# Julian's musings

## What is mathematics, really?

Greg Ashman recently published two provocative posts (the first and the second) in response to Dan Meyer’s post, claiming that “A lot of people don’t seem to understand what mathematics is”. Dan Meyer’s statement:

“Math is only objective, inarguable, and abstract for questions defined so narrowly they’re almost useless to students, teachers, and the world itself.”

formed the starting point of this. Greg’s central thesis is that mathematics uses deductive reasoning, as opposed to other subjects which use inductive reasoning. From this follows the argument that calculating things like p-values is mathematics, whereas evaluating their meaning or usefulness is science. (I encourage you to read the full posts; I have just picked a couple of points out.)

## Increasing functions and functions increasing

Here’s the graph of $y=-\dfrac{1}{x}$ for $x\ne0$.

Where is this function increasing? Is it an increasing function?

Looking at various recent examination papers, it has become clear to me that there is significant confusion between these two questions. This post is intended to bring some clarity to the situation.

At the start of this post, I will give an example of the confusion as it appears in exam questions (and probably elsewhere), and clarify what the two different phrases mean using the above example. I will then delve more deeply into the mathematics of these two things, going beyond A-level content, and use some undergraduate analysis to find equivalent conditions for them in terms of the derivatives of the functions. It is fine to skip over the technical stuff and just look at the results (theorems)!

(Exactly the same applies to the use of the term “decreasing”, but for simplicity we will focus on increasing functions in this post.)