If we start off with a consideration of the data that we perceive we recognize that numbers are useful in describing our experience. But this raises the question about what numbers themselves are and how they are even useful.
In our experience of the world, of being and existence, we notice that numbers are useful in describing counts and characteristics. Pleasing sounds are made by harmonizing/marrying together the soundwaves - modulation of sound through amplification, dimunition or cancellation. This process can be modelled by plugging numbers into functions. Functions are higher order abstractions than numbers so we will put aside our questions about the nature of functions for the moment and give our focus entirely over to numbers. But I think it would be fair to say that functions are predictive and characteristic of a feature of our experience. Counts are so familiar that it will suffice for me to say that we are always counting even if it is nothing other than the number of dollars in our bank account.
In looking at numbers what we discover is that they themselves fall into patterns and categories and structures. One such has to do with selecting which method we will use to count. We have heard of binary, decimal, hexadecimal, etc. These numbering systems presuppose the architecture of the modulus operator, a ubiquitious presence in a college math class on Number Theory. So what does this mean? What is the nature of the relationship between this level of structure and numbers.
Another area of cleavage in numbers is type. There are so many types of numbers: odd/even, positive/negative, and then the sets: natural numbers, integers, rational, irrational, real, and complex.
In abstract algebra we notice that the procedures of identity, inverse, addition/subtraction, multiplication/division are used again to define/characterise sets. So we use the language of groups, rings, homomorphism and other useful descriptive terminology.
In topology, we look at three dimensional objects like doughnuts and describe them yet again in terms of functions and other similar characteristics.
In linear algebra/vector algebras we look at functions and manipulation of functions which in turn model some aspect of reality.
In calculus we again model reality using functions and manipulating these functions to predict and add to our understanding of the world around us.
So in surveying this landscape it looks like there are techniques of manipulating functions and techniques of organizing numbers/functions using characteristics of similarity.
It seems that increasing levels of abstraction makes it easier to predict what to do with the lower levels of mathematical knowledge. So the question that comes to my mind are there other levels, higher levels of abstraction and organization which will make it trivial to understand mathematics. In other words by understanding a larger concept will the others (namely techniques of manipulation and ordering) also follow? Or maybe a better way to express my intuition is that we already have the abstractions of abstract algebra and the techniques of the other disciplines but we still don't have something which will make these features obvious and trivial. Is there something that will make all the complex techniques we have and make them irrelevant because there is an easier way to get to certain mathematical truths simple and complex.
