There are some areas of study where you can go seriously off course with self study. I think mathematics is a good example of that.
For some reason the human mind just isn't, naturally, very good at maths. By that I mean, that even the simplest of mathematical ideas and methods are not intuitive or easily obvious until you get them shown to you. Without a teacher, then most people will never get to them at all.
If you think maths is easy and obvious - as you might well do if you have had a decent maths education as a child - well it's probably because you are just not challenged any more. You've learnt all the maths you need for your daily life and for your topic area.
For example, entire civilizations rose and fell, for thousands of years, without developing an understanding of the mathematical idea of negative numbers. Right through to the sixteenth century, then European mathematicians were handicapped in their solutions of algebraic equations such as the cubic, because they didn't know how to work with negative coefficients.
Hindu mathematicians had already discovered how to do that in the ninth century, but Arabic, and then European mathematicians didn't find out until much later, even through to the late sixteenth century then having found what was a general solution to the cubic, they didn't realize this because they were unable to handle equations with negative coefficients. See Quadratic etc equations.
And many ideas that to us seem elementary when manipulating algebraic equations were not discovered until the late seventeenth century. For instance they didn't know that if a cubic equation had roots a, b and c, that the equation had to be (x-a)(x-b)(x-c) - seems obvious to us but nobody realized it. And thus of course, they also had no idea that you could check your result by substituting your values for the roots into that equation and multiplying out to see if you get the original cubic.
Other ideas that seem elementary to us but took millennia to develop include the concept of 0, place notation, and ratios (the Babylonians already had an idea of ratio, but the number at the top was always 1 or in a few special cases 2 so they didn't have the general idea of a ratio of two arbitrary numbers, just 1/n and a few special cases of 2/n).
Take the best mathematicians of our time, put them back into C16 Europe, with the education of a mathematician of that time, and almost certainly, they will never think of the idea of using negative coefficients to solve cubic equations, which to us is elementary. And many other ideas, elementary to us, are completely beyond their grasp because nobody thought of them.
And - try to teach them these ideas and you can expect considerable resistance at first.
If you've ever tried to teach maths to young children (which I did at one point in teacher training) you find that right through to secondary school they struggle with what as adults with a familiarity with maths, you'd think of as basic ideas of mathematics. All of them.
(With very few exceptions, mathematical prodigies - Ruth Lawrence was already at university completing a first class degree in maths at that age, but I didn't have anyone like that in my classes - and presumably even she had to go through those stages at some point, just much younger - would not have thought of the idea of a ratio with a number other than 1 or 2 on the top if born in ancient Sumeria, or of negative coefficients if born in C16 Europe).
Even the mathematically brilliant ones, still have to go through that process, similar really to the way our civilization went through it.
Ideas like this are very hard to understand from books. You will probably come up with some completely different way of working with equations which almost makes sense to you, and use a lot of guesswork to try to make sense of the equations you read in books.
There are some good resources for learning this. But it will probably save you many false trails and confusions to find a good mathematician to teach you.
This is true right up to research level maths. Almost nobody nowadays has even basic familiarity with all the main topic areas of research level mathematics. Even in particular areas of maths, they can't expect to have more than basic familiarity of the field. In my own area of set theory / logic / foundations, my supervisor was I think the only one I met at university who could be said to have a thorough basic understanding of the entire field of logic- and that was when he was in his sixties, having lived through most of the period of development of modern logic. Robin Gandy
Others in the field that I met had a thorough understanding of a few topics in the area of mathematical logic, and understood basic concepts throughout the field - but not a broad and also deep knowledge of the field as I could test by asking them about the particular area I specialized in which happened to be an area almost nobody else in the UK was working on (not unusual in maths when you get to postgraduate level).
I did manage to make a fair bit of headway by myself - I studied at Oxford as a mature student after having done research for some years by myself by self study in between my first degree and my research degree - that was before the days of the internet so there wasn't any possibility of going online to find other mathematicians familiar with your topic area.
But you go forward by leaps and bounds once you can talk with other mathematicians who already have a good knowledge of the field and even more so if you can find yourself a good teacher to teach you, informally or formally, who already has a good knowledge of the field.
So - I'm only talking about maths here - the disadvantages of self studying in maths are that you are much slower at getting to the same page as everyone else. And you might never "get it".
But it doesn't have to be formal study perhaps. And if you have gone through mathematics far enough to have a good understanding of algebra - and of logic as used in proof methods - and of geometry - you have a reasonable basis already. But it is still easily possible to delude yourself that you have proved something with an intricate proof running over several pages at that stage as the many false amateur proofs of Fermat's Last Theorem show.
Because that's another role of the teacher or colleagues in maths. You can produce lines of reasoning that can seem utterly convincing, but you have glided over an important point somewhere - and even looking at your equations over and over, you still don't see the mistake. In this case, if someone points out your mistake, you see it, but you are blind to your own failings. It happens so easily in maths. Eventually you may spot it.
This happens right through to advanced research level maths, for instance Andrew Wiles famous mistake in his first attempt to prove Fermat's Last Theorem when he had spent years by himself constructing his proof - this is a professional mathematician - and he finally did - but his proof had a mistake in it which he had never spotted in all that time. He managed to fix it later on but only by doing a radical change in direction in the way he proved the result.
If you have teachers and colleagues to show your work to, who are interested in what you are doing, and who are themselves also good at maths and particularly, this type of proof - there is a good chance they will spot your errors before you do. Partly because they didn't prove it themselves, so come with a fresh look at it, also they may have made and fixed similar errors before in their own work. The more eyes you can get on it the better.
But - not just with your final proof of what you think is some brilliant new theorem or approach. You need to be doing that all the way through, most of us anyway, as you learn the topic area - learning from other mathematicians and showing your work to them for correction / finding mistakes etc as you learn the topic.
And in some topic areas, especially in modern geometry springs to mind, non periodic tilings - amateurs have occasionally advanced the field with new insights. For instance Robert Ammann with his Amman tilings. But these are ones that don't need you to learn any fundamental new ideas or methods. It's basically the maths you learn as a child, with a few new tweaks and ideas.
Recreational mathematics - the maths of games - is another area where amateur mathematicians, self taught, have made new discoveries. Things like new ways to solve peg solitaire for instance, spring to mind.
Here I'm talking about maths research. Particularly for instance, trying to prove new mathematical theorems. Try geometry or recreational maths if you want a decent chance of finding new results as an amateur mathematician.
Number theory such as Fermat's last theorem may seem an obvious target, lots of simple to state results without proofs Maths in a minute: Number mysteries, but it's not as good as you'd think, mainly because it has been studied perhaps more thoroughly than just about any area of maths today. So your chance of making ground breaking discoveries as an amateur is minute.
As well as that - for some reason proofs of simple to state results in number theory often have immensely long proofs (there is no particular reason for simple to state results in maths to have short proofs). And the proofs in this topic area seem to be ones where it is especially easy to make a mistake on page 2 and never realize your error and continue for dozens of pages building one result after another on that flawed foundation.
Applying maths to other areas of science is much easier. If you've been taught well at school, have a good grasp of algebra, and depending on the topic area, geometry, and a reasonable grasp of the logic of mathematical deductions, only a little fairly elementary maths can take you a surprisingly long way. You might need some calculus for some areas of science. The maths you need for almost all science is basically just elementary school maths, plus a bit of calculus, also usually taught at school, with a very few exceptions.
This may come over as a bit biased against self learning because the question just asked for disadvantages. So that's all I've talked about here.
There are many advantages as well of course. Quora User, mentions a few in his answer. Quora User's answer to What are disadvantages of self-studying?
And all the things you can learn from others - originally were result of self learning by someone in the past.
For some reason the human mind just isn't, naturally, very good at maths. By that I mean, that even the simplest of mathematical ideas and methods are not intuitive or easily obvious until you get them shown to you. Without a teacher, then most people will never get to them at all.
If you think maths is easy and obvious - as you might well do if you have had a decent maths education as a child - well it's probably because you are just not challenged any more. You've learnt all the maths you need for your daily life and for your topic area.
For example, entire civilizations rose and fell, for thousands of years, without developing an understanding of the mathematical idea of negative numbers. Right through to the sixteenth century, then European mathematicians were handicapped in their solutions of algebraic equations such as the cubic, because they didn't know how to work with negative coefficients.
Hindu mathematicians had already discovered how to do that in the ninth century, but Arabic, and then European mathematicians didn't find out until much later, even through to the late sixteenth century then having found what was a general solution to the cubic, they didn't realize this because they were unable to handle equations with negative coefficients. See Quadratic etc equations.
And many ideas that to us seem elementary when manipulating algebraic equations were not discovered until the late seventeenth century. For instance they didn't know that if a cubic equation had roots a, b and c, that the equation had to be (x-a)(x-b)(x-c) - seems obvious to us but nobody realized it. And thus of course, they also had no idea that you could check your result by substituting your values for the roots into that equation and multiplying out to see if you get the original cubic.
Other ideas that seem elementary to us but took millennia to develop include the concept of 0, place notation, and ratios (the Babylonians already had an idea of ratio, but the number at the top was always 1 or in a few special cases 2 so they didn't have the general idea of a ratio of two arbitrary numbers, just 1/n and a few special cases of 2/n).
Take the best mathematicians of our time, put them back into C16 Europe, with the education of a mathematician of that time, and almost certainly, they will never think of the idea of using negative coefficients to solve cubic equations, which to us is elementary. And many other ideas, elementary to us, are completely beyond their grasp because nobody thought of them.
And - try to teach them these ideas and you can expect considerable resistance at first.
If you've ever tried to teach maths to young children (which I did at one point in teacher training) you find that right through to secondary school they struggle with what as adults with a familiarity with maths, you'd think of as basic ideas of mathematics. All of them.
(With very few exceptions, mathematical prodigies - Ruth Lawrence was already at university completing a first class degree in maths at that age, but I didn't have anyone like that in my classes - and presumably even she had to go through those stages at some point, just much younger - would not have thought of the idea of a ratio with a number other than 1 or 2 on the top if born in ancient Sumeria, or of negative coefficients if born in C16 Europe).
Even the mathematically brilliant ones, still have to go through that process, similar really to the way our civilization went through it.
Ideas like this are very hard to understand from books. You will probably come up with some completely different way of working with equations which almost makes sense to you, and use a lot of guesswork to try to make sense of the equations you read in books.
There are some good resources for learning this. But it will probably save you many false trails and confusions to find a good mathematician to teach you.
This is true right up to research level maths. Almost nobody nowadays has even basic familiarity with all the main topic areas of research level mathematics. Even in particular areas of maths, they can't expect to have more than basic familiarity of the field. In my own area of set theory / logic / foundations, my supervisor was I think the only one I met at university who could be said to have a thorough basic understanding of the entire field of logic- and that was when he was in his sixties, having lived through most of the period of development of modern logic. Robin Gandy
Others in the field that I met had a thorough understanding of a few topics in the area of mathematical logic, and understood basic concepts throughout the field - but not a broad and also deep knowledge of the field as I could test by asking them about the particular area I specialized in which happened to be an area almost nobody else in the UK was working on (not unusual in maths when you get to postgraduate level).
I did manage to make a fair bit of headway by myself - I studied at Oxford as a mature student after having done research for some years by myself by self study in between my first degree and my research degree - that was before the days of the internet so there wasn't any possibility of going online to find other mathematicians familiar with your topic area.
But you go forward by leaps and bounds once you can talk with other mathematicians who already have a good knowledge of the field and even more so if you can find yourself a good teacher to teach you, informally or formally, who already has a good knowledge of the field.
So - I'm only talking about maths here - the disadvantages of self studying in maths are that you are much slower at getting to the same page as everyone else. And you might never "get it".
But it doesn't have to be formal study perhaps. And if you have gone through mathematics far enough to have a good understanding of algebra - and of logic as used in proof methods - and of geometry - you have a reasonable basis already. But it is still easily possible to delude yourself that you have proved something with an intricate proof running over several pages at that stage as the many false amateur proofs of Fermat's Last Theorem show.
Because that's another role of the teacher or colleagues in maths. You can produce lines of reasoning that can seem utterly convincing, but you have glided over an important point somewhere - and even looking at your equations over and over, you still don't see the mistake. In this case, if someone points out your mistake, you see it, but you are blind to your own failings. It happens so easily in maths. Eventually you may spot it.
This happens right through to advanced research level maths, for instance Andrew Wiles famous mistake in his first attempt to prove Fermat's Last Theorem when he had spent years by himself constructing his proof - this is a professional mathematician - and he finally did - but his proof had a mistake in it which he had never spotted in all that time. He managed to fix it later on but only by doing a radical change in direction in the way he proved the result.
If you have teachers and colleagues to show your work to, who are interested in what you are doing, and who are themselves also good at maths and particularly, this type of proof - there is a good chance they will spot your errors before you do. Partly because they didn't prove it themselves, so come with a fresh look at it, also they may have made and fixed similar errors before in their own work. The more eyes you can get on it the better.
But - not just with your final proof of what you think is some brilliant new theorem or approach. You need to be doing that all the way through, most of us anyway, as you learn the topic area - learning from other mathematicians and showing your work to them for correction / finding mistakes etc as you learn the topic.
And in some topic areas, especially in modern geometry springs to mind, non periodic tilings - amateurs have occasionally advanced the field with new insights. For instance Robert Ammann with his Amman tilings. But these are ones that don't need you to learn any fundamental new ideas or methods. It's basically the maths you learn as a child, with a few new tweaks and ideas.
Recreational mathematics - the maths of games - is another area where amateur mathematicians, self taught, have made new discoveries. Things like new ways to solve peg solitaire for instance, spring to mind.
Here I'm talking about maths research. Particularly for instance, trying to prove new mathematical theorems. Try geometry or recreational maths if you want a decent chance of finding new results as an amateur mathematician.
Number theory such as Fermat's last theorem may seem an obvious target, lots of simple to state results without proofs Maths in a minute: Number mysteries, but it's not as good as you'd think, mainly because it has been studied perhaps more thoroughly than just about any area of maths today. So your chance of making ground breaking discoveries as an amateur is minute.
As well as that - for some reason proofs of simple to state results in number theory often have immensely long proofs (there is no particular reason for simple to state results in maths to have short proofs). And the proofs in this topic area seem to be ones where it is especially easy to make a mistake on page 2 and never realize your error and continue for dozens of pages building one result after another on that flawed foundation.
Applying maths to other areas of science is much easier. If you've been taught well at school, have a good grasp of algebra, and depending on the topic area, geometry, and a reasonable grasp of the logic of mathematical deductions, only a little fairly elementary maths can take you a surprisingly long way. You might need some calculus for some areas of science. The maths you need for almost all science is basically just elementary school maths, plus a bit of calculus, also usually taught at school, with a very few exceptions.
This may come over as a bit biased against self learning because the question just asked for disadvantages. So that's all I've talked about here.
There are many advantages as well of course. Quora User, mentions a few in his answer. Quora User's answer to What are disadvantages of self-studying?
And all the things you can learn from others - originally were result of self learning by someone in the past.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
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