On Neighborhoods III

"A surface is a topological space in which each point has a neighbourhood homeomorphic to the plane, ad for which any two distinct points possess disjoint neighbourhoods. […] The requirement that each point of the space should have a neighbourhood which is homeomorphic to the plane fits exactly our intuitive idea of what a surface should be. If we stand in it at some point (imagining a giant version of the surface in question) and look at the points very close to our feet we should be able to imagine that we are standing on a plane. The surface of the earth is a good example. Unless you belong to the Flat Earth Society you believe it to,be (topologically) a sphere, yet locally it looks distinctly planar. Think more carefully about this requirement: we ask that some neighbourhood of each point of our space be homeomorphic to the plane. We have then to treat this neighbourhood as a topological space in its own right. But this presents no difficulty; the neighbourhood is after all a subset of the given space and we can therefore supply it with the subspace topology." (Mark A Armstrong, "Basic Topology", 1979)

"This notion of each point in a space having a collection of 'neighbourhoods', the neighbourhoods leading in turn to a good definition of continuous function, is the crucial one. Notice that in defining neighbourhoods in a euclidean space we used very strongly the euclidean distance between points. In constructing an abstract space we would like to retain the concept of neighbourhood but rid ourselves of any dependence on a distance function. (A topological equivalence does not preserve distances.)" (Mark A Armstrong, "Basic Topology", 1979)

"Analytic functions are those that can be represented by a power series, convergent within a certain region bounded by the so-called circle of convergence. Outside of this region the analytic function is not regarded as given a priori ; its continuation into wider regions remains a matter of special investigation and may give very different results, according to the particular case considered." (Felix Klein, "Sophus Lie", [lecture] 1893)

"Subjectivists should feel obligated to recognize that any opinion (so much more the initial one) is only vaguely acceptable [...] So it is important not only to know the exact answer for an exactly specified initial problem, but what happens changing in a reasonable neighborhood the assumed initial opinion." (Bruno de Finetti, "Prevision: Ses Lois Logiques, ses Sources Subjectives", Annales de l’Institute Henri Poincaré, 1937)

"When nearest neighbor effects exist, the randomized complete block analysis [can be] so poor as to deserver to be called catastrophic. It [can not] even be considered a serious form of analysis. It is extremely important to make this clear to the vast number of researchers who have near religious faith in the randomized complete block design." (Walt Stroup & D Mulitze, "Nearest Neighbor Adjusted Best Linear Unbiased Prediction", The American Statistician 45, 1991)

"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)

"[…] topologies are determined by the way the neighborhoods are defined. Neighborhoods, not individual points, are what matter. They determine the topological structure of the parent set. In fact, in topology, the word point conveys very little information at all." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"[...] the Game of Life, in which a few simple rules executed repeatedly can generate a surprising degree of complexity. Recall that the game treats squares, or pixels, as simply on or off (filled or blank) and the update rules are given in terms of the state of the nearest neighbours. The theory of networks is closely analogous. An electrical network, for example, consists of a collection of switches with wires connecting them. Switches can be on or off, and simple rules determine whether a given switch is flipped, according to the signals coming down the wires from the neighbouring switches. The whole network, which is easy to model on a computer, can be put in a specific starting state and then updated step by step, just like a cellular automaton. The ensuing patterns of activity depend both on the wiring diagram (the topology of the network) and the starting state. The theory of networks can be developed quite generally as a mathematical exercise: the switches are called ‘nodes’ and the wires are called ‘edges’. From very simple network rules, rich and complex activity can follow." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

On Neighborhoods II

"Sciences are of a sociable disposition, and flourish best in the neighborhood of each other: nor is there any branch of learning, but may be helped and improved by assistances drawn from other arts." (William Blackstone, "Commentaries on the Laws of England" Vol. I, 1765)

"The separate atoms of a molecule are not connected all with all, or all with one, but, on the contrary, each one is connected only with one or with a few neighbouring atoms, just as in a chain link is connected with link." (Friedrich A Kekulé, "The Scientific Aims and Achievements of Chemistry", Nature 18, 1878)

"The theoretical side of physical chemistry is and will probably remain the dominant one; it is by this peculiarity that it has exerted such a great influence upon the neighboring sciences, pure and applied, and on this ground physical chemistry may be regarded as an excellent school of exact reasoning for all students of the natural sciences." (Svante Arrhenius, "Theories of Solutions", 1912)

"Theorems valid 'in the small' are those which affirm a statement about a certain neighborhood of a point without making any statement about the size of that neighborhood." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"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)

 "A good theorem will almost always have a wide-ranging influence on later mathematics, simply by virtue of the fact that it is true. Since it is true, it must be true for some reason; and if that reason lies deep, then the uncovering of it will usually require a deeper understanding of neighboring facts and principles." (Ian Richards,"Number theory", 1978)

"Cellular automata are mathematical models for complex natural systems containing large numbers of simple identical components with local interactions. They consist of a lattice of sites, each with a finite set of possible values. The value of the sites evolve synchronously in discrete time steps according to identical rules. The value of a particular site is determined by the previous values of a neighbourhood of sites around it." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)

"A characteristic of such chaotic dynamics is an extreme sensitivity to initial conditions (exponential separation of neighboring trajectories), which puts severe limitations on any forecast of the future fate of a particular trajectory. This sensitivity is known as the ‘butterfly effect’: the state of the system at time t can be entirely different even if the initial conditions are only slightly changed, i.e., by a butterfly flapping its wings." (Hans J Korsch et al, "Chaos: A Program Collection for the PC", 2008)

"A typical complex system consists of a vast number of identical copies of several generic processes, which are operating and interacting only locally or with a limited number of not necessary close neighbours. There is no global leader or controller associated to such systems and the resulting behaviour is usually very complex." (Jirí Kroc & Peter M A Sloot, "Complex Systems Modeling by Cellular Automata", Encyclopedia of Artificial Intelligence, 2009)

"The details of the shapes of the neighborhoods are not important. If the two sets of neighborhoods satisfy the equivalence criterion, then any set that is open, closed, and so on with respect to one set of neighborhoods will be open, closed, and so on with respect to the other set of neighborhoods." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

On Neighborhoods I

"A state of equilibrium in a system does not mean, further, that the system is without tension. Systems can, on the contrary, also come to equilibrium in a state of tension (e.g., a spring under tension or a container with gas under pressure).The occurrence of this sort of system, however, presupposes a certain firmness of boundaries and actual segregation of the system from its environment (both of these in a functional, not a spatial, sense). If the different parts of the system are insufficiently cohesive to withstand the forces working toward displacement (i.e., if the system shows insufficient internal firmness, if it is fluid), or if the system is not segregated from its environment by sufficiently firm walls but is open to its neighboring systems, stationary tensions cannot occur. Instead, there occurs a process in the direction of the forces, which encroaches upon the neighboring regions with diffusion of energy and which goes in the direction of an equilibrium at a lower level of tension in the total region. The presupposition for the existence of a stationary state of tension is thus a certain firmness of the system in question, whether this be its own inner firmness or the firmness of its walls." (Kurt Lewin, "A Dynamic Theory of Personality", 1935)

"A common and very powerful constraint is that of continuity. It is a constraint because whereas the function that changes arbitrarily can undergo any change, the continuous function can change, at each step, only to a neighbouring value." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"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)

"Catastrophe theory is a local theory, telling us what a function looks like  in a small neighborhood of a critical point; it says nothing about what the function may be doing far away from the singularity. Yet most of the applications of the theory [...]  involve extrapolating these rock-solid, local results to regions that may  well be distant in time and space from the singularity." (John L Casti, "Five Golden Rules", 1995)

"Continuous functions can move freely. Graphs of continuous functions can freely branch off at any place, whereas analytic functions coinciding in some neighborhood of a point P cannot branch outside of this neighborhood. Because of this property, continuous functions can mathematically represent wildly changing wind inside a typhoon or a gentle breeze." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Similarly to the graphs of continuous functions, graphs of differentiable (smooth) functions which coincide in a neighborhood of a point P can branch off outside of the neighborhood. Because of this property, differentiable functions can represent smoothly changing natural phenomena." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If a network is solely composed of neighborhood connections, information must traverse a large number of connections to get from place to place. In a small-world network, however, information can be transmitted between any two nodes using, typically, only a small number of connections. In fact, just a small percentage of random, long-distance connections is required to induce such connectivity. This type of network behavior allows the generation of 'six degrees of separation' type results, whereby any agent can connect to any other agent in the system via a path consisting of only a few intermediate nodes." (John H Miller & Scott E Page, "Complex Adaptive Systems", 2007)

"Cellular Automata (CA) are discrete, spatially explicit extended dynamic systems composed of adjacent cells characterized by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common local transition rule involving only a neighborhood of each cell." (Ramon Alonso-Sanz, "Cellular Automata with Memory", 2009) 

"Roughly speaking, a function defined on an open set of Euclidean space is differentiable at a point if we can approximate it in a neighborhood of this point by a linear map, which is called its differential (or total derivative). This differential can be of course expressed by partial derivatives, but it is the differential and not the partial derivatives that plays the central role." (Jacques Lafontaine, "An Introduction to Differential Manifolds", 2010)

On Differentiability I

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

"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)

"An essential difference between continuity and differentiability is whether numbers are involved or not. The concept of continuity is characterized by the qualitative property that nearby objects are mapped to nearby objects. However, the concept of differentiation is obtained by using the ratio of infinitesimal increments. Therefore, we see that differentiability essentially involves numbers." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] differentiation is performed by focusing on the behavior of a function near one point. A quantity obtained in this manner is essentially a local quantity. Is it possible that such local quantities can show us something very basic about global properties such as smoothness? Does there exist a place in mathematics which would enable us to study the relationship between local and global quantities?" (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"To consider differentiable functions, we must introduce a coordinate system on the plane and thereby to concentrate on the world of numbers.[...] a continuous function defined on a plane can be differentiable or nondifferentiable depending on the choice of coordinates. [...] the choice of coordinates on the plane determines which functions among the continuous functions should be selected as differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future)." (Benoît Mandelbrot, "The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward", 2004)?

"Roughly speaking, a function defined on an open set of Euclidean space is differentiable at a point if we can approximate it in a neighborhood of this point by a linear map, which is called its differential (or total derivative). This differential can be of course expressed by partial derivatives, but it is the differential and not the partial derivatives that plays the central role." (Jacques Lafontaine, "An Introduction to Differential Manifolds", 2010)

"I turn away with fright and horror from the lamentable evil of functions which do not have derivatives." (Charles Hermite, [letter to Thomas J Stieltjes])

Catastrophe Theory III

"On the plane of philosophy properly speaking, of metaphysics, catastrophe theory cannot, to be sure, supply any answer to the great problems which torment mankind. But it favors a dialectical, Heraclitean view of the universe, of a world which is the continual theatre of the battle between 'logoi', between archetypes." (René F Thom, "Catastrophe Theory: Its Present State and Future Perspectives", 1975)

"At the large scale where many processes and structures appear continuous and stable much of the time, important changes may occur discontinuously. When only a few variables are involved, as well as an optimizing process, the event may be analyzed using catastrophe theory. As the number of variables in- creases the bifurcations can become more complex to the point where chaos theory becomes the relevant approach. That chaos theory as well as the fundamentally discontinuous quantum processes may be viewed through fractal eyeglasses can also be admitted. We can even argue that a cascade of bifurcations to chaos contains two essentially structural catastrophe points, namely the initial bifurcation point at which the cascade commences and the accumulation point at which the transition to chaos is finally achieved." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)

"Catastrophe theory is a local theory, telling us what a function looks like  in a small neighborhood of a critical point; it says nothing about what the function may be doing far away from the singularity. Yet most of the applications of the theory [...]  involve extrapolating these rock-solid, local results to regions that may  well be distant in time and space from the singularity." (John L Casti, "Five Golden Rules", 1995)

"Chaos and catastrophe theories are among the most interesting recent developments in nonlinear modeling, and both have captured the interests of scientists in many disciplines. It is only natural that social scientists should be concerned with these theories. Linear statistical models have proven very useful in a great deal of social scientific empirical analyses, as is evidenced by how widely these models have been used for a number of decades. However, there is no apparent reason, intuitive or otherwise, as to why human behavior should be more linear than the behavior of other things, living and nonliving. Thus an intellectual movement toward nonlinear models is an appropriate evolutionary movement in social scientific thinking, if for no other reason than to expand our paradigmatic boundaries by encouraging greater flexibility in our algebraic specifications of all aspects of human life." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"[...] chaos and catastrophe theories per se address behavioral phenomena that are consequences of two general types of nonlinear dynamic behavior. In the most elementary of behavioral terms, chaotic phenomena are a class of deterministic processes that seem to mimic random or stochastic dynamics. Catastrophe phenomena, on the other hand, are a class of dynamic processes that exhibit a sudden and large scale change in at least one variable in correspondence with relatively small changes in other variables or, in some cases, parameters." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"Chaos and catastrophe theories directly address the social scientists' need to understand classes of nonlinear complexities that are certain to appear in social phenomena. The probabilistic properties of many chaos and catastrophe models are simply not known, and there is little likelihood that general procedures will be developed soon to alleviate the difficulties inherent with probabilistic approaches in such complicated settings." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"Fundamental to catastrophe theory is the idea of a bifurcation. A bifurcation is an event that occurs in the evolution of a dynamic system in which the characteristic behavior of the system is transformed. This occurs when an attractor in the system changes in response to change in the value of a parameter. A catastrophe is one type of bifurcation. The broader framework within which catastrophes are located is called dynamical bifurcation theory." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"Probably the most important reason that catastrophe theory received as much popular press as it did in the mid-1970s is not because of its unchallenged mathematical elegance, but because it appears to offer a coherent mathematical framework within which to talk about how discontinuous behaviors - stock market booms and busts or cellular differentiation, for instance - might emerge as the result of smooth changes in the inputs to a system, things like interest rates in a speculative market or the diffusion rate of chemicals in a developing embryo. These kinds of changes are often termed bifurcations, and playa central role in applied mathematical modeling. Catastrophe theory enables us to understand more clearly how - and why - they occur." (John L Casti, "Five Golden Rules", 1995)

"The goal of catastrophe theory is to classify smooth functions with degenerate critical points, just as Morse's Theorem gives us a complete classification for Morse functions. The difficulty, of course, is that there are a lot more ways for critical points to 'go bad' than there are for them to stay 'nice'. Thus, the classification problem is much harder for functions having degenerate critical points, and has not yet been fully carried out for all possible types of degeneracies. Fortunately, though, we can obtain a partial classification for those functions having critical points that are not too bad. And this classification turns out to be sufficient to apply the results to a wide range of phenomena like the predator-prey situation sketched above, in which 'jumps' in the system's biomass can occur when parameters describing the process change only slightly." (John L Casti, "Five Golden Rules", 1995)

"The reason catastrophe theory can tell us about such abrupt changes in a system's behavior is that we usually observe a dynamical system when it's at or near its steady-state, or equilibrium, position. And under various assumptions about the nature of the system's dynamical law of motion, the set of all possible equilibrium states is simply the set of critical points of a smooth function closely related to the system dynamics. When these critical points are nondegenerate, Morse's Theorem applies. But it is exactly when they become degenerate that the system can move sharply from one equilibrium position to another. The Thorn Classification Theorem tells when such shifts will occur and what direction they will take." (John L Casti, "Five Golden Rules", 1995)

On Continuity I (Calculus)

"Since [...] nature is a principle of motion and mutation [...] it is necessary that we should not be ignorant of what motion is [...] But motion appears to belong to things continuous; and the infinite first presents itself to the view in that which is continuous. [..] frequently [...] those who define the continuous, employ the nature or the infinite, as if that which is divisible to infinity is continuous." (Aristotle, "Physics", cca. 350 BC)

"Things [...] are some of them continuous [...] which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes,' a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite - for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end [...] and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude [...] or with multitude [...] could never be formulated. […] A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude." (Nicomachus, cca. 100 AD) 

"Since the nature of differentials […] consists in their being infinitely small and infinitely changeable up to zero, in being only quantitates evanescentes, evanescentia divisibilia, they will be always smaller than any given quantity whatsoever. In fact, some difference which one can assign between two magnitudes which only differ by a differential, the continuous and imperceptible variability of that infinitely small differential, even at the very point of becoming zero, always allows one to find a quantity less than the proposed difference." (Johann Bernoulli, cca. 1692–1702)

"There are two famous labyrinths where our reason very often goes astray. One concerns the great question of the free and the necessary, above all in the production and the origin of Evil. The other consists in the discussion of continuity, and of the indivisibles which appear to be the elements thereof, and where the consideration of the infinite must enter in." (Gottfried W Leibniz, "Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil", 1710)

"Every quantity which, keeping the same expression, increases or diminishes continually (non per saltum), is called a variable, and that which, with the same expression, keeps the same value, is called fixed or constant." (Pierre Varignon, "Eclaircissemens sur l'Analyse des Infinimens Petits", 1725)

"In fact, a similar principle of hardness cannot exist; it is a chimera which offends that general law which nature constantly observes in all its operations; I speak of that immutable and perpetual order, established since the creation of the Universe, that can be called the LAW OF CONTINUITY, by virtue of which everything that takes place, takes place by infinitely small degrees. It seems that common sense dictates that no change can take place at a jump; natura non operatur per saltion; nothing can pass from one extreme to the other without passing through all the degrees in between." (Johann Bernoulli, "Discours sur les Loix de la Communication du Mouvement", 1727)

"The Law of Continuity, as we here deal with it, consists in the idea that [...] any quantity, in passing from one magnitude to another, must pass through all intermediate magnitudes of the same class. The same notion is also commonly expressed by saying that the passage is made by intermediate stages or steps; [...] the idea should be interpreted as follows: single states correspond to single instants of time, but increments or decrements only to small areas of continuous time." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium", 1758)

"Any change involves at least two conditions, one preceding and one following, which are distinct from one another in such a way that the difference between the former and the latter can be established. Now the law of continuity prohibits the thing which is being changed to transcend abruptly from the former to the latter. It must pass through an intermediate condition which is as little distinct from the previous as from the subsequent one. And because the difference between this intermediate condition and the previous condition can be established still, there must be an intermediate condition between these two as well, and this must continue in the same way, until the difference between the previous condition and the one immediately succeeding it vanishes. As long as the set of these intermediate conditions can be established, every difference between one and the next can be established as well: hence their set must become larger than any given set if these differences shall vanish, and thus we imagine infinitely many conditions where one differs from the next to an infinitely small degree." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" [Beginnings of the Analysis of the Infinite"], 1766)

"It is held because of this law in particular, that no change may occur suddenly, but rather that every change always passes by infinitely small stages, of which the trajectory of a point in a curved line provides a first example." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" ["Beginnings of the Analysis of the Infinite"], 1766)

"Whoever wishes to extend this law [of continuity] to the real must justify his inferences by a law other than that, the suspicion remaining that he took images for things." (Abraham G Kästner, "Anfangsgründe der Analysis des Unendlichen" ["Beginnings of the Analysis of the Infinite"], 1766)

"[continuity] could be only appearance, and in this case Euler’s entire argument against the atoms would disappear; for one would be justified to apply the law of continuity only where experience shows that it agrees with the phenomena. [...]. The law of continuity thus belongs to the clothes of things which we must need rely on wherever reality seems impenetrably cloaked with it, but which we do not consider to be reality itself, and which we may still less cloak with things which do not serve us to see them." (Johann S T Gehler, "Physikalisches Wörterbuch" Bd. 4, 1798)

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

"The Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be." (Bertrand Russell, "Mysticism and Logic and Other Essays", cca. 1910)

"The course of the values of a continuous function is determined at all points of an interval, if only it is determined for all rational points of this interval." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"A common and very powerful constraint is that of continuity. It is a constraint because whereas the function that changes arbitrarily can undergo any change, the continuous function can change, at each step, only to a neighbouring value." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"As a simple trick, the discrete can often be carried over into the continuous, in a way suitable for practical purposes, by making a graph of the discrete, with the values shown as separate points. It is then easy to see the form that the changes will take if the points were to become infinitely numerous and close together." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"Analysis is primarily concerned with limit processes and continuity, so it is not surprising that mathematicians thinking along these lines soon found themselves studying (and generalizing) two elementary concepts: that of a convergent sequence of real or complex numbers, and that of a continuous function of a real or complex variable." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"Continuous functions can move freely. Graphs of continuous functions can freely branch off at any place, whereas analytic functions coinciding in some neighborhood of a point P cannot branch outside of this neighborhood. Because of this property, continuous functions can mathematically represent wildly changing wind inside a typhoon or a gentle breeze." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If we know when a sequence approaches a point or, as we say, converges to a point, we can define a continuous mapping from one metric space to another by using the property that a converging sequence is mapped to the corresponding converging sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Intuitively speaking, a visual representation associated with the concept of continuity is the property that a near object is sent to a corresponding near object, that is, a convergent sequence is sent to a corresponding convergent sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future)." (Benoît Mandelbrot, "The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward", 2004)

"Continuity is the rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable - say x - is associated with a value of a dependent variable - say y. Continuity of a function is sometimes expressed by saying that if the x-values are close together, then the y-values of the function will also be close. But if the question 'How close?' is asked, difficulties arise." (Erik Gregersen [Ed.], "Math Eplained: The Britannica Guide to Analysis and Calculus", 2011)