Ian Stewart - Collected Quotes

"An intuitive proof allows you to understand why the theorem must be true; the logic merely provides firm grounds to show that it is true." (Ian Stewart, "Concepts of Modern Mathematics", 1975)

"Many pages have been expended on polemics in favor of rigor over intuition, or of intuition over rigor. Both extremes miss the point: the power of mathematics lies precisely in the combination of intuition and rigor." (Ian Stewart, "Concepts of Modern Mathematics", 1975)

"A great many problems are easier to solve rigorously if you know in advance what the answer is." (Ian Stewart, "From Here to Infinity", 1987)

"Computer scientists working on algorithms for factorization would be well advised to brush up on their number theory." (Ian Stewart, "Geometry Finds Factor Fast", Nature Vol. 325, 1987) 

"Mathematics is good if it enriches the subject, if it opens up new vistas, if it solves old problems, if it fills gaps, fitting snugly and satisfyingly into what is already known, or if it forges new links between previously unconnected parts of the subject It is bad if it is trivial, overelaborate, or lacks any definable mathematical purpose or direction It is pure if its methods are pure - that is, if it doesn't cheat and tackle one problem while pretending to tackle another, and if there are no gaping holes in its logic It is applied if it leads to useful insights outside mathematics. By these criteria, today's mathematics contains as high a proportion of good work as at any other period, and as any other area, and much of it manages to be both pure and applied at the same time." (Ian Stewart, "The Problems of Mathematics", 1987)

"Symmetries abound in nature, in technology, and - especially - in the simplified mathematical models we study so assiduously. Symmetries complicate things and simplify them. They complicate them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They simplify them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They violate all the hypotheses of our favorite theorems, yet lead to natural generalizations of those theorems. It is now standard to study the 'generic' behavior of dynamical systems. Symmetry is not generic. The answer is to work within the world of symmetric systems and to examine a suitably restricted idea of genericity." (Ian Stewart, "Bifurcation with symmetry", 1988)

"[...] an apparently random universe could be obeying every whim of a deterministic deity who chooses how the dice roll; a universe that has obeyed perfect mathematical laws for the last ten billion years could suddenly start to play truly random dice. So the distinction is about how we model the system, and what point of view seems most useful, rather than about any inherent feature of the system itself." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"Chaos teaches us that anybody, God or cat, can play dice deterministically, while the naïve onlooker imagines that something random is going on." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"If sinks, sources, saddles, and limit cycles are coins landing heads or tails, then the exceptions are a coin landing on edge. Yes, it might happen, in theory; but no, it doesn't, in practice." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"In contrast, the system may be a pack of cards, and the dynamic may be to shuffle the pack and then take the top card. Imagine that the current top card is the ace of spades, and that after shuffling the pack the top card becomes the seven of diamonds. Does that imply that whenever the top card is the ace of spades then the next top card will always be the seven of diamonds? Of course not. So this system is random." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"In modelling terms, the difference between randomness and determinacy is clear enough. The randomness in the pack of cards arises from our failure to prescribe unique rules for getting from the current state to the next one. There are lots of different ways to shuffle a pack. The determinism of the cannonball is a combination of two things: fully prescribed rules of behaviour, and fully defined initial conditions. Notice that in both systems we are thinking on a very short timescale: it is the next state that matters - or, if time is flowing continuously, it is the state a tiny instant into the future. We don't need to consider long-term behaviour to distinguish randomness from determinacy." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"Indeed a deterministic die behaves very much as if it has six attractors, the steady states corresponding to its six faces, all of whose basins are intertwined. For technical reasons that can't quite be true, but it is true that deterministic systems with intertwined basins are wonderful substitutes for dice; in fact they're super-dice, behaving even more ‘randomly’ - apparently - than ordinary dice. Super-dice are so chaotic that they are uncomputable. Even if you know the equations for the system perfectly, then given an initial state, you cannot calculate which attractor it will end up on. The tiniest error of approximation - and there will always be such an error - will change the answer completely." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"Mathematicians are beginning to view order and chaos as two distinct manifestations of an underlying determinism. And neither state exists in isolation. The typical system can exist in a variety of states, some ordered, some chaotic. Instead of two opposed polarities, there is a continuous spectrum. As harmony and discord combine in musical beauty, so order and chaos combine in mathematical [and physical] beauty." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"It's a bit like having a theory about coins that move in space, but only being able to measure their state by interrupting them with a table. We hypothesize that the coin may be able to revolve in space, a state that is neither ‘heads’ nor ‘tails’ but a kind of mixture. Our experimental proof is that when you stick a table in, you get heads half the time and tails the other half - randomly. This is by no means a perfect analogy with standard quantum theory - a revolving coin is not exactly in a superposition of heads and tails - but it captures some of the flavour." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"Mathematicians are beginning to view order and chaos as two distinct manifestations of an underlying determinism. And neither state exists in isolation. The typical system can exist in a variety of states, some ordered, some chaotic. Instead of two opposed polarities, there is a continuous spectrum. As harmony and discord combine in musical beauty, so order and chaos combine in mathematical [and physical] beauty." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"Mathematics is a remarkable sprawling riot of imagination, ranging from pure intellectual curiosity to nuts-and-bolts utility; and it is all one thing." (Ian Stewart, "Game, Set, and Math: Enigmas and Conundrums", 1989)

"Perhaps God can play dice, and create a universe of complete law and order, in the same breath." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing – seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"The randomness of the card-shuffle is of course caused by our lack of knowledge of the precise procedure used to shuffle the cards. But that is outside the chosen system, so in our practical sense it is not admissible. If we were to change the system to include information about the shuffling rule – for example, that it is given by some particular computer code for pseudo-random numbers, starting with a given ‘seed value’ - then the system would look deterministic. Two computers of the same make running the same ‘random shuffle’ program would actually produce the identical sequence of top cards." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1989)

"The flapping of a single butterfly’s wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done." (Ian Stewart, "Does God Play Dice?", 1989)

"[…] a symmetry isn't a thing; it's a transformation. Not any old transformation, though: a symmetry of an object is a transformation that leaves it apparently unchanged." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Chaos demonstrates that deterministic causes can have random effects […] There's a similar surprise regarding symmetry: symmetric causes can have asymmetric effects. […] This paradox, that symmetry can get lost between cause and effect, is called symmetry-breaking. […] From the smallest scales to the largest, many of nature's patterns are a result of broken symmetry; […]" (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"In everyday language, the words 'pattern' and 'symmetry' are used almost interchangeably, to indicate a property possessed by a regular arrangement of more-or-less identical units […]" (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Nature behaves in ways that look mathematical, but nature is not the same as mathematics. Every mathematical model makes simplifying assumptions; its conclusions are only as valid as those assumptions." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Nature is never perfectly symmetric. Nature's circles always have tiny dents and bumps. There are always tiny fluctuations, such as the thermal vibration of molecules. These tiny imperfections load Nature's dice in favour of one or other of the set of possible effects that the mathematics of perfect symmetry considers to be equally possible." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Scientists use mathematics to build mental universes. They write down mathematical descriptions - models - that capture essential fragments of how they think the world behaves. Then they analyse their consequences. This is called 'theory'. They test their theories against observations: this is called 'experiment'. Depending on the result, they may modify the mathematical model and repeat the cycle until theory and experiment agree. Not that it's really that simple; but that's the general gist of it, the essence of the scientific method." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"To a mathematician, an object possesses symmetry if it retains its form after some transformation. A circle, for example, looks the same after any rotation; so a mathematician says that a circle is symmetric, even though a circle is not really a pattern in the conventional sense - something made up from separate, identical bits. Indeed the mathematician generalizes, saying that any object that retains its form when rotated - such as a cylinder, a cone, or a pot thrown on a potter's wheel - has circular symmetry." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"Symmetry is bound up in many of the deepest patterns of Nature, and nowadays it is fundamental to our scientific understanding of the universe. Conservation principles, such as those for energy or momentum, express a symmetry that (we believe) is possessed by the entire space-time continuum: the laws of physics are the same everywhere." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)

"A number is a process that has long ago been thingified so thoroughly that everybody thinks of it as a thing. It is just as feasible-though less familiar to most of us-to think of an operation or a function as a thing. For example, we might talk of "square root" as if it were a thing- and I mean here not the square root of any particular number, but the function itself. In this image, the square-root function is a kind of sausage machine: you stuff a number in at one end and its square root pops out at the other." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"At every turn, new vistas arise-an  unexpected river that must be crossed using stepping stones, a vast, tranquil lake, an impassable crevasse. The user of mathematics walks only the well-trod parts of this mathematical territory. The creator of mathematics explores its unknown mysteries, maps them, and builds roads through them to make them more easily accessible to everybody else." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Each of nature's patterns is a puzzle, nearly always a deep one. Mathematics is brilliant at helping us to solve puzzles. It is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity, and then using those rules and structures to explain what's going on." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Human mind and culture have developed a formal system of thought for recognizing, classifying, and exploiting patterns. We call it mathematics. By using mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"If you start with a number and form its square root, you get another number. The term for such an 'object' is function. You can think of a function as a mathematical rule that starts with a mathematical object-usually a number-and associates to it another object in a specific manner. Functions are often defined using algebraic formulas, which are just shorthand ways to explain what the rule is, but they can be defined by any convenient method. Another term with the same meaning as 'function' is transformation: the rule transforms the first object into the second. […] Operations and functions are very similar concepts. Indeed, on a suitable level of generality there is not much to distinguish them. Both of them are processes rather than things." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Mathematical 'things' have no existence in the real world: they are abstractions. But mathematical processes are also abstractions, so processes are no less 'things' than the "things" to which they are applied. The thingification of processes is commonplace." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Mathematics is not just a collection of isolated facts: it is more like a landscape; it has an inherent geography that its users and creators employ to navigate through what would otherwise be an impenetrable jungle." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"No, nature is, in its own subtle way, simple. However, those simplicities do not present themselves to us directly. Instead, nature leaves clues for the mathematical detectives to puzzle over. It's a fascinating game, even to a spectator. And it's an absolutely irresistible one if you are a mathematical Sherlock Holmes." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Patterns possess utility as well as beauty. Once we have learned to recognize a background pattern, exceptions suddenly stand out." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Proofs knit the fabric of mathematics together, and if a single thread is weak, the entire fabric may unravel." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"The entrepreneur's instinct is to exploit the natural world. The engineer's instinct is to change it. The scientist's instinct is to try to understand it - to work out what's really going on. The mathematician's instinct is to structure that process of understanding by seeking generalities that cut across the obvious subdivisions." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"The image of mathematics raised by this description of its basic objects is something like a tree, rooted in numbers and branching into ever more esoteric data structures as you proceed from trunk to bough, bough to limb, limb to twig…" (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"The ingredient that knits this landscape together is proof. Proof determines the route from one fact to another. To professional mathematicians, no statement is considered valid unless it is proved beyond any possibility of logical error. But there are limits to what can be proved, and how it can be proved. A great deal of work in philosophy and the foundations of mathematics has established that you can't prove everything, because you have to start somewhere; and even when you've decided where to start, some statements may be neither provable nor disprovable." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"[…] the meaning of the word 'solve' has undergone a series of major changes. First that word meant 'find a formula'. Then its meaning changed to 'find approximate numbers'. Finally, it has in effect become 'tell me what the solutions look like'. In place of quantitative answers, we seek qualitative ones." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"The real numbers are one of the most audacious idealizations made by the human mind, but they were used happily for centuries before anybody worried about the logic behind them. Paradoxically, people worried a great deal about the next enlargement of the number system, even though it was entirely harmless. That was the introduction of square roots for negative numbers, and it led to the 'imaginary' and 'complex' numbers. A professional mathematician should never leave home without them […]" (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"The story of calculus brings out two of the main things that mathematics is for: providing tools that let scientists calculate what nature is doing, and providing new questions for mathematicians to sort out to their own satisfaction. These are the external and internal aspects of mathematics, often referred to as applied and pure mathematics." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"There is much beauty in nature's clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"Whatever the reasons, mathematics definitely is a useful way to think about nature. What do we want it to tell us about the patterns we observe? There are many answers. We want to understand how they happen; to understand why they happen, which is different; to organize the underlying patterns and regularities in the most satisfying way; to predict how nature will behave; to control nature for our own ends; and to make practical use of what we have learned about our world. Mathematics helps us to do all these things, and often it is indispensable." (Ian Stewart, "Nature's Numbers: The unreal reality of mathematics", 1995)

"A line normally represents space, but if we reinterpret that line as representing the passage of time, then presumably we are looking at 'shapes that repeat indefinitely along a temporal line at regular intervals'. Well, nearly. 'Shape' is the wrong word for something that exists in time. What word should we use? Let's reason it out by analogy - one of the sharpest weapons in the mathematical armoury. A shape is a (more or less) complicated collection of points in space; the temporal analogue is a complicated collection of points in time. A point in time is usually called an 'event'. So instead of 'shape' we have a 'series of events'. That is, the temporal pattern corresponding to the simplest frieze is a series of events that repeats indefinitely at regular intervals of time." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"A pendulum swings in a circle, so there is a lowest point that it can possibly reach - hanging vertically downwards. Any change in potential energy must be compensated by an equal and opposite change in kinetic energy: if the bob gets higher, it has to move more slowly; if it gets lower, it has to move faster. Imagine grabbing the pendulum by its bob and raising it to some chosen height. Hold it steady. Your hand did some work, and increased the bob's potential energy. The kinetic energy is zero because the pendulum is not moving. Now let go, and from now on let the pendulum move as it wishes. Gravity pulls the bob downwards, so the potential energy decreases. Kinetic energy must increase to compensate, so the weight starts to fall. At some stage the bob must reach the lowest point of the circle: here its potential energy is zero, so all of the energy has become kinetic. Not only is it moving at this point - it is moving as fast as it can possibly go." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997) 

"A single snapshot of the pattern. A fixed spiral is elegant in form, but it does not possess symmetry in the mathematical sense. However, a rotating spiral does possess a dynamic symmetry - as does any continuously rotating object. It is a symmetry that takes place in both space and time. Start with the spiral, let time pass. It rotates to a new position. Now rotate it backwards through the appropriate angle, and it looks exactly as it did to begin with. This is a space-time symmetry: translate time and rotate space. Indeed, the rotating spiral has lots of space-time symmetries, because every period of time corresponds to some spatial rotation." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"Algebra is a kind of symbolic reasoning that works with numbers without knowing their actual values. Proofs provide a once-and-for-all guarantee that certain lines of reasoning always work." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"Any mathematical system that varies periodically in time is called an oscillator. 'Oscillate' is a fancy word for 'wobble'. Oscillators are things that wobble, things that go up and down, things that go from side to side, things that rotate. The Pattern of Tiny Feet 141 The world is full of oscillators. Physical oscillators range from galaxies, which revolve in a stately dance that takes billions of years to complete one turn, to subatomic particles, whose period of oscillation is a tiny, tiny fraction of a second." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"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)

"Dynamics involves a crucial element: time. Symmetry is about shapes, which are things in space. How can symmetries apply to time? [...] We can even put both together, and look at patterns in space and time combined. Turning space into time is a typical mathematician's trick, a kind of 'technology transfer'. However, the 'technology' involved is conceptual technology - ideas - rather than gadgetry. No mathematician would ever waste a good idea by restricting it to just one incarnation. Many physically different systems possess an underlying mathematical identity: they obey rules which, in the abstract, are exactly the same - and must therefore lead to corresponding conclusions." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"From a physical point of view, time is very different from space - and this difference is often built into mathematical systems that model the physical world. Sometimes, however, we can exploit the very general nature of mathematics, and from this point of view a spatial distance and an interval of time are simply the values of certain numerical quantities, or variables. The difference between space and time then becomes a matter of interpretation - the same underlying mathematics can have several different meanings." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"However, random walk theory also tells us that the chance that the balance never returns to zero - that is, that H stays in the lead for ever - is 0. This is the sense in which the 'law of averages' is true. If you wait long enough, then almost surely the numbers of heads and tails will even out. But this fact carries no implications about improving your chances of winning, if you're betting on whether H or T turns up. The probabilities are unchanged, and you don't know how long the 'long run' is going to be. Usually it is very long indeed." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"In fact, mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that - by some mysterious agency - capture patterns of the universe around us?" (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"It is also magical. In fact, mathematics is the closest that we humans get to true magic. How else to describe patterns in our heads that - by some mysterious agency - capture patterns of the universe around us? Mathematics connects ideas that otherwise seem totally unrelated, revealing deep similarities that subsequently show up in nature. Even though mathematics is 'just' a creation of the human mind, it has given us enormous power over the world we inhabit. In mathematics, you can set out to understand the notes played by a violin, and end up inventing television. That's real magic." (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)

"Our job is to see why the numbers turn out as they do. Along the way we shall come to understand why our 'intuition' about these numbers is badly off the mark. In fact, our intuition about probabilities often leads us astray - in many ways probability seems to be one of the areas in which human intuition is the least reliable. One consequence of this is that we tend to be unnecessarily impressed by 'coincidences' that are actually fairly likely. Another is that we adopt nonsensical strategies when betting on such things as horse races or lotteries. Our geometric intuition, based on our visual sense, is much better matched to reality. Perhaps the reason for the difference is that our evolutionary history has seldom placed us in situations where an intuitive grasp of probabilities adds much survival value. Usually a crude division into 'impossible', 'very unlikely', 'might happen', 'probably will happen', or 'certain' is enough." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"Probabilities aren't just numbers, and they aren't just frequencies-on- average. They are also rather like a substance that flows, dividing according to the likelihood of various outcomes, subdividing when several trials are performed in succession, and adding together when several outcomes are combined to give an event. This is a metaphor, but an accurate and powerful one. It is, in a sense, the metaphor that mathematicians formalise when they offer a definition of probability. In this sense, probability behaves like volume, mass, or area. The technical term is 'measure'. The technical definition of probability is 'a measure such that various nice things happen'. Probability is a quantity that flows through the conceptual maze of possible events, and it behaves just like water flowing through pipes." (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)

"The basis of many misconceptions about probability is a belief in something usually referred to as 'the law of averages', which alleges that any unevenness in random events gets ironed out in the long run. For example, if a tossed coin keeps coming up heads, then it is widely believed that at some stage there will be a predominance of tails to balance things out." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"The basis of many misconceptions about probability is a belief in something usually referred to as 'the law of averages', which alleges that any unevenness in random events gets ironed out in the long run. For example, if a tossed coin keeps coming up heads, then it is widely believed that at some stage there will be a predominance of tails to balance things out." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"The first step is to simplify the problem, by reducing it to its essential features. Mathematics always works best when its raw materials are as simple as possible: extraneous information [...] should be resolutely excluded. [...] The essential features can be captured by introducing a nice, compact notation. Symbolism isn't everything in mathematics, but it often suggests useful ideas - and here the right symbolism pays dividends." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"The 'law of averages' asserts itself not by removing imbalances, but by swamping them. Random walk theory tells us that if you wait long enough - on average, infinitely long - then eventually the numbers will balance out. If you stop at that very instant, then you may imagine that your intuition about a 'law of averages' is justified. But you're cheating: you stopped when you got the answer you wanted. Random walk theory also tells us that if you carry on for long enough, you will reach a situation where the number of H's is a billion more than the number of T's." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"The magic of mathematics is subtle and deep. It reaches far beyond the simplest and most obvious of patterns. The mathematics of imperfections is just as useful - and in many ways more beautiful - than that of perfect simplicities." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"The word 'symmetry' is used in ordinary language in at least two different ways. One carries the connotation 'having a well-balanced form' - for example, we might praise the symmetry of a painting or a sculpture, but in very general terms, implying little more than a feeling that the work is rather well designed. The second meeting of 'symmetry' is much more specific, and not as closely related to aesthetics. We say that a form has symmetry if different parts of it repeat exactly the same shape or pattern." (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)

"What is a mathematician - someone who does mathematics ? Not exactly. That's too easy an answer, and it creates too simple a maze - a circular loop of self-referential logic. No, a mathematician is more than just somebody who does mathematics. Think of it this way: what is a businessman ? Someone who does business ? Yes, but not just that. A businessman is someone who sees an opportunity for doing business where the rest of us see nothing: while we're complaining that there's no restaurant in the area, he's organising a telephone pizza delivery service. Similarly, a mathematician is someone who sees opportunities for doing mathematics that the rest of us miss." (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)

"Where did those symmetries come from? From the even more extensive set of symmetries of the (idealised) uniform state in the infinite dish. The instability of that state caused certain symmetries to be eliminated, but others persist. For target patterns, some rotations and reflections persist. For spirals, what persists is the space-time symmetries 'let time pass and then rotate back'. In a very curious sense, the patterns that we see in the spirals are evidence of other patterns that might have been - the unstable uniform state with its enormous amount of (totally boring) symmetry. They are 'caused' by something that doesn't actually happen." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"Why don't the chemicals take up the fully symmetric uniform state? Because it is unstable. Any tiny lack of uniformity grows, and destroys the uniform pattern. And in the real world there are always tiny lacks of uniformity - dust motes, bubbles, even just a few molecules vibrating because of heat. (All molecules vibrate because of heat - or more accurately 'heat' is what you get when molecules vibrate - but it only takes a few of them to trigger instability.) The instability is not intuitively obvious, but it's what happens both in the real world and in mathematical models, and here we can take it as given." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"At any rate, the role of science is not to establish some kind of factual data-bank about nature, but to help us understand nature. There are many different degrees of understanding, and many different kinds of understanding. Some aspect of nature may be not understood, or slightly understood, or fairly well understood – although it is unlikely that it will be fully understood. More subtly, it may be well understood from one point of view but an impenetrable mystery from another." (Ian Stewart & Jack Cohen, "Figments of reality: The evolution of the curious mind" 2nd Ed., 1999)

"Our brains are adaptable, general purpose machines. Instead of making specialist ‘hardwired’ reflex responses, they can adapt, learn, and change. The story of brains is one of Murphic resonance: we happen to inhabit the kind of universe in  which all the ingredients for making brains exist, and brains are such a useful trick that in such a universe they will inevitably turn up. There are good reasons why evolution might favour adaptable general-purpose nervous systems – and whatever evolution wants, and can get, it does get." (Ian Stewart & Jack Cohen, "Figments of reality: The evolution of the curious mind" 2nd Ed., 1999)

"Science observes the game of Nature, played out on the vast board of the universe, and tries to deduce the rules. Reductionism works whenever the part that is being observed resembles a dream game, where simple rules give rise to simple strategies. It breaks down when it runs into a patch of nightmare game, with simple rules but complex strategies." (Ian Stewart & Jack Cohen, "Figments of reality: The evolution of the curious mind" 2nd Ed., 1999)

"The conventional structure attributed to science is that of a body of knowledge which evolves by a process of theory and experiment. Experiments test potential weakness in theories, and theories suggest experiments to be performed. A theory that does not fit observations must be rejected, or at least modified - unless it can be shown that the observations are incorrect or have been misinterpreted, in which case an improved experiment is in order. In this view, theory and experiment seem to be on much the same footing; indeed in universities throughout the world you will find departments of theoretical physics alongside departments of experimental physics, and the two are treated by the authorities in much the same way. However, this symmetry is illusory. The role of experiments is clear and simple: it is to ensure that people do not believe things to be true merely because they want them to be. Experiments provide a ‘reality check’ that helps us avoid one of the great pitfalls of the human psyche: the belief that we know the answer before we start working on the problem." (Ian Stewart & Jack Cohen, "Figments of reality: The evolution of the curious mind" 2nd Ed., 1999)

"To many people, science is seen as a source of certainty, a box full of answers that can be trotted out when dealing with life’s many questions. Most working scientists, however, see their subject in a very different light: as a method for navigating effectively in an uncertain world. Whatever science may be, it is not just a matter of assembling ‘the facts’. The outside world seldom presents us with unequivocal facts; instead it provides a variety of indicators whose interpretation is usually open to debate." (Ian Stewart & Jack Cohen, "Figments of reality: The evolution of the curious mind" 2nd Ed., 1999)

"We lead a dual existence - in nature but not of it, perpetually reacting to our estimate of what the world will be rather than what it is right now. We mirror the world outside us with another in our heads: our perceptions of that world. It’s a distorting mirror, an imperfect representation, but to us it seems real. In a funny self-centred way we see ourselves as existing slightly to one side of the rest of the universe. We are in control of our world, we can make choices, we have minds that we can make up or change. Everything else is just following the inexorable impulses of nature." (Ian Stewart & Jack Cohen, "Figments of reality: The evolution of the curious mind" 2nd Ed., 1999)

"A mathematical circle, then, is something more than a shared delusion. It is a concept endowed with extremely specific features; it 'exists' in the sense that human minds can deduce other properties from those features, with the crucial caveat that if two minds investigate the same question, they cannot, by correct reasoning, come up with contradictory answers." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"A proof provides a cast-iron guarantee that some idea is correct. No amount of experimental evidence can substitute for that." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"As caricatured by applied mathematicians, pure math is abstract ivory-tower intellectual nonsense with no practical implications. Applied math, respond the pure-math diehards, is intellectually sloppy, lacks rigor, and substitutes number crunching for understanding. Like all good caricatures, both statements contain grains of truth, but you should not take them literally."  (Ian Stewart, "Letters to a Young Mathematician", 2006)

"By 'network' I mean a set of dynamical systems that are 'coupled together', with some influencing the behavior of others. The systems themselves are the nodes of the network- think of them as blobs - and two nodes are joined by an arrow if one of them (at the tail end) influences the other (at the head end). For example, each node might be a nerve cell in some organism, and the arrows might be connections along which signals pass from one cell to another." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Definitions pin things down, they limit the prospects for creativity and diversity. A definition, implicitly, attempts to reduce all possible variations of a concept to a single pithy phrase." (Ian Stewart, "Letters to a Young Mathematician", 2006)

 "Even when it comes to something as simple as counting, we mathematicians see the world differently from other folk." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Like many mathematicians, I get my inspiration from nature. Nature may not look very mathematical; you don't see sums written on the trees. But math is not about sums, not really. It's about patterns and why they occur. Nature's patterns are both beautiful and inexhaustible." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Math is actually very important, but because it genuinely is difficult, nearly all of the teaching slots are occupied with making sure that students learn how to solve certain types of problem and get the answers right. There isn't time to tell them about the history of the subject, about its connections with our culture and society, about the huge quantity of new mathematics that is created every year, or about the unsolved questions, big and little, that litter the mathematical landscape." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Mathematical facts fit together and lead, via logic, to new facts. A deduction is only as strong as its weakest link. To be safe, all weak links must be removed." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Mathematicians do not spend most of their time doing numerical calculations, even though calculations are sometimes essential to making progress. They do not Letters to a Young Mathematician occupy themselves with grinding out symbolic formulas, but formulas can nonetheless be indispensable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Mathematicians need proofs to keep them honest. All technical areas of human activity need reality checks. It is not enough to believe that something works, that it is a good way to proceed, or even that it is true. We need to know why it's true. Otherwise, we don't know anything at all." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Nature is always deeper, richer, and more interesting than you thought, and mathematics gives you a very powerful way to appreciate this." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Now let me explain a wonderful thing: the more mathematics you learn, the more opportunities you will find for asking new questions. As our knowledge of mathematics grows, so do the opportunities for fresh discoveries. This may sound unlikely, but it is a natural consequence of how new mathematical ideas build on older ones." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Once you understand a problem, many aspects of it suddenly become much simpler. As mathematicians the world over say, everything is either impossible or trivial." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Physicists use mathematics to study what they amusingly call the real world. It is real, in a sense, but much of physics addresses rather artificial aspects of reality, such as a lone electron or a solar system with only one planet. Physicists are often scathing about proofs, partly out of fear, but also because experiment gives them a very effective way to check their assumptions and calculations." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Sometimes in math the best way to make progress is to introduce simplifications [...]. The simplifications are not assertions about the outside world: they are ways to restrict the domain of discourse, to keep it manageable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"The further we push out the boundaries of mathematics, the bigger the boundary itself becomes. There is no danger that we will ever run out of new problems to solve." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"The mathematician's circle, with its infinitely thin circumference and a radius that remains constant to infinitely many decimal places, cannot take physical form. If you draw it in sand, as Archimedes did, its boundary is too thick and its radius too variable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Then there's the inner beauty of mathematics, which should not be underrated. Math done 'for its own sake' can be exquisitely beautiful and elegant. Not the 'sums' we all do at school; as individuals those are mostly ugly and formless, although the general principles that govern them have their own kind of beauty. It's the ideas, the generalities, the sudden flashes of insight, the realization that trying to trisect an angle with straightedge and compass is like trying to prove that 3 is an even number, that it makes perfect sense that you can't construct a regular seven-sided polygon but you can construct one with seventeen sides, that there is no way to untie an overhand knot, and why some infinities are bigger than others whereas some that ought to be bigger are actually equal [...]" (Ian Stewart, "Letters to a Young Mathematician", 2006)

"What is mathematics? It is the shared social construct created by people who are aware of certain opportunities, and we call those people mathematicians. The logic is still slightly circular, but mathematicians can always recognize a fellow spirit. Find out what that fellow spirit does; it will be one more aspect of our shared social construct." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"When you are doing math, it feels as though what you are working on is real. You can almost pick things up and turn them around, squash them and stroke them and pull them to pieces. On the other hand, you often make progress by forgetting what it all means and focusing solely on how the symbols dance." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Equations are the mathematician's way of working out the value of some unknown quantity from circumstantial evidence. ‘Here are some known facts about an unknown number: deduce the number.’ An equation, then, is a kind of puzzle, centered upon a number. We are not told what this number is, but we are told something useful about it. Our task is to solve the puzzle by finding the unknown number." (Ian Stewart, "Why Beauty Is Truth", 2007)

"The complex numbers extend the real numbers by throwing in a new kind of number, the square root of minus one. But the price we pay for being able to take square roots of negative numbers is the loss of order. The complex numbers are a complete system but are spread out across a plane rather than aligned in a single orderly sequence." (Ian Stewart, "Why Beauty Is Truth", 2007)

"When you get to know them, equations are actually rather friendly. They are clear, concise, sometimes even beautiful. The secret truth about equations is that they are a simple, clear language for describing certain ‘recipes’ for calculating things." (Ian Stewart, "Why Beauty Is Truth", 2007)

"A complex number is just a pair of real numbers, manipulated according to a short list of simple rules. Since a pair of real numbers is surely just as ‘real’ as a single real number, real and complex numbers are equally closely related to reality, and ‘imaginary’ is misleading." (Ian Stewart, "Why Beauty Is Truth", 2007)

"Equations have hidden powers. They reveal the innermost secrets of nature. […] The power of equations lies in the philosophically difficult correspondence between mathematics, a collective creation of human minds, and an external physical reality. Equations model deep patterns in the outside world. By learning to value equations, and to read the stories they tell, we can uncover vital features of the world around us." (Ian Stewart, "In Pursuit of the Unknown", 2012)

 

"There are two kinds of equations in mathematics, which on the surface look very similar. One kind presents relations between various mathematical quantities: the task is to prove the equation is true. The other kind provides information about an unknown quantity, and the mathematician’s task is to solve it - to make the unknown known." (Ian Stewart, "In Pursuit of the Unknown", 2012)

"A proof is a story told to and dissected by people who have spent much of their life learning how to read such stories and find mistakes or inconsistencies: people whose main aim is to prove the storyteller wrong, and who possess the uncanny knack of spotting weaknesses and hammering away at them until they collapse in a cloud of dust." (Ian Stewart, "Visions of Infinity", 2013)

"A symmetry of some mathematical structure is a transformation of that structure, of a specified kind, that leaves specified properties of the structure unchanged." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"Although the four colour problem is ostensibly about maps, it has no useful applications to cartography. Practical criteria for colouring maps mainly reflect political differences, and if that means that adjacent regions must have the same colour, so be it. The problem’s interest lay entirely within pure mathematics, in a new area that had only just started to develop: topology. This is ‘rubber-sheet geometry’ in which shapes can be deformed in any continuous manner. But even there, the four colour problem didn’t belong to the mainstream. It seemed to be no more than a minor curiosity." (Ian Stewart, "Visions of Infinity", 2013)

"An algebraic variety, be it real, complex, projective, or not, is a topological space. Therefore it has a shape. To find out useful things about the shape, we think like topologists and calculate the homology and cohomology groups. But the natural ingredients in algebraic geometry aren’t geometric objects like triangulations and cycles. They are the things we can most easily describe by algebraic equations." (Ian Stewart, "Visions of Infinity", 2013)

"Commutativity is a very pleasant mathematical property. Its absence is a bit of a nuisance, and this is just one of the reasons why quantising a field turns out to be tricky. Nonetheless, it can sometimes be done." (Ian Stewart, "Visions of Infinity", 2013)

"Having agreed on the axioms, a proof of some statement is a series of steps, each of which is a logical consequence of either the axioms, or previously proved statements, or both. In effect, the mathematician is exploring a logical maze, whose junctions are statements and whose passages are valid deductions. A proof is a path through the maze, starting from the axioms. What it proves is the statement at which it terminates." (Ian Stewart, "Visions of Infinity", 2013)

"Hermite’s transcendence proof is magical, a rabbit extracted with a flourish from the top hat of analysis. The rabbit is a complicated formula related to a hypothetical algebraic equation that e is assumed to satisfy. Using algebra, Hermite proves that this formula is equal to some nonzero integer. Using analysis, he proves that it must lie between - 1/2 and 1/2. Since the only integer in this range is zero, these results are contradictory. Therefore the assumption that e satisfies an algebraic equation must be false, so e is transcendental." (Ian Stewart, "Visions of Infinity", 2013)

"Homology and cohomology don’t tell us everything we would like to know about the shape of a topological space - distinct spaces can have the same homology and cohomology – but they do provide a lot of useful information, and a systematic framework in which to calculate it and use it." (Ian Stewart, "Visions of Infinity", 2013)

"Homotopy is a geometrical construction that can be carried out entirely inside the space, and it provides information on the topological type of that space. It does that using an abstract algebraic structure known as a group. A group is a collection of mathematical objects, any two of which can be combined to give another object in the group. This law of combination - often called multiplication or addition, even when it’s not the usual arithmetical operation with that name - is required to satisfy a few simple and natural conditions." (Ian Stewart, "Visions of Infinity", 2013)

"If our machines are better at some things than we are, it makes sense to use machines. Proof techniques may change, but they do that all the time anyway: it’s called ‘research’. The concept of proof does not radically alter if some steps are done by a computer. A proof is a story; a computer-assisted proof is a story that’s too long to be told in full, so you have to settle for the executive summary and a huge automated appendix." (Ian Stewart, "Visions of Infinity", 2013)

"In a loose analogy, every finite symmetry group can be broken up, in a well-defined manner, into ‘indivisible’ symmetry groups - atoms of symmetry, so to speak. These basic building blocks for finite groups are known as simple groups - not because anything about them is easy, but in the sense of ‘not made up from several parts’. Just as atoms can be combined to build molecules, so these simple groups can be combined to build all finite groups." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"In homotopy, we ask whether a given loop can be shrunk continuously to a point. In homology, we ask a different question: does the loop form the boundary of a topological disc? That is, can you fit one or more triangular patches together so that the result is a region without any holes, and the boundary of this region is the loop concerned?" (Ian Stewart, "Visions of Infinity", 2013)

"It is often difficult to reconstruct how the mathematicians of the past arrived at new discoveries, because they had a habit of presenting only the final outcome of their deliberations, not the many false steps they took along the way. This problem is often compounded, because the natural thought patterns in past ages were different from today’s." (Ian Stewart, "Visions of Infinity", 2013) 

"Many of the great mathematical problems stem from deep and difficult questions in well-established areas of the subject. They are the big challenges that emerge when a major area has been thoroughly explored. They tend to be quite technical, and everyone in the area knows they’re hard to answer, because many experts have tried and failed. The area concerned will already possess many powerful techniques, massive mathematical machines whose handles can be cranked if you’ve done your homework - but if the problem is still open, then all of the plausible ways to use those! techniques have already been tried, and they didn’t work. So either there is a less plausible way to use the tried-and-tested techniques of the area, or you need new techniques." (Ian Stewart, "Visions of Infinity", 2013) 

"Mathematical intuition is the mind’s ability to sense form and structure, to detect patterns that we cannot consciously perceive. Intuition lacks the crystal clarity of conscious logic, but it makes up for that by drawing attention to things we would never have consciously considered." (Ian Stewart, "Visions of Infinity", 2013)

"Mathematical symmetry is an idealized model. However, slightly imperfect symmetry requires explanation; it’s not enough just to say ‘it’s asymmetric’."(Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"Often the key contribution of intuition is to make us aware of weak points in a problem, places where it may be vulnerable to attack. A mathematical proof is like a battle, or if you prefer a less warlike metaphor, a game of chess. Once a potential weak point has been identified, the mathematician’s technical grasp of the machinery of mathematics can be brought to bear to exploit it." (Ian Stewart, "Visions of Infinity", 2013)

"Proof, in fact, is the requirement that makes great problems problematic. Anyone moderately competent can carry out a few calculations, spot an apparent pattern, and distil its essence into a pithy statement. Mathematicians demand more evidence than that: they insist on a complete, logically impeccable proof. Or, if the answer turns out to be negative, a disproof. It isn’t really possible to appreciate the seductive allure of a great problem without appreciating the vital role of proof in the mathematical enterprise. Anyone can make an educated guess. What’s hard is to prove it’s right. Or wrong." (Ian Stewart, "Visions of Infinity", 2013)

"Rotations and translations are global symmetries: they apply uniformly across the whole of space and time. A rotation about some axis rotates every point in space through the same angle. Gauge symmetries are different: they are local symmetries, which can vary from point to point in space. In the case of electromagnetism, these local symmetries are changes of phase. A local oscillation of the electromagnetic field has both an amplitude (how big it is) and a phase (the time at which it reaches its peak)." (Ian Stewart, "Visions of Infinity", 2013)

"Solving great problems requires a deep understanding of mathematics, plus persistence and ingenuity. It can involve years of concentrated effort, most of it apparently fruitless." (Ian Stewart, "Visions of Infinity", 2013)

"Squaring the circle sounds like a problem in geometry. That’s because it is a problem in geometry. But its solution turned out to lie not in geometry at all, but in algebra. Making unexpected connections between apparently unrelated areas of mathematics often lies at the heart of solving a great problem." (Ian Stewart, "Visions of Infinity", 2013)

"The classical theories of space, time, and matter were brought to their peak in James Clerk Maxwell’s equations for electromagnetism. This elegant system of equations unified two of nature’s forces, previously thought to be distinct. In place of electricity and magnetism, there was a single electromagnetic field. A field pervades the whole of space, as if the universe were filled with some kind of invisible fluid. At each point of space we can measure the strength and direction of the field, as if that fluid were flowing in mathematical patterns. For some purposes the electromagnetic field can be split into two components, the electric field and the magnetic field. But a moving magnetic field creates an electric one, and conversely, so when it comes to dynamics, both fields must be combined into a single more complex one." (Ian Stewart, "Visions of Infinity", 2013)

"The four colour problem is [...] equivalent to a different question: Given a network in the plane whose lines do not cross, is it possible to 4-colour the dots, so that two dots joined by a line always have different colours? The same reformulation applies with any number of colours." (Ian Stewart, "Visions of Infinity", 2013)

"The group of transformations of space-time that fixes the origin and leaves the interval invariant is called the Lorentz group after the physicist Hendrik Lorentz. The Lorentz group specifies how relative motion works in relativity, and is responsible for the theory’s counterintuitive features in which objects shrink, time slows down, and mass increases, as a body nears the speed of light." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"The most wonderful feature of elliptic functions outdoes trigonometric functions in a 

spectacular way. Not only are elliptic functions periodic: they are doubly periodic. A line is one-dimensional, so patterns can repeat in only one direction, along the line. The complex plane is twodimensional, so patterns can repeat like wallpaper: down the strip of paper, and also sideways along the wall into adjacent strips of paper. Associated with each elliptic function are two independent complex numbers, its periods, and adding either of them to the variable does not change the value of the function." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"[…] the symmetry group of the infinite logarithmic spiral is an infinite group, with one element for each real number . Two such transformations compose by adding the corresponding angles, so this group is isomorphic to the real numbers under addition." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"There are highly symmetric tiling patterns in hyperbolic geometry. For each of them, we can construct complex functions that repeat the same values on every tile. These are known as modular functions, and they are natural generalisations of elliptic functions. Hyperbolic geometry is a very rich subject, and the range of tiling patterns is much more extensive than it is for the Euclidean plane. So complex analysts started thinking seriously about non-Euclidean geometry. A profound link between analysis and number theory then appeared. Modular functions do for elliptic curves what trigonometric functions do for the circle." (Ian Stewart, "Visions of Infinity", 2013)

"We can find the minimax strategy by exploiting the game’s symmetry. Roughly speaking, the minimax strategy must have the same kind of symmetry." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"What makes a great mathematical problem great? Intellectual depth, combined with simplicity and elegance. Plus: it has to be hard. Anyone can climb a hillock; Everest is another matter entirely. A great problem is usually simple to state, although the terms required may be elementary or highly technical." (Ian Stewart, "Visions of Infinity", 2013)

"When a mathematical conjecture eventually turns out to be correct, its history often follows a standard pattern. Over a period of time, various people prove the conjecture to be true provided special restrictions apply. Each such result improves on the previous one by relaxing some restrictions, but eventually this process runs out of steam. Finally, a new and much cleverer idea completes the proof." (Ian Stewart, "Visions of Infinity", 2013)

"When we think of mathematics, what springs to mind is endless pages of dense symbols and formulas. However, those two million pages generally contain more words than symbols. The words are there to explain the background to the problem, the flow of the argument, the meaning of the calculations, and how it all fits into the evergrowing edifice of  mathematics." (Ian Stewart, "Visions of Infinity", 2013)

"A mathematical concept, then, is an organised pattern of ideas that are somehow interrelated, drawing on the experience of concepts already established. Psychologists call such an organised pattern of ideas a ‘schema’." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Although it is certainly possible to build up the whole of mathematics by axiomatic methods starting from the empty set, using no outside information whatsoever, it is also totally unintelligible to anyone who does not already understand the mathematics being built up." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Complex numbers do not fit readily into many people’s schema for ‘number’, and students often reject the concept when it is first presented. Modern mathematicians look at the situation with the aid of an enlarged schema in which the facts make sense." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"In the applications of mathematics, infinitesimal quantities are often considered as variable points on the real number line. Cauchy took this viewpoint by defining an infinitesimal to be a variable quantity that becomes arbitrarily small. In modern notation, this idea can be represented as a null sequence, which is simply a sequence that tends to zero." (Ian Stewart & David Tall,"The Foundations of Mathematics" 2nd Ed., 2015)

"It is not that the human mind cannot think logically. It is a question of different kinds of understanding. One kind of understanding is the logical, step-by-step way of understanding a formal mathematical proof. Each individual step can be checked but this may give no idea how they fit together, of the broad sweep of the proof, of the reasons that lead to it being thought of in the first place. Another kind of understanding arises by developing a global viewpoint, from which we can comprehend the entire argument at a glance. This involves fitting the ideas concerned into the overall pattern of mathematics, and linking them to similar ideas from other areas." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Newton explained naturally occurring phenomena, such as gravity and the movement of the planets, using a combination of Greek geometry and symbolic algebra to build his ideas in the calculus. Leibniz imagined quantities that could be infinitesimally small and produced a powerful symbolism for the calculus that has stood the test of time, despite widespread concerns about its logical foundation. Later giants in mathematical development focused on different aspects. Euler manipulated symbols algebraically using power series and complex numbers, and Cauchy imagined infinitesimals geometrically as variable quantities on the line or in the plane that become arbitrarily small. His approach led to major advances, using a blend of visual and symbolic methods in real and complex analysis, but it also generated significant criticism about the precise meaning. The critics had a point: the meaning had not then been fully worked out. What prevailed was more an act of faith, that everything would work out much as it always had done." (Ian Stewart & David Tall,"The Foundations of Mathematics" 2nd Ed., 2015)

"The essential quality of mathematics that binds it together in a coherent way is the use of mathematical proof to deduce new results from known ones, building up a strong and consistent theory." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"The human mind builds up theories by recognising familiar patterns and glossing over details that are well understood, so that it can concentrate on the new material. In fact it is limited by the amount of new information it can hold at any one time, and the suppression of familiar detail is often essential for a grasp of the total picture. In a written proof, the step-by-step logical deduction is therefore foreshortened where it is already a part of the reader’s basic technique, so that they can comprehend the overall structure more easily." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"When we extend the system of natural numbers and counting to embrace infinite cardinals, the larger system need not have all of the properties of the smaller one. However, familiarity with the smaller system leads us to expect certain properties, and we can become confused when the pieces don’t seem to fit. Insecurity arose when the square of a complex number violated the real number principle that all squares are positive. This was resolved when we realised that the complex numbers cannot be ordered in the same way as their subset of reals." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)