https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ [cropped-co] What's new Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao * Home * About * Career advice * On writing * Books * Applets * Mastodon * Subscribe to feed There's more to mathematics than rigour and proofs 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. 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") One can roughly divide mathematical education into three stages: 1. The "pre-rigorous" stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (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. This stage generally lasts until the early undergraduate years. 2. 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. re-doing calculus by using epsilons and deltas all over the place). 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". This stage usually occupies the later undergraduate and early graduate years. 3. 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. (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". This stage usually occupies the late graduate years and beyond. 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. (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. 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. 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". 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. (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.) 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. 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. 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). 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. One way to do this is to ask yourself dumb questions; another is to relearn your field. The ideal state to reach is when every heuristic argument naturally suggests its rigorous counterpart, and vice versa. 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". See also: * Bill Thurston's article "On proof and progress in mathematics"; * Henri Poincare's "Intuition and logic in mathematics"; * this speech by Stephen Fry on the analogous phenomenon that there is more to language than grammar and spelling; and * Kohlberg's stages of moral development (which indicate (among other things) that there is more to morality than customs and social approval). * Bloom's taxonomy of learning. Added later: It is perhaps worth noting that mathematicians at all three of the above stages of mathematical development can still make formal mistakes in their mathematical writing. However, the nature of these mistakes tends to be rather different, depending on what stage one is at: 1. Mathematicians at the pre-rigorous stage of development often make formal errors because they are unable to understand how the rigorous mathematical formalism actually works, and are instead applying formal rules or heuristics blindly. It can often be quite difficult for such mathematicians to appreciate and correct these errors even when those errors are explicitly pointed out to them. 2. Mathematicians at the rigorous stage of development can still make formal errors because they have not yet perfected their formal understanding, or are unable to perform enough "sanity checks" against intuition or other rules of thumb to catch, say, a sign error, or a failure to correctly verify a crucial hypothesis in a tool. However, such errors can usually be detected (and often repaired) once they are pointed out to them. 3. Mathematicians at the post-rigorous stage of development are not infallible, and are still capable of making formal errors in their writing. But this is often because they no longer need the formalism in order to perform high-level mathematical reasoning, and are actually proceeding largely through intuition, which is then translated (possibly incorrectly) into formal mathematical language. The distinction between the three types of errors can lead to the phenomenon (which can often be quite puzzling to readers at earlier stages of mathematical development) of a mathematical argument by a post-rigorous mathematician which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors. (In contrast, when unchecked by a solid intuition, once an error is introduced in an argument by a pre-rigorous or rigorous mathematician, it is possible for the error to propagate out of control until one is left with complete nonsense at the end of the argument.) See this post for some further discussion of such errors, and how to read papers to compensate for them. I discuss this topic further in this video with Brady "Numberphile" Haran. Share this: * Print * Email * More * * Twitter * Facebook * Reddit * Pinterest * Like Loading... Recent Comments Shu Xue Bu Jin Jin Zhi You Yan Mi Xing He Zheng Ming (2007Nian )... on There's more to mathematics th... There's more to... on There's more to mathematics th... Shu Xue Bu Zhi Shi Yan Jin He Zheng Ming . - Pian Zhi De Ma Nong on There's more to mathematics th... [] Anonymous on Polymath8b: Bounded intervals... [3b6] Mohammed Mannan on The Hilbert-Smith conjecture [] Anonymous on The Poisson-Dirichlet process,... [] Anonymous on Two announcements: AI for Math... [] Anonymous on A problem involving power... [d7f] Terence Tao on 275A, Notes 3: The weak and st... [d7f] Terence Tao on Marton's conjecture in a... [d7f] Terence Tao on Analysis I [d7f] Terence Tao on 275A, Notes 3: The weak and st... 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A success! * Polymath 13 - a success! * Non-transitive Dice over Gowers's Blog * Rota's Basis Conjecture: Polymath 12, post 3 198 comments Comments feed for this article 27 October, 2022 at 10:40 am How to build Logic Building ability - yogeshsgupta.com [...] There's more to mathematics than rigour and proofs [...] Reply 12 November, 2022 at 8:06 am erasmuse [c56] See my https://ericrasmusen.substack.com/p/ a-mathematical-diversion-with-cedars... . I do stage 3 with my 7th grade class, and it works very well. They do get the idea that there are good definitions and bad definitions for matching with real concepts. Of course, it's arithmetic I'm doing it with, not vector calculus! Reply 18 January, 2023 at 12:40 pm Solomon Ucko [487] It looks like the link ate the ellipsis. This should work: https: //ericrasmusen.substack.com/p/ a-mathematical-diversion-with-cedars Reply 20 November, 2022 at 11:33 am A Course in the Philosophy and Foundations of Mathematics << Mathematical Science & Technologies [...] [Read article online]. Also, his on a "heuristic stage" and a language metaphor for why order matters in one's formal training] [...] Reply 18 December, 2022 at 4:43 am How to Understand Any Hard Concept Intuitively (And Forever) how to - HaxTube [...] Sources (probably not exhaustive): https://betterexplained.com/ articles/developing-your-intuition-for-math/ https:// math.stackexchange.com/questions/23806/ understanding-the-intuition-behind-math There's more to mathematics than rigour and proofs [...] Reply 19 January, 2023 at 8:39 am Ask HN: Math books that made you significantly better at math? [...] 6. George Polya. Mathematics and Plausible Reasoning. An excellent expansion on Polya's ideas on How to Solve it. While the goal is to seek rigorous proofs, to get there it's powerful to be able to think based on intuition, heuristics, and plausible reasoning. A lot of math exposition is theorem/proof based and doesn't help develop these skills. In a similar vein, see also Terence Tao's classic post There's more to mathematics than rigour and proofs https://terrytao.wordpress.com/career-advice/ theres-more-to-.... [...] Reply 18 March, 2023 at 10:28 am Chasing integers [5fd] Ramanujan and many other mathematicians including possibly you do not (possibly and at least fully) fall under this three step paradigm. What explains these outliers? Reply 18 March, 2023 at 7:22 pm Solomon Ucko [487] What makes you think they didn't follow this paradigm? Reply 31 March, 2023 at 6:42 pm Approaching Type theory and Category Theory as a starting point in the study of mathematics? - Mathematics - Forum [...] question: as a non-mathematician, all tries of bridging the gap between beginners, also called by Terrence Tao "the pre-rigorous stage", and these areas seem [...] Reply 1 May, 2023 at 4:26 pm A Course in the Philosophy and Foundations of Mathematics << Mathematical Science & Technologies [...] article online]. Also, his on a "heuristic stage" and a language metaphor for why order matters in one's formal [...] Reply 25 June, 2023 at 4:33 pm Thoughts on Teaching & Learning Mathematics << Mathematical Science & Technologies [...] Terence Tao. There's more to mathematics than rigour and proofs. Available online from the author's website. [...] Reply 14 October, 2023 at 9:15 am Mathematics - A Collection of Quotes to Inspire & Delight << Mathematical Science & Technologies [...] doing the logically rigorous buildup from axiomatics (Plan A - Axiomatic). Terence Tao [Tao, 2009] makes a similar point in his discussion on getting to the stage of post-rigorous mathematics. [...] Reply 25 October, 2023 at 10:49 am What Performs it Take to Be Actually a Developer in in these times? - thats in [...] Designers maintain driving humanity onward, safeguarding lives, preventing condition and also maintaining our world safe and secure. They likewise generate technologies nobody else can possess presumed of. nevin manimala [...] Reply 26 October, 2023 at 12:11 am What Performs it Take to Be an Engineer in in these times? - web [...] Developers keep pushing mankind onward, guarding lifestyles, stopping ailment and keeping our earth secure. They additionally create advancements nobody else can possess considered. nevin manimala [...] Reply 1 January, 2024 at 3:39 am Good mathematical technique and the case for mathematical insight << Mathematical Science & Technologies [...] Medalist Terence Tao has written a short piece that describes the role of rigor and the value of mathematical technique in the training of a mathematician. In the online discussion of this article, he adds two particularly interesting remarks: the first [...] Reply 16 January, 2024 at 3:34 pm Jose Mauricio de Oliveira [50d] Hello professor Tao, The url in "Henri Poincare's Intuition and logic in mathematics;" seems to be broken, it appears it was changed to https:// mathshistory.st-andrews.ac.uk/Extras/Poincare_Intuition/ [Corrected, thanks - T.] Reply 6 March, 2024 at 3:37 am Shu Xue Bu Jin Jin Shi Yan Jin He Zheng Ming (2009) - Pian Zhi De Ma Nong [...] Xiang Qing Can Kao [...] Reply 19 June, 2024 at 3:54 am Shu Xue Bu Zhi Shi Yan Jin He Zheng Ming . - Pian Zhi De Ma Nong [...] Xiang Qing Can Kao [...] Reply 19 June, 2024 at 6:52 am There's more to mathematics than rigour and proofs haraball on June 19, 2024 at 04:51 Hacker News: Front Page - Bharat Courses [...] Article URL: https://terrytao.wordpress.com/career-advice/ theres-more-to-mathematics-than-rigour-and-proofs/ [...] Reply 19 June, 2024 at 12:17 pm Shu Xue Bu Jin Jin Zhi You Yan Mi Xing He Zheng Ming (2007Nian ) - Pian Zhi De Ma Nong [...] Xiang Qing Can Kao [...] Reply << Older Comments Leave a comment Cancel reply [ ] [ ] [ ] [ ] [ ] [ ] [ ] D[ ] For commenters To enter in LaTeX in comments, use $latex $ (without the < and > signs, of course; in fact, these signs should be avoided as they can cause formatting errors). 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