Terry Tao describes 3 stages of one's mathematics education on his web blog.

1: Pre-rigorous
2: Rigorous
3. Post-rigorous

I know how one can progress from stage 1 to stage 2 (this simply can be done by learning a book like 'How to Prove It' by Daniel Velleman or Ian Stewart's 'Foundations of Mathematics')

What can one do to progress from stage 2 to stage 3? Terry Tao states "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."

Also, are there any books that can facilitate this transition similar to how the aforementioned books can facilitate the stage 2 to stage 3 transition?

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    $\begingroup$ It's worth noting that these phases (ideally) occur several times throughout one's mathematical career, even all the way back to basic arithmetic. Pretty much any topic that can be approached formally has pre-, during, and post- phases. $\endgroup$
    – TomKern
    Commented Nov 2, 2022 at 6:52
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    $\begingroup$ @TomKern hits the nail on the head. A single mathematician does not exist in a single stage in Tao's hierarchy. Rather, when approaching any particular problem, a given mathematician may exist in any one of the stages, depending on their expertise and background. For example, I probably approach most problems in fractal geometry from a more post-rigorous point of view, but most problems in Galois theory are going to toss me right back into a pre-rigorous mindset. $\endgroup$
    – Xander Henderson
    Commented Nov 9, 2022 at 15:07

1 Answer 1


This is not as much to answer the original question (to which the answer is just that you develop any skill by trying to practice it and evaluating the results) but to tell what my understanding of a "post-rigorous thinking" is. My main point will be that it has nothing to do with "trusting your refined intuition" (Intuition is a funny thing and it deserves a separate discussion) but rather internalizing the ideas to the extent that you no longer need to apply them consciously. It is the ability to just see the answer sometimes even without immediate understanding why you see it.

What do I mean? Let's start with $4\times 9$. The first stage would be just to honestly compute it as $4+4=8, 8+4=12, 12+4=16, 16+4=20, 20+4=24$ (Erm... where am I? Looks I added $4$ five times? Ah, no, the initial one counts too, so six, actually. Three more to go) $24+4=28, 28+4=32, 32+4=36$. Hurray, $36$!. The next stage would be to use some easy anchoring points and/or properties of multiplication and proceed from them. Since it is easy to multiply by powers of $10$, one can do $4\times 9=4\times (10-1)= 4 \times 10-4=36$. Much cooler, isn't it? The final stage is you just see the answer $36$ without even knowing how your brain did count, so if somebody asks you "Why? How did you count?", you may be a bit at loss how to explain and the explanation you finally give, although formally impeccable, may have very little to do with your actual thought process. You just no longer remember which way of counting finally led to the internalization. Revisiting it (i.e., asking this question yourself) may surprise you and here you will be able to observe the distinction between "mechanical memorization" and "internalizing the idea". For, if you ended up with pure memorization of the multiplication table, you will know $4\times 9$ but not $6*99999$ or $3\times 1213122$ but if you ended up with internalizing the ideas of anchoring and split, the answers to those will be just as obvious to you, though in these cases you'll, probably, know how your brain counted.

Now two real life examples from my recent teaching experience (graduate analysis).

Problem 1: Show that there exists $\delta>0$ such that if $a,b\ge 0, a+b=1$, then $a^2+b^2\ge\delta$ (It was explicitly stated and emphasized that there was no need to come with an optimal or even numerical value!).

Students: $b=1-a$, $a^2+(1-a)^2=2a^2-2a+1=f(a), a\in [0,1]$, $f'(a)=0\Leftrightarrow 4a-2=0, a=1/2, f(a)=1/4+1/4=1/2, f(0)=f(1)=1>1/2$, so $\delta=1/2$ works (It took about 10 minutes to come up with that, believe it or not!).

I just see it. If asked to explain, I might come up with

The set $a,b\ge 1,a+b=1$ is compact, the function $a^2+b^2$ is continuous, the only way it can get to $0$ for $a,b\ge 0$ is $a=b=0$, which is not in the set, ergo...


$a+b\ge 1\Rightarrow \max(a,b)\ge 1/2\Rightarrow a^2+b^2\ge (1/2)^2\ge 1/4$.

I honestly do not know which of those really ran subconsciously in my brain or even whether it was any of those.

The difference between these ways of thinking is that with the students' approach the question "$a,b,c,d\ge 0, a^3+b^2c-5cd^6+d^{18}=1$. Conclude that $a^5+bc+d^5\ge\delta>0$" becomes insurmountable, while for me it is just as obvious as the first one.

In this case both explanations of how I might be thinking are formally correct. The next case is different in this respect.

Problem. Show that $\inf_{B\ge 0}\int_{[1,2]}\log|1-Bt|\,dt>-\infty$.

Students: Compute the integral explicitly splitting into cases (the antiderivative is elementary here), investigate the resulting function in each case, conclude.

Again, I just see it. I was really asked by one of the students "How". I evaded the response, though my initial desire was to say:

"Look, for $B\ge 2$, the integrand is positive, for $B\le 1/2$ it is bounded below by $-\log 2$, $[1/2,2]$ is a compact interval, there is no point on it where the integral diverges, so it is, probably, continuous, hence, bounded from below".

I didn't give this response, but for a while I kept it in my head always feeling that something was incongruous about it. Two things bothered me: that shaky "probably" and why I bothered to take special care of $B\le 1/2$ separately ($[0,2]$ is also compact and there is no divergence point there).

It took me some time to realize what might be a good formalization of the "obvious" here. It is

If $f:\mathbb R\to\mathbb R$ is such that $\int_\mathbb R\min(f,0)>-\infty$ and $\sup g'<-C<0$, then $\int_{\mathbb R}f\circ g\ge C^{-1}\int_\mathbb R\min(f,0)$ (because you shoot through the negative region at the speed at least $C$).

So my initial "compactness" explanation was totally off. I didn't need to take any special care of large $B$, no continuity played any role, and the only thing that I was right about (and which looked strange in that explanation) was to worry about $B$ close to $0$. And, again, my second explanation is formally impeccable but I won't give you any guarantee on whether that is what I actually thought in my subconscious brain.

I guess those two are enough to illustrate my point of view, which is that the "post-rigorous" thinking is not trusting the intuition but internalizing the ideas and techniques to the extent that you are no longer able to explain how you figure something out to the extent that when you are forced to come with an explanation, you may easily end up with a wrong one. It is like driving a car through a sharp turn and being asked "How did you choose the speed and the angle of the rotation of the steering wheel so you didn't go off the road"? In all honesty, I have no idea of how my brain determines from just seeing a curve ahead and a wet or dry asphalt which number of my speedometer I should be aiming at and by exactly how much my hands should move to rotate the steering wheel. I just do it. Can you explain what is really going on here? And it is not a "pure memorization": I have never seen this turn before. Yeah, memorization plays some role in driving: if you go over the same very road again and again, you find out that you can drive on it faster and more confidently than on an unfamiliar stretch, but you learned to drive decently, you don't panic or crash when you hit some new road either. You just exercise a bit more caution, and that's it. And you don't just "trust your feelings". Those who exercise that driving philosophy end up in a ditch more often than not. You do know exactly what you are doing. You just no longer know how you know it, but the rigor is not lost in the slightest, so I would say "post-rigorous" is a bad choice of name for this stage. I would really call it "shmarky plaft" because I have no real idea how the brain is actually operating at this level. And it is possible to achieve in any craft, mathematics being no exception. How? It just comes with experience and active learning from your errors.

If somebody can up with more bright ideas, I would be most interested in hearing them, but so far here are my two cents :-)

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    $\begingroup$ When I read your problem $1$ I immediately visualized the line segment from $(1,0)$ to $(0,1)$ in the $ab$-plane. $a^2 + b^2$ is the square distance to the origin, and is hence minimized at the point which is closet to the origin. This clearly happens at the midpoint, so we can obtain an optimal bound. $\endgroup$ Commented Nov 8, 2022 at 18:42
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    $\begingroup$ @Joseph O'Rourke (and others interested): My comments to this answer explain how I've interpreted Tao's "stage 3". $\endgroup$ Commented Nov 8, 2022 at 19:55

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