On Literature: On Mazes (From Fiction to Science-Fiction)

"A mighty maze! but not without a plan [...]" (Alexander Pope, "An Essay on Man", 1733-34)

"Language gradually varies, and with it fade away the writings of authors who have flourished their allotted time; otherwise, the creative powers of genius would overstock the world, and the mind would be completely bewildered in the endless mazes of literature." (Washington Irving, "The Sketch Book of Geoffrey Crayon", 1819–1820)

"The 'romance' of space - drivel written in the old days. When you’re not blasting, you float in a cramped hotbox, crawl through dirty mazes of greasy pipe and cable to tighten a lug, scratch your arms and bark your shins, get sick and choked up because no gravity helps your gullet get the food down." (Walter M Miller Jr., "Death of a Spaceman", 1954)

"A promise is a direction taken, a self-limitation of choice. [...] if no direction is taken, if one goes nowhere, no change will occur. One’s freedom to choose and to change will be unused, exactly as if one were in jail, a jail of one’s own building, a maze in which no one way is better than any other." (Ursula K. Le Guin, "The Dispossessed: An Ambiguous Utopia", 1974)

"The best maze is the mind.' (Ursula K Le Guin, "Mazes", 1975)

"Any path that narrows future possibilities may become a lethal trap. Humans are not threading their way through a maze; they scan a vast horizon filled with unique opportunities. The narrowing viewpoint of the maze should appeal only to creatures with their noses buried in the sand." (Frank Herbert, "Children of Dune", 1976)

"There is in Fantastica a certain place from which one can go anywhere and which can be reached from anywhere. We call it the Temple of a Thousand Doors. No one has ever seen it from outside. The inside is a maze of doors. Anyone wishing to know it must dare to enter it." (Michael Ende, "The Neverending Story", 1979)

"The truth of life is that every year we get farther away from the essence that is born within us. We get shouldered with burdens, some of them good, some of them not so good. Things happen to us. Loved ones die. People get in wrecks and get crippled. People lose their way, for one reason or another. It's not hard to do, in this world of crazy mazes. Life itself does its best to take that memory of magic away from us." (Robert McCammon, "Boy's Life", 1991)

"The maze itself is the sum of a man's life: choices he makes, dreams he hangs on to. And there at the center, there's a legendary man who had been killed over and over again countless times, but always clawed his way back to life. The man returned for the last time and vanquished all his oppressors in a tireless fury. He built a house. Around that house he built a maze so complicated, only he could navigate through it. I reckon he'd seen enough of fighting." (The Adversary" Westworld (TV series), 2016)

"We depend on nature not only for our physical survival. We also need nature to show us the way home, the way out of the prison of our own minds. We got lost in doing, thinking, remembering, anticipating–lost in a maze of complexity and a world of problems." (Eckhart Tolle, "Stillness Speaks", 2003)

On Structures: On Mazes (1950-1999)

"In mathematics […] we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of materials [….] being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations." (David Hilbert, "Geometry and the imagination", 1952)

"Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses, and of houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight regular streets and uniform houses." (Ludwig Wittgenstein, "Philosophical Investigations", 1953)

"In presenting a mathematical argument the great thing is to give the educated reader the chance to catch on at once to the momentary point and take details for granted: two trivialities omitted can add up to an impasse). The unpractised writer, even after the dawn of a conscience, gives him no such chance; before he can spot the point he has to tease his way through a maze of symbols of which not the tiniest suffix can be skipped." (John E Littlewood, "A mathematicians's miscellany", 1953)

"Man develops his way of anticipating events by construing, by scratching out his channels of thought. Thus he builds his own maze. His runways are the constructs he forms, each a two-way street, each essentially a pair of alternatives between which he can choose." (George A Kelly, "Man's construction of his alternatives", Assessment of human motives, 1958)

"With the help of physical theories we try to find our way through the maze of observed facts, to order and understand the world of our sense impressions." (Leopold Infeld, "The Evolution of Physics, Physics and Reality", 1961) 

"It is not that we propose a theory and Nature may shout NO; rather, we propose a maze of theories, and Nature may shout INCONSISTENT." (Imre Lakatos, "Falsification and the Methodology of Scientific Research Programmes", [in I. Lakatos and A. Musgrave (eds.), "Criticism and the Growth of Knowledge: Proceedings of the International Colloquium in the Philosophy of Science"] 1965)

"Depth First Search is especially appropriate for threading mazes, because it is possible to use it without having a map of the maze. It involves only local rules at nodes, plus a record of nodes and edges already used, so you can explore the graph and traverse it as you go. The name indicates the basic idea: give top priority to pushing deeper into the maze. The number of steps required is at most twice the number of passages in the maze."  (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"One of the best definitions of mathematics is 'the science of patterns'. Mathematics is how we detect, analyse, and classify regular patterns - be they numerical, geometric, or of some other kind. But what is a pattern? A pattern is a landmark in the magical maze. It's one of those things that you recognise when you see it, but it's not so easy to pin down the concept of a pattern once and for all with a neat, tidy, compact characterisation. In fact, the entire development of mathematics can be seen as a slow and erratic broadening of what we accept under the term 'pattern'." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"One very effective approach is to represent all the possible actions as a maze, and try to find a route through it. It is a logical maze rather than a real one, touched with that magic genius of mathematical transformations in which a problem that seems unassailable in one form becomes trivial in another, logically equivalent one. The idea is to represent the problem in a visual manner, using a diagram called a graph. A graph consists of a number of nodes (dots) linked by edges (lines), possibly with arrows on them. Each 'state' of the puzzle - position of the items of produce relative to the river - is represented by a node. Each 'legal' move between states is represented as an edge joining the corresponding nodes. If necessary, arrows can be added to the edges to show which is the starting state and which is the end state. The solution of the puzzle then reduces to tracing a path through its graph, starting from the initial state of the problem and finishing at the desired final state. The graph is a kind of conceptual map of the puzzle - a maze of possible states whose passages are the edges of the graph and whose junctions are its nodes." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"The analogy with threading a maze runs deeper than games and puzzles. It illuminates the whole of mathematics. Indeed, one way to think about mathematics is as an exercise in threading an elaborate, infinitely large maze. A logical maze. A maze of ideas, whose pathways represent 'lines of thought' from one idea to another. A maze which, despite its apparent complexity, has a definite 'geography', to which mathematicians are unusually attuned." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"We wrote down all the states and legal moves (here it turned out to be helpful to have a systematic notation, but that's not essential). Then we formed a graph whose nodes correspond to states and whose edges correspond to legal moves. The solution of the puzzle is then a path through the graph that joins the start to the finish. Such a path is usually obvious to the eye, provided the puzzle is sufficiently simple for the entire graph to be drawn. Puzzles of this type are really mazes, for a maze is just a graph drawn in a slightly different fashion. Metaphorically, they are logical mazes - you have to find the right sequence of moves to solve them. The graph turns the logical maze into a genuine maze, turning the metaphor into reality. The fact that solving the real maze also solves the logical maze is one of the magical features of the maze that is mathematics." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"When a book is being written, it is a maze of possibilities, most of which are never realised. Reading the resulting book, once all decisions have been taken, is like tracing one particular path through that maze. The writer's job is to choose that path, define it clearly, and make it as smooth as possible for those who follow. Mathematics is much the same. Mathematical ideas form a network. The interconnections between ideas are logical deductions. If we assume this, then that must follow - a logical path from this to that. When mathematicians try to understand a problem, they have to thread a maze of logic. The body of knowledge that we call mathematics is a catalogue of interesting excursions through the logical maze."(Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

On Structures: On Mazes (-1899)

"The confirmed prejudices of a thoughtful life are as hard to change as the confirmed habits of an indolent life; and as some must trifle away age because they trifled away youth, others must labor on in a maze of error because they have wandered there too long to find their way out." (Henry St John, "The Philosophical Works of the Late Right Honorable Henry St. John, Lord Viscount Bolingbroke", 1754)

"Observe that part of a beautiful woman where she is perhaps the most beautiful, about the neck and breasts; the smoothness; the softness; the easy and insensible swell; the variety of the surface, which is never for the smallest space the same; the deceitful maze, through which the unsteady eye slides giddily, without knowing where to fix, or whither it is carried. Is not this a demonstration of that change of surface continual and yet hardly perceptible at any point which forms one of the great constituents of beauty?" (Edmund Burke, "A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful", 1757) 

"To expect that the intricacies of science will be pierced by a careless glance, or the eminences of fame ascended without labour, is to expect a peculiar privilege, a power denied to the rest of mankind; but to suppose that the maze is inscrutable to diligence, or the heights inaccessible to perseverance, is to submit tamely to the tyranny of fancy, and enchain the mind in voluntary shackles." (Samuel Johnson, "The Rambler", 1791) 

"Caught up in the limitless maze, the fragmentation and complication of modern natural science, and yearning for the recapture of simplicity, we must forever ask ourselves: Supposing he had known nature in its present state of complexity, a basic unity withal, how would Plato have coped with it?" (Johann Wolfgang von Goethe, "Maxims and Reflections", 1822)

"The world can doubtless never be well known by theory: practice is absolutely necessary; but surely it is of great use to a young man, before he sets out for that country, full of mazes, windings, and turnings, to have at least a general map of it, made by some experienced traveler." (Philip Stanhope, "Letters Written by the Earl of Chesterfield to His Son", 1827)

"Society bristles with enigmas which look hard to solve. It is a perfect maze of intrigue." (Honoré de Balzac, "Letters of Two Brides", 1841-1842) 

"In regard to what I have said of Mill... I should not have expressed so strong an opinion had I been thinking only of his political economy. There is much that is erroneous in his Principles, and he never had an idea what capital was, but the book is not the maze of self-contradictions which his logic undoubtedly is. If you have not examined his logical theories very critically, you will hardly be aware that upon the principal points he usually holds from three to six inconsistent views all at the same time." (William S Jevons, [Letter to H. S. Foxwell] 1875)

"This maze of symbols, electric and magnetic potential, vector potential, electric force, current, displacement, magnetic force, and induction, have been practically reduced to two, electric and magnetic force." (George F Fitzgerald, The Electrician, [bookmreview] 1893)

"The voice of the sea is seductive; never ceasing, whispering, clamoring, murmuring, inviting the soul to wander for a spell in abysses of solitude; to lose itself in mazes of inward contemplation." (Kate Chopin, "The Awakening", 1899)