How to prove it

What Hacker News says

1. ForgotMyUUIDSep 2026
   I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part.

   I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them.

   Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it.

   And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.

   Show morethread
2. mna_Aug 2025
   You can do all of that without paying a monthly fee. You just need a library card (or know of a person called Anna and her archive ;) ) and a list of books. These are the ones I used:

   Precalculus by Axler

   Calculus (Ninth Edition) by Thomas

   Linear Algebra by Lay

   How To Prove It by Velleman

   Understanding Analysis by Abbott <--- I'm currently here

   Much, much, much cheaper than paying $50/month. What I've spent most on so far has been printer paper and fountain pen ink because I do exercises by hand instead of using a tablet/iPad but in total this expense has been waaaaay under $50.

   Show morethread
3. throwaway55533Jul 2024
   +1, pg is using a pretty typical argument you see in analysis/topology.

   If you want to get to real analysis/topology the typical sequence is

   1. Logic and Set theory (recommendation: How to Prove It, Velleman)

   2. Linear Algebra (don't have a good recommendation)

   3a. Real analysis (recommendation: PMA, Rudin)

   3b. Topology (recommendation: Topology, Munkres)

   I'm not sure I'd recommend learning math. It's an extremely expensive skill -- though pretty valuable in the software industry. People who go learn math are generally just drawn to it; you can't stop them even if you wanted to.

   But be aware, (1) you'll have no one to talk about math with. And (2) you'll be joining a club of all the outcasts in society, including the Unabomber.

   Show morethread
4. an_aparallelJun 2024
   I'm not really sure where i stand on this - as someone in the throws of self teaching programming (C) - and consider myself pretty smart. I've attempted learning on my own several times. Many times since 2016 and have given up on many occasions. CS50x is one of the first pieces of material which has close to what i consider sane pedagogy.

   I go back and forth between thinking Python is better for learning because "hey, less typing, less curly braces". Then thinking - great i dont know wtf is happening here - just magic i guess?? (refer XCXD comic).

   I have seen this many times in relation to education and quite simply, the way we teach, often "for speed", and "at university" - do not equate to learning anything. Learning how things works for real - takes time, and is often outside the scope of a "semester"...and quite honestly beyond our current paradigms of cordoned off disciplines.

   Programming/algorithmic thinking is more or less hardware accelerated mathematics. When i opened up Hammack's - How to Prove it - and read about sets, saw the notation - AHA moment. Similar things happened when looking at Charles Petzold's Code and seeing details of carry bits in counter chips, and comparing those to literal carry pins in gear based mechanical calculators (in Norman Bigg's "Quite Right").

   None of this comes naturally (to me) - and i think many of the learning resources are there - but are not there at the time you need them. Common comments "just do a project"..."get out of tutorial hell"...very well meaning...but even a 1000 LOC text editor is an absolute head fuck for someone who just learnt to print a triangle of hashes.

   I've reached out to the creator of the Turing Complete PC game on Steam. I'm certain once virtualising a computer which runs C in that game - will we reach the epitome of programming pedagogy :-)

   Show morethread
5. mcshicksJan 2024
   I absolutely agree that it's not for beginners, but amateurs can still have fun. I did the NNG in lean 3 and again lean 4. It didn't take me 100s of hours but I've also spent a lot of time over the last decade trying to brush up my math skills. One of the main problems with proofs is other than having a mathematician friend who's willing to check its really hard to know if you did it correctly. My son is a mathematician and I still didn't want to ask him to check my proofs! The cool thing for me about lean is I can do some form of math proofs and not have to ask somebody if I got it right.

   If you liked that you might like the natural set game as well.

   https://adam.math.hhu.de/#/g/djvelleman/stg4

   The author of the math book "How to Prove it" Daniel Velleman who I think did the set game also has a "How to Prove it with Lean" which is nice because the exercises correlate to the book.

   https://djvelleman.github.io/HTPIwL/

   I've found the lean community on zulip really open to amateurs to want to learn. I asked what I considered a "duh" type of question once I saw the answer, and the guy that helped me is kind of "the lean math" guy.

   https://leanprover.zulipchat.com/

   Show morethread
6. gsdgsdfgJan 2024
   People here mention KhanAcademy and AOPS series for self learning. I've used both when relearning Math as an adult. But there is one more resource which is absolutely terrific: Henry Sinclair Hall's books. Not only they are good (way better than the aforementioned ones), but being published in the 19th century, they are in public domain now and can be downloaded from the Internet Archive free of charge: https://archive.org/search?query=creator%3A%22Hall%2C+H.+S.+... And, also, How to Prove it by Velleman is a must read.
   Show morethread