How to think like a mathematician
by Kevin Houston · first published 2009
80% recommend20% neutral
Why HN recommends it
An accessible guide to the habits of mathematical thinking, suited to anyone seeking a digestible introduction to how mathematics is done. It serves as a good first read before proof-based or advanced material.
Summarised by Claude from recommending comments
This arsenal of tips and techniques eases new students into undergraduate mathematics, unlocking the world of definitions, theorems, and proofs.
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…theorems/axioms, set theory, epsilon/delta/continuity/limits/differentiability, natural/rationals/reals/countability etc before heading into your first proof based LA or analysis sequence: - Kevin Houston "How to Think Like a Mathematician" - Keith Devlin "Intro Mathematical Thinking" - "How to Study as a Mathematics Major" Lara Alcock (for some reason, she/Oxford Press has 2 books with seemingly identical content under Math Major and…
on “How to Study Mathematics (2017)” ↗To the above I would add: How to Think Like a Mathematician - Kevin Houston (an excellent book to read before starting) How to Read and Do Proofs - Solow The Keys to Advanced Mathematics: Recurrent Themes in Abstract Reasoning - Solow Calculus - Spivak (Actually…
on “Ask HN: How to self-learn math?” ↗…of the topics it covers use nothing more than arithmetic and simple logic. That said it is not a good general mathematics book (it is designed as a Computer Science book). A book that will help you - How to think like a Mathematician: http://www.amazon.ca/How-Think-Like-Mathematician-Undergradu...
on “Why there is no Hitchhiker’s Guide to Mathematics for Programmers” ↗…(many/most) of lecture notes and student notes up: http://www.maths.cam.ac.uk/studentreps/res/notes.html as well as study guides: http://www.maths.cam.ac.uk/undergrad/studyskills/text.pdf - books about how to think like a mathematician: Keith Devlin, Kolmogorov/Alexandrov et al did 2 Dover books, and Houston: http://www.amazon.com/How-Think-Like-Mathematician-Undergrad... and http://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199...…
on “A Software Engineer’s Adventures in Learning Mathematics” ↗What field of mathematics interests you? Or you want some digestable intro to all of them? In that case I would recommend Kevin Houston "How to Think Like a Mathematician".
on “Classic Mathematics Books for Lifelong Learners” ↗A list: https://www.topicsinmaths.co.uk/cgi-bin/sews.py?SuggestedRea... For a single suggestion, "How to Think Like a Mathematician" by Kevin Houston. A second suggestion: "A Companion to Analysis" by Tom Körner. But it depends a lot on whether you want books about math, or books of math. It sounds like you want the latter ...…
on “Ask HN: Recommend a maths book for a teenager?” ↗…in Math[2]. I think these two books complement each other nicely. Then the handful of additional (no pun intended) books that jump to mind would be: - How Mathematicians Think by William Byers[3] - How to Think Like a Mathematician by Kevin Houston[4] - Discrete Mathematics with Applications[5] by Susanna Epp - How Not To Be Wrong[6] by Jordan Ellenberg [1]: https://www.amazon.com/Universe-Speaks-Numbers-Reveals-Natur... [2]:…
on “Ask HN: Math books that made you significantly better at math?” ↗…issues in abstract word problems and strategies for understanding them. 'Introduction to Mathematical Reasoning' by Eccles was another great book to explain the meaning of mathematical statements and how to think like a mathematician. If you read some of those, and want to learn more math, start with functional programming in SML since it naturally leads to understanding proof by induction https://existentialtype.wordpress.com/2011/03/21/the-dog-tha...
on “University of Tokyo Graduate School Entrance Exam – Mathematics (2016) [pdf]” ↗Amazing books that are a joy to read: Polya's How To Solve It is great for 'how to think like a mathematician'. I love Knuth et al's Concrete Mathematics. Discrete maths plus jokes in the margins. Stillwell's Mathematics and its History - meet the main strands of mathematics, their origins, and the great mathematicians.…
on “Ask HN: How to Think Like a mathematician?” ↗…work my way through to his Differential Equations class. I'm almost done reviewing Intermediate Algebra, and this weekend will probably consist largely of wrapping that up. I'm also reading the book How To Think Like a Mathematician by Kevin Houston, so I'll probably spend some time on that.
on “Ask HN: What are you doing this weekend?” ↗…not understand it very well at that time, so I had him work through some of the math books I set aside. Before he went to college, he worked through one “Saxon Math” arithmetic book, one book titled “How to Think Like a Mathematician” by Kevin Houston (which dealt with how to do proofs), and a book titled “Fundamental Mathematics” fourth edition by Wade and Taylor (which was based on 1960s New Math). Around a third of the way through…
on “Ask HN: What book did you use to teach your children math?” ↗…‘Self-Explanation’ Strategy (pdf) - http://www.ma.rhul.ac.uk/~uvah099/Maths/HoddsAlcockInglisSel... 2) How to Read, Understand and Study Proofs - https://mikepawliuk.ca/2014/03/31/how-to-read-understand-and... 3) How to Think Like a Mathematician by Kevin Houston - https://en.wikipedia.org/wiki/How_to_Think_Like_a_Mathematic...
on “Mathematics: Its Content, Methods and Meaning” ↗