On Mathematical Thinking

"It should therewith be remembered that as mathematics studies neutral complexes, mathematical thinking is an organizational process and hence its methods, as well as the methods of all other sciences and those of any practice, fall within the province of a general tektology. Tektology is a unique science which must not only work out its own methods by itself but must study them as well; therefore it is the completion of the cycle of sciences." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

“Philosophy in its old form could exist only in the absence of engineering, but with engineering in existence and daily more active and far reaching, the old verbalistic philosophy and metaphysics have lost their reason to exist. They were no more able to understand the ‘production’ of the universe and life than they are now able to understand or grapple with 'production' as a means to provide a happier existence for humanity. They failed because their venerated method of ‘speculation’ can not produce, and its place must be taken by mathematical thinking. Mathematical reasoning is displacing metaphysical reasoning. Engineering is driving verbalistic philosophy out of existence and humanity gains decidedly thereby.” (Alfred Korzybski,  “Manhood of Humanity”, 1921)

"We can now return to the distinction between language and symbolism. A symbol is language and yet not language. A mathematical or logical or any other kind of symbol is invented to serve a purpose purely scientific; it is supposed to have no emotional expressiveness whatever. But when once a particular symbolism has been taken into use and mastered, it reacquires the emotional expressiveness of language proper. Every mathematician knows this. At the same time, the emotions which mathematicians find expressed in their symbols are not emotions in general, they are the peculiar emotions belonging to mathematical thinking." (Robin G Collingwood, "The Principles of Art", 1938)

"Figures and symbols are closely connected with mathematical thinking, their use assists the mind. […] At any rate, the use of mathematical symbols is similar to the use of words. Mathematical notation appears as a sort of language, une langue bien faite, a language well adapted to its purpose, concise and precise, with rules which, unlike the rules of ordinary grammar, suffer no exception." (George Pólya, "How to solve it", 1945)

"Geometrical intuition plays an essential role in contemporary algebro-topological and geometric studies. Many profound scientific mathematical papers devoted to multi-dimensional geometry use intensively the 'visual slang' such as, say, 'cut the surface', 'glue together the strips', 'glue the cylinder', 'evert the sphere' , etc., typical of the studies of two and three-dimensional images. Such a terminology is not a caprice of mathematicians, but rather a 'practical necessity' since its employment and the mathematical thinking in these terms appear to be quite necessary for the proof of technically very sophisticated results."(Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed [...] were made explicit when logic was formalized early in the this century [...] These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a [...] conjecture. [...] Heuristic arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. [...] Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"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)

"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)

"Musical form is close to mathematics - not perhaps to mathematics itself, but certainly to something like mathematical thinking and relationships." (Igor Stravinsky)

On Cardinality

"the law of statistical regularity lays down that the moderately large number of items chosen at random from a large group are almost sure on the average to possess the characteristics of the large group." (Willford I King, "The Elements of Statistical Method", 1912)

"The averaging of percentages themselves requires care, where the percentages are each computed on different bases, i.e. different quantities. The average is not derived by aggregating the percentages and dividing them. Instead of this, each percentage must first be multiplied by its base to bring out its relative significance to the other percentages and to the total. The sum of the resultant products is then divided by the sum of the base values [...], not merely the number of items." (Alfred R Ilersic, "Statistics", 1959)

"It is important to observe that there is an intimate connection between fuzziness and complexity. Thus, a basic characteristic of the human brain, a characteristic shared in varying degrees with all information processing systems, is its limited capacity to handle classes of high cardinality, that is, classes having a large number of members. Consequently, when we are presented with a class of very high cardinality, we tend to group its elements together into subclasses in such a way as to reduce the complexity of the information processing task involved. When a point is reached where the cardinality of the class of subclasses exceeds the information handling capacity of the human brain, the boundaries of the subclasses are forced to become imprecise and fuzziness becomes a manifestation of this imprecision." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)

"Zero is the only number that is neither positive nor negative. As such, it represents a quantity: If three is the name we give to the number of items in a trilogy, a trinity, or a triad, zero is our name for the number of items in an empty, or null set, i.e., one having no members. This is not the same as saying the set doesn't exist; in fact, we can and do make valid assertions about null sets […]" (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"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 obvious variations of the Axiom of Choice are those that restrict the cardinality of the sets in question. Other variations impose relational restrictions between the sets. When the early set theorists tried to prove the Axiom of Choice they invariably ended up showing it is equivalent to some other statement that they were unable to prove. This collection of equivalent statements has grown to an enormous size. One of its striking features is that some of the statements seem intuitively obvious while others are either wildly counterintuitive or evade any kind of evaluation." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The act of counting is governed by five principles. They describe the conditions and prerequisites that make counting possible. We call them the 'BOCIA' principles - from the words Bijection, Ordinality, Cardinality, Invariance, and Abstraction." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

On Homeomorphism II

"One of the basic tasks of topology is to learn to distinguish nonhomeomorphic figures. To this end one introduces the class of invariant quantities that do not change under homeomorphic transformations of a given figure. The study of the invariance of topological spaces is connected with the solution of a whole series of complex questions: Can one describe a class of invariants of a given manifold? Is there a set of integral invariants that fully characterizes the topological type of a manifold? and so forth." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Two figures which can be transformed into one other by continuous deformations without cutting and pasting are called homeomorphic. […] The definition of a homeomorphism includes two conditions: continuous and one- to-one correspondence between the points of two figures. The relation between the two properties has fundamental significance for defining such a paramount concept as the dimension of space." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Intuitively, two spaces that are homeomorphic have the same general shape in spite of possible deformations of distance and angle. Thus, if two spaces are not homeomorphic, they will tend to look distinctly different. Our job is to specify the difference. To do this rigorously, we need to define some property of topological spaces and show that the property is preserved under transformations by any homeomorphism. Then if one space has the property and the other one does not have the property, there is no way they can be homeomorphic." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"The definition of homeomorphism was motivated by the idea of preserving the general shape or configuration of a geometric figure. Since path components are significant characteristics of a space, it is certainly reasonable that a homeomorphism will preserve the decomposition of a space into path components. […] Suppose we are given two geometric figures that we suspect are not topologically equivalent. If both of the figures are path-connected, counting components will not distinguish the spaces. However, we might be able to remove a special subset of one of the figures and count the number of components of the remainder. If no comparable set can be removed from the other space to leave the same number of components, we will then know that the two spaces are not homeomorphic." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"The easiest way to show two figures are homeomorphic is often to construct an explicit homeomorphism between them. But what if two figures are not homeomorphic? Surely we cannot be expected to check every function between the sets and show that it is not a homeomorphism. One of the goals of the field of topology is to discover easier ways of detecting the differences between spaces that are not homeomorphic." (Robert Messer & Philip Straffin, "Topology Now!", 2006) 

"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)

"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)

On Homeomorphism I

"A manifold can be given by specifying the coordinate ranges of an atlas, the images in those coordinate ranges of the overlapping parts of the coordinate domains, and the coordinate transformations for each of those overlapping domains. When a manifold is specified in this way, a rather tricky condition on the specifications is needed to give the Hausdorff property, but otherwise the topology can be defined completely by simply requiring the coordinate maps to be homeomorphisms." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"A surface is a topological space in which each point has a neighbourhood homeomorphic to the plane, ad for which any two distinct points possess disjoint neighbourhoods. […] The requirement that each point of the space should have a neighbourhood which is homeomorphic to the plane fits exactly our intuitive idea of what a surface should be. If we stand in it at some point (imagining a giant version of the surface in question) and look at the points very close to our feet we should be able to imagine that we are standing on a plane. The surface of the earth is a good example. Unless you belong to the Flat Earth Society you believe it to,be (topologically) a sphere, yet locally it looks distinctly planar. Think more carefully about this requirement: we ask that some neighbourhood of each point of our space be homeomorphic to the plane. We have then to treat this neighbourhood as a topological space in its own right. But this presents no difficulty; the neighbourhood is after all a subset of the given space and we can therefore supply it with the subspace topology." (Mark A Armstrong, "Basic Topology", 1979)

"Showing that two spaces are homeomorphic is a geometrical problem, involving the construction of a specific homeomorphism between them. The techniques used vary with the problem. […] Attempting to prove that two spaces are not homeomorphic to one another is a problem of an entirely different nature. We cannot possibly examine each function between the two spaces individually and check that it is not a homeomorphism. Instead we look for 'topological invariants' of spaces: an invariant may be a geometrical property of the space, a number like the Euler number defined for the space, or an algebraic system such as a group or a ring constructed from the space. The important thing is that the invariant be preserved by a homeomorphism- hence its name. If we suspect that two spaces are not homeomorphic, we may be able to confirm our suspicion by computing some suitable invariant and showing that we obtain different answers." (Mark A Armstrong, "Basic Topology", 1979)

"Topology has to do with those properties of a space which are left unchanged by the kind of transformation that we have called a topological equivalence or homeomorphism. But what sort of spaces interest us and what exactly do we mean by a 'space? The idea of a homeomorphism involves very strongly the notion of continuity [...]"  (Mark A Armstrong, "Basic Topology", 1979)

"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Homeomorphism is one of the basic concepts in topology. Homeomorphism, along with the whole topology, is in a sense the basis of spatial perception. When we look at an object, we see, say, a telephone receiver or a ring-shaped roll and first of all pay attention to the geometrical shape (although we do not concentrate on it specially) - an oblong figure thickened at the ends or a round rim with a large hole in the middle. Even if we deliberately concentrate on the shape of the object and forget about its practical application, we do not yet 'see' the essence of the shape. The point is that oblongness, roundness, etc. are metric properties of the object. The topology of the form lies 'beyond them'." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"The concept of homeomorphism appears to be convenient for establishing those important properties of figures which remain unchanged under such deformations. These properties are sometimes referred to as topological, as distinguished from metrical, which are customarily associated with distances between points, angles between lines, edges of a figure, etc." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Topology studies the properties of geometrical objects that remain unchanged under transformations called homeomorphisms and deformations." (Victor V Prasolov, "Intuitive Topology", 1995)

"Note that a stable homeomorphism must preserve orientation, if X is an orientable manifold. Note also that it is easier to observe this fact than to define the term orientation." (Edwin E Moise, "Isotopies", 1997) 

Terry Gannon - Collected Quotes

"In modern mathematics there is a strong tendency towards formulations of concepts that minimise the number and significance of arbitrary choices. This crispness tends to emphasise the naturality of the construction or definition, at the expense sometimes of accessibility. Our mathematics is more conceptual today – more beautiful perhaps – but the cost of less explicitness is the compartmentalism that curses our discipline." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Like moonlight itself, Monstrous Moonshine is an indirect phenomenon. Just as in the theory of moonlight one must introduce the sun, so in the theory of Moonshine one must go well beyond the Monster. Much as a book discussing moonlight may include paragraphs on sunsets or comet tails, so do we discuss fusion rings, Galois actions and knot invariants." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine concerns the occurrence of modular forms in algebra and physics, and care is taken to avoid analytic complications as much as possible. But spaces here are unavoidably infinite-dimensional, and through this arise subtle but significant points of contact with analysis." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory. The theory has evolved to describe the relationship between finite groups, modular forms and vertex operator algebras." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is interested in the correlation functions of a class of extremely symmetrical and well-behaved quantum field theories called rational conformal field theories - these theories are so special that their correlation functions can be computed exactly and perturbation is not required." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is profoundly connected with physics (namely conformal field theory and string theory). String theory proposes that the elementary particles (electrons, photons, quarks, etc.) are vibrational modes on a string of length about 10^−33 cm. These strings can interact only by splitting apart or joining together – as they evolve through time, these (classical) strings will trace out a surface called the world-sheet. Quantum field theory tells us that the quantum quantities of interest (amplitudes) can be perturbatively computed as weighted averages taken over spaces of these world-sheets. Conformally equivalent world-sheets should be identified, so we are led to interpret amplitudes as certain integrals over moduli spaces of surfaces. This approach to string theory leads to a conformally invariant quantum field theory on two-dimensional space-time, called conformal field theory (CFT). The various modular forms and functions arising in Moonshine appear as integrands in some of these genus-1 (‘1-loop’) amplitudes: hence their modularity is manifest." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Physics reduces Moonshine to a duality between two different pictures of quantum field theory: the Hamiltonian one, which concretely gives us from representation theory the graded vector spaces, and another, due to Feynman, which manifestly gives us modularity. In particular, physics tells us that this modularity is a topological effect, and the group SL2(Z) directly arises in its familiar role as the modular group of the torus." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"The appeal of Monstrous Moonshine lies in its mysteriousness: it unexpectedly associates various special modular functions with the Monster, even though modular functions and elements of Mare conceptually incommensurable. Now, ‘understanding’ something means to embed it naturally into a broader context. Why is the sky blue? Because of the way light scatters in gases. Why does light scatter in gases the way it does? Because of Maxwell’s equations. In order to understand Monstrous Moonshine, to resolve the mystery, we should search for similar phenomena, and fit them all into the same story." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

Mathematical Trivia I: Moonshine

"The term 'moonshine' roughly means weird relations between sporadic groups and modular functions (and anything else) similar to this. It was clear to many people that this was just a meaningless coincidence." (Richard E Borcherds, "What is Moonshine?", Proceedings of the International Congress of Mathematicians, 1998)

"Moonshine concerns the occurrence of modular forms in algebra and physics, and care is taken to avoid analytic complications as much as possible. But spaces here are unavoidably infinite-dimensional, and through this arise subtle but significant points of contact with analysis." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory. The theory has evolved to describe the relationship between finite groups, modular forms and vertex operator algebras." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is interested in the correlation functions of a class of extremely symmetrical and well-behaved quantum field theories called rational conformal field theories - these theories are so special that their correlation functions can be computed exactly and perturbation is not required." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is profoundly connected with physics (namely conformal field theory and string theory). String theory proposes that the elementary particles (electrons, photons, quarks, etc.) are vibrational modes on a string of length about 10^−33 cm. These strings can interact only by splitting apart or joining together – as they evolve through time, these (classical) strings will trace out a surface called the world-sheet. Quantum field theory tells us that the quantum quantities of interest (amplitudes) can be perturbatively computed as weighted averages taken over spaces of these world-sheets. Conformally equivalent world-sheets should be identified, so we are led to interpret amplitudes as certain integrals over moduli spaces of surfaces. This approach to string theory leads to a conformally invariant quantum field theory on two-dimensional space-time, called conformal field theory (CFT). The various modular forms and functions arising in Moonshine appear as integrands in some of these genus-1 (‘1-loop’) amplitudes: hence their modularity is manifest." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Physics reduces Moonshine to a duality between two different pictures of quantum field theory: the Hamiltonian one, which concretely gives us from representation theory the graded vector spaces, and another, due to Feynman, which manifestly gives us modularity. In particular, physics tells us that this modularity is a topological effect, and the group SL2(Z) directly arises in its familiar role as the modular group of the torus." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"The appeal of Monstrous Moonshine lies in its mysteriousness: it unexpectedly associates various special modular functions with the Monster, even though modular functions and elements of Mare conceptually incommensurable. Now, ‘understanding’ something means to embed it naturally into a broader context. Why is the sky blue? Because of the way light scatters in gases. Why does light scatter in gases the way it does? Because of Maxwell’s equations. In order to understand Monstrous Moonshine, to resolve the mystery, we should search for similar phenomena, and fit them all into the same story." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"The Moonshine mystery itself is still unresolved, despite Borcherd's proof! [...] there are facts about the Monster and Moonshine that we don't understand. [...] The method leading to its discovery, brilliant though it was, gave no clue to the Monster's remarkable properties." (Mark Ronan, "Symmetry and the Monster: One of the greatest quests of mathematics", 2006)

"The term Moonshine [...] has a variety of meanings. It can refer to foolish or naive ideas, but also to the illicit distillation of spirits [...] It gave an impression of dabbling in mysterious matters that might be better left alone, but also had the useful connotation of something shining in reflected light. The true source of light is probably yet to be found, but there were further strange connections to come later [...] The Monster's connections with number theory - the Moonshine connections - had suggested it was a more beautiful and important group of symmetries than first realized. [...] The Moonshine connections between the Monster and number theory have now been placed within a larger theory, but we have yet to grasp the significance of these deep mathematical links with fundamental physics. We have found the Monster, but it remains an enigma. Understanding its full nature is likely to shed light on the very fabric of the universe." (Mark Ronan, "Symmetry and the Monster: One of the greatest quests of mathematics", 2006)

On Mind: Mirrors II

"Conscious apprehension seems to exist […] as happens in a mirror-image when the smooth and bright surface is peaceful." (Plotinus, "Enneads", cca. 270 AD)

"[…] the mind orders nothing by its own motions, but lies merely receptive under the impressions of bodies, reflecting empty images in a mirror in place of reality." (Anicius Manlius Severinus Boethius, "The Consolation of Philosophy", cca. 524)

"In the same way as regards the soul, when that kind of thing in us which mirrors the images of thought and intellect is undisturbed, we see them and know them in a way parallel to sense-perception, along with the prior knowledge that it is intellect and thought that are active. But when this is broken because the harmony of the body is upset, thought and intellect operate without an image, and then intellectual activity takes place without a mind-picture." (Plotinus, "Enneads", cca. 270 AD)

"The noetic act is without parts and has not, so to speak, come out into the open, but remains unobserved within, but the verbal expression unfolds its content and brings it out of the noetic act into the image making power, and so shows the noetic act as if in a mirror, and this is how there is conscious apprehension and persistence and memory of it." (Plotinus, "Enneads", cca. 270 AD)

"This interconnection or accommodation of all created things to each other, and each to all the others, brings it about that each simple substance has relations that express all the others, and consequently, that each simple substance is a perpetual, living mirror of the universe." (Gottfried W Leibniz,  "Monadology", 1714)

"Let the poet confine his use of individual models to what is necessary to make his subject alive and convincing. As for all the rest, let him rely on the living world as mirrored in his bosom." (Johann Wolfgang von Goethe, 1789)

"The symbol. It is the thing without being the thing, and yet the thing: an image concentrated in the mirror of the mind and yet identical with the object. How inferior is allegory by comparison. Though it may have wit and subtle conceit, it is for the most part rhetorical and conventional. It always improves in proportion to its approach to what we call symbol." (Johann Wolfgang von Goethe, "Addenda on the Paintings of Philostratus", 1820) 

"A human being, what is a human being? Everything and nothing. Through the power of thought it can mirror everything it experiences. Through memory and knowledge it becomes a microcosm, carrying the world within itself. A mirror of things, a mirror of facts. Each human being becomes a little universe within the universe!" (Guy de Maupassant, [in "The Journal of a Madman"] 1851)

"Observation is like a piece of glass, which, as a mirror, must be very smooth, and must be very carefully polished, in order that it may reflect the image pure and undistorted." (Justus von Liebig, "The Study of the Natural Sciences", 1853) 

"In her manifold opportunities Nature has thus helped man to polish the mirror of [man’s] mind, and the process continues. Nature still supplies us with abundance of brain-stretching theoretical puzzles and we eagerly tackle them; there are more worlds to conquer and we do not let the sword sleep in our hand; but how does it stand with feeling? Nature is beautiful, gladdening, awesome, mysterious, wonderful, as ever, but do we feel it as our forefathers did?" (Sir John A Thomson, "The System of Animate Nature", 1920)

"What a lost person needs is a map of the territory, with his own position marked on it so he can see where he is in relation to everything else. Literature is not only a mirror; it is also a map, a geography of the mind. Our literature is one such map, if we can learn to read it as our literature, as the product of who and where we have been. We need such a map desperately, we need to know about here, because here is where we live. For the members of a country or a culture, shared knowledge of their place, their here, is not a luxury but a necessity. Without that knowledge we will not survive." (Margaret Atwood, "Survival: A Thematic Guide to Canadian Literature", 1972)

"Even a tarnished mirror will shine like a jewel if it is polished. A mind which presently is closed by illusions originating from the innate darkness of life is like a tarnished mirror, but once it is polished it will become clear, reflecting the enlightenment of immutable truth." (Nichiren Daishonin,"On Attaining Buddhahood", 1999)

"Thinking involves reasoning about a situation, and to do that we must have some kind of dynamic "model" of the situation in our heads. Any changes we make to this mental model of the world should ideally mirror changes in the real world." (S Ian Robertson, "Problem Solving", 2001)

"'Mental models' are deeply ingrained assumptions, generalizations, or even pictures or images that influence how we understand the world and how we take action. Very often, we are not consciously aware of our mental models or the effects they have on our behavior. […] The discipline of working with mental models starts with turning the mirror inward; learning to unearth our internal pictures of the world, to bring them to the surface and hold them rigorously to scrutiny. It also includes the ability to carry on ‘learningful’ conversations that balance inquiry and advocacy, where people expose their own thinking effectively and make that thinking open to the influence of others.” (Jossey-Bass Publishers, “The Jossey-Bass Reader on Educational Leadership”, 2nd Edi. 2007)

"If intelligence is a capacity that is gradually acquired as a result of development and learning, then a machine that can learn from experience would have, at least in theory, the capacity to carry out intelligent behavior. [...] Humans have created machines that imitate us - that provide mirrors to see ourselves and measure our strength, our intellect, and even our creativity." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Language accelerates learning and creation by permitting communication and coordination. A new idea can be spread quickly if someone can explain it and communicate it to others before they have to discover it themselves. But the chief advantage of language is not communication but autogeneration. Language is a trick that allows the mind to question itself; a magic mirror that reveals to the mind what the mind thinks; a handle that turns a mind into a tool." (Kevin Kelly, "What Technology Wants", 2010)

"It is a right, yes a duty, to search in cautious manner for the numbers, sizes, and weights, the norms for everything [God] has created. For He himself has let man take part in the knowledge of these things […] For these secrets are not of the kind whose research should be forbidden; rather they are set before our eyes like a mirror so that by examining them we observe to some extent the goodness and wisdom of the Creator." (Johannes Kepler)

Mathematical Trivia II: Mirrors I

"It is impossible to disassociate language from science or science from language, because every natural science always involves three things: the sequence of phenomena on which the science is based; the abstract concepts which call these phenomena to mind; and the words in which the concepts are expressed. To call forth a concept a word is needed; to portray a phenomenon a concept is needed. All three mirror one and the same reality." (Antoine-Laurent Lavoisier, "Traite Elementaire de Chimie", 1789)

"Music is an order of mystic, sensuous mathematics. A sounding mirror, an aural mode of motion, it addresses itself on the formal side to the intellect, in its content of expression it appeals to the emotions." (James Huneker, "Chopin: The Man and His Music", 1900)

"This history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of notational problems of the present time." (Florian Cajori, "A History of Mathematical Notations", 1928)

"Given any domain of thought in which the fundamental objective is a knowledge that transcends mere induction or mere empiricism, it seems quite inevitable that its processes should be made to conform closely to the pattern of a system free of ambiguous terms, symbols, operations, deductions; a system whose implications and assumptions are unique and consistent; a system whose logic confounds not the necessary with the sufficient where these are distinct; a system whose materials are abstract elements interpretable as reality or unreality in any forms whatsoever provided only that these forms mirror a thought that is pure. To such a system is universally given the name Mathematics." (Samuel T Sanders, "Mathematics", National Mathematics Magazine, 1937)

"The only possible alternative is simply to keep to immediate experience that consciousness is a singular of which the plural is unknown; that there is only one thing and that what seems to be a plurality is merely a series of different aspects of this one thing, produced by a deception (the Indian MAJA); the same illusion is produced in a gallery of mirrors, and in the same way Gaurisankar and Mt Everest turned out to be the same peak seen from different valleys." (Erwin Schrödinger, "What Is Life?", 1944)

"Mathematical examination problems are usually considered unfair if insoluble or improperly described: whereas the mathematical problems of real life are almost invariably insoluble and badly stated, at least in the first balance. In real life, the mathematician's main task is to formulate problems by building an abstract mathematical model consisting of equations, which will be simple enough to solve without being so crude that they fail to mirror reality. Solving equations is a minor technical matter compared with this fascinating and sophisticated craft of model-building, which calls for both clear, keen common-sense and the highest qualities of artistic and creative imagination." (John Hammersley & Mina Rees, "Mathematics in the Market Place", The American Mathematical Monthly 65, 1958)

"Mathematics is a self-contained microcosm, but it also has the potentiality of mirroring and modeling all the processes of thought and perhaps all of science. It has always had, and continues to an ever increasing degree to have, great usefulness. One could even go so far as to say that mathematics was necessary for man's conquest of nature and for the development of the human race through the shaping of its modes of thinking." (Mark Kac & Stanislaw M Ulam, "Mathematics and Logic", 1968)

 "Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is 'stretched' in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)

"The impossibility of defining absolute motion can be seen as the manifestation of a symmetry known as relativistic invariance. In the same way that parity invariance tells us that we cannot distinguish the mirror-image world from our world, relativistic invariance tells us that it is impossible to decide whether we are at rest or moving steadily." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"[…] a model is a mathematical representation of the modeler's reality, a way of capturing some aspects of a particular reality within the framework of a mathematical apparatus that provides us with a means for exploring the properties of the reality mirrored in the model." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

The voyage of discovery into our own solar system has taken us from clockwork precision into chaos and complexity. This still unfinished journey has not been easy, characterized as it is by twists, turns, and surprises that mirror the intricacies of the human mind at work on a profound puzzle. Much remains a mystery. We have found chaos, but what it means and what its relevance is to our place in the universe remains shrouded in a seemingly impenetrable cloak of mathematical uncertainty." (Ivars Peterson, "Newton’s Clock", 1993) 

"The word ‘symmetry’ conjures to mind objects which are well balanced, with perfect proportions. Such objects capture a sense of beauty and form. The human mind is constantly drawn to anything that embodies some aspect of symmetry. Our brain seems programmed to notice and search for order and structure. Artwork, architecture and music from ancient times to the present day play on the idea of things which mirror each other in interesting ways. Symmetry is about connections between different parts of the same object. It sets up a natural internal dialogue in the shape." (Marcus du Sautoy,"Symmetry: A Journey into the Patterns of Nature", 2008)

Mathematical Trivia I: Monsters

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

"The orchard of science is a vast globe-encircling monster, without a map, and known to no one man; indeed, to no group of men fewer than the whole international mass of creative scientists. Within it, each observer clings to his own well-known and well-loved clump of trees. If he looks beyond, it is usually with a guilty sigh." (Isaac Asimov, "View from a Height", 1975)

"What were the needs that led me to single out a few of these monsters, calling them fractals, to add some of their close or distant kin, and then to build a geometric language around them? The original need happens to have been purely utilitarian. That links exist between usefulness and beauty is, of course, well known. What we call the beauty of a flower attracts the insects that will gather and spread its pollen. Thus the beauty of a flower is useful - even indispensable - to the survival of its species. Similarly, it was the attractiveness of the fractal images that first brought them to the attention of many colleagues and then of a wide world." (Benoît B Mandelbrot, "Fractals and an Art for the Sake of Science", Leonardo [Supplemental Issue], 1989)

"The term Moonshine [...] has a variety of meanings. It can refer to foolish or naive ideas, but also to the illicit distillation of spirits [...] It gave an impression of dabbling in mysterious matters that might be better left alone, but also had the useful connotation of something shining in reflected light. The true source of light is probably yet to be found, but there were further strange connections to come later [...] The Monster's connections with number theory - the Moonshine connections - had suggested it was a more beautiful and important group of symmetries than first realized. [...] The Moonshine connections between the Monster and number theory have now been placed within a larger theory, but we have yet to grasp the significance of these deep mathematical links with fundamental physics. We have found the Monster, but it remains an enigma. Understanding its full nature is likely to shed light on the very fabric of the universe." (Mark Ronan, "Symmetry and the Monster: One of the greatest quests of mathematics", 2006)

"The Moonshine mystery itself is still unresolved, despite Borcherd's proof! [...] there are facts about the Monster and Moonshine that we don't understand. [...] The method leading to its discovery, brilliant though it was, gave no clue to the Monster's remarkable properties." (Mark Ronan, "Symmetry and the Monster: One of the greatest quests of mathematics", 2006)

"To the average layperson, mathematics is a mass of abstruse formulae and bizarre technical terms (e.g., perverse sheaves, the monster group, barreled spaces, inaccessible cardinals), usually discussed by academics in white coats in front of a blackboard covered with peculiar symbols. The distinction between mathematics and physics is blurred and that between pure and applied mathematics is unknown. But to the professional, these are three different worlds, different sets of colleagues, with different goals, different standards, and different customs." (David Mumford, ["The Best Writing of Mathematics: 2012"] 2012)

"Infinity is a Loch Ness Monster, capturing the imagination with its awe-inspiring size but elusive nature. Infinity is a dream, a vast fantasy world of endless time and space. Infinity is a dark forest with unexpected creatures, tangled thickets and sudden rays of light breaking through. Infinity is a loop that springs open to reveal an endless spiral." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"The rigid electron is in my view a monster in relation to Maxwell's equations, whose innermost harmony is the principle of relativity." (Hermann Minkowski)