Foundations of Geometry: A Masterpiece by Hilbert - Download it for Free and Discover the Beauty of Geometry
Foundations of Geometry download
If you are interested in learning more about the fundamentals of geometry, one of the best books you can read is Foundations of Geometry by David Hilbert. This book is a classic work that presents the axiomatic approach to geometry in a clear and rigorous way. It also contains many important results and insights that have shaped the development of modern mathematics and logic. In this article, we will tell you what Foundations of Geometry is about, why it is important and relevant, and how you can download it for free.
Foundations of Geometry download
The main concepts and results of Foundations of Geometry
Foundations of Geometry was first published in 1899 as a memorial address by Hilbert at the unveiling of the Gauss-Weber monument at Göttingen. It was based on a course of lectures that Hilbert gave on Euclidean geometry at the University of Göttingen during the winter semester of 1898-1899. The book consists of five parts:
The axioms for plane geometry
The basic propositions for plane geometry
The axioms for solid geometry
The basic propositions for solid geometry
The theory of parallels
In each part, Hilbert introduces a system of axioms that define the basic concepts and relations of geometry, such as points, lines, planes, angles, congruence, similarity, etc. He then proves various propositions using these axioms in a logical and deductive way. He also discusses the relations between different systems of axioms and shows how some axioms are independent or dependent on others.
Some of the main concepts and results that Hilbert introduces or proves in Foundations of Geometry are:
The concept of geometric displacement, which is a transformation that preserves congruence and orientation.
The concept of geometric addition, which is a way to define sums and differences of segments using congruence.
The concept of geometric multiplication, which is a way to define products and quotients of segments using similarity.
The concept of an algebra of segments, which is a system that satisfies the laws of arithmetic using geometric addition and multiplication.
The proof that Desargues's theorem (which states that if two triangles are perspective from a point, then they are perspective from a line) is equivalent to the assumption that plane geometry can be embedded in space geometry.
The proof that Pasch's axiom (which states that if a line intersects two sides of a triangle, then it also intersects the third side) is independent of the other axioms for plane geometry.
The proof that the axiom of continuity (which states that between any two points there exists a third point) is independent of the other axioms for plane and solid geometry.
The proof that the parallel postulate (which states that through a point not on a line there exists one and only one line parallel to the given line) is independent of the other axioms for plane geometry.
The proof that the parallel postulate is equivalent to several other statements, such as the sum of the angles of a triangle being equal to two right angles, or the existence of similar triangles.
The proof that there exist non-Euclidean geometries, such as hyperbolic and elliptic geometry, that satisfy all the axioms for plane geometry except the parallel postulate.
The historical and mathematical context of Foundations of Geometry
Foundations of Geometry was not written in a vacuum. It was influenced by the work and ideas of many other mathematicians, especially those who were associated with Göttingen. Hilbert himself was a student and colleague of some of the most prominent figures in mathematics at the time, such as Gauss, Weber, and Riemann. He also interacted with other mathematicians who were working on similar or related topics, such as Peano, Bolyai, Lobachevsky, Klein, Poincaré, Russell, and Frege.
One of the main motivations for Hilbert to write Foundations of Geometry was to clarify and resolve some of the issues and controversies that arose from the discovery and development of non-Euclidean geometries in the 19th century. These geometries challenged the classical view of geometry as based on Euclid's Elements, which was considered to be a model of certainty and truth. They also raised questions about the nature and validity of mathematical reasoning and knowledge.
Hilbert's approach to geometry was inspired by Gauss's idea of treating geometry as a system of relations between abstract elements that are defined by axioms. He also followed Weber's method of testing the independence and compatibility of axioms by constructing models or counterexamples. He also adopted Riemann's concept of manifold as a generalization of space that can have different properties depending on the choice of metric.
Hilbert's work on geometry had a profound impact on modern mathematics and logic. It stimulated further research on axiomatic systems, consistency proofs, metamathematics, model theory, set theory, algebraic geometry, differential geometry, topology, and more. It also influenced the development of formal languages, symbolic logic, proof theory, computability theory, and foundations of mathematics.
The pedagogical and practical value of Foundations of Geometry
Foundations of Geometry is not only a historical and mathematical masterpiece, but also a valuable resource for teaching and learning geometry. It offers several advantages and benefits for students and teachers alike:
It presents geometry in a clear and rigorous way that emphasizes logical thinking and deductive reasoning.
It exposes students to different systems of geometry that have different properties and applications.
It helps students understand the role and significance of axioms, definitions, propositions, proofs, models, counterexamples, etc. in mathematics.
It illustrates the beauty and elegance of Hilbert's proofs and methods.
It encourages students to explore and discover new results and connections in geometry.
It prepares students for more advanced topics in mathematics and logic.
In addition to its pedagogical value, Foundations of Geometry also has practical value for various fields and problems. Some examples are:
The concept of geometric displacement can be used to study rigid motions, symmetries, transformations, etc.
The concept of geometric addition can be used to measure distances, angles, areas, volumes, etc.
The concept of geometric multiplication can be used to scale figures, compare ratios, find proportions, etc.
The concept of an algebra of segments can be used to perform arithmetic operations with lengths or coordinates.
The concept of non-Euclidean geometries can be used to model curved spaces or surfaces.
The concept of manifold can be used to generalize space to higher dimensions or different structures.
Conclusion
In this article, we have given you an overview of Foundations of Geometry by David Hilbert. We have explained what it is about, why it is important and relevant, and how 71b2f0854b