# How to understand mathematics on a deep level?

I've been focusing on self studying mathematics for the past couple months, and I'm currently working on discrete mathematics. Here's my attempt at a metaphor to describe my issue.

Imagine you have a jar, whatever is in the jar represents the knowledge you possess. When learning something like the natural sciences (physics, chemistry, etc.) it's like pouring a liquid into the jar. The ideas form more smoothly in your mind, and fit in more naturally with the knowledge you already have.

But when I'm studying math in particular, sometimes it feels like dumping lego bricks into that jar. It can feel jarring and unpleasant. Each theorem, definition and proof is like a singular lego brick. And you see how the pieces might fit together, but the bricks aren't quite connected to eachother yet. In fact, they just form a jumbled mess right now.

If that made sense to you, I just want to know if this is natural. As interesting I find mathematics, the jarring feeling I described earlier can make it really unpleasant to read through a book. And this can be very discouraging. It's like you find something interesting, but at the same time you find it unpleasant. How do you overcome this, is it just a matter of persistence?

• I think I might understand your question after reading the details, at least in my own way. One thing I like to do to lessen that jarring feeling and more quickly assimilate the ideas of the theorems into my mind is to reword the theorem in more layman's type terms. Being able to do this also shows me that I am really understanding the theorem's hypotheses and results. Try to summarize the theorem's hypotheses and results in one sentence that easily flows and explains what's going on. Then, when you go to recall the theorem, or are discussing it, you can recall your easier wording of it. – layman Jul 27 '15 at 3:53
• For some reason math authors often make zero effort to motivate the material or explain how someone might have thought of it. They might just throw out some very abstract definitions with no commentary. It helps me when I find books that attempt to explain the motivation and intuition. – littleO Jul 27 '15 at 4:21
• @littleO Some may find that that clutters their view of a particular idea and they would rather much prefer the bare bones of an idea. There do exist many math books still that take the reader on a journey. – Ali Caglayan Jul 27 '15 at 4:24
• This question is kind of a follow-up to the question How to begin self study of Mathematics? by the same proposer and I think this clever question should be encouraged and not censored. I strongly disagree on the "primary opinion-based" decision. The answer by @Ralph_Hiesey is based on (50 year -) expert experience, a very specific expertise. – J.-E. Pin Jul 27 '15 at 10:03

I majored in math in college over 50 years ago-- this is from my distant experience of that time.

A previous writer (littleO)had an important comment about finding the right book. Some authors seem to be trying to demonstrate their cleverness rather than trying to really convey to the reader the motivation and intuition for the math. So if one text doesn't seem to work for you, look for another.

However I also understand the jarring feeling one has about first encountering some ideas in mathematics. First-- I think you have to come to appreciate the jarring feeling--and enjoy the feeling of being challenged by it. The process of understanding doesn't necessarily happen as quickly as in the the natural sciences so you shouldn't be discouraged too soon. Quite unlike reading a novel, it usually takes a number of passes over the same material. If you're having difficulty understanding--work hard trying to understand-- then get away from it and come back to the same section a day or two later is often helpful. And also it's extremely important to actually do some exercises in a textbook. I don't think you can learn math properly without doing these.

Eventually the jarring should change--perhaps suddenly, or perhaps gradually-- into understanding, not only in a formal sense, but also an intuitive sense. That can make you want to experience more.

You might have a good intusion for science, but to understand modern science you also need the language of mathematics. Learning (other) languages can be boring but it's worth it being able to use it.

Perhaps there are more urgent disciplines than discrete math to start with? Calculus and algebra are more important at the begining in science and is more easy to melt into the scientific soup...