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June 2nd, 2010, 01:46 PM  #1 
Newbie Joined: Jun 2010 Posts: 2 Thanks: 0  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 selfstudy: 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. 
June 2nd, 2010, 07:42 PM  #2  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: What Order?
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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 hindsight. Quote:
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  
June 3rd, 2010, 09:04 AM  #3 
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  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 proofmaking techniques is a necessary addition, and discrete math often leads to a friendly introduction to that subject.

June 3rd, 2010, 01:10 PM  #4 
Newbie Joined: Jun 2010 Posts: 2 Thanks: 0  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. 
June 3rd, 2010, 03:26 PM  #5  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: What Order? Quote:
 
June 3rd, 2010, 11:26 PM  #6 
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  Re: What Order?
Ah yes, I'm so sorry! Elementary Number Theory. I suppose I rely on my textbook memory too strongly.

June 4th, 2010, 09:44 AM  #7  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: What Order? Quote:
 
June 4th, 2010, 10:09 AM  #8 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  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. 
June 7th, 2010, 11:54 AM  #9 
Senior Member Joined: Jan 2009 Posts: 345 Thanks: 3  Re: What Order?
Calculus: Graphical, Numerical & Analytical. By Finney Waits & Thomas.


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