"Definite portions of a manifoldness, distinguished by a mark or by a boundary, are called Quanta. Their comparison with regard to quantity is accomplished in the case of discrete magnitudes by counting, in the case of continuous magnitudes by measuring. Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case also we can only determine the more or less and not the how much. The researches which can in this case be instituted about them form a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness. (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)
"The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously withdrawing any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits." (George H. Lewes, "Problems of Life and Mind", 1873)
"Magnitude-notions are only possible where there is an antecedent general notion which admits of different specialisations. According as there exists among these specialisations a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specialisations are called in the first case points, in the second case elements, of the manifoldness." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)
"The great revelation of the quantum theory was that features of discreteness were discovered in the Book of Nature, in a context in which anything other than continuity seemed to be absurd according to the views held until then." (Erwin Schrödinger, "What is Life?", 1944)
"The discrete change has only to become small enough in its jump to approximate as closely as is desired to the continuous change. It must further be remembered that in natural phenomena the observations are almost invariably made at discrete intervals; the 'continuity' ascribed to natural events has often been put there by the observer's imagination, not by actual observation at each of an infinite number of points. Thus the real truth is that the natural system is observed at discrete points, and our transformation represents it at discrete points. There can, therefore, be no real incompatibility."
"Whereas the continuous symmetries always lead to conservation laws in classical mechanics, a discrete symmetry does not. With the introduction of quantum mechanics, however, this difference between the discrete and continuous symmetries disappears. The law of right-left symmetry then leads also to a conservation law: the conservation of parity." (Chen-Ning Yang, "The Law of Parity Conservation and Other Symmetry Laws of Physics", [Nobel lecture] 1957)
"We must recognize here that continuity and discontinuity may not always be clearly distinguishable. Ultimately what may matter is the precision of the perspective applied to the question at hand. Thus, if quantum mechanics is true, then reality is fundamentally discrete at a microscopic level. Nevertheless it may be useful to view it as continuous when the analysis is less precise, much as we perceive a movie to be continuous even though it actually consists of a sequence of discrete and discontinuous still photographs. Our mind draws an in-visible line between the stills creating the illusion of continuity." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)
"Finite Nature is a hypothesis that ultimately every quantity of physics, including space and time, will turn out to be discrete and finite; that the amount of information in any small volume of space-time will be finite and equal to one of a small number of possibilities. [...] We take the position that Finite Nature implies that the basic substrate of physics operates in a manner similar to the workings of certain specialized computers called cellular automata." (Edward Fredkin, "A New Cosmogony", PhysComp ’92: Proceedings of the Workshop on Physics and Computation, 1993)
"Conventional wisdom, fooled by our misleading 'physical intuition', is that the real world is continuous, and that discrete models are necessary evils for approximating the 'real' world, due to the innate discreteness of the digital computer." (Doron Zeilberger, "'Real' Analysis is a Degenerate Case of Discrete Analysis", 2001)
"Spacetime […] turns out to be discrete, described by a structure called spin foam." (Lee Smolin, “The New Humanists: Science at the Edge”, 2003)
"Art imitates nature through the incorporation of discrete symmetries, and indeed, reflection symmetry can be found throughout nature."
"[…] according to the new conception, the physically described world is built not out of bits of matter, as matter was understood in the nineteenth century, but out of objective tendencies - potentialities - for certain discrete, whole actual events to occur. Each such event has both a psychologically described aspect, which is essentially an increment in knowledge, and also a physically described aspect, which is an action that abruptly changes the mathematically described set of potentialities to one that is concordant with the increase in knowledge. This coordination of the aspects of the theory that are described in physical/mathematical terms with aspects that are described in psychological terms is what makes the theory practically useful."
"At very small time scales, the motion of a particle is more like a random walk, as it gets jostled about by discrete collisions with water molecules. But virtually any random movement on small time scales will give rise to Brownian motion on large time scales, just so long as the motion is unbiased. This is because of the Central Limit Theorem, which tells us that the aggregate of many small, independent motions will be normally distributed." (Field Cady, "The Data Science Handbook", 2017)
"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)
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