For me, it’s all about consistency. It’s spending a few minutes to a few hours every day, forever, til the day I die.
I usually can squeeze in 30 minutes to an hour every day to study something (whether it’s math or something else — right now I’m studying cinematography). Sometimes that’s in 15-minute chunks if it’s a busy day. Usually it’s before bed or while I’m eating lunch or if I have extra time on the weekends while my kids are napping.
It’s all about just doing a little bit every day. That’s been successful for me.
Grab a pen and paper, open up the first book in any of the guides, and start reading. Read for 15 minutes, or 20 minutes, or whatever time you have on your lunch break or before you go to bed or while you’re using the bathroom. Do it again the next day. And the next. And the next.
That’s how I did it. There’s no brilliance involved. It’s just jumping in. The more you do, the easier it gets.
I knew consistency was important, and did not place much importance on "brilliance", although that attribute has been showered upon me since I was little.
What I did not know, and still don't know is that studying only for a few minutes or half an hour regularly will make me good at something as advanced math.
You seem to know your stuff and I like your approach and attitude, and I will do now what I do very rarely and upon serious consideration- take a leap of faith.
I will start doing math everyday for at least half an hour, and I will see how that goes.
It also comes after many semesters of calculus, which depend upon it, and before any topology, which it depends on. Even if you are just interested in these things as tools for describing physical phenomena, there is value in placing mathematical knowledge in its natural structure.
For what it’s worth, the curriculum in this guide is modeled after the math major maps of many universities, including the one I attended (Penn). I would be curious to know what part of an undergraduate math curriculum will lead people very far astray…
It's just that it is very much a "mathematics for engineers" style course. I think very few of the subjects outlined there give you a flavor for what "real" mathematics is really all about at all (except for algebra, which you do mention).
Apart from the applied stuff you mention, the real core of a mathematics education involves, I think, 4 main areas with significant overlap
Group A:
number theory, graph theory, combinatorics
which shares concepts with
Group B:
Algebra, Topology, complex analysis, differential geometry, metric spaces...etc
which shares concepts with
Group C:
Functional analysis, measure theory
which shares concepts with
Group D:
probability and statistics.
As for the applied math that you mention, you should really need to add vector calculus and I'd highly encourage anyone to take a course on fluid mechanics (from a mathematics department instead of an engineering department) to get a real feel for vector calculus in action.
I suggest taking another look at the list and comparing it to the required courses of the undergraduate math majors at the top 20 universities in the USA.
Real analysis, complex analysis, topology, and number theory are there (topology and number theory are both listed as electives since most math programs categorize them as such). Graph theory, functional analysis, differential geometry, probability, and statistics are almost always either electives or graduate courses.
It’s funny, because most of the things you mention as “real math” are things that many math undergraduates don’t learn (not until graduate school at least) but that physics students learn as undergraduates (differential geometry, measure theory, functional analysis, etc.).
I have never met a physics student that even knows what functional analysis even is, despite it being at the core of quantum mechanics. I can't even imagine why someone in a physics department would learn about measure theory. True that any student learning General Relativity will get an introduction to differential geometry.
Except for the course ordering, it lines up almost identically with the math major at my university. I'm not sure what the other posters are going on about. Most of their preferred topics that they feel you missed are either upper division electives or graduate level.
I suspect that people forget that undergraduate programs don’t really cover very much. This true not just for math, but for pretty much every other major. I mean, think about how little of physics you learn if you only take undergraduate courses!
Yeah I was a physics major and honestly it felt like we barely scratched the surface. An undergraduate degree in physics is sufficient to be a high school physics teacher, but not to be a physicist of any sort.
though I am a little surprised that they have 1 course of differential equations in there instead of complex analysis as a required topic, as I think the latter is a better pure math topic. But it's MIT, so be it. Whether directly or indirectly, many of us learned to view math the MIT way by patiently working through foundational books like Artin and Munkres.
That said, my mention of non-introductory algebra topics probably is more of a personal idiosyncracy/interest.
If you have a solid background in calculus, I'd recommend Zill's Advanced Engineering Mathematics, which is pretty much basic math for physicists and engineers (aka for people who need to "use it").
In that case, I recommend starting out with Zill's Precalculus with Calculus Previews and then working through Stewart's Calculus: Early Transcendentals!
You cut out the middle of that paragraph, which says:
"Sadly, there is all sorts of baggage around learning it (at least in the US educational system) that is completely unnecessary and awful and prevents many people from experiencing the pure joy of mathematics. One of the lies I have heard so many people repeat is that everyone is either a “math person” or a "language person” — such a profoundly ignorant and damaging statement. Here is the truth: if you can understand the structure of literature, if you can understand the basic grammar of the English language or any other language, then you can understand the basics of the language of the universe."
I'm not sure what your point is. Are you implying that you are not contributing to the mystification and idealization of mathematics?
In other words, I do not see how you are dealing with the "baggage" of learning mathematics beyond name-dropping it. In my opinion, the mysticism is the baggage. And then the rest of the blogpost reads like a conventional curriculum within the conventional academic regime with which we associate that baggage.
I don't follow. The author dismissively ctrl-V'd a paragraph with no further explanation, and my response asking for elaboration gets shadow-buried by a mod. What?
Your post wasn't simply asking for elaboration—it was written in the cross-examining style that we particularly want to avoid here because it kills curious conversation.
If you didn't intend to come across like an interrogator trying to back an opponent into the corner, then your comment needed to be written quite differently.
As someone with a keen interest in learning Engineering part time, I found your write ups really helpful though! I enjoy learning math but like to have an angle towards a practical and useful application. It keeps me a little more motivated than pure math learning. With ADHD the concept of being able to build cooler things always keeps me going. But somewhere along the way of learning purely theoretical things for too long my brain just loses interest (not enough reward), even though I enjoy it in the moment it is hard to get to the starting line and take the first step after a while :)
Check out my physics guide: https://www.susanrigetti.com/physics. It has both the physics core curriculum AND the math essentials you need to know in order to understand the physics essentials. (And thank you!)
I usually can squeeze in 30 minutes to an hour every day to study something (whether it’s math or something else — right now I’m studying cinematography). Sometimes that’s in 15-minute chunks if it’s a busy day. Usually it’s before bed or while I’m eating lunch or if I have extra time on the weekends while my kids are napping.
It’s all about just doing a little bit every day. That’s been successful for me.