Ask HN: Best resources to gain math intuition? In high school I did the lowest tier maths and then jumped in the deep end by doing a year of electrical engineering. I’ve now done those harder math classes but I feel like there’s holes here and there. I think when I took physics it really brought out these flaws and lack of intuition. Would anyone have a good resource for building this up? Thanks |
Math is a big field so you have to understand what parts you're really interested in and how you want information to be presented so that you'll learn it. For example, I'm interested in computation aspects of mathematics (information theory, computation group theory, abstract algebra, analysis, etc.) and I always prefer a "computer programmers" intuition in how to learn these subjects. That is, understanding how to 'program it', whatever that means for the different subjects I'm interested in.
Here is a small list of intuition I've learned about finding good resources:
* Books, and sometimes textbooks, are still a valuable resource. It's still the case that having a book on a subject that has curated content is better than the random Wikipedia articles or blog posts on the subject. Use Wikipedia, obviously, and look at blog posts, but I search for books in the subject area, especially if it's a field I'm not familiar with.
* When looking for books, prefer books that have "elementary" in the title, as in "elementary introduction". The more "advanced" books are talking about the bounds of research in the area, often fussing over esoteric issues whereas the "elementary" books give the foundation of knowledge in that area.
* Ideally for me, books would have "fundamental algorithms" somewhere in the title, as these books usually are exactly what I need to understand a field.
* When reading, ideally I make sure to do the exercises or run through the proofs myself. Mathematics is not a spectator sport and a large part of it is "learn by doing". Finding good resources so those exercises are meaningful is hard but they still need to be done.
* I often check MathOverflow, MathUndeflow, Physics.Stackexchange, CStheory.Stackexchange and other accompanying sites. There are a surprising number of good answers to questions of the form "what is the motivation behind...". As the subjects get more esoteric, these questions become more infrequent these resources are still invaluable. Asking questions on these sites is also an option and usually helpful.
* In the past I've watched more in depth lectures from mathematicians, either from conferences or from things like OpenCourseWare. There's a lot of 'folklore' wisdom that's embedded with people that sometimes comes out when viewing actual researchers talk about their research that wouldn't otherwise be apparent or emphasized in papers.
* I sometimes visit blogs from mathematicians or about mathematicians.
When I was younger in college, I was fortunate to have a social group of friends who were graduate students and TAs that had an appetite for discovery and teaching. There was a lot of folklore and intuition that was taught which would have been difficult to find otherwise. I think many graduate students in mathematics essentially use their exposure to their advisor, other teachers and other students to build that intuition.
I should also mention that there isn't "one way" to learn about these subjects. I take a computational perspective because that's my preference but I'm fully aware that not everyone thinks that way. Every person has their own perspective on what's fundamental and how they learn and build intuition even if they can be grouped in to rough categorizations (though I'd be hard pressed to quantify those categorizations). I've found the way I learn and optimize for it and I unfortunately have a hard time when information isn't presented in the way I need it to be, at least initially while I'm building intuition and learning a subject for the first time.
I can't find the quote now but there was a mathematician that was talking about Erdos and how Erdos didn't have deep knowledge or at least didn't use "higher mathematics" like Lie theory or other higher abstractions. Yet Erdos was prolific in his sense with his "elementary" methods, probably because he understood his tools and the problems deeply. As an analogy, it'd be like someone who knows assembly well trying to analyze a Haskell script. The Haskell programmer might have intuition from the constructs of that language but someone who knows assembly well understands that each of the abstractions in Haskell must eventually boil down to assembly instructions and can understand it from that perspective.
I also try to employ the "20% effort for 80% gain" rule. There are usually some basic concepts so learning them as fast as possible is the goal. This also allows for maximum gain for effort spent as if the field is interesting, I can dive deeper or move onto another if it's not.
I try to avoid resources that are "TED talk" like, press releases, or other "feel good" resources, like 3Brown1Blue. These are great for being inspired by mathematics (which is important!) but are usually devoid of content. Resources like 3Brown1Blue I find especially pernicious as they couch deep understanding by regurgitating facts without providing any fundamental insight.
I tend to stay away from Springer books as they're usually dense. They might be good for reference but for initial learning I've found them to be pretty bad.
People often say "read the original papers" but I found this to be horrible advice as the original papers often are a very rough 'proto' model of the ideas presented and don't benefit from work that's been done to simplify and extract the important parts of the theory without the cruft. Often times, mathematicians have their own pet notation which further get in the way of understanding. One exception is Shannon's paper on information theory.
In no particular order, here are a list of books I've found exceptional (very much catered to my personal taste):
[0] Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnsen
[1] The Way of Analysis by Strichartz
[2] Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein
[3] Fundamental Problems of Algorithmic Algebra by Yap
[4] Fundamental Algorithms for Permutation Groups by Butler
[5] A Mathematical Theory of Communication by Shannon
[6] Complexity and Criticality by Christensen and Moloney
I have not found what I consider exceptional texts on number theory, Galois theory or cryptography.
Here are some blogs I occasional visit:
[7] https://rjlipton.wordpress.com/ - Godel's Lost Letter and P=NP
[8] https://terrytao.wordpress.com/ - Terrence Tao's blog
Here are the SO sites:
[9] https://mathoverflow.net - Math overflow
[10] https://math.stackexchange.com/ - Math "underflow"
[11] https://physics.stackexchange.com/ - Physics SE
[12] https://cstheory.stackexchange.com/ - Theoretical Computer Science SE
Math videos:
[13] https://www.msri.org/videos/dashboard - MSRI Videos
1. Do exercises from competitive-exam books written for secondary school children.
2. Try to solve as many as possible exam papers in stipulated time.
This will help you to build intuition. Intuition comes only after doing something for long time (unless you are born-genius). Memorization plays very important role in learning maths, language, music instruments. On second note, don't fear to change resources if you're not comfortable. What I mean is - in childhood if you like teacher, you excel in subject. So you have to find material which match your temperament. Wish you best luck !
It depends on why you seem to be "missing" something and your post doesn't actually give any insight into what seems to be missing for you.
For me, math intuition takes practice and a lot of visuals, graphing and diagrams. Some people get it through proofs and equations, but I feel like the way math is taught in school doesn’t usually do a great job of developing intuitions. Coding it up forces me to learn it, and playing with it and tweaking a lot along the way helps develop the intuition.
BTW, John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them."
As a mathematician, I say: what you lack is not math intuition, but math knowledge. Read really hard math textbooks and solve exercises. This is the only way known to me to get better at math.
I'd add, speaking as a math professor: don't worry too much about holes and lack of intuition. The more you learn, the bigger you will feel like your holes are!
Whatever you choose, in my opinion it should involve solving lots of problems. Subject to that, I'd recommend simply that you dive in to any math subject that attracts your attention.
I think this is relevant with regards to your perceived lack of intuition: they have a way of making you discover the solution to complex problems by introducing complexity through a succession of simple problems, in a very piecemeal manner.
I think, with modern tools, you need math intuition more than computational skills. That's why my site is all about interactive experience. There are no proofs or problems but a lot of things to tweak and drag.
I think category theory will give you an unique point of view of math, also basic logic and philosophy are very important for a good and solid foundation on math intuition.
https://arxiv.org/abs/1803.05316 along with their youtube lectures.
very basic but refreshing:
https://www.youtube.com/playlist?list=PL8dPuuaLjXtNgK6MZucdY...
if you are a programmer or developer:
https://bartoszmilewski.com/2014/10/28/category-theory-for-p... https://blog.ploeh.dk/2017/10/04/from-design-patterns-to-cat...
Another part of intuition is from Max Zorn (from Zorn's lemma statement of the axiom of choice):
"Be wise, generalize."
E.g., for the set of real numbers R and a positive integer n, a lot that goes on in the n-dimensional vector space R^n is a generalization of what can see in 3D, e.g., from solid geometry.
E.g., in both cases, a biggie is a perpendicular (orthogonal) projection and, again, the Pythagorean theorem. E.g., regression in statistics is a perpendicular projection.
Perpendicular (orthogonality) is a biggie and is a major part of, say, Fourier series. I.e., each of the sine/cosine waves used is an orthogonal axis, and to find the corresponding Fourier series coefficient just project onto that axis. The projection is an integral of a product, and that is commonly an inner product which close to just a cosine of an angle as in plane and solid geometry and a perpendicular projection and close to correlation in statistics, etc.
E.g., a huge fraction of applied math is from analysis in pure math, and from G. F. Simmons the two pillars of analysis are "continuity and linearity". Linearity generalizes enormously: The quantum mechanics super position is essentially linearity. Under meager assumptions, differentiation and integration in calculus are linear operators. In probability theory, expectation is a linear operator. The wave equation is a linear partial differential equation. Linear programming works on linear equations. Of course, in linear algebra, matrix multiplication is a linear operator. When something is not linear, it may be locally linear which can be enough to get useful results.
For more, a good lesson is to approximate: Commonly we can't get just what we want in just one step but can iterate and approximate as closely as we please. So, can use simple things, sine waves, polynomials, continuous functions, and more, as means of approximation. Such approximation gets us close to more in continuity and, in particular, completeness -- the real numbers are complete and the rational numbers are not but via iteration can approximate the reals as closely as we please. Then this generalizes: The big point about Hilbert space (as mathematicians but not always physicists define it) is completeness. A joke, partly correct, is that "calculus is the elementary consequences of the completeness property of the real number system". E.g., the integral in calculus (and its better version in measure theory) is defined in terms of an iterative approximation. So, if you are good with sine waves, polynomials, continuous functions, wavelets, and more, then you can iterate and approximate a lot, in many cases, everything there is in that case.
I think that the way to get math intuition is to learn a mathematical language, like Julia, and play around with it. Plot things. Change parameters.
Also learn a theorem prover. Maybe Agda or Coq or Lean.
but for basic things, funny enough i feel like some of these math games / apps and math puzzle apps help me recognise patterns in numbers etc. better.
that being said, i hardly use maths ,certainly not advanced maths. but i do feel things like that are somewhat fun way to keep your senses a bit sharper.
in the end, practice makes perfect, but doing maths straight up is boring as hell to me, so i try my hand at little puzzle and maths games / apps to combat the boringness.
Try to be around problems that requires a high degree of math skills to solve. let the problems drive you.
intuition is usually built by spending hundreds of hours thinking, eating, and drinking the problems you're truly interested in to arrive to a solution.
If you spend enough time doing this, you'll know exactly what areas you need to inquire about as your understanding to the problem becomes clearer and gaps start to narrow down.
If you don't know what you need to learn you probably haven't spent enough time on the problem or just read the solution which made you feel this way, not realizing that the solution was done by a person that could have spent 100x the time you spent on it and built the intuition you're asking about.
If something is to dig here is that the earlier you start the better you will be.
I guess the loophole here is simply being born with talent :P
https://betterexplained.com/articles/intuitive-understanding...
It’s on another level imo.
I'd love to filter the internet by "content that can give you insights right now that would otherwise take years of study in a specific discipline to even know exists".
Another that I saw recently on an HN comment (thanks tptacek) is https://www.youtube.com/watch?v=nfY0lrdXar8
A quick google for "3blue1brown awesome github" (my usual strategy for finding good similar content) landed me here: https://github.com/rossant/awesome-math
The awesome lists are pretty good starting points for finding good content, but there's a ton of them of variable quality, so you end up with stuff like this: https://github.com/jonatasbaldin/awesome-awesome-awesome (which has 68 forks...)
His whole channel is also on LBRY if you don't like YouTube https://beta.lbry.tv/@3Blue1Brown
May be it is distinct from others, may be you will not be 'narrowing' to the right answer within seconds -- like many folks who do Olympiads...
Just do basic things, but every other day. Get books/materials that have solutions (not just problems). use those, compare your results, and then try again.
If you feel like you do not understand 'why', you will need a particular subject area. Switch to read about applications of that area, how historically it came about and so.
And then back to problem solving, proofs, and reading other people's papers (when you can..).
It is hard work, but over time you will build up your version of the so called intuition, it will be powerful, you will be able to apply it all over the place.
Also there are a number of math forums where you can reach out, if you are really stuck and cannot figure out how a particular proof, or solution was obtained.
A common theme I have noticed is whatever it is, getting a geometric understanding of it aids intuition significantly. Others have mentioned the 3Blue1Brown videos. They are an excellent example of this.
These HN threads always bear out great resources and I've made note of (and acquired) a few of these, so I'll list them here.
Burn Maths Class - https://www.goodreads.com/book/show/26195956-burn-math-class
Book of Proof - https://www.people.vcu.edu/~rhammack/BookOfProof/
The Topology Of Numbers (number theory)- http://pi.math.cornell.edu/~hatcher/TN/TNpage.html
The Evolution of Trust (game theory) - https://ncase.me/trust/
Visual Information Theory - https://colah.github.io/posts/2015-09-Visual-Information/
Information Theory For Smarties - http://tuvalu.santafe.edu/~simon/it.pdf
Abstract Algebra - https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...
Algebra Cheatsheet - https://argumatronic.com/posts/2019-06-21-algebra-cheatsheet...
Control Theory Basics - https://www.youtube.com/user/ControlLectures/playlists
[0] https://www.amazon.com/Pure-Mathematics-Beginners-Rigorous-I...
So if you're looking for intuition, you're looking for deep familiarity which means spending the time/effort it takes to get familiar.
Resources don't help much as they're almost always about someone else's "intuition". Struggle that leads into false turns and force you to backtrack help a lot more I think.
An example - some time ago, I told some colleagues that they've been dealing with convolution even before middle school .. which surprised them. Multiplying numbers is the same operation as convolving their digit sequences. But if you didn't fool around enough with multiplication, or try to relate it to algebra, polynomials, and such, this can be hard to see.
Edit:typo fixed
Alternatively if you're really interested in intuition, you could also look at the Math Olympiads. Pick a problem, beat your head on it, finally look at the solution, repeat. There are web sites and prep books.
At the high school and college level, the Olympiads for math and CS are pretty analogous. But there's really popular semi-formal coding contests which exist outside academia which don't really have a math equivalent.
I'd say math contests are more popular among high schoolers, and semi-formal coding contests more popular among college students.
Art of Problem Solving (AoPS) [https://artofproblemsolving.com/] is a really good resource, and there's a very healthy online community.
They're also similar in how olympiads are different from the "real thing" (TM).
Academia.SE discussion about this [https://academia.stackexchange.com/questions/86451/does-the-...]
As someone who did math olympiads in high school, my 2 cents is that they're a fantastic way to learn how to solve and approach problems and gain intuition. And I'd say intuition mainly comes from solving problems.
https://mitpress.mit.edu/books/street-fighting-mathematics (free - look for pdf download link)
https://www.betterworldbooks.com/product/detail/How-to-Solve...
I think it can also be helpful to learn some things about the history of math and the historical context that different ideas came from. Here's a nice example covering complex numbers https://www.youtube.com/watch?v=T647CGsuOVU
Maybe tangential to your ends, but the Crest of the Peacock is a nice book on non-European mathematical traditions, which provides some insight into how the process of establishing and validating mathematical knowledge works in other cultures.
You might try books written by physicists or that are about mathematical physics (an author to look out for depending on your level is VI Arnold), since arguments will be of a more geometric or physical nature and appeal more to intuition. Stillwell is another author (not a physicist) that tends to write books that give context and geometric intuition
You might like playing around with Pinter & Humphreys for Algebra, or Jänich for Topology (fantastic book for building intuition around topology).
There's a linear algebra lurking everywhere in the realm of applied math (some people like to joke that machine learning is really just linear algebra) so it really is worth your time to have a firm understanding of it.
However, you have written
> I think when I took physics it really brought out these flaws and lack of intuition
which suggests you have some good practical experience with physics problem solving which has precipitated a certain feeling that you need to learn more about some kind of math. I would advise that you try to exploit this. In the same breath I want to recognize (as someone who did their BS and MS in physics) that physicists are not always so careful or explicit in how they are doing their mathematics. So learning means eventually going beyond physics sources and into a much wider world of mathematical thought. The particular things that mathematicians care about may or may not be relevant to the problem you are trying to solve in physics, and a good part of developing that intuition is to figure out which particular caveats that a mathematician expounds upon (more often than not, some esoterica about the space(s) that they are working in or the class of isomorphisms under which their results are invariant) matter physically. As you develop and intuition about these things a bonus is that you will be able to skim through mathematics resources much faster.
You don't understand mathematics,
you just get used to it - jvn
It's also a language when someone is teaching it.
Prior knowledge is assumed; so it's difficult to detect that it is a gap that's making it difficult. When I finally realized my hard-won breakthroughs were about prior material, I dropped back to do that prior level... again and again.
This thorough approach is far too time-consuming for your needs. I really hope you find a quicker way - and please tell me!
[0] https://www.amazon.com/Programmers-Introduction-Mathematics-...
Source: listen from 28:35 to 29:35 https://overcast.fm/+Soyvpq978/28:35
[0] Mathematical Proofs: A Transition to Advanced Mathematics https://g.co/kgs/stSmxJ
Im sort of taking the approach of learning the definitions of things via Anki in a real exploratory way. Just read about something that takes my fancy then I keep finding I run into definitions I've learned before and this time something that was previously gibberish made more and more sense.
Recently I read the paper posted here about how an optimal regulator needs to model what it's trying to regulate. I was amazed at how much more of it I had access to due to learning some definitions here and there.
I've picked up a Rubik's cube lately and am learning group theoretic concepts to help me solve it. I'm finding it a neat way to engage with the maths in a practical way too!
It doesn't shock me that there are holes. I noticed that some math topics are very important to engineering and physics coursework, but given short shrift in the math department. Examples are the way that complex numbers are used, and specific kinds of differential equations such as the general harmonic oscillator.
My college physics coursework actually had its own "math methods" class, intended to fill some of those gaps, and to get us prepared for the higher level physics courses.
Mathematics is a very general tool. As with any very general tool, a lot of the devil is in the details of how to use it in any particular domain.
For this reason, in-sourcing mathematics service courses is best for everyone. The very best math-adjacent departments in every field tend to do this either directly or indirectly. E.g., in the direct model, many CS departments internalize the Discrete Mathematics course and some combinatorics. And an example of the indirect model is Mathematics departments that hire Math Finance professors to cover the service load for econ/fin/bus depts.
I think this in-sourcing (either directly or indirectly) is best for everyone -- mathematics depts don't do a good job at teaching those service courses and often don't do a great job of it in any case. Unfortunately, most departments don't have the headcount (in students or faculty) for a specialized mathematics curriculum, so they have to share the math faculty with N other majors to predictable effect.
Shoot an email over to codertutor@gmail.com or text or call +1-718-360-3176 if you are interested or if you have any questions.
This is my print flyer for reference: https://i.imgur.com/Gdwa7m9.png
Looking forward to hearing from you.
I got my undergrad in math and physics. I was good at math. It wasn't until I had been teaching high school for 3-4 years when some gave me a copy of Shortcut Math by Gerard Kelly. After reading it and practicing the techniques, arithmetic made so much sense. I was able to easily add, subtract, and multiply larger numbers in my head.
Interestingly enough, many of the techniques taught in this book are also part of the common core math curriculum. It's a way to help students gain number sense.
I asked this a few weeks ago but at an off peak time and not many folks saw the question. But I know HN probably has good recommendations so trying again here. Should still be relevant and helpful to OP.
A post I like is on adding numbers 1 to 100 [2]. The staple formula is n(n+1)/2, sum of arithmetic progression. How can we intuitively arrive that this formula?
> Technique 1: Pair Numbers Pairing numbers is a common approach to this problem. Instead of writing all the numbers in a single column, let’s wrap the numbers around, like this:
1 2 3 4 5
10 9 8 7 6
An interesting pattern emerges: the sum of each column is 11. As the top row increases, the bottom row decreases, so the sum stays the same.
Because 1 is paired with 10 (our n), we can say that each column has (n+1). And how many pairs do we have? Well, we have 2 equal rows, we must have n/2 pairs.
Number of Paris x Sum of Each Pair = (n/2) (n + 1) = n(n+1) / 2
[1] https://betterexplained.com/
[2] https://betterexplained.com/articles/techniques-for-adding-t...
For me it was trying to frame as much as possible in a geometric lens. Draw pictures of anything and everything as much as possible. Graphs, number lines, 3d animations, whatever it takes.
Also, understand the "bold print" from the "fine print". I mean to say, every theorem will make a general statement, and then have a bunch of conditions where it holds. So worry first about the bold print, then try to understand the fine print.
Much of mathematics is about making the strongest and most general statements in bold print with as little fine print as possible. Basically the less fine print you have, the more important probably your result is.
There was a psychology teacher that had this theory, so when he had kids he started teaching them to play Chess as soon as possible and his two daughters became world champions
