how to develop algebraic intuition

this is a vague question but what’s a good way to develop algebraic intuition?

i have pretty good analysis intuition (dipped my toes into most grad analysis topics – measure theory, spectral theory, pdes, optimization, sdes, etc.) but for the life of me even undergrad algebra things are sometimes unintuitive to me (often when they intersect with number theory type things).

obviously i’m aware of the age old adage of just doing problems and i’ve tried – i have i think 4? algebra books ranging in difficulty from frayleigh to ashe – but i just can’t make substantial progress because i’m constantly stymied by seemingly trivial steps in proofs.

a little while ago i had the idea that maybe playing with the objects using something like magma would be useful but i can’t afford a magma subscription just for fun.

also i thought about approaching it from the other direction – cat theory through something practical like haskell. but the trouble there is (as everyone says) cat theory doesn’t make sense unless you have algebraic examples to translate to.

i’ll give an example that i should probably be embarrassed about but since i have no mathematical shame i’m not: i’ve read the definitions of group, ring, field, algebra, probably over 100 times each. i still for the life of me can never remember which ones have units and/or inverses for each of the operations (though i do know which ones have two operations) and which ones don’t. like there’s nothing about the word ring (i just now had to look it up) to me that leaps out as addition should be invertible but multiplication need not be.

it’s like i’m one of those people that immediately forgets your name after you introduce yourself to me (which is in fact true for me) but for algebraic definitions. note i do not have this problem in the least in analysis because i always have a picture of what’s going on. e.g. my last analysis class was ~10 years ago and i can define uniform and lipshitz continuity without skipping a beat (though i don’t remember absolute continuity i realize).

now i’ve gotten away with this because i’m not a math phd (CS) but still there are things i’m personally interested in (like de rham cohomology) that just demand facility with algebra.

anyway this isn’t a rant or a cry for help – i’m genuinely interested in ideas for other ways to approach this seemingly ludicrous learning … “disability”.

submitted by /u/ice109
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I absolutely love math. Convince me not to do a PhD.

1.) I love math, a lot. I can’t stop talking about it, regardless of whether the person I’m talking to cares. I’ve always loved math, I was a math Olympian as a child and helped start and run calculus club in high school. I did my undergrad degree in pure math and took all of my electives in math. I’m probably not in the top 1% or 10% of math students in my MS program in terms of natural ability or intelligence, but I am in the top 1% or 10% for enthusiasm.

2.) I took a few years off and worked in industry in applied math and I currently work in virtual reality.

3.) I have not yet applied to schools or fellowships, but I feel that I will be competitive. My GPA in my MS is 4.0, undergrad GPA is lower but sufficient, I published something as an undergrad and am doing a master’s thesis as well. I haven’t taken the GRE or subject test yet, but I tried a general GRE practice test and got an almost perfect score, and the GRE subject test doesn’t look too intimidating and I tend to test well, and I have one strong recc letter already written and think I can get at least one more strong one. So I think I could put together a competitive application. I’m not going to get into Harvard, but I don’t think getting funding would be impossible for me.

4.) I know topology is my schtick and I have discussed a few potential problems I could work on with topology professors during their office hours. I have strong coding skills and some experience publishing and I feel like I could find problems to work on that would be interesting to me and that I believe that I am capable of doing the work.

5.) I’m still young and I have no responsibilities and no life.

6.) There are internships in my area and elsewhere that hire math grad students in my subject that pay 100k a year as interns while you are in school in a specific area of AI related to what I want to study. So I may be able to make money while in school even.

7.) I feel strongly that I have a good, specific plan to make a lucrative career move out of what I study. I want to go into computer vision in AI, I already have a foundation and experience in ML and some computer vision knowledge and specific job skills. I am certain that if my plan doesn’t pan out that I am in high enough demand and know enough people in the industry that I can easily go back to something similar to what I am doing now and still make great money.

8.) I’m hard-working, my last review at work said that I am a model for productivity and my manager said I’m the most productive employee at my level and that hiring me was our team’s best decision. I think I’m really good at delivering results and I’m self-directed.

It might seem like if I have a well-paying job now, and I love it, what on Earth would I be thinking to do a PhD? But I don’t intend to quit my job tomorrow. Eventually my job will stagnate. I’ve moved up in my job quickly enough, but I will eventually hit a ceiling and stagnate. Currently I work on a R&D virtual reality project, and it’s super cool work. But eventually this project will end, and then I will have to find new work at my company, and the company I work at pays better than any other company in my city by a lot, so if I left my company I’d be taking a pay hit, but it’s unlikely I will find another cool project like this at my company because even though the company is big they don’t have a ton of work in my sphere, they hire mostly hardware and electrical engineers whereas I like more application/cloud-based work, so I feel that there will eventually come a time when my career will stagnate at this company, and I know a lot of people who work the same corporate jobs for decades but I don’t want to keep doing this forever. If my project gets cut I want to do a PhD. I think there is a not insignificant chance that my project may be cut, possibly soon. If that happens I may have a severance package which could fund me for 6 months. If my project does NOT get cut, I might postpone starting a PhD, but I feel strongly compelled to do one to the point that I feel it will haunt me if I do not.

What is the best counter-argument to say that I should not do a PhD in math?

submitted by /u/theNextVilliage
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[Becoming resentful towards math] How do I make sense of my experience with math?

I’m in my final year of a math undergraduate program. My “story” with math is very strange.

I was quite average at math all my life (I was much better in other classes). When I started out in university, I began in a major completely unrelated to math. I had to take a couple of calculus classes, eventually I became interested and started learning rigorous math on my own, and eventually decided to switch my major to math, and a few years later we’re here.

The reason for the “change of heart” is quite clear to me. University math and high school math are not the same, calling them both “math” is a crime. University math is rigorous, theoretically-driven, and revolves around proofs. High school math is about memorizing and applying formulas quickly. I have no appreciation of “being able to use formulas quickly” math, nor am I good at it. I do have an appreciation for well-built math, starting from a foundation, and building a theory on it, abstracting it, and working within it rigorously.

Throughout my undergrad, I got an A (the highest possible grade) in all my math courses, and with ease.

In a first real analysis course, I learned all the material before each test in 2 days (granted, already had lots of intuition from before) and beat the class average by 40% each time. Likewise in other courses, I slacked off routinely, and still beat the class average, in some cases by up to 50% (yes we had a 39% class average once). I’m not saying any of this in an attempt to brag (if it comes off that way, I’m sorry). What I’m saying is, I’m not bad at writing proofs. Yet, as I wrote above, I am bad at the “other” type of math, the plug and chug stuff. So my whole point with this paragraph is that to me, it seems the two are not the same. Your ability to do “basic math” quickly, and your ability to prove non-trivial things, don’t seem to be identical to me.

Come around the GRE math subject, I get a 700 (59 percentile). Ok… so apparently I’m still not particularly good at the more “high-school-like” math mentality. I’m not very good at plug and chug, I guess. Alright, then wrote the general GRE, and got 165V, 159Q (ran out of time on every quant section). Okay, so apparently I really am not good at plug-and-chug, even when it’s high school math!

Reading now online how these scores will be viewed, I’m becoming perplexed, disappointed, and also resentful. Everywhere I read, I see “If you’re applying for a math PhD, you better have perfect quant scores!”. As if doing high school math has anything to do with writing proofs?!

What’s going on here? Do I just live in a different universe, where your ability to use formulas quickly has nothing to do with “real math”? Why is it that everywhere I look, people pretend that this bullshit is actually a measure of how well you can do “real math”? If this really is a test of your ability to do grad school, I guess I better just go back to my old major.

submitted by /u/Advanced-Wrongdoer
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Trend of Discrete Idea –> Continuous Idea

I’m not sure if this is much of an idea so forgive me for my naivete, but I was reading the book “The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity” which is a biography on Cantor and an overview of the work he did on the Continuum Hypothesis and how later mathematicians picked up where he left off. It was a really amazing read and I recommend the book to anyone as it’s quite recreational.

But my main question is this idea that a discrete concept (such as the tiers of infinity) potentially being a continuous concept reminded me so much of how mathematicians of he past had similar qualms on things such as real numbers existing (let alone irrationals) or of how in fractal geometry, the idea that non-integer dimension may exist.

In mathematics, it often appears that such trends aren’t coincidental. To quote George Lucas, “It’s like poetry, so that they rhyme”. I was wondering if anyone has more examples of a concept that seemed discrete at first, but eventually turned into one (in shocking manner perhaps) of a continuous nature.

Thanks.

submitted by /u/YoungLePoPo
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