Ask HN: Best resources to gain math intuition?
This is something that I&#x27;ve struggled with myself so I can only give you advice as I&#x27;ve learned it from my perspective.<p>Math is a big field so you have to understand what parts you&#x27;re really interested in and how you want information to be presented so that you&#x27;ll learn it. For example, I&#x27;m interested in computation aspects of mathematics (information theory, computation group theory, abstract algebra, analysis, etc.) and I always prefer a &quot;computer programmers&quot; intuition in how to learn these subjects. That is, understanding how to &#x27;program it&#x27;, whatever that means for the different subjects I&#x27;m interested in.<p>Here is a small list of intuition I&#x27;ve learned about finding good resources:<p>* Books, and sometimes textbooks, are still a valuable resource. It&#x27;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&#x27;s a field I&#x27;m not familiar with.<p>* When looking for books, prefer books that have &quot;elementary&quot; in the title, as in &quot;elementary introduction&quot;. The more &quot;advanced&quot; books are talking about the bounds of research in the area, often fussing over esoteric issues whereas the &quot;elementary&quot; books give the foundation of knowledge in that area.<p>* Ideally for me, books would have &quot;fundamental algorithms&quot; somewhere in the title, as these books usually are exactly what I need to understand a field.<p>* 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 &quot;learn by doing&quot;. Finding good resources so those exercises are meaningful is hard but they still need to be done.<p>* 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 &quot;what is the motivation behind...&quot;. 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.<p>* In the past I&#x27;ve watched more in depth lectures from mathematicians, either from conferences or from things like OpenCourseWare. There&#x27;s a lot of &#x27;folklore&#x27; wisdom that&#x27;s embedded with people that sometimes comes out when viewing actual researchers talk about their research that wouldn&#x27;t otherwise be apparent or emphasized in papers.<p>* I sometimes visit blogs from mathematicians or about mathematicians.<p>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.<p>I should also mention that there isn&#x27;t &quot;one way&quot; to learn about these subjects. I take a computational perspective because that&#x27;s my preference but I&#x27;m fully aware that not everyone thinks that way. Every person has their own perspective on what&#x27;s fundamental and how they learn and build intuition even if they can be grouped in to rough categorizations (though I&#x27;d be hard pressed to quantify those categorizations). I&#x27;ve found the way I learn and optimize for it and I unfortunately have a hard time when information isn&#x27;t presented in the way I need it to be, at least initially while I&#x27;m building intuition and learning a subject for the first time.<p>I can&#x27;t find the quote now but there was a mathematician that was talking about Erdos and how Erdos didn&#x27;t have deep knowledge or at least didn&#x27;t use &quot;higher mathematics&quot; like Lie theory or other higher abstractions. Yet Erdos was prolific in his sense with his &quot;elementary&quot; methods, probably because he understood his tools and the problems deeply. As an analogy, it&#x27;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.<p>I also try to employ the &quot;20% effort for 80% gain&quot; 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&#x27;s not.<p>I try to avoid resources that are &quot;TED talk&quot; like, press releases, or other &quot;feel good&quot; 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.<p>I tend to stay away from Springer books as they&#x27;re usually dense. They might be good for reference but for initial learning I&#x27;ve found them to be pretty bad.<p>People often say &quot;read the original papers&quot; but I found this to be horrible advice as the original papers often are a very rough &#x27;proto&#x27; model of the ideas presented and don&#x27;t benefit from work that&#x27;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&#x27;s paper on information theory.<p>In no particular order, here are a list of books I&#x27;ve found exceptional (very much catered to my personal taste):<p>[0] Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnsen<p>[1] The Way of Analysis by Strichartz<p>[2] Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein<p>[3] Fundamental Problems of Algorithmic Algebra by Yap<p>[4] Fundamental Algorithms for Permutation Groups by Butler<p>[5] A Mathematical Theory of Communication by Shannon<p>[6] Complexity and Criticality by Christensen and Moloney<p>I have not found what I consider exceptional texts on number theory, Galois theory or cryptography.<p>Here are some blogs I occasional visit:<p>[7] <a href="https:&#x2F;&#x2F;rjlipton.wordpress.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;rjlipton.wordpress.com&#x2F;</a> - Godel&#x27;s Lost Letter and P=NP<p>[8] <a href="https:&#x2F;&#x2F;terrytao.wordpress.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;terrytao.wordpress.com&#x2F;</a> - Terrence Tao&#x27;s blog<p>Here are the SO sites:<p>[9] <a href="https:&#x2F;&#x2F;mathoverflow.net" rel="nofollow">https:&#x2F;&#x2F;mathoverflow.net</a> - Math overflow<p>[10] <a href="https:&#x2F;&#x2F;math.stackexchange.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;math.stackexchange.com&#x2F;</a> - Math &quot;underflow&quot;<p>[11] <a href="https:&#x2F;&#x2F;physics.stackexchange.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;physics.stackexchange.com&#x2F;</a> - Physics SE<p>[12] <a href="https:&#x2F;&#x2F;cstheory.stackexchange.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;cstheory.stackexchange.com&#x2F;</a> - Theoretical Computer Science SE<p>Math videos:<p>[13] <a href="https:&#x2F;&#x2F;www.msri.org&#x2F;videos&#x2F;dashboard" rel="nofollow">https:&#x2F;&#x2F;www.msri.org&#x2F;videos&#x2F;dashboard</a> -  MSRI Videos