This is an excerpt from There’s more to mathematics than rigour and proofs of Terrence Tao:

  1. The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. 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. 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. The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond.

Does this have any stance in the context of cognitive science? A comment in Mathematics Education says that this idea can't be broaden in other fields/jobs in any level of academic, since:

Tao is using the word rigor in a way that's fairly narrow and specific to math. Namely, rigor means proving every statement one makes or uses, in full detail, from some list of axioms. One cannot rigorously work on cars, any more than one can rigorously eat one's dinner. It also does not make sense to talk about "post-rigorous" in the context of solving geometry exercises, because "rigorous" doesn't mean being good at something.

But I still don't understand this. Sure, it does not make sense to say "eat dinner rigorously", but it is possible for a good cook to know the hidden ingredients, the recipe or the taste of it by just looking and smelling the food, and I think this is equivalent to "proving every statement one makes or uses, in full detail, from some list of axioms".

So how is this phenomenon described rigorously in cognitive science?

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    $\begingroup$ I don't think this is really answerable in the SE format. You are taking one person's opinion of mathematics, which probably not nearly everyone in mathematics even agrees with, and asking us to apply it to the cognitive sciences. It's a great discussion question, but SE is meant for Q&A rather than discussion. $\endgroup$
    – Bryan Krause
    Commented Apr 18, 2018 at 16:26
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    $\begingroup$ Despite the high upvotes, closure declined, and 2 answers from current and former mods, it looks like this question has proven itself to in fact attract exclusively opinion-based answers (the words "I think" appear in all answers, and most claims are uncited), so it's time to finally close this question. $\endgroup$
    – Arnon Weinberg
    Commented Nov 27, 2023 at 3:00

3 Answers 3


Mathematics is the

abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics)

Cognitive sciences are

the interdisciplinary, scientific studies of the mind and its processes

I think that mathematics is a way to achieve something, to put it bluntly, it's a tool. A tool to, for example, model certain brain processes. Further, math is an abstract tool; cognitive sciences is an applied science. Cognitive sciences can use math to accomplish its goals. By contrast, math can never use Cognitive Sciences to prove anything.

Now you say that rigor means

[an] emphasis [on] theory; comfortably manipulat[ing] abstract mathematical objects without focusing too much on what such objects actually “mean"

I don't think one can study the mind without focusing on what it means. Instead, studying the mind is often done exactly to try to find out how it works and what certain findings mean! Now -

Scientific rigor in general means

the quality of being believable or trustworthy

While in maths it has a specific meaning:

logical validity or accuracy

So in terms of semantics the answer is that it is indeed a matter of meaning. Rigor can mean a lot of things, which is different across disciplines.

  • $\begingroup$ Would asking whether logical validity or accuracy is believable or trustworthy out of scope of cognitive science? Related on Philosophy: Is Mathematics always correct?, Should I trust mathematics?. Another question: do you think the post-rigorous stage defined by Tao is essentially what we usually say "gut instinct"? $\endgroup$
    – Ooker
    Commented May 4, 2018 at 5:31

I can recognise similar processes in psychological science as a researcher.

For instance:

  • Throughout your life you generate a lay theory of how people work and how human psychology works. The source of all these beliefs are complex and generally not well understood by people.

  • When you study psychological science, you learn about measurement, statistics, research methodology. You also get exposed to various theories. Gradually, you are training in how to reason about how psychological knowledge is generated. This can involve many different things (e.g., how to interpret a t-test or factor analysis; how to judge a measurement instrument; how to critically evaluate a research design). The focus on these detailed steps can make it difficult to see the big picture.

  • But if you spend enough time updating your theories of human psychology based on a rigorous understanding of methods and the theories that such methods support, you can, in theory, begin to operate intuitively with this knowledge, to generate expectations and novel hypotheses.

So I think this is just an example of the idea that a true expert needs to have learnt the rules in order to effectively break the rules.

  • $\begingroup$ Do you have a term describing it? $\endgroup$
    – Ooker
    Commented Apr 21, 2018 at 2:10

I am only an undergraduante on physics, but I would like to give myself a shot to answer this, because I've read and applied some knowledge to increase my cognitive perfomance on my tests -such as chunking and retrieval routine- and I think I can contribute- with my perspective about your question.

Terence Tao affirmations look very related to how crystalized intelligence shapes the behavior towards an subject wich one has increasingly more and more experience. For a mathematician and for a physicist, for example, those subjects would be planar and tridimensional geometry, matrix operations and etc., wich are basic topics on mathematics and physics.

So, the more one works with different subjects and acquires more experience, more results of some calculations wich one did, in undergraduate years as example, can be easily recalled later on. So, the routes to work on questions and problems in your field become more optimal to a point you can "comfortably manipulate mathematical objects", because it is easier to predict the results of mathematical problems.

It reminds me the article "Recall of random and distorted chess positions: Implications for the theory of expertise" by Gobet and Simon (1996). It affirmes -based on a computer simulation- that a Chess Master has about 50 000 chunks of memory associated to patterns of chess positions. Thus, it is intuitive to also apply that notion to a experient mathematician, because both had learned and acumuled lots of results. Consequently, the mathematician can manipulate mathematical elements with proficiency such as Chess Master can do with chess positions.


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