Summary Struggles with the Continuum A Historical Perspective arxiv.org
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This article explores the historical challenges mathematicians and physicists encountered with the continuum, including Zeno's paradoxes and logical issues, stemming from the infinitely divisible nature of the real line.
Slides
Slide Presentation (12 slides)
Key Points
- Mathematicians and physicists have struggled with the continuum, facing challenges with infinity and real numbers.
- Zeno's paradoxes and convergent sequences were developed to dismiss these paradoxes.
- Gödel's theorems and the continuum hypothesis present logical problems when dealing with infinity and real numbers.
- Nonstandard analysis and constructivism aim to tackle these logical problems but are not widely adopted.
- The infinitely divisible nature of the real line in physics poses challenges in predicting the behavior of physical systems, particularly with integrals and differential equations.
- A successful theory of quantum gravity may shed light on these problems, but no experimentally verified predictions have been made.
- Newtonian gravity struggles with simultaneous collisions of three or more bodies and potential "pathological" solutions.
- Quantum mechanics provides a finite energy bound and avoids problems associated with noncollision singularities.
- Classical electrodynamics of point particles faces challenges with the Abraham-Lorentz-Dirac force law and defining the self-force exerted by the particle's own electric field.
- Quantum field theory, particularly quantum electrodynamics, incorporates both quantum mechanics and special relativity but lacks a rigorous mathematical formulation.
- General relativity relates the curvature of spacetime to the flow of energy and momentum but does not incorporate quantum mechanics.
- Singularities in black holes are not well-defined points or regions in spacetime and are still not fully understood.
Summaries
32 word summary
This article examines the historical struggles mathematicians and physicists faced with the continuum, including Zeno's paradoxes and logical problems. Challenges in physics arise from the infinitely divisible nature of the real line.
88 word summary
This article explores the historical struggles mathematicians and physicists have faced when dealing with the continuum. It discusses Zeno's paradoxes and the development of convergent sequences to dismiss them. The article also delves into logical problems related to infinity and real numbers, including Godel's theorems and the continuum hypothesis. Challenges in physics arise from the infinitely divisible nature of the real line, particularly when dealing with integrals and differential equations. Despite progress in certain areas, there is still much work needed to fully understand and address these challenges.
127 word summary
This article explores the historical struggles mathematicians and physicists have faced when dealing with the continuum. It discusses Zeno's paradoxes and the development of convergent sequences to dismiss them. The article also delves into logical problems related to infinity and real numbers, including Godel's theorems and the continuum hypothesis. Challenges in physics arise from the infinitely divisible nature of the real line, particularly when dealing with integrals and differential equations. The article mentions the struggles with the continuum in Newtonian gravity, quantum mechanics, and classical electrodynamics. It highlights the difficulties in making definite predictions and addressing infinite energy associated with point particles. Despite progress in certain areas, such as quantum mechanics and classical electrodynamics, there is still much work needed to fully understand and address these challenges.
660 word summary
This article explores the struggles mathematicians and physicists have faced when dealing with the continuum. It begins by discussing Zeno's paradoxes and the development of convergent sequences to dismiss these paradoxes. The article then delves into the logical problems that arise when dealing with infinity and real numbers, including Godel's theorems and the continuum hypothesis. It also touches on foundational work in mathematics, such as nonstandard analysis and constructivism, which aim to tackle these logical problems, but are not widely adopted or considered impactful in real-life applications.
The article highlights the challenges posed by the infinitely divisible nature of the real line in physics. It mentions the difficulties in finding theories that can consistently predict the behavior of physical systems, particularly when dealing with integrals and differential equations. These challenges are attributed to the continuum nature of spacetime itself. The article mentions that a successful theory of quantum gravity may shed light on these problems, but no experimentally verified predictions have been made by any current theory of quantum gravity.
Next, the article discusses the struggles with the continuum in Newtonian gravity. It mentions how collisions between point masses can be handled mathematically, but simultaneous collisions of three or more bodies present more difficulties. The article also mentions Xia's work on solutions where particles shoot off to infinity in a finite amount of time, raising questions about whether Newtonian gravity can make definite predictions in all cases.
The article then moves on to quantum mechanics and its role in addressing some of the challenges posed by the continuum. It explains how quantum mechanics provides a finite energy bound and avoids the problems associated with noncollision singularities. The Kato-Lax-Milgram-Nelson theorem is mentioned as a key result that helps understand how kinetic energy triumphs over potential energy in quantum systems.
The article continues by discussing the struggles in classical electrodynamics of point particles. It mentions the Abraham-Lorentz-Dirac force law and its problems, such as the inclusion of a Schott term involving the third derivative of the particle's position. The article explores the difficulties in defining the self-force exerted by the particle's own electric field and the infinite energy associated with the field of a point charge. It also discusses efforts to simplify Dirac's calculations and the progress made by Kijowski and his coauthors in formulating a well-behaved classical theory that describes charged point particles interacting with the electromagnetic field.
In conclusion, the struggles with the continuum in mathematics and physics are far from over. The challenges posed by the infinite and infinitely divisible nature of the continuum continue to impact theories and predictions. While progress has been made in certain areas, such as quantum mechanics and classical electrodynamics, there is still much work to be done to fully understand and address these challenges.
Attempts have been made to have particles interact directly with each other within the framework of special relativity, but this approach does not admit a Hamiltonian formulation in terms of particle positions and momenta, limiting its application. Quantum field theory emerges when charged particles interact electromagnetically, incorporating both quantum mechanics and special relativity. However, it poses significant challenges due to its complexity and lack of a rigorous mathematical formulation. General relativity combines gravity with relativity and describes spacetime as a 4-dimensional Lorentzian manifold, but infinities arise from the continuum nature of spacetime and are connected to black holes and the Big Bang.
Overall, struggles with the continuum have been an ongoing challenge in physics and mathematics. While progress has been made in various areas, such as finding mathematical frameworks for quantum field theory and understanding the behavior of perturbative QED, there is still much work to be done to fully understand and resolve these issues.
In general relativity, particles move along timelike geodesics, which are paths that go slower than light and bend as little as possible. A geodesic is a path in space that, when slightly varied, can only become longer. The behavior of singularities in general relativity
661 word summary
This article examines the historical struggles mathematicians and physicists have faced when dealing with the continuum. It begins by discussing Zeno's paradoxes and the development of convergent sequences to dismiss these paradoxes. The article then delves into the logical problems that arise when dealing with infinity and real numbers, including Godel's theorems and the continuum hypothesis. It also touches on foundational work in mathematics, such as nonstandard analysis and constructivism, which aim to tackle these logical problems but are not widely adopted or considered impactful in real-life applications.
The article highlights the challenges posed by the infinitely divisible nature of the real line in physics. It mentions the difficulties in finding theories that can consistently predict the behavior of physical systems, particularly when dealing with integrals and differential equations. These challenges are attributed to the continuum nature of spacetime itself. The article mentions that a successful theory of quantum gravity may shed light on these problems, but no experimentally verified predictions have been made by any current theory of quantum gravity.
Next, the article discusses the struggles with the continuum in Newtonian gravity. It mentions how collisions between point masses can be handled mathematically, but simultaneous collisions of three or more bodies present more difficulties. The article also mentions Xia's work on solutions where particles shoot off to infinity in a finite amount of time, raising questions about whether Newtonian gravity can make definite predictions in all cases.
The article then moves on to quantum mechanics and its role in addressing some of the challenges posed by the continuum. It explains how quantum mechanics provides a finite energy bound and avoids the problems associated with noncollision singularities. The Kato-Lax-Milgram-Nelson theorem is mentioned as a key result that helps understand how kinetic energy triumphs over potential energy in quantum systems.
The article continues by discussing the struggles in classical electrodynamics of point particles. It mentions the Abraham-Lorentz-Dirac force law and its problems, such as the inclusion of a Schott term involving the third derivative of the particle's position. The article explores the difficulties in defining the self-force exerted by the particle's own electric field and the infinite energy associated with the field of a point charge. It also discusses efforts to simplify Dirac's calculations and the progress made by Kijowski and his coauthors in formulating a well-behaved classical theory that describes charged point particles interacting with the electromagnetic field.
In conclusion, the struggles with the continuum in mathematics and physics are far from over. The challenges posed by the infinite and infinitely divisible nature of the continuum continue to impact theories and predictions. While progress has been made in certain areas, such as quantum mechanics and classical electrodynamics, there is still much work to be done to fully understand and address these challenges.
Attempts have been made to have particles interact directly with each other within the framework of special relativity, but this approach does not admit a Hamiltonian formulation in terms of particle positions and momenta, limiting its application. Quantum field theory emerges when charged particles interact electromagnetically, incorporating both quantum mechanics and special relativity. However, it poses significant challenges due to its complexity and lack of a rigorous mathematical formulation. General relativity combines gravity with relativity and describes spacetime as a 4-dimensional Lorentzian manifold, but infinities arise from the continuum nature of spacetime and are connected to black holes and the Big Bang.
Overall, struggles with the continuum have been an ongoing challenge in physics and mathematics. While progress has been made in various areas, such as finding mathematical frameworks for quantum field theory and understanding the behavior of perturbative QED, there is still much work to be done to fully understand and resolve these issues.
In general relativity, particles move along timelike geodesics, which are paths that go slower than light and bend as little as possible. A geodesic is a path in space that, when slightly varied, can only become longer. The behavior of singularities in general relativity
1490 word summary
This article explores the struggles that mathematicians and physicists have faced when dealing with the continuum. It begins by discussing Zeno's paradoxes and the development of convergent sequences to dismiss these paradoxes. The article then delves into the logical problems that arise when dealing with infinity and real numbers, including Gödel's theorems and the continuum hypothesis. It also touches on foundational work in mathematics, such as nonstandard analysis and constructivism, which aim to tackle these logical problems. However, these approaches are not widely adopted or considered impactful in real-life applications.
The article then shifts its focus to the challenges posed by the infinitely divisible nature of the real line in physics. It highlights the difficulties in finding theories that can consistently predict the behavior of physical systems, particularly when dealing with integrals and differential equations. These challenges are attributed to the continuum nature of spacetime itself. The article mentions that a successful theory of quantum gravity may shed light on these problems, but no experimentally verified predictions have been made by any current theory of quantum gravity.
Next, the article discusses the struggles with the continuum in Newtonian gravity. It mentions how collisions between point masses can be handled mathematically, but simultaneous collisions of three or more bodies present more difficulties. The article also mentions Xia's work on solutions where particles shoot off to infinity in a finite amount of time. These issues raise questions about whether Newtonian gravity can make definite predictions in all cases.
The article then moves on to quantum mechanics and its role in addressing some of the challenges posed by the continuum. It explains how quantum mechanics provides a finite energy bound and avoids the problems associated with noncollision singularities. The Kato-Lax-Milgram-Nelson theorem is mentioned as a key result that helps understand how kinetic energy triumphs over potential energy in quantum systems.
The article continues by discussing the struggles in classical electrodynamics of point particles. It mentions the Abraham-Lorentz-Dirac force law and its problems, such as the inclusion of a Schott term involving the third derivative of the particle's position. The article explores the difficulties in defining the self-force exerted by the particle's own electric field and the infinite energy associated with the field of a point charge. It also discusses efforts to simplify Dirac's calculations and the progress made by Kijowski and his coauthors in formulating a well-behaved classical theory that describes charged point particles interacting with the electromagnetic field.
In conclusion, the struggles with the continuum in mathematics and physics are far from over. The challenges posed by the infinite and infinitely divisible nature of the continuum continue to impact theories and predictions. While progress has been made in certain areas, such as quantum mechanics and classical electrodynamics, there is still much work to be done to fully understand and address these challenges.
In Kijowski's approach, the problem of the singular part of the electromagnetic field is resolved because the particle's acceleration is completely determined by the singular part. The radiation reaction can be seen by decomposing the electromagnetic field into external and retarded fields, which leads to the recovery of the original Abraham-Lorentz-Dirac law. However, this also allows for "pathological" solutions where particles can extract arbitrary amounts of energy from the electromagnetic field, and it is still unclear if solutions exist for all time. Classical point particles interacting with the electromagnetic field present challenges for physicists and mathematicians due to their infinitesimally small size and infinitely strong electromagnetic fields.
Attempts have been made to get rid of fields and have particles interact directly with each other within the framework of special relativity. Schwarzschild introduced a framework in 1903 where charged particles exert an electromagnetic force on each other without the use of fields. Forces are transmitted at the speed of light and depend on the motion of other particles within their future lightcones. This approach raises issues of reverse causality, but it avoids the thorny issue of how a point particle responds to its own electromagnetic field and does not describe the motion of a charged particle in an external electromagnetic field. However, this approach does not admit a Hamiltonian formulation in terms of particle positions and momenta, limiting its application.
Quantum field theory emerges when charged particles interact electromagnetically, incorporating both quantum mechanics and special relativity. It is a more complicated theory than classical physics and has become the best description of all known forces except gravity. However, quantum field theory poses significant challenges due to its complexity and the lack of a rigorous mathematical formulation. While progress has been made in constructing mathematical frameworks for quantum field theory, such as axiomatic formulations, no fully rigorous formulation has been found for interacting particles in four-dimensional spacetime.
Quantum electrodynamics (QED) is a specific quantum field theory that describes the interaction of photons and charged particles. It involves a dimensionless parameter called the fine structure constant, which determines the strength of the interaction. QED uses perturbation theory to calculate physical quantities as power series in the fine structure constant. However, it is widely believed that these power series diverge, and their convergence is not yet known. Despite this, perturbative QED has been remarkably successful in making accurate predictions, even though the power series may not converge.
General relativity combines gravity with relativity and describes spacetime as a 4-dimensional Lorentzian manifold. It relates the curvature of spacetime to the flow of energy and momentum through Einstein's equation. In general relativity, infinities arise from the continuum nature of spacetime and are connected to black holes and the Big Bang. While general relativity does not incorporate quantum mechanics, many physicists hope that quantum gravity will provide a solution to the struggles with the continuum.
Overall, struggles with the continuum have been an ongoing challenge in physics and mathematics. While progress has been made in various areas, such as finding mathematical frameworks for quantum field theory and understanding the behavior of perturbative QED, there is still much work to be done to fully understand and resolve these issues.
In general relativity, particles move along timelike geodesics, which are paths that go slower than light and bend as little as possible. A geodesic is a path in space that, when slightly varied, can only become longer. For a timelike path traced out by a particle moving slower than light, the proper time along the path is defined as the time ticked out by a clock moving along that path. Test particles are a mathematical trick for studying the geometry of spacetime, but actual particles are not test particles because they are affected by forces other than gravity and affect the geometry of the spacetime they inhabit. However, sufficiently light particles that are affected very little by forces other than gravity can be well approximated by test particles. When a small round ball consisting of many test particles is initially at rest and not affected by forces other than gravity, it will not change shape or size to first order in time. However, to second order in time it can expand or shrink, become stretched or squashed, or become an ellipsoid due to the curvature of spacetime.
Einstein's equation relates the flow of momentum in different directions to the volume of the ball. The components of the stress-energy tensor, which describe how much momentum in a certain direction is flowing through a point in spacetime, determine the flow of momentum in each direction. Einstein's equation also shows that positive energy density and positive pressure both curve spacetime in a way that makes a freely falling ball of point particles tend to shrink. When a star more than 8 times the mass of our Sun runs out of fuel, its core collapses and it may become either a neutron star or a black hole depending on its mass.
If general relativity is correct, black holes contain singularities. Singularities are not well-defined points or regions in spacetime but rather refer to incomplete timelike or null geodesics. A geodesic is incomplete if it ceases to be well-defined after a finite amount of time. The Penrose-Hawking singularity theorems state that if certain conditions are met, singularities must occur in general relativity. Weak cosmic censorship is a hypothesis proposed by Penrose that claims that spacetime singularities are always hidden from view by event horizons. However, there are counterexamples to weak cosmic censorship, and a stronger version of the hypothesis called generic weak cosmic censorship is currently being investigated.
The behavior of singularities in general relativity is still not fully understood, and there is ongoing research to study and analyze their properties. The continuum nature of spacetime poses challenges in every major theory of physics, as it leads to infinities. However, through hard work and rigorous analysis, these challenges can be addressed and predictions can be extracted from these theories. The question remains whether the continuum is an approximation to a deeper model of spacetime, and this will require further study and patience to determine.