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:

- 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.
- 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.
- 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:

- 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. - 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.
- 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.

## 175 comments

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23 August, 2016 at 2:38 am

Alex ColovicAn 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!

25 October, 2021 at 5:45 pm

AnonymousMathematicians 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.

13 October, 2016 at 2:46 am

Greg TakatsI 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.

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 […]

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… ]:”[After learning to think rigorously, comes the] ‘post-rigorous’ stage, in […]

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]. […]

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; […]

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 […]

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 […]

30 September, 2017 at 5:29 am

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20 October, 2017 at 12:00 pm

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7 November, 2017 at 12:02 pm

Defining Issues Test Results – Math 12 (Fall 2017)[…] 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 […]

12 December, 2017 at 1:03 am

njwildberger: tangential thoughtsIt 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.

18 January, 2018 at 4:27 pm

Singularity Mindset | Radimentary[…] For mathematicians, the curve is pre-rigor, rigor, post-rigor. […]

23 January, 2018 at 8:24 pm

Có Nhiều Hơn là Sự Chặt Chẽ và Chứng Minh trong Toán Học | 5[…] 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 […]

24 January, 2018 at 11:00 am

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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 […]

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 […]

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 […]

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/ […]

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 […]

21 June, 2018 at 9:31 pm

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13 August, 2018 at 1:45 am

anyoneDear 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.]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 […]

21 December, 2018 at 4:21 am

Boniface DuaneI think I’m crying. It’s that excellent.

6 January, 2019 at 2:30 pm

NisargThank you so much. It is really really very nice blog. I wish I had known this before.

9 January, 2019 at 1:43 am

Mathematical intuition – Balise's Blog[…] 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 […]

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 […]

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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 […]

17 October, 2019 at 8:22 pm

Math VaultYes 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.

25 November, 2019 at 7:08 pm

Robert DukeI 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.

26 November, 2019 at 1:50 pm

AnonymousRigor 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)

12 October, 2020 at 4:30 pm

John Gabrielhttps://www.linkedin.com/pulse/what-does-mean-concept-well-defined-john-gabriel/

25 November, 2019 at 7:11 pm

Robert DukeHopefully you can make it through all the stages.

2 February, 2020 at 8:11 am

عن الرياضيات، ورحلتها الشاقّة - فارس.[…] Clarity باستخدام الـ Formal Languages and Precise Definitions، تلك المرحلة يصنّفها Terence Tao بأنها مرحلة الـ rigorous،وربما من قرأ مقالاتي السابقة سيرى هذا […]

26 April, 2020 at 2:16 am

Aniket BhattacharyaWe are not in KaliYuga the dark age. I have tried to prove this in this video

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/ […]

21 December, 2020 at 11:46 am

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22 January, 2021 at 11:58 pm

Trin Athigapanichcan you please define what “intuition” means?

16 November, 2021 at 12:50 pm

Mohammad AzadLOL, 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!

27 April, 2021 at 11:54 pm

Quả CầuWe translated this article to Vietnamese here: Toán học còn có nhiều thứ hơn là mỗi sự chặt chẽ và chứng minh. Hope it’s fine to you.

28 April, 2021 at 11:19 am

AnonymousIt 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.)

28 April, 2021 at 5:45 pm

WolfgangI 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.

29 April, 2021 at 7:58 pm

AnonymousThis is a post about training mathematicians, not scientists. Calculus textbooks and calculus students tend to belong to the “pre-rigorous” stage.

16 August, 2021 at 9:37 am

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24 November, 2021 at 5:17 am

jamie kukuruzovicHere comes post-grad study :)

4 December, 2021 at 4:39 pm

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