How valuable is it to learn math deeply?

Reason to learn math deeply: by forcing you to master alternating quantifiers, it expands your ability to understand and handle complex arguments.

This falls, possibly, under your "developing better general reasoning skills", but I would stress it separately, because I think it's an especially transferrable skill that you get from learning rigorous math. Humans find chains of alternating quantifiers (statements like "for every x, there exists y, such that for every z...") very difficult to process. Even at length 2, people without training often confuse the meanings of forall-exist and exist-forall. To get anywhere in rigorous math, a student needs to confidently handle chains of length 4-5 without confusion or undue mental strain. This is drilled into the student during the first 1-2 years of undergraduate rigorous math, starting most notably with the epsilon-delta formalism in analysis. The reason this formalism is notoriously difficult for many students to master is precisely that it trains and drills larger chains of quantifiers than the students have hithertoo been exposed to.

(other math-y subjects have their own analogues; for example, I think the chief reason the pumping lemma is taught in CS education is to force the same sort of training)

The benefit you get is not merely the ability to easily handle statements of this sort in other, non-math fields. It seems to be (personal speculation, no hard data) more general: your ability to understand complex descriptions and arguments with multiple "points of view" improves. Understanding chains of quantifiers demands calmly keeping track of several interrelated facts in your short "reasoning memory", so to speak. With this ability improved, it's easier to keep track of complex interrelated he said-she said-they said scenarios, what-if-x-then-y-then-z-but-not-w scenarios, and so on.

Reasons not to learn math deeply, assuming you're not a research mathematician or going to become one:

1. It's effectively bottomless (unless you actually become a mathematician, and even then in many subfields you remain mostly ignorant of). That is, you'll never say "OK, now I understand the deep principles of why math is the way it is". Rather, you'll always be aware there's fascinating depth beyond what you already know. This can be frustrating.

2. If you don't use it, you'll forget it in a few years. Maybe you'll remember the basic definitions, but you'll likely forget the deep results, and definitely their proofs. If you were emotionally invested in their beauty and thought them a valuable part of your mindscape, this will frustrate and dismay you.

3. Unlike with almost any other scientific discipline, you likely won't be able to construct handwavy cool laymen-accessible metaphors of the deep stuff you found so fascinating. You can chat with outsiders about neutrons, chemical solutions, DNA or phonemes, but not about L-functions.

Random thoughts:

1. The decision that smart high school students should take calculus rather than statistics (in the U.S.) strikes me as pretty seriously misguided. Statistics has broader uses.

2. I got through four semesters of engineering calculus; that was the clear limit of my abilities without engaging in the troublesome activity of "trying." I use virtually no calculus now, and would be fine if I forgot it all (and I'm nearly there). I think it gave me no or almost no advantages. One readthrough of Scarne on Gambling (as a 12-year-old) gave me more benefit than the entirety of my calculus education.

3. I ended up as the mathiest guy around in a non-math job. But it's really my facility with numbers that makes it; my wife (who has a master's degree in math) says what I am doing is arithmetic and not math, but very fast and accurate arithmetic skills strike me as very handy. (As a prosecutor, my facility with numbers comes as a surprise to expert witnesses. Sometimes, they are sad afterward.)

4. Anecdotally, math education may make people crazy or attract crazy people disproportionately. I think that pursuit of any topic aligns your brain to think in a way conducive to that topic.

My tentative conclusions are that advanced statistics has uses in understanding the world; other serious math is fun but probably not optimal use of time, unless it's really fun. "Really fun," has value. This conclusion is based on general observation, and is hardly scientific; I may well be wrong.

Some lessons that I've learned from attempting to solve hard and tricky math problems, which I've found can be applied to problem-solving in general: (a) Focus hard and listen to confusions; (b) Your tendency to give up occurs much before the point at which you should give up; (c) Don't get stuck on one approach, keep trying many different approaches and ideas; (d) Find simpler versions of your problem; (e) Don't beat yourself up over stupid mistakes; (f) Don't be embarrassed to get help.

But of course I don't mean to say that learning math is the only way or the best way to learn these techniques.

(This discussion doesn't distinguish what could be called the rigorous and post-rigorous levels of skill, and so feels a little off (at least terminologically). At the rigorous level, which seems like what you are talking about, you know how the tools work, and can reassemble them to attack novel problems. At post-rigorous level, which seems like a better referent for "learning math deeply", you've sufficiently exercised intuitive mental models to offload most routine observations to System 1, freeing up conscious attention and allowing more ambitious intuitive inferences. Fluency as opposed to competence.)

I'd start with an anecdote from the local practice here, with regards to learning math shallowly vs with an understanding from the grounds up:

It is fairly common to derive supposed ultra low prevalences of geniuses in populations with lower mean IQs.

For example, an IQ of 160 or more is 5 SDs from 85 , but 4SDs from the 100 , so the rarity is 1/3,483,046 vs 1/31,560 , for a huge ratio of 110 times the prevalence of genius in the population with the mean IQ of 100.

This is not how it works; the higher means are a result of decreased prevalence of negative contributors - iodine deficiency, perhaps some alleles, etc. For a very extreme example, suppose that you have a population which is like US baseline, but with 50% prevalence of iodine deficiency. The mean IQ could well be 85 , but the ratio at high IQs will still be 2 rather than increase exponentially with the deviation. Of course in practice, it won't be as clear cut as this, the example is just to illustrate the point.

Figuring things like this out is not so much helped by knowing the concepts as by training and actual practice, and of course, by being trained to know where things like Gaussian distribution come from, not merely declaratively but procedurally as well. (I'm mostly speaking from the perspective of applied mathematics here).

It's interesting that the comments on this post are split in terms of whether they interpret the focus to be on math or on deeply. It's also worth noting that the term "deeply" has many different connotations. Stephen Chew, whom you link to, is using deep learning in the sense of learning something by pondering its meaning and associations. But it's very much possible for an unsophisticated to learn something deeply in the Chewish sense without acquiring a conceptual understanding of it that has transferable value. For instance, one might "deep learn" the product rule for differentiation:

(fg)' = f'g + fg'

by saying "each function gets its turn with being differentiated, and then we add up the products." This is reasonably generalizable (for instance, it generalizes to products of more than two functions, and also to product-like settings in multivariable calculus) but it doesn't necessarily help with deep conceptual understanding of the rule. On the other hand, the somewhat deeper understanding the product rule for differentiation using the chain rule for partial differentiation (see here) actually helps provide a deep sense of why the result is true.

Now, my example above in some sense disproves my claim. The reason being that the Chewish deep learning of the product rule: "each function gets it chance at being differentiated, and then we add up" -- is actually not that far off from the conceptually enlightening explanation based on the chain rule for partial differentiation. So perhaps it is true that attempting Chewish deep learning, without actually having a deep conceptual understanding, enables one to generally get quite close to the correct conceptual understanding.

I used a similar approach to learning chemistry at university level (undergraduate to PhD level, although my PhD drifted a bit from pure chemistry into computing and education). There were lots of situations where, to solve a problem, you needed the appropriate applied formula. Many (most?) students tried to memorise these formulae and the situations they applied in. I struggled to memorise them, so instead focused on how to derive the applied formula from a much smaller set of basic equations. Often there's a mental trick that makes it easier - e.g. to derive the equation for surface tension of a liquid, you think about what happens if you split a cylinder of liquid in half. I found it a lot easier to remember that sort of thing than an equation. (I've not used that particular equation since an exam in 1991 or 1992, but I can still vividly remember the mental model to derive it.)

I can't say that my approach was a better one in terms of getting good marks. When you have to answer specific questions - which you know will be drawn only from the set of situations you were taught - it's much easier and quicker to produce the appropriate equation from your memory and apply it. With my approach, you have to spend valuable exam time deriving the equation before you can apply it. (You rarely got marks for deriving the correct equation.)

However, it has been enormously useful for dealing with real world problems beyond the constrained world of exams. My basic approach - work out what I want to know, think of all the equations that might help, then do a bit of thinking and dimensional analysis to see how to get an equation to relate them - is applicable even when you've never seen the precise equation you need. Or, indeed, when nobody has ever seen it, which is often the situation when you're doing genesis research. It's also extremely helpful for understanding when your equation might not be appropriate, since you only get the equation when you've thought a fair bit about the situation you're trying to apply it to.

Also, in the real world these days, it's trivial to look up an equation. If you're doing any serious work that might require formulae, you'll almost always be sat at a computer with an internet connection. It might be marginally quicker to produce the equation from memory, but the generic skill of being able to look up the right equation is much more widely applicable. It's also, I suspect, less prone to trivial errors (e.g. misremembering a sign) for most people.

I've switched to this approach with maths over time. These days I wouldn't dream of doing any serious algebra or calculus by hand - much easier, quicker and less error-prone to stick it in to Mathematica, Wolfram Alpha or whatever. Those systems know more tricky integrals than (almost?) any human. But I don't do any research in maths, I just use maths in my research.

This ready access to information and symbolic mathematics certainly wasn't the case when the people who taught me learned their basics - it was only becoming the case for some people when I learned them (early 90s). Luckily I was one of the early ones. Sadly, it appears to be taking a long time for the world of formal education to catch on to this shift in how knowledge work happens.

My intuition is that my focus on turning things in to transferable knowledge makes it more transferable. My subjective experience is certainly that this generic approach is valuable across domains, and it feels that that's part of what's enabled me to work in many different domains. But I'm not sure I have convincing evidence that it's the case.

It definitely feels more personally satisfying to me, when I feel that I understand what I'm doing, rather than following a recipe, and have the skills to get to work on a totally-novel situation. But that's a personal preference that is not universally shared. Though my guess is that it'll be more widely shared among LWers than among the general population.

If you're in a position to choose between these approaches (learn deeply and how to apply vs memorising large amounts), I'd strongly recommend learning deeply and how to apply it. It may well not get you the best exam marks, but it'll set you up better for dealing with new situations, and that's a more valuable long-term skill.

I was very successful in my early mathematical education. I'd get As with ease, take exams early, enter mathematics competitions, etc. I had a deep understanding despite doing very little work because all the concepts seemed obvious.

I continued in the exact same way and my performance declined to the point where I was struggling to get Cs. I was now meeting concepts that were not intuitively obvious (eg. limits, proofs, complex numbers), and because of my previous success I had not developed any techniques to gain deep understanding of them. I lost all sense of enjoyment of mathematics and convinced myself that it didn't matter because a good CAS could do it all for me.

I have now started learning again, and there's one realization which has made a big difference. As a student I was always told to solve "problems". This is a terrible name and they should really be called "exercises". The questions are obviously not problems because the teacher has the answer right there in his book. If they are problems then the correct way to solve them is to copy somebody else.

Thinking about the questions as "exercises" makes it clear why you're supposed to solve them, and makes clear how much effort you should put into them. It's analogous to physical exercises -- you don't lift a weight just once and declare it solved, and you don't keep lifting the exact same weight when it becomes easy. I now take the exercises seriously and my understanding improves. I am starting to enjoy mathematics again. I wish somebody had explained this when I was a student.

A counterpoint to the "developing better reasoning skills" point above: it's known that transfer of learning from one domain to another is often very low.

In my anecdotal experience, math is the most transferable of all skills I've learnt.

Up until University where I am now I have never actually had to think hard about math problems I was presented with.

Last summer I had an epiphany in abstract algebra, and it has been hugely beneficial to see these structures everywhere in computer science. That is a lot of handy theorems you get for free.

I think that high-level pattern maching strategies are very valuable. Category theory, abstract algebra, etc.

I don't remember a period of my life where I didn't feel like I had a deep understanding of math, and so it's hard for me to separate out mathematical ability and cognitive ability.

I've also seen advice from a handful of places I respect to learn as much math as you can stand, because there often is transfer from mathematical topics to practical applications. This is much more true for engineers, physicists, and software developers than it is for people in other professions, but still suggests that the first negative consideration you raise is strong (unless it doesn't apply to you).

Reduced need for memorization (while learning math). When you understand math deeply, you see how many different mathematical problems are special cases of a single more general problem, so that in order to remember how to do all of the problems, it suffices to remember the solution to that more general problem.

I remember talking with a friend in high school about physics of electromagnetism. He had the poor fortune to take the non-calculus based version of physics, and so he had to memorize various geometries and the electric potentials they created. I was horrified- in calculus-based physics, we learned one law and then integrated as necessary.

Deduction and analogy seem like largely different reasoning processes. I suspect that what you're describing is that by learning the notation and doing enough deductive arguments, the tasks begin to become intuitive, that is, they begin to become analogical and not deductive.

Deductive thinking is conscious, deliberative, and "slow." Analogical and intuitive thinking is unconscious, nondeliberative, and "fast." So you're probably right that by learning to relegate many mathematical tasks to analogical thinking, one increases their efficiency of learning across domains.

This means that as you jump across mathematical problems you start to see that one telescoping argument looks like another, or one proof by contradiction looks like another, just like your brain has assembled an otherwise arbitrary class "tree" by taking many samples of trees across many domains, building up some kind of conditional inference algorithm for recognizing "trees."

Perhaps this comes under "Reduced need for memorization" but when someone says "deeply" I assume they mean understanding the underlying principles - specifically understanding the limitations of the tools being used:

An extremely trivial example might be how often people in businesses communicate using measures of central tendency (mean) but almost never talk about spread (standard deviation). Yet the SD is as important as the mean.

Perhaps less trivial might be that the analysis of small samples (N < 50) often use T-Statistics. This kind of test requires that your null-hypothesis is normally distributed. Often there are lots of things where this can be reasonably assumed (or has been established) however I've come across places where the assertion was at least questionable.

More advanced might knowing which tools yield a conservative result. Bonferroni for example is a method for handling multiple comparisons. However if the samples are related it can actually yield a false negative.

Learning math deeply is better than not taking math courses at all, which is better than memorizing some formulas for the exam and forgetting them afterwards.

The analogous statements are true of every other field.