On Insight (1990-1999)

 "Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through some process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics." (William P Thurston, "Mathematical education", Notices AMS 37, 1990)

"[…] new insights fail to get put into practice because they conflict with deeply held internal images of how the world works [...] images that limit us to familiar ways of thinking and acting. That is why the discipline of managing mental models - surfacing, testing, and improving our internal pictures of how the world works - promises to be a major breakthrough for learning organizations." (Peter Senge, "The Fifth Discipline: The Art and Practice of the Learning Organization", 1990)

"Science is (or should be) a precise art. Precise, because data may be taken or theories formulated with a certain amount of accuracy; an art, because putting the information into the most useful form for investigation or for presentation requires a certain amount of creativity and insight." (Patricia H Reiff, "The Use and Misuse of Statistics in Space Physics", Journal of Geomagnetism and Geoelectricity 42, 1990)

"Insight is 'mental vision,' one of the ways in which the mind escapes the limits of the obvious or the familiar." (Jennifer James, "Thinking In The Future Tense", 1991) 

"It is not surprising to find many mathematical ideas interconnected or linked. The expansion of mathematics depends on previously developed ideas. The formation of any mathematical system begins with some undefined terms and axioms (assumptions) and proceeds from there to definitions, theorems, more axioms and so on. But history points out this is not necessarily the route that creativity" (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"The chaos theory will require scientists in all fields to, develop sophisticated mathematical skills, so that they will be able to better recognize the meanings of results. Mathematics has expanded the field of fractals to help describe and explain the shapeless, asymmetrical find randomness of the natural environment." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"The scope of Theories of Everything is infinite but bounded; they are necessary parts of a full understanding of things but they are far from sufficient to reveal everything about a Universe like ours. In the pages of this book, we have seen something of what a Theory of Everything might hope to teach us about the unity of the Universe and the way in which it may contain elements that transcend our present compartmentalized view of Nature's ingredients. But we have also learnt that there is more to Everything than meets the eye. Unlike many others that we can imagine, our world contains prospective elements. Theories of Everything can make no impression upon predicting these prospective attributes of reality; yet, strangely, many of these qualities will themselves be employed in the human selection and approval of an aesthetically acceptable Theory of Everything. There is no formula that can deliver all truth, all harmony, all simplicity. No Theory of Everything can ever provide total insight. For, to see through everything, would leave us seeing nothing at all." (John D Barrow, "New Theories of Everything", 1991)

"Fitting is essential to visualizing hypervariate data. The structure of data in many dimensions can be exceedingly complex. The visualization of a fit to hypervariate data, by reducing the amount of noise, can often lead to more insight. The fit is a hypervariate surface, a function of three or more variables. As with bivariate and trivariate data, our fitting tools are loess and parametric fitting by least-squares. And each tool can employ bisquare iterations to produce robust estimates when outliers or other forms of leptokurtosis are present." (William S Cleveland, "Visualizing Data", 1993)

"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)

"[...] two related deficiencies have prevented real progress in understanding insight and its role in problem solving. First, we do not yet have a system of classifying problems into those in which insight occurs versus those in which it does not. However, only if we can isolate problems in which insight occurs will we be able to set on a firm base our theories of the mechanisms underlying insight. Second, formulation of such a taxonomic system requires that we agree on a definition of insight." (Robert W Weisberg, "Prolegomena to theories of insight in problem solving: a taxonomy of problems", 1995)

"Ideas about organization are always based on implicit images or metaphors that persuade us to see, understand, and manage situations in a particular way. Metaphors create insight. But they also distort. They have strengths. But they also have limitations. In creating ways of seeing, they create ways of not seeing. There can be no single theory or metaphor that gives an all-purpose point of view, and there can be no simple 'correct theory' for structuring everything we do." (Gareth Morgan, "Imaginization", 1997)

"Our searches for numerical order lead as often to terminal nuttiness as to profound insight." (Stephen J Gould, "Questioning the Millennium: A Rationalist's Guide to a Precisely Arbitrary Countdown", 1997)

"Cleaning up old proofs is an important part of the mathematical enterprise that often yields new insights that can be used to solve new problems and build more beautiful and encompassing theories." (Bruce Schecter, "My Brain is Open", 1998)

"We use mathematics and statistics to describe the diverse realms of randomness. From these descriptions, we attempt to glean insights into the workings of chance and to search for hidden causes. With such tools in hand, we seek patterns and relationships and propose predictions that help us make sense of the world."  (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"When scientists need to explain difficult points of theory, illustration by hypothetical example - rather than by total abstraction - works well (perhaps indispensably) as a rhetorical device. Such cases do not function as speculations in the pejorative sense - as silly stories that provide insight into complex mechanisms - but rather as idealized illustrations to exemplify a difficult point of theory." (Stephen Jay Gould, "Leonardo's Mountain of Clams and the Diet of Worms", 1998)

"We use mathematics and statistics to describe the diverse realms of randomness. From these descriptions, we attempt to glean insights into the workings of chance and to search for hidden causes. With such tools in hand, we seek patterns and relationships and propose predictions that help us make sense of the world."  (Ivars Peterson, "The Jungles of Randomness: A Mathematical Safari", 1998)

"Although mathematical notation undoubtedly possesses parsing rules, they are rather loose, sometimes contradictory, and seldom clearly stated. [...] The proliferation of programming languages shows no more uniformity than mathematics. Nevertheless, programming languages do bring a different perspective. [...] Because of their application to a broad range of topics, their strict grammar, and their strict interpretation, programming languages can provide new insights into mathematical notation." (Kenneth E Iverson, "Math for the Layman", 1999)