"All forms of complex causation, and especially nonlinear transformations, admittedly stack the deck against prediction. Linear describes an outcome produced by one or more variables where the effect is additive. Any other interaction is nonlinear. This would include outcomes that involve step functions or phase transitions. The hard sciences routinely describe nonlinear phenomena. Making predictions about them becomes increasingly problematic when multiple variables are involved that have complex interactions. Some simple nonlinear systems can quickly become unpredictable when small variations in their inputs are introduced." (Richard N Lebow, "Forbidden Fruit: Counterfactuals and International Relations", 2010)
"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are" (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points" (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)
"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)
"Today, most mathematicians have embraced the axiom of choice because of the order and simplicity it brings to mathematics in general. For example, the theorems that every vector space has a basis and every field has an algebraic closure hold only by virtue of the axiom of choice. Likewise, for the theorem that every sequentially continuous function is continuous. However, there are special places where the axiom of choice actually brings disorder. One is the theory of measure." (John Stillwell, "Roads to Infinity: The mathematics of truth and proof", 2010)
"A topological property is, therefore, any property that is preserved under the set of all homeomorphisms. […] Homeomorphisms generally fail to preserve distances between points, and they may even fail to preserve shapes." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)
"In each branch of mathematics it is essential to recognize when two structures are equivalent. For example two sets are equivalent, as far as set theory is concerned, if there exists a bijective function which maps one set onto the other. Two groups are equivalent, known as isomorphic, if there exists a a homomorphism of one to the other which is one-to-one and onto. Two topological spaces are equivalent, known as homeomorphic, if there exists a homeomorphism of one onto the other." (Sydney A Morris, "Topology without Tears", 2011)
"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)
"These days the term 'central limit theorem' is associated with a multitude of statements having to do with the convergence of probability distributions of functions of an increasing number of one- or multi-dimensional random variables or even more general random elements (with values in Banach spaces or more general spaces) to a normal distribution3 (or related distributions). In an effort to reduce ambiguity - and in view of historic developments - the denotation “central limit theorem” in the present examination will usually refer only to the 'classical' case, which deals with the asymptotic equality of distributions of sums of independent or weakly dependent random variables and of a normal distribution." (Hans Fischer, "A History of the Central Limit Theorem: From Classical to Modern Probability Theory", 2011)
"A catastrophe is a universal unfolding of a singular function germ. The singular function germs are called organization centers of the catastrophes. [...] Catastrophe theory is concerned with the mathematical modeling of sudden changes - so called 'catastrophes' - in the behavior of natural systems, which can appear as a consequence of continuous changes of the system parameters. While in common speech the word catastrophe has a negative connotation, in mathematics it is neutral." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])
"A function acts like a set of rules for turning some numbers into others, a machine with parts that we can manipulate to accomplish anything we can imagine." (David Perkins, "Calculus and Its Origins", 2012)
"A function is also sometimes referred to as a map or a mapping. This terminology is common in mathematics, but less so in physics or other scientific fields. The idea of a mapping is useful if one wants to think of a function as acting on an entire set of input values." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Analyticity can often be exploited to advantage in the study of problems of approximation, even when the objects to be approximated are functions of a real variable." (Peter D Lax & Lawrence Zalcman, "Complex proofs of real theorems", 2012)
"Classification is only one of the mathematical aspects of catastrophe theory. Another is stability. The stable states of natural systems are the ones that we can observe over a longer period of time. But the stable states of a system, which can be described by potential functions and their singularities, can become unstable if the potentials are changed by perturbations. So stability problems in nature lead to mathematical questions concerning the stability of the potential functions." (Werner Sanns, "Catastrophe Theory" [Mathematics of Complexity and Dynamical Systems, 2012])
"Functions are the most basic way of mathematically representing a relationship. [...] In mathematics, it is useful to think of a function as an action; a function takes a number as input, does something to it, and outputs a new number. [...] We are now in a position to refine our definition of a function. A function is a rule that assigns an output value f(x) to every input x. This is consistent with the everyday use of the word function: the output f(x) is a function of the input x. The output depends on the input [...]" (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"If we wish the word ‘continuous’ to prohibit jumps in a function, its definition must somehow control the vertical change of the function at a sort of microscopic level. That is, at any point on a ‘continuous’ function, the nearby points ought to be as ‘close’ as possible." (David Perkins, "Calculus and Its Origins", 2012)
"Nothing illustrates the extraordinary power of complex function theory better than the ease and elegance with which it yields results which challenged and often baffled the very greatest mathematicians of an earlier age." (Peter D Lax & Lawrence Zalcman, "Complex proofs of real theorems", 2012)
"The good news is that, as Leibniz suggested, we appear to live in the best of all possible worlds, where the computable functions make life predictable enough to be survivable, while the noncomputable functions make life (and mathematical truth) unpredictable enough to remain interesting, no matter how far computers continue to advance." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)
"Thus, much of physics can be seen as a iterative process: an object or a bunch of objects have some initial condition or seed. [...] Iterated functions are an example of what mathematicians call dynamical systems. A dynamical system is just a generic name for some variable or set of variables that change over time. There are many different types of dynamical systems - the iterated functions introduced above are just one type among many. Dynamical systems is now generally recognized as a branch of applied mathematics that studies properties of how systems change over time." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"To mathematicians who study them, moduli schemes are just as real as the regular objects in the world. […] The key idea is that an ordinary object can be studied using the set of functions on the object. […] Secondly, you can do algebra with these functions - that is, you can add or multiply two such functions and get a third function. This step makes the set of these functions into a ring. […] Then the big leap comes: If you start with any ring - that is, any set of entities that can be added and multiplied subject to the usual rules, you simply and brashly declare that this creates a new kind of geometric object. The points of the object can be given by maps from the ring to the real numbers, as in the example of the pot. But they may also be given by maps to other fields. A field is a special sort of ring in which division is possible." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)
"When confronted with multiple models, I find it revealing to pose the resulting uncertainty as a two-stage lottery. For the purposes of my discussion, there is no reason to distinguish unknown models from unknown parameters of a given model. I will view each parameter configuration as a distinct model. Thus a model, inclusive of its parameter values, assigns probabilities to all events or outcomes within the model’s domain. The probabilities are often expressed by shocks with known distributions and outcomes are functions of these shocks. This assignment of probabilities is what I will call risk. By contrast there may be many such potential models. Consider a two-stage lottery where in stage one we select a model and in stage two we draw an outcome using the model probabilities. Call stage one model ambiguity and stage two risk that is internal to a model." (Lars P Hansen, "Uncertainty Outside and Inside Economic Models", [Nobel lecture] 2013)
"There are many roads to statistical significance; if data are gathered with no preconceptions at all, statistical significance can obviously be obtained even from pure noise by the simple means of repeatedly performing comparisons, excluding data in different ways, examining different interactions, controlling for different predictors, and so forth. Realistically, though, a researcher will come into a study with strong substantive hypotheses, to the extent that, for any given data set, the appropriate analysis can seem evidently clear. But even if the chosen data analysis is a deterministic function of the observed data, this does not eliminate the problem posed by multiple comparisons." (Andrew Gelman & Eric Loken, "The Statistical Crisis in Science", American Scientist Vol. 102(6), 2014)
"Typical symbols used in mathematics are operationals, groupings, relations, constants, variables, functions, matrices, vectors, and symbols used in set theory, logic, number theory, probability, and statistics. Individual symbols may not have much effect on a mathematician’s creative thinking, but in groups they acquire powerful connections through similarity, association, identity, resemblance and repeated imagery. ¿ey may even create thoughts that are below awareness." (Joseph Mazur. "Enlightening symbols: a short history of mathematical notation and its hidden powers", 2014)
"Typical symbols used in mathematics are operationals, groupings, relations, constants, variables, functions, matrices, vectors, and symbols used in set theory, logic, number theory, probability, and statistics. Individual symbols may not have much effect on a mathematician’s creative thinking, but in groups they acquire powerful connections through similarity, association, identity, resemblance and repeated imagery. ¿ey may even create thoughts that are below awareness." (Joseph Mazur. "Enlightening symbols: a short history of mathematical notation and its hidden powers", 2014)
"Functions are ambiguous creatures - they come with multiple representations. [...] These representations break down into two categories: those that see the function as a static object - a graph or list - versus those that see it as a process - the calculator button, the input-output machine." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)
"Ironically, conventional quantum mechanics itself involves a vast expansion of physical reality, which may be enough to avoid Einstein Insanity. The equations of quantum dynamics allow physicists to predict the future values of the wave function, given its present value. According to the Schrödinger equation, the wave function evolves in a completely predictable way. But in practice we never have access to the full wave function, either at present or in the future, so this 'predictability' is unattainable. If the wave function provides the ultimate description of reality - a controversial issue!" (Frank Wilczek, "Einstein’s Parable of Quantum Insanity", 2015)
"That’s where boundary conditions come in. A boundary condition 'ties down' a function or its derivative to a specified value at a specified location in space or time. By constraining the solution of a differential equation top satisfy the boundary condition(s), you may be able to determine the value of the function or its derivatives at other locations. We say “may” because boundary conditions that are not well-posed may provide insufficient or contradictory information." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
"Why are boundary conditions important in wave theory? One reason is this: Differential equations, by their very nature, tell you about the change in a function (or, if the equation involves second derivatives, about the change in the change of the function). Knowing how a function changes is very useful, and may be all you need in certain problems. But in many problems you wish to know not only how the function changes, but also what value the function takes on at certain locations or times." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
"A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. [...] A mathematical model is never a completely accurate representation of a physical situation - it is an idealization." (James Stewart, "Calculus: Early Transcedentals" 8th Ed., 2016)
"Quantum theory can be thought of as the science of constructing wavefunctions and extracting predictions of measurable outcomes from them. […] The wavefunction is a little bit like a map - the best possible kind of map. It encodes all that can be said about a quantum system." (Hans C von Baeyer, "QBism: The future of quantum physics", 2016)
"But e’s greatest claim to fame is that when dressed up with a variable exponent, it becomes a very special function." (See box.) This function is usually written as ex, that is e raised to the x power." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"Computational systems are machines that can be described apriori and systematically, and implemented on every substrate that elicits the causal properties that are necessary to capture the respective states and transition functions." (Joscha Bach, "The Cortical Conductor Theory: Towards Addressing Consciousness in AI Models", 2017)
"Estimators are functions of the observed values that can be used to estimate specific parameters. Good estimators are those that are consistent and have minimum variance. These properties are guaranteed if the estimator maximizes the likelihood of the observations." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)
"Mathematics is the domain of all formal languages, and allows the expression of arbitrary statements" (most of which are uncomputable). Computation may be understood in terms of computational systems, for instance via defining states" (which are sets of discernible differences, i.e. bits), and transition functions that let us derive new states." (Joscha Bach, "The Cortical Conductor Theory: Towards Addressing Consciousness in AI Models", 2017)
"The elements of this cloud of uncertainty (the set of all possible errors) can be described in terms of probability. The center of the cloud is the number zero, and elements of the cloud that are close to zero are more probable than elements that are far away from that center. We can be more precise in this definition by defining the cloud of uncertainty in terms of a mathematical function, called the probability distribution." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)
"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)
"The trig functions’ input consists of the sizes of angles inside right triangles. Their output consists of the ratios of the lengths of the triangles’ sides. Thus, they act as if they contained phone-directory-like groups of paired entries, one of which is an angle, and the other is a ratio of triangle-side lengths associated with the angle. That makes them very useful for figuring out the dimensions of triangles based on limited information." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"The important point about e is that an exponential function with this base grows at a rate precisely equal to the function itself. Let me say that again. The rate of growth of ex is ex itself. This marvelous property simplifies all calculations about exponential functions when they are expressed in base e. No other base enjoys this simplicity. Whether we are working with derivatives, integrals, differential equations, or any of the other tools of calculus, exponential functions expressed in base e are always the cleanest, most elegant, and most beautiful." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"What is essentially different in quantum mechanics is that it deals with complex quantities" (e.g. wave functions and quantum state vectors) of a special kind, which cannot be split up into pure real and imaginary parts that can be treated independently. Furthermore, physical meaning is not attached directly to the complex quantities themselves, but to some other operation that produces real numbers" (e.g. the square modulus of the wave function or of the inner product between state vectors)." (Ricardo Karam, "Why are complex numbers needed in quantum mechanics? Some answers for the introductory level", American Journal of Physics Vol. 88 (1), 2020)
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