Great list. I would also add:
What is Mathematics? http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...
Great books about mathematics
51–60 of 61 posts
Re: Great books about mathematics
#52Does someone have a recommendation for a book that get deeper into mathematics by teaching actual mathematics rather than teaching about it? Many of the books I have come across leave me with more questions than answers [1]. I am interested in pure mathematics mainly including logic, set theory, category theory, etc. [1] An excellent example not mentioned by others here is also The Road to Reality by Roger Penrose. T…
What are you looking for that isn't simply a math textbook? Honestly, that's how mathematicians learn "actual" mathematical subjects, too. If you're comfortable with calculus as a subject, for example, and want a "pure mathematics" approach, I recommend Michael Spivak's Calculus . If you've never worked through a pure math textbook from start to finish, that's a good start. There's not that much interesting in the th…
We just started a study group for that book at
http://www.reddit.com/r/calculusstudygroup/
if anyone is interested.
Re: Great books about mathematics
#53Earlier quoted context omitted.
Good points. I have already gone through several levels of engineering mathematics, so do understand differential and integral calculus well. I am assuming that is that Spivak's book is about -- please correct me if I am wrong; Amazon is not showing a preview of the book. You are right in bringing countability arguments into the picture; I understand them only to some level. I would love to read a book that gives it…
Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material. Here are some exercises to give you a sense of the flavor. If you find these…
> Spivak's Calculus starts with a set of 13 axioms
I am reading third edition and it only has 12 not 13. Did that change in later editions?
Re: Great books about mathematics
#54Earlier quoted context omitted.
Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material. Here are some exercises to give you a sense of the flavor. If you find these…
> Spivak's Calculus is used as a first-year calculus textbook at lots of schools Umm... where? Not at Stanford, where we used a mainstream, much easier book. So does Princeton. Harvard is famous for having developed a "touchy-feely" calculus book. Perhaps abroad? It is typical of calculus courses in the US that the students come with fairly weak backgrounds, and a major purpose is to expose and patch holes in the stu…
>Umm... where?
Lots of places http://www.math.uga.edu/~pete/MATH2400F11.html
Re: Great books about mathematics
#55May I present one of the greatest math books for general audience: Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev. http://www.amazon.com/Mathematics-Its-Content-Methods-Meanin... Math books rarely move from the Soviet Union to west, but this did and for really good reason. Just look at the list of writers included. So far I have not seen any math books that come…
I second the book as well as the above praise for the book as one of the greatest. I have read several chapters from the book and have loved them all.
Question: I haven't read The Princeton Companion to Mathematics. How would you compare these two?
Re: Great books about mathematics
#56Does someone have a recommendation for a book that get deeper into mathematics by teaching actual mathematics rather than teaching about it? Many of the books I have come across leave me with more questions than answers [1]. I am interested in pure mathematics mainly including logic, set theory, category theory, etc. [1] An excellent example not mentioned by others here is also The Road to Reality by Roger Penrose. T…
Since you mentioned you already have a strong applied math background, try Mathematics: Form and Function by Saunders MacLane for a broad portrait of mathematics that might get you excited to read more deeply into a given subtopic. Note that while this book surveys a broad swathe of mathematics, it depends on the reader already having a pretty high level of mathematical maturity. Other than that, if you want to go de…
Thanks a lot. All the recommendations look good (previewed each on Amazon). Saunders MacLane and Spivak sound to be good starts for me.
Re: Great books about mathematics
#57May I present one of the greatest math books for general audience: Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev. http://www.amazon.com/Mathematics-Its-Content-Methods-Meanin... Math books rarely move from the Soviet Union to west, but this did and for really good reason. Just look at the list of writers included. So far I have not seen any math books that come…
I second the book as well as the above praise for the book as one of the greatest. I have read several chapters from the book and have loved them all. Question: I haven't read The Princeton Companion to Mathematics. How would you compare these two?
The Princeton Companion is more like encyclopedia. Main part of the book is articles describing 100 or so mathematical concepts in alphabetical orders. Then it has articles describing major mathematical problems.
Re: Great books about mathematics
#58Earlier quoted context omitted.
Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material. Here are some exercises to give you a sense of the flavor. If you find these…
> Spivak's Calculus is used as a first-year calculus textbook at lots of schools Umm... where? Not at Stanford, where we used a mainstream, much easier book. So does Princeton. Harvard is famous for having developed a "touchy-feely" calculus book. Perhaps abroad? It is typical of calculus courses in the US that the students come with fairly weak backgrounds, and a major purpose is to expose and patch holes in the stu…
If you took me to mean the "default" first-year calculus textbook then yes, that's not common. But it's definitely aimed at first-year college students, or at least people who haven't had prior exposure to rigorous mathematical thinking. Compare the style to, say, Spivak's Calculus on Manifolds to see what I mean.
(Edit: I just read your HN bio and know you know the stylistic differences, etc. Sorry!)
Spivak is the first-year Honors Calculus textbook at my alma mater, the University of Chicago. Harvard is also famous for having the most difficult first-year math classes that use even more advanced textbooks like Rudin's Principles of Mathematical Analysis.
My HS background in mathematics was definitely "weak," too. My senior year was the first year my school district ever offered calculus of any stripe in its entire history and I still managed to handle Spivak my first year of college. I took the AP Calculus test on my own and got a 4/5. It's not that crazy.
Re: Great books about mathematics
#59Earlier quoted context omitted.
Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material. Here are some exercises to give you a sense of the flavor. If you find these…
> Spivak's Calculus starts with a set of 13 axioms I am reading third edition and it only has 12 not 13. Did that change in later editions?
I typed it from memory, so I could be wrong. I thought the 13th axiom was the least upper bound property.
Re: Great books about mathematics
#60Earlier quoted context omitted.
Since you mentioned you already have a strong applied math background, try Mathematics: Form and Function by Saunders MacLane for a broad portrait of mathematics that might get you excited to read more deeply into a given subtopic. Note that while this book surveys a broad swathe of mathematics, it depends on the reader already having a pretty high level of mathematical maturity. Other than that, if you want to go de…
Thanks a lot. All the recommendations look good (previewed each on Amazon). Saunders MacLane and Spivak sound to be good starts for me.
It's worth mentioning that Saunders MacLane, along with Samuel Eilenberg, invented category theory.