# Courses with Least Rigor?

1. May 2, 2013

### Zophir

I'm a sophomore at NYU, currently wrapping up the Calculus sequence this semester and Linear Algebra. I'll be taking Analysis next semester and Abstract Algebra (Called Algebra I) in the spring.

Those are, I assume, very rigorous. I'm looking to balance them out with less rigorous courses. I'm not entirely sure whether I'll continue with math after my undergraduate degree, so having said that, I'm looking to get a well rounded math background, leaning towards a more applied side. I'm not a big fan of proofs and I would like to see them as little as possible in my elective math courses.

Having said all that, here is a link with course descriptions for all the undergraduate math courses. Based on the little information they provide, and your previous experiences with math courses, I would appreciate if you guys could pick out 2 or 3 courses you think would be the least rigorous, and if possible 2 or 3 that would be the most rigorous looking as well. A short little explanation as to why you chose as you did would be very helpful and deeply appreciated.

Thanks for any and all help!

2. May 2, 2013

### VantagePoint72

I get the impression you're using "rigorous" as a euphemism for "hard". Different fields of mathematics are not inherently more or less rigorous than others. Different levels—say, for engineers vs. applied mathematicians vs. pure mathematicians—may be taught to varying degrees of rigour, but that's not an intrinsic property of the material. How rigorously a course is taught depends on the instructor, and is not something anyone can tell you from the course description.

If you have an eye towards applicability, you couldn't go wrong by starting the differential equations series. PDEs especially are indispensable in applied mathematics. Again, though, how proof-based your ODE and PDE courses will be depends on the department and the instructor.

3. May 2, 2013

### micromass

What about some programming courses?? Those will prove to be very useful later on.

4. May 2, 2013

### Zophir

Didn't meant to use rigor to indicate difficulty, I meant it quite literally (to the best of my knowledge, in mathematics, rigor implies proofs and proof writing). Which courses require the least amounts of proof-writing, not which are the easiest conceptually.

I agree to a certain extent that it depends largely on the professor, but as most of the courses have several lectures, they have to follow a certain syllabus. With some knowledge, it should be possible to figure out which courses are more rigorous as opposed to less. For example, Numerical Analysis doesn't seem to have a lot of proof writing based on the description, I could be dead wrong however, as I have no experience with the subject.

Having said all that, ODE/PDE are definitely on my radar, I plan to take them as soon as my schedule permits.

Also, Micro, I'm currently enrolled in the Python course, which is required before they allow you to take the "real" comp sci courses. I'll be taking the Intro to Comp Sci course next semester, taught in Java. If I enjoy that I'll most likely take the data structures course and keep progressing through some of the comp sci track.

Sorry if there was confusion with my usage of "rigor", I mean it in the sense of proof-based courses. (For example, look at the description of Analysis, arguably near the top in terms of proof writing, "Rigorous study of functions...")

Sorry if I'm a bit scatter brained and disorganized, in the middle of a class and writing in short bursts from my phone whenever I get the chance.

Thanks for all the help so far!

5. May 2, 2013

### Vargo

In the context of undergrad math, I think "rigor" refers to a focus on Definition/Theorem/Proof both in lecture and homework. Any course that is part of the standard undergraduate curriculum (past linear algebra) will be rigorous in this sense. That includes analysis, algebra, topology, and number theory. Probably anything called discrete math as well, but as far as I know that is less standardized than the others, so it could depend on the instructor and the exact syllabus. Probability depends a lot on the instructor and the level of the course.

Having taken a quick look at the link, there appear to be quite a few classes not focused on Def/thm/proof. Any of those modeling classes would work (finance, biology, fluid dynamics, general math modeling). You've done calc. and linear algebra, so you probably have the prereqs for any of those. Mathematical statistics would probably be fine (though I suppose the emphasis on theorems and proofs could depend a lot on the instructor). Unless you only want a very superficial knowledge, the probability and stats course sounds too fluffy (in my humble opinion).

Why not ask an advisor in your math dept.? They would know the curricula well and probably the instructors too.

6. May 2, 2013

### Integral

Staff Emeritus
Maybe you should take some business classes, or maybe PE. If you don't want rigor get out of math.

Sorry just my opinion.

7. May 2, 2013

### ModusPwnd

That's a pretty snarky and rude opinion. I can see why you apologized for it, though I doubt the apology is sincere.

8. May 2, 2013

### micromass

What he said is true, though. Mathematics is by definition rigorous and filled with proofs. So if you don't like proofs, then I don't think mathematics is the right major for him. This isn't meant to be rude though. I just think that you'll hate mathematics if you don't like proofs.

Of course, there's also applied mathematics. This is much less proof-intensive. So maybe the OP would like that. But he should stay away of pure mathematics since he's absolutely going to hate it.

9. May 2, 2013

### Zophir

Didn't expect that kind of a response, don't really see how that is helpful in the slightest.

Apart from the incredibly rude, and quite inaccurate ultimatum, you should consider actually reading my post. If you don't want to read, stay out of the forums.

Just my opinion.

Now that I'm done emulating your response, I'm trying to balance out rigorous courses (namely core major requirements) with non-rigorous courses with an emphasis on applied mathematics (these would be my electives). My program doesn't make a distinction between pure or applied math so I'm choosing my electives to reflect my interest.

If there is a problem with trying to create a reasonable schedule that would give me a good mix of pure and applied mathematics, my apologies, I didn't realize that was, for some absurd reason, frowned upon.

Maybe I didn't, express myself well, that is a distinct possibility, in which case I apologize for my hostility. Having said that, even had I come in as an incredibly rude and ignorant person who doesn't belong in mathematics, given the sub-forum you're in, you should reconsider your method of providing advice.

Vargo, thanks for the advice. Ill see if I can find more info about those courses. On the topic of advising, it's a little strange here. I have a faculty advisor and he signs off my schedule but as he wasn't a student here and he doesn't have any sort of role in terms of education, he doesn't even teach most semesters, it's kind of up to me. To the best of my knowledge, there isn't a generic advisor I can approach either, ill see if I can get in contact with someone in charge of the undergraduate program.

Thanks!

10. May 2, 2013

### Zophir

I have evidently not expressed myself well so ill say this now:

I don't love proofs but I can do them and I can appreciate them and their necessity.

I would like to get a good mix of pure and applied. As most of my requirements are pure courses, I am looking to find a nice mix of applied math electives.

I apologize for the confusion on my behalf, and for the hostility in my previous response.

11. May 2, 2013

### micromass

OK, but let me ask you something. Why are you a mathematics major? What do you hope to get out of it?
I'm asking because most math majors love the abstract thinking, the proofs and the rigor. You don't seem to love it, so it raises some questions with me.

12. May 2, 2013

### VantagePoint72

Wow, bad luck then for the entire fields of applied mathematics, statistical modeling, and other closely allied fields where the emphasis is on using mathematical results rather than deriving them. OP already said he is capable of doing proofs but prefers applications of mathematics, and has clarified that by "rigour" he meant "heavily proof-based". I agree that this response is rude and entirely unhelpful.

13. May 2, 2013

### VantagePoint72

I think the extent to which senior members of PF are giving OP grief about not enjoying proof-based mathematics as much as its applications is ridiculous. There many paths through mathematics and I think this hostile attitude that if someone isn't especially keen on doing proofs that they don't belong in the field at all is backwards and, frankly, offensive to a large section of the mathematics community. Some people get great satisfaction just from understanding deep results in mathematics—even if they're not especially keen on deriving such results themselves—and that alone is a perfectly good reason to study mathematics for those who have the time and means.

micromass, I appreciate that you're just asking questions, but I don't think you're asking any useful ones. OP can explore for himself what does and doesn't interest him, and suggesting that "he's absolutely going to hate it" is baseless. Mathematics is expansive and full of niches, and you are not doing anyone any favours by telling someone you don't even know what they are and aren't suited for based on a few short posts on the internet. More importantly, as even-handed as you've been, I think you've largely undermined your own credibility by defending Integral's comments—which are nearly bullying in their tone. I think both of you should be ashamed of yourselves. PF is meant to be a welcoming place, and today you failed dismally at that.

14. May 2, 2013

### WannabeNewton

I have many friends doing math at NYU and they have described in detail their classroom experiences; be thankful you aren't a math major at UChicago, or somewhere similar, because with your distaste for proofs (which I'm not saying is a reason to quit math at all by the way, far from that!) it would be a bit of a nightmare ;). Anyways, take a look at the finance track they have at NYU, they have tons of applied math courses in that regime. One of my close high school friends is doing finance at NYU Stern and he tells me the classes are quite applied and he loves them (granted he is obsessed with finance).

15. May 2, 2013

### Zophir

A very good question, one which I don't really have a complete answer for.

I enjoy the abstract thinking. I even enjoy the proofs most of the time. I think my issue arises at the thought of an entirely proof based course. I enjoy a mix of computational problems and proofs.

Moreover, my biggest problem, I feel, is the level of detail. I lack discipline. I enjoy sketching proofs in my head, but I get annoyed sometimes at the level of detail necessary to prove something rigorously since I take many axioms and assumption for granted.

I'm a math major because I enjoy the subject. More so than any other subject so far. My problem is that I am at the expense of sounding arrogant, a jack of all trades.

I don't like the thought of having to specify a major. I enjoy having an understanding of everything and don't like limiting myself in that regard. I have no idea what I want to do with my life either so this makes choosing even more difficult.

As to why math? I really do enjoy it, and it's a challenge to myself. Given my uncertainty for the future, I also feel like it is a good fit. An undergraduate math degree doesn't limit me in any way. If I change my mind post-grad, I will have learned how to think critically, tackle problems, deal with frustration, think outside the box, and carefully craft arguments. These are all immeasurably valuable.

Sorry for the long winded response, it's not an easy question for me to answer.

Sorry for any strange typos, autocorrect has a mind of its own.

16. May 2, 2013

### Zophir

NYU does seen to have been a godsend decision for a whimsical mathematician like myself. Finance is something I've considered but I'm not sure it's something I want to commit to. It also carries the logistical difficulty of transferring in between colleges and fulfilling degree requirements junior year. I will look into it though.

17. May 2, 2013

### micromass

I'm just asking the question here. I have no intention to be hostile towards the OP. I just don't want the OP to major in something that he absolutely going to hate. Maybe I'm wrong about him going to hate it, but I can only tell you what I think.

And for the record, what Integral said is indeed rude and bullying. I never intended to defend it. I just think that deep down he has a valid point that it's strange to do mathematics without liking proofs. Whether everybody agrees with that point or not, that is a different matter.

18. May 2, 2013

### micromass

Thanks, this is very helpful to me.

It seems to me like your complaining about courses that take rigor too far. For example, proving that $0x=0$ for all real numbers x. Is that the kind of proofs you don't like?? I agree that they look pretty useless, but a proof based course is not all about that.

If you're a jack of all trades, then applied math is certainly a wise choice. Pure math goes very deep into the math, but not very broad. At some point, you start thinking that all you're doing is completely useless and that you're just making abstractions for the sake of it.

19. May 2, 2013

### vela

Staff Emeritus
Have you considered the mathematical physics course at NYU? You're unlikely to see many proofs in it, and you'd probably find it more interesting than finance courses.

20. May 2, 2013

### Zophir

It's a class I'm really interested in taking but the issue is the pre-requisite of Physics III. I did the General Physics I-II sequence as a freshman, I had no idea what I wanted to study and I was getting my Science requirement out of the way.

Unfortunately, they're not very big on letting someone who completed the general sequence take Physics III. I don't have the time to retake the Physics sequence, if they would even allow it. I've been in correspondence with them this semester to see if there is some sort of placement test I could take.

They MIGHT let me, since I have the mathematical background necessary. I've been working on upgrading my General Physics knowledge to appropriate levels, in case they do give me permission. Any recommendations on books for self-study?

I wish I knew what I wanted to study when I was a freshman, could've double majored :(!

21. May 2, 2013

### Integral

Staff Emeritus
Look, most of my education has been in applied math, I am nearly an applied mathematician. So I have been there and done that.

Sure I could have told you, like others have, that you should be taking Diff Eqs, but a good diff eq class is rigorous. At least in part there will be proofs. (Existence and uniqueness proofs are ever present)

I like to tell people looking for the easiest Engineering major, that it is Business, 'cus that is where people looking for the easy way out end up.

Again, you need to examine your goals, using rigor as criteria for classes is a bad sign. You want to be taking the classes which interest you and result in that piece of paper at the end. Rigor should not even be considered, other then, for a math major, being concerned at a lack of it.

Sometimes you need to hear things other than what you want to hear.

I again apologize for being harsh in my last post.

22. May 2, 2013

### clope023

23. May 2, 2013

### Zophir

I appreciate that you actually apologized, and you certainly mention some important points.

Having said that, there still seems to be an issue of miscommunication here. Take for example, what you just said about differential eq. The presence of rigor would not dissuade me from taking it. I'm simply trying to avoid a situation where I end up with something like Abstract Algebra, Analysis, and Topology in the same semester. Not because that's impossible, but I might not be up to it. Moreover, I enjoy a balanced schedule, complementing a pure math class with an applied math class would be my ideal.

If I had phrased my question as "Recommended Electives for applied math" or "Computation based electives with broad applications" this issue wouldn't have arisen. You're getting stuck on the language and not the idea, possibly due to my poor job of expressing it. By now, however, it should be quite clear.

As far as using rigor as a criteria for a class, how is that a bad idea? I know my limits, and 3 upper level pure math courses is beyond me, at my current level. I would prefer to avoid that, due to my mathematical interests and also from a pragmatic perspective.

In addition to all this, I can't really pick class based on interest at this point. I know so little of upper level math I barely have the slightest idea what most courses are about. I haven't don't enough either to truly know what I like. Maybe Analysis ends up being my favorite class!

You seem like a bit of a math elitist, and that's fine, but please refrain from pushing your thoughts on math education, particularly in a program that does not distinguish between pure and applied, on me.

At the risk of sounding rude, I asked for your opinion on the level of rigor of electives, not an analysis on what this choice says about my interests, dedication, or aptitude to be a math major. Yet I've received it twice, even after clarifying my intent.

Now comes the really rude part; please refrain from posting here again. This is going to derail the conversation a second time. If you wish, I could rigorously prove how that isn't helpful towards my objective, this forum, or just about anyone.

I apologize for the rudeness but again, didn't expect that attitude from my time lurking on PF, especially not in this sub-forum.

24. May 2, 2013

### Zophir

I reserve judgement on entirely proof based courses, due to lack of experience. I'll try to make sure I don't bite off more than I can chew so I'll start slowly and carefully. Who knows, maybe they'll end up being my favorite.

From what I've gathered so far, differential equations courses seem like a good idea; interesting, widely applicable, pretty important overall. Anything with the word "modeling" I assume has an emphasis on applied mathematics?

Is there anything else that sticks out as an interesting course with an emphasis on applications?

Numerical Analysis seems like it might the bill.

25. May 2, 2013

### micromass

Depends on the Diffy Eq course, really. Some Differential equation courses are really rigorous. So it's better to check beforehand whether they focus on application or theory.

Modeling and numerical analysis are always good. Programming is nice too. (although it's not on your list).