activity most resistant to the influence of technological change". After a careful reading of "Proof Technology in Mathematics Research and Teaching", however, one is convinced that the reality is quite different. In recent decades, development of tools such as automatic theorem provers has changed the work of mathematicians to the point of them questioning the very notion of mathematical demonstration. However, there is relatively little of this technological upheaval reflected in mathematics education.This book is intended for specialists in mathematics education with an interest in computer science advances as well as for researchers in the field of automated reasoning with an interest in the pedagogical and didactic implications of their work. It provides a valuable contribution to mathematics education by initiating a process of in-depth reflection on the educational value of new technological tools such as automatic theorem provers and dynamic geometric environments. Considering the wide range of subjects covered, few people will read this book from cover to cover with an unfaltering enthusiasm. There is no doubt, though, that there is something here to satisfy everyone's mathematical appetite.The first chapter of the section titled "Automatic Theorem Provers" sheds light upon the reasons why the great advances made in automated reasoning have been slow to be reflected in the teaching of mathematics. The proofs produced by the automatic theorem provers are often deemed terse to the point of being unreadable. This explains why there are still very few mathematicians making widespread use of proof assistants. In order to mitigate the impact of the significant gap between the way human express their mathematical thinking and the write-ups produced by the machine, the authors committed themselves to identify a possible source of this communication incompatibility. For humans, the raison d'être of a mathematical proof is not only to confirm the veracity of a statement. When a mathematician produces or consults a proof, he seeks above all an explanation; he seeks to base his understanding upon more elementary statements. Hence, clarity and readability are cardinal virtues. This is why the

Towards a Geometry Automated Provers Competition

“…How to measure the readability of proofs is still a research problem [2,30,31], Chou proposed a way to measure how difficult a formal proof is (using the area method) [9], de Bruijn also proposed a coefficient, the de Bruijn factor, the quotient of the size of corresponding informal proof and the size of the formal proof, could also be used as a measure of readability [37]. This is close to a Turing test for proofs: if a human cannot distinguish the proof generated automatically from a human proof, than it is readable.…”

Four Geometry Problems to Introduce Automated Deduction in Secondary Schools

“…In the recent book edited by Gila Hanna, David Reid, and Michael de Villiers, Proof Technology in Mathematics Research and Teaching [9] many issues related to the use of automated deduction methods and tools are discussed. In its four parts: Strengths and limitations of automatic theorem provers; Theoretical perspectives on computer-assisted proving; Suggestions for the use of proof software in the classroom; Classroom experience with proof software, we have a comprehensive coverage of the area.…”

Open Geometry Prover Community Project

“…The exchange of information between the server and the client is done using the JSON format. 8 The implementation of new clients to other GATP it is easy and opens the use of the information contained in TGTP from any GATP. This server/client architecture is currently being used to establish a connection between the e-learning environment Web Geometry Laboratory and the TGTP repository [27,28].…”

“…Mathematical proof is undoubtedly the cornerstone of mathematics. All mathematics practitioner know its centrality and the difficulty in mastering it [8]. The emergence, in the last years, of computing and reasoning tools, in particular automated geometry theorem provers, has enriched our experience with mathematics immensely.…”