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So you want to study mathematics

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Re: So you want to study mathematics

#171

Post by Jach »

dr_dshiv wrote: post #145

If doing math is essential to conceptual understanding and application, could the interface of math and physics be made more human-centered? For instance, the shift from Roman numerals to Arabic numerals made doing math easier. Based on your experience, might it be possible to increase accessibility by revising some of the arcane conventions of math and physics? See Brett Victor’s 2011 proposal: http://worrydream.com…

My favorite accessibility-increasing tool is the computer. Doing math shouldn't involve so much circus math, i.e. doing things just for show, since a computer does so much immediately and accurately. We already use graphing calculators, and notation-wise graphs can be a useful tool in themselves in elaborating an idea (see also Feynman Diagrams, or electric circuits), but there's so much more calculators can do, let alone actual PCs, cell phones, and web apps. By chance in 9th grade "Intermediate Algebra/Algebra 2" I had a teacher not wholly opposed to modern technology and so he only had us do a small amount of those "solve this system of equations using a 3x4 matrix by hand, showing each matrix transformation to reach the row reduced form, taking up some pages of paper" problems before he brought in a classroom set of chonky TI-92 calculators and showed us the rref() function. That Christmas I asked my mom to upgrade me from my non-graphing scientific calculator that had served since elementary school to a TI-89 Titanium that served me even through college until I learned and got used to various PC programs. The lesson that there were powerful tools around stuck with me pretty fast though, and I wrote some simple programs on the calculator for that and other classes throughout HS (mainly just automating calculator input steps, no fundamentally new algorithms the calculator couldn't already do); in HS physics I also had learned more programming and did a little simulation with pygame and it was fun to enter numbers in the program, run it, see the mass trajectory animate and show some computed values, and then do the actual experiment and get the same results. I only wish I had been shown some PC programs earlier.

I met a friend many years later who sadly was still forced to do that rref()-by-hand for even larger systems of equations in university! That left no time to actually learn anything useful in linear algebra. Madness.

https://theodoregray.com/BrainRot/ has some nice ranting about this (though it does go a bit off the rails when it starts talking about video games).

Re: So you want to study mathematics

#172

Post by ghaff »

mmmpop wrote: post #167
ghaff wrote: post #156

Earlier quoted context omitted.

Pretty much all the classic Newtonian formulae you memorize in high school physics classes can be derived (relatively) easily once you bring calculus in. A general problem with high school classes in particular--still somewhat true in certain university classes but much fewer--is that there are dependencies across classes. So you end up with a lot of "Memorize this thing because you don't have an advanced enough back…

In my high school experience, this was avoided if you had the desire and aptitude to just take Honors Physics instead. To work around the fact that many of us were still in precalc, our teacher just taught us the power rule, the relationship between slopes of tangent lines, etc., without diving into the "why" of why those things worked. That said, yeah maybe for the most basic of university of physics the whole "deri…

For tests and the like, when I was an undergrad, it was fairly common to be able to bring in a one-page "cheat sheet." The idea was that you probably could derive a lot of the stuff given time and maybe references, but you almost certainly couldn't in a 1 hour exam in addition to solving the actual problems on the test.

Re: So you want to study mathematics

#174

Post by selimthegrim »

selimthegrim wrote: post #70
Mimmy wrote: post #6

Going from Strang to D&F seems like a steep jump. The former is an applied textbook for non-mathematicians and the latter is a proof-based text for advanced undergraduate / graduate-level math students. I would suggest working through a proof-based linear algebra book in between to ease the transition. Axler's is a good one. Alternatives include Hoffman and Kunze and the more modern Friedberg, Insel, and Spence.

Strang's latest book DE&LA is disappointing, it is linear algebra and its applications with the abstraction taken out and mushed together with supplementary notes from ODE videolectures. Mattuck's ODE course is good.

I should say his penultimate minus one book, DE&LA is from 2014 and I think he has published a couple more since then. Linear Algebra and its Applications is his 1988 book.

Re: So you want to study mathematics

#175

Post by EoinB »

As a mathematics major, I find it encouraging to see non-mathematicians sing the praises of the subject. So I applaud that. But I think the author is overstating/slightly wrong about a few things, perhaps because her exposure to mathematics has been through the lens of physics.

Maybe a more humble rewording of some of her statements e.g., "Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics." would be helpful.

Her suggested curriculum doesn't include anything from Number Theory, which is a foundational part of an advanced mathematics education. It is also one of, if not the most, beautiful topics one can study in mathematics.

I find it odd to call out "Introduction to Proofs" as a topic in and of itself. Proofs aren't really a topic in the way analysis or number theory is. At advanced levels, devising theorems and theirs proofs is what mathematics is.

Re: So you want to study mathematics

#176

Post by pnin »

brimble wrote: post #31

Are there any "math for people who just want to use it" tracks in math pedagogy? I don't care a bit about proving any of it's true, or even reading others proofs of same. "Recognize which tool to apply, then apply tool", all focused on real-world use (so, yes, it wouldn't be "real" mathematics). That's the math education I'd like—try as I might, I just can't make myself care even a little about math for math's sake.…

I have some sympathy for this sentiment, and there is no doubt that the producers of mathematics could and probably should spend more time making life easier for the users and consumers of mathematics. This is true even /within/ the discipline, for very theoretical stuff. In the end this takes vastly more work than people expect. That's not an excuse, but it's true.

However, I also have a fundamental objection. I don't see how you can be an intelligent tool user without at least a little curiosity about how your tool functions. Maybe you can apply your tool, even be highly effective, in certain instances. But this is inherently brittle knowledge. When the parameters of your problem change and you don't understand your tool well enough to adapt, you're lost.

"Math for people who just want to use it" is very broad. What do you want to use it for? Physics, biology, chemistry, computer science? Sociology? Economics? There might be some shared stuff, but for all of these disciplines there is a vast space of mathematics that might be relevant.

I think Eliezer Yudkowsky's idea of a book (series) covering "The Simple Math of Everything" is fantastic. I would love to read that book.

https://www.lesswrong.com/posts/HnPEpu5eQWkbyAJCT/the-simple...

Re: So you want to study mathematics

#177

Post by pphysch »

akomtu wrote: post #141
pphysch wrote: post #25

IME (as a math-degree-haver) the value of mathematics is in improving one's ability to mentally model and reason about complicated real-world phenomena . A lot of folks lose sight of the reality and get lost in the mysticism, especially within the academic regime. > [Mathematics] is the purest and most beautiful of all the intellectual disciplines. It is the universal language, both of human beings and of the univers…

On the mysticism note, I want to add that math is perhaps the only subject that forces one to engage the upper "abstract" mind. The lower mind is concerned with modeling real world phenomenas, while the upper mind works with purely abstract things, aka the "true reality" in mysticism.

Natural language is very abstract too

Re: So you want to study mathematics

#178

Post by mrjangles »

Py-o7 wrote: post #17

This felt like it was written by a physicist or engineer. Too much emphasis on differential equations and not enough on things like topology, functional analysis and/or non-introductory parts of algebra like say representation theory.

Yep, I came here to say this.

I think it is important that anyone who wants to study math understand that real math is not at all like what you learn in a physics or engineering department. In these departments you will always hear people say things like

>"proofs are not useful, all you have to do is memorize the 'trick' they use. Once you know which trick to use, it is easy"

or you will hear them say.

>"Math isn't about understanding, it is just about learning rules and symbols on paper".

This is not mathematics. These things do happen.... in a physics and engineering department. It is, in fact, a descriptions of a physics education, not a description of a mathematics education.

For this reason I would be careful taking mathematics advice from physicists too seriously as they may, unintentionally, lead you very far astray.

Re: So you want to study mathematics

#179

Post by dunefox »

criddell wrote: post #53
ghufran_syed wrote: post #16

I went from only having done high school math 10 years ago to completing an MS in math and statistics at my local state university while working in an unrelated field. I would recommend NOT starting with calculus if you haven’t done it, instead, just learn how to do proofs - I used Chartrand “Mathematical proofs” - You don’t need to know any math beyond algebra in order to do that most of this book. If you need to re…

I looked up the Chartrand Mathematical Proofs book and it's been a while since I had to buy a textbook, but $175 for hardcover and $75 for paperback or ebook? That's nuts. If I were a student today, I'd pirate that and feel absolutely no remorse for doing so.

Thing is that I just can't read PDFs.
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