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June 2nd, 2010, 01:46 PM   #1
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What Order?

I have recently graduated from a university with undergraduate degrees in economics and philosophy, but since then have become more interested in math. I would like to on to graduate school (probably in economics) but would like a greater foundation in math first. Admittedly, my math foundation is extremely weak. But, due to my strong interest in formal logic and my girlfriend getting degrees in econ and math, I have been lightly exposed to higher level math (real and complex analysis, linear and abstract algebra, combinatorics, etc.). Don't get me wrong, I could not even begin to try and actually answer any math questions related to those subjects!

The reason I give this background is that my degrees made me realize that I understand math best first through proof, then by application. For example, even simple arithmetic was somewhat confusing to me, but when we defined numbers (complex, real, pure imaginary, rational, irrational, etc.) and proved things like associativity, commutability, and distributivity, it all seemed to make more sense. I have begun my math learning by reading Moses Richardson's Fundamentals of Mathematics (1949), which is a GREAT book for a rigorous introduction to an incredibly large range of topics in math. Also, the copy I happened to buy from the used book store was Frank Harary's personal copy (I live in Michigan, United States and he taught not far from where I live so it is plausible that it actually is his copy)! I really like this style of teaching (explicitly define and prove), but this book is directed at giving a rigorous introduction to the topics of math to those who are not math majors. Hence, it does not apply the math; so, I need to learn and apply the math.

My question is, what order should I learn the topics of math, and in what order should I learn theory, proof, application, etc.? It seems that some proof is required for application, but also application is needed for proof. In America (I only specify because after reading some posts it seems like some, or most, on this site are not American) it is standard that we learn Algebra, Trigonometry, Calculus (which is divided into four main sections), then we go on to proving and learning the logical basis of Algebra, Geometry, Calculus. But, after taking formal logic, I personally understand the lower levels of math by getting a better understanding of the logic behind it. They do not teach the logic in the lower level math in America, they just blindly teach examples and expect you to memorize tricks in order to do the calculation which leads to an inadequate understanding of the calculations.

So far I have accumulated for self-study:
Alfred North Whitehead's An Introduction to Mathematics
Peter Eccles' An Introduction to Mathematical Reasoning
Foundations of Mathematics from MIT Press in three volumes; 1. Algebra, 2. Geometry, 3. Analysis
Spivak's Hitchhiker's Guide to Calculus
Spivak's Calculus and accompanying answer book
Stewart Shapiro's Thinking About Mathematics
And other books about the philosophy of math and whatnot which I hope to get to after I have learned more about math.

The order I listed the books was the order I wish to read them, but I am unsure about reading the Foundations of Mathematics before or after I tackle Calculus. It might be nice to read volumes 1 and 2 and leave 3 till after. But, volume 1 also deals with linear and abstract algebra, which in America is left until after learning calculus.

I am really unsure about what order to read and learn these texts. I am weary of going "out of order" compared to American universities, but then again, the order they use did not work for me. As many of you know, learning math is also about passion and genuine interest, and I get that through logic. I realize that doing problems over and over again is necessary, but I need the logical foundation in order to understand it fully.

Sorry for the long question, but I will greatly appreciate any and all advice.
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June 2nd, 2010, 07:42 PM   #2
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Re: What Order?

Hello!
Quote:
Originally Posted by woodand6
The reason I give this background is that my degrees made me realize that I understand math best first through proof, then by application. For example, even simple arithmetic was somewhat confusing to me, but when we defined numbers (complex, real, pure imaginary, rational, irrational, etc.) and proved things like associativity, commutability, and distributivity, it all seemed to make more sense.
Part of the problem with math education (at least in America, but my discussions with non-American friends makes it seem pandemic), is that emphasis is placed on rote calculation, instead of conceptual understanding. The formal method (axioms, proving properties) is a powerful learning tool because it forces conceptual understanding-- you cannot do a proof by rote, without brute-frocing the search space. Your training in philosophy has very likely made you a creful enough thinker that you can work your way through the simple properties without too much difficulty... it turns out there are people who don't learn well when given a set of axioms and some definitions. (Yeah, I can't see what's so hard either. :P )

Quote:
I really like this style of teaching (explicitly define and prove), but this book is directed at giving a rigorous introduction to the topics of math to those who are not math majors. Hence, it does not apply the math; so, I need to learn and apply the math.
At the higher levels, you'll see a lot more of the "define and prove" style of teaching, and you'll see a lot less application in math books. You'll have to start looking into the econ, or physics, or engineering text books for application.

Quote:
My question is, what order should I learn the topics of math, and in what order should I learn theory, proof, application, etc.? It seems that some proof is required for application, but also application is needed for proof.
Theory -> Proof -> Application is definitely the way to learn math. Of course, it isn't nearly that cut and dry: you won't have a full grasp of the theory without writing proofs, applications will help to motivate proofs and theory, and applications will show up in the proofs. The learning process should really flow rather organically, and the only place you may have to be consciouos of where you are is in the gap between Theory/proof and application.

Quote:
They do not teach the logic in the lower level math in America, they just blindly teach examples and expect you to memorize tricks in order to do the calculation which leads to an inadequate understanding of the calculations.
I think most people on this board could complain endlessly about the sorry state of math education.

Quote:
So far I have accumulated for self-study:
Alfred North Whitehead's An Introduction to Mathematics
Peter Eccles' An Introduction to Mathematical Reasoning
Foundations of Mathematics from MIT Press in three volumes; 1. Algebra, 2. Geometry, 3. Analysis
Spivak's Hitchhiker's Guide to Calculus
Spivak's Calculus and accompanying answer book
Stewart Shapiro's Thinking About Mathematics
And other books about the philosophy of math and whatnot which I hope to get to after I have learned more about math.
Whitehead's book is surely dated. If you are just looking for a treatment of grade-school level math (and this is what it is), it's probably safe to use, but you may want to check the notation with a more recent book. If it's supposed to be a little more advanced, it is probably not a very useful book to use.

Is it actually Foundations of Mathematics or is it Fundamentals of Mathematics? Foundations typically has a certain meaning in math, and I only see Fundamentals of Mathematics from MIT Press.
Anyway: it may be better to learn calculus before this book, if only because some calculus will be assumed. This won't show up except in examples and problems (applications) until Volume 3. maybe it will show up in volume 2, but I would doubt they would approach any truly deep geometry without covering analysis first.

Were I in your position, I'd probably read volumes 1 and 2 first (if only because they're more fun), and if 3 even starts presenting problems, go to a calculus text. If you manage to get through 3 without reading a calc text, you'll find calculus really easy in hind-sight.

Quote:
but then again, the order they use did not work for me.
It seems to me that you're right about at the point where the ordering straightens out and starts to make sense, so from where you are, the order in American universities should be more or less manageable.

This all being said, a text on "discrete math", or depending on your background, one on combinatorics and one on number theory should be interesting. Based on your comments about formal logic (which I assume you studied while studying philosophy), you may be better off skipping a discrete math book, and getting a "proof book" like How to Prove It or How to Solve it.

I hope this was helpful, and not too rambly.

Cheers,
Cory
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June 3rd, 2010, 09:04 AM   #3
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Re: What Order?

I really like the idea of moving on to a discrete math book, perhaps Rosen's Intro to Number Theory. I agree that a focus on some proof-making techniques is a necessary addition, and discrete math often leads to a friendly introduction to that subject.
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June 3rd, 2010, 01:10 PM   #4
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Re: What Order?

To cknapp:

Yes, Whitehead's book is probably quite dated, but it was only a few bucks and I have heard good things. Also, I find that reading a few intro texts (and learning them well) can really provide a surprising amount of information. I will be honest, I have not really looked at it carefully. It also seems like getting used to new notation might be a good thing since it will imprint that it's not the actual symbols in the formula that make it valid, but only the meanings of them and the formula itself.

Also, you are absolutely correct; it is Fundamentals of Mathematics edited by Behnke, Bachmann, et al. and translated by Gould; not Foundations of Mathematics. Sorry for the confusion, I happened to take the 1st volume which is subtitled Foundations of Mathematics.

As for the order of them, your idea of reading 1 and 2, then attempting 3 while referencing a calc book sounds like a good option. My girlfriend will probably be able to help me look through the calc book and note topics I should review before reading an analysis text. Luckily, she is also reviewing math this summer for grad school in the fall (although her "review" is way beyond my understanding right now!). She also said that reading analysis would make calculus much easier.

I did look at How to Prove It or How to Solve it before, but I will have to consider purchasing them now.


To jason.spade:

For Rosen's Intro to Number Theory do you mean A Classical Introduction to Modern Number Theory by a Michael Rosen, because that is a graduate level textbook and I seriously doubt I could even attempt to start that. There does seem to be another book by a Kenneth Rosen titled Elementary Number Theory that seems more appropriate for me (if only because is says "elementary" and is not graduate level). Sorry, I am just unsure.

My girlfriend does have books dealing with discrete math, and some books on proofs, one I know she has is The Nuts and Bolts of Proofs, Third Edition: An Introduction to Mathematical Proofs by Antonella Cupillari. I have dealt minimally with mathematical induction and am only vaguely familiar with the specific form. But, fortunately, I am not new to formal proofs.

--------------------------

Thank you both for your comments and suggestions.
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June 3rd, 2010, 03:26 PM   #5
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Re: What Order?

Quote:
Originally Posted by woodand6
For Rosen's Intro to Number Theory do you mean A Classical Introduction to Modern Number Theory by a Michael Rosen, because that is a graduate level textbook and I seriously doubt I could even attempt to start that. There does seem to be another book by a Kenneth Rosen titled Elementary Number Theory that seems more appropriate for me (if only because is says "elementary" and is not graduate level).
Rosen's Elementary Number Theory was surely what was intended. Ireland & Rosen is hard even for a graduate class.
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June 3rd, 2010, 11:26 PM   #6
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Re: What Order?

Ah yes, I'm so sorry! Elementary Number Theory. I suppose I rely on my textbook memory too strongly.
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June 4th, 2010, 09:44 AM   #7
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Re: What Order?

Quote:
Originally Posted by jason.spade
Ah yes, I'm so sorry! Elementary Number Theory. I suppose I rely on my textbook memory too strongly.
That shows a pretty good memory, in my view. I have to get my textbooks to remember their titles at all, with very few exceptions (Enderton and Ireland & Rosen, for example).
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June 4th, 2010, 10:09 AM   #8
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Re: What Order?

I tend refer to text books by author's last name, which only rarely leads to problems... I got Herstein and Hungerford mixed up once.
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June 7th, 2010, 11:54 AM   #9
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Re: What Order?

Calculus: Graphical, Numerical & Analytical. By Finney Waits & Thomas.
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