Re: How to approach the study of mathematics?
- From: "Peter Webb" <webbfamily-diespamdie@xxxxxxxxxxxxxxx>
- Date: Mon, 17 Apr 2006 17:15:03 +1000
"Colleyville Alan" <nospam@xxxxxxxxxx> wrote in message
news:fPadnQ3t5pBusN7ZRVn-vA@xxxxxxxxxxxxxx
I would like some advice about how best to learn mathematics. Over the
last couple of years, I have developed an interest in studying financial
engineering and all major universities that teach FE require a quantitative
background. Generally, the prerequisites include calculus I, II and III,
linear algebra, ordinary and partial differential equations, a
calculus-based probability class and a calculus-based inferential
statistics class, plus a few other math courses. Some universities
suggested supplemental reading such as "Mathematical Methods in the
Physical Sciences" by Mary L. Boas and "Probability and Measure" by Patrick
Billingsley (these last two recommended by the University of Chicago).
I have seen many Herman Rubin's posts in which he states that learning how
to compute does not bring an understanding of mathematics and that all too
often students memorize formulas for the exams and cannot remember
anything two weeks later. That is certainly true for me. After not
having a math class in 30 years (I'm 52 now), I began taking elementary
math last summer and got A's in both algebra and trigonometry. I did
develop *some* understanding of the material, but way too much of even
these very elementary classes has been an exercise in memorizing and
forgetting. I would work 200+ problems per chapter and develop enough
expertise with the techniques involved, and then do well on the test, but
even if I remember the technique, most of the time it is not a deep,
intuitive knowledge.
I've also seen Herman post that analysis should be taught before calculus
and that, in general, abstract math should be taught before computational
skills. That may be true, and in my self-study I can arrange the order of
learning as well as the subject matter, but in the formal education
process, the computational classes have to be taken in a particular order.
Furthermore, I cannot put them off for two years while I pursue
independent study; there is a timetable I need to stick to.
With all of that said, I have developed a desire to learn more about math
than simply the computational classes that are required for the study of
financial engineering. I want to understand math and not merely memorize
a bunch of formulas. So my questions boil down to two general categories:
1.. What topics are required to really understand mathematics? I have
seen discussions on this board about analysis, topology, abstract algebra,
set theory, group theory, combinatorics, and a number of other topics, and
am wondering what is really an acceptable framework for developing an
understanding of the subject. I've seen the mathematical atlas
(http://www.math-atlas.org/) and I know the subject is too big to
understand everything, but I mean a reasonable grounding in math.
What do you need to know to truly understand computers?
A intimate knowledge of Microsoft applications?
C++ experience?
Assembly language?
Digital circuit design?
Photolithography ?
Maths is like that - layered. Do I understand Calculus better for having
studied basic set theory? Well, a little maybe, because it gives me comfort
that the concepts of numbers and functions has some logical basis - just as
knowing that Silicon is a semiconductor gives me faith that Excel will work.
Personally, I think that trying to understand the general concepts behind
Calculus would be a lot more harder if you hadn't already solved lots of
differential and integrals. The abstract is much easier to follow when you
know the concrete well.
Calc I is a blast (or it was when I did it 35 years ago). The fundamental
thereom of calculus and the immense power Calculus gives you is extremely
exciting. Going up through Calc II and pde's there is a lot more ad-hoc
stuff that just seems to work for some types of equations, and I didn't
enjoy it. This is some pretty serious heavy going.
For a sampling of what Maths can hold, a book often recommended in this
newsgroup - and the book which sealed my love of Maths way back then - is
Courant and Roberts "What is mathematics":
www.amazon.com/gp/product/0195105192/sr=8-1/qid=1145256729/ref=pd_bbs_1/002-5140408-7454410?%5Fencoding=UTF8
for $14.63
Read it through, and see what you are still thinking about a few weeks
later.
This book, because of its age, almost certainly does not contain anything or
much on set theory and algebraic topology (my copy isn't handy). Set theory
is quite accessible, and provides some very surprising results very early
(doubtless you have seen the endless discussions here and on sci.logic over
Cantor's diagonal proof, Godel, etc). There are also some excellent web
books on the subject. And because it deals with the very foundations of
mathematics, there is no pre-requisites other than being able to perform
symbolic manipulation.
2.. How I might more deeply and fully understand the math that I am
taking? Would it be first learning proofs and perhaps supplementing
Stewart with Apostol? Try to learn analysis simultaneously along with
calculus? Is there a preferred order of study, or method of study? Is
there a way to really get to the underlying concepts? Any ideas?
Thanks in advance
Alan
It doesn't work like that. Whilst maths is layered, you can't practically
say "OK, I will fully understand what a Real number is, then I will
understand what a function is, then I will learn about Operators, and then I
will learn how to differentiate y = sin(x)". You have to work at several
different layers at once. You go down one path until it gets too hard, and
go somewhere else. Later you will find that you come back to where you left
off, and find it much easier or more interesting now you know "why".
.
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