For the uninitiated, math often feels like a foreign language, maybe not even from this world. But it is far from that. The roots of the field lie deep in the foundations of how we perceive and understand the world.
When you look at the desk you may be sitting at right now, you understand the scene before you as an aggregation of objects, not just a complex sea of colors. There is a mug, a laptop, some paper, a pen, all of which have lots of different other properties, but the important part is that we see each item as a distinct object. In our mind, we can do lots of things with these objects. We can group some together (to form a set), we can count the items in a group, we can compare different groups with each other. All of these are mathematical operations.
What are we doing when counting a group of objects? We create an abstraction, or a model, that captures a property of this group. Let's say I got a bunch of apples that I count with my fingers. My fingers now represent the amount of apples in my possession. Why is this useful? Suppose you're asking me for two apples. By removing two fingers from my count, I can quickly determine how much apples I would be left with after giving in to your demand, without having to manipulate the apples themselves.
Now fingers and apples have (close to) nothing to do with each other. The link I constructed between my fingers and the apples is not part of the (outside) world, but is real in an idealistic sense. (An interesting application of this are modern computers. All the inside of a processing unit does is a complex manipulation of electromagnetic states. Only by assigning mathematical abstractions to these states and state changes is a computer doing useful things for us. The math is not in the computer but in the engineers that designed it.) Instead of always having to use real objects for models, math provides us the toolset to create these abstractions independently. Instead of being confined to using our fingers (or stones, sticks, ...), assigning numbers allows to be agnostic about the methods used to get to that count and what we do with it after.
In other words, math is the language of abstraction.
What makes math powerful is the ability to talk about itself, to develop the language independently of the things it is abstracting. But this tendency is also what gives mathematicians the reputation of sitting in their ivory tower, working on problems completely detached from any practicality. But we should not forget the advances made through concrete problems in other fields, that could not be tackled by the methods available at the time, specifically in physics. Math is pushed forward by both the theoretician as well as the applied mathematician.
At the same time, we should not make the error of mistaking the model for the real thing. The ideas of math seem so powerful that not just a few people see it as the fundamental fabric of the universe. I disagree with this. Instead of math being unreasonably effective due to capturing some fundamental feature of the universe, it rather seems to me that math formalizes how we as humans reason about the world.
Teaching should reflect this relationship between math and our intuitive understanding of the world. In teaching math there is always the dichotomy between teaching abstract theory and practical application. In my personal experience, neither is sufficient to get a solid understanding. Teaching practical application alone gives people the tools they supposedly need, but not the understanding to circumnavigate problems that arise when these tools stop working. For the well-trained and most gifted, theory alone might give them the ability to use all the tools they need, but that is certainly not true for everyone. Tying the theory to an intuitive understanding is often what is needed to illuminate the subject, make the pieces fall into place.
Maybe this is a personal grudge showing through, but to me the worst offender here is statistics, in the way it is classically taught at universities. It typically begins with a very abstract introduction to probability theory and then jumps straight into the application of t-tests and the like, without establishing a proper link and intuition that will help the student to quickly understand other approaches and new methods. (The times are changing though, there is now a lot of great material out there. Statistical Rethinking by Richard McElreath is one of many.)
I would like to create a handbook of mathematical abstraction. The goal of this handbook is to lay out the tools of modern mathematics in a way that connects them to human reasoning.
It should be clear by now that this project is not supposed to give the most rigorous account of the field, I certainly would not be qualified for that. It is as much a project to teach myself as to hopefully help others learn math in an efficient and hopefully fun way. (There's also a philosophical component to it. I want to explore the connection between human perception, human reasoning and mathematics I tried to argue for here. As the tools of mathematics are widely used in modern science, there are also epistemiological implications to it.)
On a technical note, I want to integrate this handbook with all the possibilities digital media offers: interactive diagrams, assisted problems for practicing, audio and/or video, et cetera.