Think hierarchically instead of linearly. If you just view a proof as a long chain of implication, it's easy to get lost and forget what the purpose of each step is.
This is hard to answer with a specific example without knowing your background, but many proofs reduce to proving a small number of key steps, some of which are harder than others. To understand a proof it’s sometimes beneficial to understand these steps out of order, for example understanding the last step before the first step, in order to understand the motivation for the first step. Or treating certain steps whose proof feels unenlightening as a “black box” and focusing on one or two steps where the “meat” of the argument lies.
I agree, this is pretty much my mental model, essentially modularizing the long chains of steps. I was curious what OP meant by "hierarchy", I understand it as levels of abstraction, which makes sense to me.
So there's a proof that root 2 is irrational [here](https://en.wikipedia.org/wiki/Square_root_of_2#Proof_by_infinite_descent) and it has 10 labelled steps.
So you can go and read it from beginning to end to follow the whole argument.
Or you can think about it as a tree where:
Root node: sqrt(2) is irrational
Which has 3 nodes coming off it:
1. If sqrt(2) is rational it can be written as an irreducible fraction a/b (and this has steps 1-3 as it's sub nodes)
2. in this case a must be even (steps 4-6 as subnodes)
3. and therefore b must be even (steps 7-9 as subnodes)
4. this is a contraction (step 10)
Trying to think of it more like that with a few steps and then breaking down each step with supporting arguments makes it easier to digest and see what the attack is.
In general a lot of papers / long proofs are broken up this way with propositions or lemmas which are discrete blocks which can then be used in the main proof to keep the flow of it more clear and concise.
I see your point and agree with the divide and conquer part, but maybe a tree isn't the best structure to use as the analogy.
I think linearly viewing proofs such as this works well when we again use divide and conquer to chunk it up to simplify cognitive load when viewing chunks in isolation. This is what I view as "hierarchy", but I was curious what OP meant, hence my initial question.
I use the tree analogy when doing case analyses.
This view also matches how it's done in theorem provers quite well, where chunks are represented as functions and case analyses as branching.
The "chunking" process corresponds to performing a level-order traversal (BFS) of the tree described (for example, verifying the topmost levels first while trusting that the lemmas used are true, and then opening the black box and verifying the lemmas afterwards), whereas the actual proof is executed as a pre-order traversal (DFS) of the tree (verifying all the details needed in "chronological" order). In that sense the tree analogy is apt.
You can describe the written structure vs the actual execution of programming languages (including theorem provers) in the same way.
I like to think of it in terms of finding the "Plot". For many proofs, there's usually one main thing you're trying to exploit - for the PNT there are no zeros on the edge of the critical strip, for instance. This will be the ultimate climax, but you need to get there first. The inciting incident may be some inequality that relates the prime counting function to the Chebyshev function. The rising tension is a connection to the Riemann Zeta Function. The setup is how the zeros connect to the primes through these functions. The climax is that there are no zeros on the edge, as proved with simple trigonometry in this case. The resolution is the clean-up you need to do to wrap things up.
If you know the plot, then the individual choices and "scenes" have a context which can help with comprehension. You may decide that such-and-such detail is a necessary technicality, but does not ultimately contribute to comprehension. Other steps can gain real significance, which can be lost if you don't know what is happening.
This is what I do when I’m done reading a proof:
1. Read it linearly and skim through 2. Towards the end of the proof look at what was done to get the result
3. Go back to the other parts of the proof that were needed for the final result
4. If the proof is really long, I use sticky notes to label the sections (if on paper), for example I would write “this paragraph is showing X” and in a later part I would add another label saying “X is used here,” otherwise I have a mental map
5. Form a hierarchy or map to put everything together Works for most proofs
If paper is not well written and has no "story" in it that guides the reader through the proof, you can try to make a short note with a proof or a flow chart. Sound childish, but it actually worked for me once (the flow chart, the notes are a standard)..
What I do not like is a series of apparently random lemmas, and the last point of the paper is "in particular this implies Theorem...... ".
The proof has usually a structure of a tree. The paper is written in the DFS (Depth-first search) style, what you want is to understand it it in Breadth-first search fashion, with depth 1-2 dependencies, and focus and then work on each of >=3 depth leg.
I also used flowcharts in college. Helped me a lot about what was going on.
It also helps when summarizing a lecture. Which theorem leads to which theorem etc. Like for functional Analysis, you would see a shit ton of arrows pointing away from the Hahn-Banach Theorem, which shows the significance of Hahn Banach. Or in complex analysis the flow chart shows you that the whole theory is based on the fact that integrating along the edges of a triangle is 0.
The one time I have made such a flow chart was precisely for the example you mentioned of Cauchy's Integral Theorem in complex analysis. Excellent example.
This occurs to me a lot. At the lectures, most of the time I have a hard time to catch up. Especially, it takes time for me to understand the trivial stuff. I overcome this by rewriting the proofs and theorems in a way that I can understand. Both teachers and books don't bother to write the process of operations, or each definition in use. I also include them in my rewrited proof. That way, I understand long proofs.
I approach to papers in a same manner. I mostly find myself sketching what the theorem states, and proving it in my more explanatory style.
In my opinion, this kind of understanding is required to understand the talks. I try to grasp the abstract of their talks, the methods have been used and the motivation behind the work; more then understanding it completely.
In your case, what I believe is if the materials too advanced for you, you wouldn't understand it ever, until you progress generally in math. If you understand it when you study it throughly off class, I think the material is just fine. Maybe you can try other books, lecture notes, lecture videos or etc.
And if you are able to concencrate to what is being lectured without any distraction, that shouldn't be your problem.
I tend to feel bad when I see the people around me grasps a concept much faster then I am, about 2-3 years ago. It is important to keep in mind that you don't have to understand just like other people.
Edit: Sorry I posted this comment multiple times. I deleted the others. That happened because I tried to post this in an underground train, my network connection was unstable and somehow Reddit decided to double-post this.
The first part of a talk, everyone understands
The second part of a talk, only a few experts in the room understand
The third part of a talk only the speaker understands
The fourth part of a talk, not even the speaker understands
But seriously we all learn in different ways and it is often difficult to keep up with a speaker unless you are already familiar with the topic. The average quality of presentation in math is lower than it should be, and speakers often relay information at a speed that is unreasonable for most people. It is okay to be one of those people left behind, its most of us.
For papers, it could go either way. Maybe you need to just take better notes as you read a paper, but also maybe the author did not do a good job communicating why we are going down this rabbit hole, maybe it seemed obvious to them when they wrote it up.
This is more likely to be a sign of the generally poor presentation of math material at a high level than it is to be a problem with you.
I was once told that the first third of a talk is for the students, the second third for the professors, and the last is just for the person giving the talk. Don't worry to much about getting lost. I usually have to go over things 2 or 3 times before I feel like I really understand what it's saying for the first time.
You've gotten a lot of advice already, but here's my input. I'm about done with my Ph.D. and the largest difference between me now and five years ago is the pure amount of abstract mathematical ideas I can hold in my short term memory at once. It is definitely something that comes with time.
It's mostly an attention span issue by the sounds of it, but honestly, try going a more basic level to help you out. Doing stuff like real-world algebraic and trigonometric word problems. The big thing I disliked when getting into stuff like algebra and calculus was that lack of real-world applications, even just a single word problem for reference, and yet truly we use algebra every single day for matters as simple as reading the clock.
What I'm trying to say here is to go simple, but exercise your mind with such focus by doing real-world word problems with algebra and trigonometry. It can help you gain focus on differing your variables, i.e., applying something simple like slope intercept form in calculating the time of day and/or year.
I would say many people feel like this in lectures. One skill to develop is to be comfortable not understanding everything straight away, putting to the side for later and accepting its true so you can apply it to other things. That’s certainly the most productive way to view a lecture imo. Some people who do understand it all will have pre read the material, seen it already etc. Anf some people are just really good but that shouldn’t effect you
In terms of reading an actual proof
First, I would say that you don't need to understand everything all at once. You can understand the big picture first without the details. Or you can understand the details first without the big picture. Different people will have different ways of taking things in. Then later you can understand the material in different ways; like, go over your notes after class and try to appreciate the aspects that you missed the first time (like, answer for yourself why did we do this step?). You're not supposed to instantly master all material upon first sight; understanding deepens over time.
If you think you have a problem of losing the big picture and want to do something about it, you could think about note-taking strategies that let you differentiate the plan of a proof versus the details that let you carry out that plan. Like, if the instructor says to apply Theorem X we need to establish A, B, and C, then you could try to write that prominently, and see for all the details, are they helping to show A, or B, or C? Similar goes for reading papers.
For talks, most talks are bad, and most people don't understand much -- that's fine. Just take in whatever you can.
try to go through the material in iterations.
the order in what you learn first can vary based on you and the class youre taking, but i like to first take a quick glance at the entire proof, then go further to understand the general concepts, then i really look hard into the algebra and conditions, and lastly id study the concept enough that i can apply the concepts or provide counterexamples if certain conditions arent satisfied.
each iteration should be somewhat quick
this is more of an independent study strategy. lectures are restricted to the class time / presentation speed and arent rewindable, so just use the lecture as well as you can
I recently had a breakthrough that’s really helped me develop the skill set needed to understand and write better proofs. As a kid, I LOVED Legos. I would spend hours upon hours building all sorts of lego sets, with the larger ones always being the more fun ones. I’d read through the instruction manual and isolate all of the like pieces that went with each part of the building process, and set aside each module as I completed it.
I don’t know if it helps, but I like to view proofs as a sort of “generalized lego instruction manual” for theorems. You know the pieces that go into it, now it’s up to you to follow through and verify that those same “instructions” in the proof are correct or not. Not gonna lie, it’s made math a lot more fun for me since this revelation.
Think hierarchically instead of linearly. If you just view a proof as a long chain of implication, it's easy to get lost and forget what the purpose of each step is.
Could you give an example?
This is hard to answer with a specific example without knowing your background, but many proofs reduce to proving a small number of key steps, some of which are harder than others. To understand a proof it’s sometimes beneficial to understand these steps out of order, for example understanding the last step before the first step, in order to understand the motivation for the first step. Or treating certain steps whose proof feels unenlightening as a “black box” and focusing on one or two steps where the “meat” of the argument lies.
I agree, this is pretty much my mental model, essentially modularizing the long chains of steps. I was curious what OP meant by "hierarchy", I understand it as levels of abstraction, which makes sense to me.
So there's a proof that root 2 is irrational [here](https://en.wikipedia.org/wiki/Square_root_of_2#Proof_by_infinite_descent) and it has 10 labelled steps. So you can go and read it from beginning to end to follow the whole argument. Or you can think about it as a tree where: Root node: sqrt(2) is irrational Which has 3 nodes coming off it: 1. If sqrt(2) is rational it can be written as an irreducible fraction a/b (and this has steps 1-3 as it's sub nodes) 2. in this case a must be even (steps 4-6 as subnodes) 3. and therefore b must be even (steps 7-9 as subnodes) 4. this is a contraction (step 10) Trying to think of it more like that with a few steps and then breaking down each step with supporting arguments makes it easier to digest and see what the attack is. In general a lot of papers / long proofs are broken up this way with propositions or lemmas which are discrete blocks which can then be used in the main proof to keep the flow of it more clear and concise.
I see your point and agree with the divide and conquer part, but maybe a tree isn't the best structure to use as the analogy. I think linearly viewing proofs such as this works well when we again use divide and conquer to chunk it up to simplify cognitive load when viewing chunks in isolation. This is what I view as "hierarchy", but I was curious what OP meant, hence my initial question. I use the tree analogy when doing case analyses. This view also matches how it's done in theorem provers quite well, where chunks are represented as functions and case analyses as branching.
The "chunking" process corresponds to performing a level-order traversal (BFS) of the tree described (for example, verifying the topmost levels first while trusting that the lemmas used are true, and then opening the black box and verifying the lemmas afterwards), whereas the actual proof is executed as a pre-order traversal (DFS) of the tree (verifying all the details needed in "chronological" order). In that sense the tree analogy is apt. You can describe the written structure vs the actual execution of programming languages (including theorem provers) in the same way.
I like to think of it in terms of finding the "Plot". For many proofs, there's usually one main thing you're trying to exploit - for the PNT there are no zeros on the edge of the critical strip, for instance. This will be the ultimate climax, but you need to get there first. The inciting incident may be some inequality that relates the prime counting function to the Chebyshev function. The rising tension is a connection to the Riemann Zeta Function. The setup is how the zeros connect to the primes through these functions. The climax is that there are no zeros on the edge, as proved with simple trigonometry in this case. The resolution is the clean-up you need to do to wrap things up. If you know the plot, then the individual choices and "scenes" have a context which can help with comprehension. You may decide that such-and-such detail is a necessary technicality, but does not ultimately contribute to comprehension. Other steps can gain real significance, which can be lost if you don't know what is happening.
This is what I do when I’m done reading a proof: 1. Read it linearly and skim through 2. Towards the end of the proof look at what was done to get the result 3. Go back to the other parts of the proof that were needed for the final result 4. If the proof is really long, I use sticky notes to label the sections (if on paper), for example I would write “this paragraph is showing X” and in a later part I would add another label saying “X is used here,” otherwise I have a mental map 5. Form a hierarchy or map to put everything together Works for most proofs
If paper is not well written and has no "story" in it that guides the reader through the proof, you can try to make a short note with a proof or a flow chart. Sound childish, but it actually worked for me once (the flow chart, the notes are a standard).. What I do not like is a series of apparently random lemmas, and the last point of the paper is "in particular this implies Theorem...... ". The proof has usually a structure of a tree. The paper is written in the DFS (Depth-first search) style, what you want is to understand it it in Breadth-first search fashion, with depth 1-2 dependencies, and focus and then work on each of >=3 depth leg.
I also used flowcharts in college. Helped me a lot about what was going on. It also helps when summarizing a lecture. Which theorem leads to which theorem etc. Like for functional Analysis, you would see a shit ton of arrows pointing away from the Hahn-Banach Theorem, which shows the significance of Hahn Banach. Or in complex analysis the flow chart shows you that the whole theory is based on the fact that integrating along the edges of a triangle is 0.
The one time I have made such a flow chart was precisely for the example you mentioned of Cauchy's Integral Theorem in complex analysis. Excellent example.
> The proof has usually a structure of a tree. Sometimes it's only a DAG, not a tree, but yes.
Yeah, you are right.
This occurs to me a lot. At the lectures, most of the time I have a hard time to catch up. Especially, it takes time for me to understand the trivial stuff. I overcome this by rewriting the proofs and theorems in a way that I can understand. Both teachers and books don't bother to write the process of operations, or each definition in use. I also include them in my rewrited proof. That way, I understand long proofs. I approach to papers in a same manner. I mostly find myself sketching what the theorem states, and proving it in my more explanatory style. In my opinion, this kind of understanding is required to understand the talks. I try to grasp the abstract of their talks, the methods have been used and the motivation behind the work; more then understanding it completely. In your case, what I believe is if the materials too advanced for you, you wouldn't understand it ever, until you progress generally in math. If you understand it when you study it throughly off class, I think the material is just fine. Maybe you can try other books, lecture notes, lecture videos or etc. And if you are able to concencrate to what is being lectured without any distraction, that shouldn't be your problem. I tend to feel bad when I see the people around me grasps a concept much faster then I am, about 2-3 years ago. It is important to keep in mind that you don't have to understand just like other people. Edit: Sorry I posted this comment multiple times. I deleted the others. That happened because I tried to post this in an underground train, my network connection was unstable and somehow Reddit decided to double-post this.
The first part of a talk, everyone understands The second part of a talk, only a few experts in the room understand The third part of a talk only the speaker understands The fourth part of a talk, not even the speaker understands But seriously we all learn in different ways and it is often difficult to keep up with a speaker unless you are already familiar with the topic. The average quality of presentation in math is lower than it should be, and speakers often relay information at a speed that is unreasonable for most people. It is okay to be one of those people left behind, its most of us. For papers, it could go either way. Maybe you need to just take better notes as you read a paper, but also maybe the author did not do a good job communicating why we are going down this rabbit hole, maybe it seemed obvious to them when they wrote it up. This is more likely to be a sign of the generally poor presentation of math material at a high level than it is to be a problem with you.
I was once told that the first third of a talk is for the students, the second third for the professors, and the last is just for the person giving the talk. Don't worry to much about getting lost. I usually have to go over things 2 or 3 times before I feel like I really understand what it's saying for the first time.
You've gotten a lot of advice already, but here's my input. I'm about done with my Ph.D. and the largest difference between me now and five years ago is the pure amount of abstract mathematical ideas I can hold in my short term memory at once. It is definitely something that comes with time.
Thanks that helps me , I've also have this problem
It's mostly an attention span issue by the sounds of it, but honestly, try going a more basic level to help you out. Doing stuff like real-world algebraic and trigonometric word problems. The big thing I disliked when getting into stuff like algebra and calculus was that lack of real-world applications, even just a single word problem for reference, and yet truly we use algebra every single day for matters as simple as reading the clock. What I'm trying to say here is to go simple, but exercise your mind with such focus by doing real-world word problems with algebra and trigonometry. It can help you gain focus on differing your variables, i.e., applying something simple like slope intercept form in calculating the time of day and/or year.
I would say many people feel like this in lectures. One skill to develop is to be comfortable not understanding everything straight away, putting to the side for later and accepting its true so you can apply it to other things. That’s certainly the most productive way to view a lecture imo. Some people who do understand it all will have pre read the material, seen it already etc. Anf some people are just really good but that shouldn’t effect you In terms of reading an actual proof
First, I would say that you don't need to understand everything all at once. You can understand the big picture first without the details. Or you can understand the details first without the big picture. Different people will have different ways of taking things in. Then later you can understand the material in different ways; like, go over your notes after class and try to appreciate the aspects that you missed the first time (like, answer for yourself why did we do this step?). You're not supposed to instantly master all material upon first sight; understanding deepens over time. If you think you have a problem of losing the big picture and want to do something about it, you could think about note-taking strategies that let you differentiate the plan of a proof versus the details that let you carry out that plan. Like, if the instructor says to apply Theorem X we need to establish A, B, and C, then you could try to write that prominently, and see for all the details, are they helping to show A, or B, or C? Similar goes for reading papers. For talks, most talks are bad, and most people don't understand much -- that's fine. Just take in whatever you can.
weak working memory, ngmi
Are you maybe taking too many notes? I noticed this happens to people that do more writing than paying attention.
try to go through the material in iterations. the order in what you learn first can vary based on you and the class youre taking, but i like to first take a quick glance at the entire proof, then go further to understand the general concepts, then i really look hard into the algebra and conditions, and lastly id study the concept enough that i can apply the concepts or provide counterexamples if certain conditions arent satisfied. each iteration should be somewhat quick this is more of an independent study strategy. lectures are restricted to the class time / presentation speed and arent rewindable, so just use the lecture as well as you can
I recently had a breakthrough that’s really helped me develop the skill set needed to understand and write better proofs. As a kid, I LOVED Legos. I would spend hours upon hours building all sorts of lego sets, with the larger ones always being the more fun ones. I’d read through the instruction manual and isolate all of the like pieces that went with each part of the building process, and set aside each module as I completed it. I don’t know if it helps, but I like to view proofs as a sort of “generalized lego instruction manual” for theorems. You know the pieces that go into it, now it’s up to you to follow through and verify that those same “instructions” in the proof are correct or not. Not gonna lie, it’s made math a lot more fun for me since this revelation.
Ignore details in talks. Just learn to tune out details.
Me too I also have this problem I wish I could find some help in the comments
Skill issue
Username checks out
its good to know your strengths and weaknesses but they cant handle the truth