The reason to explore is captured in the title of R.P. Feynman’s Book: The pleasure of finding things out.
The things to explore are things that capture our thoughts and surprise us. Be that as it may, humans have evolved a reward system so that when objects strike us as pretty, perhaps a better way of saying that, is symmetric.
When in the gamut of human experience does the ability to describe things happen?
To answer this question we perhaps need first to have looked for a language to describe the world around us.
Formal description that is nouns, dogs, cats, trees, numbers, patterns all needed for the purposes of description to have a way of codify them to an acceptable level of depth.
On patterns and numbers we have abstracted relationships of things to or perhaps better said between each other.
In order to facility this exploration we have defined the knowledge of philosophy to this subject to be named mathematics.
To capture these things we humans have divided up some portions of the universe for the purpose of discussion called domains at least in the English language.
Insider the domain of mathematics there are many discrete headings;
it also includes the study of ideas which as the infinite, the infinitesimal and many other topics, all of which fall within the domain.
Now to address the question I began with, but didn’t state as yet.
I’m asked now and again what is it I study in Mathematics, I wanted to answer this but then I decided that the word study is wrong, perhaps I don’t study, I explore small regions of some of these things and eventually perhaps I am at best a very curious onlooker.