How to “memorize” a theorem

[@JJGhostHunters]

Just finished a grueling PhD program in electrical and computer engineering. When you said there were "optional" enrichment courses that you pretty much had to complete, it reminded me of the 3 published papers that were never mentioned in my curriculum and turned out to be "required" before submitting my final dissertation.

[@autumnfox3619]

How different was it compared to MSc/MEng in ECE? I am on my 2nd year in Electrical and Computer Engineering, thinking about PhD myself

[@IsomerSoma]

I do diagrams of how the theorems (and structures in general) relate to each other and what their application is. By developing such a network of relations they not only stick, but you know the chain of theorems and concepts leading up to a theorem too. Almost no repetition needed. Adding some visuals makes such a network even stickier. Making such a diagram is quite hard as one has to chunk efficiently, intuitively and logically, one has to figure out what the most important relationships are and how to spatially arange all of this. Do what you do if you want, but imo what you seem to be doing doesnt seem to be efficient. The order memorizing then understanding and finally getting big picture is skewed. Having some idea of the big picture first makes learning the details *greatly* more efficient.

[@stevencvisuals]

ditto on this.

[@Alex-fh4my]

I wrote a comment talking about this as well, I think this approach probably isnt the best but you should consider this: 'I think the idea here is that when you first see a proof it can be difficult to "store" all of it in your short term memory. If you can't store the whole thing, you have to look at it piece by piece, which doesn't let you see the full picture. This way you at least commit the proof to memory, and then once its in your mind you can play around with it and manipulate it until you understand the bigger picture' Though overall I agree with you. This would probably be a last resort kinda thing

[@tear728]

The mind map strategy for something like math is sort of clumsy but this seems like a good idea

[@tear728]

​@@Alex-fh4mydivide and conquer

[@IsomerSoma]

@@tear728 It indeed is difficult, but the reason this is effective is the difficulty. It forces you to order your knowledge in the most simple and comprehensive way.

[@JoaoSilva-yz4zq]

I would like to add, in case you did not know, that Fatou's Lemma, the Dominated Convergence Theorem and the Monotone Convergence Theorem are all equivalent results. In other words, any of the these three results can be proved using any of the others.

[@caiodavi9829]

brasil porra!

[@Freddy-ss7mb]

Brasil zil zil zil tantantantan

[@soyoltoi]

In order to understand, you first have to have it in your head. If you can efficiently upload it into your head first, then you can set yourself up for successful understanding as you make connections between these theorems and concepts afterwards or as you go.

[@eliasmai6170]

i don't really memorize theorems. i do a lot of the exercises which make use of the theorems so that by the end, the theorem sticks in my mind.

[@_mario_2985]

Do you find these exercises online by just googling or do you get enough exercises for the major theorems in class?

[@_mario_2985]

And also how do you study smaller stuff like propositions? Often they dont have a name but some are still important to know. I often cannot really find exercises to such propositions so I just try to think of exercises of my own

[@artophile7777]

That's a really good method if you're doing lite maths but if you're taking an advanced maths course like analysis, there's just too many theorems and lemmas to do questions on, so memorization comes in handy Edit: Of course memorize only when you understand

[@jakobrhein8684]

I have the same approach, but i am not doing pure math. I just have a crash course in analysis and linear algebra called 'math for physicists 1-3' and i early gave up to understand every (any😅) proof provided in the script. There is simply not enough time for doing that, because you dont need to understand every proof for doing the exercises, which are necessary for being allowed to write the exam.

[@tobiassander3995]

But that won't help much when you're asked to state the theorem rigorously.

[@Alex-fh4my]

I think the idea here is that when you first see a proof it can be difficult to "store" all of it in your short term memory. If you can't store the whole thing, you have to look at it piece by piece, which doesn't let you see the full picture. This way you at least commit the proof to memory, and then once its in your mind you can play around with it and manipulate it until you understand the bigger picture. Typically though its much easier to remember something if it is tied to other ideas conceptually, instead of just an arbitrary abstract sequence of symbols (extract meaning vs. rote learn). But in the case of something like this I think the approach is fair, since its hard to build the conceptual understanding in a single time step

[@CoronaryArteryDisease.]

Calculus is as far as I have gone in math (well, “far” is relative, and I have done a good amount of statistics) but I briefly dabbled in proofs, and this is quite interesting to hear your approach. Maybe I should dabble some more lol

[@JohnH777]

This is very helpful, especially since I've been slacking off on studying Royden, which has a lot of the theorems you mentioned. I also wish my uni has summer enrichment courses.

[@asbjrnandersen4765]

You really do need to memorize things at least in my experience. Most of my math exams were exams where you draw a random theorem which you then need to prove to the professor. Only way to prepare for such an exam is literally remembering the proofs....

[@advaypakhale5254]

This is awesome! I do something similar. For each topic, I have an excel sheet that I fill up under two columns - question and answer. The questions might be of the form "What is the definition of ?" or "State and its proof" or really any other non-trivial trick/technique/application which I know I cannot reproduce on my own if I were asked to (because I'm mortal). In the answer column, I then give a sketch of the answer in the answer column in no more than a short paragraph, just like what you did in the video; I focus on the key details of the proof/idea that would allow me to reproduce the argument on my own reliably. I try to add to this excel sheet as dilligently as possible during the time I'm reading the module or textbook. I've found that adding as you go along is a lot more efficient as opposed to leaving all the work to the end of a module, although it comes with the drawback of not seeing the big picture which sometimes leads to focusing on the wrong details. I then load this excel sheet up into Anki (an awesome spaced repetition flashcard program) and do my flashcards whenever I get the time, like on the bus. Anki takes care of making sure that I am seeing each flashcard just as frequently as I need to in order to not forget it, courtesy of its algorithm. By the time I get to the exam for each module, I often find that many of the concepts have already stuck, and I will just run through all the flashcards one more time for good measure. Using Anki has entirely changed the way I interact with knowledge and memorisation, and I hope others will find it useful too!

[@Dr.1.]

Such helpful advice

[@oo_rf_oo8824]

When I first leant about Vitali Covering theorem (the infinite version) and Brunn Minkowski inequality which comes after it shortly, I was totally stunned; like how do you even comprehend this thing, let alone memorize it. But I guess by knowing the gist of the theorem would make it easier to pull out the prove from memories.

[@DavidGillooly]

Nice view. I do think the idea of doing it and then take a break .....is a good process that I have usued with some success....

[@Dr.1.]

Dude never stop

[@AffaanMustafa]

man i’m just tryna pass real analysis rn got my final in 2 weeks pray 4 me. then gotta take pt 2 right after if i pass and if i pass that i’m chillin until grad.

[@hunningtonnuo9213]

Did you pass?

[@LGcommaI]

Some, though of course by far not all, authors reserve 'family' for SEQUENCES of sets; in particular, the possibility to have the same set several times in a family is characteristic of the concept 'family'. (Multiplicity) I would recommend to always use the term 'hypergraph' for 'set of sets'. (This might grate of the ear of many an analyst, but ''hypergraph' is an ever more standard term.)

[@ilirdemiri2496]

May you suggest me a book, which is the best book, according you, with problems solution in metric spaces?

[@cainmorano4956]

You don't memorize, you Internalize. When you were a kid, like you said, you memorized. But now you work with a lot of content and eventually you start to remember and cold recall facts. More importantly you intuitively see relationships among the written symbols and understand the how and why across a broad network of ideas.

[@capefry8323]

Low Iq question i'm about to ask, but isn't real analysis just proofs of calculus and only calculus, or am i mistaken?

[@muhammadmustafa2946]

thanks for the vid!

[@GastroenterologyPINNs]

Are there every theorems about real analysis in one book? (or just the main theorems)

[@blinderfleck1625]

balls, a lot of balls haha

[@Math.A-level-student]

Good video 👍

[@rexraptor1094]

This is actually a good idea. I might try it out.

[@iraphii2023]

Why is real analysis a topic of study in a Ph.D.? In Germany, we cover Analysis 1, 2, and 3 in the first three semesters of the Bachelor's program, followed by more advanced mathematics during the Master's.

[@iraphii2023]

We have this theromes in the first 3 Semesters in Bachelor program, i dont understand this xd

[@iraphii2023]

For example, here is a sheet for Bachelor 2. to 3. Semester www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.010/Le/Analysis3_WS1819/Sheet3_Analysis3.pdf

[@timotejstibilj3450]

These theorems are probably just prerequisites for PhD

[@halminnesota699]

Truly the study of real analysis just stops after the courses you take! Nothing after that, real analysis is a complete field, we have brilliant German scientists to show us that thank god.

[@okoyoso]

Measure theory is usually not taught in undergrad in the US.

[@jennifertate4397]

Are "Balls" and "Spheres" the same thing?

[@erikysilvagomes5496]

I struggle with this question for a long time and got a satisfactory answer imo. For all kind of knowledge you want to acquire it is relevant to clearly define your objective, and this is not trivial. Why did anyone wants to memorize a theorem? This actually improves learning and understanding it? Even more clearly: when I know that I learned something? It is quite common for me to consider that I learned something when I was able to solve problems and explain something. So this is my objective: be able to solve problems and explain something I want to learn. Since your objective it's quite clear you can propose some techniques to reach it. The techniques must be evaluated with two criteria: specificity and generativity. The first criterion its the most relevant: the technique must be aligned with your objective. The second one helps efficiency: the more you are required to generate new information, the more efficient the technique is. In my example, the trivial strategy could be try to understand the theorem part by part and explain it for me without seeing it, or rephrasing in another words. Last, but not least I emphasize the crucial question: why would you like to memorize a theorem? This is the final objective or it is just a hidden (ineffective) step to reach the real goal (to learn)? Something that becomes clear for me is that you often don't want to memorize, but just learn, so what it "to learn" for you? P.S.: you can relate this reasoning with this paper: DOI: 10.1080/00461520.1986.9653022.

[@constantin2449]

very good Video

[@BadAss_691]

Massive respect struggling bro. Just sit all day long with pen paper and books and discover something new.

[@Stopinvadingmyhardware]

Write out the verbose explanation of each step of the theorem and derivative that actually matters.

[@GastroenterologyPINNs]

The problem is I’m under average....

[@__hannibaal__]

Never and Never Memorize Theorems; understand by practice and by inventing counter examples.

[@caiodavi9829]

bro is on elementary school giving out some tips to a phd student 💀💀

[@tekgewet]

@@caiodavi9829he even explained this in the video 💀

[@uzay394]

Wheedens measure and integration in the background… triggered *screams*

[@SteveJian]

I suppose he is holding the camera in one hand with the other hand doing the paperwork.

[@devjyoti5614]

Is there any minkowskis space in real analysis?🙃

[@mastershooter64]

sure, Minkowski space is just a metric space with the Minkowski metric :) in real analysis you would use a general metric, d(x, y) and you could set that to the Minkowski metric

[@alexeybychkov2742]

@@mastershooter64 Minkowski metric is not a metric in a strict sense, since d(x, y) = 0 implies x and y lie in the same lightcone rather than x = y, which it would do if it was a metric. So, Minkowski metric is rather a pseudo-metric (doesn't require nondegeneracy). What is mentioned in the 3rd line, Minkowski inequality, is not really related to Minkowski space (as far as I can tell). Though, it is due to the same Minkowski :)

[@finmat95]

Memorizing theorems is so stupid.

[@laplacesdemon82]

Why u never show face

[@AlexandrBorschchev]

Because not everyone wants their face on camera, stop asking this stupid question.

[@eternaldreamerofsleep]

His own privacy is to be respected

[@josh34578]

Does he need to show face to prove it exists?

[@laplacesdemon82]

Just curious 😢