"A logarithm is a mapping that allows you to multiply by adding." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)
"As an abstract mathematical function, log maps every positive real number onto a corresponding real number, and maps every product of positive real numbers onto a sum of real numbers. Of course, there can be no table for such a mapping, because it would be infinitely long. But abstractly, such a mapping can be characterized as outlined here. These constraints completely and uniquely determine every possible value of the mapping. But the constraints do not in provide an algorithm for computing such mappings for al1 the real numbers. Approximations to values for real numbers can be made to any degree of accuracy required by doing arithmetic operations on rational numbers." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)
"[…] most earlier attempts to construct a theory of complexity have overlooked the deep link between it and networks. In most systems, complexity starts where networks turn nontrivial. No matter how puzzled we are by the behavior of an electron or an atom, we rarely call it complex, as quantum mechanics offers us the tools to describe them with remarkable accuracy. The demystification of crystals-highly regular networks of atoms and molecules-is one of the major success stories of twentieth-century physics, resulting in the development of the transistor and the discovery of superconductivity. Yet, we continue to struggle with systems for which the interaction map between the components is less ordered and rigid, hoping to give self-organization a chance." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)
"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)
"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."
"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)
"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."
"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)
"Making a map is the physical production including conceptualization and design. Mapping is the mental interpretation of the world and although it must precede the map, it does not necessarily result in a map artifact. Mapping defined in mathematics is the correspondence between each element of a given set with each element of another. Similarly in linguistics emphasis is on the correspondence between associated elements of different types. For designers all drawings are maps - they represent relationships between objects, places and ideas." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 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)
"A neural-network algorithm is simply a statistical procedure for classifying inputs (such as numbers, words, pixels, or sound waves) so that these data can mapped into outputs. The process of training a neural-network model is advertised as machine learning, suggesting that neural networks function like the human mind, but neural networks estimate coefficients like other data-mining algorithms, by finding the values for which the model’s predictions are closest to the observed values, with no consideration of what is being modeled or whether the coefficients are sensible." (Gary Smith & Jay Cordes, "The 9 Pitfalls of Data Science", 2019)
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