How to study mathematics rigorously

[Matzematze]

I've come to these forums a lot, but I don't think I've ever really posted--so here goes number one.

I am a freshman in a large public university, and I just recently switched over to mathematics/economics as my majors. I was originally an engineering major, but I abandoned ship because it was way too practical for me (weird, I know...). I just finished multivariable calculus, and I did fairly well in it (A+) which prompted me to consider studying mathematics more rigorously. I never really liked math; I saw it as a bunch of rules mainly because that's how it was always taught to me. Though, I'm really interested now because I've realized mathematics is the language of every science. I've come to believe (inform me if I'm wrong) that I can penetrate into any science by picking up a book insofar as I know the underlying mathematics.

So I have a few questions: Would you say that everyone is capable of learning very high-end mathematics and proofs? If you think not everyone is capable, what percentage of the population do you think can fully understand Maxwell's equations, real analysis etc..? I do find myself struggling sometimes, but I'm not sure if this is normal or not.

How do I go about learning math on my own? Should I buy textbooks? Just solve a ton of problems until I develop the intuition? Just wait 4 years by going through college?


Any thoughts would be appreciated. Just chime in with whatever insights/personal experiences to add to the discussion.

 

[MarneMath]

I don't wish to get dragged into a debate about only super smart people can do really well in math, so I won't, but I will make a few minor points.

1)Your comment regarding being able to 'penetrate' into any science by knowing the mathematics is a bit misleading. It may true that you might have some insight on the mechanics and how resulting equations may behave, but often times it takes the physical knowledge to put reasonable constraints on the solution. Without a good grasp on the actual science, you have no idea what is reasonable.

2)Learning mathematics is a 'full contact sport'. You need to do a lot of problems, and read a lot of math. The more math you do, the more you learn, and thus the more you can do. It takes a lot of 'not knowing' how to solve a problem to advance. You'll find some problems takes you a week to get a good grasp on, other times you'll find things being absorded easily.

3)So with that said, struggling is natural. You'll often find yourself comparing yourself to other math majors and that's bad. Some people just understand some things easily, and you'll encounter this people in every class, and eventually that becomes "everyone else seems to understand this, why can't I?" Then this leds to "I must suck at math!" Reality is more along the lines that humans just pay more attention to the guy doing better than the guy struggling just as much as you do. So with that in mind, just keep focus and keep learning.

 

[3.141592]

I completely agree with MarneMath.

Regarding being able to 'penetrate' into any science by knowing the mathematics; well, you'll certainly penetrate the proverbial skin of all those people who've spent years in their discipline when you walk in, pick up a book, and tell them you don't need to bother with med school because you've a maths education. To see this more clearly, would you rather have a diagnosis by a dentist or a math major who's read about some dentistry? Ok, so you can perhaps give some insight on some aspects to some extent. Sure. But there is so much more to any science you wish to name than mathematics. In fact, if you want to say roughly that science and experiments are intertwined, then, since experiments involve measurements (as well as design), do stats instead.

You know English grammar. And English vocabulary. You can pick up any book written in English and read it. Cover to cover. You'll have more to say about it than someone with poor knowledge of grammar and a smaller vocab. And much more to say than someone who knows no English. Some of what you say will no doubt be insightful. But you can't really think you can 'penetrate' to substantial depth into any topic whatever, so long as you can read the language in which it's written?

As for what I think the percentage of people who can learn it? Pick any percentage you like, to any decimal place you like - except 0 I suppose - and it won't matter one bit. Let's not pee in the statistical pool anymore than it has been by plucking 'statistics' out of the air. (No offence, really :) )

More constructively to that point, I think someone posted on here about the cult of genius. Also, the inverse power of praise. I think you should read around those. Thinking that 'either you get it or you don't' is a fantastic way to train yourself to give up on a thing the second it gets hard. Don't have any of it. No royal road to geometry, right?

 

[Stephen Tashi]

I agree with the previous posters about math not being the skeleton key to all fields of science. In studying science, It certainly helps to know mathematics and you do see posters in various scientific topics that are stuck on problems that a math major would immediately clarify. However, being a mathematician isn't going to enable you to pick up a random science textbook and zip through it.

As to what fraction of the population can learn to do mathematical proofs - you have to define what population you are considering - the entire human population?, the population of university students? I don't know how to estimate how many people "can do" mathematical proofs, but if you look at the population of students that accomplish this task, you can get an idea about what how many "actually do".

I don't think "the average bright person" can learn to do mathematical proofs any more than the average bright person can learn to be a chess master or an expert Bridge player. That's just a guess. It is clear that the average bright person doesn't learn to do proofs - and I'm only talking about learning how to do proofs at the level of typical textbook exercises. Before you burn your bridges to engineering, you should take the course that introduces people to doing rigorous proofs. (It's often the "wash-out" course for prospective math majors.)

I also think the average bright person doesn't make much progress teaching themselves mathematics without exposure to math courses or, at least, to people who know mathematics and will discuss it. The average bright person is too confident about their own point of view. What is needed is to exposure to how people who can do abstract mathematics think. (The average person is also too slow and plodding when doing self study. They do better with the spur of homework and tests.)

 

[Matzematze]

So when I said "language of sciences" I meant it quite literally. If I picked up a higher level physics textbook and so a del operator, I wouldn't really know what to do with it. Knowing the mathematics wouldn't necessarily make me qualified to learn string theory, but at least I have a much higher chance of going through the book and understanding it. I do know what the del operator means now, and I'm sure it will come in handy in some economics textbook someday. Penetrating the depths of the subject will happen by starting at step 1 and going through the sequence necessary. I believe it would be easier to get to the end of the subject if I already know the math.

By "sciences", I really should have been specific. I don't think I can learn dentistry, forestry, or biology just because I know math. The two subjects I am really interested in are economics and physics. These two really do need some understanding of the 'language' that is math.

So I would consider myself the average bright person. I have been trying to learn how to write proofs on my own, but I just can't get motivated for long enough. I suppose you are right on that point, Tashi.

 

[homeomorphic]

So I have a few questions: Would you say that everyone is capable of learning very high-end mathematics and proofs?

I think I could rule out downs syndrome type people. Otherwise, it's hard to say. If very high-end means win the fields medal, I don't think so. If it means eventually understand enough math for an undergraduate degree, maybe. Typically, it takes 4 years. If someone were to devote their entire life to reach that modest level, just to prove it was possible, I think eventually, almost anyone would probably succeed. I don't know. The problem is that people who don't have much of a knack for it aren't interested in it enough to spend their whole lives on it, so there's no way to know what would happen if they did. As things stand, the average person has a lot of difficulty with even the most elementary math.

If you think not everyone is capable, what percentage of the population do you think can fully understand Maxwell's equations, real analysis etc..?

There's no point in guessing a percentage. However, judging by the current standards of education would be misleading in the extreme. I think people are not being educated even in the same ballpark of what they would be capable of if our educational system were as effective as it should be.

I do find myself struggling sometimes, but I'm not sure if this is normal or not.

Very normal. The bad news is that it gets 10 times worse if you get to graduate level stuff, and another 10 times worse if you get to research-level stuff. The good news is that, to an extent, you may get that much stronger. I know math professors at fancy departments who tell the tale of having failed calculus. Personally, I breezed through my upper division math classes with very little trouble, but graduate school has nearly destroyed me. But, when I started out in college, I got C's in stupid classes like differential equations and linear algebra. I made quite an epic comeback with later classes that everyone else thought were so difficult. I did okay, and at times, even pretty well, in grad school, until I had to do research, which is a whole different story. If you thrive on stress and high pressure and are very competitive or, if you are almost as interested in teaching as you are in math, academic math might be a good career for you, otherwise it complete masochism and should be avoided at all costs.

How do I go about learning math on my own? Should I buy textbooks?

I would. But the library is very useful. It's good to try different books and see which ones you like. Try Visual Complex Analysis. Best math book ever.

Just solve a ton of problems until I develop the intuition?

I often poo-poo the "just do the problems" approach, and this raises a few eyebrows because people do not bother to question whether I might have some deeper reasons for this, and they sometimes are also too quick to jump to the conclusion that I am saying you don't need to do problems. You need to do problems. Do a lot of problems. However, there are various issues with JUST doing the problems. One is that it's not just whether you do the problem or not. You can ask what the point is, what did I learn from this, what can I take away from it, is there a more enlightening way to do it, quicker way to do it, etc.

Just wait 4 years by going through college?

You can do that. I would recommend doing some stuff on your own. But there are various hazards. You might sort of wander aimlessly in mathematics, without a clear goal that you are working towards and get kind of lost. So, you have to ask what the end goal is, what are you going to do with it, and then try to pursue what will help you get towards that goal. Some random exploration might be good, but, like I said, you don't want to get lost. Also, there are a lot of bad books out there, plus a lot of books that aren't bad, but that you won't be ready for, and so on.

 

[espen180]

I mostly agree with previous posters, especially MarneMath.

I think everyone can learn mathematics, and learn how to write mathematical proofs. There is a lot of routine involved, but at the core, "practice makes perfect" goes a long way.

@Stephen: I don't think the comparison with chess masters and Bridge experts is fair. However, any person who invests enough energy into playing chess for an extended period of time will become able to plan relatively long sequences of moves using abstract reasoning. This process of searching for the best moves, and determining why they are the best moves possible, is much more similar to the process of writing mathematical proofs.
If you can play chess, then I think you at least have the potential to do fairly abstract mathematics.