Re: How to approach the study of mathematics?

Hello,

I have exactly the same situation as you. Started interested in math
from applied perspective, did take 3 courses of calculus, diff
equations, linear algebra, complex variables, probability, and
statistics, but eventually became interested in 'pure math'.

I've talked to someone about enrolling into grad level 'pure math'
program and been told that the key courses I should take, probably even
before I get enrolled, just 'to get a taste of it', is Real Analysis
and Abstract Algebra. However, I've done my research as well and
realized that the pre-requisite to both is some basics of logic, proof
techniques, and set theory. In pure math, what you do most of the time
is Proofs, Proofs, and Proofs, and then more Proofs, and you normally
prove some statements about sets (whether it is analysis or algebra),
and you use the logic to reason as you go through the proofs. So I've
spent some time learning some basic logic/proofs sets on my own, and
then went to school, to take algebra and analysis, but on undegrad
level, just to get a taste of it. The fact that I started with some
proof technique and set theory in my pocket was a tremendous help. I'm
still going through the basic staff, and if I made thorugh it, I"ll
take the same Analysis and Algebra, but on undergrad level.

You do need to have some calculus background, at least a couple of
courses, to have a motivation, so to speak, to take a course on
Analysis. Ditto for Linear Algebra before taking the course on Abstract
Algebra. Not a prerequisite, strictly speaking, but some kind of
'motivation'.

In pure math, at least on the level I do it know, you do very little
computations, and mostly proofs. I really like it, it is sort of a
chess game except with much richer context than chess. For instance,
when you take Calculus, some basic theorems are given without the
proofs. When you take Analysis, you'll spend at least half of the
course for 'setting up the proper environment', some basic axioms, and
then you end up deriving all those theorems that were given in Calculus
without a proof. That's the major motivation, at least for me. When I
took the calculus, I was always curious about where all that stuff came
from.

The textbook I've used for both studying logic/proofs/sets and then as
my textbook in undergrad course on analysis is Lay's "Analysis with
introduction to proofs". Highly recommend. You can get it and read a
few chapters to see if you have a taste for the whole thing.

.

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