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

[dijkarte]

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?

 

[OldEngr63]

Self-taught in any field is just as good as academic study, provided it produces the necessary result. That last part is the hitch -- the result. Can you be assured that, working alone and without guidance, you are going to be doing anything worthwhile? Certainly as far as learning the existing material, if you are really bright this can be done on your own although it may be difficult and you may go down some long blind alleys. The really bad part is when you get into the original research that isolation can be a major danger.

 

[dijkarte]

Well I tried Academia before at both levels, and before I wasted more time and money I'd realized that it's a business that really gives you nothing but a certificate. The ones who did well and succeeded are because they worked hard and out of their classes.
But can you give me an instance where I may be lost in teaching myself? My argument again is that if someone get the maturity and knowledge about the subject, what does the university have to offer more other than an official license? I reconfirm here I have met many many at the university or at work who got PhD.'s from well recognized unis but they don't know much outside their academic research paper.

I can list what I see as the pitfalls of the academia, and here is a few:

1 - Too expensive
2 - Too unnecessarily lengthy
3 - Inadequate resources
4 - Too many in a class that we don't get the focus of the lecturer
5 - Stressful exams and grading policies, which vary from one prof to another
7 - High percent of unrealistic research topics
8 - Staff attitude

Time to do some rework on academia to get it out of the 17th century.

Where's recognized distance learning?

Why don't we have classes and courses designated for more advanced students, with varying maturities and capabilities?

Why is it so linear? BS --> Master --> PhD or DIE

Things we need to consider.

 

[espen180]

1 - Nothing to do about this. Move to another country where it is cheaper?
2 - There is a LOT of material to cover.
3 - Could you be more spesific?
4 - I disagree. This sounds like a personal problem. Could you clarify what you mean?
5 - Exams tend to be stressful. Nothing to do about that. Still, it's the only way to effectively and economically evaluate a student. What do you have against the grading policies?
7 - Please be spesific.
8 - This must vary from university to university. I have never seen any problem here.

 

[ZapperZ]

Well I tried Academia before at both levels, and before I wasted more time and money I'd realized that it's a business that really gives you nothing but a certificate. The ones who did well and succeeded are because they worked hard and out of their classes.
But can you give me an instance where I may be lost in teaching myself? My argument again is that if someone get the maturity and knowledge about the subject, what does the university have to offer more other than an official license? I reconfirm here I have met many many at the university or at work who got PhD.'s from well recognized unis but they don't know much outside their academic research paper.

I can list what I see as the pitfalls of the academia, and here is a few:

1 - Too expensive
2 - Too unnecessarily lengthy
3 - Inadequate resources
4 - Too many in a class that we don't get the focus of the lecturer
5 - Stressful exams and grading policies, which vary from one prof to another
7 - High percent of unrealistic research topics
8 - Staff attitude

Time to do some rework on academia to get it out of the 17th century.

Where's recognized distance learning?

Why don't we have classes and courses designated for more advanced students, with varying maturities and capabilities?

Why is it so linear? BS --> Master --> PhD or DIE

Things we need to consider.

Since this is now turned into a more general issue regarding academia in general rather than just specific to mathematics, let me ask you to look at this: How many papers that have been produced recently (say, the last 50 years) had only ONE author?

I'm not saying such things don't exist, but what are the odds that one can work in isolation, especially nowadays, when the field is so vast, and so interdependent on many other things?

One is in at an academic setting NOT just to sit in a class and absorb! This is a fallacy, and this is maybe only true at the undergraduate level. But go beyond that, and the RESOURCES available to you is incomparable. You have access to people who are experts in different fields, you are introduced to colloquiums and seminars on a variety of topics, you have contacts with others that you simply do not have outside of such a setting.

Distance learning? What's that when you are doing experimental work and have to be on site to actually DO an experiment?

If you want to stick it to the mathematics program, then be specific. Otherwise, don't pass wholesale judgement on programs that you don't know anything about.

Zz.

 

[dkotschessaa]

You have the same bitterness about academia that I was carrying around a few years ago. I decided to go back after trying to do something on my own. If you're serious, you'll get over it, and get back to school. If not, your chances of success are extremely low. All that stress and hard work and dealing with professors is necessary to push yourself to learn new things. You won't do it alone. Sorry.

-DaveK

 

[OldEngr63]

Before you earn a PhD, you can rely to some extent on other people to teach you things. After you finish a PhD, almost everything you learn past that point will be self-taught. That is what a PhD is all about in the most fundamental sense; it is learning how to learn about just about anything you need to learn. Once you pass that point, you will rarely be able to find a teacher for anything you want to learn, so it all becomes self-taught learning.

That does not mean that there is not a benefit to being in a community of people doing similar work for most purposes; there clearly is, because the exchange of ideas generates more ideas. But it does mean that you can go off and work entirely alone, if need be, and still expect to do the job. But it may be slower going.

 

[DeadOriginal]

Usually a person that has someone teaching and guiding them is more likely to excel in what they are trying to learn because the path they need to take has already been created.

That isn't to say that it is impossible to learn by yourself but it is much more likely that you'll be using a lot of time making mistakes you shouldn't.

 

[dijkarte]

I'm misunderstood here. Sorry.

First let me be more specific and pick math as a theoretical subject. So yeah I don't mean anything like Medicine or Law, or even physics that need some experiments in a lab.

I agree that a basic undergraduate degree is something necessary to get started. I'm talking here about someone who already has a degree and mature enough to educate themselves. They know well about their subject of interest. They probably have a lot of related work experience as well. So I'm not talking about an average teenager who has no education beyond his high school. :D
Add to this the technology we have now to dig into all kind of resources and connect with many experts from all over the world, internet! This forum is an excellent example.
We can learn and exchange ideas here while we are at our most comfortable place, home.

Would not that make a difference?

Again what mistakes you talking about? So in Academia students don't? Oh they all got A+ I see I see. :D

 

[dkotschessaa]

Still, you would face tremendous odds, and will have to work aimlessly and un-paced on your own for an indeterminant amount of time, until you finally realize that you should have just studied somewhere in the first place.

If you already have a degree you can more easily (well, not "easily") get another undergraduate degree in 2 years or less (no liberal arts courses would probably be required.) You might even be able to spend some time studying for a graduate degree, but that would be difficult, and you would have a hard time getting recommendations. I just think that to succeed in this field you have to spend some time working very hard and very focused, and that without people pushing you beyond what you think you're capable of it is just so highly improbable that you will get anywhere.

Connections are a benefit, your support network is a benefit, the resources of a university are a benefit, your professors and colleagues are a benefit, the scrap of paper you get at the end is a benefit (perhaps of the least importance, yet still important.)

Most people that try on their own simply plod along. If you were so exceptionally talented that you didn't need school, you probably wouldn't be asking in the first place. Such people are so extremely rare.

-DaveK

 

[dijkarte]

If you were so exceptionally talented that you didn't need school, you probably wouldn't be asking in the first place. Such people are so extremely rare.

What about this:
People who are not talented enough that they need to go to school for higher education, and those who don't trust their mental abilities, are extremely many.

I see it as a self confidence issue. If it's not through Academia then we cannot do it.

Again, you cannot study Medicine because it's Medicine. And I'm not talking about undergraduate studies either, only graduate level.

 

[genericusrnme]

Imo you certainly do NOT need a university to learn in this modern age, universities are now boxes you put money and time into and get a network and a certificate out of.

Does having a certificate mean you KNOW more about something that someone else? No, it does not.

The only real aid to education a university can give you over reading from books is
1. different insights from lecturers (although reading multiple books on one subject is kind of equivelant to this)
2. structure, a university will tell you what you can learn and when to learn it, this may or may not help with your motivation and continued efforts
3. pace, a university will give you a set pace to complete work in, good if you're used to a slow pace

It really depends on what kind of person you are. If you are extremely passionate about your subject and spend most of your time learning about it then university isn't really going to do much for you (other than a certificate that tells people you have done this and a network of contacts). If you're not so much in this category then university will help you keep on track.

Either way you're probably still better off getting yourself a degree if only for the certificate that says you have a degree

 

[Number Nine]

What about this:
People who are not talented enough that they need to go to school for higher education, and those who don't trust their mental abilities, are extremely many.

I see it as a self confidence issue. If it's not through Academia then we cannot do it.

Again, you cannot study Medicine because it's Medicine. And I'm not talking about undergraduate studies either, only graduate level.

Honestly, if the only things you got out of University were the lectures, then you weren't a very good student. The only thing that my formal courses did for me was to give enough background knowledge to start learning properly in the lab. Everything worthwhile that I've learned at University has been the result of either consulting with my professors and working with them in the lab, or through my own independent studies that emerged and were largely directed by issues that arose in said labs. I don't know how you're going to teach yourself anything worthwhile if you don't have the means to determine what is and isn't worthwhile, which almost requires that you be in a research environment.

 

[dijkarte]

Thank you genericusrnme. This is exactly what I'm trying to say.

 

[dkotschessaa]

What about this:
People who are not talented enough that they need to go to school for higher education, and those who don't trust their mental abilities, are extremely many.

If phrasing it that way makes you feel justified, then ok. Sounds like you've already made up your mind and are looking for approval, rather than a realistic perspective.

Good luck.

-Dave K

 

[Vladmir]

I believe once an individual has come to an intersection of their life, where they know what their focus is and how to communicate their ideas properly...the purpose of college starts to collapse. For example, many colleges today try to give direction within a specific field. However, when individual(x) has the ability to surpass a given field(y)...a new coordinate must be set. ( A new standard of direction )

[Use (z) to link (x)&(y) to determine best fit/ratio. Once a 1:1 ratio of (x&y) has been determine along the (z) axis... (You can keep going if you choose)]

College only allows me to go so far. When I choose to push myself further, only I can determine my direction that will best fit the world today. It's a messy ordeal. People who can move at a faster rate are still bound by the rate of others.

 

[Hobin]

College only allows me to go so far.

Which is exactly why most people should go to college if they can. Because, let's be real here, you can only read so many textbooks before dying of boredom, and it's a real pain to stay motivated when learning on your own (of course, I'm sure there are people who don't have this problem, but those are probably few and far between). It's of course much more interesting to work on cool projects 'n stuff, but college is perfect for getting a good grounding for almost everyone.

 

[dijkarte]

That's what I think of college at undergraduate level, necessary direction and guidance about a specific field. Mainly it helps with establishing the necessary tools to get started. In the end you will be working after Bachelor degree, unless someone prefer to hide and stick to Academia and stay away of the real world. Then this is a different matter. When at work you will be facing variety of challenging problems sometimes open research unsolved problems and you need to do the research yourself. It's very uncommon that someone faces a problem that they did not study at college that they want to go back to pursue a graduate degree to get the problem solved and then go back to work? Unrealistic. This is what the basic degree does for you, prepares you and helps your maturity to learn new stuff by yourself. Otherwise what's the purpose of first undergraduate degree then?

 

[dkotschessaa]

Why did you pose the question if you are sufficiently satisfied you already know the answer?

 

[jbmiller]

If phrasing it that way makes you feel justified, then ok. Sounds like you've already made up your mind and are looking for approval, rather than a realistic perspective.

Good luck.

-Dave K

Agreed. One can't gain insight on the complete other side without first opening his eyes. You sound like one who can't learn from a teacher simply because you think you're automatically doing it the right way, and that your way is the only way.

 

[homeomorphic]

It's interesting to think what would have happened if I had taught myself everything. There would have been some big advantages to being self-taught, but overall, I think I have profited a lot from working towards a PhD. I still ended up teaching myself most of the stuff, anyway.

One thing was that it forced me to cover a certain amount of material in a certain time period. This was good because I used to get stuck on things too much (this was also partly bad because the pace was too fast--I could have retained a lot more if there wasn't so much information being thrown at me, and if I had time to stop and catch my breath). Better to move on, sometimes. It was also good to have to struggle with a lot of challenges that I might not have risen to by myself.

 

[victor.raum]

Here is my two cents:

I imagine that the first step toward becoming a research mathematician is to learn everything there is to know in a given field. You study and study up until the point where you're caught up with your peers, and then you can join them in collectively staring into the gaping darkness of what is yet to be discovered.

I have no knowledge of how research mathematics works, so I cannot pass judgement on whether a university is or is not a fostering environment for PhD's who are doing their work of illuminating the darkness.

However, I do have experience as an undergrad, as well as experience as an independent learner. From this I can say that a university is an awful place to get a pre-research education. In my experience universities tend teach in ways that makes their own functioning smoother, and they either don't care or are just blind to how their practices harm their students.

The the university structure I say:

I won't waste my brain power and cortisol on your tests just to prove to you that I know what I know. I'll decide when I feel comfortable with my current material, and I'll decide when I'm ready to move forward. If I ever move forward into new material too soon then it will be immediately obvious to me, and in response I can just take a step back and pick up my old material again. Learning often goes like that, you go forward into new material and then back again to cover patches in old material; in this area your ridged one-way course structure completely fails to accommodate.

I won't have you telling me which books to read. I like my copy of "Vector Calculus" by Marsden & Tromba better than any of the trash calculus books you've ever thrown at me. I prefer "University Physics" over Halliday & Resnick, and I can't find any reason why I should submit to your book selections. Researching and selecting a good textbook to purchase for a new subject is a ritual I much enjoy, and one that you completely destroy.

If my linear algebra textbook has 20 questions which all repeat the same matrix manipulations over and over, then I'll be the one deciding how many I do before I feel comfortable and stop. I won't have you telling me I need to do all 20, thanks very much.

If I'm going to pay you $40,000 per year for an education then I will be deciding what I read, I will be assigning my own daily work, and you will be at my every beck and call whenever I have questions (which will be very often). But you want me to pay you $40,000 for the "privilege" of being told what to do with every waking hour of my study time? Ha! Are you kidding me?

If I get into a really good grove with a difficult math problem then I'm not going to stop working on it because you want me to hand in a lab report by tomorrow. My immediate educational desires are more important than your artificial deadlines, so go away and let me study. I do very much enjoy chemistry, but it can wait; and likewise when I find myself especially motivated for the chemistry, the math can wait.

I will never set an alarm and wake up under-rested for any of your silly mandatory activities. Alarms and sleeplessness are for the working world, where real life external pressures sometimes necessitate just being awake regardless of being tired. Sleep and learning, however, must always go hand in hand. Any system that asks me to loose sleep in the name of education is broken, and every one of your programs which claims to be doable without sleep loss is either lying or is of dubious rigor.

Back to the books,
-Victor

 

[ZapperZ]

People are forgetting that there is a difference between learning something, versus being a "professional" on that something. It involves knowing the difference between what is interesting versus what is important! Those two are not always mutually inclusive!

Knowing what is important requires an extensive knowledge of not just the field of study, but also the state of knowledge and direction of the field. It is important to know what is considered to be important IF you want to be a professional in that field. It means that you can sell what you wish to do to the relevant bodies and institutions so that you can be PAID to do it! And to be able to know what is important, you have to interact with other people in your field, be able to establish your own reputation among your peers, and continue to listen to all the development in your field and related fields. This is not something that can be done in isolation!

I can easily point out the need for that simply by highlighting the existence of Princeton's Institute for Advanced Studies and Canada's Perimeter Institute. If these theorists can easily learn by themselves without needing others, then why establish such a center where they can talk and exchange ideas with each other? There is an expected responsibility in being called a physicist, an engineer, a mathematician, etc., and it goes BEYOND just the knowledge of a particular subject!

There is an abundant of short-sightedness in this thread that does not consider the professional aspect of a career. If you want to study and understand mathematics, for example, you can do that without going to school. However, if you want to be a mathematician, then YES. The short-sightedness here is in not understanding the difference between the two.

Zz.

 

[dkotschessaa]

Excellent, ZapperZ.

 

[OldEngr63]

@ victor raum: ha, ha, ha. That was really great fun. How long can you keep it up?

 

[dijkarte]

People are forgetting that there is a difference between learning something, versus being a "professional" on that something. It involves knowing the difference between what is interesting versus what is important! Those two are not always mutually inclusive!

Exactly that's where self learning is for, professional knowledge and mastery of the subject.

And to be able to know what is important, you have to interact with other people in your field, be able to establish your own reputation among your peers, and continue to listen to all the development in your field and related fields. This is not something that can be done in isolation!

I don't know who ever came up with this idea that self-learners are isolated from experts in the filed? We study from multiple books and references that are authored by the many experts! There's something called INTERNET! We do research and we have access to countless resources with the most recent advances in the subject. We don't live in the 17th century anymore :)

If these theorists can easily learn by themselves without needing others, then why establish such a center where they can talk and exchange ideas with each other?

Profit!

There is an expected responsibility in being called a physicist, an engineer, a mathematician, etc., and it goes BEYOND just the knowledge of a particular subject!

So is it all about licensing a title? What about buying a certificate that makes me a Doctor? :)

However, if you want to be a mathematician, then YES. The short-sightedness here is in not understanding the difference between the two.

So let's talk mathematics, is this what you trying to say?

University degree in mathematics <==> Mathematician

 

[dijkarte]

Excellent victor.raum! Very well written!

 

[alexmahone]

Here is my two cents:

I imagine that the first step toward becoming a research mathematician is to learn everything there is to know in a given field. You study and study up until the point where you're caught up with your peers, and then you can join them in collectively staring into the gaping darkness of what is yet to be discovered.

I have no knowledge of how research mathematics works, so I cannot pass judgement on whether a university is or is not a fostering environment for PhD's who are doing their work of illuminating the darkness.

However, I do have experience as an undergrad, as well as experience as an independent learner. From this I can say that a university is an awful place to get a pre-research education. In my experience universities tend teach in ways that makes their own functioning smoother, and they either don't care or are just blind to how their practices harm their students.

The the university structure I say:

I won't waste my brain power and cortisol on your tests just to prove to you that I know what I know. I'll decide when I feel comfortable with my current material, and I'll decide when I'm ready to move forward. If I ever move forward into new material too soon then it will be immediately obvious to me, and in response I can just take a step back and pick up my old material again. Learning often goes like that, you go forward into new material and then back again to cover patches in old material; in this area your ridged one-way course structure completely fails to accommodate.

I won't have you telling me which books to read. I like my copy of "Vector Calculus" by Marsden & Tromba better than any of the trash calculus books you've ever thrown at me. I prefer "University Physics" over Halliday & Resnick, and I can't find any reason why I should submit to your book selections. Researching and selecting a good textbook to purchase for a new subject is a ritual I much enjoy, and one that you completely destroy.

If my linear algebra textbook has 20 questions which all repeat the same matrix manipulations over and over, then I'll be the one deciding how many I do before I feel comfortable and stop. I won't have you telling me I need to do all 20, thanks very much.

If I'm going to pay you $40,000 per year for an education then I will be deciding what I read, I will be assigning my own daily work, and you will be at my every beck and call whenever I have questions (which will be very often). But you want me to pay you $40,000 for the "privilege" of being told what to do with every waking hour of my study time? Ha! Are you kidding me?

If I get into a really good grove with a difficult math problem then I'm not going to stop working on it because you want me to hand in a lab report by tomorrow. My immediate educational desires are more important than your artificial deadlines, so go away and let me study. I do very much enjoy chemistry, but it can wait; and likewise when I find myself especially motivated for the chemistry, the math can wait.

I will never set an alarm and wake up under-rested for any of your silly mandatory activities. Alarms and sleeplessness are for the working world, where real life external pressures sometimes necessitate just being awake regardless of being tired. Sleep and learning, however, must always go hand in hand. Any system that asks me to loose sleep in the name of education is broken, and every one of your programs which claims to be doable without sleep loss is either lying or is of dubious rigor.

Back to the books,
-Victor

I have been self-studying mathematics for the last 2 years and I wholeheartedly agree with you on all counts.

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. The best part is that during this time I've never got "fed up of mathematics", which I imagine is a common occurrence in universities. I also learned LaTeX (which I use for working out problems), which I highly doubt I would have learned at most universities. 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.

I will be going to university this fall to get a formal undergraduate degree.

 

[Number Nine]

I don't know who ever came up with this idea that self-learners are isolated from experts in the filed? We study from multiple books and references that are authored by the many experts!

This really won't help you if you want to be on the leading edge of a field. As an example:

My field of interest is computational cognitive neuroscience (the use of mathematics and computer simulation to understand the exact mechanism by which the brain generates cognitive processes). A major research topic in this field concerns the interactions between structures called the dorsolateral prefrontal cortex (DLPFC) and the basal ganglia (along with associated structures that release the neurotransmitter dopamine), and how these interactions influence the persistent activity of clusters of neurons in the DLPFC that are thought to underlie working memory (the ability to hold and manipulate information in memory for short periods; think of doing long division in your head).

The difficulties in self-study this material are two fold: The first is that the techniques (particularly the mathematical ones) in use are "home grown" in a sense; they've been developed by the researchers in parallel with the neuroscience itself, and there are really no good resources that will tell you what you need to know and how to use it (I imagine that this is true of many fields). There was a book published fairly recently that details the dynamical systems approach to neural modelling, a familiarity with which is necessary to so much as get your foot in the door, but that's pretty much it. You can read the literature, of course, but almost all of it assumes that you already understand the motivation behind the techniques.

Second, and I alluded to this in the above paragraph, the literature is woefully out of date. As far as the techniques themselves go, there are very few, if any, textbooks. Beyond that, someone who knew the techniques but was only reading the literature would still be completely behind. Going back to my working memory example above: The research in this field is currently undergoing a bit of a paradigm shift of sorts, even since some research in deepest depths of neurobiology discovered that a particular type of a neuron in a structure called the ventral tegmental area actually released glutamate as well as dopamine, combined with some recent advances in our knowledge of the time course of the effect of dopamine on the DLPFC. You will not read about this in any textbook (nor will any textbooks discuss it for many years), but if you approached this field hoping to study the influence of dopamine on working memory processes (a research area which makes up, like, 30% of cognitive neuroscience), reading the literature would set you on a path that is now considered largely irrelevant (and incorrect) by most researchers.

The utility of University is not that it offers classes; it's that it offers you the opportunity to interact with people who are on the cutting edge of your chosen field.

I also learned LaTeX (which I use for working out problems), which I highly doubt I would have learned at most universities.

Most people study LaTeX as graduates (when they start preparing papers or manuscripts). I can't imagine ever using it to work out problems; that just seems grossly inefficient.

 

[ZapperZ]

Exactly that's where self learning is for, professional knowledge and mastery of the subject.




I don't know who ever came up with this idea that self-learners are isolated from experts in the filed? We study from multiple books and references that are authored by the many experts! There's something called INTERNET! We do research and we have access to countless resources with the most recent advances in the subject. We don't live in the 17th century anymore :)



Profit!



So is it all about licensing a title? What about buying a certificate that makes me a Doctor? :)



So let's talk mathematics, is this what you trying to say?

University degree in mathematics <==> Mathematician

This has a lot of misconception.

First of all, the two institutions that I mentioned are NON PROFIT INSTITUTIONS .

Secondly, you are missing A LOT by not interacting with people . To think that you can get all the info simply by reading stuff is a fallacy. A lot of info are never published. Authors often leave out details in papers. I find that out over and over again by attending talks and seminars.

Thirdly, one of my PRL paper was born out of the result of my conversation with people during a coffee break at a conference! These intellectual interactions are something that are severely under valued and not appreciated by people who have not been in these professions. It is why people go to conferences, why we attend talks, and why there are institutions set up to get people to interact!

Again, there is a difference between someone with a knowledge of something versus someone who is a professional in that subject and expect to MAKE A LIVING out of it. This is true in most professions.

Zz.

 

[homeomorphic]

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.

 

[bcbwilla]

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!

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).

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?

 

[bcbwilla]

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!"

 

[dijkarte]

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.

 

[Number Nine]

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.