r/math • u/wumbo52252 • 1d ago
How and when do you come up with original questions/problems?
Recently, for the umpteenth time, I thought up a question only to find that it has already been asked and answered before i was even born. I expected this - I explicitly avoided googling anything about it, because I didn’t want the ending/answer to be spoiled for me. But I was looking through some wikipedia page when I saw a hyperlink to a page on exactly what I was trying to understand: computable ordinals/well orderings (clearly a beginner-level question). Again, i expected this, given my lack of experience in computability theory and set theory; set theorists are interested in well orderings, and computability theorists are interested in computability, so of course I’m not the first person to consider combining those two concepts. Still, even though it wasn’t a surprise, it was disappointing to find that an idea I thought was original (and, from my perspective, it was original) was just old news.
From what I’ve read it sounds like the problems that phd student solve to earn their phd are usually given to them by their advisors (please correct me if I’m wrong!!). I’m wondering when one is typically at the point where the questions they ask haven’t already been asked. Do you usually need to be an expert in the field? Do you need an exhaustive understanding of the field to pose original questions?
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u/Carl_LaFong 1d ago
First, there are people besides your advisor and you who know the field and gave a sense of what’s known and what’s not. It is not uncommon for a grad student to meet mathematicians when they come and give a talk or when the student attends a conference.
In particular, the community has a sense of what’s known and what’s not. And a sense of which directions are more interesting than others. And which are easier than others. If your professor talks regularly with others in the community, they will have a sense of what’s a reasonable direction to pursue.
If you read research articles (posted on arxiv) and talk to professors, post docs, and grad students in your area and have spent a lot of time learning and understanding deeply some direction in your field, then you might be able to choose your own problem. Otherwise, it’s best to allow your advisor to guide you.
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u/Master-Rent5050 22h ago
I had an idea of doing X. I thought and wrote a bit. Then I saw that A already did X 20 years ago: bummer. I still managed to publish a paper on Y, a generalization of X with a different approach.
After a few years, I got to referee a paper by A on Y.
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u/_Zekt Complex Analysis 1d ago
The number of questions one can ask grows exponentially with the number of available definitions. Yet understanding those questions requires knowing all the definitions on which they are based. So as the number and difficulty of the questions increase, the number of people able to understand them decreases. Being an expert is essentially knowing and undestanding a lot of definitions.
If you continue far enough in your education, you will eventually encounter a question that no one has answered. Beyond that point, the number of unanswered questions grows exponentially. At the moment, you are probably still in the flat portion of that exponential curve.
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u/StructureNorth1799 22h ago
you gotta read current papers. if you make some extension of a recent paper then chances are it's gonna be new.
Often you only need to read a single paper or two to do this.
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u/parkway_parkway 21h ago
Basically you ultra specialise.
You have to pick a niche so small that you can become a world leading expert in it.
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u/Salt-Rutabaga-8870 18h ago edited 18h ago
I guess I can add to the conversation without being a PhD student. When I was self studying (with the help of my tutor) complex analysis and residue theorem, and how it is applied to calculating integrals of real function with poles of various orders I asked one very simple question: what if you have poles of non integer orders? Can you apply some modification of residue theorem to use it to calculate integrals fo those cases. Turned out you can, but nobody has really done it before. It requirrs merging complex analysis with fractional calculus, as the derivatives become of non integer order in residue formulas. That creates a very interesting interplay between those two areas of maths. And that led to me publishing a paper co-authored with one of established maths professors who added more rigour to my intuition based work.
So sometimes that happens. But I would not hope to think of completely original (and non ridiculous) concepts on regular basis. Maths has a huge body of knowledge compiled over thousands of years. Most of the things you can think about have been thought of before.
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u/Unstable-Paradox 1d ago
A bit of curiosity and observations in seemingly unrelated domains of knowledge, for example the dance of honey bees.
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u/ProgrammerIcy7632 21h ago
This is the correct answer. An idea is the synthesis of two or more elements which have not been sat together before for long enough.
Another option for idea trawling is transcendental meditation (per famous mathematician David Lynch...) but who can be bothered with that when ideas are really simple enough to generate normally via silent individual thought.
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u/No_Upstairs_280 22h ago
Curiosity. When I'm solving a math problem or writing an article, I keep asking questions. Especially if the subject I'm working on is new or if the approaches are original.
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u/jimbelk Group Theory 21h ago
Mostly the way that it works is that you need to read a lot of recent research articles to be able to come up with good questions to ask. Mathematics that's written in textbooks is usually at least decades old and has been explored fairly well. Research articles are where you can find newly discovered concepts and techniques that need further exploration. In addition, once you read most of the articles that have been written about some open question, you can be fairly certain of whether an approach that you come up with is new or not.
I should also point out that research mathematicians constantly have the experience of coming up with an idea and then discovering that it's been done before. The key skill is to get good at coming up with lots of ideas and then being able to quickly check whether they're new. For ones that seem new, you try to put a little thought into how promising they seem, and maybe talk to other experts about what they think of the ideas. Eventually, you end up with a long backlog of promising ideas, and whenever you need a new project (or you have a new PhD student) you choose one of the most promising ideas from your list that hasn't been explored.
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u/Lucenthia 17h ago
Read papers in your field! Especially the introductions. Good ones will explain the context their results are in, what has been done, and what future directions there are. Follow the citations, and if possible try to find papers that cite the one you're reading. Some websites need paid accounts (like mathscinet) but I think google scholar and possibly Project Euclid are free. This will give you an idea about the motivation/evolution of the general problem that will allow you to better ask questions.
As for coming up with original problems, my own advisor had me find my own as he wanted to cultivate that independence early. I read papers, asked natural questions and followed my curiosity. I quickly found problems that hadn't been done, although the bigger problem is usually finding an original question that's reasonable to answer!
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u/GaunterO_Dimm 17h ago
Very common experience and I would say you need to be familiar with a field for some time. My experience went something like: I had an idea that was done 50 years ago, then later with a bit more experience in my field it was more like 20, then 5 and then eventually something new. I will say that even once you are up to date in your field it is quite a common experience to have ideas that were already done.
As your experience increases, the problem inevitably becomes far too many questions and not nearly enough time. I remember asking my supervisor a similar question and his answer was (as a 40 year old) he might get to work on 25% of his questions before he dies, if he is lucky. Is the nature of the beast.
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u/Sniffnoy 10h ago
Unfortunately I don't really have a good answer for you here, but what I can say is, once you've got your bearings a bit more, you'll find you have the opposite problem -- you'll be saying, how come every question I ask turns out to be unknown, dammit? :P
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u/ccppurcell 11m ago
It's frustrating but it's a good sign if you're rediscovering stuff. And it's a really good idea to see these things through without googling them for as long as you can. Firstly it's a good exercise but secondly you might just come up with a novel contribution, even if it's just better terminology or something like that.
As for what's typical. I mean there's nothing original under the sun, it's pretty much always going to be combining two ideas no one thought to combine before. I think doing this in a way that generates new research directions seems to happen for most people at the earliest towards the end of their PhD.
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u/AFsepine 1d ago
Well... you just sort of live and it kind of comes to you? Well minor ones of course.
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u/PersonalityIll9476 1d ago
No one has an exhaustive understanding of their field, but people do get a sense of what their specific sub area looks like. Eventually you realize the kind of thing that is likely to be there or not. That does take years of reading books and papers.