JESPER LÜTZEN
- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567370
- eISBN:
- 9780191717925
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567370.003.0023
- Subject:
- Physics, History of Physics
In the introduction to his book Principles of Mechanics, Heinrich Hertz emphasised that one of the advantages of his geometric formulation of his mechanics is that it throws a bright light upon ...
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In the introduction to his book Principles of Mechanics, Heinrich Hertz emphasised that one of the advantages of his geometric formulation of his mechanics is that it throws a bright light upon William Rowan Hamilton's method of treating mechanical problems by the aid of characteristic functions. Hertz developed the geometric version of the Hamilton formalism in the first kinematic book and then applied these results to the motion of free holonomic systems, and finally to the motion of unfree systems. Thus, Hertz was able to express the analytical equations of the Hamilton formalism for a conservative system, without taking the hidden system into account except through the force function U. The geometry that made his theory for the straightest distance so appealing no longer holds in his description of conservative systems. However, it is possible to introduce a different metric in configuration space, so that the geometric part of the theory also applies to conservative systems.Less
In the introduction to his book Principles of Mechanics, Heinrich Hertz emphasised that one of the advantages of his geometric formulation of his mechanics is that it throws a bright light upon William Rowan Hamilton's method of treating mechanical problems by the aid of characteristic functions. Hertz developed the geometric version of the Hamilton formalism in the first kinematic book and then applied these results to the motion of free holonomic systems, and finally to the motion of unfree systems. Thus, Hertz was able to express the analytical equations of the Hamilton formalism for a conservative system, without taking the hidden system into account except through the force function U. The geometry that made his theory for the straightest distance so appealing no longer holds in his description of conservative systems. However, it is possible to introduce a different metric in configuration space, so that the geometric part of the theory also applies to conservative systems.
Max. M Edling
- Published in print:
- 2003
- Published Online:
- January 2005
- ISBN:
- 9780195148701
- eISBN:
- 9780199835096
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195148703.003.0015
- Subject:
- Political Science, American Politics
Offers brief sketches of the institutionalization of the military and fiscal powers granted by the US Constitution, and of the uses made of them by the Federalists in the 1790s. Gives an outline of ...
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Offers brief sketches of the institutionalization of the military and fiscal powers granted by the US Constitution, and of the uses made of them by the Federalists in the 1790s. Gives an outline of the fiscal and financial policy of the Federalists, thereby assessing Alexander Hamilton's claim to have restored public credit while reducing the tax pressure on the citizens. Also looks at the fate of the Federalist program after the Federalists had lost power to the Jeffersonian Republicans.Less
Offers brief sketches of the institutionalization of the military and fiscal powers granted by the US Constitution, and of the uses made of them by the Federalists in the 1790s. Gives an outline of the fiscal and financial policy of the Federalists, thereby assessing Alexander Hamilton's claim to have restored public credit while reducing the tax pressure on the citizens. Also looks at the fate of the Federalist program after the Federalists had lost power to the Jeffersonian Republicans.
David Armstrong
- Published in print:
- 1993
- Published Online:
- November 2003
- ISBN:
- 9780198275282
- eISBN:
- 9780191598739
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198275285.003.0003
- Subject:
- Political Science, International Relations and Politics
While there are important differences between the American Revolution and other revolutions, there are also some striking similarities. The basic ideas of the American outlook on international ...
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While there are important differences between the American Revolution and other revolutions, there are also some striking similarities. The basic ideas of the American outlook on international affairs emerged in the hundred years before the revolution. Central amongst these ideas was the need to build and secure a republic in North America. A fundamental debate took place after the revolution between the different views of Alexander Hamilton and Thomas Jefferson as to the most appropriate American posture towards the rest of the world. The essential elements in this debate have continued to affect the US foreign policy.Less
While there are important differences between the American Revolution and other revolutions, there are also some striking similarities. The basic ideas of the American outlook on international affairs emerged in the hundred years before the revolution. Central amongst these ideas was the need to build and secure a republic in North America. A fundamental debate took place after the revolution between the different views of Alexander Hamilton and Thomas Jefferson as to the most appropriate American posture towards the rest of the world. The essential elements in this debate have continued to affect the US foreign policy.
Ken Binmore
- Published in print:
- 2005
- Published Online:
- January 2007
- ISBN:
- 9780195178111
- eISBN:
- 9780199783670
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195178111.003.0007
- Subject:
- Economics and Finance, Microeconomics
When do we care for others as we care for ourselves? William Hamilton showed that we should be expected to care for our family members in proportion to our degree of relationship to them. Such ...
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When do we care for others as we care for ourselves? William Hamilton showed that we should be expected to care for our family members in proportion to our degree of relationship to them. Such reasoning explains why eusociality evolved independently at least twelve times in the order Hymenoptera, which includes ants, bees, and wasps, but only three times elsewhere in the animal kingdom. It also verifies Thomas Hobbes' answer to the question: Why cannot mankind live sociably one with another as bees or ants?Less
When do we care for others as we care for ourselves? William Hamilton showed that we should be expected to care for our family members in proportion to our degree of relationship to them. Such reasoning explains why eusociality evolved independently at least twelve times in the order Hymenoptera, which includes ants, bees, and wasps, but only three times elsewhere in the animal kingdom. It also verifies Thomas Hobbes' answer to the question: Why cannot mankind live sociably one with another as bees or ants?
Oliver Johns
- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567264
- eISBN:
- 9780191717987
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567264.003.0006
- Subject:
- Physics, Atomic, Laser, and Optical Physics
The general calculus of variations discussed in the previous chapter may be used to derive variational principles in mechanics. Two different, but closely related, variational principles are ...
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The general calculus of variations discussed in the previous chapter may be used to derive variational principles in mechanics. Two different, but closely related, variational principles are presented here: Hamilton’s principle and the phase space Hamilton’s principle. Variational principles play a crucial role in quantum theory, general relativity, and theoretical physics in general. The path in configuration space that is a solution to the Lagrange equations is often referred to as the classical path. This is the path of natural motion of a mechanical system as it responds to the forces included in the potential part of the Lagrangian. This chapter discusses Hamilton’s principle expressed in Lagrangian equations, Hamilton’s principle with constraints, and phase space and Hamilton’s principle.Less
The general calculus of variations discussed in the previous chapter may be used to derive variational principles in mechanics. Two different, but closely related, variational principles are presented here: Hamilton’s principle and the phase space Hamilton’s principle. Variational principles play a crucial role in quantum theory, general relativity, and theoretical physics in general. The path in configuration space that is a solution to the Lagrange equations is often referred to as the classical path. This is the path of natural motion of a mechanical system as it responds to the forces included in the potential part of the Lagrangian. This chapter discusses Hamilton’s principle expressed in Lagrangian equations, Hamilton’s principle with constraints, and phase space and Hamilton’s principle.
JESPER LÜTZEN
- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567370
- eISBN:
- 9780191717925
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567370.003.0021
- Subject:
- Physics, History of Physics
In his book Principles of Mechanics, Heinrich Hertz treats integral principles in three parts. First, he discusses the properties of geodesics and their relation to straightest paths in a purely ...
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In his book Principles of Mechanics, Heinrich Hertz treats integral principles in three parts. First, he discusses the properties of geodesics and their relation to straightest paths in a purely geometric way. These investigations then form the basis for a discussion of integral principles applied to the motion of free systems, and finally to the motion of conservative systems. Hertz's approach differed from the usual approach in three respects. First, it was based on his geometry of systems of points. Second, it dealt with forces as an indirect result of a coupling of the visible system with a hidden system. Third, it limited the applicability of even the most general of these principles (Hamilton's principle) to special mechanical systems declaring that it was invalid in general. Hertz pointed out that Hamilton's principle and the principle of least action did not apply to non-holonomic systems.Less
In his book Principles of Mechanics, Heinrich Hertz treats integral principles in three parts. First, he discusses the properties of geodesics and their relation to straightest paths in a purely geometric way. These investigations then form the basis for a discussion of integral principles applied to the motion of free systems, and finally to the motion of conservative systems. Hertz's approach differed from the usual approach in three respects. First, it was based on his geometry of systems of points. Second, it dealt with forces as an indirect result of a coupling of the visible system with a hidden system. Third, it limited the applicability of even the most general of these principles (Hamilton's principle) to special mechanical systems declaring that it was invalid in general. Hertz pointed out that Hamilton's principle and the principle of least action did not apply to non-holonomic systems.
A.D. Neate and A. Truman
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0013
- Subject:
- Mathematics, Probability / Statistics, Analysis
This chapter summarises a selection of results on the inviscid limit of the stochastic Burgers equation emphasising geometric properties of the caustic, Maxwell set and Hamilton-Jacobi level surfaces ...
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This chapter summarises a selection of results on the inviscid limit of the stochastic Burgers equation emphasising geometric properties of the caustic, Maxwell set and Hamilton-Jacobi level surfaces and relating these results to a discussion of stochastic turbulence. It shows that for small viscosities there exists a vortex filament structure near to the Maxwell set. It is discussed how this vorticity is directly related to the adhesion model for the evolution of the early universe, and new explicit formulas for the distribution of mass within the shock are included.Less
This chapter summarises a selection of results on the inviscid limit of the stochastic Burgers equation emphasising geometric properties of the caustic, Maxwell set and Hamilton-Jacobi level surfaces and relating these results to a discussion of stochastic turbulence. It shows that for small viscosities there exists a vortex filament structure near to the Maxwell set. It is discussed how this vorticity is directly related to the adhesion model for the evolution of the early universe, and new explicit formulas for the distribution of mass within the shock are included.
Andrea Braides
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198507840
- eISBN:
- 9780191709890
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507840.003.0004
- Subject:
- Mathematics, Applied Mathematics
Homogenization problems for a general class of integrals are solved by a direct approach. Different homogenization formulas are given, both in an asymptotic form and as a cell problem (in the convex ...
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Homogenization problems for a general class of integrals are solved by a direct approach. Different homogenization formulas are given, both in an asymptotic form and as a cell problem (in the convex case). These are applied in the study of the asymptotic behaviour of Riemannian metrics and Hamilton-Jacobi equations.Less
Homogenization problems for a general class of integrals are solved by a direct approach. Different homogenization formulas are given, both in an asymptotic form and as a cell problem (in the convex case). These are applied in the study of the asymptotic behaviour of Riemannian metrics and Hamilton-Jacobi equations.
James A.R. Marshall
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691161563
- eISBN:
- 9781400866564
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161563.003.0001
- Subject:
- Biology, Evolutionary Biology / Genetics
This book demonstrates the generality of inclusive fitness theory, with particular emphasis on its fundamental evolutionary logic. It presents the basic mathematical theory of natural selection and ...
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This book demonstrates the generality of inclusive fitness theory, with particular emphasis on its fundamental evolutionary logic. It presents the basic mathematical theory of natural selection and shows how inclusive fitness theory deals with more complicated social scenarios. Topics include the Price equation, Hamilton's rule, nonadditive interactions, conditional behaviors, heritability, and maximization of inclusive fitness. This chapter provides a brief historical introduction to the problem of apparent design in biology, evolutionary explanations of this, and in particular, evolutionary explanations of individual behaviors that appear designed to benefit not the individual themselves, but other members of their species. It examines how social behaviors can be shaped by natural selection and discusses the problem of providing an evolutionary explanation of self-sacrifice by individuals, altruism in group selection, and multilevel selection theory.Less
This book demonstrates the generality of inclusive fitness theory, with particular emphasis on its fundamental evolutionary logic. It presents the basic mathematical theory of natural selection and shows how inclusive fitness theory deals with more complicated social scenarios. Topics include the Price equation, Hamilton's rule, nonadditive interactions, conditional behaviors, heritability, and maximization of inclusive fitness. This chapter provides a brief historical introduction to the problem of apparent design in biology, evolutionary explanations of this, and in particular, evolutionary explanations of individual behaviors that appear designed to benefit not the individual themselves, but other members of their species. It examines how social behaviors can be shaped by natural selection and discusses the problem of providing an evolutionary explanation of self-sacrifice by individuals, altruism in group selection, and multilevel selection theory.
James A.R. Marshall
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691161563
- eISBN:
- 9781400866564
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161563.003.0004
- Subject:
- Biology, Evolutionary Biology / Genetics
This chapter examines how the logic of inclusive fitness theory can be mathematically formalized using the Price equation, and how that formalization can be used to derive Hamilton's rule in its ...
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This chapter examines how the logic of inclusive fitness theory can be mathematically formalized using the Price equation, and how that formalization can be used to derive Hamilton's rule in its simplest form, as applied to unconditional behaviors having additive effects on fitness. Various biological phenomena, such as sex allocation and working policing within eusocial insect colonies, have been analyzed by considering what strategies maximize individuals' inclusive fitness, and how observed social behaviors should correlate with quantities such as relatedness. The chapter derives Hamilton's rule by introducing some notation for the effects of behaviors on fitnesses of individuals that interact socially, to make explicit precisely how genes (and later phenotypes) affect fitness, and to give a general form of Hamilton's rule that will apply to any (unconditional, additive) behavior regardless of its details. It shows that inclusive fitness is a genuinely novel extension of the classical fitness studied by Charles Darwin, R. A. Fisher, and others.Less
This chapter examines how the logic of inclusive fitness theory can be mathematically formalized using the Price equation, and how that formalization can be used to derive Hamilton's rule in its simplest form, as applied to unconditional behaviors having additive effects on fitness. Various biological phenomena, such as sex allocation and working policing within eusocial insect colonies, have been analyzed by considering what strategies maximize individuals' inclusive fitness, and how observed social behaviors should correlate with quantities such as relatedness. The chapter derives Hamilton's rule by introducing some notation for the effects of behaviors on fitnesses of individuals that interact socially, to make explicit precisely how genes (and later phenotypes) affect fitness, and to give a general form of Hamilton's rule that will apply to any (unconditional, additive) behavior regardless of its details. It shows that inclusive fitness is a genuinely novel extension of the classical fitness studied by Charles Darwin, R. A. Fisher, and others.
James A.R. Marshall
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691161563
- eISBN:
- 9781400866564
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161563.003.0005
- Subject:
- Biology, Evolutionary Biology / Genetics
This chapter examines what happens in nonadditive interactions when such interactions take place between relatives, and how Hamilton's rule can be extended in two different ways to accommodate such ...
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This chapter examines what happens in nonadditive interactions when such interactions take place between relatives, and how Hamilton's rule can be extended in two different ways to accommodate such nonadditivity. It first considers the selective pressures on nonadditive behaviors directed towards relatives by making use of the replicator dynamics to capture interactions within structured populations, so that on average, interactions within the population occur between relatives. It then describes two extensions to Hamilton's rule to deal with nonadditive interactions. One approach takes deviations from additivity and accounts for them all in a single synergistic coefficient. The other approach applies partial regression to keep a version of Hamilton's rule with only three parameters, in which costs and benefits vary according to the frequency of social individuals in a population. The chapter also explains the use of the Price equation to study nonadditive social interactions between relatives.Less
This chapter examines what happens in nonadditive interactions when such interactions take place between relatives, and how Hamilton's rule can be extended in two different ways to accommodate such nonadditivity. It first considers the selective pressures on nonadditive behaviors directed towards relatives by making use of the replicator dynamics to capture interactions within structured populations, so that on average, interactions within the population occur between relatives. It then describes two extensions to Hamilton's rule to deal with nonadditive interactions. One approach takes deviations from additivity and accounts for them all in a single synergistic coefficient. The other approach applies partial regression to keep a version of Hamilton's rule with only three parameters, in which costs and benefits vary according to the frequency of social individuals in a population. The chapter also explains the use of the Price equation to study nonadditive social interactions between relatives.
James A.R. Marshall
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691161563
- eISBN:
- 9781400866564
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161563.003.0006
- Subject:
- Biology, Evolutionary Biology / Genetics
This chapter examines social behaviors that are expressed conditional on the phenotype of others. David Queller argued that inclusive fitness analyses need to be done on a per-behavior basis, citing ...
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This chapter examines social behaviors that are expressed conditional on the phenotype of others. David Queller argued that inclusive fitness analyses need to be done on a per-behavior basis, citing as an example the decision over whether to reproduce directly, and whether to aid a reproductive. Queller showed that inclusive fitness predictions are only sensible when one analyzes what an individual should do, given it finds itself in a particular behavioral role. The chapter first provides an overview of implicit and explicit conditionality and presents two classic examples: William D. Hamilton's greenbeard traits and Robert Trivers's theory of reciprocal cooperation. It also introduces an extension of Hamilton's rule to deal with explicitly conditional behaviors; this extension features a measure of phenotypic assortment that appears not to be the classic genetic relatedness of Hamilton's rule.Less
This chapter examines social behaviors that are expressed conditional on the phenotype of others. David Queller argued that inclusive fitness analyses need to be done on a per-behavior basis, citing as an example the decision over whether to reproduce directly, and whether to aid a reproductive. Queller showed that inclusive fitness predictions are only sensible when one analyzes what an individual should do, given it finds itself in a particular behavioral role. The chapter first provides an overview of implicit and explicit conditionality and presents two classic examples: William D. Hamilton's greenbeard traits and Robert Trivers's theory of reciprocal cooperation. It also introduces an extension of Hamilton's rule to deal with explicitly conditional behaviors; this extension features a measure of phenotypic assortment that appears not to be the classic genetic relatedness of Hamilton's rule.
James A.R. Marshall
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691161563
- eISBN:
- 9781400866564
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161563.003.0007
- Subject:
- Biology, Evolutionary Biology / Genetics
This chapter examines four variants of Hamilton's rule and how they give different evolutionary explanations for certain social behaviors such as greenbeard traits. These variants are: HR1, which ...
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This chapter examines four variants of Hamilton's rule and how they give different evolutionary explanations for certain social behaviors such as greenbeard traits. These variants are: HR1, which extends Hamilton's rule with a synergistic coefficient capturing the deviation from additivity of fitness interactions; HR2, which deals with the conditional expression of phenotype; HR3, which is concerned with fitness as partial regression; and HR4, the geometric view of relatedness. These variants differ in how they treat the three key parameters of the original: “relatedness,” “cost,” and “benefit.” The chapter also considers how the nongenetic explanation of the evolution of altruism can actually be recast in a version with genetic relatedness, and how geometric relatedness underlies phenotypic assortment. Finally, it discusses different viewpoints on conditional behaviors.Less
This chapter examines four variants of Hamilton's rule and how they give different evolutionary explanations for certain social behaviors such as greenbeard traits. These variants are: HR1, which extends Hamilton's rule with a synergistic coefficient capturing the deviation from additivity of fitness interactions; HR2, which deals with the conditional expression of phenotype; HR3, which is concerned with fitness as partial regression; and HR4, the geometric view of relatedness. These variants differ in how they treat the three key parameters of the original: “relatedness,” “cost,” and “benefit.” The chapter also considers how the nongenetic explanation of the evolution of altruism can actually be recast in a version with genetic relatedness, and how geometric relatedness underlies phenotypic assortment. Finally, it discusses different viewpoints on conditional behaviors.
M. Gail Hamner
- Published in print:
- 2003
- Published Online:
- November 2003
- ISBN:
- 9780195155471
- eISBN:
- 9780199834266
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195155475.001.0001
- Subject:
- Religion, Philosophy of Religion
The development of pragmatism is the most important achievement in the history of American philosophy. M. Gail Hamner here examines the European roots of the movement in a search for what makes ...
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The development of pragmatism is the most important achievement in the history of American philosophy. M. Gail Hamner here examines the European roots of the movement in a search for what makes pragmatism uniquely American. She argues that the inextricably American character of the pragmatism of such figures as Charles Sanders Peirce and William James lies in its often‐understated affirmation of America as a uniquely religious country with a God‐given mission, and as populated by God‐fearing citizens. By looking at European and British thinkers whom the pragmatists read, Hamner examines how pragmatism's notions of self, nation, and morality were formed in reaction to the work of these thinkers. She finds that the pervasive religiosity of nineteenth‐century American public language underlies Peirce's and James's resistance to aspects of the philosophy and science of their non‐American colleagues. This religiosity, Hamner shows, is linked strongly to the continuing rhetorical power of American Puritanism. Claims made for and about Puritanism were advanced throughout the nineteenth century as rallying cries for specific political, social, and individual changes. It was in this religiously and politically charged environment that Peirce and James received and reinterpreted non‐American voices. Hamner traces the development of pragmatism by analyzing the concepts of consciousness, causality, will, and belief in two German thinkers (Hermann von Helmholtz and Wilhelm Wundt) and two Scottish thinkers (William Hamilton and Alexander Bain), and by examining how their ideas were appropriated by Peirce and James. The book is arranged in three main parts: Evolution of German psychology; Evolution of Scottish psychology; and Pragmatic reception of European psychology.Less
The development of pragmatism is the most important achievement in the history of American philosophy. M. Gail Hamner here examines the European roots of the movement in a search for what makes pragmatism uniquely American. She argues that the inextricably American character of the pragmatism of such figures as Charles Sanders Peirce and William James lies in its often‐understated affirmation of America as a uniquely religious country with a God‐given mission, and as populated by God‐fearing citizens. By looking at European and British thinkers whom the pragmatists read, Hamner examines how pragmatism's notions of self, nation, and morality were formed in reaction to the work of these thinkers. She finds that the pervasive religiosity of nineteenth‐century American public language underlies Peirce's and James's resistance to aspects of the philosophy and science of their non‐American colleagues. This religiosity, Hamner shows, is linked strongly to the continuing rhetorical power of American Puritanism. Claims made for and about Puritanism were advanced throughout the nineteenth century as rallying cries for specific political, social, and individual changes. It was in this religiously and politically charged environment that Peirce and James received and reinterpreted non‐American voices. Hamner traces the development of pragmatism by analyzing the concepts of consciousness, causality, will, and belief in two German thinkers (Hermann von Helmholtz and Wilhelm Wundt) and two Scottish thinkers (William Hamilton and Alexander Bain), and by examining how their ideas were appropriated by Peirce and James. The book is arranged in three main parts: Evolution of German psychology; Evolution of Scottish psychology; and Pragmatic reception of European psychology.
Alfonso Sorrentino
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164502
- eISBN:
- 9781400866618
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164502.003.0005
- Subject:
- Mathematics, Applied Mathematics
This chapter describes another interesting approach to the study of invariant sets provided by the so-called weak KAM theory, developed by Albert Fathi. This approach can be considered as the ...
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This chapter describes another interesting approach to the study of invariant sets provided by the so-called weak KAM theory, developed by Albert Fathi. This approach can be considered as the functional analytic counterpart of the variational methods discussed in the previous chapters. The starting point is the relation between KAM tori (or more generally, invariant Lagrangian graphs) and classical solutions and subsolutions of the Hamilton–Jacobi equation. It introduces the notion of weak (non-classical) solutions of the Hamilton–Jacobi equation and a special class of subsolutions (critical subsolutions). In particular, it highlights their relation to Aubry–Mather theory.Less
This chapter describes another interesting approach to the study of invariant sets provided by the so-called weak KAM theory, developed by Albert Fathi. This approach can be considered as the functional analytic counterpart of the variational methods discussed in the previous chapters. The starting point is the relation between KAM tori (or more generally, invariant Lagrangian graphs) and classical solutions and subsolutions of the Hamilton–Jacobi equation. It introduces the notion of weak (non-classical) solutions of the Hamilton–Jacobi equation and a special class of subsolutions (critical subsolutions). In particular, it highlights their relation to Aubry–Mather theory.
Thomas J. Curry
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780195145694
- eISBN:
- 9780199834129
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195145690.003.0002
- Subject:
- Religion, History of Christianity
Outlines the extent to which modern scholars have strayed from an understanding of the history surrounding the First Amendment, and elevated logic over history. Instead of recognizing religious ...
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Outlines the extent to which modern scholars have strayed from an understanding of the history surrounding the First Amendment, and elevated logic over history. Instead of recognizing religious liberty as a “natural right,” and the Constitution as a limitation on government, they have interpreted the First Amendment as augmenting the power of government. This development has brought about what both James Madison and Alexander Hamilton especially feared.Less
Outlines the extent to which modern scholars have strayed from an understanding of the history surrounding the First Amendment, and elevated logic over history. Instead of recognizing religious liberty as a “natural right,” and the Constitution as a limitation on government, they have interpreted the First Amendment as augmenting the power of government. This development has brought about what both James Madison and Alexander Hamilton especially feared.
M. Gail Hamner
- Published in print:
- 2003
- Published Online:
- November 2003
- ISBN:
- 9780195155471
- eISBN:
- 9780199834266
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195155475.003.0005
- Subject:
- Religion, Philosophy of Religion
Demonstrates the resonance between Charles Sanders Peirce (the American pragmatist) and William Hamilton (a Scottish professor of logic and metaphysics), which can be outlined in three ways: first, ...
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Demonstrates the resonance between Charles Sanders Peirce (the American pragmatist) and William Hamilton (a Scottish professor of logic and metaphysics), which can be outlined in three ways: first, both men present a philosophy that balances Kant's idealism with T.H. Reid's naturalism (Peirce calls this task a ‘critical common‐sensism’); second, they both discuss questions of faith in a manner that implies a transcendent or cosmological perspective; and third, they exhibit a focused interest in logic. However, the pragmatist always evinces slightly different priorities: while Hamilton remains a committed nominalist throughout his writings, Peirce attempts to reconcile Kant and British empiricism as part of his larger argument against nominalism; while Hamilton maintains a Calvinist trinitarianism, Peirce's musings on questions of faith direct him closer to Spinoza's panentheism; and finally, while Hamilton's logic remains an important but separate line of philosophical inquiry, Peirce develops a logic of relations that conjoins his interest in logic to his semiotic and phenomenology, and thus becomes a pervasive part of his philosophy. After giving a brief exposition of the main points of his philosophy, the discussion of Hamilton examines how relativity, conditionality, and free will inform his statements about causality, consciousness, belief, and action. Of greatest interest is how the concept of consciousness relates to the concept of belief, such that the former acts as the guarantor of the latter.Less
Demonstrates the resonance between Charles Sanders Peirce (the American pragmatist) and William Hamilton (a Scottish professor of logic and metaphysics), which can be outlined in three ways: first, both men present a philosophy that balances Kant's idealism with T.H. Reid's naturalism (Peirce calls this task a ‘critical common‐sensism’); second, they both discuss questions of faith in a manner that implies a transcendent or cosmological perspective; and third, they exhibit a focused interest in logic. However, the pragmatist always evinces slightly different priorities: while Hamilton remains a committed nominalist throughout his writings, Peirce attempts to reconcile Kant and British empiricism as part of his larger argument against nominalism; while Hamilton maintains a Calvinist trinitarianism, Peirce's musings on questions of faith direct him closer to Spinoza's panentheism; and finally, while Hamilton's logic remains an important but separate line of philosophical inquiry, Peirce develops a logic of relations that conjoins his interest in logic to his semiotic and phenomenology, and thus becomes a pervasive part of his philosophy. After giving a brief exposition of the main points of his philosophy, the discussion of Hamilton examines how relativity, conditionality, and free will inform his statements about causality, consciousness, belief, and action. Of greatest interest is how the concept of consciousness relates to the concept of belief, such that the former acts as the guarantor of the latter.
M. Gail Hamner
- Published in print:
- 2003
- Published Online:
- November 2003
- ISBN:
- 9780195155471
- eISBN:
- 9780199834266
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195155475.003.0007
- Subject:
- Religion, Philosophy of Religion
Charles Sanders Peirce was an American pragmatist whose notions of, and about community, inform not only his reformulation of ‘science’ and ontology but also the complex theory of continuity ...
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Charles Sanders Peirce was an American pragmatist whose notions of, and about community, inform not only his reformulation of ‘science’ and ontology but also the complex theory of continuity (synechism) that structures his pragmatism. Peirce's realism is examined through his theorizing of community and synechism, and an assessment of his attempt to reconcile Kant's idealism with British empiricism. One consequence of this reconciliation is a theory of truth that posits both the singleness of truth (a characteristic presumed by Hermann von Helmholtz and William Hamilton), and truth's infinite deferral in the light of the fallible modes of human perception and reasoning. Envisioning fallibilism as occurring always within and between communities of inquiry, Peirce develops the famous pragmatic supposition that truth is that which no one has a reason to disbelieve. The discussion of Peirce's realism includes his theory of generals and its resonance with Helmholtz's theory of the reality of natural laws; for both thinkers, the reality of a law (or general) differs from individual (and equally real) instances of that law, with the difference residing in Peirce's synechism, where Helmholtz attributes the reality of natural laws to the overarching action of causality.Less
Charles Sanders Peirce was an American pragmatist whose notions of, and about community, inform not only his reformulation of ‘science’ and ontology but also the complex theory of continuity (synechism) that structures his pragmatism. Peirce's realism is examined through his theorizing of community and synechism, and an assessment of his attempt to reconcile Kant's idealism with British empiricism. One consequence of this reconciliation is a theory of truth that posits both the singleness of truth (a characteristic presumed by Hermann von Helmholtz and William Hamilton), and truth's infinite deferral in the light of the fallible modes of human perception and reasoning. Envisioning fallibilism as occurring always within and between communities of inquiry, Peirce develops the famous pragmatic supposition that truth is that which no one has a reason to disbelieve. The discussion of Peirce's realism includes his theory of generals and its resonance with Helmholtz's theory of the reality of natural laws; for both thinkers, the reality of a law (or general) differs from individual (and equally real) instances of that law, with the difference residing in Peirce's synechism, where Helmholtz attributes the reality of natural laws to the overarching action of causality.
CLARE KELLAR
- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780199266708
- eISBN:
- 9780191708930
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199266708.003.0005
- Subject:
- History, History of Religion
This chapter begins by discussing that the end to England's aggressive evangelization in Scotland made for a distinct change in the religious climates of both countries during the 1550s, thus, ...
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This chapter begins by discussing that the end to England's aggressive evangelization in Scotland made for a distinct change in the religious climates of both countries during the 1550s, thus, preparing both countries for a reform. It examines the situation of the Scottish exiles and evaluates the Edwardian church. It discusses archbishops Hamilton and Cranmer's duty to create comprehensive schemes for church reform, focusing on unity and inclusiveness, and avoiding contentious extremes of opinion. It investigates the concurrent schemes of catholic reform. It also discusses several European conflicts and the hardening of the battle lines.Less
This chapter begins by discussing that the end to England's aggressive evangelization in Scotland made for a distinct change in the religious climates of both countries during the 1550s, thus, preparing both countries for a reform. It examines the situation of the Scottish exiles and evaluates the Edwardian church. It discusses archbishops Hamilton and Cranmer's duty to create comprehensive schemes for church reform, focusing on unity and inclusiveness, and avoiding contentious extremes of opinion. It investigates the concurrent schemes of catholic reform. It also discusses several European conflicts and the hardening of the battle lines.
KEITH JEFFERY
- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199239672
- eISBN:
- 9780191719493
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239672.003.0004
- Subject:
- History, Military History
Henry Wilson was clearly a rising star and returned from South Africa with a greatly enhanced reputation. Enjoying the patronage of Lord Roberts, he was stimulatingly ensconced in the War Office, ...
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Henry Wilson was clearly a rising star and returned from South Africa with a greatly enhanced reputation. Enjoying the patronage of Lord Roberts, he was stimulatingly ensconced in the War Office, right at the centre of things. He spent his first nine months working as Ian Hamilton's assistant on honours and rewards for service in South Africa, work which included preparing individual citations where soldiers were ‘mentioned in despatches’ and drawing up lengthy lists of medal-winners. It was a potential poisoned chalice. In a profession (and, indeed, society) so minutely hierarchical, army careers could be made or broken by the gaining — or not — of medals and distinctions. This chapter discusses Wilson's responsibilities in the War Office of Ireland, his frustration with the office's inefficiencies, and his role in the campaign for army reform in the country.Less
Henry Wilson was clearly a rising star and returned from South Africa with a greatly enhanced reputation. Enjoying the patronage of Lord Roberts, he was stimulatingly ensconced in the War Office, right at the centre of things. He spent his first nine months working as Ian Hamilton's assistant on honours and rewards for service in South Africa, work which included preparing individual citations where soldiers were ‘mentioned in despatches’ and drawing up lengthy lists of medal-winners. It was a potential poisoned chalice. In a profession (and, indeed, society) so minutely hierarchical, army careers could be made or broken by the gaining — or not — of medals and distinctions. This chapter discusses Wilson's responsibilities in the War Office of Ireland, his frustration with the office's inefficiencies, and his role in the campaign for army reform in the country.
