![]() ![]() this speech by Stephen Fry on the analogous phenomenon that there is more to language than grammar and spelling and.Henri Poincare’s “ Intuition and logic in mathematics“.Bill Thurston’s article “ On proof and progress in mathematics“.Then you will be able to tackle maths problems by using both halves of your brain at once – i.e., the same way you already tackle problems in “real life”. The ideal state to reach is when every heuristic argument naturally suggests its rigorous counterpart, and vice versa. One way to do this is to ask yourself dumb questions another is to relearn your field. So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. The point of rigour is not to destroy all intuition instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. (Among other things, this can impact one’s ability to read mathematical papers an overly literal mindset can lead to “compilation errors” when one encounters even a single typo or ambiguity in such a paper.) All too often, one ends up discarding one’s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one’s mathematical education. Unfortunately, this has the unintended consequence that “fuzzier” or “intuitive” thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as “non-rigorous”. It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions. (See also “ There’s more to maths than grades and exams and methods“.) But the transition from the second to the third is equally important, and should not be forgotten. The transition from the first stage to the second is well known to be rather traumatic, with the dreaded “proof-type questions” being the bane of many a maths undergraduate. This stage usually occupies the late graduate years and beyond. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the “big picture”. The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory.This stage usually occupies the later undergraduate and early graduate years. The emphasis is now primarily on theory and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. re-doing calculus by using epsilons and deltas all over the place). ![]() The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner (e.g.This stage generally lasts until the early undergraduate years. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. ![]() One can roughly divide mathematical education into three stages: ![]() For instance, the first phase is characterized by the question ‘How can we eat?’, the second by the question ‘Why do we eat?’ and the third by the question, ‘Where shall we have lunch?’ ( Douglas Adams, “ The Hitchhiker’s Guide to the Galaxy“) The history of every major galactic civilization tends to pass through three distinct and recognizable phases, those of Survival, Inquiry and Sophistication, otherwise known as the How, Why, and Where phases. ![]()
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