"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)
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