I enjoy studying mathematics as a hobby, mainly, and right now I've been pushing my way through into the land of formality and axiomatic systems. They were not joking when they said it was going to be traumatic.
Now, the question I've got is if it's possible to study mathematics on a higher level, as a hobby? I'm no genius but I'd like to believe I'm smarter than average. I primarily just enjoy playing around with ideas and discovering things, but I'm pretty awful at proofs and formality, although it's getting better, since I normally just think intuitively about these things.
The last question I have is for those that perhaps study the subject at university, and that's about the connection of intuition and formality. I realize formality is really important, but surely when trying to prove or solve something, or discovering something new, do you think formally? It's an intuitive process THEN put into formality to make sure it is correct, right?
Maths student here
It's incredibly important to have an intuitive understanding of every concept you study. If you don't, you're just learning text and algorithms.
Tadashi Tokieda put it like this: I'm not proficient at reading sheet music, but i know people who are, and they can read sheet music and "hear" the music while reading. When you just start out learning how to read sheet music, you just learn what every line means, and maybe you read something like G A B F E etc. You're not hearing the music. Similarly if you're just learning to play the piano you'll find that it's usually easier to play a song from memory than from a sheet of paper. When you get better at both, lines blur and the two start overlapping.
That's what mathematical formality is. You have to feel the maths as much as you know how to read it. When you do, proofs will come much more easily as well. Very few Theorems are strictly formal, usually they're a formalization of an idea.
>Very few Theorems are strictly formal, usually they're a formalization of an idea.
This. Often times textbooks only give the formalization and leave it up to the professor to give the real idea or intention behind it. It's also sometimes the case that our formalizations are overpowered workarounds caused by a failure to formally capture a simple insight (because the insight may not be possible to formalize. An example of this is delta-epsilon proofs in analysis introduced as a workaround for infinitesimals (delta-epsilon proofs are slightly more powerful than we're interested in on an intuitive level since they say things about arbitrarily large epsilon as well).
Learning from textbooks without a professor and classmates may be more difficult for this reason.
>been reading sheet music since I was 8
>faggot teacher refused to teach me theory
>brain not fast enough to send signals from mental map of sheet music to fingers
>need to memorize music into muscle memory by hours of rote repition/"feel" out all music
>can't "hear" notes nor rhythms
>can't even remotely blur lines
I know I'm not one of those people who can't form sounds in my head; I listen to music in my mind all the time.
>tfw destined to remain at sub-semantics/syntax-understanding level for life
Then I have a follow-up question, since a lot of more "formal" textbooks I'm reading now just generally go for the "Proposition, Proof, Analysis" structure, and just repeat that over and over, how am I to get an intuitive understanding of the subject? It's like asking someone to get an intuitive understanding of geometry after reading Euclid's Elements (possibly a bad example).
Look for words like 'morally', 'heuristic', and 'intuitively'. Do tonnes of exercises to develop an intuition. Decent professors usually make remarks during lectures to aid your intuition.
Hatcher's Algebraic Topology
Hodges' A Shorter Model Theory
Mumford's Red Book of Varieties and Schemes
Serre's Linear Representations of Finite Groups (it's his least showoff-y book, I think he tried to tone it down because it was intended for chem students)
Shafarevich's Basic Algebraic Geometry
I'm not that guy but many universities including mine have separate programs for pure, applied, general, statistics, and actuary science. A pure math students curriculum is very different from applied and totally different from the rest.
I am doing the same anon. I'd like to major in maths but I am not in the financial position to do so. Gotta stick with MechE instead. Here are some resources to help out. pic is what some /sci/ anon typed out. It's not definitive but it is a useful guideline for what to learn next.
So there's two things I would recommend.
The first thing is better than the second. Try and come up with lots of examples and then try to come up with simple questions and figure out if they're true or false. For example
>can x ever be true when y is false?
>is it always the case that..?
>what happens in x scenario?
>can I figure out a case that satisfies this? Can I figure out why it satisfies it and maybe generalize it?
Just try to keep the questions simple at first, like thought experiment level.
Hopefully somewhere along the way you'll figure out some deep insight that will suddenly make everything seem like common sense, and if not you'll at least have a bunch of examples and a better idea of what's going on.
The second thing you should do is use Google. We live on the information age. Unless you're doing graduate level math then it's something that millions of people have learned and many have struggled with. You can often find explanations, analogies, examples, etc.. online.
I say the first thing is more important because for intuition one should always have more examples than theorems and more theorems than definitions. Unfortunately books are written in the opposite manner. You may have a lot of definitions, then a handful of key theorems, then a couple examples if you're lucky. If a problem is giving you too much trouble then it's often better to move onto something else or to look it up on the internet (sometimes finding one example or getting one question answered will lead to more). Just don't become dependent on the internet, keep in mind that your ultimate goal is being able to figure stuff out on your own, not just being able to Google it.
No, I would be surprised if you finish your bachelor's.
Later undergrad courses will ask more vague questions that will require intuition. Grad courses will be structured around that type of work. Also you may have courses (and TA requirements) where you have to give talks on the material as well as answering questions from the confused audience.
Besides, why would you even want to go into a masters if you can't even produce original work? You'll need to have an interesting project proposal to even get in.
Wait, so it's important to connect the ideas you learn both rigorously AND intuitively, right? So using Calculus as an example, infinitesimals and rate of change could be an example of an intuitive understanding, while the epsilon-delta style is an example of the rigor.
Why then are books like baby Rudin and Spivak's Calculus recommended? They give no, or very, very close to no intuition behind the subject matter, and the ideas they teach.
>Now, the question I've got is if it's possible to study mathematics on a higher level, as a hobby?
It is, with enough dedication.
>connection of intuition and formality
I usually think deeply about a theorem, trying to visualize it as much as possible. I guess you could call this the intuitive part. If I understand why it must be so, I have no problem writing a proof, this comes with practice. Otherwise, I usually just try to apply a few common proof techniques and hope it works out/I figure it out. It might also be worth noting that there exist algorithms which can determine whether a given statement is true or false, as long as the statement isn't independent, so in theory you could prove anything formally, by simply following the rules of inference of whatever formal language you're working in. This is very time-consuming though, and for all intents and purposes it is currently unfeasible.
With a minor in programming and philosophy!
I can't produce original work and even new ideas; I can only repeat ideas through memorization. I've memorized my real analysis exam solutions (six of them; 2009 to 2014) and gotted a Full A+; also my university is ranked 19th in the world.
Too bad, I still want to do a Masters, and PhD; no one is going to stop my dream or I'll change universities if someone tried to block me from grad school.
You`re an idiot. If you have a go at it without any real thought and just fuck around reading sheets and "listening music in your mind" then ofc it will take forever to become a valuable player. Just do ear training and a little bit of theory until you can hear an interval between two notes and identify what that interval is. Then when you hear something you can understand what`s going on with melodies in songs and sort of learn these motions, not just when you`re playing them but when you`re listening to something. That`s the trick.
What you think intuition is is NOT intuition. Intuition is grounded in rigor. First learn your rigor well and then you can start feeling what intuition is eventually. Watch lectures in real analysis from youtube. And PLEASE do the exercises assigned rigorously.
I'm by no means a math student/hobbyist but I came across this a while ago, you might find it insightful.
I disagree in the sense that that could be an example of intuition. If you extend the reals with infinitesimals (i.e. hyperreals) then you can do all of the calculus with them rigorously. However I agree that rigor is an important component to intuition, as is formalism.
I'd argue that intuition is many things including but not limited to knowing the "intention" of a theorem or definition (i.e. what that definition or theorem is trying to capture). In this case infinitesimals and delta-epsilon have the same intention. However what they actually say is different.