June 2012
Review of Symbolic Logic;Jun2012, Vol. 5 Issue 2, p294
Academic Journal
Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.In this paper our major concern is with methodological issues of purity and thus we treat the connection to other areas of the planimetry/stereometry relation only to the extent necessary to articulate the problem area we are after.Our strategy will be as follows. In the first part of the paper we will give a rough sketch of some key episodes in mathematical practice that relate to the interaction between plane and solid geometry. The sketch is given in broad strokes and only with the intent of acquainting the reader with some of the mathematical context against which the problem emerges. In the second part, we will look at a debate (on “fusionism”) in which for the first time methodological and foundational issues related to aspects of the mathematical practice covered in the first part of the paper came to the fore. We conclude this part of the paper by remarking that only through a foundational and philosophical effort could the issues raised by the debate on “fusionism” be made precise. The third part of the paper focuses on a specific case study which has been the subject of such an effort, namely the foundational analysis of the plane version of Desargues’ theorem on homological triangles and its implications for the relationship between plane and solid geometry. Finally, building on the foundational case study analyzed in the third section, we begin in the fourth section the analytic work necessary for exploring various important claims about “purity,” “content,” and other relevant notions.


Related Articles

  • Wang Xiaotong on Right Triangles: Six Problems from 'Continuation of Ancient Mathematics' (Seventh Century AD). Tina Su-lyn Lim; Wagner, Donald B. // East Asian Science, Technology & Medicine;2014, Issue 37, p12 

    Wang Xiaotong's Jigu suanjing is primarily concerned with problems in solid and plane geometry leading to polynomial equations which are to be solved numerically using a procedure similar to Horner's Method. We translate and analyze here six problems in plane geometry. In each case the solution...

  • seg·ment.  // American Heritage Student Science Dictionary;2009, p306 

    Several definitions of the term "segment" are presented. Segment is the portion of a line between any two of its points. It can also refer to the region bounded by an arc of a circle and the chord that connects the endpoints of the arc. Segment is also the portion of a sphere included between a...

  • GEOMETRY AND TRIGONOMETRY.  // Handy Math Answer Book;2006, p165 

    The article provides information on the historical background of geometry and trigonometry. Geometry is another branch of mathematics which deals with figures and objects and it was derived from two Greek terms "gë" means earth and "metreein" means to measure. It cites its divisions such as...

  • The non-platonic and non-Archimedean noncomposite polyhedra. Timofeenko, A. // Journal of Mathematical Sciences;Nov2009, Vol. 162 Issue 5, p710 

    If a convex polyhedron with regular faces cannot be divided by any plane into two polyhedra with regular faces, then it is said to be noncomposite. We indicate the exact coordinates of the vertices of noncomposite polyhedra that are neither regular (Platonic), nor semiregular (Archimedean), nor...

  • The Gergonne and Nagel centers of an n-dimensional simplex. Hajja, Mowaffaq // Journal of Geometry;2005, Vol. 83 Issue 1/2, p46 

    In contrast with the analogous situation for a triangle, the cevians that join the vertices of a tetrahedron to the points where the faces touch the insphere (or the exspheres) are not concurrent in general. This observation led the present author and P. Walker in [4] to devise alternative...

  • TOPOLOGY OF A NEW LATTICE CONTAINING PENTAGON TRIPLES. Pop, Monica L.; Diudea, Mircea V. // Studia Universitatis Babes-Bolyai, Chemia;2010 Special Issue, p199 

    A new crystal-like network is designed by using some net operations. The topology of this hypothetical lattice is characterized by Omega polynomial and Cluj-Ilmenau CI index.

  • PLANE AND SOLID GEOMETRY.  // Education;May1923, Vol. 43 Issue 9, p587 

    The article reviews the book "Plane and Solid Geometry," by Walter B. Ford and Charles Ammerman.

  • Automatically Proving Plane Geometry Theorems Stated by Text and Diagram. Gan, Wenbin; Yu, Xinguo; Zhang, Ting; Wang, Mingshu // International Journal of Pattern Recognition & Artificial Intell;Jun2019, Vol. 33 Issue 7, pN.PAG 

    This paper presents an algorithm for proving plane geometry theorems stated by text and diagram in a complementary way. The problem of proving plane geometry theorems involves two challenging subtasks, being theorem understanding and theorem proving. This paper proposes to consider theorem...

  • FAMILIES OF PLANAR SETS HAVING STARLIKE UNION. Breen, Marilyn // Journal of Geometry;Jul1999, Vol. 65 Issue 1/2, p50 

    Analyzes the finite geometry of compact sets in a plane. Notions of visible and starshaped form segments on fixed natural number; Details on the theorem by Krasnosel'skii; Relationship between starshaped unions and nonempty intersections of compact convex sets.


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics