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What is nice about this book is it includes solutions to exercises. I wish more proof-theoretic books included exercises in the back of the book like this one d

What is nice about this book is it includes solutions to exercises. I wish more proof-theoretic books included exercises in the back of the book like this one does. Yes, you can prove something in multiple ways, and no that isn't a good excuse to not include at least one version of the proof in the back of the book. It really benefits the self-learner. The math community seems to have a lot of artificial gatekeeping to keep non-academics out. They do this in several ways. 1) They recommend textbooks to beginners that are too advanced for their skill level. 2) These textbooks do not include solutions and the advice given is that solutions would somehow rob them of the experience (don't let those that lack self-discipline ruin learning for non-traditional students that aren't in a formal classroom with access to professors and TAs). The self-learner needs some sort of feedback system. 3) They tell the self-learner that solutions aren't provided because everyone has the ability to know if their solutions are correct without having their work checked. Would you write a complicated program and write no test cases, or could you instantly know your program is bug free the first time you write it? Why is math suddenly any different? Despite popular elitist opinions, I'd recommend this book over Axler for the beginner that doesn't know linear algebra. Everyone says Axler is the perfect first book in linear algebra. It really isn't. Even Axler admits this himself in the preface. He assumes that it is a second approach to the field. But people like to be elitist and recommend books to beginners that aren't always best from a pedagogical perspective. Those are the same class of people that recommend Rudin or Spivak to someone that wants to study elementary calculus 1 material.

skh 8y ago
In my experience (math teacher at a community college for 20 years and former Ph.D. student who quit before finishing the thesis) the overwhelming majority of people can not learn mathematics properly from a textbook. They need a teacher or someone to help with problems. It's very difficult to get to a point of really understanding the definition/motivation behind concepts. Even with a teacher it's hard to understand. Here's a simple example. The distributive property says: a (b + c) = ab + ac I can teach students to expand 3(x + 2). With some practice almost all of them will get this. They'll say 3(x+2) = 3x + 3(2) = 3x + 6 Do you know how hard it is to convince people that due to the nature of equality you can reverse the steps? Some never understand that 3x + 6 = 3(x + 2) Some will never understand that 3x + 2x = 5x because of the distributive property and that combining like terms for the expression 3x + ax is the same process. I don't think there is any desire to be elitist in terms of having good books that are appropriate. It's just hard to do. We've gone through various cycles of reforming calculus, introducing old concepts in a new way. Through all of the changes one thing has remained constant in my experience. The percentage of people who can get it remains the same. I do agree with your point on providing solutions.
- release_cycle 8y ago
  You make good points. A person that doesn't understand the distributive property, going into Calculus 1 is not the person that is going to understand Rudin or Spivak on their own without guidance. Yet, people in mathematics communities would recommend Spivak or Rudin anyway and look down on them if they used Stewart or Thomas. To them if you aren't learning a subject in the fullest rigor possible, then it isn't worth learning. It doesn't matter if the person doesn't even know the basics, they blindly recommend these type of books regardless. These are excellent books once you already seen the material before, not so much for the average person that doesn't know what the distributive property is.
- roenxi 8y ago
  > the overwhelming majority of people can not learn mathematics properly from a textbook. While you are right, a related issue is that most maths textbooks are atrocious at most aspects that aren't writing pages of equations. There is almost nobody sitting at the 3-way intersection of great mathematician, great writer and great educator who can then write great textbooks. I've found books with terms like "History" and "Philosophy" in the title are much better places to learn about mathmatics; combined with wikipedia for formulas and details. Any book with excercises but no solutions or historical context has turned out to be basically useless to me, even as a reference (wikipedia is usually better for simple stuff). I've got something like 4 books on statistics on my shelf at the moment. The only one that I've actually manged to read and learn something from has been Chatterjee's Philosophy of Statistics, because it talks about what techniques were developed in context of which problem, failed alternative approaches, explains what was confusing to some of the greatest minds in the history of statistics, etc. This has been vastly more useful in setting up a framework for what the world of statistics looks like that I can attach a whole bunch of proofs and suchlike too. It has been enlightening in a way that textbooks can't really manage.
  - tejuis 8y ago
    It is easy to agree with these findings. In general it is important for humans to get answers to WHY. History answers typically to WHY questions and to some extent also examples, i.e. you see by example WHY the theory works.
Kaveren 8y ago
Completely in agreement. I really dislike when math books don't have solutions to exercises available, since I self-study. Like Diestel's Graph Theory is great, it's the book you want to read if you're serious about the subject (D.B. West's book has exercises with solutions, I suppose), but there's absolutely no solutions to be found anywhere; the hints to solutions are now relegated to the "Professional Edition" as well, you can only find hints to old editions online. Makes me question whether or not I understand the material fully, since I can't check work.
sitkack 8y ago
Would you make a book list for other mathematical subjects? I have tried following the "recommended book" route and it really isn't productive. The first person recommends A and then everyone else doesn't want to look at l33t so they concur.
- BossingAround 8y ago
  I would highly recommend ASU's College Algebra and Problem Solving [1]. They have a large comprehensive test at the beginning that determines your knowledge. Should you be missing knowledge, they'll use the same platform to teach you all the prerequisites (i.e. they won't let you take the course until you go through ALEKS, their learning platform). After that's done, this course is something like preparation for college math classes. So far, I like it a lot, thanks to ALEKS. Give it a shot; it's free. [1] https://www.edx.org/course/college-algebra-problem-solving-asux-mat117x https://www.edx.org/course/college-algebra-problem-solving-a...
avz 8y ago
> They tell the self-learner that solutions aren't provided because everyone has the ability to know if their solutions are correct without having their work checked. Would you write a complicated program and write no test cases, or could you instantly know your program is bug free the first time you write it? Why is math suddenly any different? Math is an art of making a specific type of rigorous, formal arguments that are meant to convince, beyond all doubt, another human being that one statement follows from another. If an argument like that fails to convince even the proof author themselves, then you know the argument isn't good even without comparing it to another solution. In practice, after you've written down your proof, you examine it for weak spots. If any particular step raises doubts or objections then you recursively spell it out in greater detail (or replace with an alternative approach) until eventually all doubt dissipates. At that point you don't need to compare it to another solution. In fact, as you recognized yourself, your solution is likely to be different from the book author's, so comparison isn't helpful. Similarly, a problem like "write a small application to..." in a coding book or an assignment like "write an essay on..." in a writing book don't include solutions. Code, proofs and essays are all too free-form to subject them to correctness verification by comparison with a standard. Perhaps what you have in mind are examples. All good math textbooks include worked-out proofs that illustrate the techniques. In fact these are generally the bulk of the book! Perhaps you're asking for more examples. That's fair. Proof problems OTOH should be given without solutions to encourage independent creative thought of the reader and to avoid incorrect implication that there is the single correct solution. (Disclaimer: Not a mathematician.)
citizenkeen 8y ago
So, as someone who finagled their way into an MSCS progress with no calculus... What intro calc textbook for a self-disciplined self-learnerv would you recommend?
- release_cycle 8y ago
  I'd recommend "Calculus Made Easy" for intuitive exposure. It was the book Feynman studied from. Then pick up any calculus textbook and chug through it (Thomas is good from what I've heard). Even if the questions don't have answers in the back of the book you can check your computational steps for free using: https://www.symbolab.com https://www.symbolab.com. I also recommend lots of practice, so khan academy is good for drilling + any problem set book with lots of calculus problems, such as "Schaum's 3,000 Solved Problems in Calculus" or "Essential Calculus Skills Practice Workbook". They don't have to be huge calculus text, doing problems is more important than reading through 1,000s pages of colorful examples. You can find shorter calculus books that focus primarily in drilling calculus techniques. Focus on those to nail the techniques.

skh 8y ago

In my experience (math teacher at a community college for 20 years and former Ph.D. student who quit before finishing the thesis) the overwhelming majority of people can not learn mathematics properly from a textbook. They need a teacher or someone to help with problems. It's very difficult to get to a point of really understanding the definition/motivation behind concepts. Even with a teacher it's hard to understand. Here's a simple example. The distributive property says: a (b + c) = ab + ac I can teach students to expand 3(x + 2). With some practice almost all of them will get this. They'll say 3(x+2) = 3x + 3(2) = 3x + 6 Do you know how hard it is to convince people that due to the nature of equality you can reverse the steps? Some never understand that 3x + 6 = 3(x + 2) Some will never understand that 3x + 2x = 5x because of the distributive property and that combining like terms for the expression 3x + ax is the same process. I don't think there is any desire to be elitist in terms of having good books that are appropriate. It's just hard to do. We've gone through various cycles of reforming calculus, introducing old concepts in a new way. Through all of the changes one thing has remained constant in my experience. The percentage of people who can get it remains the same. I do agree with your point on providing solutions.

release_cycle 8y ago
You make good points. A person that doesn't understand the distributive property, going into Calculus 1 is not the person that is going to understand Rudin or Spivak on their own without guidance. Yet, people in mathematics communities would recommend Spivak or Rudin anyway and look down on them if they used Stewart or Thomas. To them if you aren't learning a subject in the fullest rigor possible, then it isn't worth learning. It doesn't matter if the person doesn't even know the basics, they blindly recommend these type of books regardless. These are excellent books once you already seen the material before, not so much for the average person that doesn't know what the distributive property is.
roenxi 8y ago
> the overwhelming majority of people can not learn mathematics properly from a textbook. While you are right, a related issue is that most maths textbooks are atrocious at most aspects that aren't writing pages of equations. There is almost nobody sitting at the 3-way intersection of great mathematician, great writer and great educator who can then write great textbooks. I've found books with terms like "History" and "Philosophy" in the title are much better places to learn about mathmatics; combined with wikipedia for formulas and details. Any book with excercises but no solutions or historical context has turned out to be basically useless to me, even as a reference (wikipedia is usually better for simple stuff). I've got something like 4 books on statistics on my shelf at the moment. The only one that I've actually manged to read and learn something from has been Chatterjee's Philosophy of Statistics, because it talks about what techniques were developed in context of which problem, failed alternative approaches, explains what was confusing to some of the greatest minds in the history of statistics, etc. This has been vastly more useful in setting up a framework for what the world of statistics looks like that I can attach a whole bunch of proofs and suchlike too. It has been enlightening in a way that textbooks can't really manage.
- tejuis 8y ago
  It is easy to agree with these findings. In general it is important for humans to get answers to WHY. History answers typically to WHY questions and to some extent also examples, i.e. you see by example WHY the theory works.

Kaveren 8y ago

Completely in agreement. I really dislike when math books don't have solutions to exercises available, since I self-study. Like Diestel's Graph Theory is great, it's the book you want to read if you're serious about the subject (D.B. West's book has exercises with solutions, I suppose), but there's absolutely no solutions to be found anywhere; the hints to solutions are now relegated to the "Professional Edition" as well, you can only find hints to old editions online. Makes me question whether or not I understand the material fully, since I can't check work.

sitkack 8y ago

Would you make a book list for other mathematical subjects? I have tried following the "recommended book" route and it really isn't productive. The first person recommends A and then everyone else doesn't want to look at l33t so they concur.

BossingAround 8y ago
I would highly recommend ASU's College Algebra and Problem Solving [1]. They have a large comprehensive test at the beginning that determines your knowledge. Should you be missing knowledge, they'll use the same platform to teach you all the prerequisites (i.e. they won't let you take the course until you go through ALEKS, their learning platform). After that's done, this course is something like preparation for college math classes. So far, I like it a lot, thanks to ALEKS. Give it a shot; it's free. [1] https://www.edx.org/course/college-algebra-problem-solving-asux-mat117x https://www.edx.org/course/college-algebra-problem-solving-a...

avz 8y ago

> They tell the self-learner that solutions aren't provided because everyone has the ability to know if their solutions are correct without having their work checked. Would you write a complicated program and write no test cases, or could you instantly know your program is bug free the first time you write it? Why is math suddenly any different? Math is an art of making a specific type of rigorous, formal arguments that are meant to convince, beyond all doubt, another human being that one statement follows from another. If an argument like that fails to convince even the proof author themselves, then you know the argument isn't good even without comparing it to another solution. In practice, after you've written down your proof, you examine it for weak spots. If any particular step raises doubts or objections then you recursively spell it out in greater detail (or replace with an alternative approach) until eventually all doubt dissipates. At that point you don't need to compare it to another solution. In fact, as you recognized yourself, your solution is likely to be different from the book author's, so comparison isn't helpful. Similarly, a problem like "write a small application to..." in a coding book or an assignment like "write an essay on..." in a writing book don't include solutions. Code, proofs and essays are all too free-form to subject them to correctness verification by comparison with a standard. Perhaps what you have in mind are examples. All good math textbooks include worked-out proofs that illustrate the techniques. In fact these are generally the bulk of the book! Perhaps you're asking for more examples. That's fair. Proof problems OTOH should be given without solutions to encourage independent creative thought of the reader and to avoid incorrect implication that there is the single correct solution. (Disclaimer: Not a mathematician.)

citizenkeen 8y ago

So, as someone who finagled their way into an MSCS progress with no calculus... What intro calc textbook for a self-disciplined self-learnerv would you recommend?

release_cycle 8y ago
I'd recommend "Calculus Made Easy" for intuitive exposure. It was the book Feynman studied from. Then pick up any calculus textbook and chug through it (Thomas is good from what I've heard). Even if the questions don't have answers in the back of the book you can check your computational steps for free using: https://www.symbolab.com https://www.symbolab.com. I also recommend lots of practice, so khan academy is good for drilling + any problem set book with lots of calculus problems, such as "Schaum's 3,000 Solved Problems in Calculus" or "Essential Calculus Skills Practice Workbook". They don't have to be huge calculus text, doing problems is more important than reading through 1,000s pages of colorful examples. You can find shorter calculus books that focus primarily in drilling calculus techniques. Focus on those to nail the techniques.