This helpful workbook-style "bridge" book introduces students to the foundations of advanced mathematics, spanning the gap between a practically oriented calculus sequence and subsequent courses in algebra and analysis with a more theoretical slant.
Part 1 focuses on logic and number systems, providing the most basic tools, examples, and motivation for the manner, method, and concerns of higher mathematics. Part 2 covers sets, relations, functions, infinite sets, and mathematical proofs and reasoning.
Author Dennis Sentilles also discusses the history and development of mathematics as well as the reasons behind axiom systems and their uses. He assumes no prior knowledge of proofs or logic, and he takes an intuitive approach that builds into a formal development. Advanced undergraduate students of mathematics and engineering will find this volume an excellent source of instruction, reinforcement, and review.
About the Author
Dennis Sentilles is Professor Emeritus in the Department of Mathematics at the University of Missouri.
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A Bridge to Advanced Mathematics
By DENNIS SENTILLES
Dover Publications, Inc.Copyright © 2011 Dennis Sentilles
All rights reserved.
"Humble thyself, impotent reason"—Pascal
LOGIC, LANGUAGE AND MATHEMATICS
Despite some evidence to the contrary, each member of the species Homo sapiens seems to fancy himself as a cold, analytical and logical thinker in matters of great personal importance, and for the most part he is intuitively logical, though perhaps not always reasonable and coldly analytical, may be with good reason. Mathematics of course is famous for its objective, analytical judgment of things and as might be expected the logic of mathematics is more precise (you might claim tortuous!) than the logic of everyday affairs. This chapter is a survey of the most common patterns of logic and logical form found to be of constant use in mathematics. This will not be an in depth study of logic, but its thrust will be truly mathematical: to study the patterns of logical thought as entities in themselves, independently of any particular use or instance, but of course accompanied by examples and problems to give meaning to these patterns.
In plain fact, very little logic is used or needed to do most mathematics, and much of this is naturally though often only vaguely acquired by anyone with a strong interest in the subject. The intent herein is to remove whatever uncertainty there may be and to give to logic the closer attention it requires for use in mathematics. To serve this end it will be convenient to introduce some symbolic logic, but because a detailed capability in symbolic logic is not needed for mathematics, a lengthy study of symbolic logic will not be attempted. If the reader finds symbolic logic interesting in itself and would like to know more about it, there are some admirable and detailed investigations of it referenced at the close of this chapter. The outlook in this chapter is that logic is only one of a number of tools used to do mathematics, and that a little logic can take one a long way.
Having stated that little logic is needed to do mathematics and that one is likely to be acquainted with much of it by now anyway, what can justify any effort to learn about logic abstracted from any particular use? Would it perhaps suffice to simply begin with mathematics, believing that one will pick up logic as the need arises? Probably not. It should prove worthwhile to bring to your attention the common forms of exposition and reasoning in mathematics isolated from any particular application, so that you will obtain a grasp of logic and mathematical exposition as something in itself, independently of how it is used. This done, you should find it easier to follow the course of reasoning in abstract mathematics and be more certain of its validity. More importantly, you will be better able to focus on the mathematics itself, its content, and not be distracted by its logical form.
Now to turn this all around and assert that mathematics is the quintessence of logic! Because of the exhaustive work of others, this assertion has some validity. The Principia Mathematica was a partially successful attempt by the philosopher-mathematicians Alfred North Whitehead and Bertrand Russell to construct the basic elements of mathematics (such as the idea of "one," etc.) as logical extensions of a very few basic logical forms expressed symbolically. To paraphrase Whitehead himself, mathematics is the exhibition of unexpected relationships between two or more abstracted concepts. The mode of exhibition is called proof and the logic of mathematics furnishes a set of rules and guidelines governing the validity, the exposition and the development of proof in mathematics. Put another way, to do mathematics is to expose the logical dependence of one concept in mathematics upon another. Because of this, mathematics can be reduced to an exercise in purely logical form, but this is not the most common way of doing it.
There is more to this matter of proof and it bears on the intent of this entire text. Proof is a very special kind of thing in mathematics. In the carefully constructed world of mathematical thought, proof can meet an uncommon standard, not expected in other disciplines. It is hoped that in using this text the reader will come to an appreciation of, and an ability for, proof in mathematics, and an acquaintance with its power and limits. The standards of proof demanded are greater than one commonly pursues and are likely to be only slowly acquired. But at the same time there is more, much more, to proof than logic and the care to meet certain standards. Logic is one part of mathematical proof that can be learned. An equally important part of proof springs from the same force that causes an artist to paint, a poet to write. It is a feeling for one's subject, intuition and those other non-rational means by which one knows something is right and cannot be any other way. Logic is a part of thought that retrieves substance from ideas conceived in this way. As the would-be artist who cannot hold a steady brush cannot paint, the would-be scientist or mathematician who cannot explain and argue precisely for what he knows accomplishes little. Logic helps to give the mathematician the steady mind and sure course of reason that he needs.
2. LOGIC AND SYMBOLIC LOGIC
The logic chosen to do mathematics is basically the logic of most affairs with a few special cases tacked on to accommodate the natural evolution of mathematical thought over the past four thousand years. Nevertheless, it is the logic chosen for mathematics. It is not the way we must reason, but is the way we find it convenient to reason because our reasoning, for the most part, then coincides with the reasoning we grow into as we come to think and to use a language. The logic that certain mathematicians and logicians use would seem strange to us. They work with such logics because they find it useful, which is a point worth making now and which will be emphasized again: mathematics is what mathematicians make it, independently of (though not in ignorance of) themselves, our history, or the world around us. So, too, with the logic of mathematics. The system of logic we will study is the one that has thus far been the most useful and the least subject to philosophical dispute. It is not ordained by any deity, dictator or the constitution as the way things must be but exists as it is because it serves well. In a later section we will consider some special forms of set theoretic logic not accepted by all mathematicians as valid.
What is logic? A definition is difficult and in the end essentially impossible, but, following the suggestion of Russell and others who have studied the matter, one of the best answers is that the logic of an argument is that which is left over when the meaning of the argument has been removed. To find meaning in that, consider the following examples.
Example 1.2.1. If the policies of a government are just, then the power of enforcement is derived from the consent of the governed and is not derived from law sinstituted to enforce these policies. Therefore, if these policies do not have the consent of the governed or must be enforced by law, then they are unjust.
Example 1.2.2. If 2 ≠ 4, then green trees grow on barren hills and children are not happy. Therefore, if green trees do not grow on barren hills or children are happy, then 2 = 4.
If you will closely examine these two arguments, the first of which has some content, the second of which is frivolous, you will see that they have something in common, something that remains when all meaning is taken away. There is a form or structure which remains and it is the same in both. Both arguments have the abstracted form:
If p, then q and not r. Therefore, if not q or r, then not p
where p, q and r represent the meaningful statements in either argument in the order that they occur. This form is the logic of both arguments and indicates what was meant above about logic: logic is that which remains when all meaning has been removed. Thus the logic of an argument is the abstracted form of that argument as a thing independent of the content of the argument. The study of logic is the study of the possible form and structure of argument. Form is always a big thing in mathematics, in logic it is almost everything. This point has more use than you might expect at first; keep it in mind, it might save you some time!
The logic of mathematics certifies both arguments above as valid arguments. It must certify both (or neither) because both have the same form. This must bring into question then the place of truth in logic. If logic is concerned only with form, and the same form can occur in the juxtaposition of the most frivolous statements, or the most weighty ones, or the most absurd and the most believable, then one cannot in logical manipulations concern himself with meaning or truth. In practice this has two consequences. The first is that one must have a means of validating the logical form of an argument independently of the content of the argument, independently of the truth or falsity of its constituent parts. This means is provided by symbolic logic and the method of truth-tables, which furnish a precise and concise procedure for validating the logical form of an argument independently of its content. The second consequence is that logic cannot be used to make value judgments. For example, in Example 1.2.1, logic only certifies that the second sentence is a consequence of the first, and furnishes no basis for judgment of the validity of the first sentence. At the same time this hardly makes logic irrelevant. One can use logic to determine whether or not the first sentence in Example 1.2.1 is a true consequence of that admirably logical document, our Declaration of Independence, though one (to emphasize it again) cannot use logic to verify that document as a true statement about men and government.
There is yet more to be said about these examples. These contain all of the usual and essentail constituents of a logical argument! This need by no means be clear now but should soon become so. Each of the sentences in these examples consists of a collection of declarative phrases related and ordered by the words and, not, and if ... then.... The second sentence is a collection of the "logical opposites" of each of the phrases in the first sentence, related and ordered by the words or and if ... then.... The problem of determining the logical validity of each argument comes down to a verification that the first combination (the first sentence) of the separate declarative phrases must imply the second combination without regard to the meaning or truth of each declarative phrase. That is to say, the problem of establishing the logical validity of these is one of showing that if p, q and r represent "statements" and if the combination: if p, then q and not r, is given, then the combination: if not q or r, then not p, must follow without regard to the truth of p, q and r. That (as we will see) the second combination of p, q and r follows logically from the first is a consequence of the logic chosen to do mathematics and, again, has no further justification.
These last few paragraphs furnish a sort of overview of the task and intent of a study of logic. If you have picked up the main points and see some sense in them, use them as a guide for the sequel. If, for one reason or another, you feel this whole business is rather hazy ("what is left over when all meaning has been removed" does have an air of self-contradiction about it!), then your sensibilities are merely yet to be satisfied. In either case, the task before us is to examine the ways in which a given collection of simple statements, and often their "logical opposites," related and ordered by the relatively few words and, or and if ... then ..., can be manipulated and reordered to produce further combinations which in some precise and determinable sense (which we call logical) are related to the given statements without (for the last time!) paying any heed to the meaning of the statements themselves. To serve this last end it is both natural and useful to introduce some notation, and hence some symbolic logic, as was done earlier to point out the form of Examples 1.2.1 and 1.2.2.
3. STATEMENTS, PROPOSITIONS, DISJUNCTION AND CONJUNCTION
As will be seen in detail in Chapter 2, mathematical validity and logical validity, like any value judgment, is a relative thing and must always rest on some agreeable but ultimately unprovable basis. The matter of choosing a basis for any part of mathematics is a complicated, interesting and deceptive one: the Russell paradox is an outstanding example of this deceptiveness. In a later chapter we will consider the most common kind of basis for much of mathematics, the axiomatic method. Roughly speaking, the degree of intuition and imprecision that one is willing to live with, in the establishment of a basis for whatever system is to be developed, is directly proportional to the pace with which that development establishes "useful" results and, in return, the ease with which paradoxes can (possibly, not necessarily) creep in. Thus, if one begins with an extremely rigorous and non-intuitive basis for logic, governed by strict rules of as simple a nature as possible leaving no possibility of subjective interpretation, it will take some time to develop, within this setting, the principal useful results of mathematical logic. At the same time, the likelihood of misunderstanding and inconsistency would be virtually eliminated. We will establish the validity of the most useful forms of logical discourse on the basis of an intuitively motivated agreement on a precise meaning for the relatively few words which connect a sequence of statements into what is called a logical argument. Such a basis will suffice for the material in this book, although it is far from the epitomy of logical rigor.
In line then with this aim, we begin with the agreement that by the word statement we mean a declarative sentence. Thus
x is a number greater than 2.
is a statement, as is,
All statements in this book are false.
which indicates that one must narrow down the class of statements that can be assimilated into any sensible system of logic, for by the barest notions of intuitive logic this last statement is self-contradictory.
Certain statements have a character referred to as truth. Thus,
Fish are agile swimmers.
has a universal, if innocuous, character of truth, while
All water contains fish.
has an equivalent character of untruth (falsehood) while yet
Water pollution must end.
has a character of truth that is subjective. Other statements, such as
There is life on some other planet in our galaxy.
has an undetermined and perhaps undeterminable character of truth while the statement
Individual freedom and individual security are inversely proportional. has a ring of truth to the author, but not perhaps to the reader.
Admitting that truth, like progress, is in the eye of the beholder, mathematical logic avoids the issue by only claiming a capability to investigate the consequences of statements which already have a settled-upon character of truth or falsehood and claims no ability to determine this character. In a later chapter we will consider the nature of the truth of statements in mathematics. For the purposes of basic mathematical logic we need consider only statements to which a "truth-value" is assumed to be assigned. To indicate that we are thinking of a statement as true, we will say that the truth-value is 1, whereas, to indicate we are thinking of it as untrue, we will say that the truth-value is 0. To have a name to remind us of the nature of such statements we will use the term proposition; a proposition is a statement which has a truth-value of either 1 or 0 and not both.
Excerpted from A Bridge to Advanced Mathematics by DENNIS SENTILLES. Copyright © 2011 Dennis Sentilles. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Preface to the Second (Dover) Edition iii
Part 1 Starting Points 3
Chapter 1 Logic, Language and Mathematics 9
1 Introduction 9
2 Logic and Symbolic Logic 11
3 Statements, Propositions, Disjunction and Conjunction 13
4 The Negation of a Proposition 17
5 Logical Equivalence 20
6 The Kernel of Logical Thought: Implication 22
7 A Final Connective for Symbolic Logic: Equivalence 25
8 The Proof of Implications: Direct Method 28
9 The Proof of Implications: Indirect Methods 32
10 Tautology 36
11 Some Odds and Ends 39
12 Negation in the Mathematical Idiom 40
13 Quantifiers and Propositional Functions 44
14 Quantifiers That Aren't There and Other Sleights-of-Mind 54
Chapter 2 The Foundations of Mathematics 65
1 Introduction 65
2 Mathematics as a System of Thought 67
3 A Very Brief History of Mathematics 69
4 The Development of Non-Euclidean Geometry 75
5 The Axiomatic Method in Mathematics 89
6 The Natural Number System 98
7 The Real Number System 104
8 The Natural Numbers as a Part of (R 111
9 Infinity 117
10 Some Problems and Properties of Axiom Systems 135
Part 2 The Strategic Attack in Mathematics 147
Chapter 3 A Formally Informal Theory of Sets 155
1 Introduction 155
2 Fundamental Set Operations 161
3 Subsets 164
4 Set Theory: Second Floor 166
5 Cross Products of Sets 170
6 Operations with Arbitrarily Large Collections of Sets 173
7 The Axiom of Choice 180
Chapter 4 Topology and Connected Sets 185
1 Introduction 185
2 Basic Concepts: Open Sets and Closed Sets 193
3 The Closure of a Set 200
4 Topology Meets Set Theory 205
5 Connectedness in a Topological Space 209
6 The General Theory of Connected Sets 216
Chapter 5 Functions 227
1 Introduction 227
2 Relations and Functions 229
3 Idealizing the Function Concept 243
4 Functions Acting on Sets 248
5 Inverse and Composite Functions 257
Chapter 6 Counting the Infinite 267
1 Introduction: Counting the Finite 267
2 Extension: Counting the Infinite 272
3 Countably Infinite Sets and Uncountably Infinite Sets 275
4 Beyond the Countably Infinite: (R and the Answer to Question I 282
5 Cantor's Proof That (R Is Uncountable 291
6 The Schroder-Bernstein Theorem 294
7 The Continuum Hypothesis 296
Chapter 7 Equivalence Relations 299
1 Introduction 299
2 Equivalence Relations and Equivalence Classes 303
3 Cardinal Number 307
4 A Characterization of Open Sets in (R 309
5 A Factorization Theorem 313
Chapter 8 Continuity, Connectedness and Compactness 317
1 Introduction 317
2 Continuity, or, Distortion without Tearing 318
3 The Main Theorem: Answer to Question II 330
4 A Source of ApplicationCompactness 34
5 Euclidian n-Dimensional Space 351
6 Connectedness in En 359
7 Compactness in En 364
Appendix A 373