16 ms·

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 comfortab

by jfarmer 13y ago

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 three subjects you listed — logic, set theory, and category theory — that doesn't depend on other subjects or a prior level of mathematical maturity. Category theory was originally invented to solve and categorize problems in algebraic topology, for example. For example, I rather like mathematical logic and model theory, but you're going to have a rough time if you don't have a visceral, intuitive understanding of countability arguments and at least a handful of subjects you'd be reasoning "about." Unless you know the standard model of arithmetic, for example, how can you think about non-standard models? Without that, important results like the Löwenheim–Skolem theorem will likely seem contextless.

alok-g 13y ago
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 a formal treatment. The following has been great for example: https://news.ycombinator.com/item?id=6838917 https://news.ycombinator.com/item?id=6838917 Another one showed up on HN recently that I am still to read in full: https://news.ycombinator.com/item?id=6966695 https://news.ycombinator.com/item?id=6966695 Thanks Finally, cannot help but mention in praise, Colin Wright here on HN has been a good help before in clearing some of my doubts on the subject.
- jfarmer 13y ago
  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 exercises trivial then the textbook might not be for you. If you find them hard, well, welcome to math! :) These are all before we get to any "calculus." Here "function" means a function of the real numbers. 1. Let f be a function that satisfies the conclusions of the Intermediate Value Theorem. Prove that if f takes on each value only once then f is continuous. Generalize this to the case where f takes on each value only finitely many times. 2. Prove that if n is even, then there is no continuous function f which takes on every value exactly n times. 3. A set A of real numbers is said to be sense if every open interval contains a point of A. Prove that if f is continuous and f(x) = 0 for all numbers x in a dense set A then f(x) = 0 for all x. 4. Find a function which is continuous at every irrational point and discontinuous at every rational point (and prove it as such) Spivak's Calculus is used as a first-year calculus textbook at lots of schools, so if you find the above even a little challenging or strange-seeming then I'd recommend going through the book. The last chapter of the textbook is a rigorous construction of the real numbers from the rationals using Dedekind cuts (referenced in the first link).
  - Someone 13y ago
    "3. A set A of real numbers is said to be sense if every open interval contains a point of A." For those googling: there's a typo: sense => dense (http://en.wikipedia.org/wiki/Dense_set http://en.wikipedia.org/wiki/Dense_set)
    - jfarmer 13y ago
      Indeed! Me and my fat fingers. :)
  - alok-g 13y ago
    Sounds good. Thanks!
  - impendia 13y ago
    > 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 students' backgrounds in algebra and trigonometry.
    - dominotw 13y ago
      >Umm... where? Lots of places http://www.math.uga.edu/~pete/MATH2400F11.html http://www.math.uga.edu/~pete/MATH2400F11.html
    - jfarmer 13y ago
      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.
Estragon 13y ago
Seconding the recommendation of Spivak's Calculus. That book lit a fire in me when I was 18. Be sure to work through the exercises, that's where all the fun is.
dominotw 13y ago
We just started a study group for that book at http://www.reddit.com/r/calculusstudygroup/ http://www.reddit.com/r/calculusstudygroup/ if anyone is interested.

by jfarmer 13y ago

alok-g 13y ago

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 a formal treatment. The following has been great for example: https://news.ycombinator.com/item?id=6838917 https://news.ycombinator.com/item?id=6838917 Another one showed up on HN recently that I am still to read in full: https://news.ycombinator.com/item?id=6966695 https://news.ycombinator.com/item?id=6966695 Thanks Finally, cannot help but mention in praise, Colin Wright here on HN has been a good help before in clearing some of my doubts on the subject.

jfarmer 13y ago
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 exercises trivial then the textbook might not be for you. If you find them hard, well, welcome to math! :) These are all before we get to any "calculus." Here "function" means a function of the real numbers. 1. Let f be a function that satisfies the conclusions of the Intermediate Value Theorem. Prove that if f takes on each value only once then f is continuous. Generalize this to the case where f takes on each value only finitely many times. 2. Prove that if n is even, then there is no continuous function f which takes on every value exactly n times. 3. A set A of real numbers is said to be sense if every open interval contains a point of A. Prove that if f is continuous and f(x) = 0 for all numbers x in a dense set A then f(x) = 0 for all x. 4. Find a function which is continuous at every irrational point and discontinuous at every rational point (and prove it as such) Spivak's Calculus is used as a first-year calculus textbook at lots of schools, so if you find the above even a little challenging or strange-seeming then I'd recommend going through the book. The last chapter of the textbook is a rigorous construction of the real numbers from the rationals using Dedekind cuts (referenced in the first link).
- Someone 13y ago
  "3. A set A of real numbers is said to be sense if every open interval contains a point of A." For those googling: there's a typo: sense => dense (http://en.wikipedia.org/wiki/Dense_set http://en.wikipedia.org/wiki/Dense_set)
  - jfarmer 13y ago
    Indeed! Me and my fat fingers. :)
- alok-g 13y ago
  Sounds good. Thanks!
- impendia 13y ago
  > 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 students' backgrounds in algebra and trigonometry.
  - dominotw 13y ago
    >Umm... where? Lots of places http://www.math.uga.edu/~pete/MATH2400F11.html http://www.math.uga.edu/~pete/MATH2400F11.html
  - jfarmer 13y ago
    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.

Estragon 13y ago

Seconding the recommendation of Spivak's Calculus. That book lit a fire in me when I was 18. Be sure to work through the exercises, that's where all the fun is.

dominotw 13y ago

We just started a study group for that book at http://www.reddit.com/r/calculusstudygroup/ http://www.reddit.com/r/calculusstudygroup/ if anyone is interested.