18 October 2002

Welcome to The Mathematics Corner

by George E. Hrabovsky
President, MAST

Welcome

Hello! Welcome! I know, I know. A second column? Doesn't this guy ever sleep?

To tell you the truth, the answer is no; I am a chronic insomniac. So, here I am at 3 AM writing this column after playing tag with my pet rat for the last fifteen minutes (I won, if you are interested).

So, what is this new column about? Well, okay, it's about math. Why not just read The Mind of a Theorist? That has lots of math in it, right? Well, yes it does have a lot of math. But just like until The Mind of a Theorist came along there was no theory stuff just for theorists; now there is no math stuff just for the sake of amateur mathematicians. After having the seed planted by Sheldon some weeks ago (I know he didn't mean for ME to write it) I decided to give it a try. So, here we are.

I intend this column to be a mirror of The Mind of a Theorist, for mathematics. I do not intend to make this into a course of instruction. It will address areas of interest. It will be prejudiced, I have certain interests in math and those are what I will tend to write about. If you want me to write about something else, you will have to tell me about it.

What is Mathematics?

Those of you have read my other column know that I have spent a lot of time discoursing on what a theorist is and does. I will do the same thing here.

A mathematician's job is to invent ideas and prove them. Sound easy, right? Well, maybe not...

What does it mean to prove something? How do you go about proving an idea? For that matter, how do you go about inventing an idea to prove?

You can only invent an idea by developing expertise in the subject of interest to you. Why? It allows you to know what has been done and avoid unnecessary duplication of effort. If you can't name at least five unsolved problems in a branch of mathematics then you do not have enough expertise to make contributions except by accident. Even if you can name them, it doesn't mean you have the expertise.

The problem is that if you don't know the subject well you will not be able to prove your idea. So, before we can tackle the problem of inventing and proving ideas, we must learn how to become an expert.

We also want to avoid having to spend five years gaining the necessary expertise. So, how do we do this?

Becoming a Mathematician

I will now steal some of what I wrote in the Theorist column "Coursing Through Mathematics."

When we study mathematics it is tempting to just think of it as a collection of tools that merge or diverge depending upon the application under study. This seems to be how lower-level math courses are taught. Even in higher-level courses we rarely get to see the overall picture until it is too late. Almost always we, as students, are left to motivate the discussions and lecture topics for ourselves. For the non-mathematician this can be stultifying, and makes mathematics harder than it needs to be (and it is pretty hard at the best of times) and far more bewildering. The professors/authors look smarter than we could hope to be as they reverse the order of a derivation and seem to work magic (because they know the answer already and can supply it, then show it is true, while giving us no real way to get the answer they got from first principles). It is a pretty dismal state of affairs...

Mathematics does have a flow to it. It is designed to meet certain needs that are unique to the branch of mathematics under study, and at the same time it can look eerily similar to other branches of mathematics. Mathematics is concerned with three primary topics: mathematical objects, relationships between these objects, and showing when these objects or relationships are equivalent. Keeping these three points in mind you can learn a lot about any branch of mathematics by answering five questions:

1. What are the objects used in this branch of mathematics?
2. What are the principal relationships between the objects?
3. What are the criterion for equivalence?
4. What is the central goal of the branch of mathematics?
5. What are the principle methods used to achieve the goal?

 

These capsule answers are not sufficient to develop an understanding of mathematics. When you try to learn mathematics keep them in mind. As you read a paragraph of text ask yourself these questions over and over again. Did the paragraph reveal a new mathematica object? A new relationship between objects? A new equivalence (or condition when things are not equivalent)? A new goal? A new method?

Do not stop with these questions, though. Always demand that any claim be shown to be true clearly. Do you understand the proof of a theorem? Can you explain it to an imaginary person in front of you? Try it! If you don't understand something try looking at it from a different point of view. Try different approaches, see if they solve the difficulty. Try applying the idea to a simple case, to extreme cases, to problems where you already know the answer.

The idea here is to get you to do something other than simply reading or listening to a lecture. Mathematics is not a spectator sport. In the same way that you will never become an athlete by watching athletes (you have to actually do the work), mathematics requires that you develop expertise not just in the ideas, but also the methods.

Of course, we also need resource material to ask these questions of. Here is what I consider to be a fine course of reading to bring you up to speed in what I will call the Core Study of mathematics (these books are mostly chosen for their price first, and then their quality; none of them are substandard and I like them all--just be aware that there are better books out there). Where possible I will offer what I consider to be the best book in my library on the subject (and in those cases where that book is out of print, I will offer an alternative).

1. Silvanus P. Thompson, Martin Gardner (1998), Calculus Made Easy, St. Martin's Press. This is the classic text by Silvanus with updated notation, notes to the text added by Gardner, and three initial chapters on functions, limits, and derivatives also by Gardner. This is the best text on elementary techniques in calculus that I have EVER SEEN!!!! It is a little short on practice problems (but you can get a Schaum's outline book for that). This book explains calculus. The problem is that it is not a very mathematical book in that you are not gaining any practice in doing proofs. For a more mathematical treatment, and for lots of problems, I recommend Frank Ayres, Jr., Elliott Mendelson, (1999) Calculus, McGraw-Hill (Schaum's Outline series).

2. Now that you have experience thinking about calculus, you should study a book that addresses proofs and advanced mathematical techniques. The one I like is not inexpensive, so perhaps the best way would be to follow this column (I will be developing many proof strategies). Here is the book: John P. D'Angelo, Douglas B. West (2000), Mathematical Thinking. This is a very modern book covering the ideas of problem-solving and proofs at an intermediate level. This book assumes some familiarity with calculus.

3. Thomas A. Garrity (2002), All the Mathematics You Missed [But Need to Know for Graduate School], Cambridge University Press. This is a fabulous little book (well, it is over three hundred pages). It begins with a discussion of equivalences in various branches of mathematics. This is really very revealing and gives a strong clue as to what mathematics is all about. Then it gives a paragraph description of several branches of mathematics and describes what you should know how to do for each. Then it devotes a chapter to giving an intense overview of each of the branches. This is an excellent way to get things clearly in your head before diving into an advanced textbook. The only failing that I can see is that there are no chapters on either functional analysis or tensor analysis. Other than that (and if you understand all that is in this book, those subjects will be readily accessible), this is a truly wonderful book.

4. Seymour Lipschutz, Marc Lipson, (2001), Linear Algebra, McGraw-Hill (Schaum's Outline series). This is a very thorough treatment of linear algebra. Coupled with the introductory chapter on linear algebra in Garrity (to put everything into perspective) this book will give you most of what you need for more advanced study. Indeed, most of the other books I will mention have linear algebra as a prerequisite.

5. Robert S. Borden, (1983), A Course in Advanced Calculus, Elvesier Science Publishing Co. (reprinted by Dover Publications in 1998). This is one of the best available book on advanced calculus. A more modern treatment that I like is that of Robert S. Strichartz, (2000), The Way of Analysis, Jones and Bartlett. This book is my personal favorite, I was fortunate to find a mint-condition used copy.

6. Saunders MacLane, Garrett Birkhoff, (1993), Algebra, Chelsea Publishing Co. This book begins with a detailed study of the algebra of sets, functions, and numbers. Then it builds on these ideas to study such algebraic structures as groups, rings, vector spaces, and many more. This is my favorite book on algebra. It has LOTS of practice problems in a nice balance between actual calculations and proofs. Be prepared to work hard. If you do put the work in, it is immensely rewarding and you will come away with a deep understanding of these concepts.

7. Melvin Hausner, (1965), A Vector Space Approach to Geometry, Prentice-Hall (reprinted by Dover Publications in 1998). This book applies the concepts of linear and abstract algebra to the problem of geometry. Of course, the true classic in the field is H. S. M. Coxeter, (1969), Introduction to Geometry, John Wiley and Sons. Unfortunately, this most-excellent available text is out of print and hard to find; if you find a copy pay any price to get it, it is worth it!

8. Theodore W. Gamelin, Robert Everist Greene, (1983), Introduction to Topology, W. B. Saunders Company (reprinted by Dover Publications in 1999). A fine introduction to both point-set and algebraic topology.

9. Heinrich W. Guggenheimer, (1963), Differential Geometry, McGraw-Hill (reprinted in 1977 by Dover Publications). This is a very nice treatment of differential geometry including Lie groups and tensors. Of course, the most significant resource is Michael Spivak, (1999), A Comprehensive Introduction to Differential Geometry, Publish or Perish, Inc. This FIVE-VOLUME set is a magnum opus by someone who really cares about the subject.

10. V. I. Arnold, (1973), Ordinary Differential Equations, MIT Press (translated by Richard Silverman). This awesome tome covers the subject from a very modern point of view.

11. Donald Greenspan (1961), Introduction to Partial Differential Equations, McGraw-Hill (reprinted by Dover Publications in 2000). A very nice treatment of partial differential equations and their role in mathematical physics.

12. Stephen D. Fischer, (1990), Complex Variables, Wadsworth and Brooks (reprinted by Dover Publications in 1999). A really nice treatment of complex analysis including numerous geometrical applications and applications to differential equations. A very good book to read alongside Fischer is Tristan Needham, (2000), Visual Complex Analysis, Oxford University Press. This book is a wonderful treatment of fundamental topics, though it does not address conformal mapping or Riemann surfaces.

13. A. N. Kolmogorov, S. V. Fomin, (1970), Introductory Real Analysis, Prentice-Hall (translated by Richard Silverman), (reprinted by Dover Publications in 1975). A classic in the field and still a good read, expect to work hard.  Another fine and inexpensive book is N. L. Carothers, (2000), Real Analysis, Cambridge University Press. This is a very solid treatment covering all of the normal subjects.

14. George Bachman, Lawrence Narici, (1966), Functional Analysis, Academic Press (reprinted in 2000 by Dover Publications). This is an incredibly broad survey of the most important aspects of this topic.

If you work at it you can get through these books in four years.

Expressing Ideas Mathematically

We will begin with some important terminology:

Any idea that you can express in words or symbols constitutes a mathematical sentence.

Any mathematical sentence that is formed out of a single idea is called an atomic sentence.

Any mathematical sentence that is not atomic is called a compound sentence and is composed of several ideas.

Any mathematical sentence that is either true or false is called a proposition.

Any proposition that is always true is called a tautology.

Any proposition that is always false is called a contradiction.

Any mathematical sentence that is neither true nor false is called a paradox.

Connectives and Truth Tables

We stated before that every proposition is either true or false, this is called its truth value. We can always assign a symbol to represent any sentence, including propositions and paradoxes. If we consider two arbitrary propositions, p and q, we can make a table of all combinations of truth values,

This is called a truth table.

If we combine propositions we must connect them somehow. Several symbols have been invented for this task, these are called connectives. The first that we will consider is when p and q are true. This is called a conjunction and is represented by the symbol . We write this We can construct a truth table for the conjunction, using (1) as a basis. We will assign the value only when both and is true, and under every other situation,

T

F

F

F

The second connective that we will consider is when either p or q are true. This is called a disjunction and is represented by the symbol . We write this . We can construct a truth table for the disjunction, using (1) as a basis. We will assign the value only when either or is true (or both are true), and under every other situation,

T

T

T

F

The third connective that we will consider is when  p is not true. This is called a negation and is represented by the symbol . We write this . We can construct a truth table for the negation,

p
T	F
F	T

It is important to realize that we can prove some types of statements using a truth table. Any proposition whose truth value is T for all combinations is a tautology, so we can prove that something is a tautology. Any proposition whose truth value is F for all combinations is a contradiction, so we can prove something is a contradiction. Two statements are equivalent if they have the same truth table values, so we can prove that two statements are equivalent. Equivalent statements are combined using the symbol, would indicate that p is equivalent to q.

Thus far we have been combining propositions in pairs. Any combination of symbols and connectives is called a formula. When we combine more than a pair of propositions we always want to keep the symbols in pairs. For this purpose we will use parentheses () to enclose pairs in formulas where three propositions create a formula, we will use square brackets [] for enclosing triplets, and curly brackets {} for all others. Such formulas are called well-posed. Here is an example of a well-posed formula,

Here is an example of a formula that is not well-posed,

.

Derived Implications

There is another way of combining propositions that is very useful in mathematics. We can say that if we are given any truth value for p when q is true that implies q. This is called a conditional sentence and is written . We can make a truth table for the conditional by  assigning the value when is true or both and is false (this latter shows that since is false, so is ), and under every other situation,

T

T

F

T

We can make the statement that given q when p is true that implies . This is the converse of the conditional and is written .

We can also make the statement that given when is true that implies . This is the contrapositive of the conditional and is written .

We can also make the statement that p has a truth value if and only if q has the same truth value. Another way of saying this is that implies , and implies . This is called the biconditional sentence and is written . Here is its truth table,

T

F

F

T

The conditional sentence, the converse, the contrapositive, and biconditional sentences are all derived implications.

Some Suggested Practice Problems

1. Invent three mathematical sentences. Determine which of them (if any) are atomic sentences. Determine which of them (if any) are compound. Determine which of them (if any) are propositions. Determine which of them (if any) are tautologies. Determine which of them (if any) are contradictions. Determine which of them (if any) are paradoxes.

2. Invent three propositions.

3. Invent three tautologies, prove that they are tautologies.

4. Invent three contradictions, prove that they are contradictions.

5. Invent a paradox.

6. Create a truth table for three arbitrary propositions.

7. If we introduce the connective where either p or q is true, but not both, we can call this the exclusive or (XOR). Write the truth table for the XOR connective.

8. Invent three well-posed formulas. What are their truth tables? Hint: perform each combination of propositions as a new column in the truth table.

9. Establish the truth table for the converse and the contrapositive. What can you say about the contrapositive and the conditional sentence?

10. If we make the statement that implies and we called this the inverse sentence, what is its truth table?

11. Establish the truth tables for each of the following rules:
    
    
    
    
    
    
    

Converted by Mathematica      October 14, 2002