15 October 2004

The Return of the Mathematics Corner

George E. Hrabovsky, President, MAST

Where we have been

Since its beginning, "The Mathematics Corner" has been a column in search of a purpose. With the new emphasis on projects in The Citizen Scientist, I now feel that the purpose of this column will be to develop mathematical projects the same way that people write up experiments.  

I will begin with a short exposition of the central topic of the project being presented, then I will clearly state the work I am suggesting. The goal is NOT to do all of the projects myself. It is to provide the necessary background for the reader to do the project. 

This means that I will be starting out at a fairly elementary level. However, I must issue a warning. Even elementary projects can take surprising turns. When he was a high schooler, Andrew Wiles thought that proving Fermat's Last Theorem would be easy. After all, Fermat has written that he had discovered an amazing proof too large to write in the margins. If Fermat could do it, then it should be accessible to a high schooler. Right?  When Andrew Wiles completed the proof (the second time; the first time it was in error), the thing weighed in at well over two hundred typeset pages and involved advanced analysis, algebraic geometry, and the theory of elliptic curves.

What does it take to be a mathematician?

In the past I have written extensively on how to learn mathematics. The question must then be asked, "If I learn mathematics, am I a mathematician?"  The answer is, surprisingly, no. 

In the same way that you are not a writer just because you can write, you must DO mathematics in order to be a mathematician. There are some required skills. Just like a writer needs to understand the meaning of words, their structure, and how to combine them in sentences and paragraphs through the application of punctuation and grammar, a mathematician must formulate and prove theorems, abstract generalities from specific cases, and deduce applications from abstract concepts. 

Logic is the language of set theory, and set theory is the language of mathematics. We will be talking a LOT about proving conjectures and statements, formulating those statements and formal definitions, and trying to have a lot of fun while developing the machinery to create new mathematics.

A course in pure mathematics?

I am doing away with the pretense of not trying to teach mathematics here. The totality of these columns will, indeed, teach you to do pure mathematics. In each column I intend to cover one topic, present an idea for a project, provide guidance from previous columns (after this one), provide some advice to help you generally, and then give a list of places to look for more information.

The general flow of these columns will change in time, but will follow a definite structure. 

Logic: Mathematics relies heavily on your ability to prove what you are claiming. Do not simply accept what you are told. Ask questions and demand proof. We will begin by developing skill in proofs, and we will return to it whenever the need arises.

Set Theory: When we make statements, we will most often be generalizing our results. To do this we need to refer, at some level, to every possible case. This requires the mathematics of sets.

Algebra: When we impose rules connecting members of a set we are creating algebraic structures. These structures form the underlying basis for the equations and functions you may be familiar with. We will also be studying how different structures relate to one another.

Geometry: Some of the structures we will study have properties such as distance that are constant despite how we might manipulate the structure. These are called geometrical objects. We will study how these shapes and objects relate to each other.

Analysis: Most things change with time, and analysis gives us the tools to understand those changes. At higher levels we will discover structures built out of change itself!

Combinatorics: Here we will study the nature of how things relate to each other.

Probability Theory: This is the study of the likelihood of events happening within a set of possible events.

Mathematical Statistics: This is the study of structures made out of data.

Topology: Sometimes the structures we deal with can change their shape without losing their boundaries.

These are the primary branches of mathematics, in no particular order after the first four. Each branch has subbranches that we might get to.

How to approach all of this

Below I will outline a set of books and online resources. As you read through them, keep a few things in mind. Each branch of mathematics exists because it deals with a specific set of problems in specific ways. It is vitally important to understand the fundamental scope and purpose of each branch.

Mathematics is all about the structure of mathematical objects, the structure being represented by relationships among the objects. Keep a running list of all of the objects that you encounter. Most often these will be in the form of definitions and undefined technical terms.

One source of projects is to think up examples of any definition. Look for extreme cases, counterexamples, etc. Develop a list of relationships between objects as you encounter them. These will be undefined technical terms, unproven postulates assumed to be true, definitions, and either unproven statements or conjectures and proven theorems. Here again, a source of projects is to create examples of all of these.

Another source of projects is to invent new proofs for existing theorems (perhaps using different mathematics than existing proofs), or to prove unproven statements and conjectures. Always look for connections between ideas. When you encounter such, note them immediately. This is another source of projects. How are two areas of mathematics equivalent? How can they be combined?

Another source of projects is to note equivalent forms of objects or relationships between objects. Such equivalences expose deep insights into the structure of mathematics and also allow for the application of new methods to existing problems.

Having said all of that, how do you approach such a vast subject? Patience and one step at a time. I have found it useful to acquire a library of mathematics. This can be very expensive, but fortunately there are many freely available books on the web that are fairly easy to locate. Enter in your favorite search engine such phrases as "Free Electronic Books," or "Mathematics," followed by, "Course Lecture Notes." If you are going to pursue pure or applied mathematics, or even theoretical science, you will need at least one book (and I suggest at least three), from each of these categories:

General Mathematics Reference Books: These include such things as the "Schaum's Outline Mathematical Handbook," "The Encyclopedic Dictionary of Mathematics," and various other mathematical handbooks and encyclopedias. These will give you critical information that you can just look up. I personally own at least five handbooks and three encyclopedias (including the "Kluwer Encyclopedia of Mathematics" on CD-ROM). I also have acquired two mathematical formularies freely from the Internet.

Elementary Mathematics Books: These include arithmetic, elementary algebra, high-school-level geometry, college algebra, trigonometry, and finite mathematics. One of each should do. These might be required to give you the necessary background, but eventually they will exist just to provide specific references to postulates, definitions, and theorems. These topics form the foundations of later knowledge, so make sure that you get around to each subject. If possible, invent a set of projects based around each major idea.

Calculus: This is the first break-point topic, for it is vital that you understand the ideas of calculus. I recommend having at least two applications-oriented calculus texts and one that is more theoretical. Specifically, I like Sylvanus Thompson's book, "Calculus Made Easy," as an application-oriented book. "Schaum's Outline of Calculus" is very good. For a theoretical calculus book for those on a budget, I recommend Silverman's "Modern Calculus and Analytic Geometry" from Dover.

If money is no object, then I recommend either Apostle's two volume, "Calculus" or Courant and John's three volume "Introduction to Calculus and Analysis." There are several free electronic calculus books, too. "Applied Calculus," by Karl Dovermann of Hawaii, is very good, and I highly recommend Dan Sloughter's "Difference Equations to Differential Equations" and its companion volume, "The Calculus of Functions of Several Variables."

Transitional Mathematics Books: These books are often based on calculus, and explore in a chapter or two another branch of higher mathematics. My favorite of these is Garrity's "All the Mathematics Your Missed." I also recommend that you pick up one or more books of the Mathematical Methods for Scientists and Engineers type. For those with no budget, I recommend Gallier's, "Algebra and Analysis for Computer Science," available for free online. Sean Mauch is in the process of creating (and has an impressive 2,321 pages completed) a tome called, "Introduction to Methods of Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers." This book covers a lot of topics in great detail, and each year it grows! It is also free! James Nearing has written, "Mathematical Tools for Physics."

If you want to pay for a book and do not have a lot of money Dover has a good selection. The best (in my opinion) is Dennery and Krzywicki's "Mathematics for Physicists." I also like Collins' "Mathematical Methods for Physicists and Engineers." If you don't mind spending money, I like Riley, Hobson, and Bence's "Mathematical Methods for Physicists and Engineers;" Cantrel's "Modern Mathematical Methods for Physicists and Engineers;" Hassani's "Mathematical Methods for Students of Physics and Related Fields" and "Mathematical Physics;" Matthews and Walker's "Mathematical Methods of Physics;" and Geroch's "Mathematical Physics." My favorite (though at a very advanced level) is Prakash's" Mathematical Perspectives on Theoretical Physics." If you have three or more of these books, you might not need some of those that follow, particularly in the areas of linear algebra, ordinary differential equations, complex analysis, Fourier analysis, and partial differential equations.

Mathematical Thinking: These books describe the process of mathematical methods of proof and an introduction to set theory. I have found no suitable online material for this that does not require a lot of undergraduate level mathematics. The classic work in mathematical problem solving is George Polya's "How to Solve It." I would also recommend a book on proofs. My favorite is "A Transition to Higher Mathematics" by Smith, Eggan and St. Andre. "Mathematical Thinking" by D'Angelo and West is also quite good, with a vast array of topics covered.

Linear Algebra: This is, perhaps, more important than calculus. We know how to solve linear problems most of the time. Understanding how to do this is important. Here are the best online (free) textbooks on linear algebra: Marcel Finan's "Fundamentals of Linear Algebra," Jim Hefferon's "Linear Algebra," Sheripov's "Course of Linear Algebra and Multidimensional Geometry" (in my opinion this Sheripov is the best,) David Santos' "Linear Algebra," Jim Carrel's "Basic Concepts of Linear Algebra," and Autar Kaw's "Introduction to Matrix Algebra." For those on a budget, my favorite (and one of my favorites regardless of price range) is "Schaum's Outline of Linear Algebra." Dover also has Shilov's, "Linear Algebra." For those who want to spend a little more, I recommend Sheldon Axler's, "Linear Algebra Done Right" and Paul Halmos' "Finite-Dimensional Vector Spaces."

Combinatorics: There are three aspects of combinatorics that are important in many areas of mathematics: Counting techniques, graph theory, and optimization. The best of the online and free books are: Chen's "Discrete Mathematics," Marcel Finan's "Lecture Notes on Discrete Mathematics," (note that if you have either of these two you can safely ignore step 5 above), and "Discrete Mathematics" by Lovasz and Vestergombi. I can recommend "Schaum's outline of Combinatorics" for those on a budget, though this book does not go heavily into graph theory. For those with more money I can recommend three books: Norman Biggs' "Discrete Mathematics," Cameron's "Combinatorics," and "A Course in Combinatorics" by van Lint and Wilson (this last book is at a very high level).

Ordinary Differential Equations: Ordinary differential equations are those equations that involve one quantity changing with respect to another. Free online texts that I like are: "Ordinary Differential Equations" by Adkins and Davidson and Kutz's "Introduction to Differential Equations." For those on a budget, we have these Dover books: Dettman's "Introduction to Linear Algebra and Differential Equations" (this book also contains lots of good material on linear algebra, suitable for step 6 above), "Differential Equations with Applications" by Ritger and Rose, and "Ordinary Differential Equations" by Tenenbaum and Pollard's. There is also "Schaum's Outline of Differential Equations."

If you can spend money, I recommend Boyce and DiPrima's "Elementary Differential Equations with Boundary Value Problems," Simmons' "Differential Equations with Applications and Historical Notes," and Braun's "Differential Equations and Their Applications." At a somewhat higher level is Arnold's classic work, "Ordinary Differential Equations."

Advanced Calculus: These books cover a lot of the same material as the standard calculus texts but using the tools of linear algebra and rigorous proof. The emphasis is on demonstrating the validity of the methods of calculus. The free online books are: Goursat's three-volume set, "A Course in Mathematical Analysis," Hardy's "A Course in Pure Mathematics," Whittaker and Watson's "A Course in Modern Analysis," Chen's "Fundamentals of Analysis," Wilde's "AnalysisÑAn Introductory Course," and, my favorite, Shlomo Sternburg's "Advanced Calculus."

For those who want books on a budget, I recommend "Schaum's Outline of Advanced Calculus" and the Dover Books "Introduction to Analysis" by Rosenlicht, "Advanced Calculus of Several Variables" by C. H. Edwards, Jr., and Borden's "A Course in Advanced Calculus" (my favorite of the cheap books.) For those who can spend money, I recommend Harold Edwards' "Advanced Calculus: A Differential Forms Approach," Taylor and Mann's "Advanced Calculus," and, my favorite of this class of book (though I struggle between this and Sternburg's book), Strichartz's"The Way of Analysis."

Abstract Algebra: These books explore mathematical structures in their purest form. The free online books are: Clark's "Elementary Abstract Algebra," Arapura's "Abstract Algebra Done Concretely," Finan's "Modern Algebra," Garrett's "Intro Abstract Algebra," and Connell's "Elements of Abstract and Linear Algebra." While the latter text is not a substitute for step 6, it is for step 5. This book is also my favorite online algebra book at an undergraduate level.

For those who want to purchase books on a budget, I recommend Seth Warner's "Modern Algebra" and Allan Clark's "Elements of Abstract Algebra," both published by Dover. Clark's book is my favorite of the two. The book I recommend for those who have a little more to spend (and one that I use a lot as a reference) is MacLane and Birkhoff's "Algebra," not to be confused with their book "A Survey of Modern Algebra." I also recommend having at least one advanced treatment of abstract algebra. Free online books that cover this material are Ash's "Abstract Algebra: The Basic Graduate Year," Leonard Evans' "A Graduate Algebra Text," and Surowski's "Workbook in Higher Algebra." For hardcopy textbooks, I know of no good inexpensive text. I like Hungerford's "Algebra."

Geometry: These books cover topics including Euclidean, non-Euclidean, and projective geometries. Most base their results, to at least a small degree, on abstract algebra (specifically group theory and vector spaces). The free online books are: David Hilbert's "Foundations of Geometry," Carnes' "Geometry," and Danny Caligari's "Classical Geometry," which is my favorite. For those on a budget, I suggest four Dover books: Bruce Meserve's "Fundamental Concepts of Geometry" (this author has also written the excellent companion volume "Fundamental Concepts of Algebra), Paul Yale's "Geometry and Symmetry," which is a really good undergraduate-level course, Dan Pedoe's "Geometry: A Comprehensive Course" (at the graduate level), and Nicholas Kazarinoff's, "Ruler and the Round," which is about geometric constructions.

For those who have money to spend: Coxeter's "Introduction to Geometry," (this is my favorite geometry book), Moise's "Elementary Geometry from an Advanced Viewpoint," Roe's "Elementary Geometry," Greenberg's "Euclidean and Non-Euclidean Geometries," and Brennan, Espleen, and Gray's "Geometry."

Elementary Topology: These books are intended to develop basic notions of topological spaces and the different ways of approaching them. A free online book is "Elementary Topology: A First Course Textbook in Problems" by Viro, Ivanov, Netsvetaev, and Kharlamov. This is probably my favorite elementary level textbook regardless of price. Another good one is Sydney Morris' "Topology without Tears."

For those on a budget I can point you to several good Dover books, including Gamelin and Greene's "Introduction to Topology" and Gemignani's "Elementary Topology." There is also "Schaum's Outline of General Topology." For those with money, I recommend Buskes and van Rooij's "Topological Spaces," Adamson's "A General Topology Workbook," Borges' "Elementary Topology and Applications," and Armstrong's "Basic Topology."

Differential Geometry: These books cover the theory of curves, surfaces, and manifolds. The free online books are: Lee's "Differential Geometry, Analysis, and Physics," Koch's "Mathematics 433/533: Class Notes," Moore's "Lecture Notes on Curves and Surfaces for Mathematics 147AB," Michor's "Topics in Differential Geometry," and Shifrin's "Differential Geometry: A First Course in Curves and Surfaces."

For those on a budget I can recommend the Schaum's outline and these Dover books for a traditional treatment: Kreyszig's "Differential Geometry" and Struik's "Lectures on Classical Differential Geometry." Auslander and MacKenzie's "Introduction to Differentiable Manifolds" provides a more modern treatment, though none of these is fully modern (as Lee, Koch, or Moore are in the free books.) For those with money, I recommend O'Neill's "Elementary Differential Geometry" and Spivak's 5-volume set, "A Comprehensive Introduction to Differential Geometry."

Real Analysis: These books take the principles of advanced calculus and generalize them over abstract spaces. The free online books are: Simms' "Analysis," Hunter and Nachtergaele's "Applied Analysis," Faris's "Real Analysis: Part I" and, "Real Analysis: Part II," and Kuttler's "Basic Analysis." For those on a budget, I recommend Komogorov and Fomin's "Introductory Real Analysis." This remains one of the best real analysis books even today, and it is published by Dover. If you have money, I recommend Rudin's "Real and Complex Analysis" (this also meets the next section's requirements), Carothers' "Real Analysis," and Goldberg's "Methods of Real Analysis."

Complex Analysis: These books take the principles of real analysis and topology and extend them to the complex numbers. The free online books are: Cain's "Complex Analysis" and Ash and Novinger's "Complex Variables." For those on a budget, I can recommend two Dover books: Knopp's "Theory of Functions" and Fisher's "Complex Variables." For those who have money, I recommend Needham's "Visual Complex Analysis" and Markushevich's "Theory of Functions."

Fourier Analysis: These books describe how complicated patterns can often be broken down into combinations of simpler patterns. Free online books include: Miller's "Topics in Harmonic Analysis with Applications to Radar and Sonar," Thiele's "Classical Fourier Analysis," Brown's "Lecture Notes: Harmonic Analysis," and, one of the best, Axler, Bourdon, and Ramey's "Harmonic Function Theory."

For those on a budget, there is "Schaum's Outline of Fourier Analysis" and the Dover books "Fourier Series" by Tolstov, Sneddon's "Fourier Transforms," Wilcox and Meyers' "An Introduction to Lebesgue Integration and Fourier Series," and Katznelson's "Introduction to Harmonic Analysis." For those with money, I recommend, Champeney's "Fourier Theorems" and Stein's "Harmonic Analysis."

Functional Analysis: These books deal with special types of functions on different kinds of spaces. The free online books are: Abbas' "Functional Analysis," Hille and Phillips' "Functional Analysis and Semi-Groups," Chen's "Linear Functional Analysis," and Angenent's "Math 725 Lecture Notes." For those on a budget, there are the Dover books: Bachman and Narici's "Functional Analysis," Kolmogorov and Fomin's "Elements of the Theory of Functional Analysis," Riesz and Sz.-Nagy's "Functional Analysis," Goldberg's "Unbounded Linear Operators," Akhiezer and Glazman's "Theory of Linear Operators in Hilbert Space," and Zemanian's books, "Distribution Theory and Transform Analysis" and, "Generalized Integral Transforms."

For those with money, I recommend Oden and Demkowicz's "Applied Functional Analysis," Debnath and Mikusinski's "Introduction to Hilbert Spaces with Applications," Rudin's "Functional Analysis," Megginson's "An Introduction to Banach Space Theory," and the four-volume masterpiece by Reed and Simon, "Functional Analysis," "Fourier Analysis, Self-Adjointness," "Scattering Theory," and "Analysis of Operators."

Partial Differential Equations: Often differential equations appear where a quantity changes with respect to more than one other quantity. This is a partial differential equation. The online free books for this are: Birnir's "Elementary Partial Differential Equations and Applications," Showalter's "Hilbert Space Methods for Partial Differential Equations," and Herod's "An Introduction to Partial Differential Equations."

For those on a budget, I recommend "Schaum's Outline of Partial Differential Equations" and the Dover books: "Introduction to Partial Differential Equations" by Broman, Greenspan's "Introduction to Partial Differential Equations," Farlow's "Partial Differential Equations for Scientists and Engineers," Guenther and Lee's "Partial Differential Equations of Mathematical Physics and Integral Equations," and Gustafson's "Introduction to Partial Differential Equations and Hilbert Space Methods." For those with money to spend, I like Troutman's "Boundary Value Problems of Applied Mathematics," Zauderer's "Partial Differential Equations of Applied Mathematics," Garabedian's "Partial Differential Equations," and the quite advanced, but excellent, three-volume set by Taylor, "Partial Differential Equations I: Basic Theory, II: Linear Equations, and III: Nonlinear Equations."

OK, you have a bunch of books.  Now what?  Now think of something that is interesting.  Can't think of anything?  OK, well here is something that is of interest to me that I know little about, the so-called Riemann hypothesis. I am not even sure I know what it is. Beginning with my next column I will explore this with all of you. I will even assume that I know little beyond high school algebra and geometry.

Created by Mathematica  (11 October 2004).