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I'm one of those who realized near the end of their Bachelor's in Business Mgmt that they really really love math. I'm interested in research in pure, theoretical math. I am confident in my ability to learn advanced math on my own. I admit this is based on a small sample of experiences I've had so far with self-studying. After a year of working, I decide to do it full time.

Since I'm behind my peers, I was thinking to just learn the fundamentals (Logic, Real Analysis, Abstract Algebra and Geometry) on my own, and then apply for a Master's. The reason I want to do these on my own is that I find I understand things better at my own pace taking the time to solidify the fundamentals. I took a year of math courses on an exchange program at London School of Econ and it felt like learning disconnected facts. Something like saying object x is round, black, with 4 holes etc. vs showing you a picture a black shirt button (which I find more efficient mentally). Concepts were introduced which were too new but no time was given to "form a picture" of the objects. As a result, it felt like very "syntactic". If I understand the basic concepts used over and over in math, rather than things built out of those facts like calculus, linear algebra, applied math and other "less abstract" branches, I feel I'd do well in my Master's course. I love taking the time explore mathematical concepts on my own, seeing how things fit. I find quite often I make my own concepts (simple ones though) where the picture feels incomplete, which is why I want to do research.

I was wondering how long I should continue this approach (i.e. when have I learned enough fundamentals)? More objectively, what should aim to master on my own, before applying to a master's program? (unless you disagree this is the right approach)

PS: I'm 22, and I have financial support at the moment while I get this done. I've started with Logic and Set Theory and becoming good at proofs. I find calculus and linear algebra intro books skip over too many of the more abstract underlying concepts, hence the bottom up approach.

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  • Related – Antonio Vargas Apr 9 '15 at 22:04
  • It's an unorthodox approach but you might be a maths "natural" who needs a lot of time to get concepts into your own mental toolkit but, once there, can reason very powerfully about them. If so then set theory seems a good area to start with and I'd suggest continuing to topology. You might want to check the career of R.H.Bing. – TheMathemagician Apr 10 '15 at 1:29
  • "Becoming good at proofs" sounds like a warning flag. It is similar to "I want to become a lifeguard, I am now getting good at swimming". Proofs is 95% of mathematics after you've done the calculus and linear algebra aimed for engineers etc. If proof by induction still scare you, you need to do much more. – Per Alexandersson Sep 29 '15 at 0:50
  • I hope other people will pardon my critique, but I would object to the notion that "set theory" is foundational, for one thing. True, it is interesting, and may be helpful, but the (by now obsolete) idea that set theory is a prerequisite for "everything" is passe'. Nor are "proofs", per se, the essence of mathematics. For outsiders to "professional mathematics", the internet provides much disinformation about mathematics... unfortunately, but sociologically-understandably. – paul garrett Sep 29 '15 at 1:00
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I disagree that this is the right approach. Many grad schools in the US would require you to have a Bachelor's degree in some subject and to make up the courses you missed by not having a Bachelor's degree in math. So your best option is to get started actually completing those courses. I think that unless you have some publications in math with someone who can vouch for your skills even though you don't have the coursework, you're not going to get into a good Master's degree program without the necessary coursework. The further your history is from the required classes, the more work you're going to need to do. Some programs might admit you with the requirement that you take a few semesters of undergrad classes to catch up, but if you need more than a few catch-up courses, you're really looking at a second Bachelor's degree or working on the side through a non-degree-seeking program at your local university.

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  • Thanks for injecting a bit of realism. Yes I agree a totally unproven person could never get accepted into a master's program. I was planning on becoming good first with the basics (which I honestly feel I'd learn better on my own). Perhaps there are these options after that: 1. Join a bachelor's and graduate super early 2. Talk to professors to get some ideas on research problems i could work on. If I'm really as good as I think, then they should be able to vouch for me/my research. Then try for a master's directly. – user32892 Apr 10 '15 at 4:05
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    I think you're going to find that professors don't just give out their research problems for people to work on. Also, may not be able to start a BS in math program and graduate quickly. Most have requirements beyond mathematics that will take up a lot of your time. Your best bet, if you want to accelerate your path, would be to take the math courses you lack at a quick pace as a non-degree seeker. You'll meet a lot of folks along the way who can give you some advice about getting into a math grad program with (what will be) your unusual transcript. You might also find some research to work on. – Bill Barth Apr 10 '15 at 12:11
  • Thanks for getting back. I clearly need to revise my expectations about how much work this takes (although I have decided get it done whatever it takes). I will look at non-degree programs, and may be even a bachelor's degree. I'll have to figure out how to fit in my own self-study, but what's life without a challenge :) – user32892 Apr 11 '15 at 4:05
  • I have a bit of a disagreement with this. Namely, yes, the typical/orthodox route involves degrees certifying "a certain... <cough> competence in iconic rituals", I would claim that this barely sets the acolytes above "people on the street who are interested"... Maybe I'll have to elaborate in an "answer"... But, yes, good to raise these issues... – paul garrett Sep 29 '15 at 1:02
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I don't know how common they are in math, but in physics/astrophysics I know several people who have pursued "post-baccalaureate" work. Here is an example from my own department. The people I know who are a part of this program are from non-physics/astro backgrounds and are using this program to transition into a graduate astrophysics track.

I think if you can find a good post-baccalaureate program (or something similar) in math, it will provide what you need and what you are looking for.

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    Postbacs are not that common in math, but they exist and I agree that it would be a good idea for the OP to pursue one. – Pete L. Clark Sep 28 '15 at 22:14
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Although perhaps you might slightly discount this advice on the grounds that apparently I'm sometimes perceived as "radical" or some other dismissive + mildly perjorative modifier, I might recommend that you do study for a bit, especially if you have funding, to... drumroll... be a better scholar for the thing that you want to ... be a scholar/researcher/expert/maven.

In on-the-ground practical terms, if you can find a friendly "post-bac", this would indeed be helpful. These are the catch-up possibilities for people in the U.S. to replicate what (more narrowly educated, due to the system) students in Europe have been required to do... crazily-ironically, whether they were interested or not.

Back to specific advice: follow your interests; do not believe people/advice that urge absolute conformity to ... style, content.

The tipping point is making-a-living verus scholarship-or-whatever. Now that the Cold War is over, it is not as easy to make a living proposing ways to defeat that particular "Evil Empire", and, truly things are subtler. But not much, though perhaps even less gratifying for any of us who thought there'd be progress... nevermind...

But this does explain the situation young academics often find themselves-in. So it's not "how long do I study before trying to enter academe", but a different question about one's own practical situation.

Back to an idealized issue: if one has a good job, and spare energy to study, I'd recommend doing more of this pre-study... and put off petitioning for re-entry to academe a bit... Getting some corroboration of one's putative competence would be wise...

So, as often, the issue is not quite (as I would think) what the questioner asked... but/and the question raises the right further questions...

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