Is this a good idea for studying mathematics?

[bacte2013]

Dear Physics Forum advisers,

I am an undergraduate student in US and an inspiring applied mathematician in the fields of theoretical computer science. I recently have been reading a book called "Joy of Learning" by Hironaka Heisuke, a mathematician and Fields Medalist (1970); the book seems to be published only in Japan and South Korea, so I do not think there is no translated version in English.

In the book, Professor Heisuke remarked that whenever he asked a question to mathematics students (undergraduates and graduates), most time he got responses like "Oh, I am studying algebraic geometry", "I am studying abstract algebra", "I am studying analytic number theory", etc. He felt little disappointment since he hoped to hear ideas and theories the students formulated. He said that it would be a great way to study the mathematics by formulating one's own conjecture first, investigating about it more by reading books involving them, and developing that conjecture or changing it if it is not validated.

I was intrigued by his remark, and I would like to implement his thought to my current method of studying. My method of studying the mathematics is just reading books in topics of my interest, and formulating interesting questions as I read the books and trying to answer them. Do you think it would be a good idea to come up with my own idea before reading the books? For example, before reading a book in the abstract algebra, should I come up with idea like an existence of patterned sets or an interesting question like applications of groups and fields to cryptography, and try to think about ways I can describe those sets (groups, rings, and fields) before diving into the book? Since I am doing a undergraduate research in the theoretical computer science that involves a lot of mathematics (combinatorics, graph theory, analysis, etc.), perhaps I should try to formulate my own theories from the main project, and try to answer them too?

I really apologize for the long post, and look forward to hear back from you!

 

[micromass]

Yes, this is a very good idea and it is essentially how mathematics research works. In research you need to read a lot of papers. But you don't do some randomly. You usually have some conjecture in your mind that you want to know more about and the papers you read will be approached from that perspective.

But reading things in order to solve a specific problem is dangerous. It can read to selective reading and ignoring many things which are unrelated to that problem but which you still need to know.

So you need to take the best of both worlds: you need to do both. So a part of your time, you keep yourself busy with reading standard material. But another part of your time you actively pursue an interest of yours and you go really deep into that. How do you do the latter? Well, formulating a conjecture that you would like to solve is one thing. But it can be difficult to formulate conjectures without much advanced knowledge. But the next best thing is good too: is there something you would really like to know well? You don't need to have a conjecture, but it might be a theory or result you would really love to understand. Then work towards it.

So what do you do practically. Either you formulate a conjecture, or you identify some topic that you would really like to know. Then you ask knowledgeable people to provide you a very direct path towards that goal. And then you start studying that path with always the specific goal in your mind.

Here's an example that I have succesfully done with some students of mine. I had a student who told me he was very interested in general relativity, but that he was very confused by the math behind it. So I first identified what level math he was on. And then I provided a pathway towards general relativity. During this, we studied e.g. differential geometry. During every concept in differential geometry that we studied, we immediately thought about how it could be applicable to Relativity and what it meant physically. Sometimes this was of course not possible (immediately). But other times, it gave us a lot of cool insights.

 

[fresh_42]

Well it all depends on where you're starting at and where you want to go to. What Heisuke probably meant is that it can provide you a lot of motivation if you try to figure out how something special you're heavily interested in works. Studying math you'll often reach points where seemingly nothing makes sense or just you cannot figure out why two consecutive lines in a textbook belong together. I've never met someone smart enough to avoid those moments. Though the impetus from longing for answers cannot be underestimated. However, learning is a rather personal task and everybody has to find his own suitable way. Some need tasks to be solved, some want to understand concepts and axiomatic bases behind it and another might just capture everything with an eidetic memory. Nevertheless in studying math it helps a lot to "play with it". Trying to solve a self imposed question can urge you to "play". Whether it's sufficient for your purposes may be another question but it certainly helps a lot.

 

[bacte2013]

Yes, this is a very good idea and it is essentially how mathematics research works. In research you need to read a lot of papers. But you don't do some randomly. You usually have some conjecture in your mind that you want to know more about and the papers you read will be approached from that perspective.

But reading things in order to solve a specific problem is dangerous. It can read to selective reading and ignoring many things which are unrelated to that problem but which you still need to know.

So you need to take the best of both worlds: you need to do both. So a part of your time, you keep yourself busy with reading standard material. But another part of your time you actively pursue an interest of yours and you go really deep into that. How do you do the latter? Well, formulating a conjecture that you would like to solve is one thing. But it can be difficult to formulate conjectures without much advanced knowledge. But the next best thing is good too: is there something you would really like to know well? You don't need to have a conjecture, but it might be a theory or result you would really love to understand. Then work towards it.

So what do you do practically. Either you formulate a conjecture, or you identify some topic that you would really like to know. Then you ask knowledgeable people to provide you a very direct path towards that goal. And then you start studying that path with always the specific goal in your mind.

Here's an example that I have succesfully done with some students of mine. I had a student who told me he was very interested in general relativity, but that he was very confused by the math behind it. So I first identified what level math he was on. And then I provided a pathway towards general relativity. During this, we studied e.g. differential geometry. During every concept in differential geometry that we studied, we immediately thought about how it could be applicable to Relativity and what it meant physically. Sometimes this was of course not possible (immediately). But other times, it gave us a lot of cool insights.

Dear Professor Micromass,

Thank you very much for your detailed advice! By taking the best of both worlds, do you mean that I need to do read the books assigned for courses I am taking (is this what you mean by a "standard reading"?) and books covering the topics of my interest? My current undergraduate research involves a new theorem relating to the algorithms, and I also developed some ideas relating to it. I have been reading books treating the mathematical branches involved in my research, so I guess I am actually following some of your advice.

About the selective reading, will skipping the prerequisite be a great problem? For example, I want to learn about the extremely graph theory since it is involved in my research and I am interested in it. Should I jump directly to the books covering the extremal graph theory or should I start with the combinatorics and introductory graph-theory books? I often run into a problem of prerequisites...There are some topics that I am really interested in it, such as a complex analysis and analytic number theory, and I am very eager to learn about them. However, both topics have prerequisites involving analysis and introductory number theory...I found I like to just jump directly to the topics of my interest and learn prerequisites as I go, but I see that it can be dangerous judging from your advice about the selective reading. Do you think it is a great idea to just jump directly to the topics and learn the necessary prerequisites as I go?

Thank you very much for your time!