On Mapping XI

"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." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"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." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)

"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)

🪷On Mind: Mental Models XLI

"Invention, strictly speaking, is little more than a new combination of those images which have been previously gathered and deposited in the memory: nothing can come of nothing." (Joshua Reynolds, "Discourses on Art", [discourse] 1769) 

"Taste is the intermediate faculty which connects the active with the passive powers of our nature, the intellect with the senses; and its appointed function is to elevate the images of the latter, while it realizes the ideas of the former."(Samuel T Coleridge, "On the Principles of Genial Criticism", 1814)

"The imagination […] that reconciling and mediatory power, which incorporating the reason in images of the sense and organizing (as it were) the flux of the senses by the permanence and self-circling energies of the reason, gives birth to a system of symbols, harmonious in themselves, and consubstantial with the truths of which they are the conductors." (Samuel T Coleridge, "The Statesman's Manual", 1816) 

"Theories usually result from the precipitate reasoning of an impatient mind which would like to be rid of phenomena and replaces them with images, concepts, indeed often with mere words." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833) 

"Word and picture are correlatives which are continually in quest of each other, as is sufficiently evident in the case of metaphors and similes. So from all time what was said or sung inwardly to the ear had to be presented equally to the eye. And so in childish days we see word and picture in continual balance; in the book of the law and in the way of salvation, in the Bible and in the spelling-book. When something was spoken which could not be pictured, and something pictured which could not be spoken, all went well; but mistakes were often made, and a word was used instead of a picture; and thence arose those monsters of symbolical mysticism, which are doubly an evil." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833) 

"This language controls by reducing the linguistic forms and symbols of reflection, abstraction, development, contradiction; by substituting images for concepts. It denies or absorbs the transcendent vocabulary; it does not search for but establishes and imposes truth and falsehood." (Herbert Marcuse, "One-Dimensional Man", 1964) 

"Imagination is the outreaching of mind […] the bombardment of the conscious mind with ideas, impulses, images and every sort of psychic phenomena welling up from the preconscious. It is the capacity to ‘dream dreams and see visions’" (Rollo May, "The Courage to Create", 1975) 

"Myth is the system of basic metaphors, images, and stories that in-forms the perceptions, memories, and aspirations of a people; provides the rationale for its institutions, rituals and power structure; and gives a map of the purpose and stages of life." (Sam Keen, "The Passionate Life", 1983) 

"We must begin by distinguishing between visual mental imagery and visual perception: Visual perception occurs while a stimulus is being viewed, and includes functions such as visual recognition (i. e., registering that a stimulus is familiar) and identification (i. e., recalling the name, context, or other information associated with the object). Two types of mechanisms are used in visual perception: ‘bottom-up’ mechanisms are driven by the input from the eyes; in contrast, ‘top-down’ mechanisms make use of stored information (such as knowledge, belief, expectations, and goals). Visual mental imagery is a set of representations that gives rise to the experience of viewing a stimulus in the absence of appropriate sensory input. In this case, information in memory underlies the internal events that produce the experience. Unlike afterimages, mental images are relatively prolonged." (Stephen M Kosslyn, "Mental images and the brain", Cognitive Neuropsychology 22, 2005) 

Steven G Krantz - Collected Quotes

"A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so – and proves a theorem." (Steven Krantz, "Conformal Mappings", American Scientist Vol. 87 (5), 1999)

"Complex analysis is a rich and textured subject. It is quite old, and its history is broad and deep. [...] Basic complex analysis is startling for its elegance and clarity. One progresses very rapidly from the basics of the Cauchy theory to profound results such as the fundamental theorem of algebra and the Riemann mapping theorem." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"Conformal mappings are characterized by the fact that they infinitesimally (i) preserve angles, and (ii) preserve length (up to a scalar factor)." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"Every smoothly bounded domain in the complex plane has a Green’s function. The Green’s function is fundamental to the Poisson integral, the theory of  harmonic functions, and to the broad panorama of complex function theory. [...] The Green’s function contains information about the geometry of its domain, and it also contains information about the harmonic analysis of the domain. We can rarely calculate the Green’s function explicitly, but we can obtain enough qualitative information so that it is a potent tool." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"It is a truism that the Riemann mapping theorem allows us to transfer the complex function theory of any simply connected domain (except the plane itself) back to the unit disk, or vice versa. But many of the more delicate questions require something more. If we wish to study behavior of functions at the boundary, or growth or regularity conditions, then we must know something about the boundary behavior of the conformal mapping." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"Harmonic measure is a device for estimating harmonic functions on a domain. It has become an essential tool in potential theory and in studying the corona problem. It is useful in studying the boundary behavior of conformal mappings, and it tells us a great deal about the boundary behavior of holomorphic functions and solutions of the Dirichlet problem."(Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"Isometries are very rigid objects. They are completely determined by their first-order behavior at just one point [...]" (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"One of the most striking facts about the Poincaré metric on the disk is that it turns the disk into a complete metric space. How could this be? The boundary is missing! The reason that the disk is complete in the Poincaré metric is the same as the reason that the plane is complete in the Euclidean metric: the boundary is infinitely far away. [...] One of the important facts about the Poincaré metric is that it can be used to study not just conformal maps but all holomorphic maps of the disk. The key to this assertion is the classical Schwarz lemma." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"The Green’s function is so basic that its existence and essential properties depend only on multivariable calculus. The theory of harmonic measure, the construction of the Poisson kernel, the solution of the Dirichlet problem, and the rudiments of potential theory all depend on the Green’s function." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"The Riemann mapping theorem has been said by some to be the greatest theorem of the nineteenth century. The entire concept of the theorem is profoundly original, and its proof introduced many new ideas. Certainly normal families and the use of extremal problems in complex analysis are just two of the important techniques that have grown out of studies of the Riemann mapping theorem." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"The Schwarz lemma is one of the simplest results in all of complex function theory. A direct application of the maximum principle, it is merely a statement about the rate of growth of holomorphic functions on the unit disk."  (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"[…] a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"A proof in mathematics is a psychological device for convincing some person, or some audience, that a certain mathematical assertion is true. The structure, and the language used, in formulating that proof will be a product of the person creating it; but it also must be tailored to the audience that will be receiving it and evaluating it. Thus there is no ‘unique’ or ‘right’ or ‘best’ proof of any given result. A proof is part of a situational ethic." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"Another feature of Bourbaki is that it rejects intuition of any kind. Bourbaki books tend not to contain explanations, examples, or heuristics. One of the main messages of the present book is that we record mathematics for posterity in a strictly rigorous, axiomatic fashion. This is the mathematician’s version of the reproducible experiment with control used by physicists and biologists and chemists. But we learn mathematics, we discover mathematics, we create mathematics using intuition and trial and error. We draw pictures. Certainly, we try things and twist things around and bend things to try to make them work. Unfortunately, Bourbaki does not teach any part of this latter process." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"Heuristically, a proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"It is proof that is our device for establishing the absolute and irrevocable truth of statements in our subject." (Steven G Krantz, "The History and Concept of Mathematical Proof", 2007)

"[…] proof is central to what modern mathematics is about, and what makes it reliable and reproducible." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"The idea of Morse theory is that the topology/geometry of a manifold can be understood by examining the smooth functions (and their singularities) on that manifold." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"There are two aspects of proof to be borne in mind. One is that it is our lingua franca. It is the mathematical mode of discourse. It is our tried-and true methodology for recording discoveries in a bullet-proof fashion that will stand the test of time. The second, and for the working mathematician the most important, aspect of proof is that the proof of a new theorem explains why the result is true. In the end what we seek is new understanding, and ’proof’ provides us with that golden nugget." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"There is no other scientific or analytical discipline that uses proof as readily and routinely as does mathematics. This is the device that makes theoretical mathematics special: the tightly knit chain of reasoning, following strict logical rules, that leads inexorably to a particular conclusion. It is proof that is our device for establishing the absolute and irrevocable truth of statements […]." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"In everyday conversation, people sometimes argue about whether a statement is true or not. In mathematics there is nothing to argue about. In practice a sensible statement in mathematics is either true or false, and there is no room for opinion about this attribute." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"It can be asserted that a 'proof' [...] is a psychological device for convincing the reader that an assertionis true. However our view in this book is more rigid: a proof is a sequence

of applications of the rules of logic to derive the assertion from the axioms. There is no room for opinion here. The axioms are plain. The rules are rigid. A proof is like a sequence of moves in a game of chess. If the rules are followed, then the proof is correct, otherwise it is not." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"One mistake that students commonly make early on is that they assume that, in a topological space, any set is either open or closed. This is like meeting a blonde person and a brunette and assuming therefore that all people are either blonde or brunette. [...] It is in fact possible for a set to be both open and closed." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"One of the nice features of the metric space setting is that all topological

notions can be formulated in terms of sequences. Such is not the case in an

arbitrary topological space. [...] Whereas a sequence is modeled on the natural numbers, a net is modeled on a more general object called a directed set. The general feel of the subject

is similar to that for sequences, but it is rather more abstract." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"The Continuum Hypothesis is the assertion that there are no cardinalities strictly between the cardinality of the integers and the cardinality of the continuum (the cardinality of the reals). [...] In logical terms, we say that the Continuum Hypothesis is independent from the other axioms of set theory, in particular it is independent from the Axiom of Choice." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"The most fundamental tool in the subject of point-set topology is the homeomorphism. This is the device by means of which we measure the equivalence of topological spaces." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"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)

"If we are to be effective mathematics teachers, we should endeavor to understand students’ values and students’ goals. Not to mention their motivations." (Steven G Krantz, "A Mathematician Comes of Age", 2012)

"[...] it is one thing to posit a set of axioms for a putative discipline. It is quite another to show that those proposed axioms are not mutually contradictory. A set of axioms that is not mutually contradictory is also called a consistent axiom system. Of course, a set of mutually contradictory axioms might be such that one can easily see a contradiction or maybe a simple argument could reveal a contradiction. More worrisome is the possibility that an argument that is very clever or very long or both is needed to reveal a contradiction. Given a mathematical theory that seems to be free of internal contradictions, the way mathematicians show it is, in fact, free of contradictions is by constructing what is called a model for the theory. This involves using another mathematical theory, say set theory, to produce a concrete mathematical object that satisfies the axioms of the theory being investigated. Once a model has been constructed, then we know that if set theory itself is free of internal contradictions, then the same is true of the theory being investigated. In practice, one does not often go all the way back to set theory to construct a model - mathematicians work with higher level constructions - but, in principle, they could start with set theory and work up from there." (Steven G Krantz & Harold R Parks, "A Mathematical Odyssey: Journey from the Real to the Complex", 2014)

"An introverted mathematician is one who looks at his shoes when he talks to you. An extroverted mathematician is one who looks at your shoes when he talks to you." (Steven G Krantz, "A Primer of Mathematical Writing" 2nd Ed., 2016)

"Definitions are part of the bedrock of mathematical writing and thinking. Mathematics is almost unique among the sciences - not to mention other disciplines - in insisting on strictly rigorous definitions of terminology and concepts. Thus we must state our definitions as succinctly and comprehensibly as possible. Definitions should not hang the reader up, but should instead provide a helping hand as well as encouragement for the reader to push on." (Steven G Krantz, "A Primer of Mathematical Writing" 2nd Ed., 2016)

"One fault that all mathematicians have is this: we think that when we have said something once clearly then that is the end of it; nothing further need be said. This observation explains why mathematicians so often lose arguments. You must repeat." (Steven G Krantz, "A Primer of Mathematical Writing" 2nd Ed., 2016)

"That is the trouble with facts: they sometimes force you to conclusions that differ with your intuition." (Steven G Krantz, "A Primer of Mathematical Writing" 2nd Ed., 2016)

"The chief feature of mathematical thinking is that it is logical. Certainly there is room for intuition in mathematics, and even room for guessing. But, in the end, we understand a mathematical situation and/or solve a problem by being very logical. Logic makes the process dependable and reproducible. It shows that what we are producing is a verifiable truth." (Steven G Krantz," Essentials of Mathematical Thinking", 2018)