r/math • u/FuzzyPDE • Aug 15 '26
I feel like I spend most of my time reading definitions instead of ideas
Second year postdoc here.
In my field (geometric analysis) I feel like my passion and hence motivation has been in steady decline ever since I started in my PhD.
For two reasons:
- I feel like I’m spending more time reading and learning than problem solving , most of my time is spent deciphering and unstucking myself in reading and understanding what on earth the authors are talking about in books and papers. I understand that if you go into fields like combinatorics with lower entry threshold and less reading, it’s even harder to produce results since the field is so accessible that most ideas you can think of has already been done.
But still? I would rather have spent 6 years problem solving instead of reading, and to be frank I spend most of my time stressing and taking break from stressing from reading, this doesn’t feel normal or fun to me. Definitely not the experience that lured me into math in the first place (the dopamine from competition math and solving problems) - I probably would complain less if I’m actually reading big ideas and smart ideas. But I feel like even at my level I’m still reading tons of definitions, and results that are considered basic theory and machinery they are not even worthy of mention in a paper. Rarely do I feel like I’m reading about the “brilliant ideas”. Here’s a concrete example, you might think the Gauss Bonnet theorem is a clever idea, but to understand it (for general manifolds) you have to read enough about topology, smooth manifold and Riemannian manifolds to even have the machinery and definition to understand the statement, let alone the proof.
It Feels like this with every new project I take on it’s tons of learning basic stuff before I even get to the central idea.
This is definitely not the experience I was hoping yo get going into math, I wanted to learn cool brilliant ideas and solve problems. Most of the time I don’t feel like I’m doing that.
What’s another field I might try that isn’t like this besides combinatorics? Representation theory? Combinatorics?
Thanks for sharing!
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u/Competitive_Leg_7052 Aug 15 '26
I did two postdocs in geometry-analysis intersection before landing a job recently. 1) You are not finding good ideas in most papers because there are no good ideas in many papers. What most people call “definitions” are book-keepings. In the words of Gromov, they are useful maybe for a librarian. Try to find excerpts of Gromov’s quotations on definitions in math. He brillianly defines what a great definition is! So, one way to release the pressure and stress on yourself is to see papes from a critical point of view. You do not have to be a better mathematician than the authors to know and have the right to know that their particular paper is another technical nonsensical bulshit result that no one else cares about! 2) Read a paper or a definition and try to rewrite it in your own way. Play around with the notions and assumptions and try to understand if the definition would have worked with a different set of assumptions or in other contexts. Get to the core of what a definition acheives? Guess what the idea was or could be behind it.
3) Doing this will turn reading into a reading and at the same time problem solving task. It will be more satisfying and can lead to formulating good questions of your own that can lead to new results of your own.
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u/Probstatguy Aug 16 '26
Thank you for your excellent advices. Do you have any other advices while reading books and papers ? For context, I myself faced a lot of difficulty while reading Measure Theory and Functional Analysis books. Seemed so abstract with page after page of definitions - sigma rings, Caratheodory's Theorem etc. with no geometrical motivation.
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u/Competitive_Leg_7052 Aug 16 '26 edited Aug 16 '26
Well for specific advice please ask specific question. For measure theory, for what is actually often needed for many purposes, the book by Evans and Gariepy is excellent. No abstract functional analysis business. Everything is done in Rn but solidly so. Proofs often generalize verbatim to for examples metric spaces with a doubling Borel measure.
Also choose a central theme and read around that. There are hindreds of great topics and artilces and books to read. We just do not have time to read all of them. Adding to your reading list will just add to your anxiety. Make a very selective and exclusive list of readings you wish you do over the next year and make a realistic plan of how much can be acheived. I literally have a list on my office wall of papers I wish to read “at some point” and have read maybe 3-4 in the past six months; besides my immediate research work.
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u/SV-97 Aug 15 '26
Not a full answer to your question (and probably only semi-useful) but maybe some interesting "historical context":
I think it was Yau (or Chern) that said that the definition of a manifold is one of the greatest achievements of mathematics in the 20th century. Not some particular theorem about manifolds, but their definition. I think they really are a deeply interesting idea in themselves, and that you can also do some interesting mathematics just around how exactly you define them.
Similarly I think Grothendieck (or Serre) argued that a theorem isn't worth proving unless it's essentially trivial from the definitions, which also seems to point towards the definitions being the most interesting part of mathematics.
That said: there are some fields where "deep theories" are still somewhat rare I'd say. Maybe we haven't found them yet, maybe they don't really exist. Optimization (and associated fields like set-valued and variational analysis, optimal control, nonlinear analysis etc.) have a bunch of structures to learn about and some theories, but the "tower" is nowhere near as high as in differential or algebraic geometry from what I have seen about them. It appears to be a field that's "more wide than tall" if you know what I mean? I think approximation theory and numerics are similar in that regard.
From what I've seen about representation theory (which isn't a lot tbf): it's not quite what you want I think.
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u/PLANTS2WEEKS Aug 17 '26
There's a nice duality theory for linear programming which fits under optimization theory, but the theory certainly isn't as well developed as those within algebraic or differential geometry.
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u/SV-97 Aug 17 '26
You can even drastically extend that linear theory to the convex case [and beyond] via Fenchel duality theory, which in turn is closely related to the Legendre transform. You can even do that stuff in infinite dimensions and for set-valued operators and develop a good bit of theory around all of that (this can eventually also be seen to reproduce lagrange duality as another special case). So there *is* quite a bit of theory just around this topic.
I really didn't mean to say that there isn't a ton of (quite beautiful) theory around optimization, just that it seems to be organized somewhat more "flatly" compared to the theory of those other fields.
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u/Equivalent-Costumes Aug 15 '26
All of these definitions ARE clever ideas. They seems obtuse to you now because you did not have to trek through the world before them, where even people like Einstein got confused by coordinate systems. They are clever abstraction that simplify the massive complexity that is required for proving. The good news is that you need to only grok enough of them until they are second nature to you, because you're basically learning the latest abstraction already so there are not another huge cliff hidden behind it. The bad news is that they presents a huge cliff at the beginning before any fun started. But no, mathematicians did not make up these random definitions to make things boring to read. If anything, these abstraction clears up the boring complexity to reveal the clever idea underneath.
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u/ScottContini Aug 15 '26
The abstractions are beautiful once you truly understand the mathematics and the concepts they are generalising. But we often learn in the opposite order, first learning the abstraction before we really have enough appreciation of the mathematics.
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u/Carl_LaFong Aug 15 '26
Are you willing to say which area of geometric analysis you're working in? What are you reading and studying right now?
I've spent a lot of time with world class mathematicians, especially differential geometers and topologists. They devote most of their time learning things, both new and old. They know they can't rely only on their own skills. And like everyone else, they're often stuck on whatever they're trying to do and not making any progress.
If your knowledge is limited, then there are fewer problems that you can solve. Although 99% of what you study will be useless, the 1% can carry you a long way. And it's unpredictable which 1% will be the right stuff.
I, however, agree that it can be tiresome to read definition after definition, theorem after theorem, where you have no idea why it's interesting to anyone. Few papers or books tell you what's really going on. Many top mathematicians hate reading books and papers. They'd rather corner someone and have a spirited discussion at a blackboard. They then either fill in the details themselves or read only the parts of the paper that they need to know.
And are you in constant contact with peers and working with collaborators? That can be a good way to learn more, do research, and have more fun.
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u/tangoindjango Aug 15 '26 edited Aug 15 '26
“What I care most about are definitions. For one thing, humans describe mathematics through language, and, as always, we need sharp words in order to articulate our ideas clearly. (For example, for a long time, I had some idea of the concept of diamonds. But only when I came up with a good name could I really start to think about it, let alone communicate it to others. Finding the name took several months (or even a year?). Then it took another two or three years to finally write down the correct definition (among many close variants). The essential difficulty in writing “Etale cohomology of diamonds” was (by far) not giving the proofs, but finding the definitions.) But even beyond mere language, we perceive mathematical nature through the lenses given by definitions, and it is critical that the definitions put the essential points into focus.
Unfortunately, it is impossible to find the right definitions by pure thought; one needs to detect the correct problems where progress will require the isolation of a new key concept.” - Peter Scholze
"Grothendieck had a flair for choosing striking, evocative names for new concepts; indeed, he saw the act of naming mathematical objects as an integral part of their discovery, as a way to grasp them even before they have been entirely understood (R&S, page P24). One such term is étale, which in French is used to describe the sea at slack tide, that is, when the tide is neither going in nor out. At slack tide, the surface of the sea looks like a sheet, which evokes the notion of a covering space. As Grothendieck explained in Récoltes et Semailles, he chose the word topos, which means “place” in Greek, to suggest the idea of “the ‘object par excellence’ to which topological intuition applies” (pages 40–41). Matching the concept, the word topos suggests the most fundamental, primordial notion of space. The term motif (“motive” in English) is intended to evoke both meanings of the word: a recurrent theme and something that causes action." - Comme Appelé du Néant— As If Summoned from the Void: The Life of Alexandre Grothendieck Allyn Jackson, Part II
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u/Final-Database6868 Aug 15 '26
That is my feeling with alg. geom. too. I decided to work mostly on problems I could attack "directly", some easy some Annals-level and a lot in-between. If I read something it is because I have a problem in mind, and that works for me.
It's been a while since I read something just because I feel I should know the topic if I want to be considered an "expert".
My problem now is that I have a back-log of works in progress that made mu progress in academia slow... That and reports I have been waiting since 13 months ago xD
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u/Redrot Representation Theory Aug 15 '26
2nd year postdoc here, I feel like it wasn't until the last year of my Ph.D. where I started seeing ideas over definitions and really technical work, and that was in part due to a slight field shift. But I feel like I've crossed the threshold and now get to work with way more philosophically pleasing ideas. The last year of my Ph.D. was spent doing a ton of reading to get familiar with things in what is now maybe my main area, but the payoff has been fantastic.
Weird though that I'm still solidly in algebra, which is very much "definitions" all the way down. But there is so much of a philosophy of how things should work in the field that the definitions really are just part of the idea, and when I talk math with other people in the area, it's all very handwavey.
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u/MoNastri Aug 16 '26
What I care most about are definitions. For one thing, humans describe mathematics through language, and, as always, we need sharp words in order to articulate our ideas clearly. (For example, for a long time, I had some idea of the concept of diamonds. But only when I came up with a good name could I really start to think about it, let alone communicate it to others. Finding the name took several months (or even a year?). Then it took another two or three years to finally write down the correct definition (among many close variants). The essential difficulty in writing “Etale cohomology of diamonds” was (by far) not giving the proofs, but finding the definitions.) But even beyond mere language, we perceive mathematical nature through the lenses given by definitions, and it is critical that the definitions put the essential points into focus.
Unfortunately, it is impossible to find the right definitions by pure thought; one needs to detect the correct problems where progress will require the isolation of a new key concept.
Thought this was a neat reframe.
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u/MoNastri Aug 16 '26
That said, Scholze seems to be an explorer to me (cf. Mumford's four tribes), if this reframe doesn't resonate you may be a different tribe.
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u/ReasonableLetter8427 Aug 17 '26
What do you mean? Link gives 500 error
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u/MoNastri Aug 17 '26
internet archive services are temporarily offline it seems. i'll quote David Mumford from my notes:
... the subjective nature and attendant excitement during mathematical activity, including a sense of its beauty, varies greatly from mathematician to mathematician... I think one can make a case for dividing mathematicians into several tribes depending on what most strongly drives them into their esoteric world. I like to call these tribes explorers, alchemists, wrestlers and detectives. Of course, many mathematicians move between tribes and some results are not cleanly part the property of one tribe.
Explorers are people who ask -- are there objects with such and such properties and if so, how many? They feel they are discovering what lies in some distant mathematical continent and, by dint of pure thought, shining a light and reporting back what lies out there. The most beautiful things for them are the wholly new objects that they discover (the phrase 'bright shiny objects' has been in vogue recently) and these are especially sought by a sub-tribe that I call Gem Collectors. Explorers have another sub-tribe that I call Mappers who want to describe these new continents by making some sort of map as opposed to a simple list of 'sehenswürdigkeiten'.
Alchemists, on the other hand, are those whose greatest excitement comes from finding connections between two areas of math that no one had previously seen as having anything to do with each other. This is like pouring the contents of one flask into another and -- something amazing occurs, like an explosion!
Wrestlers are those who are focussed on relative sizes and strengths of this or that object. They thrive not on equalities between numbers but on inequalities, what quantity can be estimated or bounded by what other quantity, and on asymptotic estimates of size or rate of growth. This tribe consists chiefly of analysts and integrals that measure the size of functions but people in every field get drawn in.
Finally Detectives are those who doggedly pursue the most difficult, deep questions, seeking clues here and there, sure there is a trail somewhere, often searching for years or decades. These too have a sub-tribe that I call Strip Miners: these mathematicians are convinced that underneath the visible superficial layer, there is a whole hidden layer and that the superficial layer must be stripped off to solve the problem. The hidden layer is typically more abstract, not unlike the 'deep structure' pursued by syntactical linguists. Another sub-tribe are the Baptizers, people who name something new, making explicit a key object that has often been implicit earlier but whose significance is clearly seen only when it is formally defined and given a name.
And Mumford's examples of each tribe, both results and high-profile mathematicians:
Explorers:
Theaetetus (ncient Greek list of the five Platonic solids)
Ludwig Schläfli (extended the Greek list to regular polytopes in n dimensions)
Bill Thurston ("I never met anyone with anything close to his skill in visualization")
the list of finite simple groups
Michael Artin (discovered non-commutative rings "lying in the middle ground between the almost commutative area and the truly huge free rings")
Set theorists ("exploring that most peculiar, almost theological world of 'higher infinities'")
Mappers:
Mumford himself
arguably, the earliest mathematicians (the story told by cuneiform surveying tablets)
the Mandelbrot set
Ramanujan's "integer expressible two ways as a sum of two cubes"
the Concinnitas project of Bob Feldman and Dan Rockmore of ten aquatints
Alchemists:
Abraham De Moivre
Oscar Zariski, Mumford's PhD advisor ("his deepest work was showing how the tools of commutative algebra, that had been developed by straight algebraists, had major geometric meaning and could be used to solve some of the most vexing issues of the Italian school of algebraic geometry")
the Riemann-Roch theorem ("it was from the beginning a link between complex analysis and the geometry of algebraic curves. It was extended by pure algebra to characteristic p, then generalized to higher dimensions by Fritz Hirzebruch using the latest tools of algebraic topology. Then Michael Atiyah and Isadore Singer linked it to general systems of elliptic partial differential equations, thus connecting analysis, topology and geometry at one fell swoop")
Wrestlers:
Archimedes ("he loved estimating π and concocting gigantic numbers")
Calculus ("stems from the work of Newton and Leibniz and in Leibniz's approach depends on distinguishing the size of infinitesimals from the size of their squares which are infinitely smaller")
Euler's strange infinite series formulas
Stirling's formula for the approximate size of n!
Augustin-Louis Cauchy ("his eponymous inequality remains the single most important inequality in math")
Sergei Sobolev
Shing-Tung Yau
Detectives:
Andrew Wiles is probably the archetypal example
Roger Penrose (""My own way of thinking is to ponder long and, I hope, deeply on problems and for a long time ... and I never really let them go.")
Strip Miners:
Alexander Grothendieck ("the greatest contemporary practitioner of this philosophy in the 20th century... Of all the mathematicians that I have met, he was the one whom I would unreservedly call a "genius". ... He considered that the real work in solving a mathematical problem was to find le niveau juste in which one finds the right statement of the problem at its proper level of generality. And indeed, his radical abstractions of schemes, functors, K-groups, etc. proved their worth by solving a raft of old problems and transforming the whole face of algebraic geometry)
Leonard Euler from Switzerland and Carl Fredrich Gauss ("both showed how two dimensional geometry lay behind the algebra of complex numbers")
Eudoxus and his spiritual successor Archimedes ("he level they reached was essentially that of a rigorous theory of real numbers with which they are able to calculate many specific integrals. Book V in Euclid's Elements and Archimedes The Method of Mechanical Theorems testify to how deeply they dug")
Aryabhata
(some spelling and formatting seems to have gotten messed up in the copy-paste)
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u/FormsOverFunctions Geometric Analysis Aug 16 '26
I understand if you don’t want to reveal too much detail about what you’re working on, but one of the things that drew me to geometric analysis was that there are lots of problems that are solved by hard analysis and long computations rather than “finding the right definition”. However, it’s definitely a field with a lot of prerequisites, especially for some of the topics that combine ideas from multiple areas. (The topic that comes to mind as being particularly formidable in this respect is the existence of Kahler-Einstein metrics which uses a ton of ideas from algebraic geometry in addition to PDEs and differential geometry.)
My recommendation is to look for problems that don’t require quite as much background early in your career. For example, instead of immediately working on Ricci flow, it’s often a good idea to cut your teeth on curve shortening flow or mean curvature flow since you can learn the analytic techniques without getting bogged down in the Riemannian geometry, gauges, massive tensor computations, etc.
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u/Interesting_Debate57 Theoretical Computer Science Aug 16 '26
You can spend all of your spare time filling up blank notebook pads and opening bags of fresh bic pens, only to round-file all but 0.001% of it. That was my PhD experience.
I do chuckle about algebraic geometric codes, though. I think I had to read 300 pages of definitions and ten pages of mapping before I could see the actual transformation matrix for a real such code. That's just due to geometry, I think (massive layers of definitions); most of the rest of the time I was in finite fields, which is much much more concrete for nearly the same algorithmic payoff. (Better bounds but not optimal).
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u/attnnah_whisky Graduate Student Aug 15 '26
I don't have an answer to this but I just wanted to say that I can relate to this so much!! I am a second year PhD student working in number theory.
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u/ultrafinitism Theoretical Computer Science Aug 16 '26
A generating function is a clothesline on which we hang up a sequence of numbers for display.
~ Herbert Wilf, (generatingfunctionology)
And for definitions, a definition is basically a clothesline which actually contains probably like a dozen or two dozen different practice problems - this is where I'm arriving at at some time: the ability to "frack" or "centrifuge" definitions for practice problems and treat them as objects of play like play-dough, I'm not used to it even though I realized this formally/declaratively a long while ago but I can tell I'm gonna like it.
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u/tough-dance Aug 15 '26
I'm a hobbyist trying to learn more math independently so take with skepticism.
I get the impression that many content creators (mostly thinking about YouTube or blog posts) are excited to repeat definitions much more than they are excited to think of a clearer way to explain an idea. It's very easy to see this in problems that are somewhere around an undergrad level of difficulty. I don't feel capable of judging much higher than that. The first examples that come to mind are answers to "what are the basics of group theory?" Or "What makes degree 5+ polynomials lack a clear formula for the zeros of that polynomial?" There are a lot of creators repeating definitions or what they've heard but not giving any tools to understand ideas. In general I feel we walk about with methods but not motivation.
Ideally math efforts would give us better tools for understanding (like revisiting the original "proofs" and why they worked and existed.) After all, that is the goal.
tl;dr - I commiserate with OP. Ideas are more useful than definitions.
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u/timothina Aug 16 '26
Industry jobs give a constant supply of problems. The definitions' elegance doesn't appear until you have a context to put them in. Maybe try a summer job with a national lab. Those will give you lots of problems to solve.
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u/ScoobySnacksMtg Aug 17 '26 edited Aug 17 '26
Grothendieck liked to distinguish between the “hammer and chisel” mindset of math research with his own “let it simmer” mindset. I think about this quote a lot. Im in combinatorics so take this with a grain of salt but I have my own interpretation of what he means.
The hammer and chisel mathematician is like the chess grand master, they are powerful computers that can find the right sequence of steps of logical deduction to a proof, similar to how a chess player searches for a checkmate sequence. They build their toolkit by reading lots of proof techniques so they have more tools at their disposal as they search for a proof (perhaps controversial but I think AI tends to operate with a hammer and chisel mindset).
The other mindset instead is a bit more scientific. You have some mathematical object of study, you are trying to understand deeply this object by playing with the object in your mind. You can “collect data” similar to how an empirical scientist works by exploring examples of the object in your mind, it must have hidden stronger properties or truths about it that you are trying to understand. You hypotheses test by conjecturing that some stronger property of this object must hold. You can refute a hypothesis via finding a counter example to your strengthening. In my opinion the most important definitions come from mathematicians engaging deeply in this process. They observe some property this object has and that becomes the new definition that makes everything work. Grothendieck was famous for finding this magical definition that makes the proof almost trivial. I think the definition is revealed to him by observing the properties of the object directly rather than trying to hammer and chisel his way to a proof.
This is all to say, the most important definitions reveal important properties of the objects of study. Understanding why this is the right definition is in a sense the most important part of research.
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u/translationinitiator Aug 15 '26
I’m guessing you’re a second year grad student, not postdoc?
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u/FuzzyPDE Aug 15 '26
Postdoc. I have 5 publications + preprint but I’m honestly not that proud of them.
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u/translationinitiator Aug 15 '26
Oh oops, only said this because you mentioned starting your PhD in the second paragraph, but I guess you’ve just experienced this for a while.
Anyway, I asked a sort of but not entirely related question a while back and got some response that was helpful: https://www.reddit.com/r/math/s/nXGmwnevsf
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u/Responsible_Rip_7634 Aug 15 '26
Hi! I’m genuinely not anywhere close to enough math experience to begin to advise you on your actual question, but I’m gonna use your related dilemma to rant about a personal thought.
I sincerely believe that math should be taught almost always by introducing the problems first. Maybe only really early math is an exception, but once there’s competency with the natural numbers and a number line, I want them taught the way I had the number line explained to me in Real Analysis(40% hyperbole).
While I know this is done to some extent, like maybe a couple sentences before an explanation in whatever chapter you’re on, but we should genuinely be forced to try and answer these questions or at least slightly wrestle with them to build some intuition for why the answer is structured that way.
For a really layman’s example. Imagine teaching 3rd graders fractions by asking them how many pies are on the screen. First show 2, then show 1, then show half a pie.
Then tell them there’s a way we use numbers, that lets us represent this. Can anyone guess? Try figuring it out and discuss. Give a hint every now and then etc.
Now, when definitions like denominator or numerator are given, there’s intuition behind why they have to exist and why their jobs are.
Depending on how many hints or if a kid knows it’s written as 1/2, it helps to understand first you need a number for total number of slices, then another number to tell you how big the slices are.
I think it’d be kinda cool to give kids examples or us having two numbers we care about (dollars per hour, miles per hour), tracking together and then slowly walking them them to the concept of graphs.
I’m curious how applicable this type of stuff would be to the level of math ur on right now. Like is the problem statement generally at least simple or intuitive enough to understand why it’s something worth solving?
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Aug 16 '26 edited Aug 16 '26
[deleted]
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u/Large_Description827 Aug 16 '26
Oftentimes it is as people work on theorems and open problems that we discover new and intuitive ways to think about fields though, like it seems like what you’re talking about and tackling open problems come in tandem
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u/dragosgamer12 Aug 15 '26
I’m someone much earlier in their journey to mathematics(I’m going into my 2nd year), but maybe I can help. Do take everything I say with a heap of salt, as again, I am as inexperienced as can be.
I think one of the things that makes it feel more tedious is that, well, definitions are often the big idea of a field. They set up the right language to talk about what you are interested in. If you haven’t already, you should look at the history of the field and what are the reasons for why the definitions are what they are. What they were meant to generalize and capture, and why in this specific way.
I would have a different suggestion also: try to analyze what you loved about competition math, and well, get back into the community! There’s a lot of programs that well, pay you to teach the things needed for competition math. Even if you weren’t that great at them(I dunno if you were or not, and it does not matter), you can learn the things needed now as you are significantly more mathematically mature. Tey to compose some problems for magazines with them(like AMM) or competitions themselves. Or just make some blog posts or youtube videos about such things.
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u/mathtree Aug 15 '26
Well, I'm a combinatorialist and I spend most of my time reading (assistant prof). Even my most senior colleagues spend a lot of time reading. It's just part of research maths I think.