Do you need calculus to learn pure mathematics?

[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??

 

[Punkyc7]

You could learn Abstract without calculus.

 

[micromass]

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.

 

[BloodyFrozen]

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

 

[rasmhop]

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??

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.

 

[bcrowell]

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.

 

[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.

 

[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 textbook 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)

 

[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.

 

[micromass]

I am not saying Calculus is not interesting, but so far from my textbook 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)

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.

 

[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.

 

[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 recommend a good starter one that would be amazing.

 

[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.

 

[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.

 

[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

 

[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

 

[homeomorphic]

In my opinion, calculus was when math started to be beautiful and elegant.

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.

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

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.

 

[WannabeNewton]

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.

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.

 

[homeomorphic]

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.

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.

 

[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 let's say topology?

 

[homeomorphic]

Would an intro to proof book be the best thing to start with then?

I'm not sure. Personally, I would be inclined to start off by doing a lot of intuitive geometry, then work on proofs (the pitfall might be that it would have only tangential relevance to the usual undergraduate curriculum, but I question the usual undergraduate curriculum). Proofs are like grammar and spelling in writing.

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?

Maybe, maybe not. If it eventually becomes a breeze to do the problems in the book, that's when you know it will be a breeze to take the class.

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 let's say topology?

Probably. You can usually just talk to a professor if you want to skip prerequisite courses and they will probably let you. Assuming you know your stuff. They probably won't quiz you on it or anything, but if you don't know it well enough, you will have a really hard time in the class.

 

[TheKracken]

Could you possibly recommend a few books for me?

 

[homeomorphic]

Well, at that level, there's Geometry and the Imagination, by Hilbert. Good preview of a lot of different stuff. I don't think it has exercises. Also, Lines and Curves: A Practical Geometry Handbook is good for learning to think about geometry intuitively.

For group theory, you might try Visual Group Theory, the book by Pinter, and Symmetry, by Hermann Weyl. I haven't read any of those, so I feel silly recommending them, but if I were going to read some books about abstract algebra on a more basic level, those would be the ones I would read.

I didn't necessarily learn from good books for a lot of subjects, so I don't always have good ones to recommend. A lot of subjects are easy enough, it sort of doesn't hurt you that much if the book isn't very good if you can think on your own, though you might have a more pleasant time learning it from a better book.

 

[WannabeNewton]

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.

I do agree you can't avoid them at times; there are books on riemannian manifolds that just can't avoid computational problems in some form or another because they do bring things down to Earth in terms of local coordinates and such. I just find them to be tedious and not very stimulating mentally which sucks big time because as a physics major many of my physics classes involve doing computational problems although the physics aspect of it provides a stimulation of its own.

 

[micromass]

I do agree you can't avoid them at times; there are books on riemannian manifolds that just can't avoid computational problems in some form or another because they do bring things down to Earth in terms of local coordinates and such. I just find them to be tedious and not very stimulating mentally which sucks big time because as a physics major many of my physics classes involve doing computational problems although the physics aspect of it provides a stimulation of its own.

Solution: change your major to mathematics.

 

[WannabeNewton]

Solution: change your major to mathematics.

Well you sure are persistent haha.

 

[saminator910]

I think you can, I an a high school student, took trig, spent the summer learning a little calculus, not much derivatives, integrals. This fall I am in Pre calc, really easy since I know some calculus. I got a book on general topology, and It is really cool, no calculus, A little hard to get into, but satisfying because it isn't computational. I got back into calculus a little, mostly vector calculus, and multivariable, because I want to learn about differential topology. So the point you can get into basic pure math, then I think you will stray back into calculus, it really is a wonderful tool.

Books:

Bert Mendelson, intro to topology

Number theory, George E. Andrews

Short basic reads, no prerequisites, they open the door to more advanced concepts

 

[mathwonk]

i think i agree with a lot of what has been said. calculus is not strictly prerequisite to many undergraduate courses like linear algebra, abstract algebra, number theory, topology, elementary probability, combinatorics.

basically there are thee big divisions of math: algebra, analysis, and geometry. and calculus is in the analysis side. but there are many topics which blur the divisions, and combine the power of calculus, the most powerful tool out there, with the study of the new subject. so the more clever you are at introducing calculus into your subject, often the more successful you will be at that new subject.

calculus is strictly prerequisite to differential equations, differential geometry, differential manifolds, integral geometry, advanced statistics, infinite dimensional linear algebra (functional analysis),...

there are also subjects wherein calculus is not used logically, but is crucial psychologically, as micromass said. cohomology in algebraic topology is in a sense a generalization of integration, and derivations in field and ring theory are an algebraic generalization of derivatives. topology also delves deep into the problems hat arise in convergence theory or limits in calculus, but can be studied earlier, sometimes beneficially.

If i were recommending I would emphasize having a strong background in elementary algebra and geometry before calculus, which few students have. I.e. I would recommend euler's elements of algebra and euclid's elements (of geometry).

but anything that appeals to you is fine.

 

[BloodyFrozen]

For group theory, you might try Visual Group Theory, the book by Pinter, and Symmetry, by Hermann Weyl. I haven't read any of those, so I feel silly recommending them, but if I were going to read some books about abstract algebra on a more basic level, those would be the ones I would read.

Pinter is very friendly and a good intro. It's easy, but well-written.

 

[Sankaku]

For group theory, you might try Visual Group Theory, the book by Pinter, and Symmetry, by Hermann Weyl. I haven't read any of those, so I feel silly recommending them, but if I were going to read some books about abstract algebra on a more basic level, those would be the ones I would read.

I am always happy to recommend Pinter, but it would be hard work if the OP has no formal background with proofs. Being an Algebra book-junkie, I also have Visual Group Theory and the first 2/3 would be excellent for someone just finishing high-school. Pinter could follow at whatever point the student's maturity caught up.

One option not mentioned so far is some basic discrete mathematics. There is crossover with Algebra, Number Theory and Computer Science. It also doesn't need any calculus at the beginning.

I liked this book (get the cheaper international edition):
https://www.amazon.com/dp/0131679953/?tag=pfamazon01-20

It is also a good intro to proofs. Don't pay attention to the reviews. I suspect it is mostly unprepared CompSci students complaining about their first real encounter with math.

 

[Jjasongray]

Would an intro to proof book be the best thing to start with then?

Spivak is kind of an intro to proofs book. I would recommend starting with it. It doesn't assume you know anything at all and it sort of eases into it. Once you start limits there's a bit of a difficulty spike but once you start "getting it" the rest comes easily since limits are the hardest topic in the book and the one that everything else is based off in the subject. I took a class with it having no knowledge of proofs at all and it turned out okay.

Also, don't ignore calculus! It's really a pretty interesting subject. While I admit differentiation is very dry, integration is an art. Differential equations are also really fun in my opinion.

It's also insanely powerful in any applied subject. When I first learned calc I was mystified it worked so well! I would do problems and would wonder how the hell such simple techniques got answers to seemingly complex questions.

That's usually how calculus is taught at first, with a focus on computation and almost no proofs. That's why Spivak is a recommendation on here because it pretty much does the opposite.