Algebra vs Analysis

[CthulhuWingInTheDeep]

The description of how the two fields of Mathematics get from point A to point B is so spot on haha! Especially for Analysis when you have to do an epsilon-delta proof. I always started at point B and tried to work backwards. I remember thinking it kinda felt like I was "cheating" but one professor I had laughed at me when I told him that and said it's just how it's done.

[azimuth]

Enjoying these videos because you're giving us a nice "lay of the land" -- it's making higher math seem more approachable.

[sonstorm]

When I was studying aboard in New Zealand, I noticed that my classmates in the equivalent undergrad level were leagues ahead of me in our Analysis Course, but I seemed to have a bit more experience in our Algebra based courses. Talking to the students, grads, and professors there, I got the feeling that topics in analysis are explored and ingrained in a lot more in lower education than in the US while the US has a higher emphasis on the algebra side of things. I also noticed that the recommended order of courses for math is different as well. So I think there's some merit to that observation.

[vnever]

Man you are so damn right about the analysis part haha 😆😆😆
Glad to know it's a common thing. Backtracking is one good way to solve analysis problems.

[improlawl]

My proofs professor always said that Analysis proofs move in a U-shape, because you have to start at both sides, meet in the middle, and then write the real proof. Basically another graphical approach to your description of analysis proofs :)

[jplikesmaths]

Algebra is to do with equality and analysis is to do with inequality

[bruceybruce]

Without question, that was the most accurate description of analysis I've ever heard 😂

[carterwoodson]

I have been really struggling in my first analysis class.

I feel like this perspective will be useful for me going forward, cannot thank you enough! loving these videos

[jh_esports]

That Lara Alcock book about analysis one can see in the background is an absolute gem!

[alxjones]

I see algebra as the study of composition. It gives a framework for taking raw sets and giving us one or more ways to compose elements from that set to get another element from that same set. The ruleset for those compositions, both independently and how they interact, defines the algebraic structure. By studying all the ways elements of sets can compose, we end up with a very general theory that covers things like symmetry and patterns, as well as the mathematics objects we're typically familiar with beforehand, like numbers, polynomials, and functions.

On the other hand, I see analysis as the study of measurement. The two main things we want to measure are change and size, which correspond to the derivative and the integral of elementary calculus. In order to study change, we need some way of "smoothly moving through a set", which is what topology affords us. In order to study size, we need some kind of "meter stick" to measure more general spaces against, which is what a measure space affords us.

[nikanevskaya]

I'm undegrade in my first year, so I took Lineal Algebra 2 and this was a quite challenging class for me, and I like Analysis more, but I like Lineal Algebra because it's challenging. :)
Thank you for your great videos.

[hessianhyde]

Wow how you describe solving analysis problem is so on spot. I never really thought about styles of solving math problems, but this term I'm doing stochastic process, and the way to prove convergence is exactly like what you described, which is hilarious and sad at the same time (cuz I hate the backtracking approach).

[superiontheknight]

This is awesome! For context, I am currently studying Calculus, and have just learned about the Chain Rule(which is surprisingly easy so far and awesome!). So I definitely have quite a ways to go in order to understand a lot of this. That said, I agree with what you've said regarding understanding the terminology(speaking as someone who's only learned up to this point, and could be incorrect). From what I've seen in the past, whether it be an integral, a derivative, a series, or heck even the notation for a set, it all looks like witchcraft until you understand what it's talking about. :)

Anyways, I have been struggling with writing Proofs for the longest time. I always suspect that I'm leaving something out, or doing something incorrectly. While I didn't quite understand everything you just wrote down(because I have yet to learn what everything means as you said), it did at least shine a little bit of a light on how to go about it. It also makes way more sense how some solutions for differing problems that I've seen may have appeared(not trying to imply that's the case for all of them). They didn't appear out of thin air, but were worked partway through, only to go the end and work backwards, and use that to finish the proof. Really cool stuff! :)

So yea, I suppose this is a long winded comment to say that I'm liking what I'm seeing, and to keep up the good work! :)

[xGriffy]

„What am I trying to say…“ truly encapsulates analysis

[PhillipRhodes]

Well, , I'm not pursuing a Ph.D., so I can't answer the question about "What field do you want to get your Ph.D. in?" But I can say that my (primary) interest in maths is in service of my interests in AI and Cognitive Science and to a lesser degree, Physics. I was very inspired by the book "The Universe Speaks in Numbers" when I read it last year, and decided to really "lean in" to re-learning the maths I've forgotten, and then pushing forward into some material that I never studied back in the day (I was a C.S. major).

Right now I'm still grinding through approximately the equivalent of what an undergrad math major would study early in their journey. Calc I, Calc II, Calc III, Differential Equations, Discrete Math, Linear Algebra, Statistics, Probability, etc. But I plan to keep pushing forward, using books, Youtube videos, web resources, etc. and probably get into some of the graduate level stuff.

What you said about Algebra being all about patterns really resonated with me, as AI is also largely about patterns, albeit possibly at a different level of abstraction. But that brings me to my point - abstraction and the morphisms between objects are different levels of abstraction, are largely where my interests lie. So to the extent that some of this maths might be useful to me from an AI viewpoint, I suppose you can say that I have more interest in Algebra than Analysis. But then again, AI involves many probabilistic elements, which means Probability Theory, which in turn means Measure Theory, which means Analysis. And then I also have reason to be interested in Graph Theory, Combinatorics, and some more esoteric fields like Catastrophe Theory, and Pattern Theory, so... I guess I'm kinda in the mold of "I want to know all of it". :-)

In the end, I'll follow Bruce Lee's advice and "be like water" and just flow where the research takes me.

[Rayquesto]

As a structural engineer, I am so glad I went straight into the field than get a Phd in math. I know many of my colleagues who did take that path and this only looks insane.

When it comes to my field, they really want an MSc. Degree. So, I can see how that happens, but by far the job market was not nearly this complicated to deal with.

[Justinr-r]

Hi, you are very talented in mathematics and im not sure if you know this but is it possible to major mathematics as a premed and still be ready for the MCAT. Keep up the good work!

[RAyLV]

From the examples you gave, I found the Analysis one to be a bit more easier, but I guess the algebra one was hard to follow because I am not familiar with its jargon.

[awildstevey]

“Purest form of mathematics”

Category Theory: hold my beer…

[reefu]

Oh my god that analysis “point A to point B” analogy was so true hahah