“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

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I hope over the past few months that I have challenged your perceptions of mathematics. (After all, that is my ultimate goal.) Perhaps, I have challenged you in such a way that you want to continue to explore and pursue the subject on your own accord to a certain extent of formality that suits you best. Be warned, however, that there is a correct way and an incorrect way to learn this subject. What you have done in your math classes so far has been largely incorrect: In school, math concepts do not build on each other nor is there any emphasis placed on the “behind-the-scenes” rigour. Truly, to understand the subject requires an awareness of both, and below, I provide you insight from a great modern mathematician on how to learn this subject in the way it was meant to be learned.

As a side note, we will be drawing on these ideas during the grand finale of “Musings of a Math Nerd,” if that gives you any incentive to continue.

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“There’s More to Mathematics than Rigour and Proofs”

By: Dr. Terence Tao

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 a 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”.

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 makes formal errors because one is
*unable*to understand how the rigorous mathematical formalism actually works, and is 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 it is pointed out to them.
- Mathematicians at the post-rigorous stage of development are not infallible, and are also 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 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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Eric and Caitlin,

You are both correct. The transition from the first and second step is often the hardest to make (though the the transition from the second and third step should not be overlooked as it can have its own problems as well). When you first learn mathematics, the focus is on applications: Compute this, find this, and calculate that. You dive into computation without the theory behind it. Transitioning from phrases such as “compute” and “find” to “verify” and “prove” is naturally difficult because the emphasis on computation is lost. Mathematics has to be relearned from the perspective of verifications and proofs. To be a mathematician, however, one must be affluent with all the questions–“compute”, “verify”, or otherwise. A mathematician must apply rigour where necessary and computation where necessary.

I am glad you have taken the time to explore more of calculus, Eric. Though a relatively young mathematical sub-discipline, its implications are fundamental to our understanding of the universe. We actually are going to use calculus for the grand finale blog. If you have time, check out infinite series.

In calculus in high school, my teacher very often explained to us the respective theories behind the problems we would be solving. For example, he showed us how the original founders and developers of calculus determined the rules for finding derivatives of different kinds of expressions. Often times, we would have to explain the derivations of certain formulas on exams. But I must admit, since I never particularly enjoyed math, I always just memorized the theories and regurgitated them on tests. As a result, however, I’m probably not as efficient in calculus as I would be if I took the time to understand these theories.

The three stages that you outline actually make a lot of sense to me, given the fact that I am certainly no mathematician and I often struggle with math. The computation develops into theory, and then into fully understanding the mathematical principles. I find that it is hard to break from the first stage to the second in many cases, but that is largely due to our educational principles for learning math.

I have to thank you Ryan, as these blogs have sparked my interest over the last few weeks, even so much as prompting me to learn different facets of calculus online in my spare time.

This is a really great blog, Ryan. I have often said that the reason why RCL is my favorite class is because it isn’t about the “put in your mouth and spit out on paper” technique, but about, as you said, “the behind the scenes rigor.” It’s about true intellectual exploration. I wish that my math class were structured under the same principles as what you described. But, it’s a matter of forcing the information, in bulk, down our throats and when exam week has come and the gag reflux is triggered, the information spills onto the sheet, never to be thought of again.