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.

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12 March, 2013 at 12:40 pm

Catarina Dutilh NovaesMy (philosophical!) two cents:

http://m-phi.blogspot.nl/2013/03/terry-tao-on-rigor-in-mathematics.html

19 August, 2013 at 4:04 am

CalmariusI’m good at maths. I won several math and physics competitions on primary and high school. Although finally I’m become a programmer, I’m still interested in physics and maths, and I try to understand things mainly in physics.

The problem I see is that many things in papers and lecture notes are described in very terse, plain rigorous way. Even the maths lecture at the college was presented in this way. I often had the feeling that the lecturer is ‘showing off’, and I often thought they are playing a game of telling everything to a students in a way they have no chance to understand it.

In constrant with this, when something is described by examples, problems, analogies, intuition and induction or by telling the patterns, even understanding the most sophisticated concept becomes easy.

After the intuitive foundations are built then we can generalize to N, other fields, etc, and discover the corner cases and gotchas, and backtrack towards the axioms (or towards the desired level).

But in real life I see that things are explained the other way around: axioms, definition, definition, definition, definition, definition, theorem, proof, theorem, proof, theorem, proof, theorem, proof, definition, definition, definition, definition, definition, theorem, proof, theorem, proof, theorem, proof, theorem, proof. It’s very easy to get lost at the very beginning. Because all look like meaningless symbol folding.

It’s very easy to get lost in this. All that’s missing is: what is this all about? What can I do with this?

For me this’s something like deciphering what does a program do by starting from the silicon and learning how to build an x86 CPU, then reading the machine code.

While you can do the same by reading the nicely commented source code, without even knowing anything about the gory details of the hardware.

Personally I cannot think of differentiation without visualizing the slope or thinking of a rate of change; also I cannot think of integration without visualizing the area under the graph, or thinking of summation. Visualizing matrix determinant as an area of a parallelogram (2×2) or a parallelepiped (3×3) was a great eye opener, now I know why the determinant of 0 is so special.

I’ve also found saying ‘x and y have the same sign’ in plain Enlish much easier to understand than simply writing down ‘xy > 0’.

I like maths, I like puzzles, I like solving problems, I like understanding world. I can prove many theorems. I can use mathematical tools when I need to. But excess rigor is not for me.

Probably I’m thinking this way because I’m an engineer, and I use maths, not inventing it. But heck I discovered, reinvented many things way before I learned about it.

19 August, 2013 at 6:17 am

DanielI’m not sure that analogy to CPU hardware is accurate, although it does illustrate the point. Maybe if it were drawn to understanding the fundamentals of machine language (I mean code the computer actually understands, not sure if I’m using the right term), it would be more accurate. Because the fundamental definitions, theorems, etc are just one step away from the topics discussed, whereas hardware and software need some bridging. In any case, the analogy it’s still true.

I think one reason why the rigor is necessary is to ensure that one knows what it might be used for and, secondly, to demonstrate the manner in which one establishes a concept. This, I think, would allow the student perhaps replicate this rigorous path with a different concept (perhaps a new one). It seems this is the difference between a math student and a engineering/physics student, the former focused on the concept and the later on usage. Rigor remains just as impenetrable for us as for you, though after a while one gets used to it. It does, admittedly, have that magic trick effect (when you produce some useful result).

27 November, 2013 at 11:18 am

chris“I think one reason why the rigor is necessary is to ensure that one knows what it might be used for and, secondly, to demonstrate the manner in which one establishes a concept. ”

I think it’s quite the opposite: most “hardcore” mathematicians that put a lot of emphasis on formalism and rigor, have a hard time relating to real world concepts and show extremely poor skills at properly using math in a different fields like programming when it comes to real world software production. Actually most mathematicians make very poor programmers and engineers. Also, typical math proofs are not required in real world applications(sometimes proofs can actually be misleading), it’s real world testing that does the validation.

19 September, 2013 at 8:57 am

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24 October, 2013 at 9:48 am

Terry Tao: On Hard Work | Fahad's Academy[…] short, there is no royal road to mathematics; to get to the “post-rigorous” stage in which your intuition matches well with what one can establish rigorously, one has to […]

10 January, 2014 at 12:01 pm

whaaales | Principled memorization[…] ideal student is thus drawn into a (physically, if not mathematically) “post-rigorous stage” where conceptual fluency isn’t impeded by a need to build things up from the beginning every […]

19 January, 2014 at 7:34 pm

xingeThere is also a speech given by Atiyah about the style of Mathematics, in witch he talked about the intuitive mathematician.

20 January, 2014 at 2:12 am

Ronnie BrownOne analogy I like is that between describing a route to the station in terms of the landscape (e.g. turn tight at the oak tree and left at the traffic lights) and in terms of listing all the cracks in the pavement. One task in mathematics is to build language and notation in which to describe the “landscape”, i,e, the structures which arise and help to guide our understanding. Of course, Grothendieck was a master at this. And of course the modern language for discussing mathematical structures is category theory. Sometimes also one “knows” that something is true because it fits with so many things. I also spent 9 years on an “idea for a proof in search of a theorem”: what was lacking was a gadget (a homotopy double groupoid) to realise the idea.

I also feel there is not enough discussion about methodology. See the article: http://pages.bangor.ac.uk/~mas010/methmat.html . “The methodology of mathematics” . which has been published variously.

20 January, 2014 at 11:13 am

Tre niveauer af matematisk præcision | hanshuttel.dk[…] amerikanske matematiker Terence Tao har (som så ofte før) et interessant blogindlæg om netop dét. Det, han hæfter sig ved, er hvor svært (og nødvendigt) at nå til det tredje, post-rigorous […]

22 January, 2014 at 3:56 pm

themathmasterI always wished I was better at proofs. If I had been I may have pursued further graduate work.

Great write up.

18 February, 2014 at 10:55 pm

Bruce SmithThe link “this post” seems broken, in “See this post for some further discussion of such errors, and how to read papers to compensate for them.”

[Corrected, thanks – T.]28 February, 2014 at 2:13 pm

dy/dan » Blog Archive » [Confab] Circle-Square[…] Terrence Tao writes about that continuum here. […]

7 April, 2014 at 7:37 pm

Chapter 1: Dimension | Complex Analytic[…] My hope is that by starting the year by considering higher dimensions will help with both of these issues. I used this idea in the middle of the semester last year and the students were fascinated with the idea. (End-of-semester surveys generally had it as one of their favorite topics and one they wanted to learn more about). Furthermore many of the students bought into the idea that in order to explore an arena like higher dimensions, where we lacked a great deal of intuition, it would be necessary to think more carefully and try to formalize some of our intuition from lower dimensions. (This incidentally is what I think of when I think about mathematical rigor.) […]

25 May, 2014 at 11:38 am

SemiografoNice article. Even better for mentioning Poincaré. The Value of Science is a very influential reading to me. I always found myself a bit loser for being somewhat stuck at the stage 1 of mathematical reasoning. Poincaré gave me a little hope when saying that both theoretical and intuitive minds are important to science. Obviously there are ones that reach the stage 3, but they’re pretty rare I guess.

17 June, 2014 at 9:08 am

QuoraHow should we treat/interpret the relation between intuition and rigor in mathematical proofs?https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

4 July, 2014 at 6:28 pm

Vivek KaulHi Terry,

Should formalism be introduced strongly at the middle school and high school level? Or only people in graduate or advanced undergraduate should be introduced to formalism. I am asking as a person wanting to teach mathematics to middle school and high school students.

Vivek

14 July, 2014 at 12:04 pm

tjzagerI would love to cite this piece in a book I’m writing, but am having trouble finding the publication date. I’d appreciate any help.

[This article is not currently published other than on this blog. -T.]11 August, 2014 at 9:36 am

Toward an Understanding of Mathematics | Empathic Dynamics[…] There is some literature on various ways that students begin to understand maths. Here, I am more interested in advanced undergraduate and graduate learning rather than earlier concepts that a lot of the education literature is focused on. An example would be Terence Tao’s discussion of three stages of mathematical development. […]

14 August, 2014 at 3:36 am

How to study math to really understand it and have a healthy lifestyle with free time? | Crescent Yemen[…] stages of mathematics education to be particularly relevant for me. It sounds relevant to you, too: terrytao.wordpress.com/career-advice/… – Jesse Madnick Jun 11 ’11 at […]

30 August, 2014 at 12:32 pm

Why Do We Think The Way We Do? - My blog[…] Thought seems to be related to this sort of mental simulation, this considering of consequences and verification of intuition. Indeed, we might think of intuitive, unconscious thought as a sort of tennis partner with slower, conscious reasoning — a back and forth. The intuition provides material to the conscious mind and the conscious mind processes that information, which sculpts and corrects the intuition. […]

11 September, 2014 at 7:28 pm

David MW PowersThis is an excellent way of looking at the way we treat mathematics, but different individuals will have different areas of expertise and be in different corners for different areas of mathematics or different application areas. For those that aren’t professional mathematicians, or those that specialize in particular areas of mathematics, some facility either in formal manipulation or intuitive understanding will be either never be gained or will fall away with disuse. Weaning ourselves away from grounded applications, to developing more general understandings, models and formalisms, is key to the power of mathematics, and often means navigating uncharted waters without the benefit of intuition, but ideally will lead to develop better deeper intuitions that in pure mathematics can go beyond real world applicability.

But being able to read and write the formalism is not very useful if it can’t be tested and applied against our intuitions. Having intuitions or ownership of a model is difficult if you are not the originator and don’t have access to the originator’s intuitions. One of the biggest problems I find with other people’s papers in applied areas (Engineering, Computing, Neuroscience, …), is the tendency to reproduce and even extend formal models without understanding either the assumptions or the intuitions that underlie them. At conferences, authors who are challenged on the appropriateness and applicability of a model often show that they are unaware or unmindful of the assumptions and only understand the model in the formal mechanical sense of being able to manipulate equations, verify derivations and produce proofs within the bounds of the model. This perhaps characterizes the dangers of an applied hybrid of heuristic and rigorous approaches and pinches the square into a figure eight as these are brought together in the same individual, or even the ubiquitous paradigm of an entire field.

I mean that an individual has sufficient rigour to go through the motions at a the model level, but is applying canned heuristics at the application level, without either pre- or post-rigour intuition being in evidence, without addressing satisfaction of assumptions or performing sanity checks on conclusions. And I’m not just talking about students. In some cases, whole fields are operating in a kind of unsound limbo, because a formal but inappropriate model takes precedence over common sense intuition and understanding of the boundary cases and the impact of assumptions.

11 September, 2014 at 8:50 pm

SemiografoGood point, David. I’m almost an illiterate on pure math. However, I’m not sure if I’m illiterate on abstract thinking. I’m not even sure if I’m good on intuitions.

On the other hand, I think I’m good on language. I can do abstract reasoning when interpreting and using linguistic tools like metaphors and metonymy. I think language is underrated as a math tool. It’s as abstract as greek characters, but its mapping to _some_ intuition is more natural, at least to me. Every time I see greek letters in an explanation I translate it to language and then to some intuition, if needed, but I always do the translation step. If some concept from pure math is explained by a textual description, I tend to capture it easier and I can instantly map to a bunch of real-world applications.

I know, someone will say that natural language is ambiguous, but so is the actual academic math. How much papers do you read where there isn’t a single sentence written in plain natural language? In this case, ambiguity is not language’s fault, since the author decides doing a textual explanation because math isn’t enough to cover the entire concept. There isn’t any notation capable of motivating the reader for reading a math paper. Thus natural language is an advertising tool, but also is employed for covering gaps when pure math notation isn’t enough.

In short, I think pure math (i.e., math heavily relied on notation) is overrated as an abstract reasoning tool. You can do similar reasoning with plain natural language, so you could reach a broader audience and allow more intuitions to emerge — even pure math intuitions. Proving my hypothesis is the harder part, since I should use pure math, but I think I could contribute with some empirical results for motivating a mathematician to try something on this path. There is room for all kind of abstract thinking mathematicians.

18 October, 2014 at 10:23 pm

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23 November, 2014 at 7:57 am

LucianThere is no connection between maths and rigour. Opera singers or ballet dancers, for instance, also require insane amounts of rigour and precision in the profession of their skill. However, no one actually thinks of classical music or ballet dancing in such terms. At the same time, rather reductively and discriminately, math is perceived in this way,almost to the point of confusion. I prefer intuitive insights or visual explanations over rigorous proofs any day.23 November, 2014 at 10:43 am

Ronnie BrownThere are several ways of looking at this question.

One is the distinction between art and craft. So the final rigour in a proof is the craftmanship of a mathematician, making sure that everything works as claimed. No patches needed (though they sometimes are).

Another is that proofs are like describing a route, and this is in a certain landscape. How much detail should you give when describing a walk to the station? You do not want to describe all the cracks in the pavement, but you do want to warn of dangerous manholes.

One of the jobs of mathematicians is t build a landscape in which proofs, routes, can be found. I heard a comment of Raoul Bott on Grothendieck in 1958, that:”Grothendieck was prepared to work very hard to make proofs tautological.” There is a good aim to make it clear **why** something is true; that may need new concepts.

I once had a student criticism of may first year analysis course: “Professor Brown gives too many proofs.” So I decided next year there would be no theorems and no proofs; what they will get were “facts” and “explanations”.

However there is a kind of obligation that an “explanation” should actually explain something!

A good test of a future mathematician is not necessarily current level of performance, but do they actually want to know why something is true.

19 December, 2014 at 8:00 am

The role of proofs in mathematical writing | Gyre&Gimble[…] There’s more to mathematics than rigour and proofs. Thanks to David Roberts for this reference. […]

24 December, 2014 at 3:12 am

Mathematical thinking skills for engineering students | CL-UAT[…] also Terry Tao’s There’s more to mathematics than rigour and proofs and the anonymous answer to the question What is it like to understand advanced mathematics? […]

6 February, 2015 at 5:27 pm

Career Advice by Prof Terence Tao, Mozart of Mathematics | MScMathematics[…] 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 […]

24 February, 2015 at 8:31 pm

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9 May, 2015 at 12:50 pm

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9 May, 2015 at 12:54 pm

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9 May, 2015 at 1:00 pm

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10 May, 2015 at 7:45 am

conchafofaNice article. In some sense, I feel like (1)->(2) is a transition that is made only once in mathematical life, it is like learning “formality”. Transition (2)->(3) is done for each new subject one studies… maybe one first finds a dry descripcion of the subject (formal/axiomatic), and then tries to build intuition looking at particular cases/models.

10 May, 2015 at 5:20 pm

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21 May, 2015 at 2:40 pm

john zI stopped my intellectual pursuits after I couldn’t take more of academia, ignoring the importance of intuition and heuristic thinking, which I think is a necessary pre-analytical process, for all undergraduates (in math or any other field). I am of the belief that strong intuition, (and importantly, in any IQ range), allows for, enhancements high level problem solving ability. Intuition can be just as important as analysis. It is what sets analysis up.

21 May, 2015 at 3:40 pm

John GabrielIntuition is very dangerous, no matter how intelligent one is. Rigour is very important in mathematics. Of course there is little or no rigour in mainstream mythmatics. Analysis is not rigorous, never was rigorous, will never be rigorous.

Terry Tao is no mathematician, Fields Medal or not.

4 July, 2015 at 5:08 pm

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4 July, 2015 at 5:09 pm

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13 July, 2015 at 12:34 am

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14 August, 2015 at 10:04 pm

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19 August, 2015 at 11:42 pm

QuoraDoes being good at theoretical math/proofs rely more on verbal skill than quantitative skill?Some students at the beginning level, are lost in the abstraction and would think that proofs mainly rely on “verbal skills” . As a students learns more, he discovers that writing a proof is much like writing a story, sure one needs to know grammar a…

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5 October, 2015 at 8:58 pm

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24 January, 2016 at 12:25 am

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