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 * 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). 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 this: Like Loading... Recent Comments There's more to math... on There's more to mathematics th... [3c7] Terence Tao on 246B, Notes 3: Elliptic functi... [3c7] Terence Tao on 246B, Notes 3: Elliptic functi... [3c7] Terence Tao on The completeness and compactne... [3c7] Terence Tao on 246B, Notes 3: Elliptic functi... [] Anonymous on The Bombieri-Stepanov proof of... [a36] J on 246B, Notes 3: Elliptic functi... AMS :: Resources for... on Resources for displaced m... [d54] Aditya Guha Roy on 275A, Notes 0: Foundations of... [460] Jas, the Physicist on Continually aim just beyond yo... [a36] J on 246B, Notes 3: Elliptic functi... [a36] J on 246B, Notes 3: Elliptic functi... [a36] J on 246B, Notes 3: Elliptic functi... [cro] A puzzle inspired by... on An airport-inspired puzzle [a36] J on 246B, Notes 3: Elliptic functi... [ ] [Search] Articles by others * Andreas Blass - The mathematical theory T of actual mathematical reasoning * Gene Weingarten - Pearls before breakfast * Isaac Asimov - The relativity of wrong * Jonah Lehrer - Don't! - the secret of self-control * Julianne Dalcanton - The cult of genius * Nassim Taleb - The fourth quadrant: a map of the limits of statistics * Paul Graham - What You'll Wish You'd Known * Po Bronson - How not to talk to your kids * Scott Aaronson - Ten signs a claimed mathematical proof is wrong * Tanya Klowden - articles on astronomy * Timothy Gowers - Elsevier -- my part in its downfall * Timothy Gowers - The two cultures of mathematics * William Thurston - On proof and progress in mathematics Diversions * Abstruse Goose * BoxCar2D * Factcheck.org * Gapminder * Literally Unbelievable * Planarity * PolitiFact * Quite Interesting * snopes * Strange maps * Television tropes and idioms * The Economist * The Onion * The Straight Dope * This American Life on the financial crisis I * This American Life on the financial crisis II * What if? (xkcd) * xkcd Mathematics * 0xDE * A Mind for Madness * A Portion of the Book * Absolutely useless * Alex Sisto * Algorithm Soup * Almost Originality * AMS blogs * AMS Graduate Student Blog * Analysis & PDE * Analysis & PDE Conferences * Annoying Precision * Area 777 * Ars Mathematica * ATLAS of Finite Group Representations * Automorphic forum * Avzel's journal * Blog on Math Blogs * blogderbeweise * Bubbles Bad; Ripples Good * Cedric Villani * Climbing Mount Bourbaki * Coloquio Oleis * Combinatorics and more * Compressed sensing resources * Computational Complexity * Concrete nonsense * David Mumford's blog * Delta epsilons * DispersiveWiki * Disquisitiones Mathematicae * Embuches tissues * Emmanuel Kowalski's blog * Equatorial Mathematics * fff * Floer Homology * Frank Morgan's blog * Gerard Besson's Blog * Godel's Lost Letter and P=NP * Geometric Group Theory * Geometry and the imagination * Geometry Bulletin Board * George Shakan * Girl's Angle * God Plays Dice * Good Math, Bad Math * Graduated Understanding * Hydrobates * I Can't Believe It's Not Random! * I Woke Up In A Strange Place * Igor Pak's blog * Images des mathematiques * In theory * James Colliander's Blog * Jerome Buzzi's Mathematical Ramblings * Joel David Hamkins * Journal of the American Mathematical Society * Kill Math * Le Petit Chercheur Illustre * Lemma Meringue * Lewko's blog * Libres pensees d'un mathematicien ordinaire * LMS blogs page * Low Dimensional Topology * M-Phi * Mark Sapir's blog * Math Overflow * Math3ma * Mathbabe * Mathblogging * Mathematical musings * Mathematics Illuminated * Mathematics in Australia * Mathematics Jobs Wiki * Mathematics Stack Exchange * Mathematics under the Microscope * Mathematics without apologies * Mathlog * Mathtube * Matt Baker's Math Blog * Mixedmath * Motivic stuff * Much ado about nothing * Multiple Choice Quiz Wiki * MyCQstate * nLab * Noncommutative geometry blog * Nonlocal equations wiki * Nuit-blanche * Number theory web * Online Analysis Research Seminar * outofprintmath * Pattern of Ideas * Pengfei Zhang's blog * Persiflage * Peter Cameron's Blog * Phillipe LeFloch's blog * ProofWiki * Quomodocumque * Ramis Movassagh's blog * Random Math * Reasonable Deviations * Regularize * Research Seminars * Rigorous Trivialities * Roots of unity * Science Notes by Greg Egan * Secret Blogging Seminar * Selected Papers Network * Sergei Denisov's blog * Short, Fat Matrices * Shtetl-Optimized * Shuanglin's Blog * Since it is not... * Sketches of topology * Snapshots in Mathematics ! * Soft questions * Some compact thoughts * Stacks Project Blog * SymOmega * Tanya Khovanova's Math Blog * tcs math * TeX, LaTeX, and friends * The accidental mathematician * The Cost of Knowledge * The Everything Seminar * The Geomblog * The L-function and modular forms database * The n-Category Cafe * The n-geometry cafe * The On-Line Blog of Integer Sequences * The polylogblog * The polymath blog * The polymath wiki * The Tricki * The twofold gaze * The Unapologetic Mathematician * The value of the variable * The World Digital Mathematical Library * Theoretical Computer Science - StackExchange * Thuses * Tim Gowers' blog * Tim Gowers' mathematical discussions * Todd and Vishal's blog * Van Vu's blog * Vaughn Climenhaga * Vieux Girondin * Visual Insight * Vivatsgasse 7 * Williams College Math/Stat Blog * Windows on Theory * Wiskundemeisjes * XOR's hammer * Yufei Zhao's blog * Zhenghe's Blog Selected articles * A cheap version of nonstandard analysis * A review of probability theory * American Academy of Arts and Sciences speech * Amplification, arbitrage, and the tensor power trick * An airport-inspired puzzle * Benford's law, Zipf's law, and the Pareto distribution * Compressed sensing and single-pixel cameras * Einstein's derivation of E=mc^2 * On multiple choice questions in mathematics * Problem solving strategies * Quantum mechanics and Tomb Raider * Real analysis problem solving strategies * Sailing into the wind, or faster than the wind * Simons lectures on structure and randomness * Small samples, and the margin of error * Soft analysis, hard analysis, and the finite convergence principle * The blue-eyed islanders puzzle * The cosmic distance ladder * The federal budget, rescaled * Ultrafilters, non-standard analysis, and epsilon management * What is a gauge? * What is good mathematics? * Why global regularity for Navier-Stokes is hard Software * Detexify * GmailTeX * Inverse Symbolic Calculator * jfig * LaTeX to Wordpress * Online LaTeX Equation Editor * Sage: Open Source Mathematical Software * Subverting the system The sciences * Academic blogs * American Academy of Arts and Sciences * Australian Academy of Science * Bad Astronomy * National Academy of Science * RealClimate * Schneier on security * Science-Based Medicine * Seven warning signs of bogus science * The Royal Society * This week in evolution * Tree of Life Web Project Top Posts * There's more to mathematics than rigour and proofs * Career advice * On writing * Does one have to be a genius to do maths? * Books * 246B, Notes 3: Elliptic functions and modular forms * Higher uniformity of arithmetic functions in short intervals I. All intervals * Don't prematurely obsess on a single "big problem" or "big theory" * Learn and relearn your field * Ask yourself dumb questions - and answer them! Archives * April 2022 (2) * March 2022 (5) * February 2022 (3) * January 2022 (1) * December 2021 (2) * November 2021 (2) * October 2021 (1) * September 2021 (2) * August 2021 (1) * July 2021 (3) * June 2021 (1) * May 2021 (2) * February 2021 (6) * January 2021 (2) * December 2020 (4) * November 2020 (2) * October 2020 (4) * September 2020 (5) * August 2020 (2) * July 2020 (2) * June 2020 (1) * May 2020 (2) * April 2020 (3) * March 2020 (9) * February 2020 (1) * January 2020 (3) * December 2019 (4) * November 2019 (2) * September 2019 (2) * August 2019 (3) * July 2019 (2) * June 2019 (4) * May 2019 (6) * April 2019 (4) * March 2019 (2) * February 2019 (5) * January 2019 (1) * December 2018 (6) * November 2018 (2) * October 2018 (2) * September 2018 (5) * August 2018 (3) * July 2018 (3) * June 2018 (1) * May 2018 (4) * April 2018 (4) * March 2018 (5) * February 2018 (4) * January 2018 (5) * December 2017 (5) * November 2017 (3) * October 2017 (4) * September 2017 (4) * August 2017 (5) * July 2017 (5) * June 2017 (1) * May 2017 (3) * April 2017 (2) * March 2017 (3) * February 2017 (1) * January 2017 (2) * December 2016 (2) * November 2016 (2) * October 2016 (5) * September 2016 (4) * August 2016 (4) * July 2016 (1) * June 2016 (3) * May 2016 (5) * April 2016 (2) * March 2016 (6) * February 2016 (2) * January 2016 (1) * December 2015 (4) * November 2015 (6) * October 2015 (5) * September 2015 (5) * August 2015 (4) * July 2015 (7) * June 2015 (1) * May 2015 (5) * April 2015 (4) * March 2015 (3) * February 2015 (4) * January 2015 (4) * December 2014 (6) * November 2014 (5) * October 2014 (4) * September 2014 (3) * August 2014 (4) * July 2014 (5) * June 2014 (5) * May 2014 (5) * April 2014 (2) * March 2014 (4) * February 2014 (5) * January 2014 (4) * December 2013 (4) * November 2013 (5) * October 2013 (4) * September 2013 (5) * August 2013 (1) * July 2013 (7) * June 2013 (12) * May 2013 (4) * April 2013 (2) * March 2013 (2) * February 2013 (6) * January 2013 (1) * December 2012 (4) * November 2012 (7) * October 2012 (6) * September 2012 (4) * August 2012 (3) * July 2012 (4) * June 2012 (3) * May 2012 (3) * April 2012 (4) * March 2012 (5) * February 2012 (5) * January 2012 (4) * December 2011 (8) * November 2011 (8) * October 2011 (7) * September 2011 (6) * August 2011 (8) * July 2011 (9) * June 2011 (8) * May 2011 (11) * April 2011 (3) * March 2011 (10) * February 2011 (3) * January 2011 (5) * December 2010 (5) * November 2010 (6) * October 2010 (9) * September 2010 (9) * August 2010 (3) * July 2010 (4) * June 2010 (8) * May 2010 (8) * April 2010 (8) * March 2010 (8) * February 2010 (10) * January 2010 (12) * December 2009 (11) * November 2009 (8) * October 2009 (15) * September 2009 (6) * August 2009 (13) * July 2009 (10) * June 2009 (11) * May 2009 (9) * April 2009 (11) * March 2009 (14) * February 2009 (13) * January 2009 (18) * December 2008 (8) * November 2008 (9) * October 2008 (10) * September 2008 (5) * August 2008 (6) * July 2008 (7) * June 2008 (8) * May 2008 (11) * April 2008 (12) * March 2008 (12) * February 2008 (13) * January 2008 (17) * December 2007 (10) * November 2007 (9) * October 2007 (9) * September 2007 (7) * August 2007 (9) * July 2007 (9) * June 2007 (6) * May 2007 (10) * April 2007 (11) * March 2007 (9) * February 2007 (4) Categories * expository (292) + tricks (11) * guest blog (10) * Mathematics (831) + math.AC (8) + math.AG (42) + math.AP (112) + math.AT (17) + math.CA (179) + math.CO (182) + math.CT (8) + math.CV (37) + math.DG (37) + math.DS (86) + math.FA (24) + math.GM (12) + math.GN (21) + math.GR (88) + math.GT (16) + math.HO (12) + math.IT (13) + math.LO (51) + math.MG (45) + math.MP (28) + math.NA (23) + math.NT (183) + math.OA (22) + math.PR (102) + math.QA (6) + math.RA (42) + math.RT (21) + math.SG (4) + math.SP (48) + math.ST (9) * non-technical (173) + admin (45) + advertising (48) + diversions (6) + media (13) o journals (3) + obituary (15) * opinion (32) * paper (226) + book (19) + Companion (13) + update (21) * question (125) + polymath (85) * talk (67) + DLS (20) * teaching (188) + 245A - Real analysis (11) + 245B - Real analysis (21) + 245C - Real analysis (6) + 246A - complex analysis (11) + 246B - complex analysis (5) + 246C - complex analysis (5) + 247B - Classical Fourier Analysis (5) + 254A - analytic prime number theory (19) + 254A - ergodic theory (18) + 254A - Hilbert's fifth problem (12) + 254A - Incompressible fluid equations (5) + 254A - random matrices (14) + 254B - expansion in groups (8) + 254B - Higher order Fourier analysis (9) + 255B - incompressible Euler equations (2) + 275A - probability theory (6) + 285G - poincare conjecture (20) + Logic reading seminar (8) * travel (26) additive combinatorics almost orthogonality approximate groups arithmetic progressions Ben Green Cauchy-Schwarz Cayley graphs central limit theorem Chowla conjecture compressed sensing correspondence principle distributions divisor function eigenvalues Elias Stein Emmanuel Breuillard entropy equidistribution ergodic theory Euler equations exponential sums finite fields Fourier transform Freiman's theorem Gowers uniformity norm Gowers uniformity norms graph theory Gromov's theorem GUE hard analysis Hilbert's fifth problem hypergraphs incompressible Euler equations inverse conjecture Joni Teravainen Kaisa Matomaki Kakeya conjecture Lie algebras Lie groups linear algebra Liouville function Littlewood-Offord problem Maksym Radziwill Mobius function Navier-Stokes equations nilpotent groups nilsequences nonstandard analysis parity problem politics polymath1 polymath8 Polymath15 polynomial method polynomials prime gaps prime numbers prime number theorem random matrices randomness Ratner's theorem regularity lemma Ricci flow Riemann zeta function Roth's theorem Schrodinger equation sieve theory structure Szemeredi's theorem Tamar Ziegler ultrafilters universality Van Vu wave maps Yitang Zhang RSS The Polymath Blog * Polymath projects 2021 20 February, 2021 * A sort of Polymath on a famous MathOverflow problem 9 June, 2019 * Ten Years of Polymath 3 February, 2019 * Updates and Pictures 19 October, 2018 * Polymath proposal: finding simpler unit distance graphs of chromatic number 5 10 April, 2018 * A new polymath proposal (related to the Riemann Hypothesis) over Tao's blog 26 January, 2018 * Spontaneous Polymath 14 - A success! 26 January, 2018 * Polymath 13 - a success! 22 August, 2017 * Non-transitive Dice over Gowers's Blog 15 May, 2017 * Rota's Basis Conjecture: Polymath 12, post 3 5 May, 2017 175 comments Comments feed for this article 23 August, 2016 at 2:38 am Alex Colovic [295] An excellent description of the learning curve! This post was pointed out to me by a reader of my blog, Mr Peter Munro, as a comment to a post about my ongoing troubles (http:// www.alexcolovic.com/2016/08/before-olympiad.html#comment-form). Even though I am a chess grandmaster for quite some time, and I can safely put myself in the "post-rigorous" stage, I still find that I am very prone to formal mistakes. This has affected my performances of late, which has also affected my confidence. I think mathematicians are lucky not to have to live in a competitive environment! Reply 25 October, 2021 at 5:45 pm Anonymous [] Mathematicians do live in a competitive environment if they work in academia! It's very often the case that you are beaten to a result by someone else working on the same problem. Reply 13 October, 2016 at 2:46 am Greg Takats [bfe] I found that many IMO medalists are at the third stage when I did some research on them. I think stages 2 and 3 correspond to the conscious competence and unconscious competence stages. Reply 5 November, 2016 at 11:50 am A Numberplay Farewell - My Blog [...] Fast and Slow." As for Mapmaker -- I mean to refer to one of skills in Terence Tao's "post-rigorous reasoning," Freeman Dyson's "bird," and in Bill Thurston's "On Proof and Progress." (Reasoning [...] Reply 15 January, 2017 at 5:46 pm Understand advanced mathematics? | Since 1989 [...] know, because you know how to fill in the details. Terence Tao is very eloquent about this here [ https://terrytao.wordpress.com/ca&# 8230; ]:"[After learning to think rigorously, comes the] 'post-rigorous' stage, in [...] Reply 5 May, 2017 at 8:15 am Why I didn't understand (real) analysis. - Site Title [...] I was conversing with a professor a while back and I told him that while I took Advanced Calculus, I didn't know what a derivative was and I didn't know what a continuous function was. I didn't know how to explain it to him at that time but I think I do now. You see, calculus is a deep subject and traditionally there is a sequence that one goes through in order to have a working understanding of it. First one learns the algorithms which manipulates functions; that is, one learns how to calculate derivatives, integrals, and limits of function, all the stuff one goes through when taking elementary calculus. Then one learns the theory of those algorithms; i.e., what is a derivative? an integral? a limit? At this stage one writes proofs in order to (1) destroy bad intuition and (2) elevate good intuition (and to develop more good intuition)[1]. [...] Reply 27 May, 2017 at 5:35 pm Thinking on the page - under the sea [...] best, this is what happens in what Terry Tao calls the "rigorous" stage of mathematics education, writing, "The point of rigour is not to destroy all intuition; [...] Reply 25 July, 2017 at 7:18 am The "blogs I like" - Christopher Blake's Blog [...] the blog of Terry Tao. In Terry Tao's blog, I particularly recommend There's more to mathematics than rigour and proofs. During my PhD I read this post, and I could not put into better words the importance of [...] Reply 20 August, 2017 at 6:21 am In praise of specifications and implementations over definitions - Designer Spaces [...] in rigorous mathematics one will see a definition like the [...] Reply 30 September, 2017 at 5:29 am Metarationality: a messy introduction - drossbucket [buc] [...] At the highest levels, in fact, the emphasis on rigour is often relaxed somewhat. Terence Tao describes this as: [...] Reply 20 October, 2017 at 12:00 pm The Aesthetic Stage | Radimentary [...] like Raphael, but a lifetime to paint like a child." Terry Tao says much the same thing on the three stages of mathematical education. What I mean to say is that it would be absolutely wrong to read the rest of this essay as simply [...] Reply 7 November, 2017 at 12:02 pm Defining Issues Test Results - Math 12 (Fall 2017) [su_] [...] by them. In mathematics, this big-picture, application-focused view is often called "Post-Rigorous," roughly corresponding to Lawrence Kohlberg's "Post-Conventional" stage of [...] Reply 12 December, 2017 at 1:03 am njwildberger: tangential thoughts [201] It is sometimes useful to remember that most of mathematics historically has been applied mathematics, concentrating on real life situations, or generalizations or variants. Every so often someone comes along and says: can we establish what we are doing at this lofty level by reducing, via pure logic, to simpler stuff at a lower level? This is what the pure mathematician has been doing for the past one or two hundred years or so. Gradually the level of mathematics that we analyse very carefully gets simpler and ever more elementary, say from continuous functions, to real numbers, to (because of Dedekind cuts) infinite sets, to ... and that's where things more or less stopped in the 1920's and 30's. But with the advent of modern computers, it is gradually becoming clearer that our pure mathematics is actually not that rigorous at all. Terry's categories, while perhaps useful, hide a painful truth: most of pure mathematics does not work logically if we had to explain it to a smart computer, even though it may seem reasonable to some of us equipped with the correct "intuition". Unfortunately pretty soon we will have to adjust to the new kid(s) on the block, with zero intuition but loads of relentless computing power. Watch out pure mathematicians: Google and Deep Mind's Alpha is coming, and it is going to walk all over our intuition, just as it has for the modern Go and Chess players. Reply 18 January, 2018 at 4:27 pm Singularity Mindset | Radimentary [...] For mathematicians, the curve is pre-rigor, rigor, post-rigor. [...] Reply 23 January, 2018 at 8:24 pm Co Nhieu Hon la Su Chat Che va Chung Minh trong Toan Hoc | 5 [cro] [...] translated this article from "There's more to mathematics than rigour and proofs", since I found it to be very exhaustive, and yet very interesting and somewhat relevant to even [...] Reply 24 January, 2018 at 11:00 am Teaching Ladders | Radimentary [...] For mathematicians, the curve is pre-rigor, rigor, post-rigor. [...] Reply 17 March, 2018 at 9:35 pm New Strategies for Learning | Aha! Moments and Dumb Notes [...] I got my self pointed out to an advise from Terry Tao from this other advise, I realized that in order for me to progress faster, I need to relax a bit [...] Reply 29 March, 2018 at 11:15 am An Introduction - FRC Controls and Code [...] and "big ideas" increasing alternately and reinforcing each-other, much like Terry Tao's description of the progression of mathematical understanding. But the general principle - that [...] Reply 1 April, 2018 at 3:29 pm etc. - Aenigma Mundi Rect [...] cf https://terrytao.wordpress.com/career-advice/ theres-more-to-mathematics-than-rigour-and-proofs/ you do the math not so that you are good at proofs, or can distinguish proofs, or are confident of [...] Reply 5 April, 2018 at 7:58 am There's more to mathematics than rigour and proofs | What's new | Severud.org [...] 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... -- Read on terrytao.wordpress.com/career-advice/ theres-more-to-mathematics-than-rigour-and-proofs/ [...] Reply 20 May, 2018 at 5:27 pm I think my mathematical way of working is too technical and not creative enough. I struggle to understand the "big picture" which hinders me to become a better mathematician. Do you have any helpful advice for me? - Nevin Manimala's Blog [...] this article written by Terrence Tao, he described three stages of mathematical [...] Reply 21 June, 2018 at 9:31 pm NRIC Z Enterprises [ima] [...] problems? Note that there is more to maths than grades and exams and methods; there is also more to maths than rigour and proofs. It is also important to value partial progress, as a crucial stepping stone to a complete [...] Reply 13 August, 2018 at 1:45 am anyone [] Dear Professor Tao, The link to the article "There's more to mathematics than grades and exams and methods" appears to be broken. [Fixed now - T.] Reply 29 October, 2018 at 6:12 pm Book Review: A Philosophy of Software Design - ThreatIntel [...] doing this gives us concrete grounding when we talk software design. It's how we move into post-rigorous stage of software engineering, and know what we mean when we use terms like "interface" and [...] Reply 21 December, 2018 at 4:21 am Boniface Duane [834] I think I'm crying. It's that excellent. Reply 6 January, 2019 at 2:30 pm Nisarg [dfb] Thank you so much. It is really really very nice blog. I wish I had known this before. Reply 9 January, 2019 at 1:43 am Mathematical intuition - Balise's Blog [785] [...] of the most interesting articles I read on the topic was from Terry Tao, There's more to mathematics than rigour and proofs. He distinguishes between three "stages" in math [...] Reply 12 March, 2019 at 6:34 am Relearning advanced undergraduate/beginning graduate level mathematics - Nevin Manimala's Blog [...] advanced undergraduate level mathematics. The motivation comes form reading Terrence Tao's blog post, which [...] Reply 24 August, 2019 at 9:57 am One trick pony i madlavning - og matematik - hanshuttel.dk [...] Set pa denne made svarer de forskellige kompetenceniveauer inden for madlavning pa en made til d... og som jeg tidligere har skrevet om pa denne blog. [...] Reply 26 September, 2019 at 3:49 am Intuition and Rigour - Two sides of Math Learning | PiVerb Math Olympiad [...] is an extract from a post by Fields medalist, Mr. Terence Tao, The author makes these points in the context of higher education in math. But the idea is [...] Reply 17 October, 2019 at 8:22 pm Math Vault [7d3] Yes sir! While the valid use of deduction is obvious a cornerstone quality of mathematics, it shouldn't be emphasized to the extent of sacrificing other cornerstone qualities of mathematics. These include the need of toying, the exercise of intuition, the drive to discover, and the general process of a mathematical experience. For those who're just starting with higher math, this guide on higher math learning could be a first step towards a holistic experience. Math is both a scientific and artistic endeavor, and focusing one aspect of it exclusive to all others can lead to a skewed perspective on mathematics too. Reply 25 November, 2019 at 7:08 pm Robert Duke [2d3] I feel like set theory ruined intuition. People already knew about sets naturally because we know how to count and can think of lists. Defining sets forced people to think in one way. Also doing proof by contradiction all the time means nothing intuitively and is an easy way of proving something just to finish the proof. So we can skip all proofs by contradiction if its to provide an intuitive explanation, its just an algorithm with no meaning just to know if something is true or false. So what is the point of doing the proof? We already know if its true or false because someone else already did it. Everyone is stuck in this mindset now after set theory came out because it gives them a sense of power over true and false. However, it doesn't generate creativity, and only provides a way to prove true or false. At this point its just a game of logical symbols from set theory. I feel like we should only use rigor when we need to test our conjectures or need it as a tool to understand something. I agree proof is very important but intuition is more important. Math was perfectly fine before set theory. Also what is the point of redoing EVERYTHING that has already been done? Why does everyone use RUDIN? All we are doing is going through every proof that has already been done and memorizing all of it. I don't have time to test my ideas and make new proofs of theorems because I have to rush all the time. This way of learning only suites some people. There is no creation in this. Its like a musician who only does covers of another musician and never creates anything. This is forcing mathematicians to all be the same. I believe math will never generate its greatest accomplishments because of this. We need to let people make conjectures and back off the rigor a bit and let people explore their ideas. The curriculum for graduates needs to be broader. Why is analysis and algebra the core? There are more parts of math then this and maybe people are not interested in this area. I hate being forced to do math I don't consider interesting. Personally I would prefer to study other areas of math rigorously not analysis and algebra. So I cant be a mathematician if I don't learn the precise math the universities want me to. So then how do you discover new ideas? We only allow people to discover new ideas after they have a Ph.D. an did math the way the university made them do it. That way they discover ideas related to their goals. It seems like math gets narrower and narrower in its methods, and we will converge to only doing proofs. At that point its just a game of symbols with no numbers. And we are going back over stuff that is already known and redoing it again with sets. We already know the derivative. And we don't need sets to help with this. We need to create new ideas that are as great as the derivative instead of staying at the derivative and floating around it. Its for people who are not creative and are computers and people who memorize. The derivative is only important because it can solve problems. Math is only for solving problems. I wish I could have been a mathematician in 1600s instead of "modern" mathematics. The "modern" mathematicians want to act like they did something great but there is nothing. The derivative was the only great idea for hundreds of years. What is also funny is we use greek and latin letters to look cool even though we don't speak latin or greek. We should redo these symbols in English since we read and write in English. Thats what the ancient mathematicians did. The math was in their language. But we use their notation even though we use a different language. We are not doing math like they did with respect to notation. We are less intuitive. I think the way we learn mathematics at the universities is foolish and with its methods of learning will never shape minds into their best geniuses. Because this school system rejects people if they don't do it their way. The curriculum need to meet the learning styles of everyone. So all we are doing is generating mathematicians that are robots for proofs, and they have no creativity and therefore the only creations will be derived from proof. With this in mind new definitions will probably rarely occur and the derivative will remain the greatest discovery because of the universities lack of supporting creative mathematicians over computerized mathematicians. Reply 26 November, 2019 at 1:50 pm Anonymous [] Rigor is needed for precise definitions of new concepts and for checking proofs of claimed new theorems. It is also needed for precise objective(!) communication of mathematical derivations (since "intuitive derivation" is unreliable, imprecise and subjective) Reply 12 October, 2020 at 4:30 pm John Gabriel [7de] https://www.linkedin.com/pulse/ what-does-mean-concept-well-defined-john-gabriel/ Reply 25 November, 2019 at 7:11 pm Robert Duke [2d3] Hopefully you can make it through all the stages. Reply 2 February, 2020 at 8:11 am `n lryDyt, wrHlth lshqW@ - frs. [...] Clarity bstkhdm l Formal Languages and Precise Definitions, tlk lmrHl@ ySnWfh Terence Tao b'nh mrHl@ l rigorous,wrbm mn qr' mqlty lsbq@ syr~ hdh [...] Reply 26 April, 2020 at 2:16 am Aniket Bhattacharya [cf3] We are not in KaliYuga the dark age. I have tried to prove this in this video Reply 22 November, 2020 at 8:49 am On Economic Matters: The Need for Argumentation and Rigor | Deus Fortis [...] One may read these works for further information:https:// paulromer.net/mathiness/https://terrytao.wordpress.com/career-advice/ theres-more-to-mathematics-than-rigour-and-proofs/https:// nassimtaleb.org/2016/09/intellectual-yet-idiot/ [...] Reply 21 December, 2020 at 11:46 am Research Part 1 Reflection - English Studio Portfolio [...] https://terrytao.wordpress.com/career-advice/ theres-more-to-mathematics-than-rigour-and-proofs/ [...] Reply 22 January, 2021 at 11:58 pm Trin Athigapanich [6b5] can you please define what "intuition" means? Reply 16 November, 2021 at 12:50 pm Mohammad Azad [108] LOL, I don't know if you are serious or not but I will define intuition as: "What you think of the problem without any formal reasoning" e.g. prove that x^2 is non-negative for all real numbers. My intuition tells me that when you multiply two numbers with the same sign you won't get a negative number, but I didn't explain why rigorously! Apparently, according to the awesome and inspiring Terence Tao, Mathematicians in the "Post-rigorous" stage use their deep understanding of rigourous math to make intuition on the fly and then they can prove their intuition rigorously when needed! Reply 27 April, 2021 at 11:54 pm Qua Cau [349] We translated this article to Vietnamese here: Toan hoc con co nhieu thu hon la moi su chat che va chung minh. Hope it's fine to you. Reply 28 April, 2021 at 11:19 am Anonymous [] It seems that the creation of new concepts and theories should belong to the "post rigorous" stage (since it rely mostly on intuition and accumulated experience.) Reply 28 April, 2021 at 5:45 pm Wolfgang [d3c] I am aware that the topic in the way it is introduced here is possibly meant to be more a discussion among mathematicians and their intellectual development. However, one should, in my opinion, not forget the connection of mathematics to science. Focusing on rigor might be good and possibly indispensable for mathematicians to learn their craft. At the same time it might be bad when rigor (and the associated abstraction) is imposed to the same degree in the education of scientists. Mathematicians act a little like car manufacturers which demand that anyone who wants to drive a car has to know all the steps and details and procedures that were necessary in building it. While a car engineer make up a good driver, knowing exactly what happens with his car in every situation, this is by no means necessary to become a good driver in general. If one compares calculus books written before and after Bourbaki the former are incredible in conveying the important ideas in a less abstract manner and make them understandable comparatively easy, which for most people wanting to get the idea and apply it to more or less standard cases is just enough. In my opinion the high demands on rigor and abstraction in this context do more harm than anything else, preventing a lot of smart people using mathematics to their benefits. Reply 29 April, 2021 at 7:58 pm Anonymous [] This is a post about training mathematicians, not scientists. Calculus textbooks and calculus students tend to belong to the "pre-rigorous" stage. Reply 16 August, 2021 at 9:37 am Book pitch - Power Overwhelming [and] [...] should state now this is against common wisdom. Terrence Tao for example describes mathematical education in three parts: pre-rigorous, rigorous, post-rigorous. [...] Reply 18 August, 2021 at 3:24 am Student Diaries: An expedition through the realms of the process and the result - Raising A Mathematician Foundation [...] [1] Terence Tao's blog which can be found here [...] Reply 13 October, 2021 at 2:31 am Post-rigorous thinking in Interpreting Studies - Still Thinking [...] this blog post, mathematician Terry Tao argues that learning mathematics involves three [...] Reply 24 November, 2021 at 5:17 am jamie kukuruzovic [be4] Here comes post-grad study :) Reply 4 December, 2021 at 4:39 pm Advancing mathematics by guiding human intuition with AI | Delightful & Distinctive COLRS [...] of both rigorous formalism and good intuition that one can tackle complex mathematical problems"25. The following framework, illustrated in Fig. 1, describes a general method by which [...] Reply 25 December, 2021 at 4:41 pm Advancing mathematics by guiding human intuition with AI - Nature.com - Auto Robot Demo [cro] [...] of both rigorous formalism and good intuition that one can tackle complex mathematical problems"25. The following framework, illustrated in Fig. 1, describes a general method by which mathematicians [...] Reply 19 April, 2022 at 11:01 am There's more to mathematics than rigour and proofs - The web development company Lzo Media - Senior Backend Developer [...] Article URL: https://terrytao.wordpress.com/career-advice/ theres-more-to-mathematics-than-rigour-and-proofs/ [...] Reply << Older Comments Leave a Reply Cancel reply Enter your comment here... [ ] Fill in your details below or click an icon to log in: * * * * Gravatar Email (Address never made public) [ ] Name [ ] Website [ ] WordPress.com Logo You are commenting using your WordPress.com account. ( Log Out / Change ) Twitter picture You are commenting using your Twitter account. ( Log Out / Change ) Facebook photo You are commenting using your Facebook account. ( Log Out / Change ) Cancel Connecting to %s [ ] Notify me of new comments via email. [ ] Notify me of new posts via email. [Post Comment] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 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). See the about page for details and for other commenting policy. Blog at WordPress.com.Ben Eastaugh and Chris Sternal-Johnson. Subscribe to feed. * Follow Following + [9ecc62] What's new Join 10,490 other followers [ ] Sign me up + Already have a WordPress.com account? Log in now. * + [9ecc62] What's new + Customize + Follow Following + Sign up + Log in + Copy shortlink + Report this content + View post in Reader + Manage subscriptions + Collapse this bar Send to Email Address [ ] Your Name [ ] Your Email Address [ ] [ ] loading [Send Email] Cancel Post was not sent - check your email addresses! Email check failed, please try again Sorry, your blog cannot share posts by email. %d bloggers like this: [b]