Terence Tao - type books in other fields?Self-study Real analysis Tao or Rudin?Abstract Algebra book with exercise solutions recommendations.Books to read to understand Terence Tao's Analytic Number Theory PapersLooking for fast text-booksReferences on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence TaoLinear Algebra and Analysis books recommendationCould you recommend one among these Analysis Books for self-study based on my background?What 'intermediate' or 'advanced' good set theory books are there?Best books for self study with challenging exercises and solutions?Looking for a different type of Linear Algebra book
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Terence Tao - type books in other fields?
Self-study Real analysis Tao or Rudin?Abstract Algebra book with exercise solutions recommendations.Books to read to understand Terence Tao's Analytic Number Theory PapersLooking for fast text-booksReferences on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence TaoLinear Algebra and Analysis books recommendationCould you recommend one among these Analysis Books for self-study based on my background?What 'intermediate' or 'advanced' good set theory books are there?Best books for self study with challenging exercises and solutions?Looking for a different type of Linear Algebra book
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$begingroup$
I have looked at Tao's book on Measure Theory, and they are perhaps the best math books I have ever seen. Besides the extremely clear and motivated presentation, the main feature of the book is that there is no big list of exercises at the end of each chapter; the exercises are dispersed throughout the text, and they are actually critical in developing the theory.
Question: What are some other math books written in this style, or other authors who write in this way? I am open to any fields of math, since I will use this question in the future as a reference.
That was the question; the following is just why I think Tao's style is so great.
- When you come to an exercise, you know that you are ready for it. There is no doubt in the back of your mind that "maybe I haven't read enough of the chapter to solve this exercise"
- Similarly, there is no bad feeling of "maybe I wasn't supposed to use this more advanced theorem for this exercise, maybe I was supposed to do it from the basic definitions but I can't". It makes everything feel "fair game"
- It makes it difficult to be a passive reader
- It makes you become invested in the development of the theory, as if you are living back in 1900 and trying to develop this stuff for the first time
I think you can achieve a similar effect with almost any other book, if you try to prove every theorem by yourself before you read the proof and stuff like that, but at least for me there are some severe psychological barriers that prevent me from doing that. For example, if I try to prove a theorem without reading the proof, I always have the doubt that "this proof may be too hard, it would not be expected of the reader to come up with this proof". In Tao's book, the proofs are conciously left to you, so you know that you can do it, which is a big encouragement.
real-analysis abstract-algebra complex-analysis reference-request soft-question
asked 9 hours ago
OviOvi
13.1k10 gold badges42 silver badges120 bronze badges
$endgroup$
|
show 2 more comments
$begingroup$
I have looked at Tao's book on Measure Theory, and they are perhaps the best math books I have ever seen. Besides the extremely clear and motivated presentation, the main feature of the book is that there is no big list of exercises at the end of each chapter; the exercises are dispersed throughout the text, and they are actually critical in developing the theory.
Question: What are some other math books written in this style, or other authors who write in this way? I am open to any fields of math, since I will use this question in the future as a reference.
That was the question; the following is just why I think Tao's style is so great.
- When you come to an exercise, you know that you are ready for it. There is no doubt in the back of your mind that "maybe I haven't read enough of the chapter to solve this exercise"
- Similarly, there is no bad feeling of "maybe I wasn't supposed to use this more advanced theorem for this exercise, maybe I was supposed to do it from the basic definitions but I can't". It makes everything feel "fair game"
- It makes it difficult to be a passive reader
- It makes you become invested in the development of the theory, as if you are living back in 1900 and trying to develop this stuff for the first time
I think you can achieve a similar effect with almost any other book, if you try to prove every theorem by yourself before you read the proof and stuff like that, but at least for me there are some severe psychological barriers that prevent me from doing that. For example, if I try to prove a theorem without reading the proof, I always have the doubt that "this proof may be too hard, it would not be expected of the reader to come up with this proof". In Tao's book, the proofs are conciously left to you, so you know that you can do it, which is a big encouragement.
real-analysis abstract-algebra complex-analysis reference-request soft-question
asked 9 hours ago
OviOvi
13.1k10 gold badges42 silver badges120 bronze badges
$endgroup$
1
$begingroup$
I personaly would like something like this for algebraic topology. Someone knows a book in this style for this field?
$endgroup$
– Cornman
9 hours ago
1
$begingroup$
@Cornman: Hatcher is great. If you want something at more or less the same general level of Hatcher but a bit more specific, Bott and Tu's "Differential Forms in Algebraic Topology" is the best-written math textbook I've come across on any subject.
$endgroup$
– anomaly
8 hours ago
1
$begingroup$
@Cornman: Probably just analysis on manifolds, to deal with differential forms. (Hatcher deals with CW category; Bott and Tu mostly stays in the category of smooth manifolds, at least until they get into Eilenberg-MacLane spaces.) If memory serves, there's less homological algebra in Bott and Tu (for one thing, a lot of it is done over $mathbbR$); in Hatcher, you should probably be at least familiar with modules for the chapter on cohomology. For both, you should be familiar with basic undergrad point-set topology.
$endgroup$
– anomaly
7 hours ago
1
$begingroup$
Also, for Hatcher: Pay careful attention to the appendix on CW-complexes. That's probably the steepest learning curve to the book, but you can dip in and out of that appendix while reading the main part of the book if you don't want to wade through it all at once. You should probably be comfortable with it by the beginning of the chapter on homology, though, and it makes the chapter on the fundamental grou p easier.
$endgroup$
– anomaly
7 hours ago
1
$begingroup$
That the exercises are interspersed vs at chapter end is not necessarily a good feature, because it may give you too much of a hint as to what met methods you should apply. Later when you tackle research problems in the wild you won't have those crutches, which means you may be less well prepared than you would be using a textbook that weaned you earlier from those contrived contexts.
$endgroup$
– Bill Dubuque
7 hours ago
|
show 2 more comments
$begingroup$
I have looked at Tao's book on Measure Theory, and they are perhaps the best math books I have ever seen. Besides the extremely clear and motivated presentation, the main feature of the book is that there is no big list of exercises at the end of each chapter; the exercises are dispersed throughout the text, and they are actually critical in developing the theory.
Question: What are some other math books written in this style, or other authors who write in this way? I am open to any fields of math, since I will use this question in the future as a reference.
That was the question; the following is just why I think Tao's style is so great.
- When you come to an exercise, you know that you are ready for it. There is no doubt in the back of your mind that "maybe I haven't read enough of the chapter to solve this exercise"
- Similarly, there is no bad feeling of "maybe I wasn't supposed to use this more advanced theorem for this exercise, maybe I was supposed to do it from the basic definitions but I can't". It makes everything feel "fair game"
- It makes it difficult to be a passive reader
- It makes you become invested in the development of the theory, as if you are living back in 1900 and trying to develop this stuff for the first time
I think you can achieve a similar effect with almost any other book, if you try to prove every theorem by yourself before you read the proof and stuff like that, but at least for me there are some severe psychological barriers that prevent me from doing that. For example, if I try to prove a theorem without reading the proof, I always have the doubt that "this proof may be too hard, it would not be expected of the reader to come up with this proof". In Tao's book, the proofs are conciously left to you, so you know that you can do it, which is a big encouragement.
real-analysis abstract-algebra complex-analysis reference-request soft-question
asked 9 hours ago
OviOvi
13.1k10 gold badges42 silver badges120 bronze badges
$endgroup$
I have looked at Tao's book on Measure Theory, and they are perhaps the best math books I have ever seen. Besides the extremely clear and motivated presentation, the main feature of the book is that there is no big list of exercises at the end of each chapter; the exercises are dispersed throughout the text, and they are actually critical in developing the theory.
Question: What are some other math books written in this style, or other authors who write in this way? I am open to any fields of math, since I will use this question in the future as a reference.
That was the question; the following is just why I think Tao's style is so great.
- When you come to an exercise, you know that you are ready for it. There is no doubt in the back of your mind that "maybe I haven't read enough of the chapter to solve this exercise"
- Similarly, there is no bad feeling of "maybe I wasn't supposed to use this more advanced theorem for this exercise, maybe I was supposed to do it from the basic definitions but I can't". It makes everything feel "fair game"
- It makes it difficult to be a passive reader
- It makes you become invested in the development of the theory, as if you are living back in 1900 and trying to develop this stuff for the first time
I think you can achieve a similar effect with almost any other book, if you try to prove every theorem by yourself before you read the proof and stuff like that, but at least for me there are some severe psychological barriers that prevent me from doing that. For example, if I try to prove a theorem without reading the proof, I always have the doubt that "this proof may be too hard, it would not be expected of the reader to come up with this proof". In Tao's book, the proofs are conciously left to you, so you know that you can do it, which is a big encouragement.
real-analysis abstract-algebra complex-analysis reference-request soft-question
real-analysis abstract-algebra complex-analysis reference-request soft-question
asked 9 hours ago
OviOvi
13.1k10 gold badges42 silver badges120 bronze badges
asked 9 hours ago
OviOvi
13.1k10 gold badges42 silver badges120 bronze badges
edited 8 hours ago
Ovi
asked 9 hours ago
OviOvi
13.1k10 gold badges42 silver badges120 bronze badges
asked 9 hours ago
OviOvi
13.1k10 gold badges42 silver badges120 bronze badges
asked 9 hours ago
OviOvi
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13.1k10 gold badges42 silver badges120 bronze badges
1
$begingroup$
I personaly would like something like this for algebraic topology. Someone knows a book in this style for this field?
$endgroup$
– Cornman
9 hours ago
1
$begingroup$
@Cornman: Hatcher is great. If you want something at more or less the same general level of Hatcher but a bit more specific, Bott and Tu's "Differential Forms in Algebraic Topology" is the best-written math textbook I've come across on any subject.
$endgroup$
– anomaly
8 hours ago
1
$begingroup$
@Cornman: Probably just analysis on manifolds, to deal with differential forms. (Hatcher deals with CW category; Bott and Tu mostly stays in the category of smooth manifolds, at least until they get into Eilenberg-MacLane spaces.) If memory serves, there's less homological algebra in Bott and Tu (for one thing, a lot of it is done over $mathbbR$); in Hatcher, you should probably be at least familiar with modules for the chapter on cohomology. For both, you should be familiar with basic undergrad point-set topology.
$endgroup$
– anomaly
7 hours ago
1
$begingroup$
Also, for Hatcher: Pay careful attention to the appendix on CW-complexes. That's probably the steepest learning curve to the book, but you can dip in and out of that appendix while reading the main part of the book if you don't want to wade through it all at once. You should probably be comfortable with it by the beginning of the chapter on homology, though, and it makes the chapter on the fundamental grou p easier.
$endgroup$
– anomaly
7 hours ago
1
$begingroup$
That the exercises are interspersed vs at chapter end is not necessarily a good feature, because it may give you too much of a hint as to what met methods you should apply. Later when you tackle research problems in the wild you won't have those crutches, which means you may be less well prepared than you would be using a textbook that weaned you earlier from those contrived contexts.
$endgroup$
– Bill Dubuque
7 hours ago
|
show 2 more comments
1
$begingroup$
I personaly would like something like this for algebraic topology. Someone knows a book in this style for this field?
$endgroup$
– Cornman
9 hours ago
1
$begingroup$
@Cornman: Hatcher is great. If you want something at more or less the same general level of Hatcher but a bit more specific, Bott and Tu's "Differential Forms in Algebraic Topology" is the best-written math textbook I've come across on any subject.
$endgroup$
– anomaly
8 hours ago
1
$begingroup$
@Cornman: Probably just analysis on manifolds, to deal with differential forms. (Hatcher deals with CW category; Bott and Tu mostly stays in the category of smooth manifolds, at least until they get into Eilenberg-MacLane spaces.) If memory serves, there's less homological algebra in Bott and Tu (for one thing, a lot of it is done over $mathbbR$); in Hatcher, you should probably be at least familiar with modules for the chapter on cohomology. For both, you should be familiar with basic undergrad point-set topology.
$endgroup$
– anomaly
7 hours ago
1
$begingroup$
Also, for Hatcher: Pay careful attention to the appendix on CW-complexes. That's probably the steepest learning curve to the book, but you can dip in and out of that appendix while reading the main part of the book if you don't want to wade through it all at once. You should probably be comfortable with it by the beginning of the chapter on homology, though, and it makes the chapter on the fundamental grou p easier.
$endgroup$
– anomaly
7 hours ago
1
$begingroup$
That the exercises are interspersed vs at chapter end is not necessarily a good feature, because it may give you too much of a hint as to what met methods you should apply. Later when you tackle research problems in the wild you won't have those crutches, which means you may be less well prepared than you would be using a textbook that weaned you earlier from those contrived contexts.
$endgroup$
– Bill Dubuque
7 hours ago
1
1
$begingroup$
I personaly would like something like this for algebraic topology. Someone knows a book in this style for this field?
$endgroup$
– Cornman
9 hours ago
$begingroup$
I personaly would like something like this for algebraic topology. Someone knows a book in this style for this field?
$endgroup$
– Cornman
9 hours ago
1
1
$begingroup$
@Cornman: Hatcher is great. If you want something at more or less the same general level of Hatcher but a bit more specific, Bott and Tu's "Differential Forms in Algebraic Topology" is the best-written math textbook I've come across on any subject.
$endgroup$
– anomaly
8 hours ago
$begingroup$
@Cornman: Hatcher is great. If you want something at more or less the same general level of Hatcher but a bit more specific, Bott and Tu's "Differential Forms in Algebraic Topology" is the best-written math textbook I've come across on any subject.
$endgroup$
– anomaly
8 hours ago
1
1
$begingroup$
@Cornman: Probably just analysis on manifolds, to deal with differential forms. (Hatcher deals with CW category; Bott and Tu mostly stays in the category of smooth manifolds, at least until they get into Eilenberg-MacLane spaces.) If memory serves, there's less homological algebra in Bott and Tu (for one thing, a lot of it is done over $mathbbR$); in Hatcher, you should probably be at least familiar with modules for the chapter on cohomology. For both, you should be familiar with basic undergrad point-set topology.
$endgroup$
– anomaly
7 hours ago
$begingroup$
@Cornman: Probably just analysis on manifolds, to deal with differential forms. (Hatcher deals with CW category; Bott and Tu mostly stays in the category of smooth manifolds, at least until they get into Eilenberg-MacLane spaces.) If memory serves, there's less homological algebra in Bott and Tu (for one thing, a lot of it is done over $mathbbR$); in Hatcher, you should probably be at least familiar with modules for the chapter on cohomology. For both, you should be familiar with basic undergrad point-set topology.
$endgroup$
– anomaly
7 hours ago
1
1
$begingroup$
Also, for Hatcher: Pay careful attention to the appendix on CW-complexes. That's probably the steepest learning curve to the book, but you can dip in and out of that appendix while reading the main part of the book if you don't want to wade through it all at once. You should probably be comfortable with it by the beginning of the chapter on homology, though, and it makes the chapter on the fundamental grou p easier.
$endgroup$
– anomaly
7 hours ago
$begingroup$
Also, for Hatcher: Pay careful attention to the appendix on CW-complexes. That's probably the steepest learning curve to the book, but you can dip in and out of that appendix while reading the main part of the book if you don't want to wade through it all at once. You should probably be comfortable with it by the beginning of the chapter on homology, though, and it makes the chapter on the fundamental grou p easier.
$endgroup$
– anomaly
7 hours ago
1
1
$begingroup$
That the exercises are interspersed vs at chapter end is not necessarily a good feature, because it may give you too much of a hint as to what met methods you should apply. Later when you tackle research problems in the wild you won't have those crutches, which means you may be less well prepared than you would be using a textbook that weaned you earlier from those contrived contexts.
$endgroup$
– Bill Dubuque
7 hours ago
$begingroup$
That the exercises are interspersed vs at chapter end is not necessarily a good feature, because it may give you too much of a hint as to what met methods you should apply. Later when you tackle research problems in the wild you won't have those crutches, which means you may be less well prepared than you would be using a textbook that weaned you earlier from those contrived contexts.
$endgroup$
– Bill Dubuque
7 hours ago
|
show 2 more comments
4 Answers
4
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$begingroup$
Vakil's notes on Algebraic Geometry http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf are written in the same style. A nice contrast to the terse, traditional nature of the standard reference, Hartshorne https://www.springer.com/gp/book/9780387902449
answered 9 hours ago
bouncebackbounceback
1,0133 silver badges13 bronze badges
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add a comment |
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Another example, for functional analysis, is Lax's book, "Functional Analysis." It's a very-received and commonly-used textbook (see https://mathoverflow.net/questions/72419/a-good-book-of-functional-analysis), and it leaves many results to exercises as you read along, similar to Tao's style.
Another example, although to a lesser extent, is Abbott's introductory real analysis book "Understanding Analysis." This is a very good book with in-depth explanations and visuals. The author leaves plenty of results to exercises, and in some sections, has you construct many of the tools yourself through guided exercises (such as the sections on double sums and Fourier series).
answered 8 hours ago
cmkcmk
5,5011 gold badge9 silver badges28 bronze badges
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add a comment |
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I am not sure about your mathematical background, but you can try "How to Prove It". I am currently studying this book myself. As someone with no formal mathematical education, and someone who had been really terrified of mathematical proofs before, I think this book is exteremely well written.
Excerpt from the book's introduction:
The book begins with the basic concepts of logic and set theory, to
familiarize students with the language of mathematics and how it is
interpreted. These concepts are used as the basis for a step-by-step
breakdown of the most important techniques used in constructing
proofs. The author shows how complex proofs are built up from these
smaller steps, using detailed 'scratch work' sections to expose the
machinery of proofs about the natural numbers, relations, functions,
and infinite sets.
I believe it may be suitable for you, because:
When you come to an exercise, you know that you are ready for it.
There is no doubt in the back of your mind that "maybe I haven't read
enough of the chapter to solve this exercise"
As I've just pointed out, I don't have mathematical background, or put it bluntly, I'm quite bad at math. However, even for me, it is extremely easy to follow everything author says.
It makes it difficult to be a passive reader
Indeed it is. Besides having plenty of exercises after each chapter, there are a lot of them scattered within each chapter. Unless you devote you time and energy and solve each exercise yourself, I believe it will be pretty hard to follow anything.
It makes you become invested in the development of the theory, as if
you are living back in 1900 and trying to develop this stuff for the
first time
As you can see from the name of the book, the author's aim is teach students how to prove things. And, when trying to prove something yourself, you will definitely need to use your own reasoning and develop your own approaches to the problem.
You can check out the book here
answered 9 hours ago
NelverNelver
3849 bronze badges
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I recommend the following books, which are similar to Tao's books in that their exercises are very well planned out and form an integral part of the text itself.
- Pinter, "A Book of Abstract Algebra"
- Fong & Spivak 2017, "Seven Sketches in Compositionality:. An Introduction to Applied Category Theory"
answered 8 hours ago
Ben BrayBen Bray
3461 silver badge10 bronze badges
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4 Answers
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4 Answers
4
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$begingroup$
Vakil's notes on Algebraic Geometry http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf are written in the same style. A nice contrast to the terse, traditional nature of the standard reference, Hartshorne https://www.springer.com/gp/book/9780387902449
answered 9 hours ago
bouncebackbounceback
1,0133 silver badges13 bronze badges
$endgroup$
add a comment |
$begingroup$
Vakil's notes on Algebraic Geometry http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf are written in the same style. A nice contrast to the terse, traditional nature of the standard reference, Hartshorne https://www.springer.com/gp/book/9780387902449
answered 9 hours ago
bouncebackbounceback
1,0133 silver badges13 bronze badges
$endgroup$
add a comment |
$begingroup$
Vakil's notes on Algebraic Geometry http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf are written in the same style. A nice contrast to the terse, traditional nature of the standard reference, Hartshorne https://www.springer.com/gp/book/9780387902449
answered 9 hours ago
bouncebackbounceback
1,0133 silver badges13 bronze badges
$endgroup$
Vakil's notes on Algebraic Geometry http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf are written in the same style. A nice contrast to the terse, traditional nature of the standard reference, Hartshorne https://www.springer.com/gp/book/9780387902449
answered 9 hours ago
bouncebackbounceback
1,0133 silver badges13 bronze badges
answered 9 hours ago
bouncebackbounceback
1,0133 silver badges13 bronze badges
answered 9 hours ago
bouncebackbounceback
1,0133 silver badges13 bronze badges
answered 9 hours ago
bouncebackbounceback
1,0133 silver badges13 bronze badges
1,0133 silver badges13 bronze badges
add a comment |
add a comment |
$begingroup$
Another example, for functional analysis, is Lax's book, "Functional Analysis." It's a very-received and commonly-used textbook (see https://mathoverflow.net/questions/72419/a-good-book-of-functional-analysis), and it leaves many results to exercises as you read along, similar to Tao's style.
Another example, although to a lesser extent, is Abbott's introductory real analysis book "Understanding Analysis." This is a very good book with in-depth explanations and visuals. The author leaves plenty of results to exercises, and in some sections, has you construct many of the tools yourself through guided exercises (such as the sections on double sums and Fourier series).
answered 8 hours ago
cmkcmk
5,5011 gold badge9 silver badges28 bronze badges
$endgroup$
add a comment |
$begingroup$
Another example, for functional analysis, is Lax's book, "Functional Analysis." It's a very-received and commonly-used textbook (see https://mathoverflow.net/questions/72419/a-good-book-of-functional-analysis), and it leaves many results to exercises as you read along, similar to Tao's style.
Another example, although to a lesser extent, is Abbott's introductory real analysis book "Understanding Analysis." This is a very good book with in-depth explanations and visuals. The author leaves plenty of results to exercises, and in some sections, has you construct many of the tools yourself through guided exercises (such as the sections on double sums and Fourier series).
answered 8 hours ago
cmkcmk
5,5011 gold badge9 silver badges28 bronze badges
$endgroup$
add a comment |
$begingroup$
Another example, for functional analysis, is Lax's book, "Functional Analysis." It's a very-received and commonly-used textbook (see https://mathoverflow.net/questions/72419/a-good-book-of-functional-analysis), and it leaves many results to exercises as you read along, similar to Tao's style.
Another example, although to a lesser extent, is Abbott's introductory real analysis book "Understanding Analysis." This is a very good book with in-depth explanations and visuals. The author leaves plenty of results to exercises, and in some sections, has you construct many of the tools yourself through guided exercises (such as the sections on double sums and Fourier series).
answered 8 hours ago
cmkcmk
5,5011 gold badge9 silver badges28 bronze badges
$endgroup$
Another example, for functional analysis, is Lax's book, "Functional Analysis." It's a very-received and commonly-used textbook (see https://mathoverflow.net/questions/72419/a-good-book-of-functional-analysis), and it leaves many results to exercises as you read along, similar to Tao's style.
Another example, although to a lesser extent, is Abbott's introductory real analysis book "Understanding Analysis." This is a very good book with in-depth explanations and visuals. The author leaves plenty of results to exercises, and in some sections, has you construct many of the tools yourself through guided exercises (such as the sections on double sums and Fourier series).
answered 8 hours ago
cmkcmk
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answered 8 hours ago
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answered 8 hours ago
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$begingroup$
I am not sure about your mathematical background, but you can try "How to Prove It". I am currently studying this book myself. As someone with no formal mathematical education, and someone who had been really terrified of mathematical proofs before, I think this book is exteremely well written.
Excerpt from the book's introduction:
The book begins with the basic concepts of logic and set theory, to
familiarize students with the language of mathematics and how it is
interpreted. These concepts are used as the basis for a step-by-step
breakdown of the most important techniques used in constructing
proofs. The author shows how complex proofs are built up from these
smaller steps, using detailed 'scratch work' sections to expose the
machinery of proofs about the natural numbers, relations, functions,
and infinite sets.
I believe it may be suitable for you, because:
When you come to an exercise, you know that you are ready for it.
There is no doubt in the back of your mind that "maybe I haven't read
enough of the chapter to solve this exercise"
As I've just pointed out, I don't have mathematical background, or put it bluntly, I'm quite bad at math. However, even for me, it is extremely easy to follow everything author says.
It makes it difficult to be a passive reader
Indeed it is. Besides having plenty of exercises after each chapter, there are a lot of them scattered within each chapter. Unless you devote you time and energy and solve each exercise yourself, I believe it will be pretty hard to follow anything.
It makes you become invested in the development of the theory, as if
you are living back in 1900 and trying to develop this stuff for the
first time
As you can see from the name of the book, the author's aim is teach students how to prove things. And, when trying to prove something yourself, you will definitely need to use your own reasoning and develop your own approaches to the problem.
You can check out the book here
answered 9 hours ago
NelverNelver
3849 bronze badges
$endgroup$
add a comment |
$begingroup$
I am not sure about your mathematical background, but you can try "How to Prove It". I am currently studying this book myself. As someone with no formal mathematical education, and someone who had been really terrified of mathematical proofs before, I think this book is exteremely well written.
Excerpt from the book's introduction:
The book begins with the basic concepts of logic and set theory, to
familiarize students with the language of mathematics and how it is
interpreted. These concepts are used as the basis for a step-by-step
breakdown of the most important techniques used in constructing
proofs. The author shows how complex proofs are built up from these
smaller steps, using detailed 'scratch work' sections to expose the
machinery of proofs about the natural numbers, relations, functions,
and infinite sets.
I believe it may be suitable for you, because:
When you come to an exercise, you know that you are ready for it.
There is no doubt in the back of your mind that "maybe I haven't read
enough of the chapter to solve this exercise"
As I've just pointed out, I don't have mathematical background, or put it bluntly, I'm quite bad at math. However, even for me, it is extremely easy to follow everything author says.
It makes it difficult to be a passive reader
Indeed it is. Besides having plenty of exercises after each chapter, there are a lot of them scattered within each chapter. Unless you devote you time and energy and solve each exercise yourself, I believe it will be pretty hard to follow anything.
It makes you become invested in the development of the theory, as if
you are living back in 1900 and trying to develop this stuff for the
first time
As you can see from the name of the book, the author's aim is teach students how to prove things. And, when trying to prove something yourself, you will definitely need to use your own reasoning and develop your own approaches to the problem.
You can check out the book here
answered 9 hours ago
NelverNelver
3849 bronze badges
$endgroup$
add a comment |
$begingroup$
I am not sure about your mathematical background, but you can try "How to Prove It". I am currently studying this book myself. As someone with no formal mathematical education, and someone who had been really terrified of mathematical proofs before, I think this book is exteremely well written.
Excerpt from the book's introduction:
The book begins with the basic concepts of logic and set theory, to
familiarize students with the language of mathematics and how it is
interpreted. These concepts are used as the basis for a step-by-step
breakdown of the most important techniques used in constructing
proofs. The author shows how complex proofs are built up from these
smaller steps, using detailed 'scratch work' sections to expose the
machinery of proofs about the natural numbers, relations, functions,
and infinite sets.
I believe it may be suitable for you, because:
When you come to an exercise, you know that you are ready for it.
There is no doubt in the back of your mind that "maybe I haven't read
enough of the chapter to solve this exercise"
As I've just pointed out, I don't have mathematical background, or put it bluntly, I'm quite bad at math. However, even for me, it is extremely easy to follow everything author says.
It makes it difficult to be a passive reader
Indeed it is. Besides having plenty of exercises after each chapter, there are a lot of them scattered within each chapter. Unless you devote you time and energy and solve each exercise yourself, I believe it will be pretty hard to follow anything.
It makes you become invested in the development of the theory, as if
you are living back in 1900 and trying to develop this stuff for the
first time
As you can see from the name of the book, the author's aim is teach students how to prove things. And, when trying to prove something yourself, you will definitely need to use your own reasoning and develop your own approaches to the problem.
You can check out the book here
answered 9 hours ago
NelverNelver
3849 bronze badges
$endgroup$
I am not sure about your mathematical background, but you can try "How to Prove It". I am currently studying this book myself. As someone with no formal mathematical education, and someone who had been really terrified of mathematical proofs before, I think this book is exteremely well written.
Excerpt from the book's introduction:
The book begins with the basic concepts of logic and set theory, to
familiarize students with the language of mathematics and how it is
interpreted. These concepts are used as the basis for a step-by-step
breakdown of the most important techniques used in constructing
proofs. The author shows how complex proofs are built up from these
smaller steps, using detailed 'scratch work' sections to expose the
machinery of proofs about the natural numbers, relations, functions,
and infinite sets.
I believe it may be suitable for you, because:
When you come to an exercise, you know that you are ready for it.
There is no doubt in the back of your mind that "maybe I haven't read
enough of the chapter to solve this exercise"
As I've just pointed out, I don't have mathematical background, or put it bluntly, I'm quite bad at math. However, even for me, it is extremely easy to follow everything author says.
It makes it difficult to be a passive reader
Indeed it is. Besides having plenty of exercises after each chapter, there are a lot of them scattered within each chapter. Unless you devote you time and energy and solve each exercise yourself, I believe it will be pretty hard to follow anything.
It makes you become invested in the development of the theory, as if
you are living back in 1900 and trying to develop this stuff for the
first time
As you can see from the name of the book, the author's aim is teach students how to prove things. And, when trying to prove something yourself, you will definitely need to use your own reasoning and develop your own approaches to the problem.
You can check out the book here
answered 9 hours ago
NelverNelver
3849 bronze badges
answered 9 hours ago
NelverNelver
3849 bronze badges
answered 9 hours ago
NelverNelver
3849 bronze badges
answered 9 hours ago
NelverNelver
3849 bronze badges
3849 bronze badges
add a comment |
add a comment |
$begingroup$
I recommend the following books, which are similar to Tao's books in that their exercises are very well planned out and form an integral part of the text itself.
- Pinter, "A Book of Abstract Algebra"
- Fong & Spivak 2017, "Seven Sketches in Compositionality:. An Introduction to Applied Category Theory"
answered 8 hours ago
Ben BrayBen Bray
3461 silver badge10 bronze badges
$endgroup$
add a comment |
$begingroup$
I recommend the following books, which are similar to Tao's books in that their exercises are very well planned out and form an integral part of the text itself.
- Pinter, "A Book of Abstract Algebra"
- Fong & Spivak 2017, "Seven Sketches in Compositionality:. An Introduction to Applied Category Theory"
answered 8 hours ago
Ben BrayBen Bray
3461 silver badge10 bronze badges
$endgroup$
add a comment |
$begingroup$
I recommend the following books, which are similar to Tao's books in that their exercises are very well planned out and form an integral part of the text itself.
- Pinter, "A Book of Abstract Algebra"
- Fong & Spivak 2017, "Seven Sketches in Compositionality:. An Introduction to Applied Category Theory"
answered 8 hours ago
Ben BrayBen Bray
3461 silver badge10 bronze badges
$endgroup$
I recommend the following books, which are similar to Tao's books in that their exercises are very well planned out and form an integral part of the text itself.
- Pinter, "A Book of Abstract Algebra"
- Fong & Spivak 2017, "Seven Sketches in Compositionality:. An Introduction to Applied Category Theory"
answered 8 hours ago
Ben BrayBen Bray
3461 silver badge10 bronze badges
answered 8 hours ago
Ben BrayBen Bray
3461 silver badge10 bronze badges
answered 8 hours ago
Ben BrayBen Bray
3461 silver badge10 bronze badges
answered 8 hours ago
Ben BrayBen Bray
3461 silver badge10 bronze badges
3461 silver badge10 bronze badges
add a comment |
add a comment |
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1
$begingroup$
I personaly would like something like this for algebraic topology. Someone knows a book in this style for this field?
$endgroup$
– Cornman
9 hours ago
1
$begingroup$
@Cornman: Hatcher is great. If you want something at more or less the same general level of Hatcher but a bit more specific, Bott and Tu's "Differential Forms in Algebraic Topology" is the best-written math textbook I've come across on any subject.
$endgroup$
– anomaly
8 hours ago
1
$begingroup$
@Cornman: Probably just analysis on manifolds, to deal with differential forms. (Hatcher deals with CW category; Bott and Tu mostly stays in the category of smooth manifolds, at least until they get into Eilenberg-MacLane spaces.) If memory serves, there's less homological algebra in Bott and Tu (for one thing, a lot of it is done over $mathbbR$); in Hatcher, you should probably be at least familiar with modules for the chapter on cohomology. For both, you should be familiar with basic undergrad point-set topology.
$endgroup$
– anomaly
7 hours ago
1
$begingroup$
Also, for Hatcher: Pay careful attention to the appendix on CW-complexes. That's probably the steepest learning curve to the book, but you can dip in and out of that appendix while reading the main part of the book if you don't want to wade through it all at once. You should probably be comfortable with it by the beginning of the chapter on homology, though, and it makes the chapter on the fundamental grou p easier.
$endgroup$
– anomaly
7 hours ago
1
$begingroup$
That the exercises are interspersed vs at chapter end is not necessarily a good feature, because it may give you too much of a hint as to what met methods you should apply. Later when you tackle research problems in the wild you won't have those crutches, which means you may be less well prepared than you would be using a textbook that weaned you earlier from those contrived contexts.
$endgroup$
– Bill Dubuque
7 hours ago