"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)
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