David Wells - Collected Quotes

"All the [mathematical] models that we have considered […] have been rough and ready. They have all been obviously crude approximations, and no one supposes that they are anything more. This does not mean that they are useless - far from it - but it does mean that the answers they give to practical questions are also approximations. There is a pragmatic payoff here between the use of simple models which give good-enough answers which are good value for money, and the use of much more sophisticated models which are more powerful, but also more complex to use, perhaps requiring more advanced mathematics and the use of computers." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"By practice and experience, much of it vicarious, through studying the achievements of others, we develop the strength to tackle novel and unfamiliar situations." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Chess players recognize and applaud good play, from a single smart move to a brilliant combination to an entire game which is considered a masterpiece. Mathematicians recognize and applaud good mathematics, from clever tricks to brilliant proofs, and from beautiful conceptions to grand and deep ideas which advance our understanding of mathematics as a whole. It takes imagination and insight to discover the best moves, at chess or mathematics, and the more difficult the position, the harder they are to find. Chess players learn by experience to recognize types of positions and situations and to know what kind of moves are likely to be successful; they exploit brilliant local tactics as well as deep stategical ideas. So do mathematicians. Neither games nor mathematics play themselves - they both need a player with understanding, good ideas, judgement and discrimination to play them. To develop these essential attributes, the player must explore the game by playing it, thinking about it and analysing it. For the chess player and the mathematician, this process is scientific: you test ideas, experiment with new possibilities, develop the ones that work and discard the ones that fail. This is how chess players and mathematicians develop their tactical and strategical understanding; it is how they give meaning to chess and mathematics." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Do mathematicians know exactly what they are talking about, or don't they? This is the first bridge that mathematicians have to cross in their search for certainty, and it has to be admitted that they cannot cross it with complete confidence, so their search for absolute certainty is already compromised. They can certainly rely on the long history of their most basic ideas, and the universal agreement that these concepts 'work'. However, this is another way of saying that, if a concept is new and untried, then even mathematicians should be wary of it. They may well discover with experience that their intuitive expectations of it are false. Indeed, this is one of the great values of experimental mathematics - experiments provide the mathematician with data against which mathematicians' natural expectations can be continually tested." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Graph theory is typical of much modern mathematics. Its subject matter is not traditional, and it is not a development from traditional theories. Its applications are not traditional either. […] Graph theory is not concerned with continuous quantities. It often involves counting, but in integers, not measuring using fractions. Graph theory is an example of discrete mathematics. Graphs are put together in pieces, in chunks, rather like Meccano or Lego, or a jigsaw puzzle." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"However mathematics starts, whether it is in counting and measuring in everyday life, or in puzzles and riddles, or in scientific queries about projectiles, floating bodies, levers and balances, or magnetic lines of force, it eventually becomes detached from its roots and develops a life of its own. It becomes more powerful, because it can be applied not just to the situations in which it originated but to all other comparable situations. It also becomes more abstract, and more game-like." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Ironically, mathematicians often infer all sorts of properties about objects which they only suspect, or hope, actually exist. If their suspicions turn out to be unfounded, then they seem to end up by knowing rather a lot about something which does not exist and which might seem, therefore, not to have any properties at all." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"It is typical that there is more than one way of looking at a geometrical figure, just as there are many ways of looking at lines of algebra. Perception, 'seeing', is an essential feature of mathematics. This is obvious when we are looking for patterns - how can you possibly 'spot' a pattern if you cannot in some sense 'see' it? But it is just as true when the mathematician is looking for hidden connections, or studying a position in a mathematical game, searching for a tactical sequence, or trying to 'see' the possibilities clearly. Superficially, it might seem that it is only geometry (and related fields of mathematics) that depends on perception, but this is not so. Perception is everywhere in mathematics." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematical models are continually invoking ideas of infinitely smooth surfaces, weightless strings, weightless beams, perfectly spherical balls, projectiles flying through airless space, gases which are perfectly compressible and liquids which are perfectly incompressible, and so on. The purpose of such simplifications is, in theory, to understand the world better despite the oversimplification, which you hope either will not matter or will be corrected when you construct a second (better) model." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematicians get a different kind of pleasure from the illumination of solving a problem, when what was once mysterious and obscure is made plain. Revealing the hidden connections in a situation is delightful - like reaching the top of a mountain after a hard climb, and seeing the landscape spread out before you. All of a sudden, everything is clear! If the result is extremely simple, so much the better . To start with confusing complexity and transform it into revealing simplicity is a marvellous reward for hard work. It really does give the mathematician a 'kick'!" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematics-as-science naturally starts with mysterious phenomena to be explained, and leads (if you are successful) to powerful and harmonious patterns. Mathematics-as-a-game not only starts with simple objects and rules, but involves all the attractions of games like chess: neat tactics, deep strategy, beautiful combinations, elegant and surprising ideas. Mathematics-as-perception displays the beauty and mystery of art in parallel with the delight of illumination, and the satisfaction of feeling that now you understand." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematics depends on the ability to 'see', literally and abstractly, in novel and unexpected ways, which is another way of saying that it depends on the brilliance and subtlety of the human brain which just happens to be wonderfully adapted to this very purpose. [...] It is one of the marvels of mathematics that everything can be seen in different ways from different points of view, sometimes literally, sometimes metaphorically speaking." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mystery is found as much in mathematics as in detective stories. Indeed, the mathematician could well be described as a detective, brilliantly exploiting a few initial clues to solve the problem and reveal its innermost secrets. An especially mathematical mystery is that you can often search for some mathematical object, and actually know a lot about it, if it exists, only to discover that in fact it does not exist at all - you knew a lot about something which cannot be." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Playing the game of mathematics is much harder than investigating scientifically! To jot down some numbers, a few differences, and spot a pattern is child's play compared to playing the game of algebra." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Puzzle composers share another feature with mathematicians. They know that, generally speaking, the simpler a puzzle is to express, the more attractive it is likely to be found: similarly, simplicity is for both a desirable feature of the solution. Especially satisfying solutions are often described as 'elegant', a word that - no surprise here - is also used by scientists, engineers and designers, indeed by anyone with a problem to solve. However, simplicity is by no means the only reward of success. Far from it! Mathematicians (and scientists and others) can reasonably expect two further returns: they are (in no particular order) firstly the power to do things, and secondly the perception of connections which were never before suspected, leading in turn to the insight and illumination that mathematicians expect from their best arguments." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995) 

"'Technique' is a term used equally by chess players and mathematicians to describe sequences of moves which are standard, familiar and unoriginal. Once upon a time, the particular technique was an invention, a new discovery, but no longer. The precise sequence of moves required may never have been played before in the history of the world, yet no new ideas, no originality and no imagination are demanded, at least of the experienced player. (To the learner, of course, the most mundane sequences will appear novel and require original thought.)" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"The easiest way to discover mathematical facts and theorems is to treat mathematical objects just as if they were objects in the real world, and to make observations and do experiments, by drawing and measuring on geometrical figures, and making calculations with numbers (or in algebra) […]" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"[…] the most important techniques are the final results of some of the most significant breakthroughs in the history of mathematics, beautifully simplified and explained for the benefit of players everywhere. As such, technique is used by pure mathematicians and applied mathematicians alike, but in rather different ways. Pure mathematicians use familiar techniques on the way to discovering or proving new results; applied mathematicians use techniques to model phenomena in the real world […]" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"There is a difficulty, however, which has to be borne in mind. As long as you are behaving like a scientist, you can never be quite certain that your mathematical results are true. Therefore, scientific practice, although it is an essential tool of the mathematician, is also a source of danger. The mathematician who is fast asleep will not see the danger lurking, and will fall straight into the traps and snares that await. The mathematician who is wide awake, however, will gain the double benefit from scientific experiment and observation, of discovering all sorts of supposed facts, and gaining some idea why they might be true." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"There is no sharp dividing line between scientific theories and models, and mathematics is used similarly in both. The important thing is to possess a delicate judgement of the accuracy of your model or theory. An apparently crude model can often be surprisingly effective, in which case its plain dress should not mislead. In contrast, some apparently very good models can be hiding dangerous weaknesses." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"This process, of answering deep questions and so discovering what you really meant by what you've been talking about all the time, is very common in mathematics. It is also very important and very scientific. It has to be scientific, because you cannot prove that one definition is superior to another by logic: you can only make judgements that this definition is preferable to that. You could say that, until mathematicians have examined very deeply and scientifically the objects they talk about so glibly, then they do not really know what they are talking about." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Yet, Einstein's theories are also not the last word: quantum theory and relativity are inconsistent, and Einstein himself, proclaiming that 'God does not play dice!', rejected the basic reliance of quantum theory on chance events, and looked forward to a theory which would be deterministic. Recent experiments suggest that this view of Einstein's conflicts with his other deeply held beliefs about the nature of the physical universe. Certain it is that somewhere, beyond physicists' current horizons, are even more powerful theories of how the world is." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)