What is Mathematics ?
This question is one of the most complicating questions I've ever had to
take in mathematics, and has no simple satisfactory answer. The
Philosophy of mathematics is a complex and difficult topic not really
suitable for an introductory text.
At a naive level we can describe mathematics as a language that expresses
relationships. This includes Logic, Measurement, Algebra, Calculus and
Geometry. This language allows us to understand our universe and to solve
problems in it. When your eyes view a page of Mathematics, it looks like
a collection of symbols. Mathematics is not the symbols on the page but
what those symbols really mean.
Any two people from any corner of the Earth, if both understand
Mathematics, can view the same page and understand the implications
precisely, understand a question uniformly, or continue the discussion
without a single spoken word. There is no other such language taught
across the entire planet.
This Wikibook is dedicated to helping those who see the page of symbols
but do not hear the language of Mathematics.
[edit]The Disciplines of Mathematics
There are two disciplines contained within Mathematics: Logic and Theory.
They are separate but interdependent in that Mathematics is useless
without both. Unfortunately, many people are only taught the Theory side.
A more thorough description of both disciplines is presented later. For
now, we'll put it all on the table.
[edit]Mathematical Logic
Logic is the expression of ordered thoughts starting from axioms and
resulting in a conclusion. There are many rules and formalities for
Mathematical Logic which ensure that truth is maintained throughout the
logical argument. Once a conclusion is successfully built it can be used
with confidence as an axiom in another different logical argument.
Mathematical logic studies a set of artificial languages called logics.
These languages are thought to have theoretically interesting structures,
structures which are worthy of study both for their own sakes and for the
light that such study promises to shed on the methods of reasoning used
throughout mathematics.
[edit]Mathematical Theory
Theory deals with the abstraction of the real world into the Mathematical
world. As much as Mathematical Logic is rigorous and specific,
Mathematical Theory is abstract and generalized. There is no doubt that
this is where the fun of Mathematics shines through. Using Mathematical
Theory a person can divine how to build a house or why cell phones work,
make predictions about seemingly random events, even predict the motions
of planets, stars; and galaxies.
When Theory moves from the abstract to the real world, it is called
Applied Mathematics. These are the every day experiences people have with
Mathematics and is the small part of the realm of Mathematics with which
people usually are most familiar.
[edit]Where did Mathematics conne from?
Nobody is certain, but Mathematics may have been created even before the
first words were spoken by humans. For example, our ancestors surely
encountered some of the following problems:
Hunger
An attack by wild beasts
Aggression by another group
Our ancestors may have used some Logic to determine the best place to
find food and satisfy their hunger. They could have applied Math Theory
(applied mathematics) to determine if they had enough rocks to repel the
attacking beasts. And they could have used both Logic and Applied Math to
determine if it was best to stand and fight or run from an aggressive
group of neighbors. If you think through these situations yourself you
will realize that you can determine a course of actýon or reach a
decision without speaking one single word! Congratulations! You just
heard the first whispers of the Language of Mathematics in your head!
The ability of humans to communicate through language, and their enhanced
artistic creativity and dexterity with tools, led to the invention of
writing systems. First they were to record sounds, and words, and to
symbolize objects, yet eventually more complex thoughts such as
mathematical logic were symbolized. It is this ability to communicate
math through its written language that makes it so useful.
[edit]Things students should emphasize
Students must always remember that the mathematical language (terminology
and symbols) are just representations of mathematical thought. Often
students of math get mired in or turned off by the language, when the
focus should be more on mastering the concepts. Math is universal only in
its use of common logic and common concepts. The actual symbols (letters,
words, sentences) of the language are not as important as the thought
process.
Yet fortunately, the language itself, especially in writing, has become
highly standardized over the years, just to assist in communication. But
mathematics is valid no matter how it is represented, as long as all
terms and symbols are well-defined to the reader. Sometimes there is no
single way to express math, just as, sometimes, there is no single way to
make an argument. Ideally, in the open-marketplace of ideas, the most
efficient representation becomes the accepted canon. Yet mathematical
language, like all human languages, sometimes is entangled in traditional
notation. We humans love our traditions! But often all it takes is a
fresh new representation, to make concepts that once were confusing,
suddenly clear.
In any event, it is most important to learn the concepts, and then just
view the symbolism as a tool of communication, and a bookkeeping tool
during problem solving: and an exertion of the mind, which, for most, is
unable to keep track of all the complex threads of mathematical logic.
Students should not stress too much over memorizing pages of facts and
concepts, without also striving to understand why they are true, why
those facts must follow given the arguments, and why they make sense.
Students should focus heavily on developing the skills and practicing the
intellectual gymnastics that will enable them to think mathematically and
solve mathematically posed problems. That person who can solve a problem
from scratch, create new ideas, work things out by his or her own thought
process, is more useful to Mathematics and society than one who can
simply recall facts and figures, since these can always be looked up on
Wikipedia! That is, it is more useful to have people who can write new
Wikipedia articles, than people who can just read them!
Nevertheless, memorization is also useful, since of course nobody can
possibly have the brainpower, memory, ability, time, or patience to prove
all facts from first principles. Often, grand leaps of useful
mathematical reasoning can be made by simply proving that one (perhaps
already immensely complex and rich) axiom or fact will lead to a new
conclusion. I believe Isaac Newton (?) once said, "I can see so far,
because I stand on the shoulders of giants." There is no shame in
building on the work of others, as long as one acknowledges the source.
Not only does this give fair credit where credit is due, but it allows
things to be verified, especially if the thoughts of these "giants" are
simply implied.
One beauty of math that separates it from other disciplines like science,
is that the preconditions don't always need to be established. With math,
we can simply assume our foundations to be true, and build a new logical
structure. In fact, some math concepts are so elemental and obvious, that
often they are just implied as following from common sense! Take whole
numbers and counting, for example: it seems reasonable to assume that
most people would agree that they just "are", and don't need any
intrinsic proof of validity,