Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the Gener... more Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the General Evolution Research Group, and their 37-year partnership.
In our recent book (Abraham and Roy, 2010) we have proposed a mathematical model for the quantum ... more In our recent book (Abraham and Roy, 2010) we have proposed a mathematical model for the quantum vacuum as a model of consciousness. We emphasized that the quantum vacuum is not identical to cosmic consciousness. The word akasha is translated in English as space or ether. In the Eastern tradition ether has two phases, subtle and gross. In our model, the nodes of a network correspond to subtle ether, while the physical space created in temporal slices by condensation corresponds to gross ether. In essence, the akasha is a subtle background against which everything in the material universe becomes perceptible. In this essay we expand on the self-organization of the gross ether from a subtle (submicroscopic) cellular network, with special emphasis on the illusion of time, and the apparent paradoxes such as precognition, retro-causation, and entanglement.
We extend the cognitive theories of Husserl and Poincaé up the cosmological chain to the Platonic... more We extend the cognitive theories of Husserl and Poincaé up the cosmological chain to the Platonic world of ideas. CONTENTS 1. Introduction 2. Theories of perception
The migration of the monarch butterfly is considered from the perspective of chaos theory, and co... more The migration of the monarch butterfly is considered from the perspective of chaos theory, and compared to the evolution of human culture.
Continuing in the spirit of earlier works, we propose a mathematical model for the process of int... more Continuing in the spirit of earlier works, we propose a mathematical model for the process of internalization of ideas. This entire concept presupposes a paradigm of mind with internal and external regions, which we accept provisionally for the sake of discussion. In short, we envision a physical model comprising several excitable, continuous media in parallel planes, interconnected by a process of resonance of vibrations. The mathematical model for this physical analog is then discretised, and proposed verbatim as a computational model for the mental system. This model is typical of complex dynamical systems, as they have evolved during the last twenty years or so.
Complex dynamical systems theory and system dynamics diverged at some point in the recent past, a... more Complex dynamical systems theory and system dynamics diverged at some point in the recent past, and should reunite. This is a concise introduction to the basic concepts of complex dynamical systems, in the context of the new mathematical theories of chaos and bifurcation.
Progress in Biophysics and Molecular Biology, 2017
We examine the relationship between mysticism and mathematical creativity through case studies fr... more We examine the relationship between mysticism and mathematical creativity through case studies from the history of mathematics. CONTENTS 1. Introduction
World scientific series on nonlinear science, series A, 2013
Dynamical Systems Theory in the spirit of Poincaré (DST) has been in vogue since the controversia... more Dynamical Systems Theory in the spirit of Poincaré (DST) has been in vogue since the controversial award of a prize by King Oscar II of Sweden and Norway, on his 60th birthday, January 21, 1889, to Poincaré. DST diffused Eastward (via Stockholm, Saint Petersburg, and Moscow) to Gorky in Russia, to Japan, and Westward (via Princeton, Mexico City, and Rio), to Berkeley in California. Now in the centennial year of the premature death of Poincaré, it is time for a review of these peregrinations.
This article may be used for research, teaching, and private study purposes. Any substantial or s... more This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
The synchronization of nonlinear oscillators is well-known and is a traditional topic in complex ... more The synchronization of nonlinear oscillators is well-known and is a traditional topic in complex dynamical system theory. The synchronization of chaotic attractors is less well-known, but is of obvious interest in many applications to the sciences: physical, biological, and social. In a recent experimental study of coupled lattices of Rössler attractors, (jointly with Michael Nivala) we were surprised to discover global periodic behavior in large regimes of the parameter space. This emergent periodicity in a field of chaos may be of significance in the origin of life, and in many life processes. In this talk we will explore the emergence of global periodicity, and also the periodic windows in the bifurcation diagram of the Rössler attractor, which may be the local cause of this global behavior.
Growth processes abound in nature, and are frequently the target of modeling exercises in the sci... more Growth processes abound in nature, and are frequently the target of modeling exercises in the sciences. In this article we illustrate an agent-based approach to modeling, in the case of a single example from the social sciences: bullying..
Progress in biophysics and molecular biology, Jan 14, 2015
Is there a world of mathematics above and beyond ordinary reality, as Plato proposed? Or is mathe... more Is there a world of mathematics above and beyond ordinary reality, as Plato proposed? Or is mathematics a cultural construct? In this short article we speculate on the place of mathematical reality from the perspective of the mystical cosmologies of the ancient traditions of meditation, psychedelics, and divination.
We study evolutionary games with a continuous space of strategies A. The current state is the dis... more We study evolutionary games with a continuous space of strategies A. The current state is the distribution F of players over A. The payoff function defines an adaptive landscape for each state. Each player continuously adjusts her strategy in A towards higher ground in the landscape. Consequently the distribution F changes continuously and so the landscape changes, and the players adjust again. Assuming gradient adjustment, this interplay between landscape and state is described by a partial differential equation or, equivalently, by a dynamical system on the infinite-dimensional space F of current states. The paper illustrates these ideas using Veblen's notion of conspicuous consumption, i.e., one's payoff depends on where one's consumption x falls in the current distribution F. For simplicity we take A = [0, 1], and obtain explicit solutions to the landscape dynamics for special cases. Two Propositions characterize the long run and short run dynamics for more general cases. Using the simulation package NetLogo we illustrate both Propositions. * We are grateful to the National Science Foundation for support under grant SES-0436509. Peter Towbin and Paul Viotti provided useful comments. We also want to acknowledge Joel Yellin, with whom the first author worked out some of the ideas on Veblen consumption.
This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters... more This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters will study the related topics of tensors and differential forms. A basic operation introduced in this chapter is the Lie derivative of a function or a vector field. It is introduced in two different ways, algebraically as a type of directional derivative and dynamically as a rate of change along a flow. The Lie derivative formula asserts the equivalence of these two definitions. The Lie derivative is a basic operation used extensively in differential geometry, general relativity, Hamiltonian mechanics, and continuum mechanics.
The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ... more The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ℝn using a partition of unity subordinate to an atlas. The change-of-variables theorem guarantees that the integral is well defined, independent of the choice of atlas and partition of unity. Two basic theorems of integral calculus, the change-of-variables theorem and Stokes’ theorem, are discussed in detail along with some applications.
Manifolds have enough structure to allow differentiation of maps between them. To set the stage f... more Manifolds have enough structure to allow differentiation of maps between them. To set the stage for these concepts requires a development of differential calculus in linear spaces from a geometric point of view. The goal of this chapter is to provide this perspective.
We are now ready to study manifolds and the differential calculus of maps between manifolds. Mani... more We are now ready to study manifolds and the differential calculus of maps between manifolds. Manifolds are an abstraction of the idea of a smooth surface in Euclidean space. This abstraction has proved useful because many sets that are smooth in some sense are not presented to us as subsets of Euclidean space. The abstraction strips away the containing space and makes constructions intrinsic to the manifold itself. This point of view is well worth the geometric insight it provides.
Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the Gener... more Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the General Evolution Research Group, and their 37-year partnership.
In our recent book (Abraham and Roy, 2010) we have proposed a mathematical model for the quantum ... more In our recent book (Abraham and Roy, 2010) we have proposed a mathematical model for the quantum vacuum as a model of consciousness. We emphasized that the quantum vacuum is not identical to cosmic consciousness. The word akasha is translated in English as space or ether. In the Eastern tradition ether has two phases, subtle and gross. In our model, the nodes of a network correspond to subtle ether, while the physical space created in temporal slices by condensation corresponds to gross ether. In essence, the akasha is a subtle background against which everything in the material universe becomes perceptible. In this essay we expand on the self-organization of the gross ether from a subtle (submicroscopic) cellular network, with special emphasis on the illusion of time, and the apparent paradoxes such as precognition, retro-causation, and entanglement.
We extend the cognitive theories of Husserl and Poincaé up the cosmological chain to the Platonic... more We extend the cognitive theories of Husserl and Poincaé up the cosmological chain to the Platonic world of ideas. CONTENTS 1. Introduction 2. Theories of perception
The migration of the monarch butterfly is considered from the perspective of chaos theory, and co... more The migration of the monarch butterfly is considered from the perspective of chaos theory, and compared to the evolution of human culture.
Continuing in the spirit of earlier works, we propose a mathematical model for the process of int... more Continuing in the spirit of earlier works, we propose a mathematical model for the process of internalization of ideas. This entire concept presupposes a paradigm of mind with internal and external regions, which we accept provisionally for the sake of discussion. In short, we envision a physical model comprising several excitable, continuous media in parallel planes, interconnected by a process of resonance of vibrations. The mathematical model for this physical analog is then discretised, and proposed verbatim as a computational model for the mental system. This model is typical of complex dynamical systems, as they have evolved during the last twenty years or so.
Complex dynamical systems theory and system dynamics diverged at some point in the recent past, a... more Complex dynamical systems theory and system dynamics diverged at some point in the recent past, and should reunite. This is a concise introduction to the basic concepts of complex dynamical systems, in the context of the new mathematical theories of chaos and bifurcation.
Progress in Biophysics and Molecular Biology, 2017
We examine the relationship between mysticism and mathematical creativity through case studies fr... more We examine the relationship between mysticism and mathematical creativity through case studies from the history of mathematics. CONTENTS 1. Introduction
World scientific series on nonlinear science, series A, 2013
Dynamical Systems Theory in the spirit of Poincaré (DST) has been in vogue since the controversia... more Dynamical Systems Theory in the spirit of Poincaré (DST) has been in vogue since the controversial award of a prize by King Oscar II of Sweden and Norway, on his 60th birthday, January 21, 1889, to Poincaré. DST diffused Eastward (via Stockholm, Saint Petersburg, and Moscow) to Gorky in Russia, to Japan, and Westward (via Princeton, Mexico City, and Rio), to Berkeley in California. Now in the centennial year of the premature death of Poincaré, it is time for a review of these peregrinations.
This article may be used for research, teaching, and private study purposes. Any substantial or s... more This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
The synchronization of nonlinear oscillators is well-known and is a traditional topic in complex ... more The synchronization of nonlinear oscillators is well-known and is a traditional topic in complex dynamical system theory. The synchronization of chaotic attractors is less well-known, but is of obvious interest in many applications to the sciences: physical, biological, and social. In a recent experimental study of coupled lattices of Rössler attractors, (jointly with Michael Nivala) we were surprised to discover global periodic behavior in large regimes of the parameter space. This emergent periodicity in a field of chaos may be of significance in the origin of life, and in many life processes. In this talk we will explore the emergence of global periodicity, and also the periodic windows in the bifurcation diagram of the Rössler attractor, which may be the local cause of this global behavior.
Growth processes abound in nature, and are frequently the target of modeling exercises in the sci... more Growth processes abound in nature, and are frequently the target of modeling exercises in the sciences. In this article we illustrate an agent-based approach to modeling, in the case of a single example from the social sciences: bullying..
Progress in biophysics and molecular biology, Jan 14, 2015
Is there a world of mathematics above and beyond ordinary reality, as Plato proposed? Or is mathe... more Is there a world of mathematics above and beyond ordinary reality, as Plato proposed? Or is mathematics a cultural construct? In this short article we speculate on the place of mathematical reality from the perspective of the mystical cosmologies of the ancient traditions of meditation, psychedelics, and divination.
We study evolutionary games with a continuous space of strategies A. The current state is the dis... more We study evolutionary games with a continuous space of strategies A. The current state is the distribution F of players over A. The payoff function defines an adaptive landscape for each state. Each player continuously adjusts her strategy in A towards higher ground in the landscape. Consequently the distribution F changes continuously and so the landscape changes, and the players adjust again. Assuming gradient adjustment, this interplay between landscape and state is described by a partial differential equation or, equivalently, by a dynamical system on the infinite-dimensional space F of current states. The paper illustrates these ideas using Veblen's notion of conspicuous consumption, i.e., one's payoff depends on where one's consumption x falls in the current distribution F. For simplicity we take A = [0, 1], and obtain explicit solutions to the landscape dynamics for special cases. Two Propositions characterize the long run and short run dynamics for more general cases. Using the simulation package NetLogo we illustrate both Propositions. * We are grateful to the National Science Foundation for support under grant SES-0436509. Peter Towbin and Paul Viotti provided useful comments. We also want to acknowledge Joel Yellin, with whom the first author worked out some of the ideas on Veblen consumption.
This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters... more This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters will study the related topics of tensors and differential forms. A basic operation introduced in this chapter is the Lie derivative of a function or a vector field. It is introduced in two different ways, algebraically as a type of directional derivative and dynamically as a rate of change along a flow. The Lie derivative formula asserts the equivalence of these two definitions. The Lie derivative is a basic operation used extensively in differential geometry, general relativity, Hamiltonian mechanics, and continuum mechanics.
The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ... more The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ℝn using a partition of unity subordinate to an atlas. The change-of-variables theorem guarantees that the integral is well defined, independent of the choice of atlas and partition of unity. Two basic theorems of integral calculus, the change-of-variables theorem and Stokes’ theorem, are discussed in detail along with some applications.
Manifolds have enough structure to allow differentiation of maps between them. To set the stage f... more Manifolds have enough structure to allow differentiation of maps between them. To set the stage for these concepts requires a development of differential calculus in linear spaces from a geometric point of view. The goal of this chapter is to provide this perspective.
We are now ready to study manifolds and the differential calculus of maps between manifolds. Mani... more We are now ready to study manifolds and the differential calculus of maps between manifolds. Manifolds are an abstraction of the idea of a smooth surface in Euclidean space. This abstraction has proved useful because many sets that are smooth in some sense are not presented to us as subsets of Euclidean space. The abstraction strips away the containing space and makes constructions intrinsic to the manifold itself. This point of view is well worth the geometric insight it provides.
Uploads
Papers by Ralph Abraham