As it is almost certain that this blog will include a lot of mathematics and a lot of talk about mathematics, I decided to set myself the somewhat ridiculous task of deciding what mathematics is…
[I had most of these thoughts in the last few days and so they might be a bit rough, but I felt that I did make some sort of progress. I have also (in a radical coup) deliberately avoided the use of examples, to see if I can explain what maths is without actually showing it to you directly, because, although this post will hardly serve as a terribly practical alternative, I am often irritated by having to explain what it is I do by saying “well, imagine a 3-dimensional sphere…”]
After some (light) research, I found out that in philosophy spiel I am much more of a quasi-empiricist, a fallibilist, humanist and relativist than an absolutist or a foundationalist when it comes to my views on mathematics (both the epistemology thereof and its place in society).
What on earth do these silly terms all mean? I am no expert and am throwing them around willy-nilly to poke fun at them myself. I will explain in a rough manner what I am getting at, and with a lot of bias toward my own point.
Roughly speaking, the two extremes which define the spectrum of standpoints I am discussing are defined thus: Some people are all like “maths is an absolute and certain body of perfect objective knowledge existing in a superhuman realm”. I intend here to conjure up the (often negative) image of maths as a cold, atomistic, unforgiving, dispassionate subject: A harsh right/wrong divide, with machinistic routines to be ‘mastered’ and obfuscatory notation to be ‘understood’.
On the other hand, some people are all like “maths is a human, historical and social discipline, the certainty and universality of which has cultural limitations”. This ‘human-centric’ view is intended to conjure up the (what I think is positive) image of maths as a a practice concerned with ideas, relationships, connections and intuition which relies on a rich literature (written by people for people!) and diligent pedagogy to progress.
The descriptions of maths I have given still do not answer the question of what it actually is. Obviously I want my interpretation of what it is to fit the latter description. With said description in mind, let’s give it a go:
I will argue that mathematics is really an art form. For now, let’s all hold back from asking “But what is art?”. We’ll keep ‘art’ as a black box for now. One thing that’s for sure is that ‘art’ is much more appropriately fitted to the second and not the first of my two descriptions in the previous section.
Those of use with working eyes will be accostomed to appreciating things visually: A cute baby, a beautiful person, an awesome sunset etc. ‘Art’ in its narrowest sense, began by representing things that we could already see with our eyes and art today in its ‘art gallery’ sense, consists mostly of things which we look at with our eyes in order to appreciate.
A lot of (modern) art uses words and/or sounds as well as just images and objects. So, we generalise our definition of art to include art through words and language and art through sounds and rhythms. Call these extra art forms poetry and music. To a certain extent, these simplistic descriptions are actually quite satisfying.
The medium of mathematics is ideas. To partially paraphrase Hardy, mathematics is an intangible art form in which one makes patterns with ideas. At its purest, it is not drawn, constructed, painted, composed, recited or played. Nor is it seen, heard or felt. We create mathematical ideas in our minds and we appreciate them in our minds. In some sense this makes it the most abstract art possible. Obviously these ideas are often inspired by things from the reality we perceive or conceived of in order to help explain this reality, but the sense in which they are mathematical is bound to the fact that they have a purely abstract existence.
A Very Human Art Form
What, then, is the ‘maths’ we are used to seeing? i.e. What are textbooks and symbols and numbers and problem sheets all about? Well, once one is happy with the idea that maths is an art form made with ideas, other facets of the subject can be made to fall easily into place.
Part of the making of mathematics is in linking together ideas by reasoning that one ‘follows’ from another. You start with one concept and think “what happens if I manipulate the idea in this specific way?”…..”How does the idea behave?”. Once you think you know, you might be led to make a mental note of the consequences of the manipulation you have performed. But now, you wish to make another manipulation, and another, and another. Wouldn’t it be easier if you could record the state of your mathematical idea at each stage? This is one of things notation is used for. Due to the abstractness of your idea, language may not fit the bill for the purposes of denoting and recording it. With sophisticated enough notation, you are able to deal with more complex ideas and more ideas at a time because you can write them down and refer back to them etc. However, these symbols are not mathematics, they just denote mathematical ideas.
Through a combination of language and mathematical notation, we can communicate these ideas effectively, although, to be honest, this process often seems inefficient. As many mathematicians will know, seeing something in your mind is very different from communicating the idea to another mathematician. It is a testament to the very human nature of the art that the mathematical community puts the emphasis it does on writing up your new ideas and disseminating them effectively.
[I think I’ll stop here for now. Examples of mathematics (all preached from this viewpoint of course!) will be there for all to see in subsequent posts and I also intend to discuss the interpretation of other facets of modern maths from this standpoint in the future. Recommended essays: