Beginning Mathematics/What is Mathematics?
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.
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.
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.
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.
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.