A
liberal arts education is founded upon the principle that one should learn
about the principle achievements of the human race. Learning about the achievements of the human race involves
reading, opining, reflecting, and doing.
It is in the doing that one begins to experience some of the richest and
most satisfying components of the liberal arts tradition. This is because reading or hearing about
principles is all well and good, but it does not enable a person to go and
discover, create, or invent and add to the canon of human knowledge. Indeed, the mere recitation of facts and
principles of the canon is more an exercise for a PBS documentary than for a
college course. Nonetheless, the appreciation of that which has come before is
a part of the college experience.
A modern
mathematics education is founded upon the same principles. It is concerned with a particular aspect of
the canon: the discernment between the veracity or lack thereof of claims. A
sagacious man is one who is willing to entertain the possibility that his ideas
are wrong. He is brave enough to investigate this possibility with an objective
framework which outlines a scheme to arrive at a conclusion such that the claim
under investigation is shown to be true, false, or non determinable based on
the knowledge available.
Thus,
one particular component of the canon that is central to mathematics is the
search for truth. Suppose a person
tells you, the sky is green. You would likely say that that was not true
based on the empirical evidence that the sky is not green, but blue or gray on
most occasions. However, can it ever be
green? Such claims are not what
mathematics is about. Mathematics is
not so concerned with temporal questions, questions of perceptions, questions
of opinion, but with questions which transcend such a temporal realm. That is not to say that temporal questions
are not asked in mathematics, but at this beginning level we will concern
ourselves with more general claims.
Indeed
mathematics is many things. It is a
language that formalises abstract concepts and thoughts. It is a collection of knowledge about
relations, measurements, figures, processes, objects, quantities, etc. It is a method of knowing. I do not think there is one definition which
does justice to the subject, thus I shall not burden the reader with
rudimentary mental reflections on what it is or is not. I will simply allow for the subject to exist
and assume the reader is interested in unlocking its secrets and in so doing
understand the subject better than when he entered this course.
Some of
the principles of mathematics that we will study in this course will be
nuanced, but all of them will have as their nature a characteristic such that
they will be useful in later studies, they will be applicable in multiple
situations, and they will be provocative.
We will study some of the major findings of the past 150 years or so and
will endeavour to do so such that we not only discuss the findings but
discover, create, or invent some of the findings ourselves. By doing such we will better understand the
principles of mathematics and truly know
mathematics rather than witness
mathematics. So much of the educational
experience of students today is founded upon a climate of witnessing: use of a
computer or calculator, listening to a lecture, watching a video, etc.
I do not accept this as a real liberal arts
educational experience. Watching another person do mathematics is not knowing
mathematics just as watching Andy Roddick play tennis does not make me a tennis
player. This analogy illustrates what
is ahead for the student enrolled in this course. You will be expected to do
mathematics, not watch mathematics being done.
You will be expected to be conversant in the language of mathematics,
not have a translator by your side to punch in words and get definitions. You will be expected to be comfortable with
the processes and methods illustrated and discussed and the only way I know to
be capable of doing such is by doing homework.
Real
learning takes place not in the classroom, but in other settings colloquially
referred to as at home; hence, homework.
Homework for this course requires memorisation of some symbols, axioms,
definitions, lemmas, theorems, and corollaries (oft called rote), involves
working multiple problems (oft called drill exercises to reinforce key
methods), and tackling complex
problems. The time spent in class is
designated to come together as a community of scholars and reflect on what we
understand, revise and extend that which we may not be truly clear on, and
build on that which we have mastered.
A solution to a problem, a
proof of the veracity of a claim or a counter-argument is correct not by the
mindless memorisation of an algorithm, the copying of a solution from a book or
notes, by punching buttons on a calculator, or by authority of the instructor
but because it is reasoned correctly.
We shall learn to reason properly, to write out results cogently, and to
critically review arguments which claim truth.
If this
is not to ones liking, then perhaps the college experience is not for him or
at the very least perhaps mathematics is not for him. To such a student my best advise is seek out other avenues of
learning or expression because it is a waste of your time, it is a waste of my
time, and it is a waste of other students time to be in a class from which you
receive nothing, are miserable in, and wish not to be enrolled. Life (and mathematics) is a banquet, and if
you dont like the menu, leave the
restaurant and find a cuisine more appealing to your pallet!
This
text is not merely a collection of exercises such that it is referenced only
when attempting homework. It should be
read before class commences. It should
be read after the class discussion for reinforcement. It should be referenced
when doing homework. Many mathematics
texts are written in such an obtuse manner that reading them can be a dreadful
experience. I hope such is not the case
with this text. However, in my career,
I have found there were times that some of the most difficult texts were better
than no text, there is no rule that states one cannot reference another text,
and if you find a problem with the exposition, then by all means please tell me
- - this is a work in progress. However,
please note that I intentionally have attempted to write this text on a
college level, I am not interested in dumbing down the material or in any way
talking down to the student, and am not amenable to revising this tenet.
The justification for the
existence of this text is complex, but can be summed up by noting that there
are a number of introductory texts available on the market which give an
overview of advanced mathematics and are intended to be exercises in the transition
for a student to upper level mathematics.
I opine that many of them are quite acceptable, but do not contain all
of the material that we will discuss in the semestre, many of them are written
on such an elementary level as to be laughable and are not fitting to be part
of our course, and many are written at too advanced a level such that the
prerequisite knowledge assumed is more advanced than can be justified in the
United States of the twenty-first century.
So, like the bears porridge, I hope this text is just right.[1] I assume that a student has learned
elementary school arithmetic: addition, subtraction, multiplication, division,
manipulation of fractions, decimals, and has been exposed to bases other than
ten. I assume that a student can properly manipulate algebraic expressions, can
prove basic claims in geometry, has an understanding of trigonometry,
functions, graphs, conic sections, natural, prime, rational, irrational, real,
and complex numbers. I assume a student can solve linear, quadratic compound,
absolute, and higher order equations and inequalities. Nonetheless, if a
subject is sketchy, if the memory of such is difficult to recall, or if such
was not discussed in a students primary or secondary education, then it is the
students responsibility to remediate himself; he should review said material
as soon as the problem has surfaced.
Mathematics is quite
literally a subject that is built like a building. It rests upon certain fundamentals - - the foundation, and can
only be mastered if said foundation is strong.
There are many short cuts that one can follow that work for a day, a
week, a year, or longer but eventually come back to haunt the individual
because the required skill or understanding that is central to concept that the
short cut bypassed is not understood by the student; hence, the short cut was
not a favour but a burden in the long run.
So it is with learning mathematics,
better to take time now and understand than try to rush through homework
and not really understand but to get it done. Mathematics is sequenced in a way
such that subsequent discussion is predicated upon previous topics.
So too in this course we
shall see that the course is laid out sequentially - - each topic follows (most
often) from previous. Further, mathematics is not a subject that
is joined by unary ideas so that each is unique and independent of other
topics. Hence, a refinement such that
one topic rests with discussion of others is not possible. So even though the chapters are designated
one, two, three, etc. let us consider for example material from set theory and logic. The two are not independent, so elementary set theory concepts
and logic concepts will be introduced simultaneously. Further refinement and expansion will be relegated to individual
chapters or sections.
Please
note that terms such as is, are, must, right, and wrong are used liberally
throughout the text. This is due to the
nature of pure mathematics. Assuming a
set of axioms and using logic we can deduce certain truths (based on the
axioms). So when you read a statement
such as, 3 ³ 2 by the law of addition -
- since the statement 3 > 2 is true it must
follow that 3 > 2 Ϊ 3 = 2, understand that is
predicated upon the axioms (of the reals).
Contents
Chapter
1: An Introduction to Logic
§ 1.1 Statements, negations conjunctions, disjunctions and truth tables . 2
Exercises ... 7
§ 1.2 Conditional and biconditional statements . 8
Exercises . . 12
§ 1.3 The Laws of logic . 15
Exercises ... 22
§ 1.4 Arguments, argument forms, proofs, and counterexamples . .. 23
Exercises . .. 34
§ 1.5 More on logic proofs .... . 35
Exercises ... 38
§ 1.6 More on fallacies .. .... . 39
Chapter
2: An Introduction to the Basic Concepts
of Sets, Syllogistic Logic, and Quantification
§ 2.1 Basic notation and concepts for sets .. .. .. 43
Exercises . .. 50
§ 2.2 Venn diagrammes and other illustrations for sets .. .. .. 53
Exercises .. .. 64
§ 2.3 An introduction to syllogistic logic and basic quantification.. .. .. 66
Exercises .. . . 74
§ 2.4 More on syllogistic logic and two place quantification .. .. .. . 77
Exercises . 82
§ 2.5 Logic and Deduction .. .. .. .. 85
Exercises . 88
§ 2.6 A Treatise on Deductive Logic, Sets, and Mathematics .. .. .. 90
Chapter 3: An
Introduction to Axiom Systems and Mathematical Induction
§ 3.1 Basic rational for axiom systems . .. ... 95
Exercises .. 99
§ 3.2 Some axiom systems .. .. ..... 100
Exercises .. .. ... ..... 109
§ 3.3 Formal Arithmetic .. . .. 111
Exercises 121
§ 3.4 The Peano Axioms. .. .. .. . 125
Exercises 134
§ 3.5 Mathematical induction. .. .. 140
Exercises .... 149
§ 3.6 On the foundation of pure, applied, and computational mathematics ... 153
[1] As in class lectures, my humour will infect this text (apologies). It is the case that try as I might, I simply cannot resist a corny joke or two, but with the help of my assistant, I will try to minimise the instances of the jokes and will ensure that they in no way compromise the pedagogical value of the material.