On the Unavoidable Uncertainty of Truth in Dynamic Geometry Proving

Botana, Francisco; Recio, Tomás

doi:10.1007/s11786-016-0246-4

Cited by 5 publications

(9 citation statements)

References 6 publications

(4 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, it is possible to cut the right angle in three using only straightedge and compass, but apart from a null set, it is not possible to do it for any angle. We face here a well known problem in rigidity theory or in mechanical geometry theorem proving where an assertion can be either generally true (that is true except for degenerate cases) or generally false, or even, true on parts/false on parts [Botana and Recio(2016)]. This also should be taken into consideration especially when algebra is involved.…”

Section: Definition and Discussionmentioning

confidence: 99%

On the Mechanization of Straightedge and Compass Constructions

Schreck

2019

J Syst Sci Complex

View full text Add to dashboard Cite

The geometric constructions obtained with only straightedge and compass are famous and play a special role in the development of geometry. On the one hand, the constructibility of figures is a key ingredient in Euclid geometry and, on the other hand, unconstructibility gave birth to famous open problems of the ancient Greece which were unlocked only in the nineteenth century using discoveries in algebra. This paper discusses the mechanization of straightedge and compass constructions. It focuses on the algebraic approaches and presents two methods which are implemented; one is due to Lebesgue and the other one was jointly designed by Gao and Chou. Some links between the algebraic approach of constructions and synthetic geometry are described.

show abstract

Section: Definition and Discussionmentioning

confidence: 99%

On the Mechanization of Straightedge and Compass Constructions

Schreck

2019

J Syst Sci Complex

View full text Add to dashboard Cite

show abstract

“…Moreover, as shown by Botana and Recio, even for simple theorems, the interpretation can be non-trivial [BR16]. Our formalization, by providing a formal link between the synthetic axioms and the algebraic equations, paves the way for storing standardized, structured, and rigorous geometric knowledge data based on an explicit axiom system [CW13].…”

Section: Introductionmentioning

confidence: 93%

Formalization of the arithmetization of Euclidean plane geometry and applications

Boutry

Braun

Narboux

2019

Journal of Symbolic Computation

View full text Add to dashboard Cite

This paper describes the formalization of the arithmetization of Euclidean plane geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of the book by Schwabhäuser Szmielew and Tarski Metamathematische Methoden in der Geometrie. We defined the arithmetic operations geometrically and proved that they verify the properties of an ordered field. Then, we introduced Cartesian coordinates, and provided characterizations of the main geometric predicates. In order to prove the characterization of the segment congruence relation, we provided a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems. To obtain the characterizations of the geometric predicates, we adopted an original approach based on bootstrapping: we used an algebraic prover to obtain new characterizations of the predicates based on already proven ones. The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry. Indeed, without a "back-translation" from algebra to geometry, algebraic methods only prove theorems about polynomials and not geometric statements. However, thanks to the arithmetization of geometry, the proven statements correspond to theorems of any model of Tarski's Euclidean geometry axioms. To illustrate the concrete use of this formalization, we derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Gröbner basis method. Moreover, we solve a challenge proposed by Beeson: we prove that, given two points, an equilateral triangle based on these two points can be constructed in Euclidean Hilbert planes. Finally, we derive the axioms for another automated deduction method: the area method.

show abstract

“…Another typical example of the difficulties involved in handling this concept is Example 7 in [5], concerning Euler's formula regarding the radii of the inner and outer circles of a triangle with vertices (−1, 0), (1, 0), (u [1], u [2]). Here the dimension of the hypotheses variety is expected to be 2 (referring to the two coordinates of the only free vertex of the triangle), but applying the algebraic definition it turns out to be three.…”

Section: Fig 2 a Simple Examplementioning

confidence: 99%

“…Luckily, such warning sign-even without actually involving algorithmic factorization, but human reflection, instead-leads to the discovery of new (surely for the user, but, sometimes, for the scientific community as well) theorems. For instance, it has already allowed one of the authors of this paper to work out some contributions, such as a converse to Varignon Theorem [2], or the generalization of the Steiner-Lehmus Theorem (c.f. Example 9 in [5], fully described in [9]).…”

Section: Motivation For a New Frameworkmentioning

confidence: 99%

“…Some explicit examples of this situation have been provided by one of the authors of this paper as early as 1998 in [10]; further examples appeared in [11] or in sections 2.3 and 2.4. of [12], reproducing in English the example, in Spanish, from [10]). A recent paper, specifically devoted to discuss and to present such cases is [2]. But it was not, until even more recently, at [14], that a new terminology to describe such cases has been introduced, labelling as generally true on components or, simply, as true on components, those statements that are true in some, and false in some other, non-degenerate components: i.e.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Detecting truth, just on parts

2018

View full text Add to dashboard Cite

We introduce and discuss, through a computational algebraic geometry approach, the automatic reasoning handling of propositions that are simultaneously true and false over some relevant collections of instances. A rigorous, algorithmic criterion is presented for detecting such cases, and its performance is exemplified through the implementation of this test on the dynamic geometry program GeoGebra.Keywords automatic deduction in geometry, automatic geometry theorem proving · automatic geometry theorem discovery · elementary geometry · Gröbner basis · zero divisor · true on parts, false on parts · true on components · dynamic geometry Partially supported by the Spanish Research

show abstract

On the Unavoidable Uncertainty of Truth in Dynamic Geometry Proving

Cited by 5 publications

References 6 publications

On the Mechanization of Straightedge and Compass Constructions

On the Mechanization of Straightedge and Compass Constructions

Formalization of the arithmetization of Euclidean plane geometry and applications

Detecting truth, just on parts

Contact Info

Product

Resources

About