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I hesitated to ask this question since it may not relevant to this website, but I saw a question of same category here but still couldn't find an answer ...

Since childhood, I've had a sharp enthusiasm for Maths, studying books from an elementary level to ones suited for the level of a Masters of Science student at Melbourne University. I am currently 15, but the formal classroom setting feels like a waste of my time. Because of my passion for Maths, going through the formal procedure of education is making me severely depressed...

Here is my question:

What careers which do not require an academic degree (all research positions require a PhD) might be open (in Australia) to a self-learner who wants to do research on Pure Mathematics?

Salaries for a postdoc position are ~80k in Australia, but I am looking for a research career even of a ~15k salary because that would be enough money to survive while doing research. I have yet to published any paper to impress some institution to believe in me, if it helps at all.

EDIT - Thank you all very much for your very informative answers. To be more precise, I would like to write my main reasons of the 'horror' of university:

  1. Solving exercises or studying the texts in self-study themselves can (although difficult) be stopped for the duration of university affairs (classes, ...) but I believe that research can't be frequently stopped by other irrelevant things even with university affairs [university classes will be Math, but not be the research that I want to engross in, at least for first few years], since being creative requires to be much more engrossed (to put it simpler: 1 2-day is more efficient than 2 1-day for research).

  2. Even with prestigious universities it involves a lot of bureaucracy, which they may find necessary to obtain a degree but are irrelevant to Mathematics.

  3. Being very deep concentrated and having manic passion causes a lot sensitivity, that's why we have some recluse people (in Arts and Science) and most of them are suffering from Bipolar Disorder caused by their manic-enthusiasm. While there are many people with great achievements living in society with high-degrees UNFORTUNATELY I don't think it's impossible for me.

    3a. An example of negativity in university education is the focus on institutional ranking. Combined with job-seaerching being the main purpose of study for some, it could impact someone who just seeks the beauty of Mathematics faith in seeking a degree.

  4. University costs a lot for a Bachelors of Science, much more than it would to be a mathematician in solitude.

EDIT 2 - In my question, I have asked that from where I can get financial support for basic life expenditure, however many answers include enrollment in universities which costs A LOT for just BSc (supposing that I can survive with my mentioned conditions in first EDIT).

Thank you, everyone. I highly appreciate the brilliant advise in many of your answers, however my main question remains how would I pursue research in Mathematics without a degree? I know the path that is recommended for pursuing a research career, but the cost of university and not having peace of mind are too great of an expense for me.

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    ... and if you really are ahead of your class mates, many university professors are willing to help you design your own program, such as by designing special classes for you, or by letting you get in on their research. – Dave Clarke Jun 9 '15 at 11:16
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    One thing it's important to understand: University is not just "Advanced High School", it's a very, very different animal, and you'll be around a different group of people. Many people who hated highschool loved university (myself included) – LindaJeanne Jun 9 '15 at 13:40
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    Don't get ahead of yourself. You're very young, and when you're a bit older you will realize how little you used to know and how much humility you should have shown. This is not anything personal, just a universal fact about wisdom: the more you know, the more you know of your ignorance. Read Plato's "The Apology." – mbsq Jun 9 '15 at 16:02
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    Note that studying at an elite university does not necessarily involve taking classes. In at least some (e.g. Cambridge), the only requirement is that you sit the examinations. Class attendance is entirely optional, and if you can pass the exam without attending class, that's fine. If you are really as advanced as you suggest, professors will be happy to guide you in more advanced study, and probably fight with each other to have you work with them. Don't underestimate the value of having experienced, intelligent people to discuss your research with. – avid Jun 9 '15 at 21:51
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    @MKR I'm starting to get the feeling that no matter how brilliant you are, your reluctance to work with other people and your certainty of your own superior mathematical skill (even if that self-perception is accurate) threaten to overshadow your talents and impede your career. – Kyle Strand Jun 13 '15 at 4:57

11 Answers 11

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The question you seem to be asking is "Can I be paid to do research, without any formal education, if I am very good?" The likely answer is no. I know many serious researchers across countless fields, and the only ones I am aware of who can make a living without any formal training are inventors who got quite lucky in terms of their ideas and their marketability. Many great artists, mathematicians, and scientists did other things to survive while doing the work they were truly passionate for, and we romanticise many very inspirational people. Research is probably much more a function of diligence and persistence, even when things are depressingly boring, than it is of intelligence or genius. I would suggest reading some of the good bits of advice many modern prodigies have given on the subject. Terence Tao in particular has several good pieces on accelerated education, working hard, and being a genius.

Yes, school can be terribly dull if you are quick. But being able to succeed in that environment is an invaluable human skill, which will prove useful in a research career when looking for funding, or when setting up research groups and similar things, as well as really showing that you have a sense of humility. We can complain about how slow and tedious bureaucracy can be, but if you really are quite good, it is worth your time to learn and understand how to survive in it. Mathematics is not simply something devoid of social interaction or human involvement. Pure and applied mathematics stem from human ideas and are inherently connected to social and cultural concepts, and these connections are often a part of what traditional formal education gives us. I would absolutely advise you to go further than what you see in a classroom, do independent and guided research as early as possible, and test out of whatever classes you can. But skipping the process entirely will handicap your ability to contribute meaningfully to mathematical research.

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    The key thing here is guidance, I think, as research divorced from human interaction is the surest path to self-delusion. Community brings social and political challenges, but it also brings peer review. One of the things I've learned as I slowly become an expert in various aspects of my field is that I've had to grow out of several misunderstandings along the way. Perhaps you will make fewer mistakes than I did, but, without guidance, it might take a long time to even find out if you have. The human challenges are often harder than the technical, but I've found it worth the effort. – Dan Bryant Jun 9 '15 at 15:04
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Be very wary of doing mathematical research on your own. Even if you read all the books there are, you will most likely go off on a tangent and reinvent a theory that has been known in math for decades or centuries - simply because you never heard about it and did not know the commonly accepted names of the structures you have been working with. Or, as others point out, you might work on something that may be true but is of zero interest to anyone else.

If all you are looking for is the joy of doing mathematics, that may well be fine (although even then I'd say using up-to-date tools and theory would probably be more satisfying, just as I'd prefer to code on a modern machine and not an antique, even if it does run a compiler). However, if you want to make a difference in the mathematical world, possibly publish your findings, then you will need to interact with other mathematicians.

And this is where a formal education will help. A math degree will show other mathematicians that you at least have learned the fundamentals. When you contact a mathematician, you will need to differentiate yourself from a random crank. There are far too many home-educated random cranks in mathematics, and you are lucky if they self-identify by claiming that they can solve the general quintic equation by radicals. Having a degree in math will make it easier for you to get other mathematicians to talk to you. In addition, your advisor will have decades more experience than you and will be able to steer you in fruitful directions.

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    Good point about the cranks. There are so many people out there that have solved science... – Davidmh Jun 9 '15 at 13:38
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    There was a guy down the pub that could solve quintic equations... it was radical. – Ali Caglayan Jun 9 '15 at 15:35
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    "you might work on something that may be true but is of zero interest to anyone else" I don't know very many mathematicians that don't do this. – zibadawa timmy Jun 10 '15 at 2:12
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    @zibadawatimmy: do you know very many mathematicians at all? Most mathematicians publish articles about their work, that at least a referee and an editor deemed interesting, and are asked to give talks about their work. There is a distinction to be made between "work that only interests a few specialists" and "work that interests no one else than you". – Benoît Kloeckner Jun 10 '15 at 8:41
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    Theorem: All natural numbers are boring. Proof: For the sake of contradiction, suppose some natural numbers are interesting. Let x be the smallest interesting natural number. Who cares? QED. – JeffE Jun 11 '15 at 1:46
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Research is demanding (well, at least good research is), and a career in research requires discipline, among other things. I understand being bored in classes, but if you don't have the discipline to go through a standard academic program, it doesn't bode well for a research career.

One option is: if the courses really are too easy for you, you may be able to arrange to take more advanced courses instead (AP, IB, possibly at a university), or go to university early.

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    +1 for "if you don't have the discipline .." 99% perspiration. – mac389 Jun 10 '15 at 11:22
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There certainly exist math classes that would not bore you; they might simply be more advanced classes than the ones you're in right now. It is difficult to answer precisely without knowing more about your situation (notably which country you live in, how far you are in your studies right know), but I don't think your goal should be to avoid PhD: doing a PhD is both the gatekeeper to research positions and really about doing research, so if you are dedicated to math your goal should be to do a great PhD, not to avoid it. Without a PhD, as mentioned you can still do math by yourself while working on a side job, but this has several drawbacks. You cannot expect to get a paid research position, even with very low salary (by the way there exist low-salary research positions that require a PhD). Also, note that even if you are very bright, actually advancing the knowledge of humanity takes a great deal of work and effort. Be prepared for the path to be long, the tests to be tough, and be prepared to fail more often than you succeed.

Let me give an example of possible path in a specific case. I assume you are indeed great at math and knowledgeable (beware that some people are delusional about this, but let this point apart). If you where a French high-school student (or European with fluent French), I would then suggest you take a look at math (and physics) tests for entering Écoles Normales Supérieures. If you are able to do great at them, then you might be able to enter these schools younger than most students, and there you will find a very favorable environment which would be very different from what you know. There is hardly any chance you would get bored for long. If the tests are too difficult for you, then work toward them and see whether you can move forward one class, and aim to enter a good "classe prepa" to prepare for these tests as soon as you reasonably can (though for this you will need support from your teachers).

If you are willing to move or already in the right place, the same kind of advice can work, to look at what it takes to enter elite universities around the world, which are used to accomodate young brilliant student (I heard that Don Zagier could not do its undergraduate study at Cambridge because he was 12 or 13 at the time and they had a policy of not taking undergrad before 16, but that then he settled for MIT and came back to Cambridge for graduate studies, at the age of 15).

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Can you do research without formal education? Sure. Will it be of any relevance? Perhaps not so much.

There are a few advantages that a formal education gives you. First, it sort of forces you to sit through a series of topics that you may find boring, uninteresting, or irrelevant at the moment; but at some point they may turn out to be useful. Conversely, it also gives you easy access to other branches that may be quite difficult to get on on your own, but may prove useful later. For example, in a course on Theoretical Physics tailored for Mathematicians you may discover a certain trick to solve a particular problem that may pop up outside of Physics.

These extra courses also provide a broader view and understanding. Mathematics are useless unless they serve some sort of purpose, even if it is inside Mathematics. Knowing a bit of everything can help you find applications and identify potentially interesting problems. If you are unlucky, you may find yourself working for years in a theory that may be correct, but essentially useless for anybody else.

Related to the previous point is that it is easier to get expert supervision. At the beginning of your career, you work with a professor, that is the one giving you the broader understanding of the field, useful references, contacts, etc. that only years of experience can teach you.

You said you read a MSc level book, but have you fully understood it? How can you be sure of that? As an undergrad I thought I had master partial differential equations, and I totally aced the exam, only to find that my understanding was flawed, and all the answers were wrong. If I had not had an external evaluation, I would have taken me a long time to realise, and the fall would have been big.

And lastly, you have a lot to learn to do research. When I finished my master's project, I thought I had done quite a few clever things; but also realised, with hindsight, that I had done a few pretty dumb things that I hope won't do again in the future (nothing bad, just a few weeks of wasted effort). A few months into my PhD project I look back and see other mistakes. Learning how to do research takes time, no matter how smart you are. Your undergrad will provide a low risk environment to make some of these beginner mistakes that will teach you a lot. I, for example, would be wary of paying someone that has not had this basic experience, because I will have to fund their mistakes.

All of these things you can, of course, circumvent without an undergrad; but I don't think it is a wise strategy to reject it up front. You don't know how a BSc is, and you don't even fully know yourself. My advise is that you take it easy, learn as much as possible, and see where it leads. You don't have to give up research, as you can always continue it while studying; and if you are truly natural at Maths, most of the coursework won't take long (and if it doesn't, maybe you were not so good on your own).

Lastly, a word of caution: I have seen some people who had done some research in High School drop off university because they wanted to do the "real stuff", and it turn out that the path there is rougher than it seems. They discovered the hard way that there are easy things you can do in the flashy parts of research; but you need much more to actually be able to fully do it on your own. Don't become one of them.

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    @MKR you can't judge a book by the cover, and you can't judge a several years degree by two lectures. There is much more to this, like interacting with your peers, teaching and learning from them, and learning how to get the most out of bad lecturers. – Davidmh Jun 15 '15 at 6:17
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    This is very true. I remember at the end of my bachelor's degree, I thought I was already a fantastic writer and researcher and thought that the PhD would just sharpen the skills a bit, so to speak. I learned SO much about research and writing in my PhD program, more than I ever realized I would. You don't know what you don't know. – roseofjuly Jul 20 '15 at 15:41
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+50

I want to answer this question from a different angle, by addressing two things you say in your original question that I think are untrue and kind of show why you need a broader education.

For one, you say that universities require you to take a lot of irrelevant classes to earn a degree (or at least that's what I think you are trying to say - part of my point). A lot of those classes you think are "irrelevant," though, are necessary to broaden your thinking and will make you into a more well-rounded professional researcher. At the most basic level, the way that mathematicians communicate with one another is through scientific journal articles. In order to do that, you need to know how to write well in English, and your university English and composition classes will assist you with that. Moreover, mathematics is often developed in hopes of applications to other fields - the sciences for sure, but also the social sciences - and an understanding of the conventions and needs of those fields will enrich your understanding of how mathematics ties the world together, even if you are interested in the purest of math. I didn't get a true appreciation of mathematics until I studied my own chosen field (psychology) more deeply.

Secondly, you espouse the idea that research can't be "frequently stopped by other irrelevant things." But the way research works in the modern world...it usually IS stopped by other irrelevant things - and relevant things too. Modern scientists spend a lot of time teaching classes, advising students, writing grants, and serving on committees in their field and at their university. No one is going to hire you to to purely be a mathematician and sit around and think 8-12 hours a day - you'll also have other tasks to complete. One of the values of formal education is that you learn how to balance all of these commitments and still think deeply and do great research. Creativity does require engrossment for some period of time, but most scholars have to stay productive over long periods of time while handling other responsibilities.

  • I had set the bounty for another answer, but I found esp. last paragraph of your answer more helpful and I highly appreciate it. :) – MKR Jul 25 '15 at 18:08
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I would advise trying to complete qualifications like those in the UK. You can take A-levels without going to classes and with only minimal coursework. You could just take them without preparation to get a good high school level qualification. The next thing in the UK would be to get into a place like Oxford or Cambridge, you could do well in the STEP paper and get admitted. Then completing a maths degree at Oxbridge would be doable. The Oxford /Cambridge maths programs have very little in the way of compulsory course work. You could take the exams and work with world class researchers for the majority of the time.

In short, University courses with minimal compulsory class hours and course work do exist and you may be able to successfully complete these quickly and relatively painlessly.

Importantly taking some form of higher ed qualification will help you to gain respect and credibility when trying to publish or work with other mathematicians.

2

Yes, you can do mathematic research on your own and without a formal education. There is a term for people who do such things. They're called autodidacts. Many great contributions can be attributed to such people. I could name many notable autodidacts but in relevancy to your question, Srinivasa Ramanuja, Gottfried Wilhelm Leibniz, and Walter Pitts may provide inspiration.

My experience with self-learning began with aviation. Out of high school I wanted to learn to fly, but I couldn’t afford to go to a "pilot mill". I read the books and found a mentor. I would study at night and we would fly in the day. Eventually I made it to my commercial license, but decided airline piloting wasn't the career for me.

Then I decided to go to University for physics and engineering. I left after two years because of my frustrations with the system. I contacted companies like Dassault systems and Adobe and asked for student versions of their software. I acquired "less than authentic" versions of Comsol multi physics and Matlab. I read books and watched videos on YouTube and learned to use the software. Eventually I was proficient enough to get a mechanical/systems engineering job. I was able to get the job without a degree because I proved that I could do the job of several people. Progressively, I took on the roles of researcher, designer, engineer, and programmer. Now I both design and build the robotics for the purpose of manufacturing specialty custom products, and I love doing it.

Next, I began studying chemistry in my spare time. Again, I read the material and watched YouTube. I invested money and bought the equipment to build a laboratory. (This is the part when you’re usually told “don’t try this at home”… but all I will say is that curiosity can be as useful as it is precarious).

I'm 26 years old and I will be applying for my first patent in 2 days (self-drafted) and forming my own company all thanks to self-taught knowledge.

However, you should know that this path is not easy. It requires enormous dedication. Most nights I come home and teach myself what is required to succeed the next day. Add in family, friends, a significant other and life in general, and the pressure can build quite quickly. You have to stay positive. There are many times I've questioned taking the path less traveled as they say. There will likely be many zetetic influences, especially in the beginning of your endeavor. Social skeptics seem to be a universal constant.

One more thing worth noting - Always be open minded and on the lookout for mentors. I've had several mentors whom I learned a great deal from, and many of them entered my life unexpectedly.

So in conclusion, if you have the aptitude and the ambition then go for it, and give it your all. What's the worst that could happen? You may lose money? You may have to go back to school? But no matter what, you'll acquire new perspectives in life. You'll still net knowledge and neither of those can be can be considered a zero gain.

P.s.

Some relevant quotes:

"The only thing that interferes with my learning is my education." ~ Albert Einstein

"I loved education, which is why I spent as little time as possible in school." ~ Karl Hess

"It is a miracle that curiosity survives formal education." ~ Albert Einstein

“I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as startling." ~ Srinivasa Ramanujan

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    On the other hand, you are probably not Ramanujan. – JeffE Jun 9 '15 at 22:00
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    You do understand that those Youtube videos you watched for chemistry or the chemistry books you read, were actually prepared by people with university degrees or higher and not some unknown experimenting on their mother's basement. That alone proves that university education works (even if it did not work for you) since it makes people capable of passing knowledge even through books or videos. – Alexandros Jun 10 '15 at 19:17
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    @CaptainCodeman Of course! And I'm sure many people did tell him just that. But that doesn't contradict the fact that most people that idiots on the internet say will never amount to anything are not actually under-appreciated world-class genius autodidacts. – JeffE Jun 11 '15 at 1:42
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    None of your examples are modern. – user18072 Jun 11 '15 at 15:40
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    @CaptainCodeman I am not arguing that he probably won't be successful mathematician. I am arguing that he is probably not a once-in-a-century under-appreciated world-class genius autodidact. Surely the difference is obvious. Even if you are good at soccer, but you are probably not Lionel Messi. – JeffE Jun 12 '15 at 13:12
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First, I agree with many of the other answers that advocate for formal study to help you. And I would like to caution you against thinking that classrooms are entirely a waste of time. Far too often I've come up with novel solutions to problems specifically because I took an unrelated concept and applied it to the problem at hand. All education is beneficial.

All that said, there are a few careers I can think of that would value heavy math research (although not "pure" in the strictest sense):

  1. Finance - there is piles of research being done in finance to help produce better predictive analytics. It might be difficult to get your foot in the door there, but that sort of industry rewards success, and has a clear measurable way of evaluating it. If you can make people money, they won't care about your degree.
  2. Games - the gaming industry has a long track record of discounting degrees. There's also a good amount of work being done on algorithms here to better handle problems in rendering, network prediction, AI, etc.

Both of these options require application to some degree, even if the research itself is purer than most. For your situation, it is a benefit since your employers will care about results, not your educational background.

1

You need a Bachelors degree. Having worked in government labs (USA) there are many researchers that have a BS in math, physics, or engineering, and nothing beyond that. (It doesn't work like that for chemistry or biology, generally.)

You shouldn't focus so much on the education... the true purposes are 1) clear the minimum hurdle of the bureaucracy in order to apply for the job you want 2) prove to the employeer that you're valuable. The BS in Math can do both of those things. It's essentially impossible to do that without some kind of certification.

...enrollment in universities which costs A LOT for just BSc...

That's what loans are for. Again referencing those government labs: I may be green with envy, but some of those people were hideously overpaid. You'll have no problem paying off the loan, even if you don't receive a scholarship.

If you really want to avoid school, there are 1) accelerated programs 2) internship programs for credit while you're doing your degree 3) if you become well-liked by a professor, there are normally directed research credits you can take (so you can avoid class). I did all of these.

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The degree isn't a big issue as you think it is. Just continue studying math on your own. When you've mastered most of the math curriculum, you just contact some university and ask if you can do fast track exams for most subjects. If you show some exceptional talent in some subject, then you'll likely catch the attention of some professor, who will put severe pressure on you to become his/her Ph.D student. You'll produce a lot of work together, you'll travel all around the World, visiting many conferences where you present your work.

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    You have seen too many times the "Good Will Hunting" movie. The world does not work like that. – Alexandros Jun 9 '15 at 18:04
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    This can happen, but it is significantly more complex than this really. In particular, most potential prodigies who intentionally seek to avoid the normal flow of things rarely make it. Yes, nontraditional approaches can work out, but they must be done very carefully, if the individual really wants to have a chance to make an impact. – Andrew Krause Jun 9 '15 at 18:05
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    But what I'm suggesting is petty much the normal flow for students who are far ahead — [citation needed] Speaking as someone who was far ahead, and who now holds a faculty position... No, it really is not. – JeffE Jun 9 '15 at 22:02
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    The problem with this answer (in my opinion, obviously) is not so much that what it describes cannot be done but rather that what it describes is an unnecessarily difficult path to the goal. It's kind of like saying "Don't worry if the idea of buying a car fills you with dread. You can learn to design and build your own car from scratch!" True but...not easier, except under a very specific set of edge conditions that probably deserve to be addressed directly. – Pete L. Clark Jun 10 '15 at 5:06
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    @AdamMosheh Yes, I am saying it's good to fail. – JeffE Jun 12 '15 at 13:08

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