Ask HN: How do I learn math/physics in my thirties?
I have (had) a fairly good grasp of calculus and trigonometry and did a fairly good job working on a number of problems in high school. But over the past 12-13 years, I've really not had any need to flex my math muscles other than a problem here or there at work. Otherwise it's the same old enterprise software development.
I follow a bunch of folks on the internet and idolize them for their multifaceted personalities - be it math, programming/problem solving, physics, music etc. And these people had a natural flair for math/physics which was nurtured by their environment which made them participate in IOI/ACPC etc. in high school and undergrad which unfortunately I didn't get a taste of. I can totally see that these are the folks who have high IQs and they can easily learn a new domain in a few months if they were put in one.
Instead of ruing missed opportunities, I want to take it under my stride in my thirties to learn math/physics so as to become better at it. I might not have made an effort till now, but I hopefully have another 40 years to flex my muscles. I believe I'm a little wiser than how I was a few years back, so I'm turning to the community for help.
How do I get started? I'm looking to (re)learn the following - calculus, linear algebra, constraint solving, optimization problems, graph theory, discrete math and slowly gain knowledge and expertise to appreciate theoretical physics, astrophysics, string theory etc.
159 comments
[ 7.4 ms ] story [ 3688 ms ] threadIf you learn best in a classroom, you may have a local college that teaches math in the evenings. (I got my Master's in Statistics that way.)
If you learn best in small chunks, Khan Academy has differential and integral calculus and linear algebra, to start you out.
If you learn best from books... there are hundreds of great textbooks.
Best wishes to you. Keep up a lifetime of learning!
My learning actually accelerated in my 30s because knowledge pays compound interest -- the more knowledge you have, the faster it is to acquire new knowledge. Assuming one has continued to pursue learning, someone in their 30s would have built up a significant enough semantic tree to pin new knowledge to.
Most people find it hard to learn in their 30s because they lack the energy, environment (+kids, +spouse, etc.) or internal drive that provides them the impetus. Others find it hard to learn because of bad habits and a poor foundation (their semantic tree wasn't that well built up in their youth). But their actual abilities (even memory) haven't actually degraded all that much.
And of course, there are some who find it hard because they have reached the limits of their cognitive abilities (un-PC as it sounds, this is a real thing). You have to know if this is the case. Most of the time it is not.
I would start by building up a good foundation. Learn the basics well but don't get hung up on understanding every little detail.
Chunk your learning and use your little victories to drive you (brain hack: humans are a sucker for little victories). Use the Feynman method (learn by teaching).
Drill yourself with exercises rather than trying to understand everything -- math is one of those things where it is easier to learn hands-on by working on problems BEFORE understanding the definitions fully... understanding comes later (the patterns will emerge once your semantic tree is solid). It's a process of cognitive dissonance where you actively wrestle with problems rather than passively work through them.
People who try to understand math by reading alone (or by watching videos) tend to fail in real life -- they tend to be able to recite definitions but their ability to execute on their knowledge is weak.
This is a standard rookie mistake, and the reason why so many American kids are weaker at math compared to their Asian counterparts. Drilling--even if mindless at frst--really does help, especially when you're starting out on a new subject. It helps you develop muscle memory which in turn gives you confidence to move to the next level.
(from https://github.com/sindresorhus/awesome)
Constraint solving and optimization problems aren't things I self studied, but you can find a variety of resources to help with those based on how you learn best. For me, I did them by taking a class and relying heavily on my textbooks.
Jokes aside, try https://www.coursera.org/ or https://www.khanacademy.org/
I used to do all my work (solutions to problems, notes) using pen and (plain! not lined) paper. However I realized a couple of years ago that becoming fluent in LaTeX was a better option for me. The reason is that, with the proof neatly typeset, and the ability to re-work and edit repeatedly without making a mess, I found that I think more precisely and systematically. I still do scratch work on paper, but writing up a clean copy as I go is very beneficial.
In addition to those reasons, the other hugely important one is that my notes are now in git, I can grep them, and they don't add to the pile of objects that must be dealt with when moving to a new home.
For best results you need to make a nice LaTeX set up. I use the Skim PDF reader so that it autorefreshes on file save, and set up a Makefile and make it so the PDF is recompiled on every file save. But whatever works for you, I'm sure there are easier setups.
The downside of course is that computers are very capable distraction vehicles, you need a bit of discipline to sit at one and study / do this sort of work at the same time for prolonged periods. Pulling out the ethernet cable can help but may not be sufficient depending on one's level of discipline and access to offline distractions.
A lot of the old methods of learning actually work and so the advice is sound to strictly adhere to them when you're having struggles. Certain modern enhancements are worth a qualified mention though.
[0] https://lamport.azurewebsites.net/pubs/proof.pdf
Because that requires learning a formal proof-verification language. I'm certainly interested in that, but it is a distraction from learning undergraduate mathematics.
> If you're working with graphical concepts, why not code them up, or use a drawing program (or hey, a graphing calculator) rather than pulling out a ruler and such (and maybe learning to draw at all if you don't know how)?
> If you have sloppy handwriting (as I'm sure many of us here do), why not type in something you'll always be able to read later? (Along with whomever you show it to -- I did a lot of college homework using LaTeX. With macros I could do things way more efficiently, with comments I could go back and see what I was thinking at a misstep (if I wrote anything).)
I'm confused; my post was advocating using software, so I'm unclear why you're suggesting I use software.
> A lot of the old methods of learning actually work and so the advice is sound to strictly adhere to them when you're having struggles.
What is that, a flat contradiction of my post?
Very strange, maybe you meant to reply to a different post?
> it is a distraction from learning undergraduate mathematics
Arguably so is LaTeX. But it's desirable that students (or just people learning the same material, later) spend some of their undergraduate time learning new things, right? And not just because it's new, but hopefully because it's better. Learning new/different things is just a small step further beyond learning old things with new/different assistants. And maybe some things will have to be cut out, like 17th century prose-proofs (edit: and even just moving to structured proofs without full formal tools is an improvement...), or square roots by hand (http://www.theodoregray.com/BrainRot/)
It is painful, but I don't think there is any easy way of actually learning without just sitting down and doing problems. Have you considered auditing a course at a community college? Very few people (myself included) are motivated enough to work enough problems without the threat of assigned homework. You need to do enough problems on a topic that you are no longer struggling, then do 4-6 more. Those last problems are, IMO, the most important, they actually cement the concepts in long term memory.
As far as books, I can recommend Schaums Outlines for good examples of worked-through example problems.
Edit:fixed typos
Plus, you can get really, REALLY good deals on used college textbooks (some of which are still in pristine - as in never even been opened - condition).
My favorite undergraduate students when I was TA'ing were all students who had returned to school after spending time in the world. They knew why they were there, knew that the material was worth learning, and asked lots of questions.
Go get it -- start small, don't stop.
The colloquia are usually very subject specific. When I did my PhD in condensed matter physics, depending on the speaker, sometimes it could be 10 minutes into the lecture when it delves into narrow field-specific material I don’t understand (eg, a speaker talking about particle physics or astrophysics). Good speakers and even non-physicists can follow the whole talk regardless of subject matter. But good speakers are rare. And colloquia are typically for benefit of the department (students + faculty), so will quickly gloss over fundamentals into the real meat.
So you might learn about super cool research happening, which is great. But you aren’t likely to learn key fundamentals.
These videos are frankly better explanation of college-level math concepts than most college classes.
Also, now you probably care much more about the intuitions of mathematics over the raw mechanics of it. Once again, this channel perfectly exemplifies this concept.
One interesting option that computer programmers have to understand linear algebra that most people do not is that you could go do some stuff with computer graphics. Grab three.js, and then once you've followed some tutorial somewhere to get a triangle on the screen, start doing things, but do them manually, including implementing matrix multiplication yourself. Modern graphics has moved so far up the stack nowadays that you probably shouldn't say that you "know 3D graphics" after that exercise, because you'll know 3D graphics circa 1995. But you will have a much better intuition for linear algebra, and those videos will either make sense, or be trivially obvious to you.
(One of the reasons why the linear algebra videos can be so helpful is that it has historically been very easy to take an entire class on the topic and just grind numbers, without ever getting to that level of intuition. Differential equations, if you took physics that did not use them, can have a very similar problem, where you just grind through problems for a semester with no motivation.)
I think I understand what you're saying -- one needs more than just cool videos and cool intuition. You need to do exercises. This is a point made multiple times in those very videos.
But, the intuition provided in those videos is absolutely excellent. As an example, look at the explanation of change-of-basis in the linear algebra I series.
The videos suggest pausing and trying to figure the next bit out yourself and a couple of the videos do end with a suggestion to prove something yourself.
[1] https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...
One thing a friend of mine said, which I think has been very good advice, is to get several books on a single topic. Eventually every author will lose you and you'll get stuck; having alternate discussions will help you get through it. This is easy to do and inexpensive if you pick up some Dover math books, but I've been making heavy use of the local academic library. The math books you want to read are not in high demand at the library! You can get three or four and see if any of them are good enough to warrant a purchase later on, because you probably won't get through them in a month or whatever.
I find that for many topics there is a really good text. Calculus by Spivak is a great example, it straddles the line between calculus-in-college and analysis. Every topic seems to have a few really good books like this one, and there are often books that will take a totally different approach, like H Jerome Keisler's nonstandard calculus book using infinitesimals.
I used to see it as a real problem that I was learning math outside class, but more and more I see it as a benefit, because you can pick up the stuff you want at the resolution you want and benefit from the best books rather than whatever the publishers are bribing professors to use. Going at your own pace, you're not going to go through as much stuff as quickly, but you will actually _really_ learn it. I've spent the last three weeks or so thinking about the construction of the real numbers... in a classroom setting, you would be forced to get through this quickly to get on with the rest of the curriculum, even if you aren't interested in the rest of it.
I think both our roads are eventually going to lead us to differential geometry, and the only thing I know about that is that there appears to be a very good book on Amazon (Tapp, Differential Geometry of Curves and Surfaces), and that you may want to avoid older books that use the older notation for it. I have heard great things about the book Gravitation, but I'm totally afraid of it, not ready to go there yet. Also check out Physics from Symmetry, that book looks amazing to me but I haven't read it yet, just flipped through the contents, but it might be exactly what you're after, since it discusses the math right before applying it to specific areas of physics.
It's all about doing it like we did in high school. Pen, graph paper, and maybe a calculator. Drilling down the practice will help with theory.
Maybe figure out an actual destination and then devise a plan to get there. Deep diving into math and physics just for the sake of learning etc seems to be cargo-culting. Lawyers also sound smart until you realize they write like they do intentionally to keep people from figuring them out.
> I follow a bunch of folks on the internet and idolize them for their multifaceted personalities - be it math, programming/problem solving, physics, music etc. And these people had a natural flair for math/physics which was nurtured by their environment which made them participate in IOI/ACPC etc.
Sounds like they are good story-tellers along with whatever else they do. Have you tried putting anything out for others to consume? If you want to be like these people, then it would be good to start with writing / shipping things. If you have been doing that already, then post some links. ;)
I would recommend two outstanding textbooks. Halliday and Resnick, early editions , printed in the late 60s and 70s. If you can do all the odd problems in this two volume set, you are an educated person, regardless of your greater aspirations. Edward Purcell’s Berkeley Physics Series Second Volume on Electricity and Magnetism. Might be the best undergraduate physics textbook ever written. Did you know that magnetism arises from electrostatics and relativistic length contraction? It’s right there. You should also get yourself a copy of Feynman’s Lectures on Physics. Warning. Read it for intuition, motivation, the story of Mr. Bader, and entertainment. It’s at much too advanced a point of view to help you solve nuts and bolts physics exercises, which is what you must do. One final warning. Every one of us sits at a desk with a powerful internet-connected computer. Don’t do this. Even get a calculator to avoid this. Of course, when you are stumped you’ll want to see how a topic has been treated by others. Do it in another room.
It will stick with you forever.
It's terrifying that it takes 4 printings before the answers should be considered trustworthy...
Publishing a perfect book is difficult on par with writing code. Hell, Knuth is incredibly popular and crowdsources his error-checking, and TAOCP is still in its third edition.
Reading this made me nostalgic for my days as a physics undergrad.
What is wrong with airconditioning?
The library on my uni when I was in Math undergrad did not have AC at the beggining but was the only place where I could do any work, it was extremely difficult and I am sure impacted my progress.
But yeah, the idea of studying in a really cold room "makes sense" to me, and this might be why.
Actually, that sounds quite nice.
"The reader who has read the book but cannot do the exercises has learned nothing." -- J.J. Sakurai
(Incidentally, I tried reading Sakurai's Modern Quantum Mechanics on my own once and was immediately curb stomped. Lots of prep work required for that one...)
Any particular reason to recommend the old editions over the latter ones?
The recent ones are less "textbook." The older ones are FILLED with information with graphics here and there but it's mostly text. The recent ones are very graphical so I would assume it has less total information. With that said, it's possible that there are techniques for learning that were not considered in the older texts.
It is possible to look at samples online for you to compare if you want to see the difference. I do recommend getting the book if you decide to use it but that's just a personal preference.
Is it that the text you’re referring to? We used the 5th edition in my physics course this year. It was a tough textbook to learn from but I feel like I learned a ton.
[0]: https://www.amazon.com/Statistics-4th-David-Freedman/dp/0393...
There's also this free book, no answers though you could stackexchange if really stuck. I finished most of Apostol before starting it https://infinitedescent.xyz/
A final thing: it's really worth doing. If you long for maths; it's likely it'll conceptually take you places you won't go without it. Do it!
While I agree with everything else, I'd have to vehemently disagree with this. Studies [1] have shown that warm temperatures severely diminish our performance on complex mental tasks.
As some examples [2]:
> Sales for scratch tickets, which require buyers to choose between many different options, fell by $594 with every 1° Fahrenheit increase in temperature. Sales for lotto tickets, which require fewer decisions on the part of the buyer, were not affected.
> participants were asked to proofread an article while they were in either a warm (77°) or a cool (67°) room. Participants in warm rooms performed significantly worse than those in cool rooms, failing to identify almost half of the spelling and grammatical errors (those in cool rooms, on the hand, only missed a quarter of the mistakes).
[1] https://www.bauer.uh.edu/vpatrick/docs/Influence%20of%20Warm... [2] https://www.scientificamerican.com/article/warm-weather-make...
It's an introduction to mathematics from a programmer's standpoint, with a big focus on taste and that second level of intuition beyond rote manipulation and memorization.
Includes chapters on sets, graphs, calculus, linear algebra, and more! Each chapter has an application (a working Python implementation) of the ideas in the chapter. The applications range from physics to economics to machine learning and cryptography. One chapter even implements a Tensorflow-like neural network.
There's also a mailing list: https://jeremykun.com/2016/04/25/book-mailing-list/
Pick a book, pick a pace to work through it, and spend a few months going through it. Do the exercises in the back of each chapter, work through the solutions, and ask around if you still can't figure it out. Persistance and routine are key here.
As for books, I like Stewart's Calculus, Lay's Linear Algebra, and Hammack's Book of Proof.
For physics, I don't know what your background is. Giancoli is a popular undergrad freshman year book, where as griffith's electrodynamics is a bit more advanced.
For the programmers in the house, it would be like claiming you can code python if you’ve never coded before but watched some videos of expert teachers explaining the structure of python programming.
I often see people here on HN saying classroom lectures are more important than homework. But in my experience the real intuitive learning (at least for math and physics) doesn’t come until you’ve spent some time banging your head trying to figure things out solving problems (ie, applying the theory).
One text (Wangsness E&M maybe) had a great student quote, roughly “I understand the principles but I can’t do the problems.”
Also with physics, you need some tricks that are usually done in derivations or calculations (cause not everything can just be solved) and those do not necessarily appear in a book - but they do in lectures.
Edit: the book gives you the fundamentals though on which you work. also, they will teach you the necessary abstraction - the first thing standing in my way of a degree in physics was my intuition and need to picture stuff. working a lot with differential geometry now, I've got some of that visualization back but with linear algebra even and quantum mechanics it can stand in your way.
Susskind is an eminence - he was Feynman's buddy back in the day. And he's entertaining as hell.
Here's also Gerard 't Hooft's (Nobel laureate) list of concepts and books to master. If you finish that -in several years- you will be a qualified theoretical physicist. Whereas Susskind will give you more of an overview. http://www.goodtheorist.science/