Problems with a humanistic background

[Kolmin]

I am not sure on how I should put what I feel are my doubts in this thread, so I will start from the scratch, trying to give you the bigger picture. Obviously, I hope that's the right section of the forum.

I had a mainly humanistic background. It means that I literally grow up spending my spare time mostly on novels, philosophical books and so on. This is what I would call my mothertongue: I feel language (not English, cause I am not a native English speaker) as most of you feel integrals and so on.

At the same time I was always interested in maths, in particular in logic at the beginning, due to my philosophical interests. The interesting side is that, due to my stubbornness, an amazing interest and some peculiar choices (related to my interest in economics), my life (in terms of potential job and so on) came to be mainly math-oriented so to speak.

The problem is that this is not my mothertongue, but instead it is a second language I gradually acquired with a lot of efforts and problems without having any help (I am completely self-thaught). So what I would like to find out is if my efforts on maths are basically doomed. Indeed, I have in particular two doubts (or problems):

1) I hardly remember mathematical things (concepts, structure and so on). Basically, I always need to start from the scratch. I always need to remember why something is something else and it seems I cannot impress it in my memory (at least that's my feeling - maybe I overemphasize it...), even things that I read many times. I understand them, I can explain them to anybody, I can apply them, but give me a month and I will have the feeling that I have forgotten them. They are somewhere, but I need an external input to remember them. So, is it a real problem?
Interestingly, I have what can be considered a very good memory in almost any other field you can think about. I explain this lack of memory in maths with the fact that when I started to study it, I was focusing more on remembering than understanding (bad high school teachers), and that was a clearly idiotic choice (but nobody was there to give me decent advice). So I assume that at a certain point, when I decided to seriously studied it, I unconsciously (and smartly) bypassed memory in order to focus completely on comprehension.
Just consider also that I think that to remember something, you need to express it and discuss it (it is a sort of active process) and having nobody to discuss about maths, I completely lack this part of the learning/memorizing process.
Still the question stands: is it such a problem in terms of long-term studying (and career) project?

2) A bit more technical question. Let’s imagine that in an article to describe a certain economic structure an author uses Polish spaces. Now, I think “Ok, let’s figure out what Polish spaces are?”. So I find out that they are not present in the index of most of the topology or infinite dimensional analysis books I have. Then I find that they are in the middle of the second volume of Bourbaki’s Topology. The problem is: should I read the whole book? Can I really understand and use (!) something that comes to me out of nowhere simply cause I have read quickly the definition without deeply knowing most of the background?
The point is: in fields in which math is ancillary, do you first get the idea and then translate it in mathematical terms trying to find out the most adequate one or you need to be a math-savant from the beginning?
In other words, how can somebody who is not mathematically driven work backward to build up mathematical definitions? Moreover, is it possible?

In a certain way I think that both questions drive towards the last on: should I pursue a career on economics (which is now strongly related to mathematics) if I am not completely trained in mathematics?

Thanks a lot for any comment or feedback.

 

[MarneMath]

1)This is more common than you may believe. To keep this brief, unless you use something often or teach it a lot, I find a lot of math memory just becomes more vague. For example, I have a solid handle on most statistical tools, but if you ask me to state green theorem, i'll look at you like you're speaking swedish. The important aspect of this is this though, once you tell me what green theorem is again, I will go "oh yea" and it'll (slowly) all come back to me. So even if something becomes vague after a while, that's cool as long as you can refresh yourself. This is why many professionals have reference books.

2)As for your second question, this depends. Let's say I'm working in Algebraic Statistics and come across the term "Wold's decomposition" and I go "hmm what is this?" I will usually look it up in a book or online and if I can understand the definition, I can more than likely apply it to my knowledge; however, if I feel confused still, then I'll read the book or section that I'm not familiar with. This comes with experience. The less you know about mathematics, the more you'll find that you need to read the entire book or books.

 

[homeomorphic]

Just consider also that I think that to remember something, you need to express it and discuss it (it is a sort of active process) and having nobody to discuss about maths, I completely lack this part of the learning/memorizing process.

That would help, but it is not necessary. I hardly discuss math with anyone, yet I remember books full of it. This is not because I have an exceptional memory. My memory seems to be pretty good, but not out of the ordinary. The trick is just to know how to review, to structure your knowledge so that it reinforces itself (a web of connected ideas, rather than a random collection of facts), to use visualization, mental rehearsal, and to understand things deeply. Maybe that's easier said than done, but there it is.

Still the question stands: is it such a problem in terms of long-term studying (and career) project?

You have to ask yourself what the point of working so hard to learn stuff is if you just end up forgetting it all, anyway. Some forgetting is okay. I'm not sure how important it is to remember. I guess I would say if you remember more, you can do more, although memory isn't everything. But, generally, people don't know how to use their memories, so your competition probably forgets a lot of stuff.

Actually, Atiyah said he chose math because he had a bad memory, but with math, he could just derive everything again if he forgot it. Feynman also said he just remembered the facts and then invented explanations for them if he needed to remember them. On the other end of the spectrum, Steven Hawking is known to have an incredible memory, being able to remember thousands of pages of math. Seems to me, if you have the memory, you can put it to use, but people who don't remember things that well can also get by, if they can find ways to reconstruct their knowledge by re-deriving it. There may be something to be said for having a lot of stuff at your fingertips, though. If you have to derive it again, it may not be at your fingertips.

Personally, I get a certain kind of pleasure from going over all this knowledge that I have in my mind. Also, it fits in with my general goal of trying to make as many of nature's secrets obvious to me as I can. If you forget why they are obvious, then, they aren't obvious anymore, until I recall why it's obvious again.

 

[Kolmin]

The less you know about mathematics, the more you'll find that you need to read the entire book or books.

The trick is just to know how to review, to structure your knowledge so that it reinforces itself (a web of connected ideas, rather than a random collection of facts), to use visualization, mental rehearsal, and to understand things deeply.

True and enlightening.

Thanks a lot for your feedbacks and sorry for this late reaction.


PS: IMO, this...

I get a certain kind of pleasure from going over all this knowledge that I have in my mind

...is not that far from the active process I thought about. In some way, you discuss about math with somebody (as Woody Allen would put "nobody else than yourself, somebody that in the end you respect a lot"). Simply to do that, you also need to assume you trust your mathematical knowledge and feeling.