Petra SchwerWhat does it mean (to you) to understand a piece of mathematics?
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Bigfood@PetraSchwer
Do you need to solve this to get a room in the student dorm?
@Bigfood Go for it!
Boyd Stephen Smith Jr.@PetraSchwer If I "recognize" it in another context, OR if I feel I can recapitulate all but the "hairiest" / most verbose parts of a proof from memory.
Martin EscardoI don't know what it means to understand a piece of mathematics.
But I tell my students that if they don't find any mistakes in my lecture notes, this is a symptom that they don't understand the material. To understand it, they must find the mistakes and be able to fix them on their own. 🙂
(I use GitLab for the lecture material, and the students are encouraged to do pull requests to fix mistakes in the the lecture notes, and they do.)
@MartinEscardo that is a good recommendation. A bit like using some properties of an object without understanding its definition.
Friedrich Wilhelm Grafea lot, in the cases I try to ... 😅
and think understand a little bit, when grasping a shade of say flawlessness, compactness, self containedness of a respective proof
Fabian Transchel@PetraSchwer It obviously depends on a multitude of aspects:
Whether the statement is "useful" for me in a contextual or professional way.
Whether the line of reasoning is elegant w.r.t. my own "maths aesthetics".
Whether it matches my intuition about how the concepts in question "should be" related - i.e. whether I "like" the result 😛
How about you?
mathegudrun (she/her)@PetraSchwer funny that as answer to your question I was solely thinking about feelings I have when some pieces in a proof Fall into the right place…
@mathegudrun I love that! Understanding is something very personal (and human ) I think.
Andrew Hastings@PetraSchwer An experience I remember clearly from studying quantum physics as an undergraduate was being able to relate the mathematics to a physical phenomenon - even if it wasn't something that I could observe directly. Understanding those relationships really helped the mathematics slot into place, and vice versa.
Atif@PetraSchwer on first thought, "understand" has two obvious meanings for me: 1. plain understand and 2. internalize
if I understand a piece of mathematics, that means I'm not nervous about it, there's no sense that something's missing
To internalize, on the other hand, means that I can use it freely, or reconstruct it from an emotion or a very small amount of informations or starting assumptions. In essence, if I succeed in effectively compressing something, while being confident that I can fully and reliably unpack it later
Dr. Victoria Grinberg@PetraSchwer one sees that I am a physicist: it's about being able to imagine what it means and how it works, possibly using a (physics) metaphor.
Partner is a mathematician and our very difficult approaches to the same analysis lectures (literally same prof, same script three years apart) just blew my mind.
Markus Redeker 🐘@PetraSchwer Understanding a piece of mathematics means that can work with it and modify it freely.
When I want to understand a proof, I try to create my own version of it. A version that connects with my existing knowledge in the best possible way, not containing proofs of facts that are clear to me anyway, highlighting the for me most important ideas, and with explanations of stuff that was clear to the author but not for me.
Understanding a definition means to have a mental image of the defined concept (because my brain works this way), and to know why the definition was done this and no other way — once again it needs not to be a historical explanation, just something that gives meaning to me.
“What I cannot transform I do not understand.”