What does it mean to master a particular subject?

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So I tend to here something like "(insert famous scientist) mastered calculus at the age of 13". What exactly is that suppose to mean? Surly you can be good at a subject and get an A in the course, but still be far from a master. To those of you who are pursuing higher education or especially the professors on here, do you think you've truly mastered any subjects? If so, when did you come to realize that you had mastered it.
 

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Astronuc
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So I tend to here something like "(insert famous scientist) mastered calculus at the age of 13". What exactly is that suppose to mean? Surly you can be good at a subject and get an A in the course, but still be far from a master. To those of you who are pursuing higher education or especially the professors on here, do you think you've truly mastered any subjects? If so, when did you come to realize that you had mastered it.
Usually, mastery implies that one would have a demonstrable proficiency in the subject, and that one not only demonstrates knowledge of the theory/subject, but one applies the theory/subject to new problems. Demonstrating true mastery is usually accomplished through original work, which is usually the basis of a PhD.
 
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Usually, mastery implies that one would have a demonstrable proficiency in the subject, and that one not only demonstrates knowledge of the theory/subject, but one applies the theory/subject to new problems. Demonstrating true mastery is usually accomplished through original work, which is usually the basis of a PhD.
I think there's two criteria for mastery. One is the more objective form, which Astronuc described above, and one is a more subjective form; do YOU feel that you've mastered a subject? Or, even, do YOU think that someone who has by societies standards mastered a subject (e.g., PhD), actually has mastered that subject? IMO, it's the subjective standard that trumps all--if you yourself don't feel like you've mastered a subject, then I think you haven't mastered it. If you feel you've mastered it, then I think you have, no matter what degree you have or don't have.
 
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To qualify my above statement, I'm saying that the word "mastery" is a slippery term, and one aspect of it is definitely subjective. Do I feel I have mastered GR? No. Do I feel I have mastered addition, yes! Does that mean I've mastered avery nuance of addition or abstraction thereof that some advanced mathemetican may throw at me? I don't know. All I know is that I feel I have mastered it, and that's good enough for me. I don't feel that I have mastered GR, and I want to learn more so that I can feel like I've mastered it. In a non-mathematical example, do I feel like I have mastered tying my shoe? Yes. Does that mean I'm the best shoe ty-er in the world? No. Could my shoe tying technique be better? Probably. Would some expert shoe ty-er consider me a master at shoe tying? Probably not. Does that matter? No. I don't care what you say, I feel like I've mastered tying my shoes. That's a subjective sense of mastery, and IMO it has all the validity of holding the quality of "mastery." So, although a silly example, I think the same idea holds relevance to the less salient aspects of mastery and "qualification" in what we can refer to as mastery in the sciences and especially in the arts.
 
AlephZero
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I also like a quote from Prof Lamb (author of several standard texts on statics, dynamics, fluid mechanics, etc) at Cambridge university:
At the age of 16, I understood calculus perfectly. Now I an 66, I am very much less sure about any of it.
But pretty much the same thing had already been written in 1252 AD:
Before I studied Zen, mountains were mountains, and water was water.
After studying Zen for some time, mountains were no longer mountains, and water was no longer water.
But now, after studying Zen longer, mountains are just mountains, and water is just water.
 

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