Which is more difficult: pure or applied mathematics?

[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?

 

[chiro]

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?

Applied mathematicians usually take specifications of some particular problem, then use the appropriate mathematics to do some analysis and then typically communicate the main results and more importantly the interpretation of what the results mean for their target audience.

A few examples:

An actuary might be asked to go through the design of a new insurance product to see how probable the company might be to go bankrupt. So the actuary away, probably uses an industrial strength modeling program, does some analysis, and then gives a presentation outlining his findings and recommendations in plain english terms. Most of the board members have probably done at most some calculus and a business statistics course.

Another example could be an analyst working in fisheries. The analyst might be asked what his recommendations for the fishery are to maximize their intake of fish, but do so in a way that they will regenerate more fish in time for the next harvest. So in this example the analyst uses a difference equation to model fish intake and out-take at a particular time and presents their findings to management giving their recommendations.

There are a lot of examples, but I think its important to realize that in a lot of situations, applied mathematicians are not just mathematicians, they're communicators. It's no use deriving a great formula if you can't break it down for others less literate in math.

 

[Liquid7800]

I think thinking of new (unsolved), practical/applied problems or pioneering/discovering a new pure mathematics field require equal amounts of creativity, computer science ties very closely between the two fields.

No matter the field though----someone on these forums said it somewhere a while back, "its one thing to solve problems in a field and quite another to contribute to a field"

 

[Outlined]

Pure math is more about problem solving, theorem proving and mathematical reasoning.

Applied math is about learning procedures/recipes to solve problems, so a part of the thinking has already been done for you. You have to make modifications to the problem so that it fits the right recipe/procedure.

The 1st one is more difficult than the 2nd one.

 

[jasonRF]

Pure math is more about problem solving, theorem proving and mathematical reasoning.

Applied math is about learning procedures/recipes to solve problems, so a part of the thinking has already been done for you. You have to make modifications to the problem so that it fits the right recipe/procedure.

The 1st one is more difficult than the 2nd one.

It seems that "applied math" means different things to different folks. A friend of mine got his PhD in applied math. He was required to take all of the core analysis, algebra and topology courses with the pure math PhDs. After the core, then he took the courses on probability theory, statistics, math finance, etc. His education was NOT about learning recipes!

jason