User:Sean Kelly/Mathematics

New introduction for Mathematics (and Category:Mathematics)
Here is my proposal for a new introduction for these sections: (in progress, please add)

Mathematics is most commonly thought of as the study of numbers and equations. Most students get a foundation in mathematics by first learning Arithmetic, the study of computation with Natural numbers. Students then progress to elementary algebra, where they learn to work with variables, functions, and the Real numbers. Finally, students learn Geometry, which also gives them an introduction to logic and proof.

Mathematics builds on these basic mechanics to form a broad discipline, much like how literature is built on top of the principles of language. Modern mathematics is most simply split into two branches: Analysis, which includes Calculus, and  Algebra.


 * Numbers: If two teams are playing football, then each player can shake the hand of someone from the other team. Everyone can do this at the same time, and no one will be left out.  A number is a way of measuring quantity.  We say that the two teams have the same number of players because there is some nice way to line them up (a "bijective map", handshakes).  We can imagine a team with one extra player, or two, and so forth.  These numbers are called Natural numbers.  The study of counting natural numbers is Arithmetic.
 * However, we can get more complex in our counting. For example, what is the number of whole numbers?  There are so many that the answer cannot be written as a whole number itself.  As another example, we can ask how many valid English sentences there are.  Since we can make a sentence of any length ('I have fifty cats named Muffin, and Butters, and Joseph...') the answer, again, is not a whole number.  We can now ask, are there as many English sentences as there are whole numbers?  Take any number, and you can make it into a sentence, say, 'I love the number five hundred six'.  Take any sentence, and you


 * Relations: mathematics is also concerned with how things are related. For example, we say "Joe and I are as tall as each other, but neither of us are as tall as the Empire State Building."  We can also use these relations to arrive at intuitive conclusions, for example, "Superman can leap over the Empire State Building, therefore Superman can leap over me."  The study of relations in space is Geometry.  The study of relations to arrive at conclusions is Logic.
 * Sets and structure: A set is a collection of elements, such as numbers, ideas, skyscrapers, or other sets. Sets are the basic building block of modern mathematics, and tie together relations and numbers.  We use sets to describe otherwise complex ideas, for example, "the set of all multiples of four (4, 8, 12, ...) contained in the set of all even numbers (2, 4, 6, 8,...)."

Reasons
Discussion of this topic already exists, at Talk:Mathematics and Talk:Mathematics
 * 1) I don't want to scare people away by the first sentence of this article.  Though I know we all agree that defining mathematics as the study of numbers isn't quite right, it is nevertheless what people will be looking for when the visit the page on mathematics.
 * 2) While none of the above is the "right" definition, they are not wrong.  It's intended to be as informal as possible without being false.
 * 3) The first paragraph needs to be simple and interesting to a wide audience.  This aims at that (notice the culture-neutral mention of "football").
 * 4) If people want a formal definition of mathematics, they can read further down.
 * 5) IMHO, the first word in any formal definition of mathematics should be 'relations'.  Though I guess 'abstract structures' is good too.