Mathematica Why do people say mathematical analysis is the most important?

[copperboy]

Why do people say that Mathematical Analysis and Abstract Algebra are the two most important when it comes to the foundation of math? Maybe there are people before me have already asked such questions and I'd like to find the answers from other people, or discuss with others on this forum, or even perhaps more lucky, find some examples which are provided by you members on this forum that show how do those two fundamental subjects Mathematical Analysis and Abstract Algebra function in the future math learning or studying.
Thank you! I am a new comer and this is my first thread.

 

[HallsofIvy]

I can't answer your question, because I wouldn't say that!

I certainly wouldn't say that any branch of mathematics is "more important" than any other.

I might be inclined to say that all branches of mathematics can be loosely grouped into the two umbrellas, Abstract Algebra and Topology (i.e. algebra and geometry- discrete and continuous mathematics).

I would then say that Mathematical Analysis uses concepts from both: In order to define the "derivative" we need we need to be able to do algebraic things: subtract values of the functions and divide by a scalar. We then need to take a limit which is a topological concept.

It is true that most applications of mathematics (if that is why YOU mean by "important") are from Mathematical Analysis.

 

[copperboy]

I say those two subjects are more important when it comes to the FOUNDATION of math because my teacher told me so in class. I don't know why and my teacher doesn't want to tell us and she just said you'll see that yourself when you learn more. So I posted here to ask those who have already learned enough.

 

[CrankFan]

When people talk of the "foundation" of mathematics I think it's usually in reference to set theory, but maybe copperboy's instructor is using it in a different sense. He could mean that it's essential that a student establishes a good foundation in mathematics by first studying abstract algebra and analysis; which is more a statement about how he thinks a student of mathematics should develop.

 

[gazzo]

analysis and abstract algebra may not be the most important, but they help don't they? After reworking the foundations with mathematical rigour you can look at everything in a new light.

Meh.

 

[copperboy]

Can anyone here give me an example on how analysis and abstract algebra be used in further study of math?

 

[fourier jr]

a lot of grad schools have phd candidacy exams on analysis, algebra & topology that everybody has to pass, no matter what subject they are planning on studying. i guess that because those are the 3 most fundamental areas.

 

[ReyChiquito]

Can anyone here give me an example on how analysis and abstract algebra be used in further study of math?

Right now I am taking a course about topological methods in non linear analysis. Basically uses lots of functional analysis combined with lots of algebraic topology to understand the behavior and properties of non linear operators, like non linear partial differential equations. Its based on something called degree theory.