Reddit mentions of How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)

you should definitely give harder problems multiple tries, you might not be able to solve them right away. Go back and re-learn the concepts the problem needed. Sometimes you might need to use a concept which you are not familiar with at the moment. I recommend reading "How to Solve it " by G. Polya
https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069116407X

It covers different problem solving approaches in a agnostic fashion.

There are many but they depend on the logic you want to solve.

Good general problem solving, How to Solve It by Polya is a standard in math departments. Much of that information applies to solving any rational problem. A lot of it is not as relevant to a programmer and it will be well more than most people would need but if you are going to get a wrench why not get fully stocked a toolbox?

Standard binary logic, !(a and b) = (!a or !b) type stuff? I don't have any specific recommendations but look to philosophy sections for books on logic. Philosophy literally wrote the book on that topic before math latched on to it. Most math books on the topic will be particularly unwieldy and overly broad to what a programmer might need.

Any of those books is likely to go well beyond what you actually need. None of them are programming focused, programmers tend to learn this stuff by example or practice debugging is a great cause-effect based teacher and if you practice you will learn. Can you be more specific about what logic you need to improve and what level of skill you feel you have?

Someone else recommended learning math from the pre-calc up. I would second that if it is an option but that can be a very long road and some people just shut down at math. I know some great programmers who failed college freshmen level math classes. I know many other really intelligent and capable people who do not believe 'they can math' so I have tried to offer another path.


Edit: I meant to include with the amazon link, look for older editions on any textbooks and evaluate whether the comments and reviews indicate significant change worth the new edition price. Books on logic don't really change much but sometimes they will reword examples or update them to be better. I didn't remember to note that until the amazon price bot replied to me.

George Pólya (1887–1985) was a professor of mathematics at ETH Zürich and Stanford University, and made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He also contributed significantly to heuristics and mathematical education, to the point where one of his books is still being reprinted and actively referenced as still fundamentally relevant over 70 years later.

And for his life's work, he's remembered, if at all, as "one of von Neumann's lecturers".

Learning proofs can mean different things in different contexts. First, a few questions:

1. What's your current academic level? (Assuming, of course, you're still a student, rather than trying to learn mathematical proofs as an autodidact.)
   
   The sort of recommendations for a pre-university student are likely to be very different from those for a university student. For example, high school students have a number of mathematics competitions that you could consider (at least in The United States; the structure of opportunities is likely different in other countries). At the university level, you might want to look for something like a weekly problem solving seminar. These often have as their nominal goal preparing for the Putnam, which can often feel like a VERY ambitious way to learn proofs, akin to learning to swim by being thrown into a lake.
   
   As a general rule, I'd say that working on proof-based contest questions that are just beyond the scope of what you think you can solve is probably a good initial source of problems. You don't want something so difficult that it's simply discouraging. Further, contest questions typically have solutions available, either in printed books or available somewhere online.
   
   
2. What's your current mathematical background?
   
   This may be especially true for things like logic and very elementary set theory.
   
   
3. What sort of access do you have to "formal" mathematical resources like textbooks, online materials, etc.?
   
   Some recommendations will make a lot more sense if, for example, you have access to a quality university-level library, since you won't have to spend lots of money out-of-pocket to get copies of certain textbooks. (I'm limiting my recommendations to legally-obtained copies of textbooks and such.)
   
   
4. What resources are available to you for vetting your work?
   
   Imagine trying to learn a foreign language without being able to practice it with a fluent speaker, and without being able to get any feedback on how to improve things. You may well be able to learn how to do proofs on your own, but it's orders of magnitude more effective when you have someone who can guide you.
   
   
5. Are you trying to learn the basics of mathematical proofs, or genuinely rigorous mathematical proofs?
   
   Put differently, is your current goal to be able to produce a proof that will satisfy yourself, or to produce a proof that will satisfy someone else?
   
   
6. What experience have you already had with proofs in particular?
   
   Have you had at least, for example, a geometry class that's proof-based?
   
   
7. How would you characterize your general writing ability?
   
   Proofs are all about communicating ideas. If you struggle with writing in complete, grammatically-correct sentences, then that will definitely be a bottleneck to your ability to make progress.
   
   ---
   
   With those caveats out of the way, let me make a few suggestions given what I think I can infer about where you in particular are right now.
   
   

• The book How to Prove It: A Structured Approach by Daniel Velleman is a well-respected general introduction to ideas behind mathematical proof, as is How to Solve It: A New Aspect of Mathematical Method by George Pólya.
  
  
• Since you've already taken calculus, it would be worth reviewing the topic using a more abstract, proof-centric text like Calculus by Michael Spivak. This is a challenging textbook, but there's a reason people have been recommending its different editions over many decades.
  
  
• In order to learn how to write mathematically sound proofs, it helps to read as many as you can find (at a level appropriate for your background and such). You can find plenty of examples in certain textbooks and other resources, and being able to work from templates of "good" proofs will help you immeasurably.
  
  
• It's like the old joke about how to get to Carnegie Hall: practice, practice, practice.
  
  Learning proofs is in many ways a skill that requires cultivation. Accordingly, you'll need to be patient and persistent, because proof-writing isn't a skill one typically can acquire passively.
  
  ---
  
  How to improve at proofs is a big question beyond the scope of what I can answer in a single reddit comment. Nonetheless, I hope this helps point you in some useful directions. Good luck!

generatingfunctionology

or

How To Solve It by Polya

There are some really good books that you can use to give yourself a solid foundation for further self-study in mathematics. I've used them myself. The great thing about this type of book is that you can just do the exercises from one side of the book to the other and then be confident in the knowledge that you understand the material. It's nice! Here are my recommendations:

First off, three books on the basics of algebra, trigonometry, and functions and graphs. They're all by a guy called Israel Gelfand, and they're good: Algebra, Trigonometry, and Functions and Graphs.

Next, one of two books (they occupy the same niche, material-wise) on general proof and problem-solving methods. These get you in the headspace of constructing proofs, which is really good. As someone with a bachelors in math, it's disheartening to see that proofs are misunderstood and often disliked by students. The whole point of learning and understanding proofs (and reproducing them yourself) is so that you gain an understanding of the why of the problem under consideration, not just the how... Anyways, I'm rambling! Here they are: How To Prove It: A Structured Approach and How To Solve It.

And finally a book which is a little bit more terse than the others, but which serves to reinforce the key concepts: Basic Mathematics.

After that you have the basics needed to take on any math textbook you like really - beginning from the foundational subjects and working your way upwards, of course. For example, if you wanted to improve your linear algebra skills (e.g. suppose you wanted to learn a bit of machine learning) you could just study a textbook like Linear Algebra Done Right.

The hard part about this method is that it takes a lot of practice to get used to learning from a book. But that's also the upside of it because whenever you're studying it, you're really studying it. It's a pretty straightforward process (bar the moments of frustration, of course).

If you have any other questions about learning math, shoot me a PM. :)

These are a couple of nice old books about mathematical thinking:

• What is Mathematics? by Courant, Robbins, and Stewart
  
• How to Solve It by Polya

Recentemente estive procurando algo interessante pra ler e me deparei com várias recomendações do livro How to solve it: A New Aspect of Mathematical Method.


Um livro extremamente denso mas com muito conteúdo é o Mathematics: Its Content, Methods and Meaning. Comecei a ler esse livro, mas outras atividades me fizeram dar uma pausa. Vou tentar voltar a ele e colocar como meta terminar antes de 2020 rs.


Já li alguns livros explicando a origem dos números. Mas, de todos que li, Os números é imbatível.

Saying literally anything is better than saying nothing. Just keep your mouth moving. Verbalize your thought process. Something I think a lot of people misunderstand about this sort of question is that there usually isn't a specific right answer: they're mainly trying to evaluate how you think about problem solving. It's probably fine if your response is along the lines of "I'm not entirely sure how I'd go about that, but here's how I'd start thinking about the problem." A good way to tackle this sort of thing is to identify ambiguities in the problem statement and start suggesting assumptions you could make to concretize them, and look for ways to break large problems down into subproblems. Even if you can't solve the problem, at least show that you have some idea how to get started.

EDIT: Another book you might find useful is George Polya - How to Solve It. Classic book on general strategies for problem solving. Just google "Polya's Method" for an overview.

i meant real life problems, directly to real life, that's what i mean by problems, there's not really any better words to use

i haven't -- https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069116407X/

what are 3 of the most significant and helpful things you learned from the book tho?

> For those who prefer video lectures, Skiena generously provides his online. We also really like Tim Roughgarden’s course, available from Stanford’s MOOC platform Lagunita, or on Coursera. Whether you prefer Skiena’s or Roughgarden’s lecture style will be a matter of personal preference.
>
> For practice, our preferred approach is for students to solve problems on Leetcode. These tend to be interesting problems with decent accompanying solutions and discussions. They also help you test progress against questions that are commonly used in technical interviews at the more competitive software companies. We suggest solving around 100 random leetcode problems as part of your studies.
>
> Finally, we strongly recommend How to Solve It as an excellent and unique guide to general problem solving; it’s as applicable to computer science as it is to mathematics.
>
>
>
> [The Algorithm Design Manual](https://teachyourselfcs.com//skiena.jpg) [How to Solve It](https://teachyourselfcs.com//polya.jpg)> I have only one method that I recommend extensively—it’s called think before you write.
>
> — Richard Hamming
>
>
>
> ### Mathematics for Computer Science
>
> In some ways, computer science is an overgrown branch of applied mathematics. While many software engineers try—and to varying degrees succeed—at ignoring this, we encourage you to embrace it with direct study. Doing so successfully will give you an enormous competitive advantage over those who don’t.
>
> The most relevant area of math for CS is broadly called “discrete mathematics”, where “discrete” is the opposite of “continuous” and is loosely a collection of interesting applied math topics outside of calculus. Given the vague definition, it’s not meaningful to try to cover the entire breadth of “discrete mathematics”. A more realistic goal is to build a working understanding of logic, combinatorics and probability, set theory, graph theory, and a little of the number theory informing cryptography. Linear algebra is an additional worthwhile area of study, given its importance in computer graphics and machine learning.
>
> Our suggested starting point for discrete mathematics is the set of lecture notes by László Lovász. Professor Lovász did a good job of making the content approachable and intuitive, so this serves as a better starting point than more formal texts.
>
> For a more advanced treatment, we suggest Mathematics for Computer Science, the book-length lecture notes for the MIT course of the same name. That course’s video lectures are also freely available, and are our recommended video lectures for discrete math.
>
> For linear algebra, we suggest starting with the Essence of linear algebra video series, followed by Gilbert Strang’s book and video lectures.
>
>
>
> > If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
>
> — John von Neumann
>
>
>
> ### Operating Systems
>
> Operating System Concepts (the “Dinosaur book”) and Modern Operating Systems are the “classic” books on operating systems. Both have attracted criticism for their writing styles, and for being the 1000-page-long type of textbook that gets bits bolted onto it every few years to encourage purchasing of the “latest edition”.
>
> Operating Systems: Three Easy Pieces is a good alternative that’s freely available online. We particularly like the structure of the book and feel that the exercises are well worth doing.
>
> After OSTEP, we encourage you to explore the design decisions of specific operating systems, through “{OS name} Internals” style books such as Lion's commentary on Unix, The Design and Implementation of the FreeBSD Operating System, and Mac OS X Internals.
>
> A great way to consolidate your understanding of operating systems is to read the code of a small kernel and add features. A great choice is xv6, a port of Unix V6 to ANSI C and x86 maintained for a course at MIT. OSTEP has an appendix of potential xv6 labs full of great ideas for potential projects.
>
>
>
> [Operating Systems: Three Easy Pieces](https://teachyourselfcs.com//ostep.jpeg)
>
>
>
> ### Computer Networking
>
> Given that so much of software engineering is on web servers and clients, one of the most immediately valuable areas of computer science is computer networking. Our self-taught students who methodically study networking find that they finally understand terms, concepts and protocols they’d been surrounded by for years.
>
> Our favorite book on the topic is Computer Networking: A Top-Down Approach. The small projects and exercises in the book are well worth doing, and we particularly like the “Wireshark labs”, which they have generously provided online.
>
> For those who prefer video lectures, we suggest Stanford’s Introduction to Computer Networking course available on their MOOC platform Lagunita.
>
> The study of networking benefits more from projects than it does from small exercises. Some possible projects are: an HTTP server, a UDP-based chat app, a mini TCP stack, a proxy or load balancer, and a distributed hash table.
>
>
>
> > You can’t gaze in the crystal ball and see the future. What the Internet is going to be in the future is what society makes it.
>
> — Bob Kahn
>
> [Computer Networking: A Top-Down Approach](https://teachyourselfcs.com//top-down.jpg)
>
>
>
> ### Databases
>
> It takes more work to self-learn about database systems than it does with most other topics. It’s a relatively new (i.e. post 1970s) field of study with strong commercial incentives for ideas to stay behind closed doors. Additionally, many potentially excellent textbook authors have preferred to join or start companies instead.
>
> Given the circumstances, we encourage self-learners to generally avoid textbooks and start with the Spring 2015 recording of CS 186, Joe Hellerstein’s databases course at Berkeley, and to progress to reading papers after.
>
> One paper particularly worth mentioning for new students is “Architecture of a Database System”, which uniquely provides a high-level view of how relational database management systems (RDBMS) work. This will serve as a useful skeleton for further study.
>
> Readings in Database Systems, better known as the databases “Red Book”, is a collection of papers compiled and edited by Peter Bailis, Joe Hellerstein and Michael Stonebreaker. For those who have progressed beyond the level of the CS 186 content, the Red Book should be your next stop.
>
> If you insist on using an introductory textbook, we suggest Database Management Systems by Ramakrishnan and Gehrke. For more advanced students, Jim Gray’s classic Transaction Processing: Concepts and Techniques is worthwhile, but we don’t encourage using this as a first resource.
>

> (continues in next comment)

This! Learn to problem solve!