## What is mathematics?

Math isn’t what you thought it was.

You might think that math goes something like this:

Calculate or simplify the following expressions:

1.  $535\times 45$
2. $12^3$
3. the derivative of $f(x)=3x^4 + 25x^3 + 5\mathrm{sin}(x)$
4. $(x+5)^3$

Well, that all sucks.  No one wants to do any of that.  From this, one might think that math is just remembering the right formula, or the right steps, and then having the patience and focus to follow these steps for several minutes without making a mistake.  It involves tediously writing numbers and letters all over the paper in exactly the way that your teacher told you would work.  We often have no real idea as to why we’re doing this, or any knowledge of the vast logical constructions that make it all possible—we’re just calculating.  This post will explain that there is another kind of math—one that focuses primarily on the abstract foundations and the deep logical reasoning that goes in to establishing the math that we’re used to doing every day.  To begin exploring this “other” type of math first-hand, head on over to the lessons!

Two types of math.

I am going to refer to the above kind of math as “calculation”.  I.e., when we’re answering questions 1-4 above, we’re not doing our “mathematics” homework, but rather our “calculation” homework.  I will reserve the term “real mathematics” for what I mean by this “other world” of math.

What is “real” mathematics?   There are volumes written about this question, and I will only spend a few sentences on it here.  As you spend time reading through the lessons and exposing yourself to what I call “real” math, you’ll form your own idea of what mathematics is.  We can, however, make a little progress right here and now.

“Real” mathematics (or just mathematics) is the art of creating, understanding, and exploring the relationships between various mathematical* structures.  Mathematics is the art of making mathematics, by exploring and  understanding mathematical structures.  (Numbers are one such structure, but there are infinitely many more.  We will explore mathematical structures more in depth later, and many examples of such structures can be found on this site.)

There is indeed a good amount of what I’m calling “real” mathematics hidden inside of a standard high school math curriculum, and those who like math tend to be drawn to these bits.  But these parts of the curriculum are often drowned out by the emphasis on memorization, rote learning, and repetition.  Questions 1-4 above lie on top of a very elegant logical framework, but we often completely ignore this in order to get on with our calculations.

What do mathematicians do?

A mathematician does not spend his day calculating $2+2$ or “the derivative of $f(x)=3x^4 + 25x^3 + 5\mathrm{sin}(x)$”.  He or she instead thinks about why and precisely when it is the case that $2+2=4$, or what a derivative really means and what it can be used for.  The more math you learn, the more you realize that $2+2$ does not always equal 4 (this will be explained when we describe “clock arithmetic”).  The remarkable logical foundation on which the true statement “$2+2=4$” stands was built by mathematicians who were doing “real” mathematics.  The actual process of calculating the result of $2+2$ (and other similar calculations) is important for students to learn, but it is not the kind of thinking that goes into much of mathematics.

Everyone, from high school students to professional mathematicians, hates calculating derivatives.  The difference is that professional mathematicians have been exposed to the logical and rigorous foundations that make these calculations possible.  They’ve studied what derivatives are, where they come from, and what they lead to, and these kinds of considerations are what keeps them in the business of studying math.  Additionally, they find a deeper and more subtle beauty in the calculations themselves.  This site aims to introduce its readers to this type of math—what I’m calling “real” (as opposed to calculational)—and to explore some of the gorgeous subtleties that come along with it.

*You might be worried that I used the term “mathematical” in the definition of mathematics, but I have reason to do this.  A mathematical structure is a thing, and I am using it to describe what mathematics is as an art form.  In other words, there would be no problem saying that “painting is the art of making paintings”, and that is essentially what I’ve done here.

So true.