# Do you need calculus to learn pure mathematics?

1. Oct 6, 2012

### TheKracken

I have only taken mathematics up to trig and I was curious if I would be able to start reading books on more advanced topics like Abstract Alegebra and topology??

2. Oct 6, 2012

### Punkyc7

You could learn Abstract with out calculus.

3. Oct 6, 2012

### micromass

Staff Emeritus
Why would anybody even want to skip calculus?? In my opinion, calculus was when math started to be beautiful and elegant.

Anyway, you can do abstract algebra without calculus. For example, a book like Fraleigh should be possible to work through. But it will be hard.

You can't do topology at all without calculus. The motivations for topological terms all come from calculus and analysis.

4. Oct 6, 2012

### BloodyFrozen

It's good to have some (rigorous) calculus/LA because it is then where you start to get acquainted with "real math" (proofs).

5. Oct 6, 2012

### rasmhop

If you are mathematically mature enough and have no problem with proofs and abstract arguments, then there should be no problem in starting with abstract algebra and topology. Most students are not however.

At some point you will need calculus, but it is entirely possible to pick it up later and from a more advanced viewpoint. One of the first places where you will need calculus is in differential geometry because that subject pretty much studies objects on which one can do calculus locally, so the point of the subject (at least at a basic level) is to take what you learned in calculus and see how much you can extend to spaces that locally looks like the one you have seen.

So it can be done, but it probably shouldn't.

@micromass: I disagree. A good topology book (such as Munkres') will provide motivation enough to understand the concepts. You don't need to have studied calculus and analysis to know what an open and closed interval is, or what it means for a function to be continuous in an intuitive sense. I personally did topology without having taken calculus, and today I think it would have been much more efficient to do it the other way around, but it was by no means impossible.

6. Oct 6, 2012

### bcrowell

Staff Emeritus
You might want to look at Topology Now! by Messer.

The standard one-year course in calculus is probably 20% review, 20% cool stuff, and 60% techniques that are fundamentally unimportant (e.g., trig substitutions). For that reason, I can see why you might want to do other stuff without first slogging through a year of calc.

7. Oct 6, 2012

### WannabeNewton

If you are asking this to ultimately understand physical theories such as GR or QFT (guessing because of your other posts) then you will absolutely need calculus to understand differentiable manifolds as was stated above.

8. Oct 6, 2012

### TheKracken

Well, I am equally interested in physics but I am not looking to study these math subjects for physics reasons, but rather personal gratification. I will be taking my calculus sequence starting in january and I thought it would be nice to learn some other more interesting math topics. I am not saying Calculus is not interesting, but so far from my text book it just seems like a bunch of simple calculations and applying different methods to get a derivative ( I have not gone looking at integrals yet)

9. Oct 6, 2012

### WannabeNewton

Yes Calc 1 - 3 can be extremely dry in general. If you find it unbearably boring then you could always go through Spivak "Calculus". It is the holy grail of proof based single - variable calculus \ introduction to analysis. Also try to enroll in honors calculus if your school offers it.

10. Oct 6, 2012

### micromass

Staff Emeritus
You're just studying from the wrong calculus book. Why don't you go through a book like Spivak. That book is much better than simple calculations and finding derivatives.

11. Oct 6, 2012

### PKDfan

I just want to add in another recommendation for Spivak. I've already taken the standard Calc 1-3 sequence and I agree it can be quite boring. On the other hand, I'm working through Spivak right now and it's an absolute blast. It's very difficult at times, but it's extremely fun and rewarding.

12. Oct 6, 2012

### TheKracken

Since you guys are telling me abstract alegebra doesn't need calculus but is very proof based, along with spivak calculus is very proof based, would it be a good idea to get an intro to proof book? As I have no experience with proofs. If you could recomend a good starter one that would be amazing.

13. Oct 6, 2012

### PKDfan

How To Prove It by Daniel Velleman is what I used to get acquainted with proofs. It's pretty good, but working through the whole book might be overkill. Definitely work through the first few chapters and the chapter on induction, though.

14. Oct 6, 2012

### Angry Citizen

I wouldn't recommend anyone skipping calculus. It is a truly beautiful subject that really enriches your mathematical intuition. Yes, even calculating derivatives and integrals by hand enriches your mathematical intuition.

15. Oct 7, 2012

### TheKracken

I have no plans to skip Calculus but rather look into more pure mathematics in my own time. I am taking Calculus I in january

16. Oct 7, 2012

### SolsticeFire

Well if you study Calculus from Spivak or the like, it will be very close to the pure mathematical gratification you are looking for.

Good Luck

SolsticeFire

17. Oct 7, 2012

### homeomorphic

That's not fair to the Greeks and geometry. Ruler and compass constructions are awesome. The discovery of the platonic solids was such a beautiful fact that they attributed a mystical significance to it.

Well, you can't do formal point-set topology very easily, although the subject has no formal prerequisites. I think if someone wrote a good exposition of it, though, you could do SOME topology. For example, I could probably teach high school students to compute the Jones polynomial (though, truth be told, I don't really *understand* the Jones polynomial myself half as well as I would like). Practically speaking, though, it might be hard to find suitable books. I'm not familiar enough with them to say. Intuitive topology by Prasolov might fit the bill, for example.

I think it might be a good idea for a lot of people to skip ahead in math, just to find out that math can actually be interesting because the normal curriculum is mostly kind of boring, until you get to upper division stuff (multi-variable calc and linear algebra can be okay, and other subjects might be okay if they were taught differently). However, you can't get too far without running into the need for calculus.

If you want to get comfortable with proofs, maybe naive set theory would be a good subject to get some practice in, so you can separate out the skill of writing formal proofs from most of the other more conceptual difficulties.

It's good to get a lot of practice with calculus computations. I think you need to do some of those, too. I never read Spivak, so I don't know if he neglects that.

18. Oct 7, 2012

### WannabeNewton

He has some loosely computational things like induction proofs where you have to show some derivative formula or such. There are almost none but that is, in my opinion, for the better because honestly who likes computations haha.

19. Oct 7, 2012

### homeomorphic

Funny, I was just having this argument with a friend, who has a pretty similar outlook on math to mine, with a few very fundamental differences. I was on the other side of the argument, then, saying I don't like computations. He objected that he didn't like how mathematicians these days don't want to get their hands dirty. He solved a problem his big shot adviser couldn't solve because he computed some examples. He also said if you look at the notebooks of the great mathematicians, they have reams of computations.

I like avoiding computations by being clever, and I think computations can sometimes be unenlightening as far as understanding things goes (when trying to understand something deeply, I try to ban myself from doing any calculations where possible). But, you shouldn't be afraid to get your hands dirty, either.

20. Oct 8, 2012

### TheKracken

Would an intro to proof book be the best thing to start with then??? I would love to be ahead of the game in terms of my mathematics degree ( I plan to get a minor or *maybe* a double major) If I have a good basis in writting proofs and I start learning abstract alegebra and topology then when I get to these classes it should be breeze???

Another question; if I were to study these topics extensivly enough would it be possible to skip the class and move on to the next level of lets say topology???