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

[Robert1986]

Yes learning it at the same time while teaching it.

Yeah, but there is a critical difference. You see, the more math classes you take, and the mor experience you have doing math, the easier it is for you to learn other areas of math. For example, I took Algebra then Analysis. They were about equal as far as "inherent difficultness" but I had a much easier time with Analysis. Why? Algebra had increased my level of mathematical maturity a little. Then Analysis increased it a little more, and so on. I would imagine that this more or less continues as one progresses as a mathematician.


Another critical difference is that there is a VERY big difference in teaching a class and writing a book. I feel confident that I could teach, say, Calculus or ODEs or Linear Algebra or something, however, I DO NOT think that I could write a (good) book on any of those things; I simply do not have the experience.

 

[Pyrrhus]

Another critical difference is that there is a VERY big difference in teaching a class and writing a book. I feel confident that I could teach, say, Calculus or ODEs or Linear Algebra or something, however, I DO NOT think that I could write a (good) book on any of those things; I simply do not have the experience.

Yes. ZapperZ already brought this point. It is different LEARNING a topic, and being a PROFESSIONAL at a specific topic. You cannot believe that if you read a book by James Steward on Calculus, you will be doing research just as good as James Stewart? This is a naive claim. Even graduate students doing PhDs have to read, and learn way more beyond the book in order to do any significant contribution.

 

[Pyrrhus]

(And of course there is the patron saint of self-taught mathematicians, Ramanujan.)

Yes, but even Ramanujan did not publish on his own. He had to contact british mathematicians, and ended up with Hardy. It is pure naivety to think that you can isolate yourself, and read books, and do original work these days. You need to be on top with the current advances through seminars, conferences, access to journals...

 

[dijkarte]

Any non academia advocate out there? :D

I start to realize the need for university better, and I will consider going through the lengthy antiquated rituals of academia. Actually I feel now opening my own university instead of writing some papers, or a book. :)

 

[Robert1986]

Any non academia advocate out there? :D

I start to realize the need for university better, and I will consider going through the lengthy antiquated rituals of academia. Actually I feel now opening my own university instead of writing some papers, or a book. :)

It depends on what your goals are. If your goal is to write articles and do research, you have basically three options: 1) Be a professor and do what profs do 2) Have some sort of research job in industry/gubment 3) Have a job doing something completely unrelated to math research and do research as a hobby.

In 1 & 2 you MUST have Ph.D. (well, a Master's would get you by in some colleges and SOME research positions). In 3, you will spend 40+ years working 40+ hours at a job you (probably) don't like (or don't like as much) just because you wanted to avoid 4 years of grad school when you were in your twenties. I bet 40 year old dijkarte would want to kick 20 year old dijkarte's butt!


So, if you want to do math research, the most sensible thing is to stop whining, realize that professional mathematicians know more than you and get a Ph.D.!


On the other hand, if you are just interested in knowing about math, and you want to do something completely unrelated to math research, then self-teaching isn't the worst thing you could do. But, no one is going to want to use your books because you won't have the credential.

 

[TMFKAN64]

Yes, but even Ramanujan did not publish on his own. He had to contact British mathematicians, and ended up with Hardy.

Oh, I agree. And that fact that we have to reach back nearly 100 years to find an example of a self-taught mathematician making important contributions (and still needing assistance to publish!) should illustrate exactly *how* difficult the non-academic route is.

 

[victor.raum]

On the other hand, if you are just interested in knowing about math, and you want to do something completely unrelated to math research, then self-teaching isn't the worst thing you could do. But, no one is going to want to use your books because you won't have the credential.

Indeed, I like that someone finally mentioned that you don't necessarily have to pursue research as your end goal, in which case independent study is perfectly viable so long as it gains you the raw skills you need to accomplish your goals.

I do disagree, however, about the notion that people will not care about your work if you don't have a formal degree. I've personally never bothered to check the credentials on any of the textbook authors I read. For all I knew when I purchased my copies of Spivak and Feynman the authors could have been college drop outs. I read a few pages and enjoyed their writing styles, so I bought their books. I imagine most people buying books, at least outside of a college environment, will judge you based on the quality of what you produce, and not by the diplomas you hang on your wall.

 

[dkotschessaa]

dijkarte, I felt the same as you awhile ago, which is why I'm so blunt about this. I made all the same claims you did and said all the same phrases about it being about a piece of paper, a money making business, etc. etc. I realized it was just some gripe I had about academia because I felt slighted by it. The first time I tried to go to college I got screwed out of my education because of finances. So I felt many ways to justify my lack of education and how I didn't need it.

Had lots of arguments with my wife about it, who has a masters degree. I told her about all the idiots with comp sci degrees I had to put up with every day at my job (helpdesk and network admin.)

But anyway, that was all venting. I had to majorly put my ego aside, suck it up, and go back. It's been terrific. I know you already have one bachelors, but if you go back, it'll be very different. You have more experience. Your relationships with colleagues and professors will be different. If you've done a lot of studying already they'll be pretty impressed. But it'll still kick your *** at times - especially graduate studies, and you need that *** kicking to push yourself to do things even you didn't think were possible. And you're going to hate it some times and complain about it a lot. This is all necessary for growth. It's largely an ego thing.

-DaveK

 

[dkotschessaa]

Indeed, I like that someone finally mentioned that you don't necessarily have to pursue research as your end goal, in which case independent study is perfectly viable so long as it gains you the raw skills you need to accomplish your goals.

I do disagree, however, about the notion that people will not care about your work if you don't have a formal degree. I've personally never bothered to check the credentials on any of the textbook authors I read. For all I knew when I purchased my copies of Spivak and Feynman the authors could have been college drop outs. I read a few pages and enjoyed their writing styles, so I bought their books. I imagine most people buying books, at least outside of a college environment, will judge you based on the quality of what you produce, and not by the diplomas you hang on your wall.

The people that published their books knew their credentials for damn sure.

 

[Robert1986]

Indeed, I like that someone finally mentioned that you don't necessarily have to pursue research as your end goal, in which case independent study is perfectly viable so long as it gains you the raw skills you need to accomplish your goals.

I do disagree, however, about the notion that people will not care about your work if you don't have a formal degree. I've personally never bothered to check the credentials on any of the textbook authors I read. For all I knew when I purchased my copies of Spivak and Feynman the authors could have been college drop outs. I read a few pages and enjoyed their writing styles, so I bought their books. I imagine most people buying books, at least outside of a college environment, will judge you based on the quality of what you produce, and not by the diplomas you hang on your wall.

Yes, and you probably don't know where your doctors went to med school, where your accountant went to school, where the architect who designed your house studied, etc. Why? Because there are professional organsations that do this for you (and, of course, you pay for this service as a portion of the bill you pay the person.) The same thing goes for textbook authors. If Spivak hadn't been positively reviewed by a bunch of profs. then you wouldn't have it. And to get to the point where manuscripts are being sent to profs., it has to be OK'd by some publisher. And for this to happen, chances are the author is going to have some credential to let an editor know that his book might be worth reading.

 

[homeomorphic]

The people that published their books knew their credentials for damn sure.

But that is one of the wonders of the internet. So long as you are not concerned with turning a profit, you're fine (not a promising proposition, anyway, although people like Stewart got rich that way). I will have a PhD, so I will have credentials, but that is my plan, anyway. I expect no profits from my expository work, which I eventually plan to make available online.

 

[twofish-quant]

There are a few things that are being confused here.

1) Any sort of fundamental research involves a lot of self-teaching.
2) Self-teaching is a lot easier to do in a university. You get access to libraries, and if you knock on someone's office door they'll talk to you. Also universities have a ton of staff that will handle all of the random administrivia.
3) You have to have eat. That means that someone has to give you money. This is non-trivial.

 

[homeomorphic]

You have to have eat. That means that someone has to give you money. This is non-trivial.

If working at walmart and living in a shack is what I have to do in order to have sufficient time for my hobbies, so be it. I don't know that it's that hard to get enough money to get by. The question is what you're willing to live with.

 

[dijkarte]

In 3, you will spend 40+ years working 40+ hours at a job you (probably) don't like (or don't like as much) just because you wanted to avoid 4 years of grad school when you were in your twenties. I bet 40 year old dijkarte would want to kick 20 year old dijkarte's butt!

Weird! I did not know I've been working, me and my colleagues, something unrelated that we don't like because we don't have graduate degrees...
But anyway we kicked many 20-year old PhDs' butts and made sure they are not in the professional since they could not show any competence in terms of related knowledge, experience, personality and social skills. We had experience with a few and they sucked! They just sucked!, unfortunately.
So get out of your cave and start thinking life. Get some real work experience in your field. This is where you get the real knowledge and see things in action, whether it's math, physics, engineering or whatever branch of sciences and arts.

So telling me without this PhD I suck and I'm a loser is purely idiotic.

Telling me that my prospective work will not get any attention or no one will be looking at it is dumb and immature.

No wonder why I never met someone with PhD who is working outside the campus. They nerd themselves at 20s and age there.

 

[jbmiller]

I give myself credit for knowing this topic would raise a tremendous amount of controversy and objection.

 

[twofish-quant]

If working at walmart and living in a shack is what I have to do in order to have sufficient time for my hobbies, so be it.

That's the problem. Jobs that are time flexible require you to work long hours to meet basic necessities. Jobs that allow you to to meet basic necessities are not time flexible.

I don't know that it's that hard to get enough money to get by.

I do. It gets worse if you want to raise a family. Also even without a family, you need time off to attend conferences, read journals, get e-mail access, etc. etc. Once you are away from the university, then you realize that there are a lot of "trivial" things that aren't so trivial. Like a research library.

Self-study is wonderful, but the whole point of a university is to make it possible to self-study.

 

[twofish-quant]

So get out of your cave and start thinking life. Get some real work experience in your field.

Some of us do. I have a Ph.D., a job that pays scarily high amounts of money. I would *love* to do professional research on the side, but I can't. Part of the reason that I'm interested in this topic is that I have a pretty good idea, why I can't, and if anyone wants to do it, then listening to the barriers might be useful since people can think of ways around it.

My current career goals are to make a ton of money, retire, and then I'll find some university and just shack up there. I'll work as a janitor if they give me an office and library privileges.

The good news is that it's going to happen someday. I'm counting down the seconds until I hit 59 1/2 and can pull money out of my 401(k). If I'm lucky it will happen sooner.

So telling me without this PhD I suck and I'm a loser is purely idiotic.

It may be, but part of getting things done in the "real world" involves dealing with idiots. There's a huge amount of politics involved in getting stuff done in science. Most of it is good politics, but it takes a ton of time and effort, which you don't have if you are moon lighting.

Telling me that my prospective work will not get any attention or no one will be looking at it is dumb and immature.

It also might be true. Writing something that gets people's attention is *hard*. There are a ton of papers and journal articles, and writing something that cries out "read me" takes a lot of time and effort.

No wonder why I never met someone with PhD who is working outside the campus. They nerd themselves at 20s and age there.

Most physics and math Ph.D.'s work in industry.

 

[twofish-quant]

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.

Except why do you want to? If you are stubborn and reasonably committed it's not that hard to get a Ph.D. in math or physics. The reason that there aren't that many self-taught mathematicians or physicists, and that anyone that's crazy enough to want to do that, figures out that it's 100x easier to just go to graduate school.

Being poor is not an excuse, since the graduate school pays you.

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.

Success in academia (as with anything else) is mostly about connections. If you are not connected to the research community, you aren't going to get anywhere. Now it's not particularly difficult to get connected.

In order for you to write a decent paper, you basically have to show that you are doing something new, and showing that you are doing something new means showing that you understand everything everyone else has done on your topic of interest.

In an informal forum, I can say "the sky is blue." If I make a statement like this in a formal paper, then I have to have five citations and references about the color of the sky. Four people say it's blue. One person says it's red, but he was on Mars. Going into a paper and getting all of the citations right, take a ton of time and effort.

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.

I don't see the point in trying unless you've applied first. Universities need cheap indentured servants to teach math and physics, so it's not that particularly hard to get into graduate school.

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.

There's a difference between "not a lot" and none.

 

[Robert1986]

If working at walmart and living in a shack is what I have to do in order to have sufficient time for my hobbies, so be it. I don't know that it's that hard to get enough money to get by. The question is what you're willing to live with.

Well, I once had this attitude. However, I have now been living like a broke college student for 4 or so years and, frankly, I've had it. I work a full time job that I don't really like. Yes, I have had enough time to pursue my hobbies (at this point, it has just been school and, on occasion, the Braves) but I don't get paid a whole lot and I am tired when I eventually get to do my hobby. It makes FAR more sense for me to live like a broke college kid for a few more years and get a Ph.D. BTW, when I begin grad school, the school will be paying me, and so even though I'll still be broke, I won't have to work at the job I don't like.

 

[Robert1986]

Weird! I did not know I've been working, me and my colleagues, something unrelated that we don't like because we don't have graduate degrees...
But anyway we kicked many 20-year old PhDs' butts and made sure they are not in the professional since they could not show any competence in terms of related knowledge, experience, personality and social skills. We had experience with a few and they sucked! They just sucked!, unfortunately.
So get out of your cave and start thinking life. Get some real work experience in your field. This is where you get the real knowledge and see things in action, whether it's math, physics, engineering or whatever branch of sciences and arts.

You seem to not be able to read. If you would kindly re-read my post you will find that I clearly said that if you want to just learn math then self-study isn't bad.

So telling me without this PhD I suck and I'm a loser is purely idiotic.

I agree 100%! When the heck did I EVER say this?

Telling me that my prospective work will not get any attention or no one will be looking at it is dumb and immature.

More immature than lying about what others said, as you have done?

No wonder why I never met someone with PhD who is working outside the campus. They nerd themselves at 20s and age there.

Uhh. This is incredibly ignorant. Look at this: http://www.bls.gov/ooh/Math/Mathematicians.htm#tab-3

Only 16% of mathematicians work at a university! 37% work in the gov't and 26% in research. According to the same bls site:

For jobs as a mathematician with the federal government, candidates need at least a bachelor's degree with a major in mathematics

So you might get by in the gov't with a B.S. but who knows what you'll be doing. For research in the private sector, or at a college, however

In private industry, mathematicians typically need an advanced degree, either a master's degree or a doctorate. ... For a position as a professor of mathematics in a college or university, a doctorate is usually required.

So, you are very wrong. Not to mention the fact that the owner of the company that I work for has a Ph.D. Now, you have had a bad experience with some Ph.D. and think that they are all nuts and therefore you aren't going to pursue it. To me, it sounds like an excuse to be lazy.

Citation:
Bureau of Labor Statistics, U.S. Department of Labor, Occupational Outlook Handbook, 2012-13 Edition, Mathematicians,
on the Internet at http://www.bls.gov/ooh/math/mathematicians.htm (visited April 18, 2012).

 

[twofish-quant]

Lets say I'm authoring a book on DEs while learning the subject. After finishing the book, would it be reviewed and published?

The nice thing about the internet is that if you have lecture notes on differential equations, you can put it on a website and have anyone who wants to download it. "Publishing" something is trivially easy. The problem that you'll have is go convince someone to download your notes rather than notes that someone else has put online.

Now if you want to make money publishing textbooks, that's a whole different animal.

 

[dkotschessaa]

But that is one of the wonders of the internet. So long as you are not concerned with turning a profit, you're fine (not a promising proposition, anyway, although people like Stewart got rich that way). I will have a PhD, so I will have credentials, but that is my plan, anyway. I expect no profits from my expository work, which I eventually plan to make available online.

Well, it's the double edged sword of the Internet.

When I check an internet source, I am grateful that I didn't have to pay for it or run to the library to get a hard copy. I am also extra, extra careful in checking the credentials of the person who wrote the article. I would hope anybody would be. Here is a case where the "piece of paper" is very helpful. It's a paper trail back to where they did their studies. Where did they study? Who did they collaborate and study with?

Is that an assurance of the quality of the material? Heck no. The system is easily corruptible. People skate by on money and connections. You can hire somebody else to write your Phd. or Masters thesis for you. etc. (Of course you still have to defend them. I hope that's something). But it's the system we have, and I'm glad it's there, rather than nothing.

-DaveK

 

[dkotschessaa]

Weird! I did not know I've been working, me and my colleagues, something unrelated that we don't like because we don't have graduate degrees...
But anyway we kicked many 20-year old PhDs' butts and made sure they are not in the professional since they could not show any competence in terms of related knowledge, experience, personality and social skills. We had experience with a few and they sucked! They just sucked!, unfortunately.
So get out of your cave and start thinking life. Get some real work experience in your field. This is where you get the real knowledge and see things in action, whether it's math, physics, engineering or whatever branch of sciences and arts.

So telling me without this PhD I suck and I'm a loser is purely idiotic.

Telling me that my prospective work will not get any attention or no one will be looking at it is dumb and immature.

No wonder why I never met someone with PhD who is working outside the campus. They nerd themselves at 20s and age there.

You're making some pretty wild inferences and jumping to conclusions. You need to get a reign on your logic skills before you tackle math.

 

[twofish-quant]

Well, it's the double edged sword of the Internet.

The problem with the internet is that there is a ton of useful information out there, but it's all mostly not organized very well. For example, if I have a college textbook, I have this organized set of problems grouped by chapter with answers available. That stuff *is* available on the internet, but it's scattered in fifty different locations.

I am also extra, extra careful in checking the credentials of the person who wrote the article. I would hope anybody would be.

I actually have a general disrespect for credentials. When I read papers on the Los Alamos Preprint Server, I generally *don't* check the credentials of the person writing it because it's largely irrelevant to the quality of the paper. There are idiots with Ph.D. degrees, and some competent people without them. A lot of excellent papers are authored by graduate students. Conversely, I know of two *Nobel prize winners* and one former *university president* that happen to be totally loony when you get them to talk on a certain topic. One time, I was reading Ap.J. and I came across an article that was totally loony. I looked at the name, and it was a rather famous Nobel Prize winner. It was still a loony paper.

In my personal life, I've found that the world has its fair share of incompetent lawyers, doctors, and auto mechanics, so I've found that I've always had to teach myself some very basic law, medicine, and auto repair so that I have some clue as to whether the person that I'm seeing is competent or not. Once I've satisfied myself that said person is competent, then I can sort of trust them.

Universities are useful *not* because they give you a credential. The credential is actually unimportant in the grand scheme of things. Universities are important because they provide a lot of support facilities so that you can sit down and *think*. If I could figure out how to make the economics work, I'd perfer it if universities didn't issue degrees at all.

One reason going to graduate school is important is that you get to learn what standards are. I know what a decent paper in astrophysics looks like, and I also know that I don't have time to write one, and I'm not going to upload crap onto Los Alamos with my name on it. I'd be embarrassed.

Here is a case where the "piece of paper" is very helpful. It's a paper trail back to where they did their studies. Where did they study? Who did they collaborate and study with?

I've never found that to be very important. I know what good pizza taste like, and I know what a decent astronomy paper looks like. I don't care how you get it. One reason I'm skeptical that the OP will get anywhere is that without having gone into academia, you really don't know what good research and bad research looks like, and that gives you nothing to aim for.

The system is easily corruptible. People skate by on money and connections. You can hire somebody else to write your Phd. or Masters thesis for you.

In fact, you really can't. You have to defend the dissertation. It's academic cage fighting. You get locked into a room with five professors and they proceed to rip your work to shreds and do everything they can to make you look and feel like an idiot. If you can hold your ground, then you get the degree.

The system like any other system is corruptible, but it's not *easily* corruptible.

But it's the system we have, and I'm glad it's there, rather than nothing.

I really don't think that it's the system that we have.

Also, there is the Picasso effect. If you look at the early work of Pablo Picasso, it's rather conventional painting. Picasso had to master conventional painting before tearing convention to shreds. I think the same holds true with "conventional" academia. I think it's pretty essential for anyone that wants to tear apart the academic system to have a lot of experience with academia so that they know what it is they are tearing apart.

 

[dijkarte]

You're making some pretty wild inferences and jumping to conclusions. You need to get a reign on your logic skills before you tackle math.

Probably you need to work on your inter-personal and social skills first. This will help you a lot understanding mathematics better :)

To me, it sounds like an excuse to be lazy.

I would rather be lazy and have a well paid professional job with my own office than having a PhD and enslaved at Walmart.

 

[bp_psy]

I don't think that this thread would have survived as long in the GD.

 

[NewtonianAlch]

I don't think it's impossible, and generally the people who are sure of themselves and the path they know they should take don't talk about it in the open, they just on with it. Judging by your uncertainty, I think you should stick to academia, then again...

 

[NewtonianAlch]

The problem with the internet is that there is a ton of useful information out there, but it's all mostly not organized very well. For example, if I have a college textbook, I have this organized set of problems grouped by chapter with answers available. That stuff *is* available on the internet, but it's scattered in fifty different locations.



I actually have a general disrespect for credentials. When I read papers on the Los Alamos Preprint Server, I generally *don't* check the credentials of the person writing it because it's largely irrelevant to the quality of the paper. There are idiots with Ph.D. degrees, and some competent people without them. A lot of excellent papers are authored by graduate students. Conversely, I know of two *Nobel prize winners* and one former *university president* that happen to be totally loony when you get them to talk on a certain topic. One time, I was reading Ap.J. and I came across an article that was totally loony. I looked at the name, and it was a rather famous Nobel Prize winner. It was still a loony paper.

In my personal life, I've found that the world has its fair share of incompetent lawyers, doctors, and auto mechanics, so I've found that I've always had to teach myself some very basic law, medicine, and auto repair so that I have some clue as to whether the person that I'm seeing is competent or not. Once I've satisfied myself that said person is competent, then I can sort of trust them.

Universities are useful *not* because they give you a credential. The credential is actually unimportant in the grand scheme of things. Universities are important because they provide a lot of support facilities so that you can sit down and *think*. If I could figure out how to make the economics work, I'd perfer it if universities didn't issue degrees at all.

One reason going to graduate school is important is that you get to learn what standards are. I know what a decent paper in astrophysics looks like, and I also know that I don't have time to write one, and I'm not going to upload crap onto Los Alamos with my name on it. I'd be embarrassed.



I've never found that to be very important. I know what good pizza taste like, and I know what a decent astronomy paper looks like. I don't care how you get it. One reason I'm skeptical that the OP will get anywhere is that without having gone into academia, you really don't know what good research and bad research looks like, and that gives you nothing to aim for.



In fact, you really can't. You have to defend the dissertation. It's academic cage fighting. You get locked into a room with five professors and they proceed to rip your work to shreds and do everything they can to make you look and feel like an idiot. If you can hold your ground, then you get the degree.

The system like any other system is corruptible, but it's not *easily* corruptible.



I really don't think that it's the system that we have.

Also, there is the Picasso effect. If you look at the early work of Pablo Picasso, it's rather conventional painting. Picasso had to master conventional painting before tearing convention to shreds. I think the same holds true with "conventional" academia. I think it's pretty essential for anyone that wants to tear apart the academic system to have a lot of experience with academia so that they know what it is they are tearing apart.

Great post.

 

[homeomorphic]

That's the problem. Jobs that are time flexible require you to work long hours to meet basic necessities. Jobs that allow you to to meet basic necessities are not time flexible.

As a grad student, I make very little money, and the only reason I'm poorer than I'd like is because I loaned a lot of money to someone. Otherwise, I would feel very comfortable with what I'm making, though most people would consider my salary to be very low. Average would be like 30 k. I would consider myself to be rolling in dough if I made 30 k. So, for people with lower standards, it's not that hard. It's nice to have a little extra, just in case, though. So, there's also a risk factor involved.

I don't know that it's that hard to get enough money to get by.

I do. It gets worse if you want to raise a family. Also even without a family, you need time off to attend conferences, read journals, get e-mail access, etc. etc. Once you are away from the university, then you realize that there are a lot of "trivial" things that aren't so trivial. Like a research library.

Self-study is wonderful, but the whole point of a university is to make it possible to self-study.

Starting a family is optional. The library is helpful, but with internet access, plus buying the occasional book or journal, you can get a lot of what you need, but probably not all of it.

 

[dcpo]

A formal mathematical education is not a theoretical prerequisite for publishing maths papers, as anyone can send a paper to a journal, and if it meets the standards of that journal it'll probably be published, and if you publish enough papers in high quality journals you will be taken seriously by the community. In practice though formal education to Ph.D level is usually necessary, because most people, even talented people, will not develop their abilities to do and write mathematics without time in that kind of environment. There will of course be exceptions to this, but I question why anyone with a serious interest in producing new mathematics wouldn't just go to grad school, where they would be paid for their efforts. Anyone with the ability required to home educate themselves to research level should find the formal requirements almost trivial.