…and how do we understand it?
Mathematics is a variety of different things, or rather it is a set of overlapping ways of approaching and mastering a body of understanding, activity and thought. Here are some of the ways we look at Mathematics.
To give yourself some idea of what being mathematical is about, try a new activity in some aspect of Maths that you haven’t done before. You may find something new on The Edge page, but if you are acquainted with all of these, I am sure you can find something on or outside the edge of your knowledge. The point is to try and observe your own learning. What processes are you going through as you “do” the maths. I suggest the following aspects or ways of looking at what you are doing.
Primarily, understanding maths is doing maths. It is about the different ways of doing things rather than the body of knowledge that defines what it means to be taking part in Mathematics. You need to be doing something if you want be mathematical. of course there are restrictions on what you can do, but you must do something.
A formal language
The language of Mathematics is different to languages like English and Chinese. There are things are are strictly allowed and there are things that are strictly not. It is the formal nature of the language that often courses confusion and errors in learners. However over emphasis on the formality, and some teachers are only concerned with practicing formal exercises, prevents understanding of the beauty, creativity and utility of Mathematics.
I once phrased the expression that “Mathematics is the only creative discipline. All others are either representational or random.” It was at the entrance to maths exhibition we put on for an Arts Festival. The statement was meant to be provocative, but the ideas behind it are still clear to me. Most disciplines with an element of formality, such as the sciences or even economics and geography use their formality to represent and explore the real world. They represent the real world. Most creative domains such as the arts and literature lack the formality to be truly disciplined. There are no rules which dictate which note must follow a series of notes, or which sentence cannot follow the one before. Random is a strong word to describe these decisions, but mathematicians know that a random variable can take a variety of values within constraints.
Mathematics on the other hand is both totally disciplined and entirely creative lacking both the random element where you can do what you like and the restriction to a representation of the real world.
Mathematics, although it is not merely representative, has great value as a tool in the world at large. It is obviously used in the natural sciences and the human sciences. Less obviously, it has a role to play in the arts, but listen to Bach and Bartok or read Borges. See my friend and colleague Suman Vaze’s beautiful paintings all based on Mathematics.
The subtle questions have to do with the relationship between the formality and the tool. If the formal language can lead us to such ideas as multidimensional hyper cubes that appear to have no relation to our real world, why do they become so useful in describing it. If complex numbers were dismissed as “an amphibian between being and non-being” (Leibniz), how do they become so crucial to building our modern world 400 years later?
Read the play “It ain’t what you prove…” to see the tensions between an intuitive grasp and a formal understanding. Formality does not always lead us to uncontrovertible truth. See “The Objects of Mathematics” to look at the relationship between the formal creations and our real world images of them.