Hypothesis In Geometry

Unlocking the Secrets of Geometry: A Deep Dive into Hypotheses

Geometry, the study of shapes, sizes, and spaces, is built upon a foundation of logical reasoning. At the heart of this reasoning lies the hypothesis in geometry, a statement proposed as a possible explanation for a phenomenon or a starting point for deductive reasoning. This comprehensive guide will demystify the role of hypotheses in geometry, exploring their definition, types, applications, and importance in proving geometric theorems. We'll move beyond basic definitions, examining real-world applications and tackling common misconceptions. Prepare to gain a profound understanding of how hypotheses drive geometric discovery and proof.

What is a Hypothesis in Geometry?

A hypothesis in geometry, simply put, is an educated guess or a tentative statement that you assume to be true before proving it. Unlike a theorem, which is a proven statement, a hypothesis requires rigorous testing and proof through logical deduction. It serves as a premise or assumption from which we derive conclusions using established axioms, postulates, and previously proven theorems. Think of it as the "if" part of an "if-then" statement (conditional statement). For example, "If two lines are parallel, then their corresponding angles are congruent" – "two lines are parallel" is the hypothesis.

Types of Hypotheses in Geometry

While the core concept remains consistent, hypotheses in geometry can manifest in several forms:

Conditional Hypotheses: These are the most common type, expressed in the "if-then" format (conditional statement). They state a condition ("if" part) and a consequence ("then" part). Proving the "then" part (conclusion) relies on demonstrating the truth of the "if" part (hypothesis).

Existential Hypotheses: These hypotheses assert the existence of a specific geometric figure or property. For instance, "There exists a triangle with angles measuring 60°, 60°, and 60°." Proving this involves constructing such a triangle or demonstrating its existence through logical argumentation.

Universal Hypotheses: These claim a property holds true for all members of a set. An example: "All equilateral triangles are equiangular." Demonstrating the truth of a universal hypothesis requires proving the property applies to every possible instance within the defined set.

The Role of Hypotheses in Geometric Proofs

Hypotheses are the cornerstones of geometric proofs. They provide the initial assumptions or conditions upon which the entire proof structure rests. The process typically involves:

1. Stating the Hypothesis: Clearly articulating the initial assumption or condition.
2. Using Axioms and Postulates: Applying fundamental geometric truths (axioms and postulates) to build a logical chain of reasoning.
3. Employing Previously Proven Theorems: Utilizing established theorems to support the deductive process.
4. Reaching a Conclusion: Demonstrating the truth of the statement initially proposed as a hypothesis, thereby transforming it into a proven theorem.

Failure to accurately state or correctly utilize the hypothesis often leads to flawed proofs and incorrect conclusions.

Real-World Applications of Hypotheses in Geometry

Geometry isn't just about abstract shapes and proofs; it has significant practical applications. Hypotheses play a crucial role in these applications:

Engineering and Architecture: Engineers and architects rely heavily on geometric principles to design stable and efficient structures. Hypotheses guide the design process, allowing engineers to test the feasibility of various designs before construction begins. For example, a hypothesis might be: "If the bridge's support structures are positioned at this angle, then it will withstand a certain load."

Computer Graphics and Game Development: Creating realistic and dynamic computer graphics involves complex geometric calculations. Hypotheses are used to model and simulate the behavior of objects and light, ensuring accurate rendering and realistic animations.

Cartography and Surveying: Mapping the Earth's surface requires precise geometric measurements and calculations. Hypotheses are crucial in ensuring the accuracy of maps and surveys, particularly when dealing with large-scale projects.

Misconceptions about Hypotheses in Geometry

Several misconceptions frequently arise concerning hypotheses in geometry:

Confusing Hypothesis with Conclusion: A common mistake is treating the hypothesis as the conclusion. Remember, the hypothesis is the starting assumption; the conclusion is what is proven based on that assumption.

Ignoring Counter-Examples: A hypothesis can be disproven by finding a single counter-example. If even one instance contradicts the hypothesis, the entire statement is false.

Assuming the Hypothesis is Always True: A hypothesis is proposed as true but needs to be proven true before being accepted as a theorem.

Example of a Geometric Hypothesis and its Proof

Let's illustrate the process with a simple example:

Hypothesis: If two angles are vertical angles, then they are congruent.

Proof:

1. Diagram: Draw two intersecting lines forming four angles. Label them A, B, C, and D. Angles A and C are vertical angles, as are B and D.
2. Known Facts: Angles A and B are supplementary (they add up to 180°), as are angles B and C, C and D, and D and A.
3. Deduction: Since A + B = 180° and B + C = 180°, then A + B = B + C. Subtracting B from both sides yields A = C.
4. Conclusion: We have proven that vertical angles A and C are congruent. This same logic applies to angles B and D.

Therefore, the hypothesis "If two angles are vertical angles, then they are congruent" is proven true.

Book Outline: "Mastering Geometric Hypotheses"

I. Introduction: Defining hypotheses in geometry, their importance, and the structure of the book.

II. Types of Hypotheses: A detailed exploration of conditional, existential, and universal hypotheses, with examples.

III. Geometric Proofs and Hypotheses: Step-by-step guides on constructing geometric proofs using hypotheses, with numerous worked examples.

IV. Applications of Hypotheses: Real-world applications of hypotheses in various fields (engineering, computer graphics, etc.).

V. Common Mistakes and Misconceptions: Addressing common errors in understanding and utilizing hypotheses in geometry.

VI. Advanced Topics: Exploring more complex types of hypotheses and their use in advanced geometric proofs (optional).

VII. Conclusion: Summarizing key concepts and encouraging further exploration of geometric reasoning.

(Detailed Article Explaining Each Point of the Outline - This section would expand on the outline above, adding substantial detail and illustrative examples to each chapter. Due to word count limitations, this detailed expansion is omitted here. Each section would require several hundred words to comprehensively cover the topic.)

FAQs on Hypotheses in Geometry

1. What is the difference between a hypothesis and a theorem? A hypothesis is a proposed statement, while a theorem is a proven statement.

2. Can a hypothesis be false? Yes, a hypothesis can be false. If a counter-example is found, the hypothesis is disproven.

3. How do you prove a hypothesis in geometry? Through logical deduction, using axioms, postulates, and previously proven theorems.

4. What are the different types of geometric proofs? Common methods include direct proof, indirect proof (proof by contradiction), and proof by exhaustion.

5. What role do diagrams play in geometric proofs? Diagrams help visualize the problem and aid in understanding the relationships between geometric figures.

6. Can a single counter-example disprove a universal hypothesis? Yes, one counter-example is sufficient to disprove a universal hypothesis.

7. How are hypotheses used in real-world applications? They form the basis of design, modeling, and problem-solving in various fields.

8. What are some common mistakes to avoid when working with hypotheses? Confusing hypothesis with conclusion, ignoring counter-examples, and assuming the hypothesis is always true.

9. Where can I find more resources to learn about geometric proofs and hypotheses? Textbooks, online courses, and educational websites offer extensive resources.

1. Euclidean Geometry Axioms and Postulates: A deep dive into the foundational principles of Euclidean geometry.
2. Understanding Geometric Theorems: Exploring key theorems and their proofs.
3. Types of Geometric Proofs: A detailed explanation of different proof methods.
4. Non-Euclidean Geometries: An introduction to geometries that deviate from Euclidean principles.
5. Geometric Constructions with Compass and Straightedge: Learning how to construct geometric shapes using basic tools.
6. Trigonometry and its Applications in Geometry: Exploring the connection between trigonometry and geometric problem-solving.
7. Solid Geometry and its Applications: An introduction to three-dimensional geometric figures and their properties.
8. Analytic Geometry and Coordinate Systems: Applying algebraic methods to solve geometric problems.
9. Solving Geometric Word Problems: Practical strategies for tackling word problems involving geometric concepts.

hypothesis in geometry: On the Hypotheses Which Lie at the Bases of Geometry Bernhard Riemann, 2016-04-19 This book presents William Clifford’s English translation of Bernhard Riemann’s classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann’s reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann’s work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann’s times to contemporary research. Not only mathematicians and historians of the mathematical sciences, but also readers from other disciplines or those with an interest in physics or philosophy will find this work both appealing and insightful.
hypothesis in geometry: Science and Hypothesis Henri Poincaré, 2017-11-16 Science and Hypothesis is a classic text in history and philosophy of science. Widely popular since its original publication in 1902, this first new translation of the work in over a century features unpublished material missing from earlier editions. Addressing errors introduced by Greenstreet and Halsted in their early 20th-century translations, it incorporates all the changes, corrections and additions Poincaré made over the years. Taking care to update the writing for a modern audience, Poincaré's ideas and arguments on the role of hypotheses in mathematics and in science become clearer and closer to his original meaning, while David J. Stump's introduction gives fresh insights into Poincaré's philosophy of science. By approaching Science and Hypothesis from a contemporary perspective, it presents a better understanding of Poincare's hierarchy of the sciences, with arithmetic as the foundation, geometry as the science of space, then mechanics and the rest of physics. For philosophers of science and scientists working on problems of space, time and relativity, this is a much needed translation of a ground-breaking work which demonstrates why Poincaré is still relevant today.
hypothesis in geometry: Science and Hypothesis Henri Poincare, 2023-10-01 Science and Hypothesis by Henri Poincaré is a seminal work in the philosophy of science that explores the nature of scientific reasoning, the role of hypotheses, and the limits of knowledge. Poincaré challenges conventional views on scientific methodology, offering insightful reflections on the process of discovery and the relationship between theory and observation. Key Points: Poincaré critically examines the foundations of scientific inquiry, questioning the notion of absolute certainty and highlighting the inherent uncertainty and provisional nature of scientific knowledge, emphasizing the role of intuition, creativity, and imagination in scientific breakthroughs. The book delves into the concept of hypothesis, exploring its construction, testing, and revision within the scientific framework, while also addressing the significance of simplicity, coherence, and empirical evidence in the formulation of scientific theories. Science and Hypothesis is a thought-provoking work that challenges readers to reconsider their assumptions about the nature of scientific inquiry, offering a nuanced understanding of the interplay between theory and observation, and the ongoing quest for knowledge and understanding in the scientific endeavor.
hypothesis in geometry: Non-Euclidean Geometry Henry Parker Manning, 2021-01-19 Non-Euclidean Geometry by Henry Parker Manning is a comprehensive exploration of geometrical systems that deviate from Euclidean geometry, challenging traditional notions of space, distance, and parallel lines. Manning introduces readers to the fascinating world of non-Euclidean geometries, providing insights into their development, principles, and applications. Key Points: Manning introduces readers to the groundbreaking works of mathematicians like Nikolai Lobachevsky, János Bolyai, and Carl Friedrich Gauss, who pioneered the development of non-Euclidean geometries, revolutionizing our understanding of geometric principles and expanding the boundaries of mathematical thought. The book delves into the different types of non-Euclidean geometries, such as hyperbolic and elliptic geometries, presenting their distinctive properties, axioms, and geometric constructions. Manning explores the implications of these alternative geometries on concepts such as angles, triangles, and the nature of space itself. Non-Euclidean Geometry offers readers a captivating journey into the realm of abstract mathematics, challenging preconceived notions of geometric truth and illuminating the beauty and diversity of mathematical systems. It is a valuable resource for mathematicians, students, and anyone fascinated by the profound exploration of the nature of space and geometry.
hypothesis in geometry: Handbook of the History and Philosophy of Mathematical Practice Bharath Sriraman,
hypothesis in geometry: The Geometry Code Bruce Rawles, 2012-04 Integrate practical insights from modern physics, ancient Hermetic Laws, non-dual meta-physics, transpersonal psychology, and humor, as tools for undoing conflicting beliefs we've dreamed ourselves into. The seven Hermetic laws are explored in depth and demonstrate how a mindfulness that embraces 'other' as 'self' can reverse the typical misapplication of these inescapable laws of Mentalism, Correspondence, Vibration, Polarity, Rhythm, Cause & Effect and Generation. Ubiquitous geometric symbols, paired to each of these laws - the circle, vesica piscis, sine wave, line, spiral, fractal and yin-yang - and their countless commonplace variations, seen from the vantage point of shared interests, reflect these ideas. The inspired use of natural law restores attributes of life, love, strength, purity, beauty, perfection and gratitude to our awareness.
hypothesis in geometry: Mechanical Theorem Proving in Geometries Wen-tsün Wu, 2012-12-06 There seems to be no doubt that geometry originates from such practical activ ities as weather observation and terrain survey. But there are different manners, methods, and ways to raise the various experiences to the level of theory so that they finally constitute a science. F. Engels said, The objective of mathematics is the study of space forms and quantitative relations of the real world. Dur ing the time of the ancient Greeks, there were two different methods dealing with geometry: one, represented by the Euclid's Elements, purely pursued the logical relations among geometric entities, excluding completely the quantita tive relations, as to establish the axiom system of geometry. This method has become a model of deduction methods in mathematics. The other, represented by the relevant work of Archimedes, focused on the study of quantitative re lations of geometric objects as well as their measures such as the ratio of the circumference of a circle to its diameter and the area of a spherical surface and of a parabolic sector. Though these approaches vary in style, have their own features, and reflect different viewpoints in the development of geometry, both have made great contributions to the development of mathematics. The development of geometry in China was all along concerned with quanti tative relations.
hypothesis in geometry: Rational Reconstructions of Modern Physics Peter Mittelstaedt, 2010-11-05 Newton’s classical physics and its underlying ontology are loaded with several metaphysical hypotheses that cannot be justified by rational reasoning nor by experimental evidence. Furthermore, it is well known that some of these hypotheses are not contained in the great theories of modern physics, such as the theory of relativity and quantum mechanics. This book shows that, on the basis of Newton’s classical physics and by rational reconstruction, the theory of relativity as well as quantum mechanics can be obtained by partly eliminating or attenuating the metaphysical hypotheses. Moreover, it is shown that these reconstructions do not require additional hypotheses or new experimental results.
hypothesis in geometry: The Philosophy of Michael Dummett B.F. McGuinness, G. Oliveri, 2013-04-17 The present collection of papers derives from a philosophy conference organised in the Sicilian town of M ussomeli in September 1991. The con ference aimed at providing an analysis of certain aspects of the thought of Michael Dummett, whose contributions have been very influential in several aspects of the philosophical debate continuing within the analyt ical tradition. Logic, the philosophy of mathematics, the interpretation of Frege's philosophy, and metaphysics are only some of the areas within which Dummett's ideas have been fruitful over the years. The papers contained in this book, and Dummett's replies, will, it is hoped, not merely offer a partial reconstruction of a philosopher's life work, but provide an exciting and challenging vantage point from which to look at some of the main problems of contemporary philosophy. The First International Philosophy Conference of M ussomeli - this is what the conference was called - was an extraordinary event in many ways. The quality of the papers presented, the international reputa tion of many of the participants, the venue itself, together with the unavoidable, and sometimes quite funny, organisational hiccups, made that meeting memorable. Perhaps principally memorable was the warmth and sympathy of the people of Mussomeli who strongly supported and encouraged this initia tive. A special mention is also due to the City Council Administrators, who spared no effort to make the Conference a success.
hypothesis in geometry: Encyclopedia of the Enlightenment Michel Delon, 2013-12-04 This acclaimed translation of Michel Delon's Dictionnaire Europen des Lumires contains more than 350 signed entries covering the art, economics, science, history, philosophy, and religion of the Enlightenment. Delon's team of more than 200 experts from around the world offers a unique perspective on the period, providing offering not only factual information but also critical opinions that give the reader a deeper level of understanding. An international team of translators, editors, and advisers, under the auspices of the French Ministry of Culture, has brought this collection of scholarship to the English-speaking world for the first time.
hypothesis in geometry: Foundations and Fundamental Concepts of Mathematics Howard Eves, 2012-04-10 Third edition of popular undergraduate-level text offers historic overview, readable treatment of mathematics before Euclid, Euclid's Elements, non-Euclidean geometry, algebraic structure, formal axiomatics, sets, more. Problems, some with solutions. Bibliography.
hypothesis in geometry: Encyclopedia of Mathematics Education Louise Grinstein, Sally I. Lipsey, 2001-03-15 This single-volume reference is designed for readers and researchers investigating national and international aspects of mathematics education at the elementary, secondary, and post-secondary levels. It contains more than 400 entries, arranged alphabetically by headings of greatest pertinence to mathematics education. The scope is comprehensive, encompassing all major areas of mathematics education, including assessment, content and instructional procedures, curriculum, enrichment, international comparisons, and psychology of learning and instruction.
hypothesis in geometry: Best Practices on Advanced Condition Monitoring of Rail Infrastructure Systems Stefano Bruni, Serdar Dindar, Sakdirat Kaewunruen, 2022-03-14 This Research Topic eBook comprises Volume I and Volume II of Best Practices on Advanced Condition Monitoring of Rail Infrastructure Systems.
hypothesis in geometry: The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method Henri Poincaré, 2022-08-01 DigiCat Publishing presents to you this special edition of The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method by Henri Poincaré. DigiCat Publishing considers every written word to be a legacy of humankind. Every DigiCat book has been carefully reproduced for republishing in a new modern format. The books are available in print, as well as ebooks. DigiCat hopes you will treat this work with the acknowledgment and passion it deserves as a classic of world literature.
hypothesis in geometry: The Expanding Universe Georges Lemaître, 1985
hypothesis in geometry: Library of Congress Subject Headings Library of Congress, 2011
hypothesis in geometry: Library of Congress Subject Headings Library of Congress. Cataloging Policy and Support Office, 2009
hypothesis in geometry: Companion to the History of Modern Science G N Cantor, G.N. Cantor, J.R.R. Christie, M.J.S. Hodge, R.C. Olby, 2006-09-07 * A descriptive and analytical guide to the development of Western science from AD 1500, and to the diversity and course of that development first in Europe and later across the world * Presented in clear, non-technical language * Extensive indexes of Subjects and Names `Indeed a companion volume whose 67 essays give pleasure and instruction ... an ambitious and successful work.' - Times Literary Supplement `This work is an essential resource for libraries everywhere. For specialist science libraries willing to keep just one encyclopaedic guide to history, for undergraduate libraries seeking to provide easily accessible information, for the devisers of university curricula, for the modern social historian or even the eclectic scientist taking a break from simply making history, this is the book for you.' - Times Higher Education Supplement `A pleasure to read with a carefully chosen typeface, well organized pages and ample margins ... it is very easy to find one's way around. This is a book which will be consulted widely.' - Technovation `This is a commendably easy book to use.' - British Journal of the History of Science `Scholars from other areas entering this field, students taking the vertical approach and teachers coming from any direction cannot fail to find this an invaluable text.' - History of Science Journal
hypothesis in geometry: Mathematical Thought From Ancient to Modern Times, Volume 3 Morris Kline, 1990-03-01 This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the disciplines origins in Babylon and Egypt, the creation of geometry and trigonometry by the Greeks, and the role of mathematics in the medieval and early modern periods. Volume 2 focuses on calculus, the rise of analysis in the 19th century, and the number theories of Dedekind and Dirichlet. The concluding volume covers the revival of projective geometry, the emergence of abstract algebra, the beginnings of topology, and the influence of Godel on recent mathematical study.
hypothesis in geometry: The Logic of Scientific Discovery Karl Raimund Popper, 2002 When first published in 1959, this book revolutionized contemporary thinking about science and knowledge. It remains one of the most widely read books about science to come out of the 20th century.
hypothesis in geometry: Nature Sir Norman Lockyer, 1894
hypothesis in geometry: The Significance of the Hypothetical in the Natural Sciences Michael Heidelberger, Gregor Schiemann, 2009 Naturwissenschaftler und Philosophen haben im Lauf der Wissenschaftsgeschichte unterschiedliche Auffassungen vom Hypothesencharakter empirischer Theorien entwickelt. Der Band widmet sich drei verschiedenen Epochen, in denen der Erkenntnisoptimismus erfolgreicher Wissenschaftspraxis auf ein wachsendes Bewusstsein der Grenzen naturwissenschaftlicher Einsicht trifft: der Frühen Neuzeit (Kopernikus, Kepler, Bacon, Galilei, Descartes, Boyle, Newton, Locke, mit einem Rückblick auf die mittelalterlichen Autoren Maimonides und Gersonides), dem mechanistischen Weltbild des 19. Jahrhunderts (Herschel, Whewell, Mill, C. G. J. Jacobi, Carl Neumann, Boutroux, Ch. S. Peirce, mit einem Rückblick auf Lagrange und d'Alembert) und dem 20. Jahrhundert mit dem Aufkommen der modernen Physik (Hertz, Poincaré, Vaihinger, Duhem, Heisenberg, Popper). Abgerundet wird der Band durch Studien zur Gegenwartsdiskussion des wissenschaftlichen Realismus und den Chancen einer hypothetischen Metaphysik der Natur.
hypothesis in geometry: Research Anthology on Innovative Research Methodologies and Utilization Across Multiple Disciplines Management Association, Information Resources, 2021-12-30 Research methodology is as old as academia itself. Research methodology shifts in strategy as it crosses different disciplines and theories. This, too, is true with the shifting landscape of research opportunities and technologies available to global researchers. To achieve the most accurate and substantial research, it is important to be knowledgeable of emerging research methodologies. The Research Anthology on Innovative Research Methodologies and Utilization Across Multiple Disciplines discusses the most recent global research innovations made across multiple fields. This anthology further discusses how these research methodologies can be applied to a variety of specific fields. Covering topics such as creative thinking, qualitative research, and the research method landscape, this book is essential for students and faculty of higher education, scientists, researchers, sociologists, computer scientists, and academicians.
hypothesis in geometry: Euclid Vindicated from Every Blemish Gerolamo Saccheri, 2014-08-07 This first complete English language edition of Euclides vindicatus presents a corrected and revised edition of the classical English translation of Saccheri's text by G.B. Halsted. It is complemented with a historical introduction on the geometrical environment of the time and a detailed commentary that helps to understand the aims and subtleties of the work. Euclides vindicatus, written by the Jesuit mathematician Gerolamo Saccheri, was published in Milan in 1733. In it, Saccheri attempted to reform elementary geometry in two important directions: a demonstration of the famous Parallel Postulate and the theory of proportions. Both topics were of pivotal importance in the mathematics of the time. In particular, the Parallel Postulate had escaped demonstration since the first attempts at it in the Classical Age, and several books on the topic were published in the Early Modern Age. At the same time, the theory of proportion was the most important mathematical tool of the Galilean School in its pursuit of the mathematization of nature. Saccheri's attempt to prove the Parallel Postulate is today considered the most important breakthrough in geometry in the 18th century, as he was able to develop for hundreds of pages and dozens of theorems a system in geometry that denied the truth of the postulate (in the attempt to find a contradiction). This can be regarded as the first system of non-Euclidean geometry. Its later developments by Lambert, Bolyai, Lobachevsky and Gauss eventually opened the way to contemporary geometry. Occupying a unique position in the literature of mathematical history, Euclid Vindicated from Every Blemish will be of high interest to historians of mathematics as well as historians of philosophy interested in the development of non-Euclidean geometries.
hypothesis in geometry: Can Theories be Refuted? Sandra Harding, 2012-12-06 According to a view assumed by many scientists and philosophers of science and standardly found in science textbooks, it is controlled ex perience which provides the basis for distinguishing between acceptable and unacceptable theories in science: acceptable theories are those which can pass empirical tests. It has often been thought that a certain sort of test is particularly significant: 'crucial experiments' provide supporting empiri cal evidence for one theory while providing conclusive evidence against another. However, in 1906 Pierre Duhem argued that the falsification of a theory is necessarily ambiguous and therefore that there are no crucial experiments; one can never be sure that it is a given theory rather than auxiliary or background hypotheses which experiment has falsified. w. V. Quine has concurred in this judgment, arguing that our statements about the external world face the tribunal of sense experience not indi vidually but only as a corporate body. Some philosophers have thought that the Duhem-Quine thesis gra tuitously raises perplexities. Others see it as doubly significant; these philosophers think that it provides a base for criticism of the foundational view of knowledge which has dominated much of western thought since Descartes, and they think that it opens the door to a new and fruitful way to conceive of scientific progress in particular and of the nature and growth of knowledge in general.
hypothesis in geometry: Object Representation in Computer Vision II Jean Ponce, Andrew Zisserman, 1996-09-25 This book constitutes the strictly refereed post-workshop proceedings of the second International Workshop on Object Representation in Computer Vision, held in conjunction with ECCV '96 in Cambridge, UK, in April 1996. The 15 revised full papers contained in the book were selected from 45 submissions for presentation at the workshop. Also included are three invited contributions based on the talks by Takeo Kanade, Jan Koenderink, and Ram Nevatia as well as a workshop report by the volume editors summarizing several panel discussions and the general state of the art in the area.
hypothesis in geometry: Encyclopaedia of Mathematics Michiel Hazewinkel, 2013-12-01 This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.
hypothesis in geometry: The Structure of Being International Society for Neoplatonic Studies, 1982-01-01 Neoplatonism has sometimes been seen as a species of mysticism. This volume shows that Neoplatonism has, on the contrary, a characteristic and definable structure. It presents the logic of Neoplatonism and carefully distinguishes it from the logic of other forms of philosophy.
hypothesis in geometry: The Search for Certainty Frank J. Swetz, 2012-01-01 Self-contained and authoritative, this history of mathematics is suited to those with no math background. Its absorbing, entertaining essays focus on the era from 1800 to 2000. Contributors include Henri Poincaré, Judith V. Grabiner, and H. S. M. Coxeter, who discuss topics ranging from logic and infinity to Fermat's Last Theorem.
hypothesis in geometry: A System of Logic, Ratiocinative and Inductive John Stuart Mill, 1898
hypothesis in geometry: Imaging Industrial Flows A Plaskowski, M.S Beck, R Thorn, Tom Dyakowski, 1995-01-01 The first authoritative account of the industrial potential of tomographic imaging techniques, Imaging Industrial Flows: Applications of Electrical Process Tomography provides an overview of the instrumentation used in process tomography. Assuming only a basic knowledge of instrumentation, electronics, and computing systems, the authors concisely describe the principles behind the operation of image reconstruction techniques and the application of electrical tomography to measuring flows in pipelines and other industrial equipment. This reference will help you improve the efficiency of measurement and control equipment, monitor industrial effluents, and increase the efficiency and safety of process plant equipment, in particular, in the chemical engineering, oil technology, and flow measurement instrumentation industries.
hypothesis in geometry: Lectures on the Philosophy of Mathematics Joel David Hamkins, 2021-02-02 An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.
hypothesis in geometry: What It Is Like To Perceive J. Christopher Maloney, 2018-06-15 Naturalistic cognitive science, when realistically rendered, rightly maintains that to think is to deploy contentful mental representations. Accordingly, conscious perception, memory, and anticipation are forms of cognition that, despite their introspectively manifest differences, may coincide in content. Sometimes we remember what we saw; other times we predict what we will see. Why, then, does what it is like consciously to perceive, differ so dramatically from what it is like merely to recall or anticipate the same? Why, if thought is just representation, does the phenomenal character of seeing a sunset differ so stunningly from the tepid character of recollecting or predicting the sun's descent? J. Christopher Maloney argues that, unlike other cognitive modes, perception is in fact immediate, direct acquaintance with the object of thought. Although all mental representations carry content, the vehicles of perceptual representation are uniquely composed of the very objects represented. To perceive the setting sun is to use the sun and its properties to cast a peculiar cognitive vehicle of demonstrative representation. This vehicle's embedded referential term is identical with, and demonstrates, the sun itself. And the vehicle's self-attributive demonstrative predicate is itself forged from a property of that same remote star. So, in this sense, the perceiving mind is an extended mind. Perception is unbrokered cognition of what is real, exactly as it really is. Maloney's theory of perception will be of great interest in the philosophy of mind and cognitive science.
hypothesis in geometry: Science and Hypothesis Henri Poincare, 2016-03-31 Science and Hypothesis is a study written in 1902, by the French mathematician, Henri Poincaré. It was designed with non-specialist readers in mind, and contains information on mathematics, space, physics and biology. The main theme of this work is that the absolute truth of science is non-existent. It postulates that many scientific beliefs are closer to convenient conventions than valid explanations. The chapters of this book include: Number and Magnitude, On the Nature of Mathematical Reasoning, Mathematical Magnitude and Experiment, Space, Non-Euclidean Geometries, Space and Geometry, Experiment and Geometry, etcetera. Many vintage texts such as this are increasingly scarce and expensive, and it is with this in mind that we are republishing this book now, in an affordable, high-quality, modern edition. It comes complete with a specially commissioned biography of the author.
hypothesis in geometry: Poincaré, Philosopher of Science María de Paz, Robert DiSalle, 2014-04-11 This volume presents a selection of papers from the Poincaré Project of the Center for the Philosophy of Science, University of Lisbon, bringing together an international group of scholars with new assessments of Henri Poincaré's philosophy of science—both its historical impact on the foundations of science and mathematics, and its relevance to contemporary philosophical inquiry. The work of Poincaré (1854-1912) extends over many fields within mathematics and mathematical physics. But his scientific work was inseparable from his groundbreaking philosophical reflections, and the scientific ferment in which he participated was inseparable from the philosophical controversies in which he played a pre-eminent part. The subsequent history of the mathematical sciences was profoundly influenced by Poincaré’s philosophical analyses of the relations between and among mathematics, logic, and physics, and, more generally, the relations between formal structures and the world of experience. The papers in this collection illuminate Poincaré’s place within his own historical context as well as the implications of his work for ours.
hypothesis in geometry: The Scientific Legacy of Poincare Éric Charpentier, Etienne Ghys, Annick Lesne, 2010 Henri Poincare (1854-1912) was one of the greatest scientists of his time, perhaps the last one to have mastered and expanded almost all areas in mathematics and theoretical physics. In this book, twenty world experts present one part of Poincare's extraordinary work. Each chapter treats one theme, presenting Poincare's approach, and achievements.
hypothesis in geometry: Mathematical Expeditions Reinhard Laubenbacher, David Pengelley, 2013-12-01 The stories of five mathematical journeys into new realms, pieced together from the writings of the explorers themselves. Some were guided by mere curiosity and the thrill of adventure, others by more practical motives. In each case the outcome was a vast expansion of the known mathematical world and the realisation that still greater vistas remain to be explored. The authors tell these stories by guiding readers through the very words of the mathematicians at the heart of these events, providing an insightinto the art of approaching mathematical problems. The five chapters are completely independent, with varying levels of mathematical sophistication, and will attract students, instructors, and the intellectually curious reader. By working through some of the original sources and supplementary exercises, which discuss and solve -- or attempt to solve -- a great problem, this book helps readers discover the roots of modern problems, ideas, and concepts, even whole subjects. Students will also see the obstacles that earlier thinkers had to clear in order to make their respective contributions to five central themes in the evolution of mathematics.
hypothesis in geometry: Encyclopaedia of Mathematics M. Hazewinkel, 2013-12-01
hypothesis in geometry: The Data Parallel Programming Model Guy-Rene Perrin, Alain Darte, 1996-09-11 This monograph-like book assembles the thorougly revised and cross-reviewed lectures given at the School on Data Parallelism, held in Les Menuires, France, in May 1996. The book is a unique survey on the current status and future perspectives of the currently very promising and popular data parallel programming model. Much attention is paid to the style of writing and complementary coverage of the relevant issues throughout the 12 chapters. Thus these lecture notes are ideally suited for advanced courses or self-instruction on data parallel programming. Furthermore, the book is indispensable reading for anybody doing research in data parallel programming and related areas.
hypothesis in geometry: What is Geometry? Giandomenico Sica, 2006