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I don’t understand why people recommend books like “a programmers introduction to mathematics”. They’re advertised as books for “programmers” or people that kn

by jeff76 8y ago

I don’t understand why people recommend books like “a programmers introduction to mathematics”. They’re advertised as books for “programmers” or people that know little math, yet they hide solutions from the reader. The reader that doesn’t know proper proofs or deep mathematics likely isn’t the same ones that know if their solutions are correct. Programmers like to write code with test cases. We don’t like to write code once and trust there are no bugs. So, why would we want to write mathematics any differently? How is a beginner math student even going to know if their answers are correct? I’m sure someone reading this will say “you’re robbing the reader” if you provide solutions. I don’t agree with that. That’s a bit of gatekeeping because not everyone has access to a TA or professor and we’d really like to learn this stuff and know if we’re on the right track. Are there any actual math text with solutions that are better than the one advertised in this list?

mesaframe 8y ago
I've seen book authors talking about solutions to their book. And the two most talked about reasons are 1. Morale Stuff 2. If they provide their solutions online then the Book will not get adapted by professors.
- jeff76 8y ago
  I understand but sometimes there are books targeted for the “self-learner” that provide 0 solutions and are the type of books that save off major theorems to be proven in the exercises. That’s completely fine. Don’t advertise the books for self-learners or “programmers”. I would like to see actual books targeted at these demographics that provide solutions and aren’t concerned about being adapted in a classroom. There is a market for those that graduated university a long time ago and would like to learn math outside of a classroom. These sort of people don’t want to see solutions because they have to turn their work in for a grade, but because they want to check their work and get feedback. Outside of having to consult with TAs or professors they don’t have access to.
  - mbrudd 8y ago
    There ought to be (and probably are!) analogues of StackExchange or MathOverflow where people can post and discuss solutions for problems in specific books. Would be interesting to know about existing options.
cttet 8y ago
Mathematics is a very broad topic. Which part are you interested in?
- jeff76 8y ago
  Any books in: Introduction to proofs, Abstract algebra, Topology, Linear algebra
  - jimmy1 8y ago
    I liked the Better Explained books, particularly for Calculus, as it really explained the "why" For Linear Algebra, "Linear Algebra Done Right" -- way better than my college lin alg course. For "general mathematics", I like books that read almost like novels to really grasp the "why" of mathematics, so these are more to embellish your general understanding of "what is the point" type questions -- so things here like Euclid's Window come to mind, and "An Imaginary Tale: The story of square root minus one" will help explain complex numbers in more detail than you ever cared. Reading about the history of mathematics and the writings of some of the greats, like Rene Descartes, Lehonard Euler, Gottfried Wilhelm Leibniz, Issac Newton (the amazing thing is all these greats lived within a century of each other)
  - mesaframe 8y ago
    For linear algebra I'll suggest Jim heffron's book. He provides solutions to questions also http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/
  - tzs 8y ago
    For abstract algebra, take a look at Pinter's "A Book of Abstract Algebra" [1]. It's a Dover republication, so is not expensive. It has a lot of exercises...in many chapters more than half the pages are exercises. It does not provide answers to every exercise--maybe 10% tops--but a lot of the exercises are small and should not be any problem. These are often in a group, where he takes something that would be one hard exercise in another book and breaks it down almost to the level it would be if were part of the main text, leaving just small thing for you to fill in as exercises. There are a few recurring themes throughout the exercises, where he applies the material of the chapter to some specific application in several exercises (e.g., error correcting codes if I recall correctly), and subsequent chapters continue with those themes in their exercises. [1] https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 https://www.amazon.com/Book-Abstract-Algebra-Second-Mathemat...
Buttons840 8y ago
I recognize math as a source of useful ideas. I want a high level overview of what math can do, so that I may draw ideas and solutions from it as needed. I find that many proofs are intuitive, although I can't prove them myself; that's OK, I am happy with my fuzzy and fallible intuition here, and it serves me well enough. This is not unlike mathmeticians who often suspect something is true before proving it. Test cases are not proofs. I don't want to write mathematics, I want to write code and maybe draw some fuzzy and fallible human inspiration from mathematics.
- deleted 8y ago
  [deleted]
fogetti 8y ago
This! I completely agree! Also who cares if the solutions are online? If the professors want to use the book in classrooms then they should come up with their own exercises instead of basically stealing the book's questions. I completely agree that any book with exercises without solutions is completely useless. I didn't even have access to professors or any teaching assistants even DURING my undergraduate years. So the assumption that you can discuss the questions with your professor doesn't even hold water.
- Kihashi 8y ago
  > If the professors want to use the book in classrooms then they should come up with their own exercises instead of basically stealing the book's questions. I always wondered why there weren't supplement books/websites of practice problems that were separate from the text book. Seems like an easy way to make money. I guess because it makes it harder to pass on to the students.
  - mattkrause 8y ago
    There are! Some books have “solution manuals” or “companions” with worked solutions to a subset of the problems. There are also those big compendia of “5000 problems in linear algebra” or whatever. It’s true that these are most common for big, intro-to-intermediate classes, and less common elsewhere, but there are a lot kicking around....
  - alex7o 8y ago
    There are a lot of "problem books" where I live in Bulgaria.
chasote 8y ago
The author seems to have taken this desire into consideration too but I don't think he has figured out the way he wants to solve it. I did find this: https://github.com/pim-book/exercises https://github.com/pim-book/exercises but it doesn't seem to have taken hold quite yet. Maybe if you and others are so inclined, a nice community effort can solve this issue together and maybe that interactive discussion helps mediate some of the other reasons people give for not providing the solutions up front.
svat 8y ago
Every time there is a post here about a mathematics book, or even only only tangentially connected (as here), someone complains about the lack of solutions to exercises. I definitely sympathize with your position. But I think a few things are worth pointing out. Firstly, it's useful to remember that the purpose of a mathematics book is to teach mathematics, and conversely the purpose of studying one is to learn mathematics, not to solve its exercises. That is, the exercises are not a goal in themselves — they are an additional gift offered by the author in addition to the mathematical content. (Many famous mathematical books don't even have exercises... writing a book is already a lot of effort, coming up with good exercises is additional work, and including solutions is a bit more on top of that.) Moreover, even if a book has exercises, they are never enough, and it is the reader's job to make up many more of their own. (I think readers' giving so much weight to exercises that happen to be written in the book come from school/college mathematics education and testing.) [1]: https://academia.stackexchange.com/questions/56739/why-dont-graduate-math-texts-have-solutions-to-their-exercises https://academia.stackexchange.com/questions/56739/why-dont-... [2]: https://math.stackexchange.com/questions/57889/hardy-wrights-intro-to-number-theory-is-highly-praised-but-has-no-exercises https://math.stackexchange.com/questions/57889/hardy-wrights... ("A random sample of 10 maths books I have handy shows that 4 doesn't have exercises.") [3]: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm https://www.ocf.berkeley.edu/~abhishek/chicmath.htm (search for "no exercises") To put it in programming terms: the exercises are the tests, the way to verify that your understanding of the mathematics has no bugs in it. If you're trying to verify that your solutions to exercises are correct, you're in a position akin to writing tests for your tests: it can be useful, but it's a second-order concern. Just as tests should be as “dumb” and “obvious” as possible, if there's any doubt at all that you've solved an exercise correctly, that itself is an indication that your understanding of the mathematics isn't complete, and you need to go back and engage with the material (think deeply, write your own exercises/tests, etc) until it becomes obvious. Of course this takes a lot more effort than plowing through exercises and verifying one's solutions. Programmers have another advantage over other mathematics students: you can test your understanding of the mathematics by writing actual programs. For example, if you're reading an elementary number theory book and learn about the Chinese Remainder Theorem, you can write an actual program/function for finding solutions to a system of modular congruences. As computers require a level of precision greater than human readers, this will force your understanding to become really sharp, as you have to deal with all the corner cases as well. (No doubt there are areas of mathematics for which this is hard to do, but most of the undergraduate curriculum can fit, and if you find something that's hard then maybe making it “programmable” can be your unique contribution.) All that said, it's definitely comforting to have solutions to exercises: there's a boost in motivation from being told that your answer is correct and you can progress to the next section, and though there's a cost here (if you needed to be told your solution is correct then maybe you should actually spend more time understanding the mathematics instead of going to the next section — but then again maybe not everyone really wants to understand the mathematics that well), I think that's probably the main thing that's missed when books don't have solutions. As a self-learner, motivation is most of the challenge, and solutions can definitely help there. --- Personally, if I look back at books I went through a good fraction of as a self-learner (and loved), there's an even mix: • Concrete Mathematics and Generatingfunctionology have complete solutions. • Burton's Elementary Number Theory has “Hints”, and “Answers” to selected exercises (i.e. those where the answer was a number, not a proof — these are not much work for an author to include). • The Art of Computer Programming (obviously I've read only a tiny bit of it) has problem ratings and sometimes very terse solutions / outlines of solutions. • Uspensky and Heaslet's Elementary Number Theory (which I read before Burton) and Analytic Combinatorics have no solutions at all. I don't think solutions to exercises have made a substantial difference in my engagement with these books, but who knows.
crispyambulance 8y ago
> So, why would we want to write mathematics any differently? I think you're stretching the unit-test-to-mathematics analogy a little too far. I haven't looked at the book in question, but generally, you don't just "know if" your solution is correct, you prove it, or at least consider examples that can demonstrate if your solution works in some conceivable cases. That's how "unit tests" work in mathematics. If the book has armed you with enough knowledge to work though a problem to the end, it has done it's job. The real test will come as you build on that knowledge later on in the book or in real-life applications. Moreover, it's not like this stuff is obscure knowledge. You can find help online or by looking at other books or connecting with others.
- dwringer 8y ago
  As I read it, the point is not about using unit tests to verify correctness, but as an interactive means of learning the problem inside and out prior to finding a complete solution. The trick is in learning what you don't know, and one can do that faster with regular feedback. There does seem to be a delicate balance between letting a student "stew" for a bit to really learn how to find and be confident in solutions, and preventing the student from heading down a wrong course. Books can never provide real feedback, I suppose, but it's a nice option to have solutions available even if students must be trusted to independently exhaust the available resources for attaining them first. Often having the solution is at least helpful as a "sanity check" when it comes to mathematics - I feel it is all too easy for beginners to make proofs on hidden and untenable assumptions and not necessarily realize it.
- jeff76 8y ago
  The point isn’t we need to do formal verification to test correctness. It’s that you probably wouldn’t write a program first pass and assume it is but free. So, why would you assume beginners that don’t know how to write proofs would write correct proofs without feedback to check?
  - crispyambulance 8y ago
    Absolute beginners may not able to write an actual proof that their answer is correct, but at least they can verify that it satisfies the problem (depending on the actual problem). In any case, since the study of mathematics is cumulative. Even if the student can't prove or verify that their answer to a specific problem is correct they will eventually reach a point, often within the same problem-set, where inability to solve a problem will force them to encounter their gaps/misconceptions. The most important thing in a mathematics text, as with any other text, is lucidity of writing and the preparation of the student.
throwawaymath 8y ago
There’s no profound reason why authors don’t typically include solutions. It’s not gatekeeping and it’s not to appeal to an idealistic sense of the Right Way to learn mathematics. The real reason is twofold: 1. Authors write textbooks for universities and professors, not students. They write for the academic paradigm they’re familiar with, in which the professor is primarily teaching a student. The textbook is simply used to arrange a course, and professors would rather not have solutions available in the textbook. 2. Authors don’t earn much money from textbooks and textbooks aren’t weighed very heavily for tenure. There isn’t much of an incentive for them to write textbooks with solutions for the reason stated above, and it’s actually significantly more work to include solutions to your exercises. To put it very bluntly, there are extremely few non-students in the market for math textbooks, which is already a high-effort and low-return market for authors as it is. To get such a textbook you’d need someone who: knows the material extremely well, is very good at exposition, has a lot of time to invest in this as a passion project and doesn’t mind if it’s not used much or at all in well-known universities.
qznc 8y ago
We do have theorem checker and proof assistants these days. Are there books to learn math with Isabelle or Coq?
- throwawaymath 8y ago
  Not really, because it's extremely nontrivial to formally verify the proofs of most theorems.
dkarl 8y ago
Seeing a correct proof may or may not help you understand whether your proof is correct. What you're asking for is more like code review, not something you can package with a book.
- jeff76 8y ago
  Disagree. Some proofs are very straightforward and standard. Those are easily checkable with a solution guide.
  - dkarl 8y ago
    When you hit upon a proof like that, you know it. If you're struggling and need feedback, then your solution is probably more complicated than the elegant one in the book, and the question you're asking is, okay, my solution isn't great, but is it even a solution? There's a reason these books declare some level of "mathematical maturity" as a prerequisite. By the time you get to certain content you have to either be comfortable checking a proof (maybe set it aside for a few days like you would any other writing) or have human help.
    - jeff76 8y ago
      Think you missed my entire point of including solutions for the beginner, i.e., someone that lacks math maturity and the reasoning of how easily such a person could fool themselves into thinking their thought process is valid.
      - dkarl 8y ago
        Someone who is a beginner will be working on easy material with a textbook that has very mild expectations of "mathematical maturity." Textbooks mostly (with a few intentional exceptions) calibrate the mathematical maturity they require according to how much work a student will have done in order to be ready to tackle the book's subject matter. Some people might think they can be ready to do problem sets in an advanced math textbook while skipping the work that would have given them the maturity it requires, but they are kidding themselves. They can't advance on the spectator track and then make a horizontal move over to actually doing the math. There's nothing wrong with people passively reading the material if they enjoy it, but if they want to tackle the problems, they are going to have to work through the earlier material by solving problems as well, and if they do that, they'll find that their sophistication increases along with the demands on it.
kevintb 8y ago
You clearly haven’t read the book or even clicked on the link. It’s written by Jeremy Kun, a fantastic math professor who’s written several excellent posts on his blog on how to intuitively grasp mathematical concepts.

by jeff76 8y ago

mesaframe 8y ago

I've seen book authors talking about solutions to their book. And the two most talked about reasons are 1. Morale Stuff 2. If they provide their solutions online then the Book will not get adapted by professors.

jeff76 8y ago
I understand but sometimes there are books targeted for the “self-learner” that provide 0 solutions and are the type of books that save off major theorems to be proven in the exercises. That’s completely fine. Don’t advertise the books for self-learners or “programmers”. I would like to see actual books targeted at these demographics that provide solutions and aren’t concerned about being adapted in a classroom. There is a market for those that graduated university a long time ago and would like to learn math outside of a classroom. These sort of people don’t want to see solutions because they have to turn their work in for a grade, but because they want to check their work and get feedback. Outside of having to consult with TAs or professors they don’t have access to.
- mbrudd 8y ago
  There ought to be (and probably are!) analogues of StackExchange or MathOverflow where people can post and discuss solutions for problems in specific books. Would be interesting to know about existing options.

cttet 8y ago

Mathematics is a very broad topic. Which part are you interested in?

jeff76 8y ago
Any books in: Introduction to proofs, Abstract algebra, Topology, Linear algebra
- jimmy1 8y ago
  I liked the Better Explained books, particularly for Calculus, as it really explained the "why" For Linear Algebra, "Linear Algebra Done Right" -- way better than my college lin alg course. For "general mathematics", I like books that read almost like novels to really grasp the "why" of mathematics, so these are more to embellish your general understanding of "what is the point" type questions -- so things here like Euclid's Window come to mind, and "An Imaginary Tale: The story of square root minus one" will help explain complex numbers in more detail than you ever cared. Reading about the history of mathematics and the writings of some of the greats, like Rene Descartes, Lehonard Euler, Gottfried Wilhelm Leibniz, Issac Newton (the amazing thing is all these greats lived within a century of each other)
- mesaframe 8y ago
  For linear algebra I'll suggest Jim heffron's book. He provides solutions to questions also http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/
- tzs 8y ago
  For abstract algebra, take a look at Pinter's "A Book of Abstract Algebra" [1]. It's a Dover republication, so is not expensive. It has a lot of exercises...in many chapters more than half the pages are exercises. It does not provide answers to every exercise--maybe 10% tops--but a lot of the exercises are small and should not be any problem. These are often in a group, where he takes something that would be one hard exercise in another book and breaks it down almost to the level it would be if were part of the main text, leaving just small thing for you to fill in as exercises. There are a few recurring themes throughout the exercises, where he applies the material of the chapter to some specific application in several exercises (e.g., error correcting codes if I recall correctly), and subsequent chapters continue with those themes in their exercises. [1] https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 https://www.amazon.com/Book-Abstract-Algebra-Second-Mathemat...

Buttons840 8y ago

I recognize math as a source of useful ideas. I want a high level overview of what math can do, so that I may draw ideas and solutions from it as needed. I find that many proofs are intuitive, although I can't prove them myself; that's OK, I am happy with my fuzzy and fallible intuition here, and it serves me well enough. This is not unlike mathmeticians who often suspect something is true before proving it. Test cases are not proofs. I don't want to write mathematics, I want to write code and maybe draw some fuzzy and fallible human inspiration from mathematics.

deleted 8y ago
[deleted]

fogetti 8y ago

This! I completely agree! Also who cares if the solutions are online? If the professors want to use the book in classrooms then they should come up with their own exercises instead of basically stealing the book's questions. I completely agree that any book with exercises without solutions is completely useless. I didn't even have access to professors or any teaching assistants even DURING my undergraduate years. So the assumption that you can discuss the questions with your professor doesn't even hold water.

Kihashi 8y ago
> If the professors want to use the book in classrooms then they should come up with their own exercises instead of basically stealing the book's questions. I always wondered why there weren't supplement books/websites of practice problems that were separate from the text book. Seems like an easy way to make money. I guess because it makes it harder to pass on to the students.
- mattkrause 8y ago
  There are! Some books have “solution manuals” or “companions” with worked solutions to a subset of the problems. There are also those big compendia of “5000 problems in linear algebra” or whatever. It’s true that these are most common for big, intro-to-intermediate classes, and less common elsewhere, but there are a lot kicking around....
- alex7o 8y ago
  There are a lot of "problem books" where I live in Bulgaria.

chasote 8y ago

The author seems to have taken this desire into consideration too but I don't think he has figured out the way he wants to solve it. I did find this: https://github.com/pim-book/exercises https://github.com/pim-book/exercises but it doesn't seem to have taken hold quite yet. Maybe if you and others are so inclined, a nice community effort can solve this issue together and maybe that interactive discussion helps mediate some of the other reasons people give for not providing the solutions up front.

svat 8y ago

Every time there is a post here about a mathematics book, or even only only tangentially connected (as here), someone complains about the lack of solutions to exercises. I definitely sympathize with your position. But I think a few things are worth pointing out. Firstly, it's useful to remember that the purpose of a mathematics book is to teach mathematics, and conversely the purpose of studying one is to learn mathematics, not to solve its exercises. That is, the exercises are not a goal in themselves — they are an additional gift offered by the author in addition to the mathematical content. (Many famous mathematical books don't even have exercises... writing a book is already a lot of effort, coming up with good exercises is additional work, and including solutions is a bit more on top of that.) Moreover, even if a book has exercises, they are never enough, and it is the reader's job to make up many more of their own. (I think readers' giving so much weight to exercises that happen to be written in the book come from school/college mathematics education and testing.) [1]: https://academia.stackexchange.com/questions/56739/why-dont-graduate-math-texts-have-solutions-to-their-exercises https://academia.stackexchange.com/questions/56739/why-dont-... [2]: https://math.stackexchange.com/questions/57889/hardy-wrights-intro-to-number-theory-is-highly-praised-but-has-no-exercises https://math.stackexchange.com/questions/57889/hardy-wrights... ("A random sample of 10 maths books I have handy shows that 4 doesn't have exercises.") [3]: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm https://www.ocf.berkeley.edu/~abhishek/chicmath.htm (search for "no exercises") To put it in programming terms: the exercises are the tests, the way to verify that your understanding of the mathematics has no bugs in it. If you're trying to verify that your solutions to exercises are correct, you're in a position akin to writing tests for your tests: it can be useful, but it's a second-order concern. Just as tests should be as “dumb” and “obvious” as possible, if there's any doubt at all that you've solved an exercise correctly, that itself is an indication that your understanding of the mathematics isn't complete, and you need to go back and engage with the material (think deeply, write your own exercises/tests, etc) until it becomes obvious. Of course this takes a lot more effort than plowing through exercises and verifying one's solutions. Programmers have another advantage over other mathematics students: you can test your understanding of the mathematics by writing actual programs. For example, if you're reading an elementary number theory book and learn about the Chinese Remainder Theorem, you can write an actual program/function for finding solutions to a system of modular congruences. As computers require a level of precision greater than human readers, this will force your understanding to become really sharp, as you have to deal with all the corner cases as well. (No doubt there are areas of mathematics for which this is hard to do, but most of the undergraduate curriculum can fit, and if you find something that's hard then maybe making it “programmable” can be your unique contribution.) All that said, it's definitely comforting to have solutions to exercises: there's a boost in motivation from being told that your answer is correct and you can progress to the next section, and though there's a cost here (if you needed to be told your solution is correct then maybe you should actually spend more time understanding the mathematics instead of going to the next section — but then again maybe not everyone really wants to understand the mathematics that well), I think that's probably the main thing that's missed when books don't have solutions. As a self-learner, motivation is most of the challenge, and solutions can definitely help there. --- Personally, if I look back at books I went through a good fraction of as a self-learner (and loved), there's an even mix: • Concrete Mathematics and Generatingfunctionology have complete solutions. • Burton's Elementary Number Theory has “Hints”, and “Answers” to selected exercises (i.e. those where the answer was a number, not a proof — these are not much work for an author to include). • The Art of Computer Programming (obviously I've read only a tiny bit of it) has problem ratings and sometimes very terse solutions / outlines of solutions. • Uspensky and Heaslet's Elementary Number Theory (which I read before Burton) and Analytic Combinatorics have no solutions at all. I don't think solutions to exercises have made a substantial difference in my engagement with these books, but who knows.

crispyambulance 8y ago

> So, why would we want to write mathematics any differently? I think you're stretching the unit-test-to-mathematics analogy a little too far. I haven't looked at the book in question, but generally, you don't just "know if" your solution is correct, you prove it, or at least consider examples that can demonstrate if your solution works in some conceivable cases. That's how "unit tests" work in mathematics. If the book has armed you with enough knowledge to work though a problem to the end, it has done it's job. The real test will come as you build on that knowledge later on in the book or in real-life applications. Moreover, it's not like this stuff is obscure knowledge. You can find help online or by looking at other books or connecting with others.

dwringer 8y ago
As I read it, the point is not about using unit tests to verify correctness, but as an interactive means of learning the problem inside and out prior to finding a complete solution. The trick is in learning what you don't know, and one can do that faster with regular feedback. There does seem to be a delicate balance between letting a student "stew" for a bit to really learn how to find and be confident in solutions, and preventing the student from heading down a wrong course. Books can never provide real feedback, I suppose, but it's a nice option to have solutions available even if students must be trusted to independently exhaust the available resources for attaining them first. Often having the solution is at least helpful as a "sanity check" when it comes to mathematics - I feel it is all too easy for beginners to make proofs on hidden and untenable assumptions and not necessarily realize it.
jeff76 8y ago
The point isn’t we need to do formal verification to test correctness. It’s that you probably wouldn’t write a program first pass and assume it is but free. So, why would you assume beginners that don’t know how to write proofs would write correct proofs without feedback to check?
- crispyambulance 8y ago
  Absolute beginners may not able to write an actual proof that their answer is correct, but at least they can verify that it satisfies the problem (depending on the actual problem). In any case, since the study of mathematics is cumulative. Even if the student can't prove or verify that their answer to a specific problem is correct they will eventually reach a point, often within the same problem-set, where inability to solve a problem will force them to encounter their gaps/misconceptions. The most important thing in a mathematics text, as with any other text, is lucidity of writing and the preparation of the student.

throwawaymath 8y ago

There’s no profound reason why authors don’t typically include solutions. It’s not gatekeeping and it’s not to appeal to an idealistic sense of the Right Way to learn mathematics. The real reason is twofold: 1. Authors write textbooks for universities and professors, not students. They write for the academic paradigm they’re familiar with, in which the professor is primarily teaching a student. The textbook is simply used to arrange a course, and professors would rather not have solutions available in the textbook. 2. Authors don’t earn much money from textbooks and textbooks aren’t weighed very heavily for tenure. There isn’t much of an incentive for them to write textbooks with solutions for the reason stated above, and it’s actually significantly more work to include solutions to your exercises. To put it very bluntly, there are extremely few non-students in the market for math textbooks, which is already a high-effort and low-return market for authors as it is. To get such a textbook you’d need someone who: knows the material extremely well, is very good at exposition, has a lot of time to invest in this as a passion project and doesn’t mind if it’s not used much or at all in well-known universities.

qznc 8y ago

We do have theorem checker and proof assistants these days. Are there books to learn math with Isabelle or Coq?

throwawaymath 8y ago
Not really, because it's extremely nontrivial to formally verify the proofs of most theorems.

dkarl 8y ago

Seeing a correct proof may or may not help you understand whether your proof is correct. What you're asking for is more like code review, not something you can package with a book.

jeff76 8y ago
Disagree. Some proofs are very straightforward and standard. Those are easily checkable with a solution guide.
- dkarl 8y ago
  When you hit upon a proof like that, you know it. If you're struggling and need feedback, then your solution is probably more complicated than the elegant one in the book, and the question you're asking is, okay, my solution isn't great, but is it even a solution? There's a reason these books declare some level of "mathematical maturity" as a prerequisite. By the time you get to certain content you have to either be comfortable checking a proof (maybe set it aside for a few days like you would any other writing) or have human help.
  - jeff76 8y ago
    Think you missed my entire point of including solutions for the beginner, i.e., someone that lacks math maturity and the reasoning of how easily such a person could fool themselves into thinking their thought process is valid.
    - dkarl 8y ago
      Someone who is a beginner will be working on easy material with a textbook that has very mild expectations of "mathematical maturity." Textbooks mostly (with a few intentional exceptions) calibrate the mathematical maturity they require according to how much work a student will have done in order to be ready to tackle the book's subject matter. Some people might think they can be ready to do problem sets in an advanced math textbook while skipping the work that would have given them the maturity it requires, but they are kidding themselves. They can't advance on the spectator track and then make a horizontal move over to actually doing the math. There's nothing wrong with people passively reading the material if they enjoy it, but if they want to tackle the problems, they are going to have to work through the earlier material by solving problems as well, and if they do that, they'll find that their sophistication increases along with the demands on it.

kevintb 8y ago

You clearly haven’t read the book or even clicked on the link. It’s written by Jeremy Kun, a fantastic math professor who’s written several excellent posts on his blog on how to intuitively grasp mathematical concepts.