The way of analysis

What Hacker News says

1. abetuskApr 2025
   I'm not a practitioner, so read with some skepticism, but here's my list:

   * Calculus

   * Real Analysis

   * Statistical Mechanics

   * Probability

   I'm not sure I have any good recommendations for Calculus, but for real analysis, I would recommend "The Way of Analysis" by Strichartz [0].

   I don't have good recommendations for books on statistical mechanics, as I haven't found a book that isn't entrenched in coming from a physics perspective and teaches the underlying methods and algorithms. The best I can recommend is "Complexity and Criticality" by Christensen and Moloney [1], but it's pretty far afield of statistical mechanics and the like. Simulating percolation, the Ising model and ricepiles uses a lot of the same methods as financial simulation (MCMC, etc.).

   For probability, I would recommend "Probability and Computing" by Mitzenmacher and Upfal [2], "Probability ..." by Durrett [3] and Feller Vol. 1 and 2 [4] [5] for reference.

   I also would recommend "Frequently asked questions in Quantitative Finance" by Wilmott [6].

   Also know that there's a quantitative finance SO [7] that might be helpful.

   [0] https://www.amazon.com/Analysis-Revised-Jones-Bartlett-Mathe...

   [1] https://www.amazon.com/COMPLEXITY-CRITICALITY-Imperial-Colle...

   [2] https://www.amazon.com/Probability-Computing-Randomization-P...

   [3] https://www.amazon.com/Probability-Theory-Examples-Durrett-H...

   [4] https://www.amazon.com/Introduction-Probability-Theory-Appli...

   [5] https://www.amazon.com/Introduction-Probability-Theory-Appli...

   [6] https://www.amazon.com/Frequently-Asked-Questions-Quantitati...

   [7] https://quant.stackexchange.com/

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2. hintymadJan 2023
   For what level and in which area? Books like _Methods of mathematical physics_ can be both too hard and irrelevant to your needs. For starters,

   To become a better problem solver with high-school level maths:

     - Polya's How to Solve It.
     - Books of your choice about math contests.
     - Concrete Maths. I understand that this book is taught in college, but it requires very little advanced maths, and its techniques are hugely useful for high school students too.
   
   

   To hone my intuitions. I learned it the hard way that college maths were different from high school math: in high school, my teachers painstakingly drilled intuitions into us with very targeted explanations and tons of well designed exercises. In college, we won't get such luxury. So, it's really up to us to understand mathematical concepts intuitively before diving into technical details. For that matter, the following books helped me a lot:

     - The visual series. Visual Complex Analysis and Visual Group Theory, for instance
     - Pinter's A book of Abstract Algebra
     - Strichartz's The Way of Analysis
     - Linear Algebra Through Geometry by Wermer. The book offers a comprehensive geometric interpretation to linear algebra concepts. It's especially helpful for me to understand quadratic forms.
   
   

   To understand Analysis better. This area is vast, so I'll skip recommendations of excellent text books:

      - _Counterexamples in Analysis_. Those counterexamples in Analysis play a huge role in helping me truly appreciate the intricacies of Analysis. Similarly, books like _Counterexamples in Probability and Real Analysis_ are of great help too.
      - The Way of Analysis by Robert S. Strichartz. This books is AMAZING for laymen like me. You'd want someone to *explain* how concepts emerge, and how intuitions evolve.
   

   To become good at maths by doing maths, so the following books used to help me a lot:

     - Problems and Proofs in Real Analysis
     - Putnam and Beyond. I still suck at maths, but those well designed problems in Putnam really taught me how to seek insights in higher maths.
     - Piotr's Problems in Mathematical Analysis. But really, any problem books that challenge you will do. I'd recommend you find problem sets from the website of university courses. They cover essential techniques, and will not be as overwhelming as the books.

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3. hintymadJul 2021
   Speaking of books about calculus, I like Robert Strichartz's The Way of Analysis. Robert develops intuition and motivation of key concepts in an amazingly detailed way. He often starts with an intuitive but flawed idea, and then gradually revises the idea to a mature definition or proof. The whole book focuses on "why": why real numbers are defined that way, why limit uses the delta-epsilon notation, why we need to topology, etc and etc. Reading the book is more like an adventure than following a set path. The author explains his approach eloquently: "Mathematics is more than a collection of theorems, definitions, problems and techniques; it is a way of thought....My goal in writing this book is to communicate the mathematical ideas of the subject to the reader. I have tried to be generous with explanations....I believe that what you learn through a process of struggle is more likely to stick with you than what you learn without effort....Understanding mathematics is a complex process. It involves not only following the details of an argument and verifying its correctness, but seeing the overall strategy of the argument, the role played by every hypothesis, and understanding how different theorems and definitions fit together to create the whole. It is a long-term process; in a sense, you cannot appreciate the significance of the first theorem until you have learned the last theorem.”
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4. abetuskAug 2019
   This is something that I've struggled with myself so I can only give you advice as I've learned it from my perspective.

   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

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5. colejhudsonMar 2018
   Hey! Not too long ago I did this myself. Real analysis is somewhat unique in mathematics for the intuitive writing it's attracted.

   The most clearly written is John Baylis's 'What Is Mathematical Analysis?'. I _strongly_ recommend you read this. It's 125 pages of clarity and intuition building.

   More rigorous alternatives would be: 1. Elias M. Stein's 'Real Analysis: Measure Theory, Integration, and Hilbert Spaces'. 2. Robert S. Strichartz 'The Way of Analysis'

   Personally I would use the Baylis book and the other two as reference.

   Additional resources include the top voted mathoverflow.com and math.stackexchange.com answers. Beyond useful.

   Lastly, I have a twitter account (@math_twitr) that indexes (mostly) academic mathematicians. You might want to look through my follows there, they regularly post useful materials.

   Addendum: Hm, in addition, I think you might want to look through this Amazon wishlist of mine consisting of those math books recommended for clarity by those who should know: https://www.amazon.com/hz/wishlist/ls/2B6H3IG4PS0R1?&sort=de...

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6. rindalirMay 2022
   Some "pop"-sci books that have inspired me:

   1."Uncle Petros & Goldback's Conjecture" & "Logicomix" (both by Apostolos Doxiadis),

   2."Chaos" by James Gleick,

   3. "The Man Who Loved Only Numbers" by Paul Hoffman

   4. "A Mathematician's Apology" by G.H. Hardy

   5. "The Indian Clerk" (fiction) by David Leavitt (about GH Hardy & Ramanujan)

   6. "Genius at Play" by Siobhan Roberts (about John Conway). Normally I stay away from pop-math books with the words "The mind of..." or "Genius" in the title but this is a delightful book.

   Some "actual" math books: 1. Nonlinear Dynamics & Chaos (Strogatz)

   2. the Way of Analysis (Strichartz) -- note, this one is controversial as to its quality as a math book but I really like the presentation style

   3. The Nature of Computation (Moore & Mertens)

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