Ask HN: How do I learn math/physics in my thirties?

[mosconaut]

I'm in my early thirties and I feel I've not really made any significant effort in learning math/physics beyond the usual curriculum at school. I realize I didn't have the need for it and didn't have the right exposure (environment/friends) that would have inculcated in me these things. And perhaps I was lazy as well all these years to go that extra mile.

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.

[aj7]

There is one sure way, and it’s a test of your fortitude. You find a a college textbook with the answers to the even-numbered problems in the back. You sit down in a warm or hot room, and solve them. If the textbook is in its 4th printing or so, the answers are correct. On a few, you’ll have to work for hours. Now here is a very, very, important point. All the learning occurs on the problems you struggle with. In the blind alleys. A lot of learning in physics comprises paring down your misconceptions until the correct methodology, often surprisingly simple, appears. Then, you understand how to apply the basic laws to the problem at hand, which is what physics is. I’ll emphasize the point by stating it’s converse. A problem you can solve easily and quickly yields zero knowledge.

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.

[Promarged]

I agree with this suggestion. It took me a year to slowly absorb the entire book of Statistics [0] including solving all exercises. It's just like walking to school but there is no external supervision. I made a rule to complete one chapter every evening including exercises and sticked to it.

[0]: https://www.amazon.com/Statistics-4th-David-Freedman/dp/0393...

[alehul]

> sit down in a warm or hot room

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...

[wodenokoto]

> You sit down in a warm or hot room

What is wrong with airconditioning?

[phreeza]

Seconded. I really believe there are no shortcuts to doing lots of problems. If you can afford it, getting a physics grad student to discuss problems that stumped you every now and then might also have quite good ROI, talking to physicists might also help convey some of the physics mindset(?).

Reading this made me nostalgic for my days as a physics undergrad.

[nunya213]

I'll second this idea having survived a Physics BS doing just this. I'd also strongly recommend a series of books called Schaum's Outlines, they vary in quality but cover many advanced topics and have hundreds of solved problems in them.

[usgroup]

Defs agree with op. I learned the more advanced maths I use daily in my thirties. It took about 3 years of exactly ops method. In my case, I found it motivating to take exams because it gives you a bit of skin in the game; forces you to prioritise your study at some point.

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!

[hackermailman]

This is also what I did, going straight to the exercises except I used Calculus I by Apostol which covers some Linear Algebra. Perfect book if you need to redo math skills you've forgotten though plenty of times I had to Wikipedia, Khan Academy, and math.stackexchange in the beginning.

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/

[skellera]

https://books.google.com/books/about/Physics.html?id=8MvvAAA...

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.

[3pt14159]

I almost never went to class in university (Waterloo Engineering) and this is how I did it. The best is not letting them explain the concept to you first. Try to invent the math as you go along by covering the explanatory pages with pieces of paper and reading only one line at a time.

It will stick with you forever.

[zokier]

> Halliday and Resnick, early editions , printed in the late 60s and 70s

Any particular reason to recommend the old editions over the latter ones?

[commandlinefan]

> If the textbook is in its 4th printing or so, the answers are correct

It's terrifying that it takes 4 printings before the answers should be considered trustworthy...

[j2kun]

I'm writing a book called "A Programmer's Introduction to Mathematics". Would you like to see a draft? Shoot me an email at [email protected]

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/

[ColinWright]

I can recommend anything Jeremy does, sight unseen. Take him up on his offer.

[e0m]

Watch the 3Blue1Brown YouTube channel: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

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.

[haskellandchill]

These videos are posted any time linear algebra is mentioned. I find it almost comical at this point. What good is this intuition? I struggle to understand the value that these videos bring, but I'm not saying there is no value. I'm just lost, and kind of jealous because I need a deep understanding of linear algebra for work.

[jeffreyrogers]

I second this. I was refreshing some of the linear algebra I learned in college recently and his videos gave far more insight into what linear algebra is actually about than I was taught in college. If everyone studying linear algebra in school watched his videos before taking the course they would have a much easier time learning it. His calculus series is of similarly high quality and I would imagine his other videos are too.

[Jaxan]

It is important to really pause the videos at some points and do the "exercises". Doing the exercises is always a very important part of learning (I believe this holds in any field). Do them with pen and paper, not just in your head.

[gexla]

I'm a web developer. In my 20's I used to love picking things up just for the of it. These days I'm more of a fan of JIT learning. Push the edges of my map as I go. I'm still constantly learning, but it's more iterative. Building on conquered territory and shifting the borders as needed (always outward, but the focus on which parts of the border to push changes.) Previously I was more like a crazed monkey and never holding any ground. I still feel the importance of occasionally sneaking outside my borders and going deep into enemy territory, but those are constrained efforts. Invade, gather booty, sift for intelligence value and then decide if it's worth a more serious invasion.

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. ;)

[Hasz]

Anyone who tells you can "learn" math and pyhsics by just watching videos is lying to you. There is no substitute for actually doing lots of problems.

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.

[thisisananth]

Take a look at brilliant.org. They teach mathematics in small chunks with practice. https://brilliant.org/courses/#math-advanced https://brilliant.org/courses/calculus-done-right/ https://brilliant.org/courses/linear-algebra/

[otaviogood]

People say you have to "do" math to learn it. Usually they make it sound like you need to do the exercises in the books. I think that doing just that can be boring and demotivating.

I would suggest finding projects that can motivate you and help you exercise your math. Some suggestions of mathy things I regularly work on for fun:

1. Make a video game. If it's a 3d game, you'll have to do your matrices, dot products, trigonometry, etc.

2. shadertoy.com - This is a community site where people just program cool looking graphics for fun. All the code is open, so you can learn from it. Similar to game programming but without the mathless overhead. :)

3. Machine learning projects - I love writing various machine learning things, but the project that has been a great ML playground has been my self driving toy car. It gives me plenty of opportunities to explore many aspects of machine learning and that helps drive my math knowledge. My car repo is here: https://github.com/otaviogood/carputer but a much easier project is donkeycar.com. ML will touch on linear algebra, calculus, probabilities/statistics, etc.

The most important thing for learning is to be inspired and have fun with what you're learning. :)

[rosencrantz]

Get a real pen and paper, get a real physical book, sit and solve problems with pen and paper for hours every day for a few months. Then you will pass the exams.

[hathawsh]

Exactly. Every time I tutor someone in math, I tell them to use up at least a sheet of paper for every interesting question. When they do, their skills improve quickly. Saving paper is a false economy when it comes to math.

[Clubber]

I don't know why this is downvoted, but the process of writing and working problems on paper, at least for me, helps cement the knowledge.

[aj7]

This is exactly right. I elaborate on the method in my response.

[Ian_Paul]

I can't speak to all of the things you want to learn, but I've learned some of them on my own. For calculus and linear algebra I'd go with Khan Academy, especially since it seems like all you need is a refresher for calculus. Graph theory and discrete math I did with MIT EdX courses. Their discrete course is pretty nice and I found it very easy to follow along with.

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.

[Jemmeh]

+1 to Khan Academy. Explanations are super clear. Their website allows you to work through practice problems too, which I think is the most important thing.

[Panoramix]

For physics, I believe you fall in Leonard Susskind's target audience. You can get his book, or even better, watch his large amount of lectures: https://www.youtube.com/watch?v=iJfw6lDlTuA

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/

[doall]

To quickly review a broad range of math up until 1st or 2nd year of university, I really recommend Khan Academy https://www.khanacademy.org/math . I am currently using it to brush up my math skills for machine learning.

Before going to Khan Academy, I started reading a rigorous math textbook, but my motivation didn't last long. You really need high motivation to complete a rigorous textbook, but Khan Academy is different and I am finally able to continuously improve my math skills.

The best thing I like about Khan Academy is the large amount and instant feedback of exercises that you don't get from regular textbooks. I really wished that Khan Academy was there when I was a kid.

To get deep knowledge of math, I think that rigorous textbooks are the way to go, but before those and to prepare for them, I really recommend Khan Academy.

[chipuni]

Being in your thirties has little to do with learning. How you learn is much more important than your age.

If 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!

[wenc]

Very true.

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.

[tucaz]

I could say I'm in the same boat. Always wanted to learn such things, but never found the motivation to do so in an effective way.

I picked up various books and different learning strategies along the years but couldn't move forward cause I could not see any practical use for what I was trying to learn.

Fast forward a few years and now I'm learning both physics and mathematics.

What changed is I started working with 3D development for the furniture industry and a while later I got interested in woodworking.

Started doing some woodworking projects and had to learn some basic geometry and trigonometry to calculate cuts.

Now I'm interested in mechanical machines and electrical machines. To be able to build my own machines I have to learn some physics and other branches of mathematics and that's what I have been doing for the past months.

I probably cannot work with formal physics or mathematics but I was able to learn a lot of the concepts behind the formulas and calculations and I believe that is much more important, at least, at first.

The bottom line is you need to find something that motivates you and make you want to learn. That's how it worked for me.

[Tomminn]

Honestly, despite all the crap universities get, taking an undergraduate degree with a double major in physics and maths is an awesome way to do this. You'll meet people who are similarly passionate, be naturally competitive with them which is a motivating force not to be underestimated, and you'll meet a diverse set of teachers who each will have some awesome insights into these fields and you'll get to see first-hand how they think about solving problems.

Physics, and to a lesser extent maths[1], are topics where the top 1% of ability are actually concentrated at universities. My advice would be find a cheap university nearby and start enrolling in courses. If you're bright, motivated and take ownership of your own learning, the faculty will love interacting with you. If you're doing it to learn, don't sweat about the prestige of the place. There are people everywhere who will be much better than you at this stuff, and in some ways it's extremely motivating if you feel like with some hard work you can surpass some of your teachers, and it's extremely motivating when the best teachers recognize you as having more potential than the average student. You're never gonna feel either of these things at MIT.

[1] The problem with maths in academia is that it's massively biased toward proofs of mathematics and not use of mathematics. I've met very few PhD mathematicians who are even as close to as good at applying appropriate mathematics to problems then someone with a PhD in physics who consider themselves >50% theorist. PhD mathematicians are wonderfully knowledgeable if you say "tell me about this field of mathematics" and it's a field they know. But there is a certain extent to which they like to work by building things on a frictionless ice world, and get uncomfortable if asked to build something on the rough ground of the real world.

[illlogic2]

Hi,

I'm 30 and trying to relearn the math courses I did in college (Computer Science degree) and more. I am currently using Standford & MIT's open couseware. I feel like I am moving slower than I would if I were in a course but able to grasp the material better at this rate... I made good grades in my math courses but like you, I didn't have to use them in software engineering that much. I would like to get into a field that requires a stronger grasp of mathematics but also has a need for programming and computation (maybe machine learning or computational biology). I feel like I'm getting tired of being a software engineer (defense contractor) at a small company and looking for something higher level

Calculus (with a pdf version of the text book): https://ocw.mit.edu/resources/res-18-001-calculus-online-tex...

Linear Algebra (text book link: https://www.amazon.com/exec/obidos/ASIN/0980232716/ref=as_at...) https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...

Optimization course & book link (Stanford) https://web.stanford.edu/~boyd/cvxbook/

Statistics: http://greenteapress.com/wp/think-stats-2e/

[dkural]

I'm taking you at your word and assuming you truly want to reach the cutting edge of knowledge and learn things like QFT, Gauge Theory, String Theory, etc. alongside the math needed for it.

This is a long-term project, so I'd recommend by starting a bit with "learning about learning". There is a great, fairly short Coursera course called "learning how to learn".

Things not covered in the above course: - Your learning ability is not actually much lower in your 30s than it was in your 20s. You have a relatively benign rate of learning decline, until your late 50s / early 60s, when it drops quite a bit. You can still learn a lot, but it's meaningfully harder. (learning rate != thinking rate / creation rate!!)

You'll likely be able to write good papers into your late 60s, and perhaps 70s. There are exceptions and people who do significant work even later, but that's more unlikely.

When I started learning stuff again in my late 20s, I felt frustrated because I'd take a couple courses over a year, and by the time the year's over, I'd forget the first one. We all know what this feels like - we've forgotten most of what we've learned in college that we don't use in our profession.

When I was a teenager, I had great recall for things I've learned only once or twice. I didn't realize this was unusual and thought I got very bad at learning. In fact, the vast majority of people, including high IQ people, will need spaced repetition and study to retain things for a long period.

I'd recommend the following "schedule" to absorb things into y memory permanently. (By "learn" I mean read, do problems, write summaries.. it's a wide range):

Days to repeat: 0 (initial learning), 1, 6, 15, 37, 93, 234, 586, 1464, 3662, 9155

This would suggest interleaving classes instead of learning things sequentially for optimal time management. It's also a bigger time investment than people usually think of upfront, but pays dividends later on as the material builds-up like a cathedral of knowledge.

Like other commenters I'll also repeat: Do problems, problems, problems. The struggle is where the learning happens.

On the other hand, I wouldn't worry too much about super-high IQ etc. I don't think it's a strict requirement to have an extraordinary IQ to learn grad school physics and math.

[AIX2ESXI]

Great post. Rather thought provoking. Though I dropped out of college in my late teens, I started taking classes again 17 years later later am doing much better than I did before. As a sysadmin, I was always reading all sorts of subjects and pursuing different hobbies that further expanded my knowledge. The one subject I have been having issues is with Math, but that is due to lack of effort and stretching myself too thin.

[MockObject]

That is a fascinating number series. Is it taught in the Coursera course you mentioned?

[markhkim]

Taste, you say? The Princeton Companion is your friend: https://press.princeton.edu/titles/8350.html https://press.princeton.edu/titles/10592.html

[EliRivers]

I took a Masters of Mathematics with the Open University in my thirties.

The (my) short answer is grind. Get a good textbook on subject of interest, start reading, start scribbling, start answering the questions. That's how I did it. Three or four days a week, two to six hours a day, grind grind grind GRIND GRIND GRIND GRIND. It's geology; time and pressure.

Now and then, when really stuck, finding someone who can illuminate a point for you is worth it, but that helped me a lot, lot less than one might think.

Anticipate not understanding large chunks of it. Anticipate pressing on anyway. Anticipate not being able to answer many questions. Anticipate having to read three or four different treatments of the same thing in order to get a real understanding. Anticipate that some of it you will never understand. Anticipate that watching youtube is not a substitute. Anticipate that the sheer information density of well-written text means you might spend an hour on a single page. Anticipate questions taking you six hours to solve, leaving your table and floor strewn with the history of your consciousness. If you're prepared for all that, and it's a price you're willing to pay, there is no reason to not simply start now. Pick up the first good textbook, start grinding now. Time and pressure. It's so easy to waste time preparing to start learning; beyond making sure it's a decent textbook and getting some pencils and paper in a quiet room, the only preparation is accepting that this is going to be a long grind, and embracing it.

[hitechnomad]

I'm doing my OU Masters in Maths now, in my 40s. It is definitely hard, but I'm enjoying it. Personally I struggle to learning things thoroughly unless I'm working in the subject, or I have exams to do. My own learning workflow is to flip through a chapter to get an overview, then read/re-read it thoroughly, then go through the exercises on the chapters quickly looking at the answers. Read the chapter again, then try doing the exercises without help. I've found YouTube pretty good for getting the intuition behind some ideas.

[agentultra]

Read some books, practice exercises, and find an area of interest.

Start with some liberal-arts introduction to a particular topic of interest and delve in.

I often find myself recommending Introduction to Graph Theory [0]. It is primarily aimed at liberal arts people who are math curious but may have been damaged or put off by the typical pedagogy of western mathematics. It will start you off by introducing some basic material and have you writing proofs in a simplistic style early on. I find the idea of convincing yourself it works is a better approach to teaching than to simply memorize formulas.

Another thing to ask yourself is, what will I gain from this? Mathematics requires a sustained focus and long-term practice. Part of it is rote memorization. It helps to maintain your motivation if you have a reason, a driving reason, to continue this practice. Even if it's simply a love of mathematics itself.

For me it was graphics at first... and today it's formal proofs and type theory.

Mathematics is beautiful. I'm glad we have it.

[0] https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathe...

Update: I also recommend keeping a journal of your progress. It will be helpful to revisit later when you begin to forget older topics and will help you to create a system for keeping your knowledge fresh as you progress to more advanced topics.

[asafira]

Here's how I would do it, my 2 cents:

1) Find a good source of information --- typically, this is either very good lectures (like on youtube), a good textbook, or good lecture notes.

2) Do problems. There is a fairly large gap between those that just watch the lectures and those that have sat down and try to go through each and every step of the logic, and that's what everyone here (on HN) is pointing out when they similarly mention doing problems.

2b) Have solutions to those problems. I make this a separate point because it's important to spend quality time on a problem yourself before looking at the solutions. At the end of the day, if you read the problems and then the solution right away, that's much closer to reading the textbook itself instead of the more rigorous learning one goes through when trying things themselves.

If you were to ask me what textbooks or lectures I recommend, I think that's a more personal question than many here might guess. What topics are you most interested in? Are you really just solely interested in a solid background? How patient are you when doing problems?

Regardless, I'll give my two cents for textbooks anyway. In no particular order:

1) Griffiths E&M: https://www.amazon.com/Introduction-Electrodynamics-David-J-...

2) Axler, Linear Algebra Done Right: https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Ma...

Good luck!