We agree that if you want actually to be able to do the math then you need to do the exercises.

Do we agree that if you don't do the exercises then you probably won't actually be able to do the math?

You are discussing learning about the math, and not eventually being able to do it, because you say that you don't care about becoming a mathematician, therefore you don't need to do the math. Fair enough.

But my reading is that that's not what this thread is about. This thread, and the original submission, is about learning how to do the math.

> i do not see the exercises as essential for anyone other than practicing mathematicians.

I think you're wrong. Knowing how to actually do the math has proven useful to many people for whom it is a tool in their craft/job/employment. Learning Linear Algebra properly, being able to actually do it rather than just talk about it, can be enormously useful in Machine Learning.

>> That, to me, implies learning to do, not learning about.

> here's a fantastic explanation of the topological proof of Abel-Ruffini ... would you say that I don't understand that proof if i haven't done any exercises related to it? and therefore would you say I didn't learn any math by having watched that video?

Understanding a single proof implies very little about one's ability to actually do the math. I've met many people who are math enthusiasts and who have watched hundreds of math videos. They say they understand all of what they've seen, and yet they are unable to do the simplest proofs, or the most elementary calculations.

My experience of people's abilities is that if they haven't done the exercises, they usually can't actually do the math.

But you complain about the length of my replies, so I'll stop. I think I've made my position clear, and I think I understand what you're saying, even if I don't agree with it.

>You are discussing learning about the math

You keep repeating this but you're evading the question about abel-ruffini and the question about whether reading a history book is "learning about history" as opposed to learning history.

You're making a weird distinction. People learn in different ways. Some by doing exercises and some by just playing with the objects. I wonder how you think actual research mathematicians learn new math from papers that don't include exercises lol.

You edited your response.

>I've met many people who are math enthusiasts and who have watched hundreds of math videos

There's a difference between watching numberphile or whatever and essentially watching a lecture on a proof. Very few people are watching/consuming rigorous expositions. I think that's the difference not the lack of exercise.

Learning about history is not the same as then being able to do research in history, nor being able to apply the principles learned from it in context. So no, reading a history book is learning about history, not necessarily being able to "do history".

> You're making a weird distinction.

As someone who has done a PhD, done research in math, done research in computing, worked in research and development in industry, taught math, and headed a team doing research in technology, this is a distinction that I can clearly see. My inability to explain it to you is regrettable.

> People learn in different ways.

Yes they do.

> Some by doing exercises and some by just playing with the objects.

Doing the exercises is playing with the objects to try to answer specific questions. Good exercises are carefully constructed to help the reader learn how those objects work in an efficient manner.

> I wonder how you think actual research mathematicians learn new math from papers that don't include exercises lol.

In my experience research mathematicians learn now math from papers by, in essence, constructing their own exercises based on what they're reading. In general it takes significant experience and training to be able to do that.

Clearly you don't think one needs to do the exercises subsequently to be able to do the math. Good for you.

I disagree.

>As someone who has done a PhD, done research in math, done research in computing, worked in research and development in industry, taught math

Me too so now what? I don't think your credentials give you any real authority but just make you look like you're gatekeeping.

>Doing the exercises is playing with the objects to try to answer specific questions.

Great so then we're in agreement: playing with the object is doing the exercise.

The funny thing is that at one time I actually did all of the exercises in volume 1 of apóstol's calculus. You know what effect on me it had? I was so bored I didn't read volume 2. And today I'd still need to look up the trig substitutions to do a vexing integral.

> I don't think your credentials give you any real authority ...

It wasn't intended to, it was to provide a context for my opinion.

So let me state my opinion as clearly as I can, and then I'll leave it.

* Math is a "contact sport" ... you have to engage with it;

* Reading books is not, of itself, engaging with the math;

* Watching math videos is not, of itself, engaging with the math;

* Well designed exercises are a valuable resource;

* If you can easily do an exercise, skip ahead;

* If you can't do an exercise, persist (for a time);

* Ignoring the exercises is ignoring a resource;

* For the vast majority of people, doing the exercises is an efficient way to engage with the material;

* To say "ignore the exercises" is, for the vast majority of people, an invitation to not bother engaging with the subject;

* Doing all the exercises is probably a waste. Doing none of them is an invitation to end up with a superficial overview of the subject, and no real understanding.

See? It's pretty hard. This is what I've been dealing with for the last 20 years of on and off trying to get through the bigger Rudin book and a couple others.

Just reading doesn't get much at all. Not even a superficial overview. I tried it. It's essentially a meaningless combination of words after a certain point.

Reading extremely thoroughly is actually marginally useful. Stopping to think, do all these assumptions matter, why, what if one of them changes, etc, pencil in hand, making notes, testing things out. I've managed to "understand" the topics when doing this, and so far it's been the highest ROI method. But it does still leave one feeling like something is missing. Just because you can sight read music doesn't mean you're an expert on the piano.

Doing exercises is a huge jump on investment, and the return on that investment is a bit questionable from my experience. A couple reasons: first you don't know if you did them right. If you did them wrong then that's negative ROI. Second you don't know what a "reasonable" workload is. It varies by author. Is it three problems per chapter, is it all of them, are some orders of magnitude more difficult than others? Without some guidance it's hard to know if your difficulties are due to not understanding basic material, or due to that problem being a challenge geared toward Putnam medalists. So they may cause you to question your understanding and thus mentally roadblock you unnecessarily. And finally with proofs (and this may be a me thing), it's pretty easy to say "I guess this is okay(?)" and move on, even if you're not sure. Since nobody is ever going to review it, and it's just a homework problem, it's very very hard to will oneself to make sure every assumption is correct and you're not missing anything, even if you feel like there's a good chance you are. Or perhaps I just don't have the constitution to do so.

So while I think doing exercises is necessary for a deeper understanding, I don't know whether the ROI is worth it outside of a classroom perspective. You need feedback for exercises to be beneficial. At least, I feel like I do.

Finally, is even taking a class that useful if the end state is that two years from then you'll have forgotten most of it and so what was the point. Can you claim knowledge of a subject that you've never actually used beyond some homework problems and exam questions, or is this still a superficial understanding? Having an ends where that knowledge gets used seems critical.

I feel like I have some knowledge but I don't feel like I'm there yet. But I don't know if I know where there is. Maybe that's the biggest challenge. Does completing a Ph.D. even get you to there? No idea. But, I guess it's up to the individual to decide what they want out of it. Nobody can determine that for you.