Mathematical documents faithfully computerised: the grammatical and text & symbol aspects of the MathLang framework (original) (raw)
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2014
b Agri ibrahim Cecen Universitesi,Agri,Turkiye Oz Bu arastirmada, matematik egitimcilerinin, ogretmenlerin, ogretmen adaylarinin, universite ve ortaogretim ogrencilerinin ispata yonelik gorusleri, ispat yapabilme duzeyleri, ispat yapma surecinde yasadiklari zorluklari, ispat yapmada kullandiklari yontemleri, stratejileri, yaklasimlari, ispat cesitleri, ispatin matematik egitimindeki rolu ve ispatla ilgili surecler uzerine yapilmis olan arastirmalardan bazilarinin derlenmesine calisilmistir. Arastirma sonunda, incelenen calismalarin daha cok ogrencilerin ve ogretmen adaylarinin ispat yapma surecine odaklanildigi gorulmustur. Oysa ogretmenlerin ispata yonelik goruslerinin, algilarinin, ispat yapma surecinde kullandiklari yontemlerin, stratejilerin, yaklasimlarin ve ispat yapma becerilerinin bilinmesi onem tasimaktadir. Bu nedenle, ileride bu konuda calisma yapacak olan arastirmacilara, ogretmenlerin ispat yapma duzeyleri, ispat yapma sirasinda kullandiklari yontemler, stratejiler, yak...
Mathematical reasoning: writing and proof
Choice Reviews Online
There are no changes in content between version 1.0 of this book and version 1.1. The only changes are the addition of the Note to Students immediately following the Table of Contents, and the Creative Commons License has been changed to the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. License This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. The graphic that appears throughout the text shows that the work is licensed with the Creative Commons, that the work may be used for free by any party so long as attribution is given to the author(s), that the work and its derivatives are used in the spirit of "share and share alike," and that no party other than the author(s) may sell this work or any of its derivatives for profit.
The role of symbolic language in the transformation of mathematics
Philosophica, 2012
One important factor to be considered in the process of algebrization of mathematics is the emergence of symbolic language in the seventeenth century. Focussing on three works, In Artem analyticen Isagoge (1591) by François Viète (1540-1603); Cursus Mathematicus (1634-1637-1642) by Pierre Hérigone (1580-1643) and Geometriae Speciosae Elementa (1659) by Pietro Mengoli (1626/7-1686), in this article we analyse two relevant aspects of symbolic language: the significance of the notation in the symbolic language and the role of Hérigone's new symbolic method. This analysis allows us to better understand the role played by this circulation of ideas in the formative process of symbolic language in mathematics.. 1 Viète published several works for showing the usefulness of this analytic art. On Viète's works see: Viète (1970) and Giusti (1992). 2 Hérigone's algebra consists of 20 chapters and includes: 1: Several definitions and notations. 2, 3: Operations involving simple and compound algebraic expressions. 4: Operations involving ratios. 5: Proofs of several theorems. 6, 7: Rules for dealing with equations, which are the same as those in Viète's work [These rules were: the reduction of fractions to the same denominator ("isomerie"), the reduction of the coefficient of the highest degree ("parabolisme"), the depression of the degree ("hypobibasme") and the transposition of terms ("antithese")]. 8: An examination of theorems by "poristics". 9: Rules of the "rhetique" or exegetic in equations up to the second degree. 10-13: Solutions of several problems and geometric questions using proofs (determined by means of analysis). 14: Solutions of several "ambiguous" equations. 15: Solutions of problems concerning squares and cubes, referred to as Diophantus' problems. 16-19: Calculation of irrational numbers. 20: Several solutions of "affected" (negative sign) powers.
Gradual Computerisation/Formalisation of Mathematical Texts into Mizar
2007
We explain in this paper the gradual computerisation process of an ordinary mathematical text into more formal versions ending with a fully formalised Mizar text. The process is part of the MathLang-Mizar project and is divided into a number of steps (called aspects). The first three aspects (CGa, TSa and DRa) are the same for any MathLang-TP project where TP is any proof checker (e.g., Mizar, Coq, Isabelle, etc). These first three aspects are theoretically formalised and implemented and provide the mathematician and/or TP user with useful tools/automation. Using TSa, the mathematician edits his mathematical text just as he would use L A T E X, but at the same time he sees the mathematical text as it appears on his paper. TSa also gives the mathematician easy editing facilities to help assign to parts of the text, grammatical and mathematical roles and to relate different parts through a number of mathematical, rethorical and structural relations. MathLang would then automatically produce CGa and DRa versions of the text, checks its grammatical correctness and produce a dependency graph between the parts of the text. At this stage, work of the first three aspects is complete and the computerised versions of the text, as well as the dependency graph are ready to be processed further. In the MathLang-Mizar project, we create from the dependency graph, the roles of the nodes of the graph, and the (preamble) of the CGa encoding, a Mizar Formal Proof Sketch (FPS) skeleton. The stage at which the text is transformed into a Mizar FPS skeleton has only been explained through transformation hints, and is yet to be theoretically developed into an aspect that can be implemented and developed into a partially-automated tool. Finally, the Mizar FPS skeleton of the text is transformed (currently by hand as any Mizar expert would do and without any computerised tools) into a correct Mizar FPS and then into a fully formalised Mizar version of the text. Although we have tested our process on a number of examples, we chose to illustrate it in this paper using Barendregt's version of the proof of Pythagoras' theorem. We show how this example text is transformed into its fully formalised Mizar version by passing through the first three computerised aspects and the transformation hints to obtain Mizar FPS version. This version is then developed by hand into a fully formalised Mizar version.
1971. Discourse Grammars and the Structure of Mathematical Reasoning I: Mathematical Reasoning and Stratification of Language, Journal of Structural Learning 3, #1, 55–74. This is the first in a series of three articles which investigates some of the interrelations among three areas: first, linguistic work inspire by the ideas of Zellig Harris; second, logical investigations concerning the nature of mathematical reasoning; and third, mathematical education. The main concern is to relate basic linguistic concepts and hypotheses to the study of deductive reasoning and, then, to suggest applications to mathematical education. Mathematical education is taken to be the area of study which attempts to understand teaching and learning of mathematics and also to improve mathematical teaching in practice. The global plan of the series is as follows. Part I presents a certain view of mathematical reasoning together with a discussion of the interrelationship between the structure of mathematical discourse and the structure of normal English discourse. The second article outlines the nature of a theory of proof and includes a discussion of the utility of such a theory in mathematical education. The final article in the series develops several basic ideas involved in the construction of a usable theory of proof. NOTE ADDED December 2015. This paper, which was started almost 50 years ago, contains several themes that I developed extensively over the years in dozens of other papers. My terminology has been refined: e.g. today I would replace many occurrences of ‘proof’ by ‘deduction’. I was still being guided by the ideas and terminology of my teachers Martin Davis, Leon Henkin, and Raymond Smullyan who were guided by practices of their teacher Alonzo Church. It took me until 1986 to get over the deleterious parts of their influence. Anyway, despite the flaws I am warning you of, I am proud of what is achieved in these papers. I hope you enjoy them while finding fault with them.