On Phase Spaces

 "Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. (Edward N Lorenz, "Deterministic Nonperiodic Flow", Journal of the Atmospheric Science 20, 1963)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"[…] physicists have come to appreciate a fourth kind of temporal behavior: deterministic chaos, which is aperiodic, just like random noise, but distinct from the latter because it is the result of deterministic equations. In dynamic systems such chaos is often characterized by small fractal dimensions because a chaotic process in phase space typically fills only a small part of the entire, energetically available space." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"When a system has more than one attractor, the points in phase space that are attracted to a particular attractor form the basin of attraction for that attractor. Each basin contains its attractor, but consists mostly of points that represent transient states. Two contiguous basins of attraction will be separated by a basin boundary." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Chaos appears in both dissipative and conservative systems, but there is a difference in its structure in the two types of systems. Conservative systems have no attractors. Initial conditions can give rise to periodic, quasiperiodic, or chaotic motion, but the chaotic motion, unlike that associated with dissipative systems, is not self-similar. In other words, if you magnify it, it does not give smaller copies of itself. A system that does exhibit self-similarity is called fractal. [...] The chaotic orbits in conservative systems are not fractal; they visit all regions of certain small sections of the phase space, and completely avoid other regions. If you magnify a region of the space, it is not self-similar." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"One of the reasons we deal with the pendulum is that it is easy to plot its motion in phase space. If the amplitude is small, it's a two-dimensional problem, so all we need to specify it completely is its position and its velocity. We can make a two-dimensional plot with one axis (the horizontal), position, and the other (the vertical), velocity." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing - seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing – seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Roughly spoken, bifurcation theory describes the way in which dynamical system changes due to a small perturbation of the system-parameters. A qualitative change in the phase space of the dynamical system occurs at a bifurcation point, that means that the system is structural unstable against a small perturbation in the parameter space and the dynamic structure of the system has changed due to this slight variation in the parameter space." (Holger I Meinhardt, "Cooperative Decision Making in Common Pool Situations", 2012)

"The impossibility of predicting which point in phase space the trajectory of the Lorenz attractor will pass through at a certain time, even though the system is governed by deterministic equations, is a common feature of all chaotic systems. However, this does not mean that chaos theory is not capable of any predictions. We can still make very accurate predictions, but they concern the qualitative features of the system’s behavior rather than the precise values of its variables at a particular time. The new mathematics thus represents the shift from quantity to quality that is characteristic of systems thinking in general. Whereas conventional mathematics deals with quantities and formulas, nonlinear dynamics deals with qualities and patterns." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"Bifurcation is a qualitative, topological change of a system’s phase space that occurs when some parameters are slightly varied across their critical thresholds. Bifurcations play important roles in many real-world systems as a switching mechanism. […] There are two categories of bifurcations. One is called a local bifurcation, which can be characterized by a change in the stability of equilibrium points. It is called local because it can be detected and analyzed only by using localized information around the equilibrium point. The other category is called a global bifurcation, which occurs when non-local features of the phase space, such as limit cycles (to be discussed later), collide with equilibrium points in a phase space. This type of bifurcation can’t be characterized just by using localized information around the equilibrium point."  (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

Curved Spaces

"The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon [...] when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms." (Jean-Baptiste-Joseph Fourier, "Théorie Analytique de la Chaleur", 1822)

"I hold: 1) that small portions of space are, in fact, of a nature analogous to little hills on a surface that is on the average fiat; namely, that the ordinary laws of geometry are not valid in them; 2) that this property of being curved or distorted is constantly being passed on from one portion of space to another after the manner of a wave; 3) that this variation of the curvature of space is what really happens in the phenomenon that we call the motion of matter, whether ponderable or ethereal; 4) that in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity." (William K Clifford, "On the Space Theory of Matter", [paper delivered before the Cambridge Philosophical Society, 1870)

[...] the time stream is curved helically in some higher dimension. In your case, a still further distortion brought two points of the coil into contact, and a sort of short circuit threw you into the higher curve. (Robert H Wilson, "A Flight Into Time", Wonder Stories, 1931)

"Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth. […] No force of gravitation is […] needed to impress curvature on world-lines; the curvature is inherent in the space […]" (James H Jeans," The Growth of Physical Science", 1947) 

"Space-time is curved in the neighborhood of material masses, but it is not clear whether the presence of matter causes the curvature of space-time or whether this curvature is itself responsible for the existence of matter." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

"The mathematicians and physics men Have their mythology; they work alongside the truth, Never touching it; their equations are false But the things work. Or, when gross error appears, They invent new ones; they drop the theory of waves In universal ether and imagine curved space." (Robinson Jeffers," The Beginning and the End and Other Poems, The Great Wound", 1963)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system's parameters, making a fluid more visous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system. (James Gleick, "Chaos: Making a New Science", 1987)

"Bodies like the earth are not made to move on curved orbits by a force called gravity; instead, they follow the nearest thing to a straight path in a curved space, which is called a geodesic. A geodesic is the shortest (or longest) path between two nearby points." (Stephen Hawking, "A Brief History of Time", 1988)

"Nonlinear systems (the graph of at least one relationship displays some curved feature) are notoriously more difficult to comprehend than linear systems, that is, they are more complex. Consequently they are also more difficult to control. This is exemplified by the volumes of elegant mathematics that have been developed in the search for optimal control of linear systems. (Robert L Flood & Ewart R Carson, "Dealing with Complexity: An introduction to the theory and application of systems", 1988)

"Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system's parameters, making a fluid more visous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system." (James Gleick, "Chaos: Making a New Science", 1987)

"String theory promises to take a further step beyond that taken by Einstein's picture of force subsumed within curved space and time geometry. Indeed, string theory contains Einstein's theory of gravitation within itself. Loops of string behave like the exchange particles of the gravitational forces, or 'gravitons' as they are called in the point-particle picture of things. But it has been argued that it must be possible to extract even the geometry of space and time from the characteristics of the strings and their topological properties. At present, it is not known how to do this and we merely content ourselves with understanding how strings behave when they sit in a background universe of space and time." (John D. Barrow, "Theories of Everything: The Quest for Ultimate Explanation", 1991)

"[...] if we consider a topological space instead of a plane, then the question of whether the coordinates axes in that space are curved or straight becomes meaningless. The way we choose coordinate systems is related to the way we observe the property of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Spacetime tells matter how to move; matter tells spacetime how to curve." (John A Wheeler, "Geons, Black Holes and Quantum Foam: A Life in Physics" , 1998)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Apparent Impossibilities that Are New Truths […] irrational numbers, imaginary numbers, points at infinity, curved space, ideals, and various types of infinity. These ideas seem impossible at first because our intuition cannot grasp them, but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Mathematical language is littered with pejorative and mystical terms - such as irrational, imaginary, surd, transcendental - that were once used to ridicule supposedly impossible objects. And these are just terms applied to numbers. Geometry also has many concepts that seem impossible to most people, such as the fourth dimension, finite universes, and curved space - yet geometers (and physicists) cannot do without them. Thus there is no doubt that mathematics flirts with the impossible, and seems to make progress by doing so." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we (or our computers) do concrete calculations." (Frank Wilczek, "The Lightness of Being: Mass, Ether, and the Unification of Forces", 2008)

On Topology V

"In mathematics, logic, linguistics, and other abstract disciplines, the systems are not assigned to objects. They are defined by an enumeration of the variables, their admissible values, and their algebraic, topological, grammatical, and other properties which, in the given case, determine the relations between the variables under consideration." (George Klir, "An approach to general systems theory", 1969)

"Because of its foundation in topology, catastrophe theory is qualitative, not quantitative. Just as geometry treated the properties of a triangle without regard to its size, so topology deals with properties that have no magnitude, for example, the property of a given point being inside or outside a closed curve or surface. This property is what topologists call 'invariant' -it does not change even when the curve is distorted. A topologist may work with seven-dimensional space, but he does not and cannot measure (in the ordinary sense) along any of those dimensions. The ability to classify and manipulate all types of form is achieved only by giving up concepts such as size, distance, and rate. So while catastrophe theory is well suited to describe and even to predict the shape of processes, its descriptions and predictions are not quantitative like those of theories built upon calculus. Instead, they are rather like maps without a scale: they tell us that there are mountains to the left, a river to the right, and a cliff somewhere ahead, but not how far away each is, or how large." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Homeomorphism is one of the basic concepts in topology. Homeomorphism, along with the whole topology, is in a sense the basis of spatial perception. When we look at an object, we see, say, a telephone receiver or a ring-shaped roll and first of all pay attention to the geometrical shape (although we do not concentrate on it specially) - an oblong figure thickened at the ends or a round rim with a large hole in the middle. Even if we deliberately concentrate on the shape of the object and forget about its practical application, we do not yet 'see' the essence of the shape. The point is that oblongness, roundness, etc. are metric properties of the object. The topology of the form lies 'beyond them'." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Since geometry is the mathematical idealization of space, a natural way to organize its study is by dimension. First we have points, objects of dimension O. Then come lines and curves, which are one-dimensional objects, followed by two-dimensional surfaces, and so on. A collection of such objects from a given dimension forms what mathematicians call a 'space'. And if there is some notion enabling us to say when two objects are 'nearby' in such a space, then it's called a topological space." (John L Casti, "Five Golden Rules", 1995)

"One of the basic tasks of topology is to learn to distinguish nonhomeomorphic figures. To this end one introduces the class of invariant quantities that do not change under homeomorphic transformations of a given figure. The study of the invariance of topological spaces is connected with the solution of a whole series of complex questions: Can one describe a class of invariants of a given manifold? Is there a set of integral invariants that fully characterizes the topological type of a manifold? and so forth." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Topology studies those characteristics of figures which are preserved under a certain class of continuous transformations. Imagine two figures, a square and a circular disk, made of rubber. Deformations can convert the square into the disk, but without tearing the figure it is impossible to convert the disk by any deformation into an annulus. In topology, this intuitively obvious distinction is formalized." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"[...] there is no area of mathematics where thinking abstractly has paid more handsome dividends than in topology, the study of those properties of geometrical objects that remain unchanged when we deform or distort them in a continuous fashion without tearing, cutting, or breaking them." (John L Casti, "Five Golden Rules", 1995)

"At first, topology can seem like an unusually imprecise branch of mathematics. It’s the study of squishy play-dough shapes capable of bending, stretching and compressing without limit. But topologists do have some restrictions: They cannot create or destroy holes within shapes. […] While this might seem like a far cry from the rigors of algebra, a powerful idea called homology helps mathematicians connect these two worlds. […] homology infers an object’s holes from its boundaries, a more precise mathematical concept. To study the holes in an object, mathematicians only need information about its boundaries." (Kelsey Houston-Edwards, "How Mathematicians Use Homology to Make Sense of Topology", Quanta Magazine, 2021) [source]

"In geometry, shapes like circles and polyhedra are rigid objects; the tools of the trade are lengths, angles and areas. But in topology, shapes are flexible things, as if made from rubber. A topologist is free to stretch and twist a shape. Even cutting and gluing are allowed, as long as the cut is precisely reglued. A sphere and a cube are distinct geometric objects, but to a topologist, they’re indistinguishable." (David E Richeson, "Topology 101: The Hole Truth", 2021) [source]

Reality (From Fiction to Science-Fiction)

"The horror of the Same Old Thing is [...] an endless source of heresies in religion, folly in counsel, infidelity in marriage, and inconstancy in friendship. The humans live in time, and experience reality successively. To experience much of it, therefore, they must experience many different things; in other words, they must experience change. And since they need change, the Enemy (being a hedonist at heart) has made change pleasurable to them." (C. S. Lewis, "The Screwtape Letters", 1942)

"It is the normal lot of people who must live this life [in space] to be - by terrestrial standards—insane. Insanity under such conditions is a useful and logical defense mechanism, an invaluable and salutary retreat from reality." (Charles L Harness, "The Paradox Men", 1949)

"It seemed as if the structure of reality trembled for an instant, and that behind the world of the senses he caught a glimpse of another and totally different universe [...]" (Arthur C Clarke, "The City and the Stars", 1956)

"No live organism can continue for long to exist sanely under conditions of absolute reality; even larks and katydids are supposed, by some, to dream." (Shirley Jackson, "The Haunting of Hill House", 1959)

"Reality? It is only the illusion we can agree upon." (James Gunn, "The Joy Makers", 1961)

"When dreams become more important than reality, you give up travel, building, creating." (Gene Roddenberry, "Star Trek" ["The Menagerie"], 1966)

"The whole of modern so-called existence is an attempt to deny reality insofar as it exists."John Brunner, "Stand on Zanzibar", 1968)

"Reality, to me, is not so much something that you perceive, but something you make." (Philip K Dick, "The Android and the Human", 1972)

"The theory changes the reality it describes." (Philip K Dick, "Flow My Tears the Policeman Said", 1974)

"We exist in time. Time is what binds molecules to make your brown eyes, your yellow hair, your thick fingers. Time changes the structures, alters hair or fingers, dims the eyes, immutably mutating reality. Time, itself unchanging, is the cosmic glue, the universal antisolvent that holds our worlds together." (Marta Randall, "Secret Rider", 1976)

"There was no substitute for reality; one should beware of imitations." (Arthur C Clarke, "The Fountains of Paradise", 1979)

"Reality is that which when you stop believing in it, it doesn’t go away." (Philip K Dick, "Valis", 1981)

"The basic tool for the manipulation of reality is the manipulation of words. If you can control the meaning of words, you can control the people who must use the words." (Philip K Dick, "How to Build a Universe That Doesn’t Fall Apart Two Days Later", 1985)

"It is now known to science that there are many more dimensions than the classical four. Scientists say that these don’t normally impinge on the world because the extra dimensions are very small and curve in on themselves, and that since reality is fractal most of it is tucked inside itself. This means either that the universe is more full of wonders than we can hope to understand or, more probably, that scientists make things up as they go along." (Terry Pratchett, Pyramids, 1989)

"It is always hard when reality intrudes on belief." (Alan D Foster, "Cyber Way", 1990)

"The dreams of people are in the machines, a planet network of active imaginations hooked into their made-up, make-believe worlds. Artificial reality is taking over; it has its own children." (Storm Constantine, "Immaculate", 1991)

On Continuity VII (Spacetime)

"Time and space are divided into the same and equal divisions. Wherefore also, Zeno’s argument, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time, is fallacious. For there are two senses in which the term ‘infinte’ is applied both to length and to time and in fact to all continuous things: either in regard to divisibility or in regard to number. Now it is not possible to touch things infinite as to number in a finite time, but it is possible to touch things infinite in regard to divisibility; for time itself is also infinite in this sense."  (Aristotle, "Physics", cca. 350 BC)

"Time with its continuity logically involves some other kind of continuity than its own. Time, as the universal form of change, cannot exist unless there is something to undergo change, and to undergo a change continuous in time, there must be a continuity of changeable qualities." (Charles S Peirce, "The Law of Mind", 1892)

"Motion consists merely in the occupation of different places at different times, subject to continuity. [...] There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number, which is the limit of a certain set of quotients." (Bertrand Russell's, "Principles of Mathematics", 1903)

"The discovery of Minkowski […] is to be found […] in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude, v-1*ct, proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space-coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry." (Albert Einstein,"Relativity: The Special and General Theory", 1920)

"The scene of action of reality is not a three-dimensional Euclidean space but rather a four-dimensional world, in which space and time are linked together indissolubly. However deep the chasm may be that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavors to crystallize out of direct experience. It is a four-dimensional continuum, which is neither 'time' nor 'space'. Only the consciousness that passes on in one portion of this world experiences the detached piece which comes to meet it and passes behind it as history, that is, as a process that is going forward in time and takes place in space." (Hermann Weyl, "Space, Time, Matter", 1922)

"In classical science, it was strange to find that action [...] should yet present the artificial aspect of an energy in space multiplied by a duration. As soon, however, as we realise that the fundamental continuum of the universe is one of space-time and not one of separate space and time, the reason for the importance of the seemingly artificial combination of space with time in the expression for the action receives a very simple explanation. Henceforth, action is no longer energy in a volume of space multiplied by a duration; it is simply energy in a volume of the world, that is to say, in a volume of four-dimensional space-time." (Aram D'Abro, "The Evolution of Scientific Thought from Newton to Einstein", 1927)

"[…] evolution is only one aspect of the order of nature, of the relations of cause and effect, of continuity of space and time, which pervade the universe and enable us to comprehend its simplicity of plan, its complexity of detail." (William D Matthew, Natural History Vol. 25 (2), 1925)

"[…] the universe is not a rigid and inimitable edifice where independent matter is housed in independent space and time; it is an amorphous continuum, without any fixed architecture, plastic and variable, constantly subject to change and distortion. Wherever there is matter and motion, the continuum is disturbed. Just as a fi sh swimming in the sea agitates the water around it, so a star, a comet, or a galaxy distorts the geometry of the spacetime through which it moves." (Lincoln Barnett, "The Universe and Dr. Einstein", 1948)

"The whole fabric of the space-time continuum is not merely curved, it is in fact totally bent." (Douglas N Adams, "The Restaurant at the End of the Universe", 1980)

"Time goes forward because energy itself is always moving from an available to an unavailable state. Our consciousness is continually recording the entropy change in the world around us. [...] we experience the passage of time by the succession of one event after another. And every time an event occurs anywhere in this world energy is expended and the overall entropy is increased. To say the world is running out of time then, to say the world is running out of usable energy. In the words of Sir Arthur Eddington, 'Entropy is time's arrow'." (Jeremy Rifkin, "Entropy", 1980)

"The view of science is that all processes ultimately run down, but entropy is maximized only in some far, far away future. The idea of entropy makes an assumption that the laws of the space-time continuum are infinitely and linearly extendable into the future. In the spiral time scheme of the timewave this assumption is not made. Rather, final time means passing out of one set of laws that are conditioning existence and into another radically different set of laws. The universe is seen as a series of compartmentalized eras or epochs whose laws are quite different from one another, with transitions from one epoch to another occurring with unexpected suddenness." (Terence McKenna, "True Hallucinations", 1989)

"Symmetry is bound up in many of the deepest patterns of Nature, and nowadays it is fundamental to our scientific understanding of the universe. Conservation principles, such as those for energy or momentum, express a symmetry that (we believe) is possessed by the entire space-time continuum: the laws of physics are the same everywhere." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Continuity is only a mathematical technique for approximating very finely grained things. The world is subtly discrete, not continuous." (Carlo Rovelli, "The Order of Time", 2018)

"It is not possible to think of duration as continuous. We must think of it as discontinuous: not as something that flows uniformly but as something that in a certain sense jumps, kangaroo-like, from one value to another. In other words, a minimum interval of time exists. Below this, the notion of time does not exist - even in its most basic meaning." (Carlo Rovelli, "The Order of Time", 2018)

"Space. The continual becoming: invisible fountain from which all rhythms flow and to which they must pass. Beyond time or infinity." (Frank L Wright)

On Complex Numbers XVI

"When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power."  (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

 "√-1 is take for granted today. No serious mathematician would deny that it is a number. Yet it took centuries for √-1 to be officially admitted to the pantheon of numbers. For almost three centuries, it was controversial; mathematicians didn't know what to make of it; many of them worked with it successfully without admitting its existence. […] Primarily for cognitive reasons. Mathematicians simply could not make it fit their idea of what a number was supposed to be. A number was supposed to be a magnitude. √-1 is not a magnitude comparable to the magnitudes of real numbers. No tree can be √-1 units high. You cannot owe someone √-1 dollars. Numbers were supposed to be linearly ordered. √-1 is not linearly ordered with respect to other numbers." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"From a formal perspective, much about complex numbers and arithmetic seems arbitrary. From a purely algebraic point of view, i arises as a solution to the equation x^2+1=0. There is nothing geometric about this - no complex plane at all. Yet in the complex plane, the i-axis is 90° from the x-axis. Why? Complex numbers in the complex plane add like vectors. Why? Complex numbers have a weird rule of multiplication […]" (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"[…] i is not a real number-not ordered anywhere relative to the real numbers! In other words, it does not even have the central property of ‘numbers’, indicating a magnitude that can be linearly compared to all other magnitudes. You can see why i has been called imaginary. It has almost none of the properties of the small natural numbers-not subitizability, not groupability, and not even relative magnitude. If i is to be a number, it is a number only by virtue of closure and the laws of arithmetic." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"The complex plane is just the 90° rotation plane-the rotation plane with the structure imposed by the 90° Rotation metaphor added to it. Multiplication by i is 'just' rotation by 90°. This is not arbitrary; it makes sense. Multiplication by-1 is rotation by 180°. A rotation of 180° is the result of two 90° rotations. Since i times i is -1, it makes sense that multiplication by i should be a rotation by 90°, since two of them yield a rotation by 180°, which is multiplication by -1." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"Negative numbers posed some of the same quandaries that the imaginary numbers did to Renaissance mathematicians - they didn’t seem to correspond to quantities associated with physical objects or geometrical figures. But they proved less conceptually challenging than the imaginaries. For instance, negative numbers can be thought of as monetary debts, providing a readily grasped way to make sense of them." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Raising e to an imaginary-number power can be pictured as a rotation operation in the complex plane. Applying this interpretation to e raised to the 'i times π' power means that Euler’s formula can be pictured in geometric terms as modeling a half-circle rotation." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The fact that multiplying positive 4i times positive 4i yields negative 16 seems like saying that the friend of my friend is my enemy. Which in turn suggests that bad things would happen if i and its offspring were granted citizenship in the number world. Unlike real numbers, which always feel friendly toward the friends of their friends, the i-things would plainly be subject to insane fits of jealousy, causing them to treat numbers that cozy up to their friends as threats. That might cause a general breakdown of numerical civility." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Basis real and imaginary numbers have eternal and necessary reality. Complex numbers do not. They are temporal and contingent in the sense that for complex numbers to exist, we first have to carry out an operation: adding basis real and imaginary numbers together. Complex numbers therefore do not exist in their own right. They are constructed. They are derived. Symmetry breaking is exactly where constructed numbers come into existence. The very act of adding a sine wave to a cosine wave is the sufficient condition to create a broken symmetry: a complex number. The 'Big Bang', mathematically, is simply where a perfect array of basis sine and cosine waves start entering into linear combinations, creating a chain reaction, an 'explosion', of complex numbers - which corresponds to the 'physical' universe." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

On Manifolds II (Geometry II)

"In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent; the former belongs to the extent relations, the latter to the measure-relations. That space is an unbounded threefold manifoldness, is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed and the possible positions of a sought object are constructed, and which by these applications is forever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesies starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)

"In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system." (Charles P Steinmetz, [New York Times interview] 1911)

"That branch of mathematics which deals with the continuity properties of two- (and more) dimensional manifolds is called analysis situs or topology. […] Two manifolds must be regarded as equivalent in the topological sense if they can be mapped point for point in a reversibly neighborhood-true (topological) fashion on each other." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"In her manifold opportunities Nature has thus helped man to polish the mirror of [man’s] mind, and the process continues. Nature still supplies us with abundance of brain-stretching theoretical puzzles and we eagerly tackle them; there are more worlds to conquer and we do not let the sword sleep in our hand; but how does it stand with feeling? Nature is beautiful, gladdening, awesome, mysterious, wonderful, as ever, but do we feel it as our forefathers did?" (Sir John A Thomson, "The System of Animate Nature", 1920)

"An 'empty world', i. e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy." (Howard P Robertson, "On Relativistic Cosmology", 1928)

"Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but 'logical relations' or 'artificial manifolds'. They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"We must [...] maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

On Manifolds V (Geometry III)

"Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physicists, its effect in the field of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential to guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension it does not lose any of its peculiar character as time. [...] Musical tones can be ordered according to volume and pitch and are thus brought into a two dimensional manifold. Similarly colors can be determined by the three basic colors red, green and blue… Such an ordering does not change either tones or colors; it is merely a mathematical expression of something that we have known and visualized for a long time. Our schematization of time as a fourth dimension therefore does not imply any changes in the conception of time. [...] the space of visualization is only one of many possible forms that add content to the conceptual frame. We would therefore not call the representation of the tone manifold by a plane the visual representation of the two dimensional tone manifold." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"The sequence of numbers which grows beyond any stage already reached by passing to the next number is a manifold of possibilities open towards infinity, it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties […]" (Hermann Weyl, "Mathematics and Logic", 1946)

"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)

"The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves." (Shiing-Shen Chern, "Differential geometry, its past and its future", 1970)

"[...] a manifold is a set M on which 'nearness' is introduced (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"It is commonly said that the study of manifolds is, in general, the study of the generalization of the concept of surfaces. To some extent, this is true. However, defining it that way can lead to overshadowing by 'figures' such as geometrical surfaces." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"One could also question whether we are looking for a single overarching mathematical structure or a combination of different complementary points of view. Does a fundamental theory of Nature have a global definition, or do we have to work with a series of local definitions, like the charts and maps of a manifold, that describe physics in various 'duality frames'. At present string theory is very much formulated in the last kind of way." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann)

On Continuity II (Topology)

"Things are called continuous when the touching limits of each become one and the same and are contained in each other. Continuity is impossible if these extremities are two. […] Continuity belongs to things that naturally in virtue of their mutual contact form a unity. And in whatever way that which holds them together is one, so too will the whole be one."(Aristotle, "Physics", cca. 350 BC)

"When what surrounds, then, is not separate from the thing, but is in continuity with it, the thing is said to be in what surrounds it, not in the sense of in place, but as a part in a whole. But when the thing is separate or in contact, it is immediately ‘in’ the inner surface of the surrounding body, and this surface is neither a part of what is in it nor yet greater than its extension, but equal to it; for the extremities of things which touch are coincident." (Aristotle, "Physics", cca. 350 BC)

"I hold: 1) that small portions of space are, in fact, of a nature analogous to little hills on a surface that is on the average fiat; namely, that the ordinary laws of geometry are not valid in them; 2) that this property of being curved or distorted is constantly being passed on from one portion of space to another after the manner of a wave; 3) that this variation of the curvature of space is what really happens in the phenomenon that we call the motion of matter, whether ponderable or ethereal; 4) that in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity." (William K Clifford, "On the Space Theory of Matter", [paper delivered before the Cambridge Philosophical Society, 1870)

"That branch of mathematics which deals with the continuity properties of two- (and more) dimensional manifolds is called analysis situs or topology. […] Two manifolds must be regarded as equivalent in the topological sense if they can be mapped point for point in a reversibly neighborhood-true (topological) fashion on each other." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"In topology we are concerned with geometrical facts that do not even involve the concepts of a straight line or plane but only the continuous connectiveness between points of a figure." (David Hilbert, "Geometry and Imagination", 1952)

"General or point set topology can be thought of as the abstract study of the ideas of nearness and continuity. This is done in the first place by picking out in elementary geometry those properties of nearness that seem to be fundamental and taking them as axioms." (Andrew H Wallace, "Differential Topology: First Steps", 1968)

"The major strength of catastrophe theory is to provide a qualitative topology of the general structure of discontinuities. Its major weakness is that it frequently is not associated with specific models allowing precise quantitative prediction, although such are possible in principle." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)

"[...] if we consider a topological space instead of a plane, then the question of whether the coordinates axes in that space are curved or straight becomes meaningless. The way we choose coordinate systems is related to the way we observe the property of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"The property of smoothness includes the property of continuity. The notion of a topological space was born from the development of abstract algebra as a universal notion for the property of continuity." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"A continuous function preserves closeness of points. A discontinuous function maps arbitrarily close points to points that are not close. The precise definition of continuity involves the relation of distance between pairs of points. […] continuity, a property of functions that allows stretching, shrinking, and folding, but preserves the closeness relation among points." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"Topology is the study of geometric objects as they are transformed by continuous deformations. To a topologist the general shape of the objects is of more importance than distance, size, or angle." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"[…] topology is the study of those properties of geometric objects which remain unchanged under bi-uniform and bi-continuous transformations. Such transformations can be thought of as bending, stretching, twisting or compressing or any combination of these." (Lokenath Debnath, "The Legacy of Leonhard Euler - A Tricentennial Tribute", 2010)

"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"Topology is an elastic version of geometry that retains the idea of continuity but relaxes rigid metric notions of distance." (Samuel Eilenberg)

On Spacetime (From Fiction to Science-Fiction)

"Some people who talk about the Fourth Dimension do not know they mean it. It is only another way of looking at Time. There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it." (Herbert G Wells, "The Time Machine: An Invention", 1895)

"[...] the time stream is curved helically in some higher dimension. In your case, a still further distortion brought two points of the coil into contact, and a sort of short circuit threw you into the higher curve." (Robert H Wilson, "A Flight Into Time", Wonder Stories, 1931)

"Man has natural three-dimensional limits, and he also has four-dimensional ones, considering time as an extension. When he reaches those limits, he ceases to grow and mature, and forms rigidly within the mold of those limiting walls. It is stasis, which is retrogression unless all else stands still as well. A man who reaches his limits is tending toward subhumanity. Only when he becomes superhuman in time and space can immortality become practical." (Henry Kuttner & Catherine L Moore [aka Lewis Padgett], "Time Enough", 1946)

"Space and Time aren’t real, apart. And they aren’t really different. They fade one into the other all around us." (Jack Williamson, "The Legion of Space", 1947)

"There are and have been worlds and cultures without end, each nursing the proud illusion that it is unique in space and time. There have been men without number suffering from the same megalomania; men who imagined themselves unique, irreplaceable, irreproducible. There will be more [...] more plus infinity." (Alfred Bester, "The Demolished Man", 1953)

"There is a fifth dimension beyond that which is known to Man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man’s fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call ... The Twilight Zone." (Rod Serling, "The Twilight Zone" [TV series] 1959)

"The present, as every schoolboy knows, is only the surface of the space-time sea, and a living spacewhale can dive beneath this surface and sojourn in times past, can return, if it so desires, to the primordial moment when the cosmos was born." (Robert F Young, "Starscape with Frieze of Dreams", 1970)

"An infinity of universes swim in superspace, all passing through their own cycles of birth and death; some are novel, others repetitious; some produce macrolife, others do not; still others are lifeless. In time, macrolife will attempt to reach out from its cycles to other space-time bubbles, perhaps even to past cycles, which leave their echoes in superspace, and might be reached. In all these ambitions, only the ultimate pattern of development is unknown, drawing macrolife toward some future transformation still beyond its view. There are times when the oldest macrolife senses that vaster intelligences are peering in at it from some great beyond [...]" (George Zebrowski, "Macrolife", 1979)

"The whole fabric of the space-time continuum is not merely curved, it is in fact totally bent." (Douglas N Adams, "The Restaurant at the End of the Universe", 1980)

"Time and space were themselves players, vast lands engulfing the figures, a weave of future and past. There was no riverrun of years. The abiding loops of causality ran both forward and back. The timescape rippled with waves, roiled and flexed, a great beast in the dark sea." (Gregory Benford, "Timescape", 1980)

"The dimension of the imagination is much more complex than those of time and space, which are very junior dimensions indeed." (Terry Pratchett, "The Colour of Magic", 1983)

"History too has an inertia. In the four dimensions of spacetime, particles (or events) have directionality; mathematicians, trying to show this, draw what they call 'world lines' on graphs. In human affairs, individual world lines form a thick tangle, curling out of the darkness of prehistory and stretching through time: a cable the size of Earth itself, spiraling round the sun on a long curved course. That cable of tangled world lines is history. Seeing where it has been, it is clear where it is going - it is a matter of simple extrapolation." (Kim S Robinson, "Red Mars", 1992)

On Spacetime (2000-2019)

"Neither space nor time has any existence outside the system of evolving relationships that comprises the universe. Physicists refer to this feature of general relativity as background independence." (Lee Smolin, "Three Roads to Quantum Gravity", 2000)

"The relational picture of space and time has implications that are as radical as those of natural selection, not only for science but for our perspective on who we are and how we came to exist in this evolving universe of relations." (Lee Smolin, "Three Roads to Quantum Gravity", 2000)

"Time is described only in terms of change in the network of relationships that describes space." (Lee Smolin, "Three Roads to Quantum Gravity", 2000)

"In string theory one studies strings moving in a fixed classical spacetime. […] what we call a background-dependent approach. […] One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves - at the rate predicted by general relativity to the […] accuracy of eleven decimal place - tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up." (Lee Smolin, "Loop Quantum Gravity", The New Humanists: Science at the Edge, 2003)

"Spacetime […] turns out to be discrete, described by a structure called spin foam." (Lee Smolin, “The New Humanists: Science at the Edge”, 2003)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Space and time capture the imagination like no other scientific subject. For good reason. They form the arena of reality, the very fabric of the cosmos." (Brian Greene, "The Fabric of the Cosmos", 2004)

"The space and time of the universe that we humans inhabit contain symmetries. These are almost obvious yet subtle, even mysterious. Space and time form the stage upon which the dynamics - that is, the motion and interactions of the physical systems, atoms, atomic nuclei, protozoa, and people - are played out. The symmetries of space and time control the dynamics of the physical interactions of matter." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event." (Friedel Weinert," The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries", 2005) 

"Mathematicians call the infinite curvature limit of spacetime a singularity. In this picture, then, the big bang emerges from a singularity. The best way to think about singularities is as boundaries or edges of spacetime. In this respect they are not, technically, part of spacetime itself." (Paul Davies," Cosmic Jackpot: Why Our Universe Is Just Right for Life", 2007) 

"We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we (or our computers) do concrete calculations." (Frank Wilczek, "The Lightness of Being: Mass, Ether, and the Unification of Forces", 2008)

"The hypothesis underlying all approaches to the landscape is that there is a cosmological setting in which different regions or epochs of the universe can have different effective laws. This implies the existence of spacetime regions not directly observable […] These regions must either be in the past of our big bang, or far enough away from us to be causally unrelated." (Lee Smolin," A perspective on the landscape problem", 2012)

"One of the most crucial developments in theoretical physics was the move from theories dependent on fixed, non-dynamical background space-time structures to background-independent theories, in which the space-time structures themselves are dynamical entities. [...] Even today, many physicists and philosophers do not fully understand the significance of this development, let alone accept it in practice. One must assume that, in an empty region of space-time, the points have no inherent individuating properties - nor indeed are there any spatio-temporal relations between them - that do not depend on the presence of some metric tensor field. [...] Thus, general relativity became the first fully dynamical, background- independent space-time theory." (John Stachel, "The Hole Argument", 2014)

"For a moving object, time contracts. Not only is there no single time for different places - there is not even a single time for any particular place. A duration can be associated only with the movement of something, with a given trajectory." (Carlo Rovelli, "The Order of Time", 2018)

"Granularity is ubiquitous in nature: light is made of photons, the particles of light. The energy of electrons in atoms can acquire only certain values and not others. The purest air is granular, and so, too, is the densest matter. Once it is understood that Newton’s space and time are physical entities like all others, it is natural to suppose that they are also granular. Theory confirms this idea: loop quantum gravity predicts that elementary temporal leaps are small, but finite." (Carlo Rovelli, "The Order of Time", 2018

"Spacetime is a physical object like an electron. It, too, fluctuates. It, too, can be in a 'superposition' of different configurations." (Carlo Rovelli, "The Order of Time", 2018)

"The basic units in terms of which we comprehend the world are not located in some specific point in space. They are - if they are at all - in a where but also in a when. They are spatially but also temporally delimited: they are events." (Carlo Rovelli, "The Order of Time", 2018)

Hermann Minkowski - Collected Quotes

"By laying down the relativity postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of energy (and statements concerning the form of the energy) alone." (Hermann Minkowski, "The Fundamental Equations for Electromagnetic Processes in Moving Bodies", 1907)

"Integers are the fountainhead of all mathematics." (Hermann Minkowski, "Diophantische Approximationen: eine einfuhrung in die zahlen Theorie", 1907)

"The equations of Newton's mechanics exhibit a two-fold invariance. Their form remains unaltered, firstly, if we subject the underlying system of spatial coordinates to any arbitrary change of position ; secondly, if we change its state of motion, namely, by imparting to it any uniform translatory motion ; furthermore, the zero point of time is given no part to play. We are accustomed to look upon the axioms of geometry as finished with, when we feel ripe for the axioms of mechanics, and for that reason the two invariances are probably rarely mentioned in the same breath. Each of them by itself signifies, for the differential equations of mechanics, a certain group of transformations. The existence of the first group is looked upon as a fundamental characteristic of space. The second group is preferably treated with disdain, so that we with un-troubled minds may overcome the difficulty of never being able to decide, from physical phenomena, whether space, which is supposed to be stationary, may not be after all in a state of uniform translation. Thus the two groups, side by side, lead their lives entirely apart. Their utterly heterogeneous character may have discouraged any attempt to compound them. But it is precisely when they are compounded that the complete group, as a whole, gives us to think." (Hermann Minkowski, "Space and Time" ["Raum und Zeit"], [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. But I still respect the dogma that both space and time have independent significance. A point of space at a point of time, that is a system of values x, y, z, t, I will call a world-point." (Hermann Minkowski, "Space and Time" ["Raum und Zeit"], [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." (Hermann Minkowski, "Space and Time" ["Raum und Zeit"], [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The whole world appears resolved into such world-lines. And I should like to say beforehand that, according to my opinion, it would be possible for the physical laws to find their fullest expression as correlations of these world-lines." (Hermann Minkowski, "Space and Time" ["Raum und Zeit"], [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. But I still respect the dogma that both space and time have independent significance. A point of space at a point of time, that is a system of values x, y, z, t, I will call a world-point." (Hermann Minkowski, "The Principle of Relativity: A Collection of Original Memoirs on the, Special and General Theory of Relativity, Space and Time", Nature vol. 113, 1924)

"The rigid electron is in my view a monster in relation to Maxwell's equations, whose innermost harmony is the principle of relativity [...] the rigid electron is no working hypothesis, but a working hindrance. Approaching Maxwell's equations with the concept of the rigid electron seems to me the same thing as going to a concert with your ears stopped up with cotton wool. We must admire the courage and the power of the school of the rigid electron which leaps across the widest mathematical hurdles with fabulous hypotheses, with the hope to land safely over there on experimental-physical ground." (Hermann Minkowski [in Arthur I Miller, "Albert Einstein's Special Theory of Relativity", 1981)

Ernst Mach - Collected Quotes

"The aim of natural science is to obtain connections among phenomena. Theories, however, are like withered leaves, which drop off after having enabled the organism of science to breathe for a time." (Ernst Mach, "Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit", 1871)

"Historical investigation not only promotes the understanding of that which now is, but also brings new possibilities before us, by showing that which exists to be in great measure conventional and accidental. From the higher point of view at which different paths of thought converge we may look about us with freer vision and discover routes before unknown." (Ernst Mach, "The Science of Mechanics", 1883)

"[…] not only a knowledge of the ideas that have been accepted and cultivated by subsequent teachers is necessary for the historical understanding of a science, but also that the rejected and transient thoughts of the inquirers, nay even apparently erroneous notions, may be very important and very instructive. The historical investigation of the development of a science is most needful, lest the principles treasured up in it become a system of half-understood prescripts, or worse, a system of prejudices." (Ernst Mach, "The Science of Mechanics", 1883)

"Purely mechanical phenomena do not exist […] are abstractions, made, either intentionally or from necessity, for facilitating our comprehension of things. The science of mechanics does not comprise the foundations, no, nor even a part of the world, but only an aspect of it." (Ernst Mach, "The Science of Mechanics", 1883)

"A person who knew the world only through the theatre, if brought behind the scenes and permitted to view the mechanism of the stage’s action, might possibly believe that the real world also was in need of a machine-room, and that if this were once thoroughly explored, we should know all. Similarly, we, too, should beware lest the intellectual machinery, employed in the representation of the world on the stage of thought, be regarded as the basis of the real world." (Ernst Mach, "The Science of Mechanics; a Critical and Historical Account of Its Development", 1893) 

"In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. […] One condition, therefore, though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equilibrium, is thus applied by symmetry." (Ernst Mach, "The Science of Mechanics: A Critical and Historical Account of Its Development", 1893)

"Properly speaking the world is not composed of 'things' as its elements, but colors, tones, pressures, spaces, times, in short what we ordinarily call individual sensations." (Ernst Mach, "The Science of Mechanics", 1893)

"That branch of physics which is at once the oldest and the simplest and which is therefore treated as introductory to other departments of this science, is concerned with the motions and equilibrium of masses. It bears the name of mechanics." (Ernst Mach, "The Science of Mechanics: A Critical and Historical Account of Its Development", 1893)

"The atomic theory plays a part in physics similar to that of certain auxiliary concepts in mathematics: it is a mathematical model for facilitating the mental reproduction of facts. Although we represent vibrations by the harmonic formula, the phenomena of cooling by exponentials, falls by squares of time, etc, no one would fancy that vibrations in themselves have anything to do with circular functions, or the motion of falling bodies with squares." (Ernst Mach, "The Science of Mechanic", 1893)

"In algebra we perform, as far as possible, all numerical operations which are identical in form once for all, so that only a remnant of work is left for the individual case. The use of the signs of algebra and analysis, which are merely symbols of operations to be performed, is due to the observation that we can materially disburden the mind in this way and spare its powers for more important and more difficult duties, by imposing all mechanical operations upon the hand." (Ernst Mach, "The Economical Nature of Physical Enquiry", Popular Scientific Lectures, 1895)

"Strange as it may sound, the power of mathematics rests upon its evasion of all unnecessary thought and on its wonderful saving of mental operation. Even those arrangement-signs which we call numbers are a system of marvelous simplicity and economy. When we employ the multiplication-table in multiplying numbers of several places, and so use the results of old operations of counting instead of performing the whole of each operation anew; when we consult our table of logarithms, replacing and saving thus new calculations by old ones already performed; when we employ determinants instead of always beginning afresh the solution of a system of equations; when we resolve new integral expressions into familiar old integrals; we see in this simply a feeble reflexion of the intellectual activity of a Lagrange or a Cauchy, who, with the keen discernment of a great military commander, substituted new operations for whole hosts of old ones. No one will dispute me when I say that the most elementary as well as the highest mathematics are economically-ordered experiences of counting, put in forms ready for use." (Ernst Mach, "Popular Scientific Lectures", 1895)

"The aim of research is the discovery of the equations which subsist between the elements of phenomena." (Mach Ernst, 1898)

"[…] scientific research is somewhat like unraveling complicated tangles of strings, in which luck is almost as vital as skill and accurate observation." (Ernst Mach, "Knowledge and Error: Sketches on the Psychology of Enquiry", 1905)

"A symbolical representation of a method of calculation has the same significance for a mathematician as a model or a visualisable working hypothesis has for a physicist. The symbol, the model, the hypothesis runs parallel with the thing to be represented. But the parallelism may extend farther, or be extended farther, than was originally intended on the adoption of the symbol. Since the thing represented and the device representing are after all different, what would be concealed in the one is apparent in the other." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)

"Geometry, accordingly, consists of the application of mathematics to experiences concerning space. Like mathematical physics, it can become an exact deductive science only on the condition of its representing the objects of experience by means of schematizing and idealizing concepts." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)

"Physiological, and particularly visual, space appears as a distortion of 'geometrical space when derived from the metrical data of geometrical space. But the properties of continuity and threefold manifoldness are preserved in such a transformation, and all the consequences of these properties may be derived without recourse to physical experience, by our representative powers solely." (Ernst Mach, "Space and Geometry: In the Light of physiological, phycological and physical inquiry", 1906)

"Physics shares with mathematics the advantages of succinct description and of brief, compendious definition, which precludes confusion, even in ideas where, with no apparent burdening of the brain, hosts of others are contained." (Ernst Mach)

"Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses." (Ernst Mach)

William K Clifford - Collected Quotes

"I hold: 1) that small portions of space are, in fact, of a nature analogous to little hills on a surface that is on the average fiat; namely, that the ordinary laws of geometry are not valid in them; 2) that this property of being curved or distorted is constantly being passed on from one portion of space to another after the manner of a wave; 3) that this variation of the curvature of space is what really happens in the phenomenon that we call the motion of matter, whether ponderable or ethereal; 4) that in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity." (William K Clifford, "On the Space Theory of Matter", [paper delivered before the Cambridge Philosophical Society, 1870) 

"Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space." (William K Clifford, "On the Space Theory of Matter", [paper delivered before the Cambridge Philosophical Society, 1870) 

"Causation is defined by some modern philosophers as unconditional uniformity of succession, e.g., existence of fire follows from putting a lighted match to the fuel." (William K Clifford, "Energy and Force", 1873)

"Remember that [scientific thought] is the guide of action; that the truth which it arrives at is not that which we can ideally contemplate without error, but that which we may act upon without fear; and you cannot fail to see that scientific thought is not an accompaniment or condition of human progress, but human progress itself." (William K Clifford, "Lectures and Essays", 1879)

"[...] scientific thought does not mean thought about scientific subjects with long names. There are no scientific subjects. The subject of science is the human universe; that is to say, everything that is, or has been, or may be related to man." (William K Clifford, "Lectures and Essays", 1879)

"The aim of scientific thought, then, is to apply past experience to new circumstances; the instrument is an observed uniformity in the course of events. By the use of this instrument it gives us information transcending our experience, it enables us to infer things that we have not seen from things that we have seen; and the evidence for the truth of that information depends on our supposing that the uniformity holds good beyond our experience." (William K Clifford, "Lectures and Essays", 1879)

"The scientific discovery appears first as the hypothesis of an analogy; and science tends to become independent of the hypothesis." (William K Clifford, "Lectures and Essays", 1879)
“Force is not a fact at all, but an idea embodying what is approximately the fact.” (William K Clifford et al, “The Common Sense of the Exact Sciences”, 1885)

“We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra.” (William K Clifford et al, "The Common Sense of the Exact Sciences", 1885)

"We may conceive our space to have everywhere a nearly uniform curvature, but that slight variations of the curvature may occur from point to point, and themselves vary with the time. These variations of the curvature with the time may produce effects which we not unnaturally attribute to physical causes independent of the geometry of our space. We might even go so far as to assign to this variation of the curvature of space 'what really happens in that phenomenon which we term the motion of matter'." (William K Clifford et al, "The Common Sense of the Exact Sciences", 1885)

"Every one knows there are mathematical axioms. Mathematicians have, from the days of Euclid, very wisely laid down the axioms or first principles on which they reason. And the effect which this appears to have had upon the stability and happy progress of this science, gives no small encouragement to attempt to lay the foundation of other sciences in a similar manner, as far as we are able." (William K Clifford et al, "Scottish Philosophy of Common Sense", 1915)

"The name philosopher, which meant originally 'lover of wisdom,' has come in some strange way to mean a man who thinks it is his business to explain everything in a certain number of large books. It will be found, I think, that in proportion to his colossal ignorance is the perfection and symmetry of the system which he sets up; because it is so much easier to put an empty room tidy than a full one." (William K Clifford)

Mental Models XXXVII

"This world is finite, the other infinite, 
reality is blocked by form and image." (Jalaluddin Rumi, "Masnavi-ye Ma ‘navi" Vol. I ["Spritual Verses"], 1262-1264)

"There is nothing in us which we must attribute to our soul except our thoughts. These are of two principal kinds, some being actions of the soul and others its passions. Those I call its actions are all our volitions […] the various perceptions or modes of knowledge present in us may be called its passions in the general sense, for it is often not our soul which makes them such as they are, and the soul always receives them from the things that are represented by them." (René Descartes, "Les passions de l’âme" ["Passions of the Soul"], 1649)

"Modern mathematics, that most astounding of intellectual creations, has projected the mind's eye through infinite time and the mind's hand into boundless space." (Nicholas M Butler, "What Knowledge is of Most Worth?", 1895)

"Thus the essence of memory is not constituted by the image, but by having immediately before the mind an object which is recognised as past." (Bertrand Russell, "Theory of Knowledge" , 1913)

"The paradox of reality is that no image is as compelling as the one which exists only in the mind's eye." (Shana Alexander, "Talking Woman", 1976)

"[…] new insights fail to get put into practice because they conflict with deeply held internal images of how the world works [...] images that limit us to familiar ways of thinking and acting. That is why the discipline of managing mental models - surfacing, testing, and improving our internal pictures of how the world works - promises to be a major breakthrough for learning organizations." (Peter Senge, "The Fifth Discipline: The Art and Practice of the Learning Organization", 1990)

"I believe that philosophy is part of literature, and not the reverse. Writing is not possible without images. Yet, images don't have to be descriptive; they can be concepts […]. Concepts are mental images." (Paul Virilio, [interview with Louise Wilson] 1994)

"[...] images are probably the main content of our thoughts, regardless of the sensory modality in which they are generated and regardless of whether they are about a thing or a process involving things; or about words or other symbols, in a given language, which correspond to a thing or process. Hidden behind those images, never or rarely knowable by us, there are indeed numerous processes that guide the generation and deployment of those images in space and time. Those processes utilize rules and strategies embodied in dispositional representations. They are essential for our thinking but are not a content of our thoughts.” (Antonio R Damasio, “Descartes' Error. Emotion, Reason, and the Human Brain”, 1994)

"Presumably, one can become a mathematical genius only if one has an outstanding capacity for forming vivid mental represen­tations of abstract mathematical concepts - mental images that soon turn into an illusion, eclipsing the human origins of mathematical objects and endowing them with the semblance of an independent existence." (Stanislas Dehaene," The Number Sense: How the Mind Creates Mathematics", 2011) 

"We call it 'explanation', but it is 'description' which distinguishes us from earlier stages of knowledge and science. We describe better - we explain just as little who came before us [...] We operate with nothing but things which do not exist, with lines, planes, bodies, atoms, divisible time, divisible space - how should explanation even be possible when we first make everything into an image, into our image!" (Friedrich W Nietzsche)

On Spacetime (-1799)

"In time you will know all with certainty; Time is the only test of honest men, one day is space enough to know a rogue." (Sophocles, "Oedious the King", 429 BC)

"The objection we are dealing with argues from the standpoint of an agent that presupposes time and acts in time, but did not institute time. Hence the question about 'why God's eternal will produces an effect now and and not earlier' presupposes that time exists; for 'now' and 'earlier' are segments of time. With regard to the universal production of things, among which time is also to be counted, we should not ask, 'Why now and not earlier?' Rather we should ask: 'Why did God wish this much time to intervene?' And this depends on the divine will, which is perfectly free to assign this or any other quantity to time. The same may be noted with respect to the dimensional quantity of the world. No one asks why God located the material world in such and such a place rather than higher up or lower down or in some other position; for there is no place outside the world. The fact that God portioned out so much quantity to the world that no part of it would be beyond the place occupied in some other locality, depends on the divine will. However, although there was no time prior to the world and no place outside the world, we speak as if there were. Thus we say that before the world existed there was nothing except God, and that there is no body lying outside the world. But in thus speaking of 'before' and 'outside,' we have in mind nothing but time and place as they exist in our imagination." (Thomas Aquinas, "Compendium Theologiae" ["Compendium of Theology"], cca. 1265 [unfinished])

"There is a very different relationship between [...] space and duration. For we do not ascribe various durations to the different parts of space, but say that all endure together. The moment of duration is the same at Rome and at London, on the Earth and on the stars, and throughout the heavens. And just as we understand any moment of duration to be diffused throughout all spaces, according to its kind, without any thought of its parts, so it is no more contradictory that Mind also, according to its kind, can be diffused through space without any thought of its parts." (Isaac Newton, De Gravitatione et Aequipondio Fluridorum, cca. 1664-1668) 

"All our dignity consists of thought. It is from there that we must be lifted up and not from space and time, which we could never fill. So let us work on thinking well. That is the principle of morality." (Blaise Pascal, "Pensées", 1670)

"Not at all as far as its absolute, intrinsic nature is concerned. [,,,] Whether things run or stand still, whether we sleep or wake, time flows in its even tenor (aequo tenore tempus labitur). Even if all the stars would have remained at the places where they had been created, nothing would have been lost to time (nihil inde quicquam tempori decessisset). The temporal relations of earlier, afterwards, and simultaneity, even in that tranquil state, would have had their proper existence (prius, posterius, simul etiam in illo transquillo statu fuisset in se)."  (Isaac Barrow, "Lectiones Geometricae", 1672)

"Just as space existed before the world was created and even now there exists an infinite space beyond the world (with which God coexists) [...] so time exists before the world and simultaneously with the world (prius mundo et simul cum mundo)." (Isaac Barrow, "Lectiones Geometricae", 1672)

"Time absolutely is quantity, admitting in some manner the chief affections of quantity, equality, inequality, and proportion; nor do I believe there is anyone but allows that those things existed equal times, which rose and perished simultaneously." (Isaac Barrow, "Lectiones Geometricae", 1672)

"Although time, space, place, and motion are very familiar to everyone, it must be noted that these quantities are popularly conceived solely with reference to the objects of sense perception." (Isaac Newton, "The Principia: Mathematical Principles of Natural Philosophy", 1687)

"To measure motion, space is as necessary to be considered as time....[They] are made use of to denote the position of finite: real beings, in respect one to another, in those infinite uniform oceans of duration and space." (John Locke, "An Essay Concerning Human Understanding", 1689)

"If a plurality of states of things is assumed to exist which involves no opposition to each other, they are said to exist simultaneously. Thus we deny that what occurred last year and this year are simultaneous, for they involve incompatible states of the same thing.  If one of two states which are not simultaneous involves a reason for the other, the former is held to be prior, the latter posterior. My earlier state involves a reason for the existence of my later state. And since my prior state, by reason of the connection between all things, involves the prior state of other things as well, it also involves a reason for the later state of these other things and is thus prior to them. Therefore whatever exists is either simultaneous with other existences or prior or posterior." (Gottfried W Leibniz, "Initium rerum Mathematicarum metaphysica", 1715)

"Time is the order of existence of those things which are not simultaneous. Thus time is the universal order of changes when we do not take into consideration the particular kinds of change. Duration is magnitude of time. If the magnitude of time is diminished uniformly and continuously, time disappears into moment, whose magnitude is zero. Space is the order of coexisting things, or the order of existence for things which are simultaneous." (Gottfried W Leibniz, "Initium rerum Mathematicarum metaphysica", 1715)

"Time, matter, space - all, it may be, are no more than a point." (Denis Diderot, "Lettre sur les aveugles", 1749)

"Any point has a real mode of existence, through which it is where it is; & another, due to which it exists at the time when it does exist. These real modes of existence are to me real time & space ; the possibility of these modes, hazily apprehended by us, is, to my mind, empty space & again empty time, so to speak ; in other words, space & imaginary time." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Further we believe that GOD Himself is present everywhere throughout the whole of the undoubtedly divisible space that all bodies occupy; & yet He is onefold in the highest degree & admits not of any composite nature whatever. Moreover, the same idea seems to depend on an analogy between space & time. For, just as rest is a conjunction with a continuous series of all the instants In the interval of time during which the rest endures; so also this virtual extension is a conjunction of one instant of time with a continuous series of all the points of space throughout which this one-fold entity extends virtually. Hence, just as rest is believed to exist in Nature, so also are we bound to admit virtual extension; & if this is admitted, then it will be possible for the primary elements of matter to be simple, & yet not absolutely non-extended." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Nature is the system of laws established by the Creator for the existence of things and for the succession of creatures. Nature is not a thing, because this thing would be everything. Nature is not a creature, because this creature would be God. But one can consider it as an immense vital power, which encompasses all, which animates all, and which, subordinated to the power of the first Being, has begun to act only by his order, and still acts only by his concourse or consent. […] Time, space and matter are its means, the universe its object, motion and life its goal." (Georges-Louis L de Buffon, "Histoire Naturelle, Générale et Particulière, Avec la Description du Cabinet du Roi", 1764)