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I am doing a Ph.D. in Applied Mathematics at a U.S. university and after taking Professor X's class this Spring, he has agreed to be my advisor. About a month ago, Professor X asked me to read a particular recent paper of his and prove a conjecture.

I have been reading his paper, and I can understand the logic in his proofs from one line to the next. However, I fail to see the big picture. I would have no answer to the questions such as "What is the motivation behind coming up with this theorem?" or "What is the motivation behind the proof of this lemma?". As a result, I feel like I don't even have the slightest idea of how to prove the conjecture he assigned me.

I think I know the answer to my dilemma: just keep reading and trying until you understand. However, it is possible that he has something else to add. Perhaps he might recommend me to read another paper for more background or something like that.

Would it be immature and not worthy of a PhD student to ask this question of my advisor?

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    Hurray, also the mathematicians miss the big picture of mathematics then! Try to revert the roles: you are the leading researcher on topic X, you lived through the topic for 10/15 years and now one one of your young unexperienced minion (slaves), while solving equation X.333.a(bis) asks you "what is the point of X?". How would you take that question? would you use your experience as a claw to bash your minion or as a light to show the way to your tutored student?
    – EarlGrey
    May 28 at 14:54
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    @EarlGrey: "Hurray, also the mathematicians miss the big picture of mathematics then!" As a mathematician I'll add the (somewhat - but only somewhat - tongue-in-cheek) comment that it seems to be somekind of cultural fetish in mathematics not to explain what you are doing... ;-) May 28 at 20:04
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    Have you tried reading the papers cited in the paper he's given you yet?
    – nick012000
    May 29 at 6:49
  • @EarlGrey True. On the other hand, we Americans have a cultural obsession with talking about ourselves and our ideas. Any time that someone asked me to explain my own work, would easily be the best day of my week. May 29 at 12:54
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    If you have a good advisor, they will actually really like that you are asking this kind of question! It shows you are curious and want to understand "the big picture" beyond the specific task you've been assigned.
    – Andrew
    May 29 at 15:34
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I don't think your situation is at all unusual and it is probably a good thing to talk over with your advisor in general terms. There is a hierarchy of mathematical understanding that goes from simple computational ability at the bottom to something that I won't try to describe here.

But about 3/4 of the way up that hierarchy is true insight into a subfield or problem area. This is about the time in your career when it would be natural to be developing that, but, I hope, your advisor can assist you in this.

At some point you have a a-ha moment when things fit together and you suddenly see the big picture and how the parts fit together.

You can try to tough it out on your own, but if you explore the topic in general, not necessarily related to this particular paper, you might get a boost.

And, the fact that you are asking if such questions are immature seems to indicate that you are within reach of the next step.

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    I wish you had tried to describe it. May 29 at 20:32
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    @HashimAziz, the highest level, which is also a form of insight is when you can recognize things that "might be true" but haven't yet been proven true. A look into the future about what might be with a more or less valid intuition about its likely truth. Such a person will always have more or less fruitful lines of inquiry, but more, rather than less. I hadn't quite reached that stage when I finished my doctorate, but did a bit later. Random poking at the boundaries of knowledge isn't likely to be successful. You need more.
    – Buffy
    May 29 at 21:52
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This is a fantastic question to ask. It's where your advisor's expertise really has the chance to make a difference, and it's much more interesting to answer than "Why does line 4 follow from line 3"? Long term it's your hope to understand the big picture as well.

The best time to ask questions like this is after you've first taken a stab at it yourself: you'll be in a much better position to understand the answers! As you've done this already, I'd encourage you to proceed.

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    Thank you for your answer. My meeting is today, and since I was not planning on asking this, I did not prepare anything concrete about this. I just have a general feeling of "I really don't see the big picture" because I don't understand the big picture of how to prove ANY of the lemmas/theorems. Do you think it's a good idea to just ask him for general advice today, or should I wait until I come up with something concrete?
    – user56202
    May 28 at 15:03
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    @user56202 Sorry I didn't respond to this by the time of your meeting. But really the advice I'd give is (1) do your best to prepare (whcih you did!), (2) when you ask questions, show interest in what your advisor tells you and try to follow up (which I assume you'll be doing!), and (3) try not to worry too much about the rest. If your questions sound "immature" in some way, well, you're fairly new to research, that's to be expected.
    – academic
    May 28 at 23:00
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    No problem, thank you for your follow up! In case you're curious about what happened: I asked my advisor the general question, and he recommend me a (family of) papers to look at for motivation. So it turned out well!
    – user56202
    May 29 at 0:57
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These questions are usually addressed in the introduction of the paper. Papers are written with graduate students and professional academics in mind, and some of those readers are likely to have similar questions about the motivation.

If you can't find the answer in the paper itself, this is an excellent question to ask your advisor.

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    Thank you for your answer. The introduction does contain some high-level motivation, such as "we took the algorithm from reference [25] and made this and that modification". However, I am having trouble with the motivation of most of the technical steps of the proofs; and I don't believe that this sort of motivation is provided in any paper. Do you think it's appropriate to ask about details like this?
    – user56202
    May 28 at 14:37
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    @user56202 I believe it is :-)
    – Clément
    May 28 at 14:47
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There's a boundary somewhere between 'go and read more yourself' and 'ask for additional support'. This boundary depends on the problem you're studying and your own ability in that area, and on how much time your supervisor or colleague or friend is willing to spend discussing with you. It's there in many aspects of life, as well as being a part of studying for a PhD.

You can learn where that boundary is by trying; try reading one time and try asking another time. Pay attention to how well you managed to learn on your own and how willing your supervisor was to help you out. You could also ask your supervisor 'I'm trying to learn when to read more and when to ask for more help from you, do you have any advice?'

As you move further along, 'ask for additional support' may start to become more like 'discuss the problem with another expert'.

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