Which is more difficult: pure or applied mathematics?

[okunyg]

How would you compare applied and pure mathematics? Is one more difficult than the other? What is the general difference? What do you think is the most interesting field of those two? Can someone who is bad at using applied math, be more capable of pure math and vice versa?

("Theoretical" mathematics might be a better term instead of "pure" mathetmatics.)

 

[Kummer]

How would you compare applied and pure mathematics?

Pure mathematics is more like art. Pure mathematicians work on building a foundation for a theory. One nice feature about pure mathematics is that it is free from argument. When a mathematician makes a discovery there is no opposition, as in science. And his theory stands the test of time, unlike science where one law is shown to be wrong in special cases. But once a foundation is build (like complex analysis) applied mathematicians take its result and use it to solve important problems.

Is one more difficult than the other?

Pure math is much more difficult. Classes in applied math consist of memorizing the steps to solve problems. However, classes in pure math involve proofs, which implies a good understanding of the subject matter is required.

What is the general difference?

In pure math you need to justify everything you do. Which can sometimes make a simple argument long and complicated.

What do you think is the most interesting field of those two?

I like algebraic number theory.

Can someone who is bad at using applied math, be more capable of pure math and vice versa?

It is easier for someone in pure math to learn applied math rather than someine in applied math to learn pure math.

("Theoretical" mathematics might be a better term instead of "pure" mathetmatics.)

What is theoretical mathematics?

 

[Kurdt]

Its such a subjective question. If you're asking because you're about to make a decision as to which you'd rather do, then give them both a try and find out for yourself.

 

[Kummer]

I forgot to add. Even though I like pure much much more. Applied is also fun.

 

[ice109]

snip

i think maybe you're a little biased because applied math is not memorizing steps, it is applying math, it is taking predictive abilities of math and employing them in solving problems. I am sure an applied mathematician does not come out of school having memorized every single problem scenario that they will be faced with in their career.

 

[Werg22]

i think maybe you're a little biased because applied math is not memorizing steps, it is applying math, it is taking predictive abilities of math and employing them in solving problems. I am sure an applied mathematician does not come out of school having memorized every single problem scenario that they will be faced with in their career.

True, applied mathematicians are no biologists!

 

[ice109]

True, applied mathematicians are no biologists!

:rofl::rofl:

 

[HallsofIvy]

But how does one apply a mathematician?

 

[matt grime]

One nice feature about pure mathematics is that it is free from argument. When a mathematician makes a discovery there is no opposition, as in science

If only that were true. Mathematics is plagued by opinion (predominantly over what is 'good') just as much as the next subject. I know you mean more the 'factual correctness' of a proof, but even those can be disputed and not accepted at the high end where proofs are complicated, long and sometimes not even understood by a single person. E.g. the four colour theorem or the classification of finite simple groups.

 

[matt grime]

im sure an applied mathematician does not come out of school having memorized every single problem scenario that they will be faced with in their career.

Really? Cos the feeling I got from teaching engineering students that being taught every conceivable example and no theory was precisely what they wanted.

 

[ice109]

Really? Cos the feeling I got from teaching engineering students that being taught every conceivable example and no theory was precisely what they wanted.

that is a reflection of poor students, not the mission statement of an engineering school

 

[matt grime]

Since it is the opinions of students that seemt to really matter these days, I don't see your point.

 

[ice109]

Since it is the opinions of students that seemt to really matter these days, I don't see your point.

what? bad students don't become good engineers, good engineers do not algorithmically solve problems, they are creative. bad engineers design toilet seats and cardboard boxes

 

[daniel_i_l]

I can't understand anything that I don't find to be 100% logical and consistent. That's why I like pure math better.

 

[Kummer]

That is why I do not except the 4 color theorem nor the classification of finite simple groups. Though the finite group classification would be nice to have as a tool, I must live without it.

 

[okunyg]

I think it's really fascinating, but will I as an average student (maybe even worse) even achieve the slightest success in studies of pure math?

 

[Werg22]

Pure mathematicians do not think of themselves as students. I don't know what they are exactly, but definitely not students in the classical sense of the word.

 

[okunyg]

Well, let's put it this way: will an average high school student be able to understand pure math?

 

[morphism]

Well, let's put it this way: will an average high school student be able to understand pure math?

Sure. Why not?

 

[okunyg]

Sure. Why not?

Since pure math is said to be very hard, I thought only the most prominent students were able to learn it.

 

[JohnDuck]

Since pure math is said to be very hard, I thought only the most prominent students were able to learn it.

There's no such thing as academic predestination.

 

[okunyg]

There's no such thing as academic predestination.

So you're saying: with enough time and commitment, most people are able to learn pure math at university-level?

 

[matt grime]

Possibly - though 'enuogh' covers a multitude of sins.

 

[Kummer]

will an average high school student be able to understand pure math?

I started learning very abstract math while still in high school. So I am sure you can do the same. (Just be careful you do not buy yourself and advanced book).

 

[threetheoreom]

your right

Possibly - though 'enuogh' covers a multitude of sins.

 

[JohnDuck]

So you're saying: with enough time and commitment, most people are able to learn pure math at university-level?

Maybe. What I meant was, it's not a sure bet--the fact that you were a poor or mediocre high school student doesn't preclude you from doing well in a difficult field of study in college. It's possible to turn around. (And vice versa--I've seen many high school stars fall from grace.)

 

[Albereo]

I think it's really fascinating, but will I as an average student (maybe even worse) even achieve the slightest success in studies of pure math?

YES. Don't listen to the pretentious people here telling you that pure math is true and difficult and everything else is book-keeping.

Here's the thing: your abilities to be a good student have nothing to do with your intellectual capabilities, creativity, or ingenuity. Some of the most intelligent people (generally mathematicians) that I know can hardly scrape by with a C, because they're too busy pondering their own questions or becoming deeply involved with a single problem to actually getting around to doing the proofs required for a class. And test-taking? Forget it.

But here's the secret: mathematics isn't actually done that way. You must seek out the questions that are important, rigorously understand how your mind is making various assumptions, and have flashes of insight that allow you to arrive at conclusions. I believe the people who are best at this are the people who have tremendous amounts of creativity, the ability to think abstractly, and a constant sort of desire to build things up from their foundations.

For example. Riemann and Poincare laid the foundations for non-Euclidean geometries; i.e. geometries on surfaces that are not flat planes. (Whereas Euclid missed the axiom that he envisioned all of his points and lines and angles as existing on planes). These ideas do not have any foundation in the problem-solving methods and computations of applied mathematics. So it really depends on how you think. There is always the chance that you might be better at pure math than applied math. No guarantees, but you shouldn't just discount yourself because you struggle with applied math.

That being said, you do have to develop a sense of the language of mathematics - all of its symbolism and the way that you can communicate abstract ideas beyond any shadow of ambiguity.

 

[nonequilibrium]

Confused: the way you guys are talking, implies a diference in (undergraduate) studies between pure and applied mathematics. If so, how come I have not been aware of this? 0_o We just have "mathematics"

Edit: "just" ;)

 

[snipez90]

There shouldn't be a huge difference, since both types of mathematicians need the same basic tools. Anyways I tend to avoid discussions about pure vs. applied mathematics because most of the time they're not particularly meaningful. Most of the time people don't even bother giving a sufficiently precise definition of applied mathematics to work with. For instance some people think physics is applied mathematics, but the distinction seems partly artificial, as a good portion of theoretical physics seems like it could simply be treated as a branch of pure mathematics. Thus I'm of the opinion that the distinction is primarily based on research focus, but I certainly don't expect this to hold in general.

 

[Albereo]

Confused: the way you guys are talking, implies a diference in (undergraduate) studies between pure and applied mathematics. If so, how come I have not been aware of this? 0_o We just have "mathematics"

Edit: "just" ;)

As is the same with my institution. However, some schools actually have separate departments, or at least separate emphases, for "pure" math versus applied math. Applied math puts more of an emphasis on those fields which are relevant to forming the theoretical backbone of the various sciences. Pure math looks at developing and extending the framework of mathematics itself - looking at things that are (as yet) without direct utility.

Hopefully that gives an adequate definition of the two fields. I do agree that the distinction isn't really worth talking over (like the distinction between physical chemistry and chemical physics might not be), but in the original context of the poster's question, it might be worth at least roughly defining what we're talking about.

 

[brydustin]

Its only more "difficult" because pure mathematics is avoided until higher education... although, really there should be no reason not to teach a bit pure mathematics in more elementary settings (whose to say that children can't learn what a set is)? Pure is simply less familiar to the average person and when you become an adult you become less ready to try new things, as a result the applied math feels more comfortable. Also, a typical course in Numerical Methods at university level is much, much more advanced in its own subject than say an elementary real analysis course the same year; this is because a first course in analysis essentially says, "forget what you thought you knew, its time to treat math properly, from ground up"... whereas Numerical Methods has a rich theory in its own right. I'm sure there are several pure mathematicians that know very little about working a computer, writing algorithms, and actually USING their own theories in "Real" life.

 

[HallsofIvy]

Really? Cos the feeling I got from teaching engineering students that being taught every conceivable example and no theory was precisely what they wanted.

Since it is the opinions of students that seemt to really matter these days, I don't see your point.

Calm down, Matt! Take a deep breath, a tall bourbon, and a long weekend!

(If you think engineering students are bad, try teaching "Math for Economics and Business Administration Majors"! That was a course in the school of business admin I one volunteered to teach. The catalog description talked about "operations research" and "partial differential equations" but there was NO math prereqisite, not even pre-calculus!)

 

[HallsofIvy]

Confused: the way you guys are talking, implies a diference in (undergraduate) studies between pure and applied mathematics. If so, how come I have not been aware of this? 0_o We just have "mathematics"

Edit: "just" ;)

Where I was an undergraduate, there were specific "pure" and "applied" tracks. In some colleges there are even "industrial mathematics" tracks.

I will say that when Kummer, earlier, talks about "applied mathematics" courses as only teaching "methods" he is probably thinking only of "service courses" for, say engineers, physicists, economists, etc. There are some very rich, complex courses in "applications of mathetics" as well as "applicable mathematics" (which are quite different topics).

 

[discrete*]

Interesting topic.

I find some areas of applied mathematics to be fascinating; however, I'm much more suited for doing pure maths. What I mean is, I love the idea of something like mathematical biology, but when it comes down to actually doing something applied like this, I lack skill and interest in the work.

Pure maths for me is almost like art. It's a creative outlet for me, which I need because I'm very creative by nature. Also, it's worth noting that I hated math and did terrible in it all throughout high school. it wasn't until I stepped into a calculus class and saw a proof that I realized how interesting and beautiful mathematics was. Looking back, I should be thankful that I had a calc I professor that introduced theorems and proofs at the outset.

Another reason that I like pure maths is logic and the connections that mathematics has with philosophy. The one area of applied maths that I enjoy doing on the same level that I enjoy pure maths is cryptography and cryptanalysis. I just started getting into this type of stuff, and right now it's only through self-study, but I am hoping to take a course on the subject soon enough and expand my interests.

 

[dankshu]

Forgive me, I only took some courses in math and am not a mathematician, but what do applied mathematicians actually do? I look at the programs for applied mathematicians and it seems that, although they're heavy in computer programming work, they have to take a bunch of analysis classes and proof based numerical analysis, differential equations, statistics/probability, etc.

That's for university-wise, but what about in the workplace? My previous assumption was that they are basically engineers, but what would be the point of taking analysis classes, then? Surely there's some use in them after college... and I don't know what people in mathematical finance do either.

I assume you need to know the methods from the inside out in order to apply them, unlike an engineer?