Rudy Rucker - Collected Quotes

"In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it." (Rudy Rucker, "Infinity and the Mind: The science and philosophy of the infinite", 1982)

"The study of infinity is much more than a dry academic game. The intellectual pursuit of the absolute infinity is, as Georg Cantor realized, a form of the soul's quest for God. Whether or not the goal is ever reached, an awareness of the process brings enlightenment." (Rudy Rucker, "Infinity and the Mind: The science and philosophy of the infinite", 1982)

"At the most elemental level, reality evanesces into something called Schröedinger's Wave Function: a mathematical abstraction which is best represented as a pattern in an infinite-dimensional space, Hilbert Space. Each point of the Hilbert Space represents a possible state of affairs. The wave function for some one physical or mental system takes the form of, let us say, a coloring in of Hilbert Space. The brightly colored parts represent likely states for the system, the dim parts represent less probable states of affairs." (Rudy Rucker, "The Sex Sphere", 1983)

"The space of our universe is the hypersurface of a vast expanding hypersphere." (Rudy Rucker, "The Sex Sphere", 1983)

"What is the shape of space? Is it flat, or is it bent? Is it nicely laid out, or is it warped and shrunken? Is it finite, or is it infinite? Which of the following does space resemble more: (a) a sheet of paper, (b) an endless desert, (c) a soap bubble, (d) a doughnut, (e) an Escher drawing, (f) an ice cream cone, (g) the branches of a tree, or (h) a human body?" (Rudy Rucker, "The Fourth Dimension: Toward a Geometry of Higher Reality", 1984)

"A photon is a wavy yet solid little package that can zip through empty space without the benefit of any invisible jelly vibrating underfoot." (Rudy Rucker, "The Fourth Dimension: Toward a Geometry of Higher Reality", 1984)

"The world is colors and motion, feelings and thought [...] and what does math have to do with it? No much, if "math" means being bored in high school, but in truth mathematics is the one universal science. Mathematics is the study of pure pattern, and everything in the cosmos is a kind of pattern." (Rudy Rucker, "Mind Tools", 1987)

On Homology

"Speaking roughly, a homology theory assigns groups to topological spaces and homomorphisms to continuous maps of one space into another. To each array of spaces and maps is assigned an array of groups and homomorphisms. In this way, a homology theory is an algebraic image of topology. The domain of a homology theory is the topologist’s field of study. Its range is the field of study of the algebraist. Topological problems are converted into algebraic problems." (Samuel Eilenberg &Norman E Steenrod, "Foundations of Algebraic Topology", 1952)

"The philosophical emphasis here is: to solve a geometrical problem of a global nature, one first reduces it to a homotopy theory problem; this is in turn reduced to an algebraic problem and is solved as such. This path has historically been the most fruitful one in algebraic topology. (Brayton Gray, "Homotopy Theory", Pure and Applied Mathematics Vol. 64, 1975)

"The various homology and cohomology theories appear as complicated machines, the end product of which is an assignment of a graded group to a topological space, through a series of processes which look so arbitrary that one wonders why they succeed at all." (Jean Dieudonné, "A History of Algebraic and Differential Topology, 1900 - 1960", 1989)

"Homology theory introduces a new connection between invariants of manifolds. Continuing the "physical" analogy, we say that a homology theory studies the intrinsic structure of a manifold by breaking it into a system of portions arranged simply, or, more precisely, in a standard way. Then, given certain rules for glueing the portions together, the theory obtains the whole manifold. The main problem consists in proving the resultant geometric quantities that are independent of the decomposition and glueing (i.e., proving the topological invariance of the characteristics)." (Michael IMonastyrsky, "Topology of Gauge Fields and Condensed Matter", 1993)

"Homology theory studies properties of manifolds by decomposing them into simpler parts. The structure of these parts can be investigated easily by introducing algebraic characteristics associated with these decompositions. The main difficulty lies in proving that the corresponding characteristics of the decomposition, in fact, do not depend on the particular choice of the decomposition but are rather a topological invariant of the manifold itself." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Although it is not difficult to count the holes in a real pretzel in your hand, prior to eating it, when a surface pops out of an abstract mathematical construction it can be very difficult to figure out its properties, such as how many holes it has. The cohomology groups can help us to do so." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Topology is a child of twentieth century mathematical thinking. It allows us to consider the shape and structure of an object without being wedded to its size or to the distances between its component parts. Knot theory, homotopy theory, homology theory, and shape theory are all part of basic topology. It is often quipped that a topologist does not know the difference between his coffee cup and his donut - because each has the same abstract 'shape' without looking at all alike." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"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)

"Homology translates this world of vague shapes into the rigorous world of algebra, a branch of mathematics that studies particular numerical structures and symmetries. Mathematicians study the properties of these algebraic structures in a field known as homological algebra. From the algebra they indirectly learn information about the original topological shape of the data. Homology comes in many varieties, all of which connect with algebra." (Kelsey Houston-Edwards, "How Mathematicians Use Homology to Make Sense of Topology", Quanta Magazine, 2021)

"Mathematicians extract a shape’s homology from its chain complex, which provides structured data about the shape’s component parts and their boundaries - exactly what you need to describe holes in every dimension. […] The definition of homology is rigid enough that a computer can use it to find and count holes, which helps establish the rigor typically required in mathematics. It also allows researchers to use homology for an increasingly popular pursuit: analyzing data." (Kelsey Houston-Edwards, "How Mathematicians Use Homology to Make Sense of Topology", Quanta Magazine, 2021)

On Tensors I

"The conception of tensors is possible owing to the circumstance that the transition from one co-ordinate system to another expresses itself as a linear transformation in the differentials. One here uses the exceedingly fruitful mathematical device of making a problem 'linear' by reverting to infinitely small quantities." (Hermann Weyl, "Space - Time - Matter", 1922)

"The field equation may [...] be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium - in complete analogy with […] the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish." (Howard P Robertson, "Geometry as a Branch of Physics", 1949)

"The physicist who states a law of nature with the aid of a mathematical formula is abstracting a real feature of a real material world, even if he has to speak of numbers, vectors, tensors, state-functions, or whatever to make the abstraction." (Hilary Putnam, "Mathematics, matter, and method", 1975)

"Maxwell's equations […] originally consisted of eight equations. These equations are not 'beautiful'. They do not possess much symmetry. In their original form, they are ugly. […] However, when rewritten using time as the fourth dimension, this rather awkward set of eight equations collapses into a single tensor equation. This is what a physicist calls 'beauty', because both criteria are now satisfied.  (Michio Kaku, "Hyperspace", 1995)

 "(…) the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Nothing else has a baggage-free description. In other words, we all live in a gigantic mathematical object - one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical - including you." (Max Tegmark, "Our Mathematical Universe: My Quest for the Ultimate Nature of Reality", 2014)

"Curvature is a central concept in differential geometry. There are conceptually different ways to define it, associated with different mathematical objects, the metric tensor, and the affine connection. In our case, however, the affine connection may be derived from the metric. The 'affine curvature' is associated with the notion of parallel transport of vectors as introduced by Levi-Civita. This is most simply illustrated in the case of a two- dimensional surface embedded in three- dimensional space. Let us take a closed curve on that surface and attach to a point on that curve a vector tangent to the surface. Let us now transport that vector along the curve, keeping it parallel to itself. When it comes back to its original position, it will coincide with the original vector if the surface is flat or deviate from it by a certain angle if the surface is curved. If one takes a small curve around a point on the surface, then the ratio of the angle between the original and the final vector and the area enclosed by the curve is the curvature at that point. The curvature at a point on a two-dimensional surface is a pure number." (Hanoch Gutfreund, "The Road to Relativity", 2015) 

"In geometric and physical applications, it always turns out that a quantity is characterized not only by its tensor order, but also by symmetry." (Hermann Weyl, 1925)

"Ultra-modern physicists [are tempted to believe] that Nature in all her infinite variety needs nothing but mathematical clothing [and are] strangely reluctant to contemplate Nature unclad. Clothing she must have. At the least she must wear a matrix, with here and there a tensor to hold the queer garment together." (Sydney Evershed)

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]

On Fractals II

"A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales [...]" (Benoît Mandelbrot, "The Fractal Geometry of Nature", 1982)

"A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." (Benoît Mandelbrot, "The Fractal Geometry of Nature", 1982)

"In the mind's eye, a fractal is a way of seeing infinity." (James Gleick, "Chaos: Making a New Science, A Geometry of Nature", 1987)

"One reason nature pleases us is its endless use of a few simple principles: the cube-square law; fractals; spirals; the way that waves, wheels, trig functions, and harmonic oscillators are alike; the importance of ratios between small primes; bilateral symmetry; Fibonacci series, golden sections, quantization, strange attractors, path-dependency, all the things that show up in places where you don’t expect them [...] these rules work with and against each other ceaselessly at all levels, so that out of their intrinsic simplicity comes the rich complexity of the world around us. That tension - between the simple rules that describe the world and the complex world we see - is itself both simple in execution and immensely complex in effect. Thus exactly the levels, mixtures, and relations of complexity that seem to be hardwired into the pleasure centers of the human brain - or are they, perhaps, intrinsic to intelligence and perception, pleasant to anything that can see, think, create? - are the ones found in the world around us." (John Barnes, "Mother of Storms", 1994)

"We are approaching a more fluid state. I have talked about cultural boiling. The idea of the phase-transition period which, in fractal mathematics, is the chaotic flux between one state and another. [...] Culturally, and as a species, we are approaching a phase-transition. I don’t know quite what that means, on a human level." (Alan Moore, [interview], 1998)

"If financial markets aren't efficient, then what are they? According to the 'fractal market hypothesis', they are highly unstable dynamic systems that generate stock prices which appear random, but behind which lie deterministic patterns." (Steve Keen, "Debunking Economics: The Naked Emperor Of The Social Sciences", 2001)

"Do I claim that everything that is not smooth is fractal? That fractals suffice to solve every problem of science? Not in the least. What I'm asserting very strongly is that, when some real thing is found to be un-smooth, the next mathematical model to try is fractal or multi-fractal. A complicated phenomenon need not be fractal, but finding that a phenomenon is 'not even fractal' is bad news, because so far nobody has invested anywhere near my effort in identifying and creating new techniques valid beyond fractals. Since roughness is everywhere, fractals - although they do not apply to everything - are present everywhere. And very often the same techniques apply in areas that, by every other account except geometric structure, are separate." (Benoît Mandelbrot, "A Theory of Roughness", 2004) 

"Only at the edge of chaos can complex systems flourish. This threshold line, that edge between anarchy and frozen rigidity, is not a like a fence line, it is a fractal line; it possesses nonlinearity." (Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)

[fractal:] "A fragmented geometric shape that can be split up into secondary pieces, each of which is approximately a smaller replica of the whole, the phenomenon commonly known as self similarity." (Khondekar et al, "Soft Computing Based Statistical Time Series Analysis, Characterization of Chaos Theory, and Theory of Fractals", 2013)

"If you have a hammer, use it everywhere you can, but I do not claim that everything is fractal." (Benoît Mandelbrot)

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 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)

Isaac Asimov - Collected Quotes

"Old men tend to forget what thought was like in their youth; they forget the quickness of the mental jump, the daring of the youthful intuition, the agility of the fresh insight. They become accustomed to the more plodding varieties of reason, and because this is more than made up by the accumulation of experience, old men think themselves wiser than the young." (Isaac Asimov, "Pebble in the Sky", 1950)

"Words are a pretty fuzzy substitute for mathematical equations." (Isaac Asimov, "Foundation and Empire", 1952)

"[…] infinity is not a large number or any kind of number at all; at least of the sort we think of when we say 'number'. It certainly isn't the largest number that could exist, for there isn't any such thing." (Isaac Asimov, "Asimov on Numbers", 1959)

"Scientific theories have a tendency to fit the intellectual fashions of the time." (Isaac Asimov, "The Weighting Game", 1962)

"The mysteries of the universe and the questions that scientists strive to answer never come to an end. For that we should be grateful. A universe in which their were no mysteries for curious men to ponder would be a very dull universe indeed."(Isaac Asimov, "The Search for the Elements", 1962

"[...] the orchard of science is a vast globe-encircling monster, without a map, and known to no one man; indeed, to no group of men fewer than the whole international mass of creative scientists. Within it, each observer clings to his own well-known and well-loved clump of trees. If he looks beyond, it is usually with a guilty sigh." (Isaac Asimov, "View from a Height", 1963)

"There is not a discovery in science, however revolutionary, however sparkling with insight, that does not arise out of what went before." (Isaac Asimov, "Adding a Dimension", 1964)

"[…] it took men about five thousand years, counting from the beginning of number symbols, to think of a symbol for nothing." (Isaac Asimov, "Of Time and Space and Other Things", 1965)

"If it is exciting to probe the unknown and shed light on what was dark before, then more and more excitement surely lies ahead of us." (Isaac Asimov, "The Universe: From Flat Earth to Quasar", 1966)

"The easiest way to solve a problem is to deny it exists." (Isaac Asimov, "The Gods Themselves", 1972)

"The history of science is full of revolutionary advances that required small insights that anyone might have had, but that, in fact, only one person did." (Isaac Asimov, "The Three Numbers", Ellery Queen's Mystery Magazine, 1974)

"Self-education is, I firmly believe, the only kind of education there is. The only function of a school is to make self-education easier; failing that, it does nothing." (Isaac Asimov, "Science Past, Science Future", 1975)

"If entropy must constantly and continuously increase, then the universe is remorselessly running down, thus setting a limit (a long one, to be sure) on the existence of humanity. To some human beings, this ultimate end poses itself almost as a threat to their personal immortality, or as a denial of the omnipotence of God. There is, therefore, a strong emotional urge to deny that entropy must increase." (Isaac Asimov," Asimov on Physics", 1976) 

"People are entirely too disbelieving of coincidence. They are far too ready to dismiss it and to build arcane structures of extremely rickety substance in order to avoid it. I, on the other hand, see coincidence everywhere as an inevitable consequence of the laws of probability, according to which having no unusual coincidence is far more unusual than any coincidence could possibly be." (Isaac Asimov, "The Planet That Wasn't", 1976)

"The force of gravity-though it is the first force with which we are acquainted, and though it is always with us, and though it is the one with a strength we most thoroughly appreciate-is by far the weakest known force in nature. It is first and rearmost." (Isaac Asimov, 1976)

"It is one thing to be able to make predictions. It is another to listen to the predictions you have made and to act upon them." (Isaac Asimov, "The Road to Infinity", 1979)

"How often people speak of art and science as though they were two entirely different things, with no interconnection. An artist is emotional, they think, and uses only his intuition; he sees all at once and has no need of reason. A scientist is cold, they think, and uses only his reason; he argues carefully step by step, and needs no imagination. That is all wrong. The true artist is quite rational as well as imaginative and knows what he is doing; if he does not, his art suffers. The true scientist is quite imaginative as well as rational, and sometimes leaps to solutions where reason can follow only slowly; if he does not, his science suffers." (Isaac Asimov, "The Roving Mind", 1983)

"If arithmetical skill is the measure of intelligence, then computers have been more intelligent than all human beings all along. If the ability to play chess is the measure, then there are computers now in existence that are more intelligent than any but a very few human beings. However, if insight, intuition, creativity, the ability to view a problem as a whole and guess the answer by the 'feel' of the situation, is a measure of intelligence, computers are very unintelligent indeed. Nor can we see right now how this deficiency in computers can be easily remedied, since human beings cannot program a computer to be intuitive or creative for the very good reason that we do not know what we ourselves do when we exercise these qualities." (Isaac Asimov, "Machines that Think", 1983)

"If we assume the existence of an omniscient and omnipotent being, one that knows and can do absolutely everything, then to my own very limited self, it would seem that existence for it would be unbearable. Nothing to wonder about? Nothing to ponder over? Nothing to discover? Eternity in such a heaven would surely be indistinguishable from hell." (Isaac Asimov, "'X' Stands for Unknown", 1984)

"Science does not promise absolute truth, nor does it consider that such a thing necessarily exists. Science does not even promise that everything in the Universe is amenable to the scientific process."(Isaac Asimov, "'X' Stands for Unknown", 1984)

"Science is a process. It is a way of thinking, a manner of approaching and of possibly resolving problems, a route by which one can produce order and sense out of disorganized and chaotic observations. Through it we achieve useful conclusions and results that are compelling and upon which there is a tendency to agree." (Isaac Asimov, "'X' Stands for Unknown", 1984)

"The process of science [...] involves a slow forward movement through the reachable portions of the Universe - a gradual unfolding of parts of the mystery." (Isaac Asimov, "'X' Stands for Unknown", 1984)

"Human beings are very conservative in some ways and virtually never change numerical conventions once they grow used to them. They even come to mistake them for laws of nature." (Isaac Asimov, "Foundation and Earth", 1986)

"A hypothesis may be simply defined as a guess. A scientific hypothesis is an intelligent guess." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"A scientist is as weak and human as any man, but the pursuit of science may ennoble him even against his will."  (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"A subtle thought that is in error may yet give rise to fruitful inquiry that can establish truths of great value." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Any increase in knowledge anywhere helps pave the way for an increase in knowledge everywhere." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Anyone who writes about science must know about science, which cuts down competition considerably." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Computers are better than we are at arithmetic, not because computers are so good at it, but because we are so bad at it."  (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Experimentation is the least arrogant method of gaining knowledge. The experimenter humbly asks a question of nature." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Facts are a heap of bricks and timber. It is only a successful theory that can convert the heap into a stately mansion." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"In mathematics, we say 'suppose' all the time and see if we can end up with something patently untrue or self-contradictory [...]" (Isaac Asimov, "Prelude to Foundation" 1988)

"It is hard to describe the exact route to scientific achievement, but a good scientist doesn't get lost as he travels it." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Physics is the basic science. One can easily argue that all other sciences are specialized aspects of physics." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Religion cannot object to science on moral grounds. The history of religious intolerance forbids it." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Religion considers the Universe deterministic and science considers it probabilistic - an important distinction." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Science can amuse and fascinate us all, but it is engineering that changes the world." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Science in the service of humanity is technology, but lack of wisdom may make the service harmful."(Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Science must be taught well, if a student is to understand the coming decades he must live through."(Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Scientific apparatus offers a window to knowledge, but as they grow more elaborate, scientists spend ever more time washing the windows." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"The law of conservation of energy tells us we can't get something for nothing, but we refuse to believe it." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"There is an art to science, and science in art; the two are not enemies, but different aspects of the whole." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"There is very little flexibility in the behavior of the Universe. What it does once, it does again." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"What makes it so hard to organize the environment sensibly is that everything we touch is hooked up to everything else." (Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

"Theories are not so much wrong as incomplete." (Isaac Asimov, "The Relativity of Wrong", 1988)

"Scientific theories can always be improved and are improved. That is one of the glories of science. It is the authoritarian view of the Universe that is frozen in stone and cannot be changed, so that once it is wrong, it is wrong forever." (Isaac Asimov, "The Nearest Star", 1989)

"I believe that scientific knowledge has fractal properties; that no matter how much we learn, whatever is left, however small it may seem, is just as infinitely complex as the whole was to start with. That, I think, is the secret of the Universe." (Isaac Asimov, "Essay 400: A Way of Thinking, "The Magazine of Fantasy and Science Fiction", 1994)

"Science with all its faults has brought education and the arts to more people - a larger percentage - than has ever existed before science. In that respect it is science that is the great humanizer. And, if we are going to solve the problems that science has brought us, it will be done by science and in no other way." (Isaac Asimov, "Essay 400: A Way of Thinking, "The Magazine of Fantasy and Science Fiction", 1994)

"Individual science fiction stories may seem as trivial as ever to the blinder critics and philosophers of today - but the core of science fiction, its essence [...] has become crucial to our salvation if we are to be saved at all." (Isaac Asimov)

"Once we learn to expect theories to collapse and to be supplanted by more useful generalizations, the collapsing theory becomes not the gray remnant of a broken today, but the herald of a new and brighter tomorrow." (Isaac Asimov)