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Foundations of Geometry

Foundations of Geometry presents students with a challenge to explain and to construct geometric proofs with the axiomatic development of a consistent mathematical system and identify or describe consistent mathematical proofs. Students will be able to write and communicate proofs of geometric statements using direct or indirect arguments; and will be able to discuss the historical development of new geometries. Students will also be able to explain differences between Euclidean and Non-Euclidean geometries. An introduction to the Foundations of Geometry can be organized into the following topics:

Truth Table

In this topic we take a careful look at making mathematical statements and determining their validity. Our main tool will be the use of a truth table . But first we will consider what a mathematical statement is and how to write clear and precise mathematical statements. Then we learn how to do computations with statements so that we can determine tautologies, contradictions, and contingencies. We also construct several truth table examples including three tautologies, three contradictions, and three contingencies.

Axiomatic Method

The problem is to erect the entire structure of a geometry upon the simplest foundation possible; i.e. to choose a minimum number of undefined elements and relations, and a set of axioms (postulates) concerning them, with the property that all of neutral geometry can be logically deduced from these without further appeal to intuition. This deductive logical approach is called the axiomatic method .

Logic Problems

Before starting the axiomatic method in any area of mathematics certain basic logic principles and a well-defined starting point must be established. The following list of basic logic problems concerns constructing truth tables, working with quantifiers, and writing mathematical statements.

Geometry Terms

This topic details Hilbert's undefined geometry terms and preliminary definitions which can be used to provide the basis for traditional Euclidean geometry.

Geometry Postulates

Here is a list of geometry postulates starting with the basics of incidence, betweeness, congruence and continuity; just as Hilbert partitioned his postulates for geometry into five groups (connection, order, congruence, continuity, and parallels). Hilbert's postulates went a long way towards helping people understand that the axiomatic approach to mathematics was a viable and much needed mathematical enterprise in its own right. Many axiomatic systems have been created and even recreated for subjects that are widely understood in the mathematical community. Hilbert's postulates are close in spirit to Euclid's postulates because they do not associate real numbers with segments; i.e. there is no way to measure a line segment.

Geometry Theorems

Here is a list of geometry theorems which start the beginning of neutral geometry (Euclidean geometry without using a parallel postulate). To be sure, none of the following geometry theorems use the parallel postulate.

David Hilbert Books

Geometry and the Imagination (CHEL/87.H) (AMS Chelsea Publishing) Buy New: $39.00 New (4) Used (4) from $39.00This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer--after more than half (more)

Hilbert List Price: $29.95 Buy New: $18.88 You Save: $11.07 (37%) New (19) Used (15) from $15.00If the life of any 20th century mathematician can be said to be a history of mathematics in his time, it is that of David Hilbert. To the enchanted young mathematicians and physicists who flocked to study (more)

Principles of Mathematical Logic List Price: $29.00 Buy New: $27.54 You Save: $1.46 (5%) New (6) Used (4) from $24.90David Hilbert was particularly interested in the foundations of mathematics. Among many other things, he is famous for his attempt to axiomatize mathematics. This now classic text is his treatment of symbolic (more)

The Foundations of Geometry Buy New: $15.99 New (1) from $15.99The Foundations of Geometry (more)

David Hilbert's Lectures on the Foundations of Geometry, 1891-1902 (German Ed... List Price: $99.00 Buy New: $65.00 You Save: $34.00 (34%) New (18) Used (10) from $65.00This volume contains six sets of notes for lectures on the foundations of geometry held by Hilbert in the period 1891-1902. It also reprints the first edition of Hilbert?s celebrated Grundlagen der Geometrie (more)

David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917-193... List Price: $119.00 Buy New: $98.77 You Save: $20.23 (17%) New (1) from $98.77The bulk of this volume consists of six sets of notes for lectures Hilbert gave (often in collaboration with Bernays) on the foundations of mathematics between 1917 and the early 1930s. The notes detail (more)

The Theory of Algebraic Number Fields List Price: $115.00 Buy New: $95.21 You Save: $19.79 (17%) New (13) Used (7) from $93.22This book is a translation into English of Hilbert's "Theorie der algebraischen Zahlkrper" best known as the "Zahlbericht", first published in 1897, in which he provided an elegantly integrated overview (more)

Readings on Color, Vol. 2: The Science of Color List Price: $42.00 Buy New: $38.95 You Save: $3.05 (7%) New (7) Used (7) from $28.99"Color is an absolutely fascinating topic, one I happen to think is a beautiful and productive microcosm for cognitive science. These volumes will serve as useful resources for anyone interested in philosophy (more)

The Honors Class: Hilbert's Problems and Their Solvers List Price: $39.00 Buy Used: $1.22 You Save: $37.78 (97%) New (10) Used (19) from $1.22In a memorable address given at the International Congress of Mathematicians in Paris in 1900, David Hilbert, perhaps the most respected mathematician of his time, developed a blueprint for mathematical (more)

An Introduction to Hilbert Space and Quantum Logic (Problem Books in Mathemat... Buy New: $71.00 New (1) Used (8) from $60.00Historically, nonclassical physics developed in three stages. First came a collection of ad hoc assumptions and then a cookbook of equations known as "quantum mechanics". The equations and their philosophical (more)

Foundations of Geometry Books

Finite Mathematics (9th Edition) List Price: $138.67 Buy New: $74.95 You Save: $63.72 (46%) New (37) Used (38) from $74.95Key Message: Lial, Greenwell, and Ritchey continue their tradition of integrating relevant, realistic applications with current data sources to provide an application-oriented text for students majoring (more)

Foundations of Geometry List Price: $69.33 Buy New: $61.24 You Save: $8.09 (12%) New (16) Used (15) from $45.50For sophomore/junior-level courses in Geometry; especially appropriate for students that will go on to teach high-school mathematics. This text comfortably serves as a bridge between lower-level mathematics (more)

Fractal Geometry: Mathematical Foundations and Applications List Price: $70.00 Buy New: $54.74 You Save: $15.26 (22%) New (17) Used (9) from $54.74Since its original publication in 1990, Kenneth Falconer's Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. It introduces the general (more)

Conceptual Mathematics: A First Introduction to Categories List Price: $65.00 Buy New: $54.50 You Save: $10.50 (16%) New (15) Used (13) from $44.00The idea of a "category"--a sort of mathematical universe--has brought about a remarkable unification and simplification of mathematics. Written by two of the best-known names in categorical logic, Conceptual (more)

Kiselev's Geometry / Book II. Stereometry List Price: $35.00 Buy New: $25.00 You Save: $10.00 (29%) New (2) from $25.00The book is an English adaptation of a classical Russian grade school-level text in solid Euclidean geometry. It contains the chapters Lines and Planes, Polyhedra, Round Solids, which include the traditional (more)

Categories for the Working Mathematician (Graduate Texts in Mathematics) List Price: $69.95 Buy New: $54.54 You Save: $15.41 (22%) New (23) Used (12) from $51.00Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural (more)

Foundations of Differentiable Manifolds and Lie Groups (Graduate Texts in Mat... List Price: $69.95 Buy New: $48.98 You Save: $20.97 (30%) New (20) Used (12) from $47.91Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. It includes differentiable manifolds, tensors (more)

Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace List Price: $26.00 Buy Used: $2.07 You Save: $23.93 (92%) Used (14) from $2.07 Through Euclid's Window Leonard Mlodinow brilliantly and delightfully leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the latest notions of hyperspace. (more)

Space and Geometry: In the Light of Physiological, Psychological and Physical... List Price: $9.95 Buy New: $5.58 You Save: $4.37 (44%) New (10) Used (2) from $5.50These three essays by an eminent scientist explore the nature, origin, and development of our concepts of space from the points of view of the senses, history, and physics. They examine the subject from (more)

The 4th Dimension: Toward a Geometry of Higher Reality List Price: $17.95 Buy Used: $2.05 You Save: $15.90 (89%) Used (30) from $2.05The 4th Dimension: Toward a Geometry of Higher Reality (more)