On Functions I

"Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing, e.g., a quality. For whatever ratio is found to exist between intensity and intensity, in relating intensities of the same kind, a similar ratio is found to exist between line and line, and vice versa." (Nicole Oresme, "Tractatus de configurationibus qualitatum et motuum" ["A treatise on the uniformity and difformity of intensities"], 1352) [definition of a functional relationship between two variables]

"Here, we call function of a variable magnitude, a quantity formed in whatever manner with that variable magnitude and constants." (Johann I Bernoulli, 1718)

"A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities. […] Functions are divided into algebraic and transcendental. The former are those made up from only algebraic operations, the latter are those which involve transcendental operations."(Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

"Those quantities that depend on others in this way, namely, those that undergo a change when others change, are called functions of these quantities. This definition applies rather widely and includes all ways in which one quantity could be determined by another." (Leonhard Euler, "Foundations of differential calculus, with applications to finite analysis and series", 1755)

"I take the word 'mapping' in the widest possible sense; any point of the spherical surface is represented on the plane by any desired rule, so that every point of the sphere corresponds to a specified point in the plane, and inversely." (Leonhard Euler, "On the representation of Spherical Surfaces onto the Plane", 1777)

"When variable quantities are so tied to each other that, given the values of some of them, we can deduce the values of all the others, we usually conceive these various quantities expressed in terms of several of them, which then bear the name independent variables; and the remaining quantities expressed in terms of the independent variables, are what we call functions of these same variables." (Augustin-Louis Cauchy, "Cours d’analyse de l’École Royale Polytechnique", 1821)

"[…] a function of the variable x will be continuous between two limits a and b of this variable if between two limits the function has always a value which is unique and finite, in such a way that an infinitely small increment of this variable always produces an infinnitely small increment of the function itself." (Augustin-Louis Cauchy, "Mémoire sur les fonctions continues" ["Memoir on continuous functions"], 1844)

"If we designate by z a variable magnitude, which may take successively all possible real values, then, if to each of these values corresponds a unique value of the indeterminate magnitude w, we say that w is a function of z. […] This definition does not stipulate any law between the isolated values of the function, this is evident, because after this function has been dealt with for a given interval, the way it is extended outside this interval remains quite arbitrary." (Bernhard Riemann, 1851)

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)

"[…] 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)

"[…] 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)

"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)

"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 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)

"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)

"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)