Self-Taught vs. Academic: The Need for Formal Education in Mathematics

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Discussion Overview

The discussion centers on the necessity of formal education in mathematics compared to self-taught approaches. Participants explore the merits and drawbacks of academia, the potential for self-directed learning, and the implications for professional development and research publication.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • Some participants question the necessity of formal education for becoming a mathematician, suggesting that self-taught individuals can achieve similar results if they possess maturity and dedication.
  • Others argue that while self-study can be effective, it may lead to isolation and challenges in original research without guidance.
  • A participant expresses frustration with academia, describing it as a business that primarily offers certificates rather than substantial knowledge, and lists various pitfalls such as high costs and inadequate resources.
  • Some participants emphasize the importance of academic resources, networking, and expert guidance available in formal education settings, arguing that these are crucial for advanced learning and research.
  • There is a suggestion that the structure of academic programs is too linear and does not accommodate varying levels of maturity and capability among students.
  • A participant notes that after earning a PhD, much of the learning becomes self-directed, highlighting the importance of learning how to learn independently.
  • Another participant acknowledges that while self-learning is possible, having a mentor or teacher can significantly enhance the learning process and reduce the likelihood of making mistakes.
  • Some participants clarify that they are discussing individuals who already have a degree and relevant experience, rather than those without prior education.

Areas of Agreement / Disagreement

Participants express a range of views, with no clear consensus on the necessity of formal education versus self-study. Some advocate for the value of academic environments, while others critique them as ineffective or outdated.

Contextual Notes

Participants highlight various assumptions about the effectiveness of self-study versus formal education, the role of mentorship, and the specific context of mathematics as a theoretical subject. Limitations in the discussion include the lack of specific examples of successful self-taught mathematicians and the varying definitions of what constitutes adequate learning.

  • #31
Yes, once you get to research level stuff, you start to realize a lot of stuff isn't very well documented. So, it helps to have experts to talk to. I haven't been very good about doing that. But that's how I learned something about Teichmuller theory. Little student seminar every week that gets to the key ideas in the most direct way.

So, in order for it to work, you have to know how to take advantage of it.
 
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  • #32
alexmahone said:
How I did it: I went to MIT's maths website and bought the undergraduate books that they use, one by one. From each exercise of each book, I solved one problem (more or less) of each type. Whenever I had trouble, I posted on maths forums and usually got my doubts clarified. In these 2 years, I've finished roughly two-thirds of the undergraduate material, and am incredibly happy with my progress.
That's some impressive motivation. Well done!

alexmahone said:
I also learned LaTeX (which I use for working out problems), which I highly doubt I would have learned at most universities.
You know, it is possible to learn stuff that is not required at a university while studying at a university. As mentioned, most people learn Latex at one point or another to prepare papers. I learned Latex for this reason (to prepare a paper for publication).

alexmahone said:
The only downside is that I never memorized any theorems or formulas, and so I have to go back to my books every time someone asks me a question.
Is this what you think they do at universities, memorize things? Are you sure you really learned math?
 
  • #33
homeomorphic said:
Yes, once you get to research level stuff, you start to realize a lot of stuff isn't very well documented.
Lol, yeah. I was a little freaked out when I started noticing this.

"There's no wikipedia page for this! AAAAHHH!" :smile:
 
  • #34
The idea is how much uni gives you professional knowledge over self-taught? Does it always work for all, majority? I've been interacting with PhDs at my work and nothing special about their intellectual abilities or even professional knowledge.

BTW "research" is now another good profit business. So what we call research at graduate level is either unrealistic, inaccurate, or not related. I see it as a literature.

That's why I don't buy the graduate studies especially research ones.
 
  • #35
The only downside is that I never memorized any theorems or formulas, and so I have to go back to my books every time someone asks me a question.

That's the only downside?
Analogously, the only downside to my algebra class is that we didn't cover groups. The only downside to my linear algebra course is that we didn't cover the Jordan form. The only downside to my analysis course is that we didn't learn epsilon-delta.

That's not exactly trivial.
 
  • #36
dijkarte said:
The idea is how much uni gives you professional knowledge over self-taught? Does it always work for all, majority? I've been interacting with PhDs at my work and nothing special about their intellectual abilities or even professional knowledge.

BTW "research" is now another good profit business. So what we call research at graduate level is either unrealistic, inaccurate, or not related. I see it as a literature.

That's why I don't buy the graduate studies especially research ones.

Then I don't understand the reason why you started this thread. You obviously know everything there is to know and you have made up your mind.

Were you really looking for a place to vent? Then you've found the wrong forum.

Zz.
 
  • #37
The only downside to my linear algebra course is lack of motivation. Not really, it was dense enough and the << profanity deleted by Mods >> made it harder for students who were all at elementary level, never seen matrices before.

The only downside to my probability and statistics class is that the average of the class was "F," which is impossible for a class to have such an average but the "teacher" was good at it.

The only down side to my course XXX is that the teacher YYY was an << profanity deleted by Mods >>.

The only downside to my wasted time at university is the huge amount of wasted money.
 
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  • #38
bcbwilla said:
That's some impressive motivation. Well done!

Thank you!

You know, it is possible to learn stuff that is not required at a university while studying at a university. As mentioned, most people learn Latex at one point or another to prepare papers. I learned Latex for this reason (to prepare a paper for publication).

I was talking about learning LaTeX as an undergraduate to work out problems. I guess it's possible to learn stuff that is not required if the course isn't too hectic. Unfortunately many university schedules are tight, leaving the student with little time for recreation, let alone learning outside stuff.

Is this what you think they do at universities, memorize things?

At universities, students are forced to memorize theorems, formulas and proofs before their exams. Although this is quite boring, I must admit it has some benefits.

Are you sure you really learned math?

Just because I said that I need to refer to my textbooks every now and then, doesn't mean I didn't learn maths. The fact that I'm able to move forward to the next section after doing the exercise of the previous section, and to the next book after finishing the previous book, is ample evidence that I did. Do you agree? If not, what's your definition of "learning maths"?
 
  • #39
If you don't see math research as useful, you should watch this (and part 2, etc.),



In some ways, he leaves me unsatisfied. I think we need to do more to connect with the applications and other fields. Math these days is getting pretty obscure for outsiders to be able to come in and use it.
 
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  • #40
Were you really looking for a place to vent? Then you've found the wrong forum.

Well it's the right place to vent, this is the spirit of connecting to people, be able to communicate and exchange experiences and details :)
 
  • #41
Number Nine said:
That's the only downside?
Analogously, the only downside to my algebra class is that we didn't cover groups. The only downside to my linear algebra course is that we didn't cover the Jordan form. The only downside to my analysis course is that we didn't learn epsilon-delta.

That's not exactly trivial.

What's your point?
 
  • #42
alexmahone said:
Just because I said that I need to refer to my textbooks every now and then, doesn't mean I didn't learn maths. The fact that I'm able to move forward to the next section after doing the exercise of the previous section, and to the next book after finishing the previous book, is ample evidence that I did. Do you agree? If not, what's your definition of learning maths?
Sorry, what I meant to imply was that if you "really learned math", you'd see that it's not about memorizing formulas. I certainly don't remember all of the formulas in my textbooks. Again, sorry, I didn't mean for that comment to sound insulting. :smile:
 
  • #43
bcbwilla said:
Sorry, what I meant to imply was that if you "really learned math", you'd see that it's not about memorizing formulas. I certainly don't remember all of the formulas in my textbooks. Again, sorry, I didn't mean for that comment to sound insulting. :smile:

No worries. :smile:

But as I said, I think memorizing a few theorems and formulas is essential.
 
  • #44
dijkarte said:
The only downside to my linear algebra course is lack of motivation. Not really, it was dense enough and the << profanity deleted by Mods >> made it harder for students who were all at elementary level, never seen matrices before.

The only downside to my probability and statistics class is that the average of the class was "F," which is impossible for a class to have such an average but the "teacher" was good at it.

The only down side to my course XXX is that the teacher YYY was an << profanity deleted by Mods >>

The only downside to my wasted time at university is the huge amount of wasted money.

:smile:

Sorry to hear.
 
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  • #45
The best teacher is yourself if you can do it and there are many who can. Remember, there are group learners and independent learners. Group learners always feel more comfortable and confident in learning within a group, let's say social learners. Independent learners on the other had are those who don't need to be in a group to effectively learn and discover.
 
  • #46
dijkarte,

Could you write a paper now in mathematics, and get it publish in a peer-reviewed journal?

if you can, please do so, and come back and let us know how we waste our time studying PhDs.
 
  • #47
Could you write a paper now, and get it publish in a peer-reviewed journal?

if you can, please do so, and come back and let us know how we waste our time studying PhDs.

Sure I will. But just to let you know I'm not underestimating your PhDs or anything but you seem to think that if I don't have a PhD then I cannot do it. People are different.
 
  • #48
People without PhDs publish papers all of the time. They are called "graduate students".

More seriously, the odds are heavily stacked against you. It isn't completely impossible for a self-taught mathematician to publish good work. (And of course there is the patron saint of self-taught mathematicians, Ramanujan.)

But as is often the case, there is an easy way to do things, and a hard way. You are definitely choosing the hard way.
 
  • #49
dijkarte said:
Is Academia really necessary to become a mathematician? What about self-taught mathematicians? If someone has the maturity to learn by themselves, why need to go to university and spend time and money to sit in an overcrowded class with who knows what kind of lecture you get...?

The only advantage I see is that someone cannot teach at the university unless they have a related graduate degree. But what about publishing math papers? Do we need to have this graduate academia license to publish something or author a book?

I believe self taught is the best process.But it should be a major problem when you will try to publish your papers.
 
  • #50
After reading through this thread, I just want to say that it is definitely possible to do research mathematics without a graduate or undergraduate degree. However, so is getting hit by lightning, or being mauled by a cow. The odds are not in your favor, and you will be at a disadvantage compared to someone who did attend graduate school. For example, you will find it harder to get timely feedback from professors, keeping on the right track, or developing a research topic. That's not to say it isn't impossible to do what I suggested, but like I said, anything is possible. Also, while you can learn everything in a graduate program without actually going to graduate school(very hard but not impossible), there is a bit of beaurocracy going on. No one likes to admit the elephant in the room, but success in academia is not completely based on merit but on connections. Without these connections from attending a graduate program or having people back up your research, it will be hard for your paper to get published.

Ultimately, it seems like your main point is that you think it is possible to obtain a graduate education and write a research paper without going to grad school. What you propose sounds good on paper, but we all know that what sounds good on paper may not really in fact be a good solution due to unforeseen technicalities. Also, please keep in mind that in this day and age, there are not a lot of good, quality, research papers published by someone without a graduate education. I agree with you on many points that material in graduate school can be learned if self taught, but empirical evidence suggests that the failure rate for going down such a path is high. Let the buyer beware.
 
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  • #51
Lets say I'm authoring a book on DEs while learning the subject. After finishing the book, would it be reviewed and published?
 
  • #52
dijkarte said:
Lets say I'm authoring a book on DEs while learning the subject. After finishing the book, would it be reviewed and published?

Err... Knowing this, I probably would prefer another book on DEs. You see, I prefer to read a book from someone's who's actually mastered a subject *before* writing a book about it. But maybe that's just me.
 
  • #53
dijkarte said:
Lets say I'm authoring a book on DEs while learning the subject. After finishing the book, would it be reviewed and published?

No, and no, but you can self-publish it and get some of your friends to review it (or even review it yourself under another name), if that keeps your ego happy.

Employers, publishers, etc stay in business by betting with the odds. Almost everybody who claims they are a self-taught expert in some field but has no formal qualifications turns out to be deluded, a crackpot, or both. The guaranteed costs of wasting time with 9,999 of those completely outweighs the possible benefits of discovering the next Srinivasa Ramanujan.
 
  • #54
I completely agree. So this is one of my points, when a professor is allowed to teach a subject to graduate students while he's learning it. So He goes over night read, and next day teaches, so he is ahead of his student by literally reading a few pages of a book.
This happened and I had this experience. So this is okay by academia. Fair enough.
So in conclusion, you cannot do it without academia and it's risky, "crackpot", and "delusive", better not to, and etc. etc. etc. because it's...outside academia?
Define water. It's liquid substance called water? :)
 
  • #55
dijkarte said:
I completely agree. So this is one of my points, when a professor is allowed to teach a subject to graduate students while he's learning it. So He goes over night read, and next day teaches, so he is ahead of his student by literally reading a few pages of a book.
There is a difference between reviewing a subject before teaching it, and learning it for the first time right before teaching it.
 
  • #56
Yes learning it at the same time while teaching it.
 
  • #57
I don't think you could learn Chinese (say) from a book alone. You would need to hear it and speak it in a context. Same with maths. It has a language, and universities are places where you can learn that language with other people doing the same thing. Of course there are difficulties to be overcome, but they are small compared with working on your own.
 
  • #58
Well as I mentioned I'm not working from scratch :), I already have the qualification, age and maturity.
 
  • #59
Hobin said:
Err... Knowing this, I probably would prefer another book on DEs. You see, I prefer to read a book from someone's who's actually mastered a subject *before* writing a book about it. But maybe that's just me.

Where's the fun in that?!? I mean, it is already hard enough to find good math books written by actual mathematicians, I'm sure someone who isn't a mathematician will do a wonderful job. No, I'd much rather read a book by someone who doesn't know what he's talking about. :)
 
  • #60
I think my point was in half taken in consideration by most. However, let me complete it dijkarte once you publish such a paper, and you have gone through all the hoops, and have moved all the obstacles that you need to publish it, you will understand why is so impractical, and in cases naive to plan to do what you want to do. You could do a PhD, and have the PhD paid by someone else (TA, RA, etc.), and end up in a better position than spending your own time reading books, AND JOURNAL PAPERS (the cutting edge will be in papers, not in books).
 

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