Philosophy of physics

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In philosophy, the philosophy of physics deals with conceptual and interpretational issues in modern physics, many of which overlap with research done by certain kinds of theoretical physicists. Historically, philosophers of physics have engaged with questions such as the nature of space, time, matter and the laws that govern their interactions, as well as the epistemological and ontological basis of the theories used by practicing physicists. The discipline draws upon insights from various areas of philosophy, including metaphysics, epistemology, and philosophy of science, while also engaging with the latest developments in theoretical and experimental physics.

Contents

Contemporary work focuses on issues at the foundations of the three pillars of modern physics:

Other areas of focus include the nature of physical laws, symmetries, and conservation principles; the role of mathematics; and philosophical implications of emerging fields like quantum gravity, quantum information, and complex systems. Philosophers of physics have argued that philosophical analysis clarifies foundations, interprets implications, and guides theory development in physics.

Philosophy of space and time

The existence and nature of space and time (or space-time) are central topics in the philosophy of physics. [1]

Time

Time, in many philosophies, is seen as change. Wooden hourglass.jpg
Time, in many philosophies, is seen as change.

Time is often thought to be a fundamental quantity (that is, a quantity which cannot be defined in terms of other quantities), because time seems like a fundamentally basic concept, such that one cannot define it in terms of anything simpler. However, certain theories such as loop quantum gravity claim that spacetime is emergent. As Carlo Rovelli, one of the founders of loop quantum gravity has said: "No more fields on spacetime: just fields on fields". [2] Time is defined via measurement—by its standard time interval. Currently, the standard time interval (called "conventional second", or simply "second") is defined as 9,192,631,770 oscillations of a hyperfine transition in the 133 caesium atom. (ISO 31-1). What time is and how it works follows from the above definition. Time then can be combined mathematically with the fundamental quantities of space and mass to define concepts such as velocity, momentum, energy, and fields.

Both Newton and Galileo, [3] as well as most people up until the 20th century, thought that time was the same for everyone everywhere. The modern conception of time is based on Einstein's theory of relativity and Minkowski's spacetime, in which rates of time run differently in different inertial frames of reference, and space and time are merged into spacetime. Einstein's general relativity as well as the redshift of the light from receding distant galaxies indicate that the entire Universe and possibly space-time itself began about 13.8 billion years ago in the Big Bang. Einstein's theory of special relativity mostly (though not universally) made theories of time where there is something metaphysically special about the present seem much less plausible, as the reference-frame-dependence of time seems to not allow the idea of a privileged present moment.

Time travel

Some theories, most notably general relativity, suggest that suitable geometries of spacetime, or certain types of motion in space, may allow time travel into the past and future. Concepts that aid such understanding include the closed timelike curve.

Albert Einstein's special theory of relativity (and, by extension, the general theory) predicts time dilation that could be interpreted as time travel. The theory states that, relative to a stationary observer, time appears to pass more slowly for faster-moving bodies: for example, a moving clock will appear to run slow; as a clock approaches the speed of light its hands will appear to nearly stop moving. The effects of this sort of time dilation are discussed further in the popular "twin paradox". Although these results are experimentally observable, an intrinsic aspect of Einstein' theory is an equation applicable to the operation of GPS satellites and other high-tech systems used in daily life.

A second, similar type of time travel is permitted by general relativity. In this type a distant observer sees time passing more slowly for a clock at the bottom of a deep gravity well, and a clock lowered into a deep gravity well and pulled back up will indicate that less time has passed compared to a stationary clock that stayed with the distant observer.

Many in the scientific community believe that backward time travel is highly unlikely, because it violates causality [4] i.e. the logic of cause and effect. For example, what happens if you attempt to go back in time and kill yourself at an earlier stage in your life (or your grandfather, which leads to the grandfather paradox)? Stephen Hawking once suggested that the absence of tourists from the future constitutes a strong argument against the existence of time travel— a variant of the Fermi paradox, with time travelers instead of alien visitors. [4]

Space

Space is one of the few fundamental quantities in physics, meaning that it cannot be defined via other quantities because there is nothing more fundamental known at present. Thus, similar to the definition of other fundamental quantities (like time and mass), space is defined via measurement. Currently, the standard space interval, called a standard metre or simply metre, is defined as the distance traveled by light in a vacuum during a time interval of 1/299792458 of a second (exact).

In classical physics, space is a three-dimensional Euclidean space where any position can be described using three coordinates and parameterised by time. Special and general relativity use four-dimensional spacetime rather than three-dimensional space; and currently there are many speculative theories which use more than four spatial dimensions.

Philosophy of quantum mechanics

Quantum mechanics is a large focus of contemporary philosophy of physics, specifically concerning the correct interpretation of quantum mechanics. Very broadly, much of the philosophical work that is done in quantum theory is trying to make sense of superposition states: [5] the property that particles seem to not just be in one determinate position at one time, but are somewhere 'here', and also 'there' at the same time. Such a radical view turns many common sense metaphysical ideas on their head. Much of contemporary philosophy of quantum mechanics aims to make sense of what the very empirically successful formalism of quantum mechanics tells us about the physical world.

Uncertainty principle

The uncertainty principle is a mathematical relation asserting an upper limit to the accuracy of the simultaneous measurement of any pair of conjugate variables, e.g. position and momentum. In the formalism of operator notation, this limit is the evaluation of the commutator of the variables' corresponding operators.

The uncertainty principle arose as an answer to the question: How does one measure the location of an electron around a nucleus if an electron is a wave? When quantum mechanics was developed, it was seen to be a relation between the classical and quantum descriptions of a system using wave mechanics.

"Locality" and hidden variables

Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories given some basic assumptions about the nature of measurement. "Local" here refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields cannot propagate faster than the speed of light. "Hidden variables" are putative properties of quantum particles that are not included in the theory but nevertheless affect the outcome of experiments. In the words of physicist John Stewart Bell, for whom this family of results is named, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local." [6]

The term is broadly applied to a number of different derivations, the first of which was introduced by Bell in a 1964 paper titled "On the Einstein Podolsky Rosen Paradox". Bell's paper was a response to a 1935 thought experiment that Albert Einstein, Boris Podolsky and Nathan Rosen proposed, arguing that quantum physics is an "incomplete" theory. [7] [8] By 1935, it was already recognized that the predictions of quantum physics are probabilistic. Einstein, Podolsky and Rosen presented a scenario that involves preparing a pair of particles such that the quantum state of the pair is entangled, and then separating the particles to an arbitrarily large distance. The experimenter has a choice of possible measurements that can be performed on one of the particles. When they choose a measurement and obtain a result, the quantum state of the other particle apparently collapses instantaneously into a new state depending upon that result, no matter how far away the other particle is. This suggests that either the measurement of the first particle somehow also interacted with the second particle at faster than the speed of light, or that the entangled particles had some unmeasured property which pre-determined their final quantum states before they were separated. Therefore, assuming locality, quantum mechanics must be incomplete, as it cannot give a complete description of the particle's true physical characteristics. In other words, quantum particles, like electrons and photons, must carry some property or attributes not included in quantum theory, and the uncertainties in quantum theory's predictions would then be due to ignorance or unknowability of these properties, later termed "hidden variables".

Bell carried the analysis of quantum entanglement much further. He deduced that if measurements are performed independently on the two separated particles of an entangled pair, then the assumption that the outcomes depend upon hidden variables within each half implies a mathematical constraint on how the outcomes on the two measurements are correlated. This constraint would later be named the Bell inequality. Bell then showed that quantum physics predicts correlations that violate this inequality. Consequently, the only way that hidden variables could explain the predictions of quantum physics is if they are "nonlocal", which is to say that somehow the two particles are able to interact instantaneously no matter how widely they ever become separated. [9] [10]

Multiple variations on Bell's theorem were put forward in the following years, introducing other closely related conditions generally known as Bell (or "Bell-type") inequalities. The first rudimentary experiment designed to test Bell's theorem was performed in 1972 by John Clauser and Stuart Freedman. [11] More advanced experiments, known collectively as Bell tests , have been performed many times since. To date, Bell tests have consistently found that physical systems obey quantum mechanics and violate Bell inequalities; which is to say that the results of these experiments are incompatible with any local hidden variable theory. [12] [13]

The exact nature of the assumptions required to prove a Bell-type constraint on correlations has been debated by physicists and by philosophers. While the significance of Bell's theorem is not in doubt, its full implications for the interpretation of quantum mechanics remain unresolved.

Interpretations of quantum mechanics

In March 1927, working in Niels Bohr's institute, Werner Heisenberg formulated the principle of uncertainty thereby laying the foundation of what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg had been studying the papers of Paul Dirac and Pascual Jordan. He discovered a problem with measurement of basic variables in the equations. His analysis showed that uncertainties, or imprecisions, always turned up if one tried to measure the position and the momentum of a particle at the same time. Heisenberg concluded that these uncertainties or imprecisions in the measurements were not the fault of the experimenter, but fundamental in nature and are inherent mathematical properties of operators in quantum mechanics arising from definitions of these operators. [14]

The term "Copenhagen interpretation" is somewhat loosely defined, as many physicists and philosophers of physics have advanced similar but not identical views of quantum mechanics. It is principally associated with Heisenberg and Bohr, despite their philosophical differences. [15] [16] Features common to Copenhagen-type interpretations include the idea that quantum mechanics is intrinsically indeterministic, with probabilities calculated using the Born rule, and the principle of complementarity, which states that objects have certain pairs of complementary properties that cannot all be observed or measured simultaneously. [17] Moreover, the act of "observing" or "measuring" an object is irreversible, and no truth can be attributed to an object, except according to the results of its measurement. Copenhagen-type interpretations hold that quantum descriptions are objective, in that they are independent of any arbitrary factors in the physicist's mind. [18] : 85–90 

The Everett, or many-worlds interpretation of quantum mechanics claims that the wave-function of a quantum system is telling us claims about the reality of that physical system. It denies wavefunction collapse, and claims that superposition states should be interpreted literally as describing the reality of many-worlds where objects are located, and not simply indicating the indeterminacy of those variables. This is sometimes argued as a corollary of scientific realism, [19] which states that scientific theories aim to give us literally true descriptions of the world.

One issue for the Everett interpretation is the role that probability plays on this account. The Everettian account is completely deterministic, whereas probability seems to play an ineliminable role in quantum mechanics. [20] Contemporary Everettians have argued that one can get an account of probability that follows the Born rule through certain decision-theoretic proofs, [21] but there is as yet no consensus about whether any of these proofs are successful. [22] [23] [24]

Physicist Roland Omnès noted that it is impossible to experimentally differentiate between Everett's view, which says that as the wave-function decoheres into distinct worlds, each of which exists equally, and the more traditional view that says that a decoherent wave-function leaves only one unique real result. Hence, the dispute between the two views represents a great "chasm". "Every characteristic of reality has reappeared in its reconstruction by our theoretical model; every feature except one: the uniqueness of facts." [25]

Philosophy of thermal and statistical physics

The philosophy of thermal and statistical physics is concerned with the foundational issues and conceptual implications of thermodynamics and statistical mechanics. These branches of physics deal with the macroscopic behavior of systems comprising a large number of microscopic entities, such as particles, and the nature of laws that emerge from these systems like irreversibility and entropy. Interest of philosophers in statistical mechanics first arose from the observation of an apparent conflict between the time-reversal symmetry of fundamental physical laws and the irreversibility observed in thermodynamic processes, known as the "arrow of time" problem. Philosophers have sought to understand how the asymmetric behavior of macroscopic systems, such as the tendency of heat to flow from hot to cold bodies, can be reconciled with the time-symmetric laws governing the motion of individual particles.

Another key issue is the interpretation of probability in statistical mechanics, which is primarily concerned with the question of whether probabilities in statistical mechanics are epistemic, reflecting our lack of knowledge about the precise microstate of a system, or ontic, representing an objective feature of the physical world. The epistemic interpretation, also known as the subjective or Bayesian view, holds that probabilities in statistical mechanics are a measure of our ignorance about the exact state of a system. According to this view, we resort to probabilistic descriptions only due to the practical impossibility of knowing the precise properties of all its micro-constituents, like the positions and momenta of particles. As such, the probabilities are not objective features of the world but rather arise from our ignorance. In contrast, the ontic interpretation, also called the objective or frequentist view, asserts that probabilities in statistical mechanics are real, physical properties of the system itself. Proponents of this view argue that the probabilistic nature of statistical mechanics is not merely a reflection of our ignorance but an intrinsic feature of the physical world, and that even if we had complete knowledge of the microstate of a system, the macroscopic behavior would still be best described by probabilistic laws.

History

Aristotelian physics

Aristotelian physics viewed the universe as a sphere with a center. Matter, composed of the classical elements, earth, water, air, and fire, sought to go down towards the center of the universe, the center of the Earth, or up, away from it. Things in the aether such as the Moon, the Sun, planets, or stars circled the center of the universe. [26] Movement is defined as change in place, [26] i.e. space. [27]

Newtonian physics

The implicit axioms of Aristotelian physics with respect to movement of matter in space were superseded in Newtonian physics by Newton's First Law of Motion. [28]

Every body perseveres in its state either of rest or of uniform motion in a straight line, except insofar as it is compelled to change its state by impressed forces.

"Every body" includes the Moon, and an apple; and includes all types of matter, air as well as water, stones, or even a flame. Nothing has a natural or inherent motion. [29] Absolute space being three-dimensional Euclidean space, infinite and without a center. [29] Being "at rest" means being at the same place in absolute space over time. [30] The topology and affine structure of space must permit movement in a straight line at a uniform velocity; thus both space and time must have definite, stable dimensions. [31]

Leibniz

Gottfried Wilhelm Leibniz, 1646–1716, was a contemporary of Newton. He contributed a fair amount to the statics and dynamics emerging around him, often disagreeing with Descartes and Newton. He devised a new theory of motion (dynamics) based on kinetic energy and potential energy, which posited space as relative, whereas Newton was thoroughly convinced that space was absolute. An important example of Leibniz's mature physical thinking is his Specimen Dynamicum of 1695. [32]

Until the discovery of subatomic particles and the quantum mechanics governing them, many of Leibniz's speculative ideas about aspects of nature not reducible to statics and dynamics made little sense.

He anticipated Albert Einstein by arguing, against Newton, that space, time and motion are relative, not absolute: [33] "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is, that I hold it to be an order of coexistences, as time is an order of successions." [34]

See also

Related Research Articles

The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics, stemming from the work of Niels Bohr, Werner Heisenberg, Max Born, and others. The term "Copenhagen interpretation" was apparently coined by Heisenberg during the 1950s to refer to ideas developed in the 1925–1927 period, glossing over his disagreements with Bohr. Consequently, there is no definitive historical statement of what the interpretation entails.

<span class="mw-page-title-main">Many-worlds interpretation</span> Interpretation of quantum mechanics

The many-worlds interpretation (MWI) is a philosophical position about how the mathematics used in quantum mechanics relates to physical reality. It asserts that the universal wavefunction is objectively real, and that there is no wave function collapse. This implies that all possible outcomes of quantum measurements are physically realized in some "world" or universe. In contrast to some other interpretations, the evolution of reality as a whole in MWI is rigidly deterministic and local. Many-worlds is also called the relative state formulation or the Everett interpretation, after physicist Hugh Everett, who first proposed it in 1957. Bryce DeWitt popularized the formulation and named it many-worlds in the 1970s.

<span class="mw-page-title-main">Einstein–Podolsky–Rosen paradox</span> Historical critique of quantum mechanics

The Einstein–Podolsky–Rosen (EPR) paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen which argues that the description of physical reality provided by quantum mechanics is incomplete. In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing these hidden variables. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces, and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.

<span class="mw-page-title-main">Quantum mechanics</span> Description of physical properties at the atomic and subatomic scale

Quantum mechanics is a fundamental theory in physics that describes the behavior of nature at and below the scale of atoms. It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science.

The de Broglie–Bohm theory, also known as the pilot wave theory, Bohmian mechanics, Bohm's interpretation, and the causal interpretation, is an interpretation of quantum mechanics. It postulates that in addition to the wavefunction, an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all particles is defined by a guiding equation. The evolution of the wave function over time is given by the Schrödinger equation. The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992).

An interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments, there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, local or non-local, which elements of quantum mechanics can be considered real, and what the nature of measurement is, among other matters.

Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measurement. "Local" here refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields cannot propagate faster than the speed of light. "Hidden variables" are putative properties of quantum particles that are not included in quantum theory but nevertheless affect the outcome of experiments. In the words of physicist John Stewart Bell, for whom this family of results is named, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

<span class="mw-page-title-main">John Stewart Bell</span> Northern Irish physicist (1928–1990)

John Stewart Bell FRS was a physicist from Northern Ireland and the originator of Bell's theorem, an important theorem in quantum physics regarding hidden-variable theories.

In physics, a hidden-variable theory is a deterministic physical model which seeks to explain the probabilistic nature of quantum mechanics by introducing additional variables.

In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. A theory that includes the principle of locality is said to be a "local theory". This is an alternative to the concept of instantaneous, or "non-local" action at a distance. Locality evolved out of the field theories of classical physics. The idea is that for a cause at one point to have an effect at another point, something in the space between those points must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence.

The transactional interpretation of quantum mechanics (TIQM) takes the wave function of the standard quantum formalism, and its complex conjugate, to be retarded and advanced waves that form a quantum interaction as a Wheeler–Feynman handshake or transaction. It was first proposed in 1986 by John G. Cramer, who argues that it helps in developing intuition for quantum processes. He also suggests that it avoids the philosophical problems with the Copenhagen interpretation and the role of the observer, and also resolves various quantum paradoxes. TIQM formed a minor plot point in his science fiction novel Einstein's Bridge.

In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule. For example, a quantum particle like an electron can be described by a quantum state that associates to each point in space a complex number called a probability amplitude. Applying the Born rule to these amplitudes gives the probabilities that the electron will be found in one region or another when an experiment is performed to locate it. This is the best the theory can do; it cannot say for certain where the electron will be found. The same quantum state can also be used to make a prediction of how the electron will be moving, if an experiment is performed to measure its momentum instead of its position. The uncertainty principle implies that, whatever the quantum state, the range of predictions for the electron's position and the range of predictions for its momentum cannot both be narrow. Some quantum states imply a near-certain prediction of the result of a position measurement, but the result of a momentum measurement will be highly unpredictable, and vice versa. Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about "local hidden variables" within quantum systems.

In quantum mechanics, the measurement problem is the problem of definite outcomes: quantum systems have superpositions but quantum measurements only give one definite result.

<span class="mw-page-title-main">Bohr–Einstein debates</span> Series of public disputes between physicists Niels Bohr and Albert Einstein

The Bohr–Einstein debates were a series of public disputes about quantum mechanics between Albert Einstein and Niels Bohr. Their debates are remembered because of their importance to the philosophy of science, insofar as the disagreements—and the outcome of Bohr's version of quantum mechanics becoming the prevalent view—form the root of the modern understanding of physics. Most of Bohr's version of the events held in the Solvay Conference in 1927 and other places was first written by Bohr decades later in an article titled, "Discussions with Einstein on Epistemological Problems in Atomic Physics". Based on the article, the philosophical issue of the debate was whether Bohr's Copenhagen interpretation of quantum mechanics, which centered on his belief of complementarity, was valid in explaining nature. Despite their differences of opinion and the succeeding discoveries that helped solidify quantum mechanics, Bohr and Einstein maintained a mutual admiration that was to last the rest of their lives.

In physics, complementarity is a conceptual aspect of quantum mechanics that Niels Bohr regarded as an essential feature of the theory. The complementarity principle holds that certain pairs of complementary properties cannot all be observed or measured simultaneously, for examples, position and momentum or wave and particle properties. In contemporary terms, complementarity encompasses both the uncertainty principle and wave-particle duality.

Quantum mechanics is the study of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the moon. Classical physics is still used in much of modern science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain. The desire to resolve inconsistencies between observed phenomena and classical theory led to a revolution in physics, a shift in the original scientific paradigm: the development of quantum mechanics.

The ensemble interpretation of quantum mechanics considers the quantum state description to apply only to an ensemble of similarly prepared systems, rather than supposing that it exhaustively represents an individual physical system.

The history of quantum mechanics is a fundamental part of the history of modern physics. The major chapters of this history begin with the emergence of quantum ideas to explain individual phenomena—blackbody radiation, the photoelectric effect, solar emission spectra—an era called the Old or Older quantum theories. Building on the technology developed in classical mechanics, the invention of wave mechanics by Erwin Schrödinger and expansion by many others triggers the "modern" era beginning around 1925. Paul Dirac's relativistic quantum theory work lead him to explore quantum theories of radiation, culminating in quantum electrodynamics, the first quantum field theory. The history of quantum mechanics continues in the history of quantum field theory. The history of quantum chemistry, theoretical basis of chemical structure, reactivity, and bonding, interlaces with the events discussed in this article.

A hallmark of Albert Einstein's career was his use of visualized thought experiments as a fundamental tool for understanding physical issues and for elucidating his concepts to others. Einstein's thought experiments took diverse forms. In his youth, he mentally chased beams of light. For special relativity, he employed moving trains and flashes of lightning to explain his most penetrating insights. For general relativity, he considered a person falling off a roof, accelerating elevators, blind beetles crawling on curved surfaces and the like. In his debates with Niels Bohr on the nature of reality, he proposed imaginary devices intended to show, at least in concept, how the Heisenberg uncertainty principle might be evaded. In a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon known as quantum entanglement.

References

  1. ↑ Maudlin, Tim (2012). Philosophy of Physics: Space and Time. Princeton University Press. p. xi. ISBN   978-0691143095 . Retrieved 3 October 2017. ...the existence and nature of space and time (or space-time) is a central topic.
  2. ↑ Rovelli, C. (2004). Quantum Gravity. Cambridge Monographs on Mathematical Physics. p. 71.
  3. ↑ Roger Penrose, 2004. The Road to Reality: A Complete Guide to the Laws of the Universe. London: Jonathan Cape. ISBN   0-224-04447-8 (hardcover), 0-09-944068-7 (paperback).
  4. 1 2 Bolonkin, Alexander (2011). Universe, Human mmortality and Future Human Evaluation. Elsevier. p. 32. ISBN   978-0-12-415810-8. Extract of page 32
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  17. ↑ Omnès, Roland (1999). "The Copenhagen Interpretation". Understanding Quantum Mechanics. Princeton University Press. pp. 41–54. doi:10.2307/j.ctv173f2pm.9. S2CID   203390914. Bohr, Heisenberg, and Pauli recognized its main difficulties and proposed a first essential answer. They often met in Copenhagen ... 'Copenhagen interpretation has not always meant the same thing to different authors. I will reserve it for the doctrine held with minor differences by Bohr, Heisenberg, and Pauli.
  18. ↑ Omnès, R. (1994). The Interpretation of Quantum Mechanics. Princeton University Press. ISBN   978-0-691-03669-4. OCLC   439453957.
  19. ↑ David Wallace, 'The Emergent Multiverse', pp. 1–10
  20. ↑ David Wallace, 'The Emergent Multiverse', pp. 113–117
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  25. ↑ Omnès, Roland (2002). "11". Quantum philosophy : understanding and interpreting contemporary science (in French). Arturo Spangalli (transl.) (1st paperback ed.). Princeton: Princeton University Press. p. 213. ISBN   978-1400822867.
  26. 1 2 Tim Maudlin (2012-07-22). Philosophy of Physics: Space and Time: Space and Time (Princeton Foundations of Contemporary Philosophy) (p. 3). Princeton University Press. Kindle Edition. "Because it is a sphere, Aristotle's universe contains a geometrically privileged center, and Aristotle makes reference to that center in characterizing the natural motions of different sorts of matter. 'Upward', 'downward', and 'uniform circular motion' all are defined in terms of the center of the universe."
  27. ↑ Tim Maudlin (2012-07-22). Philosophy of Physics: Space and Time: Space and Time (Princeton Foundations of Contemporary Philosophy) (p. 4). Princeton University Press. Kindle Edition. "Aristotle adopts the concept of space, and the correlative concept of motion, that we all intuitively employ."
  28. ↑ Tim Maudlin (2012-07-22). Philosophy of Physics: Space and Time: Space and Time (Princeton Foundations of Contemporary Philosophy) (pp. 4–5). Princeton University Press. Kindle Edition. "Newtonian physics is implicit in his First Law of Motion: Law I : Every body perseveres in its state either of rest or of uniform motion in a straight line, except insofar as it is compelled to change its state by impressed forces. 1 This single law smashes the Aristotelian universe to smithereens."
  29. 1 2 Tim Maudlin (2012-07-22). Philosophy of Physics: Space and Time: Space and Time (Princeton Foundations of Contemporary Philosophy) (pp. 5). Princeton University Press. Kindle Edition.
  30. ↑ Tim Maudlin (2012-07-22). Philosophy of Physics: Space and Time: Space and Time (Princeton Foundations of Contemporary Philosophy) (pp. 9–10). Princeton University Press. Kindle Edition. "Newton believed in the existence of a spatial arena with the geometrical structure of E3. He believed that this infinite three-dimensional space exists at every moment of time. And he also believed something much more subtle and controversial, namely, that identically the same points of space persist through time."
  31. ↑ Tim Maudlin (2012-07-22). Philosophy of Physics: Space and Time: Space and Time (Princeton Foundations of Contemporary Philosophy) (p. 12). Princeton University Press. Kindle Edition. "...space must have a topology, an affine structure, and a metric; time must be one-dimensional with a topology and a metric; and, most importantly, the individual parts of space must persist through time.
  32. ↑ Ariew and Garber 117, Loemker §46, W II.5. On Leibniz and physics, see the chapter by Garber in Jolley (1995) and Wilson (1989).
  33. ↑ Rafael Ferraro (2007). Einstein's Space-Time: An Introduction to Special and General Relativity. Springer. p. 1. ISBN   978-0-387-69946-2.
  34. ↑ See H. G. Alexander, ed., The Leibniz-Clarke Correspondence , Manchester: Manchester University Press, pp. 25–26.

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